**Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse**

Eisuke Miura

*National Institute of Advanced Industrial Science and Technology (AIST) Tsukuba central 2, 1-1-4 Umezono, Tsukuba, Ibaraki Japan*

#### **1. Introduction**

22 Femtosecond–Scale Optics

Stan2007 C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Tsukamoto, A. Itoh, A. Kiriklyuk, and

Ziel1965 J. P. van der Ziel, P. S. Pershan, and L. Malmstrom, Phys. Rev. Lett. 15, 190

T. Rasing, Phys. Rev. Lett. 94, 237601 (2007). Far1846 M. Faraday, Phil. Trans. R. Soc. London 136, 104 (1846).

Waey2006 B. Van Waeyenberge *et al.*, Nature 444, 461 (2006).

(1965).

With recent progress in ultrashort ultraintense laser technologies such as chirped pulse amplification (CPA) (Strickland & Mourou, 1985), the peak power of a laser pulse is increasing year by year, and the focused intensity of 1021 W/cm<sup>2</sup> has been achieved (Aoyama et al., 2003; Perry et al., 1999). When the focused intensity of a laser pulse is higher than 10<sup>18</sup> W/cm2, quiver velocity of an electron is close to the speed of light in such a high electromagnetic field. Various nonlinear phenomena are caused by the relativistic effect of the electron motion. Self-focusing, higher harmonic generation, and so on, which are well-known phenomena in nonlinear optics, have been observed in laser-plasma interactions.

An ultrashort ultraintense laser pulse propagating in a plasma can excite a plasma wave by the nonlinear force of a high electromagnetic field, called the ponderomotive force. A longitudinal electric field is formed by the plasma wave, and electrons trapped in the potential of the plasma wave can be accelerated. This is the concept of laser-driven plasma-based electron acceleration (LPA) (Tajima & Dawson, 1979). The longitudinal accelerating electric field of the plasma wave is higher than 100 GV/m, which is a thousand times higher than that of present radio-frequency (rf) accelerators. Such a high accelerating field enables us to realize compact electron accelerators and/or obtain extremely high energy electrons. Furthermore, the electron pulse duration is extremely short, of the order of tens of femtoseconds, because the wavelength of the accelerating field, that is the plasma wave, is of the order of tens of micrometers. Next-generation electron accelerators with such unique characteristics will be realized using LPA.

Since the concept was proposed, various experimental and theoretical studies have been conducted (Esarey et al., 2009; 1996). Pioneering works of the proof-of-principle such as generation of a high accelerating field and energetic electron beams have been so far presented (Joshi et al., 1984; Kitagawa et al., 1992; Malka et al., 2002; Modena et al., 1995; Nakajima et al., 1995). However, the energy spectra of the electron beams were Maxwell-like distributions, and the beam qualities were far from those required for various applications. In 2004, a major breakthrough was brought about with the generation of well-collimated electron beams with a narrow energy spread, that is quasi-monoenergetic electron (QME) beams (Faure et al., 2004; Geddes et al., 2004; Mangles et al., 2004; Miura et al., 2005). This result is a significant step toward the realization of a laser electron accelerator.

In this chapter, we provide the overview of the present status of research on LPA. First, we briefly describe the principle of LPA. Second, we present recent results of works conducted at

out and the space charge force starts to pull the electrons back. In turn, the charge excess is formed by the electrons pulled back near the laser propagation axis. The electrons are pushed out again. By the repetition of this process, plasma oscillation is driven. The plasma frequency

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 25

 *e*2*ne ε*0*me*

Because the plasma oscillation is driven together with the propagation of the laser pulse, the phase velocity of the plasma wave *vp* is equal to the group velocity of the laser pulse given by

> <sup>1</sup> <sup>−</sup> *ne nc*

Electrons trapped in the plasma wave can be accelerated almost to the speed of light, because the phase velocity of the plasma wave is close to the speed of light in an underdense plasma

where *δn* is the modulation of the electron density. For example, assuming *ne* = 1019 cm−<sup>3</sup> and *δn* = 0.3, the electric field is 100 GV/m, corresponding to more than a thousand times as

To explain the principle of LPA, typical particle-in-cell (PIC) simulation results are shown in Fig. 1. The initial electron density of a plasma is 1.7 <sup>×</sup> 1019 cm−3, and the laser intensity of 800-nm light corresponds to the normalized vector potential of 5. A laser pulse propagates along the *x*−axis and is polarized along the *y*−axis. The snapshots of electron density distribution (a) and (b), the distributions of the electric field along the *y*−axis corresponding to the laser field (c) and (d), and the distributions of the electron density (solid curve) and the electric field (dashed curve) along the *x*−axis (e) and (f) are shown for different laser propagation lengths. As shown in Figs. 1(a) and (e), the period of the low-density and high-density parts is formed behind the laser pulse along the propagation axis, and a plasma wave is excited. As shown in Fig. 1(e), the peak longitudinal field along the *x*−axis, that is the

The situation, in which a laser pulse drives a plasma wave behind itself, is similar to that in which a boat drives a wake on a sea. Then, LPA is also called laser wakefield acceleration

As described above, because the phase velocity of the plasma wave is close to the speed of light, the velocity of electrons should be also close to the speed of light for trapping into a plasma wave. To feed accelerated electrons, an electron gun was used as an external electron source (Amiranoff et al., 1995; Clayton et al., 1993; Ebrahim, 1994). In some experiments, hot electrons produced from a laser-irradiated solid target were used (Mori, Sentoku, Kondo, Tsuji, Nakanii, Fukumochi, Kashihara, Kimura, Takeda, Tanaka, Norimatsu, Tanimoto, Nakamura, Tampo, Kodama, Miura, Mima & Kitagawa, 2009; Nakajima et al., 1995). However, it has been demonstrated that electrons in a plasma can be almost

*meωpc*

. (4)

. (5)

*<sup>e</sup>* , (6)

*ω<sup>p</sup>* =

*vp* = *c*

In the linear region, a longitudinal electric field of a plasma wave *Ex* is give by

*Ex* = *δn*

*ωp* for the electron density *ne* is given by

accelerating fields of present rf accelerators.

accelerating field in LPA, reaches 700 GV/m.

**2.3 Trapping and acceleration of electrons**

(*ne* � *nc*).

(LFWA).

the National Institute of Advanced Industrial Science and Technology (AIST). Generation of QME beams is mainly presented. We also present particle-in-cell simulations to discuss the mechanism and the conditions of QME beam generation. Using a femtosecond electron pulse obtained by LPA, a compact, all-optical, ultrashort X-ray source can be realized on the basis of laser Compton scattering. Third, we present X-ray generation by laser Compton scattering using a laser-accelerated electron beam. Finally, we briefly review recent progress toward a next step and future prospects.

#### **2. Principle of laser-driven plasma-based electron acceleration**

#### **2.1 Electron motion in a electromagnetic field**

Let us consider an electron motion in a electromagnetic field. The equation of motion for a free electron of charge *e* in an electromagnetic (laser) field is given by the Lorentz equation

$$\frac{d\mathbf{p}}{dt} = -\varepsilon(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \, , \tag{1}$$

where **p**=*γme***v**, **v**, **E**, and **B** are the momentum and velocity of an electron, and the electric and magnetic field, respectively. Here, *me* and *γ* are the electron mass and the relativistic Lorentz factor. In the case of a weak electromagnetic field, **v** × **B** component is negligible, and Eq. 1 is simplified to *me d***v** *dt* = −*e***E**. For a linearly polarized laser field, the electron quiver velocity *vq* is given by *vq* = *eEL*/*meωL*, where *EL* and *ω<sup>L</sup>* are the amplitude and frequency of the laser field. The ratio of the electron quiver velocity to the speed of light *c* is defined by

$$a\_0 = \frac{eE\_L}{m\_\epsilon \omega\_L c} = 8.5 \times 10^{-10} \lambda\_L [\mu \text{m}] \sqrt{(I\_L[\text{W/cm}^2])}\,\text{}\tag{2}$$

where *λ<sup>L</sup>* and *IL* are the laser wavelength and intensity. This gives an expression for the normalized vector potential *a*0. Relativistic effects are brought about in an electron motion in a laser field yielding *a*<sup>0</sup> ≥ 1. This region is called relativistic region. For example, the intensity of 2.2 <sup>×</sup> 1018 W/cm<sup>2</sup> gives *<sup>a</sup>*<sup>0</sup> <sup>=</sup> 1 for 800-nm laser light.

By averaging <sup>1</sup> <sup>2</sup>*mevq* <sup>2</sup> over one oscillation period of a field, the electron quiver energy is defined by

$$\mathcal{U}I\_P[\text{eV}] = \frac{1}{2} m\_\varepsilon \langle v\_q^2 \rangle = \frac{\varepsilon^2 E\_L^2}{4m\_\varepsilon \omega\_L^2} = 9.3 \times 10^{-14} (\lambda\_L[\mu\text{m}])^2 I\_L[\text{W/cm}^2] \,. \tag{3}$$

This is an expression for the ponderomotive potential *UP*. The ponderomotive potential results in a force **FP**=−∇*UP*, that is the ponderomotive force. The ponderomotive force is directed along the intensity gradient of a laser pulse envelope, and perpendicular to the laser propagation direction. The force pushes electrons out of the region of the laser pulse, and becomes the driving force for exciting a plasma wave in a plasma.

#### **2.2 Excitation of plasma wave**

Let us consider the propagation of an ultrashort ultraintense laser pulse in a low density plasma. The electron density of the plasma is much lower than the critical density *nc* given by *nc* = *ε*0*meω<sup>L</sup>* 2/*e*2, where *ε*<sup>0</sup> is the vacuum permittivity. Electrons in the plasma are pushed out of the region of the laser pulse and separated from ions by the ponderomotive force. A local charge separation is formed, because it is regarded that ions are at rest in the short time close to the laser pulse duration. After some time, the laser pulse overtakes electrons pushed 2 Will-be-set-by-IN-TECH

the National Institute of Advanced Industrial Science and Technology (AIST). Generation of QME beams is mainly presented. We also present particle-in-cell simulations to discuss the mechanism and the conditions of QME beam generation. Using a femtosecond electron pulse obtained by LPA, a compact, all-optical, ultrashort X-ray source can be realized on the basis of laser Compton scattering. Third, we present X-ray generation by laser Compton scattering using a laser-accelerated electron beam. Finally, we briefly review recent progress toward a

Let us consider an electron motion in a electromagnetic field. The equation of motion for a free electron of charge *e* in an electromagnetic (laser) field is given by the Lorentz equation

where **p**=*γme***v**, **v**, **E**, and **B** are the momentum and velocity of an electron, and the electric and magnetic field, respectively. Here, *me* and *γ* are the electron mass and the relativistic Lorentz factor. In the case of a weak electromagnetic field, **v** × **B** component is negligible, and Eq. 1

*vq* is given by *vq* = *eEL*/*meωL*, where *EL* and *ω<sup>L</sup>* are the amplitude and frequency of the laser

where *λ<sup>L</sup>* and *IL* are the laser wavelength and intensity. This gives an expression for the normalized vector potential *a*0. Relativistic effects are brought about in an electron motion in a laser field yielding *a*<sup>0</sup> ≥ 1. This region is called relativistic region. For example, the intensity

This is an expression for the ponderomotive potential *UP*. The ponderomotive potential results in a force **FP**=−∇*UP*, that is the ponderomotive force. The ponderomotive force is directed along the intensity gradient of a laser pulse envelope, and perpendicular to the laser propagation direction. The force pushes electrons out of the region of the laser pulse, and

Let us consider the propagation of an ultrashort ultraintense laser pulse in a low density plasma. The electron density of the plasma is much lower than the critical density *nc* given by

out of the region of the laser pulse and separated from ions by the ponderomotive force. A local charge separation is formed, because it is regarded that ions are at rest in the short time close to the laser pulse duration. After some time, the laser pulse overtakes electrons pushed

2/*e*2, where *ε*<sup>0</sup> is the vacuum permittivity. Electrons in the plasma are pushed

field. The ratio of the electron quiver velocity to the speed of light *c* is defined by

*meωLc* <sup>=</sup> 8.5 <sup>×</sup> <sup>10</sup>−10*λL*[*μ*m]

2

*dt* = −*e***E**. For a linearly polarized laser field, the electron quiver velocity

<sup>2</sup> over one oscillation period of a field, the electron quiver energy is

<sup>4</sup>*meωL*<sup>2</sup> <sup>=</sup> 9.3 <sup>×</sup> <sup>10</sup>−14(*λL*[*μ*m])<sup>2</sup> *IL*[W/cm2] . (3)

*dt* <sup>=</sup> <sup>−</sup>*e*(**<sup>E</sup>** <sup>+</sup> **<sup>v</sup>** <sup>×</sup> **<sup>B</sup>**) , (1)

(*IL*[W/cm2]) , (2)

**2. Principle of laser-driven plasma-based electron acceleration**

*d***p**

next step and future prospects.

is simplified to *me*

By averaging <sup>1</sup>

defined by

*nc* = *ε*0*meω<sup>L</sup>*

**2.1 Electron motion in a electromagnetic field**

*d***v**

<sup>2</sup>*mevq*

*UP*[eV] = <sup>1</sup>

**2.2 Excitation of plasma wave**

2 *me*�*vq*

*<sup>a</sup>*<sup>0</sup> <sup>=</sup> *eEL*

of 2.2 <sup>×</sup> 1018 W/cm<sup>2</sup> gives *<sup>a</sup>*<sup>0</sup> <sup>=</sup> 1 for 800-nm laser light.

<sup>2</sup>� <sup>=</sup> *<sup>e</sup>*<sup>2</sup>*EL*

becomes the driving force for exciting a plasma wave in a plasma.

out and the space charge force starts to pull the electrons back. In turn, the charge excess is formed by the electrons pulled back near the laser propagation axis. The electrons are pushed out again. By the repetition of this process, plasma oscillation is driven. The plasma frequency *ωp* for the electron density *ne* is given by

$$
\omega\_p = \sqrt{\frac{e^2 n\_\varepsilon}{\varepsilon\_0 m\_\varepsilon}}\,. \tag{4}
$$

Because the plasma oscillation is driven together with the propagation of the laser pulse, the phase velocity of the plasma wave *vp* is equal to the group velocity of the laser pulse given by

$$
v\_p = c\sqrt{1 - \frac{n\_\varepsilon}{n\_\varepsilon}}.\tag{5}$$

Electrons trapped in the plasma wave can be accelerated almost to the speed of light, because the phase velocity of the plasma wave is close to the speed of light in an underdense plasma (*ne* � *nc*).

In the linear region, a longitudinal electric field of a plasma wave *Ex* is give by

$$E\_x = \delta n \frac{m\_\varepsilon \omega\_p c}{e} \,, \tag{6}$$

where *δn* is the modulation of the electron density. For example, assuming *ne* = 1019 cm−<sup>3</sup> and *δn* = 0.3, the electric field is 100 GV/m, corresponding to more than a thousand times as accelerating fields of present rf accelerators.

To explain the principle of LPA, typical particle-in-cell (PIC) simulation results are shown in Fig. 1. The initial electron density of a plasma is 1.7 <sup>×</sup> 1019 cm−3, and the laser intensity of 800-nm light corresponds to the normalized vector potential of 5. A laser pulse propagates along the *x*−axis and is polarized along the *y*−axis. The snapshots of electron density distribution (a) and (b), the distributions of the electric field along the *y*−axis corresponding to the laser field (c) and (d), and the distributions of the electron density (solid curve) and the electric field (dashed curve) along the *x*−axis (e) and (f) are shown for different laser propagation lengths. As shown in Figs. 1(a) and (e), the period of the low-density and high-density parts is formed behind the laser pulse along the propagation axis, and a plasma wave is excited. As shown in Fig. 1(e), the peak longitudinal field along the *x*−axis, that is the accelerating field in LPA, reaches 700 GV/m.

The situation, in which a laser pulse drives a plasma wave behind itself, is similar to that in which a boat drives a wake on a sea. Then, LPA is also called laser wakefield acceleration (LFWA).

#### **2.3 Trapping and acceleration of electrons**

As described above, because the phase velocity of the plasma wave is close to the speed of light, the velocity of electrons should be also close to the speed of light for trapping into a plasma wave. To feed accelerated electrons, an electron gun was used as an external electron source (Amiranoff et al., 1995; Clayton et al., 1993; Ebrahim, 1994). In some experiments, hot electrons produced from a laser-irradiated solid target were used (Mori, Sentoku, Kondo, Tsuji, Nakanii, Fukumochi, Kashihara, Kimura, Takeda, Tanaka, Norimatsu, Tanimoto, Nakamura, Tampo, Kodama, Miura, Mima & Kitagawa, 2009; Nakajima et al., 1995). However, it has been demonstrated that electrons in a plasma can be almost

acceleration length is mainly limited by dephasing (phase slippage) of electrons trapped in a plasma wave. The dephasing is brought about by the difference between the phase velocity of a plasma wave and the velocity of accelerated electrons. Accelerated electrons in a plasma wave outrun the acceleration phase of the plasma wave, and enter the deceleration phase. In

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 27

� *nc ne*

> <sup>2</sup> *nc ne*

*λp* . (7)

. (8)

*Ld* <sup>=</sup> *<sup>c</sup>λp*/2 *c* − *vp*

*Wmax* = 4*mec*

electrons with an energy of 350 MeV can be accelerated in only 1.8-mm length.

For example, when the laser wavelength is 800 nm and the electron density is 1019 cm−3, the dephasing length and the maximum energy gain are 1.8 mm and 350 MeV. In other words,

The acceleration length is also limited by the diffraction of a laser beam and/or the energy depletion of a laser pulse for driving a plasma wave. The limit of the laser propagation length by the diffraction is not serious, because it is possible to achieve a longer propagation length by guiding a laser pulse using relativistic self-focusing, a capillary discharge plasma and so on. If the pump depletion length is defined as the length in consuming half of the initial laser pulse energy for driving a plasma wave, the pump depletion length is comparable to the dephasing length (Esarey et al., 1996). Then, the limit of the acceleration length by dephasing

The description in this section is concentrated in the liner region. However, as shown in Fig. 1, most of phenomena in LPA are in the nonlinear region, and the treatment is important. The overview of analytical description in the nonlinear region has been provided in detailed

Since the middle of 1990's, the generation of energetic electron beams based on a self-injection scheme has been demonstrated (Modena et al., 1995; Nakajima et al., 1995; Umstadter, Chen, Maksimchuk, Mourou & Wagner, 1996). Although collimated electron beams were generated, the energy spectra of the electron beams were Maxwell-like distributions and the energy spreads were large. These experiments were based on self-modulated laser wakefield acceleration (SM-LWFA) using a laser pulse with a picosecond duration and a high-density plasma with an electron density close to 1020 cm−3. In such cases, the laser pulse length is longer than the plasma wavelength. The plasma wave with a large amplitude is excited via Raman scattering and/or the self-modulation instability and can accelerate trapped electrons. Heating the plasma occurs, when the plasma wave grows. In addition, the trapped electrons in the plasma wave interact with the laser field and can be also accelerated by the laser field directly, that is so-called direct laser acceleration (DLA) (Gahn et al., 1999). The combination of these electron acceleration mechanisms can lead to the broad electron energy spectrum. However, in 2004, the generation of QME beams with a narrow energy spread was demonstrated by four groups (Faure et al., 2004; Geddes et al., 2004; Mangles et al., 2004; Miura et al., 2005). After that, many groups have so far reported the generation of QME beams (Hidding et al., 2006; Hosokai, Kinoshita, Ohkubo, Maekawa, Uesaka, Zhidkov, Yamazaki, Kotaki, Kando, Nakajima, Bulanov, Tomassini, Giulietti & Giulietti, 2006; Hsieh

**2.4 Generation of quasi-monoenergetic electron beam with a narrow energy spread**

the linear region, the dephasing length *Ld* is given by

Then, the maximum energy gain is given by

review reports (Esarey et al., 2009; 1996).

is dominant.

Fig. 1. Typical simulation results to explain the principle of LPA. Snapshots of electron density distribution on the *x* − *y* plain (a) and (b), the distributions of the electric field along the *y*−axis corresponding to the laser field (c) and (d), and the distributions of the electron density (solid curve) and the electric field (dashed curve) corresponding to the accelerating field along the *x*−axis (e) and (f) are shown for different laser propagation lengths.

automatically injected and trapped into a plasma wave. This scheme is called self-injection scheme or self-trapping scheme. In the self-injection scheme, wave-breaking plays an important role (Bulanov et al., 1997; Decker et al., 1994). As shown in Fig. 1(e), a plasma wave with a large amplitude is excited, and the steeping of the wave occurs. As the laser pulse propagates further, the amplitude of the plasma wave becomes larger, and the wave finally breaks. This phenomenon is seen as whitecaps at the ocean. Electrons slip from the wave and are trapped into the plasma wave. As shown in Figs. 1(b) and (f), electrons are trapped and accelerated in the first period of the plasma wave and form an electron bunch.

The plasma wavelength *λ<sup>p</sup>* = 2*πc*/*ωp*, which is the wavelength of the accelerating field, is the order of tens of micrometers. In the case shown in Fig. 1, the plasma wavelength is ∼ 10 *μ*m. Electrons are trapped in the narrow region of the acceleration phase and the length of the electron bunch is a few micrometers as shown in Fig. 1(f). This electron bunch is regarded as an ultrashort electron pulse with a duration of tens of femtoseconds. Thus, a femtosecond electron pulse can be obtained by LPA.

The energy gain *Wmax* is simply given by the product of the accelerating field and the acceleration length. In the linear region, the accelerating field is given by Eq. 6. The acceleration length is mainly limited by dephasing (phase slippage) of electrons trapped in a plasma wave. The dephasing is brought about by the difference between the phase velocity of a plasma wave and the velocity of accelerated electrons. Accelerated electrons in a plasma wave outrun the acceleration phase of the plasma wave, and enter the deceleration phase. In the linear region, the dephasing length *Ld* is given by

$$L\_d = \frac{c\lambda\_p/2}{c - v\_p} \simeq \frac{n\_c}{n\_c} \lambda\_p \,. \tag{7}$$

Then, the maximum energy gain is given by

4 Will-be-set-by-IN-TECH

Position, y [μm]

20 30 40 50 195 205 215 225

automatically injected and trapped into a plasma wave. This scheme is called self-injection scheme or self-trapping scheme. In the self-injection scheme, wave-breaking plays an important role (Bulanov et al., 1997; Decker et al., 1994). As shown in Fig. 1(e), a plasma wave with a large amplitude is excited, and the steeping of the wave occurs. As the laser pulse propagates further, the amplitude of the plasma wave becomes larger, and the wave finally breaks. This phenomenon is seen as whitecaps at the ocean. Electrons slip from the wave and are trapped into the plasma wave. As shown in Figs. 1(b) and (f), electrons are trapped and

The plasma wavelength *λ<sup>p</sup>* = 2*πc*/*ωp*, which is the wavelength of the accelerating field, is the order of tens of micrometers. In the case shown in Fig. 1, the plasma wavelength is ∼ 10 *μ*m. Electrons are trapped in the narrow region of the acceleration phase and the length of the electron bunch is a few micrometers as shown in Fig. 1(f). This electron bunch is regarded as an ultrashort electron pulse with a duration of tens of femtoseconds. Thus, a femtosecond

The energy gain *Wmax* is simply given by the product of the accelerating field and the acceleration length. In the linear region, the accelerating field is given by Eq. 6. The

Fig. 1. Typical simulation results to explain the principle of LPA. Snapshots of electron density distribution on the *x* − *y* plain (a) and (b), the distributions of the electric field along the *y*−axis corresponding to the laser field (c) and (d), and the distributions of the electron density (solid curve) and the electric field (dashed curve) corresponding to the accelerating

field along the *x*−axis (e) and (f) are shown for different laser propagation lengths.

accelerated in the first period of the plasma wave and form an electron bunch.

Position, x [μm]

electron pulse can be obtained by LPA.

**(e) (f)**

**(a) (b)**

**(c) (d)**

0

0.3

0.6


Electric field Ex [TV/m]

10

Electron density [1019 cm-3]

15

20


15 10


0

5


Electric field Ey [TV/m]

0

5



10

Electron density [1019 cm-3]

15

20


15 10


0

5


Electric field Ey [TV/m]

Position, y [μm]

0

5

Position, x [μm]

0

0.3

0.6


Electric field Ex [TV/m]

0 1 4 10 20 30 Electron density [1019 cm-3]

70 50



$$\mathcal{W}\_{\text{max}} = 4m\_{\varepsilon}c^{2}\frac{n\_{\varepsilon}}{n\_{\varepsilon}}\,. \tag{8}$$

For example, when the laser wavelength is 800 nm and the electron density is 1019 cm−3, the dephasing length and the maximum energy gain are 1.8 mm and 350 MeV. In other words, electrons with an energy of 350 MeV can be accelerated in only 1.8-mm length.

The acceleration length is also limited by the diffraction of a laser beam and/or the energy depletion of a laser pulse for driving a plasma wave. The limit of the laser propagation length by the diffraction is not serious, because it is possible to achieve a longer propagation length by guiding a laser pulse using relativistic self-focusing, a capillary discharge plasma and so on. If the pump depletion length is defined as the length in consuming half of the initial laser pulse energy for driving a plasma wave, the pump depletion length is comparable to the dephasing length (Esarey et al., 1996). Then, the limit of the acceleration length by dephasing is dominant.

The description in this section is concentrated in the liner region. However, as shown in Fig. 1, most of phenomena in LPA are in the nonlinear region, and the treatment is important. The overview of analytical description in the nonlinear region has been provided in detailed review reports (Esarey et al., 2009; 1996).

#### **2.4 Generation of quasi-monoenergetic electron beam with a narrow energy spread**

Since the middle of 1990's, the generation of energetic electron beams based on a self-injection scheme has been demonstrated (Modena et al., 1995; Nakajima et al., 1995; Umstadter, Chen, Maksimchuk, Mourou & Wagner, 1996). Although collimated electron beams were generated, the energy spectra of the electron beams were Maxwell-like distributions and the energy spreads were large. These experiments were based on self-modulated laser wakefield acceleration (SM-LWFA) using a laser pulse with a picosecond duration and a high-density plasma with an electron density close to 1020 cm−3. In such cases, the laser pulse length is longer than the plasma wavelength. The plasma wave with a large amplitude is excited via Raman scattering and/or the self-modulation instability and can accelerate trapped electrons. Heating the plasma occurs, when the plasma wave grows. In addition, the trapped electrons in the plasma wave interact with the laser field and can be also accelerated by the laser field directly, that is so-called direct laser acceleration (DLA) (Gahn et al., 1999). The combination of these electron acceleration mechanisms can lead to the broad electron energy spectrum.

However, in 2004, the generation of QME beams with a narrow energy spread was demonstrated by four groups (Faure et al., 2004; Geddes et al., 2004; Mangles et al., 2004; Miura et al., 2005). After that, many groups have so far reported the generation of QME beams (Hidding et al., 2006; Hosokai, Kinoshita, Ohkubo, Maekawa, Uesaka, Zhidkov, Yamazaki, Kotaki, Kando, Nakajima, Bulanov, Tomassini, Giulietti & Giulietti, 2006; Hsieh

pulse duration of 60 fs, and a center wavelength of 800 nm was obtained after a vacuum pulse compressor. Due to the priority to optimize a pulse shape of a main laser pulse, the spectrum control of a colliding laser pulse was not sufficient. Then, the pulse duration of the colliding laser pulse was little bit longer than that of the main laser pulse. The colliding laser pulse was used for X-ray generation by laser Compton scattering shown in Sec. 4. The third pulse referred as probe laser pulse with an energy of 0.5 mJ, a pulse duration of 60 fs, and a center wavelength of 800 nm was used for plasma diagnostics such as shadowgraph and

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 29

The experimental setup for electron acceleration is shown in Fig. 2. A p-polarized main laser pulse (400 mJ, 50 fs, 800 nm) was focused onto the edge of a helium gas jet using an *f* /6 off-axis parabolic mirror with a focal length of 300 mm. The laser spot size in vacuum was 9 *μ*m at the full-width-at-half-maximum (FWHM), and the Rayleigh length was 230 *μ*m. The energy concentration within the *e*−<sup>2</sup> spot was 58% of the total laser energy. The peak intensity was 5.8 <sup>×</sup> 1018 W/cm2, corresponding to the normalized vector potential of 1.6. The helium gas jet was ejected from a supersonic nozzle with a conical shape driven by a pulsed valve. The focal position was set at 1 mm above the nozzle exit. The diameter of the nozzle exit was 0.7 mm and the Mach number was 5.3. The density profile of the gas jet was measured with a Jamin interferometer. The density at the center was <sup>∼</sup> <sup>5</sup> <sup>×</sup> 1018 cm−3, corresponding to the

> CCD camera

B

Electron energy spectrometer

Electron beam

Lens

The energy spectrum of an electron beam was measured by an electron energy spectrometer with a dipole magnet. The measured energy range was from 1 to 80 MeV. An energy-resolved electron image was recorded on an imaging plate [(IP), Fujifilm: BAS-SR]. To estimate the absolute charge of electrons , the sensitivity curve reported in Ref.(Tanaka et al., 2005) was used. In addition, we have developed an absolutely-calibrated in-situ observation system for the electron energy spectrum. A Gd2O2S : Tb phosphor screen (Mitsubishi chemical: DRZ-HIGH) coupled with a CCD camera was used as an electron detector. The sensitivity of the detection system was calibrated using an IP as the reference detector (Masuda et al., 2008). The absolute electron energy spectrum was in-situ observed with a single shot at a

Lens

CCD camera

Imaging Plate

Phosphor screen

or

Lens

interferogram. **Experiment setup**

electron density of <sup>∼</sup> 1019 cm−3.

Main laser pulse

repetition rate of 1 Hz.

Fig. 2. Experimental setup for electron acceleration.

Plasma

Gas jet

CCD camera

Interference filter

Probe laser pulse

et al., 2006; Maksimchuk et al., 2007; Mori et al., 2006; Yamazaki et al., 2005). Most experimental results on QME beam generation have been explained by the acceleration in the highly nonlinear broken-wave regime, that is so-called bubble acceleration regime (Pukhov & Meyer-ter Vehn, 2002). In simulation results shown in Fig. 1, the conditions are close to those of the bubble acceleration regime. When an ultraintense laser pulse propagates in a plasma, it undergoes the self-focusing and the longitudinal pulse compression. Such laser pulse drives a highly nonlinear plasma wave and expels a large amount of electrons. Then, the electron cavitated region (bubble) is formed behind the laser pulse. The radially expelled electrons move along the boundary of the bubble and collide at the rear vertex of the bubble. Transverse wave-breaking (Bulanov et al., 1997) occurs and a large amount of electrons is injected into the bubble at the rear vertex. The field produced by the injected electrons terminates further electron injection. Electrons are trapped at the fixed narrow phase of the accelerating field. Thus, the QME beam with a narrow energy spread can be obtained. Because a large amount of electrons is instantaneously injected, the QME beam with high charge can be obtained. The electrons trapped near the rear vertex of the bubble are located behind the laser pulse. Because the trapped electrons are free from the laser pulse, an electron beam with low emittance can be obtained. These phenomena have been thoroughly investigated by many theoretical and numerical studies (Geissler et al., 2006; Gordienko & Pukhov, 2005; Kostyukov et al., 2004; Lu et al., 2007). For such bubble formation, one of the required conditions is that the spatially transverse and longitudinal sizes of the laser pulse match the plasma wavelength. To satisfy this condition, the experiments have been conducted using a focusing mirror with the long focal length yielding the spot size of ∼ 10 *μ*m and an ultraintense laser pulse with a few tens of femtoseconds yielding the pulse length of ∼ 10 *μ*m for a low density plasma around or less than 1019 cm−<sup>3</sup> yielding the plasma wavelength of <sup>∼</sup> <sup>10</sup> *<sup>μ</sup>*m.

## **3. Experimental and numerical studies on laser-driven plasma-based electron acceleration**

In this section, recent results of works conducted in the National Institute of Advanced Industrial Science and Technology (AIST) (Masuda & Miura, 2008; Miura & Masuda, 2009) are presented as an example of experimental and numerical studies on LPA.

#### **3.1 Experimental conditions**

#### **Laser system**

A Ti:sapphire laser system with a repetition rate of 10 Hz based on CPA method was used for experiments. In our laser system, two intense laser pulses were available. A 20-fs laser pulse from a Kerr-lens mode-locked oscillator was stretched to 400 ps through an Öffner type pulse stretcher, which was an aberration-free type stretcher(Cheriaux et al., 1996). To control a laser pulse shape, an acousto-optic programmable dispersive filter (Dazzler: Fastlite)(Verluise et al., 2000) was installed after the pulse stretcher. The laser pulse was amplified by a regenerative amplifier and multi-pass amplifiers. To suppress and control a prepulse, a pulse cleaner composed of a Pockels cell and a thin film polarizer was set between the regenerative and the first muti-pass amplifiers. After the first multi-pass amplifier, the laser pulse was split into three pulses. The first pulse referred as main laser pulse was amplified by a multi-pass amplifier, and a laser pulse with an energy of 750 mJ, a pulse duration of 35-50 fs, and a center wavelength of 800 nm was obtained after a vacuum pulse compressor. The main laser pulse was used for electron acceleration. The second pulse referred as colliding laser pulse was amplified by two multi-pass amplifiers, and a laser pulse with an energy of 150 mJ, a pulse duration of 60 fs, and a center wavelength of 800 nm was obtained after a vacuum pulse compressor. Due to the priority to optimize a pulse shape of a main laser pulse, the spectrum control of a colliding laser pulse was not sufficient. Then, the pulse duration of the colliding laser pulse was little bit longer than that of the main laser pulse. The colliding laser pulse was used for X-ray generation by laser Compton scattering shown in Sec. 4. The third pulse referred as probe laser pulse with an energy of 0.5 mJ, a pulse duration of 60 fs, and a center wavelength of 800 nm was used for plasma diagnostics such as shadowgraph and interferogram.

#### **Experiment setup**

6 Will-be-set-by-IN-TECH

et al., 2006; Maksimchuk et al., 2007; Mori et al., 2006; Yamazaki et al., 2005). Most experimental results on QME beam generation have been explained by the acceleration in the highly nonlinear broken-wave regime, that is so-called bubble acceleration regime (Pukhov & Meyer-ter Vehn, 2002). In simulation results shown in Fig. 1, the conditions are close to those of the bubble acceleration regime. When an ultraintense laser pulse propagates in a plasma, it undergoes the self-focusing and the longitudinal pulse compression. Such laser pulse drives a highly nonlinear plasma wave and expels a large amount of electrons. Then, the electron cavitated region (bubble) is formed behind the laser pulse. The radially expelled electrons move along the boundary of the bubble and collide at the rear vertex of the bubble. Transverse wave-breaking (Bulanov et al., 1997) occurs and a large amount of electrons is injected into the bubble at the rear vertex. The field produced by the injected electrons terminates further electron injection. Electrons are trapped at the fixed narrow phase of the accelerating field. Thus, the QME beam with a narrow energy spread can be obtained. Because a large amount of electrons is instantaneously injected, the QME beam with high charge can be obtained. The electrons trapped near the rear vertex of the bubble are located behind the laser pulse. Because the trapped electrons are free from the laser pulse, an electron beam with low emittance can be obtained. These phenomena have been thoroughly investigated by many theoretical and numerical studies (Geissler et al., 2006; Gordienko & Pukhov, 2005; Kostyukov et al., 2004; Lu et al., 2007). For such bubble formation, one of the required conditions is that the spatially transverse and longitudinal sizes of the laser pulse match the plasma wavelength. To satisfy this condition, the experiments have been conducted using a focusing mirror with the long focal length yielding the spot size of ∼ 10 *μ*m and an ultraintense laser pulse with a few tens of femtoseconds yielding the pulse length of ∼ 10 *μ*m for a low density plasma around or less

than 1019 cm−<sup>3</sup> yielding the plasma wavelength of <sup>∼</sup> <sup>10</sup> *<sup>μ</sup>*m.

**acceleration**

**Laser system**

**3.1 Experimental conditions**

**3. Experimental and numerical studies on laser-driven plasma-based electron**

are presented as an example of experimental and numerical studies on LPA.

In this section, recent results of works conducted in the National Institute of Advanced Industrial Science and Technology (AIST) (Masuda & Miura, 2008; Miura & Masuda, 2009)

A Ti:sapphire laser system with a repetition rate of 10 Hz based on CPA method was used for experiments. In our laser system, two intense laser pulses were available. A 20-fs laser pulse from a Kerr-lens mode-locked oscillator was stretched to 400 ps through an Öffner type pulse stretcher, which was an aberration-free type stretcher(Cheriaux et al., 1996). To control a laser pulse shape, an acousto-optic programmable dispersive filter (Dazzler: Fastlite)(Verluise et al., 2000) was installed after the pulse stretcher. The laser pulse was amplified by a regenerative amplifier and multi-pass amplifiers. To suppress and control a prepulse, a pulse cleaner composed of a Pockels cell and a thin film polarizer was set between the regenerative and the first muti-pass amplifiers. After the first multi-pass amplifier, the laser pulse was split into three pulses. The first pulse referred as main laser pulse was amplified by a multi-pass amplifier, and a laser pulse with an energy of 750 mJ, a pulse duration of 35-50 fs, and a center wavelength of 800 nm was obtained after a vacuum pulse compressor. The main laser pulse was used for electron acceleration. The second pulse referred as colliding laser pulse was amplified by two multi-pass amplifiers, and a laser pulse with an energy of 150 mJ, a The experimental setup for electron acceleration is shown in Fig. 2. A p-polarized main laser pulse (400 mJ, 50 fs, 800 nm) was focused onto the edge of a helium gas jet using an *f* /6 off-axis parabolic mirror with a focal length of 300 mm. The laser spot size in vacuum was 9 *μ*m at the full-width-at-half-maximum (FWHM), and the Rayleigh length was 230 *μ*m. The energy concentration within the *e*−<sup>2</sup> spot was 58% of the total laser energy. The peak intensity was 5.8 <sup>×</sup> 1018 W/cm2, corresponding to the normalized vector potential of 1.6. The helium gas jet was ejected from a supersonic nozzle with a conical shape driven by a pulsed valve. The focal position was set at 1 mm above the nozzle exit. The diameter of the nozzle exit was 0.7 mm and the Mach number was 5.3. The density profile of the gas jet was measured with a Jamin interferometer. The density at the center was <sup>∼</sup> <sup>5</sup> <sup>×</sup> 1018 cm−3, corresponding to the electron density of <sup>∼</sup> 1019 cm−3.

Fig. 2. Experimental setup for electron acceleration.

The energy spectrum of an electron beam was measured by an electron energy spectrometer with a dipole magnet. The measured energy range was from 1 to 80 MeV. An energy-resolved electron image was recorded on an imaging plate [(IP), Fujifilm: BAS-SR]. To estimate the absolute charge of electrons , the sensitivity curve reported in Ref.(Tanaka et al., 2005) was used. In addition, we have developed an absolutely-calibrated in-situ observation system for the electron energy spectrum. A Gd2O2S : Tb phosphor screen (Mitsubishi chemical: DRZ-HIGH) coupled with a CCD camera was used as an electron detector. The sensitivity of the detection system was calibrated using an IP as the reference detector (Masuda et al., 2008). The absolute electron energy spectrum was in-situ observed with a single shot at a repetition rate of 1 Hz.

beam parameters should be defined for judging QME beam generation. We set the following four criterion values. First, the peak energy is higher than 10 MeV. Second, the number of electrons in the electron energy spectrum is higher than 10<sup>10</sup> /MeV/sr at the monoenergetic peak. Third, the ratio of the peak to the background at the monoenergetic peak is higher than 2. Forth, the relative energy spread is less than 50%. When all of these conditions are fulfilled, we consider that a QME beam has generated. According to the criteria, QME beams are obtained in 10 shots among 14 shots in the case of Fig. 4. The probability of QME beam generation is 70%, if the shot-to-shot fluctuations of the peak energy, the number of electrons, and so on are ignored. In our previous experiment (Masuda et al., 2007), the probability of

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 31

Electron energy [MeV]

conditions as those giving the result shown in Fig. 3. The probability of QME beam

Fig. 4. Electron energy spectra obtained by sequential 14 shots under the same experimental

QME beam generation was a few tens of percent. The probability in the present experiment increased by a factor of 3, as compared with our previous experiment. It has been pointed out that there is a threshold energy of a laser pulse for stable generation of QME beams, which depends on the electron density of the plasma and the laser pulse duration (Mangles et al., 2007). Our present experimental conditions are in the range for stable generation predicted by

Figure 5 shows the distribution of the peak energy and the charge in the monoenergetic peak of QME beams for about 50 shots obtained under the same conditions as those giving the result shown in Fig. 3. A QME beam with a peak energy of up to 75 MeV was produced. A QME beam containing up to 88 pC in the peak at the energy of 48 MeV was also produced. In this case, the total energy of electrons in the peak was 4.2 mJ. This means that the efficiency of energy conversion from the laser pulse to the electron beam was 1%. Table 1 lists mean values and standard deviations of the parameters of QME beams obtained in about 50 shots shown in Fig. 5. The beam pointing means the angular deviation of the beam center from the laser propagation axis in the vertical direction. Although the probability of QME beam generation increases, shot-to-shot fluctuations in the charge and the energy spread are still large. The suppression of the fluctuations in the beam parameters is a key issue toward a next step. Figure 6 shows the dependence of electron energy spectra on the electron density of the plasma, where the laser power is fixed at 8.3 TW. The electron density was varied by

0 20 40 60 80

Number of electrons

scaling in Ref. (Mangles et al., 2007).

generation is 70%.

[1011 /MeV/sr]

2.0 1.5

0.5 0

1.0

Shot number

5

0

10

15

To observe the propagation of a laser pulse and the formation of a preformed plasma, a side-scattered light image and a shadowgraph image were observed simultaneously with the measurement of an electron energy spectrum. The side-scattered light image was observed through an interference filter of 800 nm from the top of the gas jet. The shadowgraph image was observed with a 60-fs, 800-nm probe laser pulse. The spatial resolutions in both the measurements were approximately 40 *μ*m.

Contrast ratios of femtosecond prepulses, generated in a regenerative amplifier, preceding the main pulse by more than 4 ns were always kept to be less than 10−<sup>6</sup> using a pulse cleaner. Our another work has demonstrated that the number of accelerated electrons was dramatically decreased for low contrast ratios of the femtosecond prepulses, although the laser power and the electron density of the plasma were different from those for the present experiment (Masuda & Miura, 2010). Then, the femtosecond prepulses were suppressed.

#### **3.2 Quasi-monoenergetic electron beam generation**

Figure 3(a) shows an energy-resolved electron image of a typical QME beam from a plasma with an electron density of 1.6 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.3-TW laser pulse. The image was recorded on an IP with a single shot. The small spot indicates the generation of the monoenergetic beam with a narrow energy spread and a small divergence angle. The divergence angle in the vertical direction was estimated to be 7 mrad at the FWHM from the vertical size of the spot. The electron energy spectrum is shown in Fig. 3(b). A monoenergetic peak was observed at the energy of 38 MeV with the relative energy spread of 19%. Here, the relative energy spread is defined as the ratio of the energy spread at the FWHM to the peak energy. The observed energy spread was little bit larger than the instrumental resolution of several percent. The total number of electrons in the monoenergetic peak was 3.1 <sup>×</sup> 108, corresponding to the charge of 50 pC.

Figure 4 shows the electron energy spectra obtained by sequential 14 shots under the same conditions as those giving the result shown in Fig. 3. All the spectra were obtained with a single shot. To evaluate the stability of QME beam generation, the criterion values of principal

Fig. 3. (a) Typical energy-resolved electron image recorded on an IP with a single shot and (b) electron energy spectrum of a QME beam obtained from a plasma with an electron density of 1.6 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.3-TW laser pulse.

8 Will-be-set-by-IN-TECH

To observe the propagation of a laser pulse and the formation of a preformed plasma, a side-scattered light image and a shadowgraph image were observed simultaneously with the measurement of an electron energy spectrum. The side-scattered light image was observed through an interference filter of 800 nm from the top of the gas jet. The shadowgraph image was observed with a 60-fs, 800-nm probe laser pulse. The spatial resolutions in both the

Contrast ratios of femtosecond prepulses, generated in a regenerative amplifier, preceding the main pulse by more than 4 ns were always kept to be less than 10−<sup>6</sup> using a pulse cleaner. Our another work has demonstrated that the number of accelerated electrons was dramatically decreased for low contrast ratios of the femtosecond prepulses, although the laser power and the electron density of the plasma were different from those for the present experiment (Masuda & Miura, 2010). Then, the femtosecond prepulses were suppressed.

Figure 3(a) shows an energy-resolved electron image of a typical QME beam from a plasma with an electron density of 1.6 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.3-TW laser pulse. The image was recorded on an IP with a single shot. The small spot indicates the generation of the monoenergetic beam with a narrow energy spread and a small divergence angle. The divergence angle in the vertical direction was estimated to be 7 mrad at the FWHM from the vertical size of the spot. The electron energy spectrum is shown in Fig. 3(b). A monoenergetic peak was observed at the energy of 38 MeV with the relative energy spread of 19%. Here, the relative energy spread is defined as the ratio of the energy spread at the FWHM to the peak energy. The observed energy spread was little bit larger than the instrumental resolution of several percent. The total number of electrons in the monoenergetic peak was 3.1 <sup>×</sup> 108,

Figure 4 shows the electron energy spectra obtained by sequential 14 shots under the same conditions as those giving the result shown in Fig. 3. All the spectra were obtained with a single shot. To evaluate the stability of QME beam generation, the criterion values of principal

1.0

1.5

2.0

0.5

0

Number of electrons [1011 /MeV/sr]

Fig. 3. (a) Typical energy-resolved electron image recorded on an IP with a single shot and (b) electron energy spectrum of a QME beam obtained from a plasma with an electron density of

0 20 40 60 80

**(b)**

Electron energy [MeV]

measurements were approximately 40 *μ*m.

corresponding to the charge of 50 pC.

**(a)**

Divergence [arb. units]

**3.2 Quasi-monoenergetic electron beam generation**

Electron energy [MeV] 0.2 10 20 40 60 80 2

1.6 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.3-TW laser pulse.

beam parameters should be defined for judging QME beam generation. We set the following four criterion values. First, the peak energy is higher than 10 MeV. Second, the number of electrons in the electron energy spectrum is higher than 10<sup>10</sup> /MeV/sr at the monoenergetic peak. Third, the ratio of the peak to the background at the monoenergetic peak is higher than 2. Forth, the relative energy spread is less than 50%. When all of these conditions are fulfilled, we consider that a QME beam has generated. According to the criteria, QME beams are obtained in 10 shots among 14 shots in the case of Fig. 4. The probability of QME beam generation is 70%, if the shot-to-shot fluctuations of the peak energy, the number of electrons, and so on are ignored. In our previous experiment (Masuda et al., 2007), the probability of

Fig. 4. Electron energy spectra obtained by sequential 14 shots under the same experimental conditions as those giving the result shown in Fig. 3. The probability of QME beam generation is 70%.

QME beam generation was a few tens of percent. The probability in the present experiment increased by a factor of 3, as compared with our previous experiment. It has been pointed out that there is a threshold energy of a laser pulse for stable generation of QME beams, which depends on the electron density of the plasma and the laser pulse duration (Mangles et al., 2007). Our present experimental conditions are in the range for stable generation predicted by scaling in Ref. (Mangles et al., 2007).

Figure 5 shows the distribution of the peak energy and the charge in the monoenergetic peak of QME beams for about 50 shots obtained under the same conditions as those giving the result shown in Fig. 3. A QME beam with a peak energy of up to 75 MeV was produced. A QME beam containing up to 88 pC in the peak at the energy of 48 MeV was also produced. In this case, the total energy of electrons in the peak was 4.2 mJ. This means that the efficiency of energy conversion from the laser pulse to the electron beam was 1%. Table 1 lists mean values and standard deviations of the parameters of QME beams obtained in about 50 shots shown in Fig. 5. The beam pointing means the angular deviation of the beam center from the laser propagation axis in the vertical direction. Although the probability of QME beam generation increases, shot-to-shot fluctuations in the charge and the energy spread are still large. The suppression of the fluctuations in the beam parameters is a key issue toward a next step.

Figure 6 shows the dependence of electron energy spectra on the electron density of the plasma, where the laser power is fixed at 8.3 TW. The electron density was varied by

0 20 40 60 80 Electron energy [MeV]

Fig. 6. The electron energy spectra observed for different electron densities of 1.3 <sup>×</sup> 1019 (dotted curve), 1.6 <sup>×</sup> 1019 (solid curve), and 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> (dash-dotted curve),

Shot number

10

15

20

Number of electrons

Fig. 7. Series of electron energy spectra obtained by about 20 consecutive shots at a repetition rate of 1 Hz for the different extraction times of the main pulse at the pulse cleaner: (a)

[1010 /MeV/sr]

0.5 0

1.0

**(b)**

 1.5

0 20 40 60 80

Electron energy [MeV]

15

10

Shot number

5

0

Figure 7(a) shows a series of electron energy spectra obtained by 21 consecutive shots at a repetition rate of 1 Hz from a plasma with an electron density of 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.5-TW laser pulse. The electron energy spectra were obtained by optimizing the extraction time of the main pulse at the pulse cleaner in addition to the optimization of the electron density. Here, this time is defined as the extraction time of the main pulse, *t*ext = 0 ns. Figure 7(b) shows a series of electron energy spectra obtained by 17 consecutive shots at a repetition rate of 1 Hz for the different extraction time *t*ext = −2.5 ns at the same laser power and electron density. The length of the nanosecond prepulse is longer, as the extraction time becomes earlier. As shown in Fig. 7(a), in most of shots, a monoenergetic peak was observed. In contrast, when the length of the nanosecond prepulse was long, monoenergetic peaks in the low energy range were observed only in a few shots among 17 consecutive shots as shown in Fig. 7(b). No monoenergetic peaks in the high energy range were observed. Furthermore,

respectively. To obtain QME beams, plasma density control is important.

1.9x10<sup>19</sup> 1.3x1019 cm-3 cm-3

1.6x1019 cm-3

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 33

Number of electrons [/MeV/sr]

Electron energy [MeV]

*t*ext = 0 ns and (b) *t*ext = −2.5 ns.

20 40 60 80

0

5

Number of electrons

[1011 /MeV/sr]

0.5 0

1.0

0

**(a)**

 1.5 10<sup>8</sup>

10<sup>9</sup>

10<sup>10</sup>

10<sup>11</sup>

10<sup>12</sup>

Fig. 5. Histogram showing the distribution of the peak energy and the charge in the monoenergetic peak of QME beams for about 50 shots obtained under the same conditions as those giving the result shown in Fig. 3.


Table 1. Statistics of the parameters of QME beams obtained in about 50 shots shown in Fig. 5.

controlling the gas jet density. As described above, QME beams were generated at an electron density of 1.6 <sup>×</sup> 1019 cm−3, as shown by the solid curve. At a lower electron density of 1.3 <sup>×</sup> 1019 cm−3, high-energy electrons were not observed, as shown by the dotted curve. On the other hand, at a higher density of 1.9 <sup>×</sup> 1019 cm−3, the electron energy spectrum was a Maxwell-like distribution, and no clear monoenergetic peaks were seen, as shown by the dash-dotted curve. QME beams can be generated only in the narrow electron density region around 1.6 <sup>×</sup> 1019 cm−3. This result shows that plasma density control is crucially important for the generation of QME beams.

#### **3.3 Effect of a nanosecond prepulse on quasi-monoenergetic electron beam generation**

There are many issues that must be controlled for stable generation of QME beams in laser-plasma interactions. Among them, the control of a prepulse and a preformed plasma is one key issue. In this subsection, we show the effects of a nanosecond prepulse on QME beam generation and propagation of an intense laser pulse.

The experimental conditions have been already shown in Sec. 3.1. The identical off-axis parabolic mirror and super sonic nozzle were used. As described in Sec. 3.1, the contrast ratios of femtosecond prepulses to a main pulse were kept to be less than 10−6. The length of a nanosecond prepulse caused by amplified spontaneous emission (ASE) was controlled by varying the extraction time of the main pulse at the pulse cleaner.

**Dependence of electron beam characteristics on length of nanosecond prepulse**

10 Will-be-set-by-IN-TECH

0

30

Charge [pC]

60

90

90

Parameters Mean ± Standard deviation

Peak energy 49 ± 15 MeV Relative energy spread 22 ± 12 % Charge in the monoenergetic peak 24 ± 20 pC Divergence angle in the vertical direction (FWHM) 7.1 ± 4.0 mrad Beam pointing in the vertical direction ±7.8 mrad

0

30

Peak energy [MeV]

Fig. 5. Histogram showing the distribution of the peak energy and the charge in the

60

monoenergetic peak of QME beams for about 50 shots obtained under the same conditions as

Table 1. Statistics of the parameters of QME beams obtained in about 50 shots shown in Fig. 5.

controlling the gas jet density. As described above, QME beams were generated at an electron density of 1.6 <sup>×</sup> 1019 cm−3, as shown by the solid curve. At a lower electron density of 1.3 <sup>×</sup> 1019 cm−3, high-energy electrons were not observed, as shown by the dotted curve. On the other hand, at a higher density of 1.9 <sup>×</sup> 1019 cm−3, the electron energy spectrum was a Maxwell-like distribution, and no clear monoenergetic peaks were seen, as shown by the dash-dotted curve. QME beams can be generated only in the narrow electron density region around 1.6 <sup>×</sup> 1019 cm−3. This result shows that plasma density control is crucially important

**3.3 Effect of a nanosecond prepulse on quasi-monoenergetic electron beam generation** There are many issues that must be controlled for stable generation of QME beams in laser-plasma interactions. Among them, the control of a prepulse and a preformed plasma is one key issue. In this subsection, we show the effects of a nanosecond prepulse on QME

The experimental conditions have been already shown in Sec. 3.1. The identical off-axis parabolic mirror and super sonic nozzle were used. As described in Sec. 3.1, the contrast ratios of femtosecond prepulses to a main pulse were kept to be less than 10−6. The length of a nanosecond prepulse caused by amplified spontaneous emission (ASE) was controlled by

**Dependence of electron beam characteristics on length of nanosecond prepulse**

0 2

6

Number of events

those giving the result shown in Fig. 3.

for the generation of QME beams.

beam generation and propagation of an intense laser pulse.

varying the extraction time of the main pulse at the pulse cleaner.

4

8

10

Fig. 6. The electron energy spectra observed for different electron densities of 1.3 <sup>×</sup> 1019 (dotted curve), 1.6 <sup>×</sup> 1019 (solid curve), and 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> (dash-dotted curve), respectively. To obtain QME beams, plasma density control is important.

Figure 7(a) shows a series of electron energy spectra obtained by 21 consecutive shots at a repetition rate of 1 Hz from a plasma with an electron density of 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> produced by an 8.5-TW laser pulse. The electron energy spectra were obtained by optimizing the extraction time of the main pulse at the pulse cleaner in addition to the optimization of the electron density. Here, this time is defined as the extraction time of the main pulse, *t*ext = 0 ns. Figure 7(b) shows a series of electron energy spectra obtained by 17 consecutive shots at a repetition rate of 1 Hz for the different extraction time *t*ext = −2.5 ns at the same laser power and electron density. The length of the nanosecond prepulse is longer, as the extraction time becomes earlier. As shown in Fig. 7(a), in most of shots, a monoenergetic peak was observed. In contrast, when the length of the nanosecond prepulse was long, monoenergetic peaks in the low energy range were observed only in a few shots among 17 consecutive shots as shown in Fig. 7(b). No monoenergetic peaks in the high energy range were observed. Furthermore,

Fig. 7. Series of electron energy spectra obtained by about 20 consecutive shots at a repetition rate of 1 Hz for the different extraction times of the main pulse at the pulse cleaner: (a) *t*ext = 0 ns and (b) *t*ext = −2.5 ns.

pulse at the focal position. When the length of a nanosecond prepulse is long, a preformed plasma is observed around the vacuum focal position as denoted by arrows, although the image in Fig. 9(a) is not clear. As shown in Fig. 9(b), the length of the preformed plasma is approximately 200 *μ*m, which is close to the Rayleigh length. Because the time delay of the probe pulse is −1 ps, the preformed plasma may be produced by a picosecond prepulse. In another experiment using an 8-TW laser pulse, formation of a preformed plasma was observed in a shadowgraph image taken at several tens of picoseconds before the arrival of the main pulse for *t*ext = −2.5 ns. Then, we think that the preformed plasma shown in Fig. 9

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 35

0 1000 2000 3000 Position [μm]

Fig. 9. (a) Typical shadowgraph image and (b) horizontal profile of the image along the laser propagation axis for the extraction time, *t*ext = −2.5 ns. The image is taken at 1 ps before the arrival of the main pulse at the focal position. A preformed plasma is observed as denoted by

Figure 10 shows typical side-scattered light images observed for the different extraction times: (a) *t*ext = 0 ns and (b) *t*ext = −2.5 ns, respectively. The laser pulse propagates from left to right in the images. The dotted circle indicates the nozzle exit position. Both the images were observed through filters with the same attenuation and the laser energies were almost same shown in Fig. 8. When the nanosecond prepulse was suppressed (Fig. 10(a)), the image was similar to that suggesting the formation of a plasma channel reported in Refs. (Masuda et al., 2007; Miura et al., 2005). In contrast, when the length of the nanosecond prepulse was long, strong side-scattered light was observed as shown in Fig. 10(b). The image was quite different from that reported in Refs. (Masuda et al., 2007; Miura et al., 2005). The main pulse is strongly scattered, because the density profile of the preformed plasma is unsuitable for guiding the laser pulse. The remnant energy of the main pulse is insufficient to excite the plasma wave with a large amplitude. Then, no energetic electrons are accelerated. The side-scattered light image shown in Fig. 10(b) suggests that a certain fraction of the energy of the main pulse is

Nozzle Exit

was produced by a nanosecond prepulse.

50

100

150

CCD counts [arb. unit]

scattered by the preformed plasma.

the arrows.

200

**(a)**

**(b)**

500 μm

the scale of the number of electrons in Fig. 7(b) is one-tenth of that in Fig. 7(a). The probability of the generation of QME beams was estimated according to the criteria described in Sec. 3.2. QME beams are obtained in 17 shots among 21 shots in the case of Fig. 7(a). The probability of QME beam generation is 81%, if the shot-to-shot fluctuations of the peak energy, the number of electrons, and so on are ignored. The probability of QME beam generation is 0 in the case of Fig. 7(b) according to the criteria described in Sec. 3.2.

Figure 8 shows the dependence of the probability of the QME beam generation (closed circles) and the laser energy (open triangles) on the extraction time of the main pulse at the pulse cleaner. Each data point of the probability was obtained with more than 10 consecutive shots at a repetition rate of 1 Hz. The electron density of the plasma was 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> for all data points. Notably, a few data points overlap at *t*ext = −2.5, −1.5, and +1.5 ns, when the probability is 0. The data points for the probability of QME beam generation are divided into three groups. When the length of the nanosecond prepulse is long (*t*ext ∼ −2 ns), the probability is quite low. In contrast, when the nanosecond prepulse is suppressed (*t*ext ∼ 0 ns), the probability dramatically increases and is close to 90%. This result shows that it is necessary to suppress a nanosecond prepulse for generation of QME beams. When the extraction time is too late (*t*ext = +1.5 ns), the probability becomes quite low again. This is due to the reduction in the laser energy as shown in Fig. 8.

Fig. 8. Dependence of the probability of QME beam generation (closed circles) and the laser energy (open triangles) on the extraction time of the main pulse at the pulse cleaner. With the earlier extraction time, the length of a nanosecond prepulse is longer. For generation of QME beams, the suppression of the nanosecond prepulse is necessary.

The optimum electron density for QME beam generation in the experiment presented in this subsection was slightly higher than that of the results in shown in Sec. 3.2, although the experimental conditions were very close. The small difference of the laser conditions may cause the difference of the optimum electron density.

#### **Plasma diagnostics**

Figures 9(a) and (b) show a shadowgraph image and a horizontal profile of the image along the laser propagation axis for *t*ext = −2.5 ns, respectively. The laser pulse propagates from left to right in the image. The black area is the shadow of the nozzle and the nozzle exit position is shown by the white area. The image was taken at 1 ps before the arrival of the main 12 Will-be-set-by-IN-TECH

the scale of the number of electrons in Fig. 7(b) is one-tenth of that in Fig. 7(a). The probability of the generation of QME beams was estimated according to the criteria described in Sec. 3.2. QME beams are obtained in 17 shots among 21 shots in the case of Fig. 7(a). The probability of QME beam generation is 81%, if the shot-to-shot fluctuations of the peak energy, the number of electrons, and so on are ignored. The probability of QME beam generation is 0 in the case

Figure 8 shows the dependence of the probability of the QME beam generation (closed circles) and the laser energy (open triangles) on the extraction time of the main pulse at the pulse cleaner. Each data point of the probability was obtained with more than 10 consecutive shots at a repetition rate of 1 Hz. The electron density of the plasma was 1.9 <sup>×</sup> 1019 cm−<sup>3</sup> for all data points. Notably, a few data points overlap at *t*ext = −2.5, −1.5, and +1.5 ns, when the probability is 0. The data points for the probability of QME beam generation are divided into three groups. When the length of the nanosecond prepulse is long (*t*ext ∼ −2 ns), the probability is quite low. In contrast, when the nanosecond prepulse is suppressed (*t*ext ∼ 0 ns), the probability dramatically increases and is close to 90%. This result shows that it is necessary to suppress a nanosecond prepulse for generation of QME beams. When the extraction time is too late (*t*ext = +1.5 ns), the probability becomes quite low again. This is due to the reduction

> Extraction time of main pulse [ns] -3 -2 -1 0 1 2

Fig. 8. Dependence of the probability of QME beam generation (closed circles) and the laser energy (open triangles) on the extraction time of the main pulse at the pulse cleaner. With the earlier extraction time, the length of a nanosecond prepulse is longer. For generation of QME

The optimum electron density for QME beam generation in the experiment presented in this subsection was slightly higher than that of the results in shown in Sec. 3.2, although the experimental conditions were very close. The small difference of the laser conditions may

Figures 9(a) and (b) show a shadowgraph image and a horizontal profile of the image along the laser propagation axis for *t*ext = −2.5 ns, respectively. The laser pulse propagates from left to right in the image. The black area is the shadow of the nozzle and the nozzle exit position is shown by the white area. The image was taken at 1 ps before the arrival of the main

300

400

Laser energy [mJ]

500

600

of Fig. 7(b) according to the criteria described in Sec. 3.2.

0

beams, the suppression of the nanosecond prepulse is necessary.

cause the difference of the optimum electron density.

**Plasma diagnostics**

20

40

60

Probability of quasi-monoenergetic

electron beam generation [%]

80

100

in the laser energy as shown in Fig. 8.

pulse at the focal position. When the length of a nanosecond prepulse is long, a preformed plasma is observed around the vacuum focal position as denoted by arrows, although the image in Fig. 9(a) is not clear. As shown in Fig. 9(b), the length of the preformed plasma is approximately 200 *μ*m, which is close to the Rayleigh length. Because the time delay of the probe pulse is −1 ps, the preformed plasma may be produced by a picosecond prepulse. In another experiment using an 8-TW laser pulse, formation of a preformed plasma was observed in a shadowgraph image taken at several tens of picoseconds before the arrival of the main pulse for *t*ext = −2.5 ns. Then, we think that the preformed plasma shown in Fig. 9 was produced by a nanosecond prepulse.

Fig. 9. (a) Typical shadowgraph image and (b) horizontal profile of the image along the laser propagation axis for the extraction time, *t*ext = −2.5 ns. The image is taken at 1 ps before the arrival of the main pulse at the focal position. A preformed plasma is observed as denoted by the arrows.

Figure 10 shows typical side-scattered light images observed for the different extraction times: (a) *t*ext = 0 ns and (b) *t*ext = −2.5 ns, respectively. The laser pulse propagates from left to right in the images. The dotted circle indicates the nozzle exit position. Both the images were observed through filters with the same attenuation and the laser energies were almost same shown in Fig. 8. When the nanosecond prepulse was suppressed (Fig. 10(a)), the image was similar to that suggesting the formation of a plasma channel reported in Refs. (Masuda et al., 2007; Miura et al., 2005). In contrast, when the length of the nanosecond prepulse was long, strong side-scattered light was observed as shown in Fig. 10(b). The image was quite different from that reported in Refs. (Masuda et al., 2007; Miura et al., 2005). The main pulse is strongly scattered, because the density profile of the preformed plasma is unsuitable for guiding the laser pulse. The remnant energy of the main pulse is insufficient to excite the plasma wave with a large amplitude. Then, no energetic electrons are accelerated. The side-scattered light image shown in Fig. 10(b) suggests that a certain fraction of the energy of the main pulse is scattered by the preformed plasma.

150 160 170

0 25 50 75 100 Electron energy [MeV]

components, as seen in Fig. 3(a).

**generation**

**Preformed plasma model in simulation**


Angle [mrad]

**(d)**

0

Position, y [μm]


140

10

Electron energy [MeV]

Number of electrons [arb. unit]

0

20

40

80 60

0

Position, y [μm]



Angle [mrad]

**(f)**

540 550 560 570

<sup>100</sup> **(a)**

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 37

0 25 50 75 100 Electron energy [MeV]

Fig. 11. 2D-PIC simulation results for an electron density of 1.6 <sup>×</sup> 1019 cm−3. Snapshots of the electron distribution on the x-y plane (a)-(c) and the number of electrons as functions of the energy and the angle of the momentum vector (d)-(f) are shown for three different positions.

The generation of a monoenergetic bunch with a peak energy of 90 MeV, a relative energy spread of 6% at the FWHM, and a divergence angle of 28 mrad at the FWHM is predicted. The predicted values of the monoenergetic component are fairly close to the experimental results. As seen in Fig. 11(e), only the monoenergetic component has a small divergence angle. In contrast, the low-energy component has a large divergence. The simulation result explains the difference in the beam divergence between the monoenergetic and the low-energy

Simulations were conducted for electron densities of 1.3 and 1.9 <sup>×</sup> 1019 cm−3. For all the densities, including 1.6 <sup>×</sup> 1019 cm−3, electron injection occurs and a monoenergetic electron bunch is formed in the first period of the plasma wave, immediately after the amplitude of the plasma wave reaches the maximum. However, the growth rate of the plasma wave depends on the electron density, and the laser propagation lengths required for electron injection are different for each electron density. The growth rate is larger as the electron density is higher. At an electron density of 1.3 <sup>×</sup> 1019 cm−3, the laser propagation length required for electron injection is longer than the gas jet length of 1 mm in the experiment. Because the electron injection does not occur inside the gas jet, high-energy electrons are not observed. In contrast, at an electron density of 1.9 <sup>×</sup> 1019 cm−3, the electron injection occurs too early, and the trapped electrons enter the deceleration phase, so the energy dissipates at the end of the gas jet. Then, a monoenergetic peak is not observed. The simulation results also explain the density dependence of the electron energy spectra shown in Fig. 6 (Masuda & Miura, 2009).

**3.4.2 Analysis on effect of a nanosecond prepulse on quasi-monoenergetic electron beam**

To investigate effects of a nanosecond prepulse on QME beam generation shown in Sec. 3.3, 2D-PIC simulations were also conducted. The dynamics both of a preformed plasma produced by a nanosecond prepulse and the propagation of a femtosecond laser pulse in a millimeter-scale plasma can not be treated simultaneously in the PIC simulation, because


Angle [mrad]

**(e)**

Position, x [μm] Position, x [μm]

**(b)**

0

Position, y [μm]


10

Position, x [μm]

1020 1030 1040 1050

0 25 50 75 100 Electron energy [MeV]

10

**(c)**

Fig. 10. Typical side-scattered light images observed for the different extraction times: (a) *t*ext = 0 ns and (b) *t*ext = −2.5 ns. The both images are observed through filters with the same attenuation. When the length of the nanosecond prepulse is long, the strong side-scattered light is observed.

## **3.4 Analysis on electron acceleration using two-dimensional particle-in-cell simulations 3.4.1 Analysis on quasi-monoenergetic electron beam generation**

#### **Conditions of two-dimensional particle-in-cell simulations**

To investigate in detail the generation of QME beams shown in Sec. 3.2, we have developed a two-dimensional particle-in-cell (2D-PIC) simulation code using a moving window technique. The moving window had 127 <sup>×</sup> <sup>127</sup>*μ*m<sup>2</sup> size with 2000 <sup>×</sup> 1000 cells containing 5 particles per cell. Simulations were conducted on the *x*-*y* plane. The laser propagation direction and the polarization direction were set in the *x*- and the *y*-axes. The laser pulse had a transverse profile with a TEM00 Hermite-Gaussian mode and a cosine temporal envelope. The laser pulse energy, the spot size, and the pulse duration were 400 mJ, 9 *μ*m at the FWHM and 50 fs at the FWHM, respectively. The electron density profile along the *x*-axis was set to be similar to the measured density profile of a gas jet. The center of the gas jet in the experiment was set at *x* = 0. The initial electron density raised at *x* = −960 *μ*m, and smoothly increased to the maximum of 1.6 <sup>×</sup> 1019 cm−<sup>3</sup> at *<sup>x</sup>* <sup>=</sup> <sup>−</sup><sup>350</sup> *<sup>μ</sup>*m. To investigate the interaction for a long propagation length, it was assumed that the electron density was constant from the position where the density reached the maximum value. The focal position in vacuum was set at *x* = −300 *μ*m. The density profile was uniform along the *y*-axis.

#### **Simulation results**

Figure 11 shows snapshots of the electron distribution on the x-y plane (a)-(c) and the number of electrons as functions of the energy and the angle of the momentum vector (d)-(f) for three different positions. In Figs. 11(a)-(c), the leading edge of the laser pulse is near the right edge and each electron is colored by its energy. At *x* = 170 *μ*m, electrons are not trapped in a plasma wave, because the amplitude of the plasma wave is small. The high-energy electrons are not observed in Fig. 11(d). After some time, trapping of electrons occurs due to sufficient growth of the plasma wave, and a monoenergetic electron bunch is trapped and accelerated in the first period of the plasma wave at *x* = 570 *μ*m, as seen in Figs. 11(b) and (e). The fine structure in the monoenergetic bunch seen in Fig. 11(b) is due to the interaction of electrons with the laser field. At *x* = 1050 *μ*m, the trapped electrons enter the deceleration phase and dissipate the energy. Then, the monoenergetic component disappears, as seen in Fig. 11(f). This result shows that the extraction position of electrons from the plasma is important for QME beam generation. The predicted optimum laser propagation length from the focal position is 870 *μ*m, which is close to the gas jet length of approximately 1 mm in the experiment. This suggests that the generation of a QME beam is brought about by matching the laser propagation length with the gas jet length.

14 Will-be-set-by-IN-TECH

Fig. 10. Typical side-scattered light images observed for the different extraction times: (a) *t*ext = 0 ns and (b) *t*ext = −2.5 ns. The both images are observed through filters with the same attenuation. When the length of the nanosecond prepulse is long, the strong

**3.4 Analysis on electron acceleration using two-dimensional particle-in-cell simulations**

To investigate in detail the generation of QME beams shown in Sec. 3.2, we have developed a two-dimensional particle-in-cell (2D-PIC) simulation code using a moving window technique. The moving window had 127 <sup>×</sup> <sup>127</sup>*μ*m<sup>2</sup> size with 2000 <sup>×</sup> 1000 cells containing 5 particles per cell. Simulations were conducted on the *x*-*y* plane. The laser propagation direction and the polarization direction were set in the *x*- and the *y*-axes. The laser pulse had a transverse profile with a TEM00 Hermite-Gaussian mode and a cosine temporal envelope. The laser pulse energy, the spot size, and the pulse duration were 400 mJ, 9 *μ*m at the FWHM and 50 fs at the FWHM, respectively. The electron density profile along the *x*-axis was set to be similar to the measured density profile of a gas jet. The center of the gas jet in the experiment was set at *x* = 0. The initial electron density raised at *x* = −960 *μ*m, and smoothly increased to the maximum of 1.6 <sup>×</sup> 1019 cm−<sup>3</sup> at *<sup>x</sup>* <sup>=</sup> <sup>−</sup><sup>350</sup> *<sup>μ</sup>*m. To investigate the interaction for a long propagation length, it was assumed that the electron density was constant from the position where the density reached the maximum value. The focal position in vacuum was set at

Figure 11 shows snapshots of the electron distribution on the x-y plane (a)-(c) and the number of electrons as functions of the energy and the angle of the momentum vector (d)-(f) for three different positions. In Figs. 11(a)-(c), the leading edge of the laser pulse is near the right edge and each electron is colored by its energy. At *x* = 170 *μ*m, electrons are not trapped in a plasma wave, because the amplitude of the plasma wave is small. The high-energy electrons are not observed in Fig. 11(d). After some time, trapping of electrons occurs due to sufficient growth of the plasma wave, and a monoenergetic electron bunch is trapped and accelerated in the first period of the plasma wave at *x* = 570 *μ*m, as seen in Figs. 11(b) and (e). The fine structure in the monoenergetic bunch seen in Fig. 11(b) is due to the interaction of electrons with the laser field. At *x* = 1050 *μ*m, the trapped electrons enter the deceleration phase and dissipate the energy. Then, the monoenergetic component disappears, as seen in Fig. 11(f). This result shows that the extraction position of electrons from the plasma is important for QME beam generation. The predicted optimum laser propagation length from the focal position is 870 *μ*m, which is close to the gas jet length of approximately 1 mm in the experiment. This suggests that the generation of a QME beam is brought about by matching

**(a) (b)**

500 μm

**3.4.1 Analysis on quasi-monoenergetic electron beam generation Conditions of two-dimensional particle-in-cell simulations**

*x* = −300 *μ*m. The density profile was uniform along the *y*-axis.

the laser propagation length with the gas jet length.

side-scattered light is observed.

**Simulation results**

Fig. 11. 2D-PIC simulation results for an electron density of 1.6 <sup>×</sup> 1019 cm−3. Snapshots of the electron distribution on the x-y plane (a)-(c) and the number of electrons as functions of the energy and the angle of the momentum vector (d)-(f) are shown for three different positions.

The generation of a monoenergetic bunch with a peak energy of 90 MeV, a relative energy spread of 6% at the FWHM, and a divergence angle of 28 mrad at the FWHM is predicted. The predicted values of the monoenergetic component are fairly close to the experimental results. As seen in Fig. 11(e), only the monoenergetic component has a small divergence angle. In contrast, the low-energy component has a large divergence. The simulation result explains the difference in the beam divergence between the monoenergetic and the low-energy components, as seen in Fig. 3(a).

Simulations were conducted for electron densities of 1.3 and 1.9 <sup>×</sup> 1019 cm−3. For all the densities, including 1.6 <sup>×</sup> 1019 cm−3, electron injection occurs and a monoenergetic electron bunch is formed in the first period of the plasma wave, immediately after the amplitude of the plasma wave reaches the maximum. However, the growth rate of the plasma wave depends on the electron density, and the laser propagation lengths required for electron injection are different for each electron density. The growth rate is larger as the electron density is higher. At an electron density of 1.3 <sup>×</sup> 1019 cm−3, the laser propagation length required for electron injection is longer than the gas jet length of 1 mm in the experiment. Because the electron injection does not occur inside the gas jet, high-energy electrons are not observed. In contrast, at an electron density of 1.9 <sup>×</sup> 1019 cm−3, the electron injection occurs too early, and the trapped electrons enter the deceleration phase, so the energy dissipates at the end of the gas jet. Then, a monoenergetic peak is not observed. The simulation results also explain the density dependence of the electron energy spectra shown in Fig. 6 (Masuda & Miura, 2009).

#### **3.4.2 Analysis on effect of a nanosecond prepulse on quasi-monoenergetic electron beam generation**

#### **Preformed plasma model in simulation**

To investigate effects of a nanosecond prepulse on QME beam generation shown in Sec. 3.3, 2D-PIC simulations were also conducted. The dynamics both of a preformed plasma produced by a nanosecond prepulse and the propagation of a femtosecond laser pulse in a millimeter-scale plasma can not be treated simultaneously in the PIC simulation, because



**quasi-monoenergetic electron beam**

**4.1 Ultrashort X-ray source based on laser Compton scattering**

Position, y [μm]

(b)


Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 39

Fig. 12. Spatial evolution of transverse envelopes of the laser electric field on the *x*-*y* plane in (a) no preformed plasma case and (b) preformed plasma case. Scattering of the laser pulse increases and the laser intensity on the propagation axis decreases in preformed plasma case.

In contrast, in preformed plasma case, the growth rate of the plasma wave is smaller than that in no preformed plasma case due to the low peak amplitude of the laser field as shown in Fig. 12(b). The time, when the amplitude of the plasma wave reaches the maximum and the electron injection occurs, is delayed and the required laser propagation length becomes longer than that in no preformed plasma case. A monoenergetic peak is formed at *x* ∼ 700 *μ*m as shown in Fig. 13(b). However, the monoenergetic peak is not clear, because the amount of injected electrons is small. The optimum extraction position of a monoenergetic electron bunch (*x* ∼ 700 *μ*m) is outside the gas jet region in the experiment. This means that the acceleration length is extremely short in the experiment. In preformed plasma case, a QME beam is not obtained, because the required condition, that is matching the laser propagation length with the gas jet length, is not satisfied due to the low growth rate of the plasma wave. The simulation results using a simple model for a preformed plasma qualitatively explain the

experimental results of effects of a nanosecond prepulse on QME beam generation.

**4. X-ray generation by laser Compton scattering using laser-accelerated**

An ultrashort electron pulse with a duration of the order of a few tens of femtoseconds can be obtained in LPA. In addition, there is the potential for realizing compact electron accelerators.

0

1

2

Laser amplitude, eEy/cmeωL

3

4

Position, y [μm]

(a)

their time scales are significantly different. In the present simulation, the preformed plasma is formed by giving a density modification of the initial electron and ion distributions.

It is assumed that a preformed plasma with a transversely hollow density profile is formed due to the expansion during a few nanoseconds. Singly ionized helium ions are distributed inside a preformed plasma region, and neutral helium atoms are distributed in the calculation region except for the preformed plasma region. The initial electron density is zero in the entire calculation region. The PIC simulation code includes the optical field ionization process. The leading edge of a laser pulse ionizes neutral atoms and singly ionized ions up to doubly ionized state. The electron density inside the preformed plasma region, where singly ionized helium ions are initially distributed, is half of that in the region, where neutral helium atoms are initially distributed. As a result, a preformed plasma with a transversely hollow density profile is formed. The main body of a laser pulse propagates in the preformed plasma. The details in a model to create a preformed plasma with a hollow density profile in the calculation have been described in Ref. (Masuda & Miura, 2010).

It is found that the longitudinal size of the preformed plasma is approximately 200 *μ*m from the shadowgraph image as shown in Fig. 9(b). Then, the longitudinal size of the preformed plasma region is assumed to be 230 *μ*m, corresponding to the Rayleigh length. Although the observed transverse size of the preformed plasma is limited by the spatial resolution, the transverse size is assumed to be 9 *μ*m, corresponding to the laser spot diameter. The preformed plasma region is set at the vacuum focal position of the laser pulse, because the preformed plasma is observed around the vacuum focal position as shown in Fig. 9(a). Except for the preformed plasma conditions, simulation conditions are the same as those shown in Sec. 3.4.1. The case above described is referred as preformed plasma case. In no preformed plasma case, neutral helium atoms are initially distributed in the entire calculation region and the initial electron density is zero.

#### **Simulation results**

Figures 12(a) and (b) show spatial evolutions of transverse envelopes of the laser electric field represented by the normalized vector potential on the *x*-*y* plane in no preformed plasma case and preformed plasma case. The spatial evolutions are obtained from the transverse envelopes of the peak amplitude of the laser electric field for every 255 time steps. In no preformed plasma case, the laser pulse focused at *x* = −300 *μ*m propagates by forming a narrow channel over the Rayleigh length due to the relativistic self-focusing effect. A part of the laser energy is scattered away from the channel. In preformed plasma case, the scattering of the laser pulse increases as shown in Fig. 12(b). This result explains the experimental observation of the strong side-scattered light image shown in Fig. 10(b). Due to the increase in the laser scattering, the peak intensity of the laser pulse in the channel decreases in preformed plasma case.

Figures 13(a) and (b) show the electron energy spectra as a function of the position of the laser pulse in no preformed plasma case and preformed plasma case. The spatial evolutions are obtained from the electron energy spectra calculated for the electrons with the angle of the momentum vector with respect to the laser propagation direction in the range of ±50 mrad in the moving window for every 255 time steps. In no preformed plasma case, a clear monoenergetic peak is formed at *x* ∼ 600 *μ*m and the maximum peak energy reaches ∼ 80 MeV as shown in Fig. 13(a). This position (*x* ∼ 600 *μ*m), which is the optimum extraction position of a monoenergetic electron bunch, is close to the end of the gas jet in the experiment. In no preformed plasma case, a QME beam is produced, because matching the length of the laser propagation with the gas jet length is achieved. This result is quite similar to that shown in Sec. 3.4.1.

16 Will-be-set-by-IN-TECH

their time scales are significantly different. In the present simulation, the preformed plasma is

It is assumed that a preformed plasma with a transversely hollow density profile is formed due to the expansion during a few nanoseconds. Singly ionized helium ions are distributed inside a preformed plasma region, and neutral helium atoms are distributed in the calculation region except for the preformed plasma region. The initial electron density is zero in the entire calculation region. The PIC simulation code includes the optical field ionization process. The leading edge of a laser pulse ionizes neutral atoms and singly ionized ions up to doubly ionized state. The electron density inside the preformed plasma region, where singly ionized helium ions are initially distributed, is half of that in the region, where neutral helium atoms are initially distributed. As a result, a preformed plasma with a transversely hollow density profile is formed. The main body of a laser pulse propagates in the preformed plasma. The details in a model to create a preformed plasma with a hollow density profile in the calculation

It is found that the longitudinal size of the preformed plasma is approximately 200 *μ*m from the shadowgraph image as shown in Fig. 9(b). Then, the longitudinal size of the preformed plasma region is assumed to be 230 *μ*m, corresponding to the Rayleigh length. Although the observed transverse size of the preformed plasma is limited by the spatial resolution, the transverse size is assumed to be 9 *μ*m, corresponding to the laser spot diameter. The preformed plasma region is set at the vacuum focal position of the laser pulse, because the preformed plasma is observed around the vacuum focal position as shown in Fig. 9(a). Except for the preformed plasma conditions, simulation conditions are the same as those shown in Sec. 3.4.1. The case above described is referred as preformed plasma case. In no preformed plasma case, neutral helium atoms are initially distributed in the entire calculation region and

Figures 12(a) and (b) show spatial evolutions of transverse envelopes of the laser electric field represented by the normalized vector potential on the *x*-*y* plane in no preformed plasma case and preformed plasma case. The spatial evolutions are obtained from the transverse envelopes of the peak amplitude of the laser electric field for every 255 time steps. In no preformed plasma case, the laser pulse focused at *x* = −300 *μ*m propagates by forming a narrow channel over the Rayleigh length due to the relativistic self-focusing effect. A part of the laser energy is scattered away from the channel. In preformed plasma case, the scattering of the laser pulse increases as shown in Fig. 12(b). This result explains the experimental observation of the strong side-scattered light image shown in Fig. 10(b). Due to the increase in the laser scattering, the peak intensity of the laser pulse in the channel decreases in preformed

Figures 13(a) and (b) show the electron energy spectra as a function of the position of the laser pulse in no preformed plasma case and preformed plasma case. The spatial evolutions are obtained from the electron energy spectra calculated for the electrons with the angle of the momentum vector with respect to the laser propagation direction in the range of ±50 mrad in the moving window for every 255 time steps. In no preformed plasma case, a clear monoenergetic peak is formed at *x* ∼ 600 *μ*m and the maximum peak energy reaches ∼ 80 MeV as shown in Fig. 13(a). This position (*x* ∼ 600 *μ*m), which is the optimum extraction position of a monoenergetic electron bunch, is close to the end of the gas jet in the experiment. In no preformed plasma case, a QME beam is produced, because matching the length of the laser propagation with the gas jet length is achieved. This result is quite similar to that shown

formed by giving a density modification of the initial electron and ion distributions.

have been described in Ref. (Masuda & Miura, 2010).

the initial electron density is zero.

**Simulation results**

plasma case.

in Sec. 3.4.1.

In contrast, in preformed plasma case, the growth rate of the plasma wave is smaller than that in no preformed plasma case due to the low peak amplitude of the laser field as shown in Fig. 12(b). The time, when the amplitude of the plasma wave reaches the maximum and the electron injection occurs, is delayed and the required laser propagation length becomes longer than that in no preformed plasma case. A monoenergetic peak is formed at *x* ∼ 700 *μ*m as shown in Fig. 13(b). However, the monoenergetic peak is not clear, because the amount of injected electrons is small. The optimum extraction position of a monoenergetic electron bunch (*x* ∼ 700 *μ*m) is outside the gas jet region in the experiment. This means that the acceleration length is extremely short in the experiment. In preformed plasma case, a QME beam is not obtained, because the required condition, that is matching the laser propagation length with the gas jet length, is not satisfied due to the low growth rate of the plasma wave. The simulation results using a simple model for a preformed plasma qualitatively explain the experimental results of effects of a nanosecond prepulse on QME beam generation.

## **4. X-ray generation by laser Compton scattering using laser-accelerated quasi-monoenergetic electron beam**

#### **4.1 Ultrashort X-ray source based on laser Compton scattering**

An ultrashort electron pulse with a duration of the order of a few tens of femtoseconds can be obtained in LPA. In addition, there is the potential for realizing compact electron accelerators.

a laser-accelerated electron beam with a Maxwell-like energy distribution. (Schwoerer et al., 2006). In this section, X-ray generation by laser Compton scattering using a laser-accelerated

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 41

Figure 14 shows the experimental setup. Laser pulses for electron acceleration and laser Compton scattering are referred as main laser pulse and colliding laser pulse. A p-polarized main laser pulse (700 mJ, 40 fs, 800 nm) was focused onto the edge of a helium gas jet using an *f* /14 off-axis parabolic mirror with a focal length of 720 mm. The laser spot diameter in vacuum was 13 *μ*m at the FWHM. The energy concentration within the *e*−<sup>2</sup> spot was 50% of the total laser energy. The peak intensity was 4.7 <sup>×</sup> 1018 W/cm2, corresponding to the normalized vector potential of 1.4. The gas jet was ejected from a supersonic nozzle with a conical shape. The diameter of the nozzle exit was 1.6 mm and the Mach number was 5. The

CCD CCD camera

Dipole magnet

20°

Fig. 14. Experimental setup for X-ray generation by laser Compton scattering using

CCD camera

A p-polarized colliding laser pulse (140 mJ, 100 fs, 800 nm) was focused around the exit of the main laser pulse from the gas jet using an *f* /6 off-axis parabolic mirror with a focal length of 300 mm. The laser spot diameter in vacuum was 9 *μ*m at the FWHM. The incident angle of the colliding laser pulse was 20◦ to the propagation axis of the main laser pulse. X-rays produced by laser Compton scattering were emitted on the coaxial direction of an electron beam. The electron beam was bended by a magnetic field and spatially separated from the X-rays. Both X-rays and electrons were simultaneously incident on a Gd2O2S : Tb phosphor screen (DRZ-HGH) through a 115-*μm*-thick Al filter. The images of X-rays and energy-resolved electrons were observed with a CCD camera. Electrons with an energy higher than 35 MeV were detected. The charge can be estimated, because the sensitivity of the detection system was calibrated for electrons (Masuda et al., 2008). The photon energy range of detected X-rays was higher than 10 keV from the calculated sensitivity of the phosphor

For synchronized collision of a colliding laser pulse with a laser-accelerated electron beam, a time-resolved shadowgraph image along the propagation axis of the main laser pulse was observed using a 60-fs probe laser pulse. A side-scattered light image through an interference filter of 800 nm, that is Thomson-scattered light image, was also observed from the top of the

Colliding laser pulse

filter Phosphor

B-field

<sup>B</sup> X-ray

Electron beam

screen

Al filter

quasi-monoenergetic electron beam is described.

focal position was set at 1 mm above the nozzle exit.

camera

Interference

Main laser pulse

laser-accelerated electron beam.

screen.

gas jet.

Plasma

Gas jet

Probe laser pulse

**4.2 Experimental conditions**

Fig. 13. Electron energy spectra as a function of the position of the laser pulse in (a) no preformed plasma case and (b) preformed plasma case. In no preformed plasma case, a clear monoenergetic peak with an energy of 80 MeV is formed at the position, *x* ∼ 600 *μ*m, which corresponds to the end of the gas jet in the experiment. At the same position, a clear monoenergetic peak is not observed in preformed plasma case.

The set of such unique characteristics enables us to realize compact, all-optical, ultrashort radiation sources in the wavelength range from extreme ultraviolet to X-ray. One of them is an X-ray source based on laser Compton scattering, that is scattering of photons by energetic electrons. Laser Compton scattering X-ray sources have attracted much attention for their potential of applications to medical imaging (Ikeura-Sekiguchi et al., 2008), because they produce a well-collimated, quasi-monochromatic X-ray beam, the photon energy of which is tunable, in a small facility, as compared with synchrotron radiation sources.

So far, generation of a femtosecond X-ray pulse by laser Compton scattering has been demonstrated by using a femtosecond laser pulse and a picosecond electron pulse from rf accelerators (Schoenlein et al., 1996; Yorozu et al., 2002). The X-ray pulse duration is determined by the interaction time between the laser and electron pulses. To obtain a femtosecond X-ray pulse, 90◦ scattering geometry should be adopted for a picosecond electron pulse. In contrast, 180◦ scattering (head-on collision) geometry can be adopted for a femtosecond electron pulse obtained by LPA. There are some advantages of using 180◦ scattering geometry. The photon energy of X-rays is higher than that for 90◦ scattering geometry, even though the electron energy and the wavelength of a laser pulse are the same. In addition, the X-ray yield is higher than that for 90◦ scattering geometry, even though the charge of an electron beam and the energy of a laser pulse are the same. Generation of X-rays with energies around 1 keV by laser Compton scattering has been demonstrated using a laser-accelerated electron beam with a Maxwell-like energy distribution. (Schwoerer et al., 2006). In this section, X-ray generation by laser Compton scattering using a laser-accelerated quasi-monoenergetic electron beam is described.

#### **4.2 Experimental conditions**

18 Will-be-set-by-IN-TECH


The set of such unique characteristics enables us to realize compact, all-optical, ultrashort radiation sources in the wavelength range from extreme ultraviolet to X-ray. One of them is an X-ray source based on laser Compton scattering, that is scattering of photons by energetic electrons. Laser Compton scattering X-ray sources have attracted much attention for their potential of applications to medical imaging (Ikeura-Sekiguchi et al., 2008), because they produce a well-collimated, quasi-monochromatic X-ray beam, the photon energy of which

So far, generation of a femtosecond X-ray pulse by laser Compton scattering has been demonstrated by using a femtosecond laser pulse and a picosecond electron pulse from rf accelerators (Schoenlein et al., 1996; Yorozu et al., 2002). The X-ray pulse duration is determined by the interaction time between the laser and electron pulses. To obtain a femtosecond X-ray pulse, 90◦ scattering geometry should be adopted for a picosecond electron pulse. In contrast, 180◦ scattering (head-on collision) geometry can be adopted for a femtosecond electron pulse obtained by LPA. There are some advantages of using 180◦ scattering geometry. The photon energy of X-rays is higher than that for 90◦ scattering geometry, even though the electron energy and the wavelength of a laser pulse are the same. In addition, the X-ray yield is higher than that for 90◦ scattering geometry, even though the charge of an electron beam and the energy of a laser pulse are the same. Generation of X-rays with energies around 1 keV by laser Compton scattering has been demonstrated using

Fig. 13. Electron energy spectra as a function of the position of the laser pulse in (a) no preformed plasma case and (b) preformed plasma case. In no preformed plasma case, a clear monoenergetic peak with an energy of 80 MeV is formed at the position, *x* ∼ 600 *μ*m, which corresponds to the end of the gas jet in the experiment. At the same position, a clear

is tunable, in a small facility, as compared with synchrotron radiation sources.

10<sup>0</sup>

10<sup>1</sup>

10<sup>2</sup>

Number of electrons [arb. unit]

10<sup>3</sup>

10<sup>4</sup>

monoenergetic peak is not observed in preformed plasma case.

Electron energy [MeV]

(a)

Electron energy [MeV]

(b)

Figure 14 shows the experimental setup. Laser pulses for electron acceleration and laser Compton scattering are referred as main laser pulse and colliding laser pulse. A p-polarized main laser pulse (700 mJ, 40 fs, 800 nm) was focused onto the edge of a helium gas jet using an *f* /14 off-axis parabolic mirror with a focal length of 720 mm. The laser spot diameter in vacuum was 13 *μ*m at the FWHM. The energy concentration within the *e*−<sup>2</sup> spot was 50% of the total laser energy. The peak intensity was 4.7 <sup>×</sup> 1018 W/cm2, corresponding to the normalized vector potential of 1.4. The gas jet was ejected from a supersonic nozzle with a conical shape. The diameter of the nozzle exit was 1.6 mm and the Mach number was 5. The focal position was set at 1 mm above the nozzle exit.

Fig. 14. Experimental setup for X-ray generation by laser Compton scattering using laser-accelerated electron beam.

A p-polarized colliding laser pulse (140 mJ, 100 fs, 800 nm) was focused around the exit of the main laser pulse from the gas jet using an *f* /6 off-axis parabolic mirror with a focal length of 300 mm. The laser spot diameter in vacuum was 9 *μ*m at the FWHM. The incident angle of the colliding laser pulse was 20◦ to the propagation axis of the main laser pulse. X-rays produced by laser Compton scattering were emitted on the coaxial direction of an electron beam. The electron beam was bended by a magnetic field and spatially separated from the X-rays. Both X-rays and electrons were simultaneously incident on a Gd2O2S : Tb phosphor screen (DRZ-HGH) through a 115-*μm*-thick Al filter. The images of X-rays and energy-resolved electrons were observed with a CCD camera. Electrons with an energy higher than 35 MeV were detected. The charge can be estimated, because the sensitivity of the detection system was calibrated for electrons (Masuda et al., 2008). The photon energy range of detected X-rays was higher than 10 keV from the calculated sensitivity of the phosphor screen.

For synchronized collision of a colliding laser pulse with a laser-accelerated electron beam, a time-resolved shadowgraph image along the propagation axis of the main laser pulse was observed using a 60-fs probe laser pulse. A side-scattered light image through an interference filter of 800 nm, that is Thomson-scattered light image, was also observed from the top of the gas jet.

**4.4 X-ray generation**

in the vertical direction is observed.

been demonstrated at the same time.

**5. Recent progress toward next step**

section of scattering on the scattered angle of X-rays.

X-rays produced by laser Compton scattering were observed, when a QME beam with a considerably high charge was obtained. Figure 16(a) shows an image of X-rays. The image was obtained with a single shot. From the energy-resolved electron image simultaneously observed with the image shown in Fig. 16(a), the peak energy and the charge in the monoenergetic peak of the QME beam were estimated to be 50 MeV and 30 pC, respectively. Figure 16(b) shows the intensity profile of the image in the vertical direction. The divergence angle in vertical direction was 5 mrad at the FWHM. The divergence angle in horizontal direction was 7 mrad at the FWHM. In laser Compton scattering, a collimated X-ray beam can be obtained. The divergence angle of the X-ray beam is given by ∼ 1/*γ*. Here, *γ* is the Lorentz factor of an electron energy. The divergence of an X-ray beam is estimated to be approximately 10 mrad from the observed peak energy of 50 MeV. The observed divergence angle was close to the predicted value from the electron energy. The maximum photon energy

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 43

<sup>0</sup> <sup>200</sup> <sup>400</sup> 600 800 <sup>0</sup>

Intensity [arb. unit]

20

10

40

50

30

Vertical position [pixel]

Fig. 16. (a) Image of X-rays produced by laser Compton scattering and (b) the vertical profile of the image. A well-collimated X-ray beam with a divergence angle of 5 mrad at the FWHM

of the X-rays was estimated to be 60 keV from the peak energy of the QME beam and the interaction angle between the electron and laser pulses. The X-ray yield was also estimated to be approximately 105 photons/pulse from the charge of the QME beam and the irradiation conditions of the colliding laser pulse by including the dependence of the differential cross

The allowance range of the delay between the main and colliding laser pulses for X-ray generation was investigated. The allowance range was approximately 100 fs, which was close to the duration of the colliding laser pulse. This result suggests that a pulse duration of a QME beam is nearly equal to or less than 100 fs. The generation of an ultrashort electron pulse has

The present status of LPA research has been presented as an example of works conducted at the AIST. In addition to our works, various works on improvement of the performance of electron beams have been so far conducted, and characterization and applications of electron

beams have been also demonstrated. In this section, these woks are briefly reviewed.

(a) (b) 70

60

The conditions for electron acceleration were different from those shown in Sec. 3.2. In prior, we confirmed that QME beams with a peak energy from 50 to 100 MeV and a charge in the monoenergetic peak of several tens of picocoulombs were obtained from a plasma with an electron density of 1.7 <sup>×</sup> 1019 cm−<sup>3</sup> (Miura et al., 2011).

#### **4.3 Synchronized collision of femtosecond laser pulse with electron beam**

For X-ray generation by laser Compton scattering, synchronized collision of a colliding laser pulse with a laser-accelerated electron beam is required. A laser-accelerated electron beam is emitted in the coaxial direction of a main laser pulse and temporally close to a main laser pulse. Then, achieving the synchronized collision of the colliding laser pulse with the laser-accelerated electron beam is equivalent to achieving the synchronized collision of the main and colliding laser pulses. Figures 15(a)-(c) show typical shadowgraph images observed for different delay times of the probe laser pulse to the main laser pulse: (a) -1.33 ps, (b) -0.67 ps, and (c) 0 ps, respectively. The main laser pulse propagated from right to left, and the

Fig. 15. (a)-(c) Shadowgraph images observed for different delay times of a probe laser pulse to a main laser pulse: (a) -1.33 ps, (b) -0.67 ps, and (c) 0 ps. (d) Thomson-scattered light image when the synchronized collision of the main and colliding laser pulses is achieved. The bright spot indicates the collision point of the two laser pulses.

colliding laser pulse propagated from left to right. The ionization fronts of the two laser pulses approached each other with varying the delay of the probe laser pulse. As seen in Fig. 15(c), the ionization fronts of the two laser pulses overlapped, and two laser pulses collided. The optical path length and the focal position in the vertical direction of the colliding laser pulse were adjusted using the shadowgraph images.

The collision point on the horizontal plane was set using a Thomson-scattered light image observed from the top of the gas jet as shown in Fig. 15(d). The dotted circle shows the position of the nozzle exit with 1.6-mm diameter. In Fig. 15(d), the main laser pulse propagated from top to bottom and the colliding laser pulse propagated from lower right to upper left in the direction of 20◦ to the main laser propagation axis. As shown by the thin arrow, a bright spot was observed, only the synchronized collision of the two laser pulses was achieved. It is supposed that the bright spot indicates the collision point of the two laser pulses. The collision point was set at the edge of the nozzle exit, which was near the extraction position of an electron beam from a plasma.

#### **4.4 X-ray generation**

20 Will-be-set-by-IN-TECH

The conditions for electron acceleration were different from those shown in Sec. 3.2. In prior, we confirmed that QME beams with a peak energy from 50 to 100 MeV and a charge in the monoenergetic peak of several tens of picocoulombs were obtained from a plasma with an

For X-ray generation by laser Compton scattering, synchronized collision of a colliding laser pulse with a laser-accelerated electron beam is required. A laser-accelerated electron beam is emitted in the coaxial direction of a main laser pulse and temporally close to a main laser pulse. Then, achieving the synchronized collision of the colliding laser pulse with the laser-accelerated electron beam is equivalent to achieving the synchronized collision of the main and colliding laser pulses. Figures 15(a)-(c) show typical shadowgraph images observed for different delay times of the probe laser pulse to the main laser pulse: (a) -1.33 ps, (b) -0.67 ps, and (c) 0 ps, respectively. The main laser pulse propagated from right to left, and the

Colliding Main Main

Colliding 1 mm Collision

Fig. 15. (a)-(c) Shadowgraph images observed for different delay times of a probe laser pulse to a main laser pulse: (a) -1.33 ps, (b) -0.67 ps, and (c) 0 ps. (d) Thomson-scattered light image when the synchronized collision of the main and colliding laser pulses is achieved. The

colliding laser pulse propagated from left to right. The ionization fronts of the two laser pulses approached each other with varying the delay of the probe laser pulse. As seen in Fig. 15(c), the ionization fronts of the two laser pulses overlapped, and two laser pulses collided. The optical path length and the focal position in the vertical direction of the colliding laser pulse

The collision point on the horizontal plane was set using a Thomson-scattered light image observed from the top of the gas jet as shown in Fig. 15(d). The dotted circle shows the position of the nozzle exit with 1.6-mm diameter. In Fig. 15(d), the main laser pulse propagated from top to bottom and the colliding laser pulse propagated from lower right to upper left in the direction of 20◦ to the main laser propagation axis. As shown by the thin arrow, a bright spot was observed, only the synchronized collision of the two laser pulses was achieved. It is supposed that the bright spot indicates the collision point of the two laser pulses. The collision point was set at the edge of the nozzle exit, which was near the extraction position of

(d)

Collision point

electron density of 1.7 <sup>×</sup> 1019 cm−<sup>3</sup> (Miura et al., 2011).

(a)

(b)

(c)

point

were adjusted using the shadowgraph images.

an electron beam from a plasma.

bright spot indicates the collision point of the two laser pulses.

**4.3 Synchronized collision of femtosecond laser pulse with electron beam**

X-rays produced by laser Compton scattering were observed, when a QME beam with a considerably high charge was obtained. Figure 16(a) shows an image of X-rays. The image was obtained with a single shot. From the energy-resolved electron image simultaneously observed with the image shown in Fig. 16(a), the peak energy and the charge in the monoenergetic peak of the QME beam were estimated to be 50 MeV and 30 pC, respectively. Figure 16(b) shows the intensity profile of the image in the vertical direction. The divergence angle in vertical direction was 5 mrad at the FWHM. The divergence angle in horizontal direction was 7 mrad at the FWHM. In laser Compton scattering, a collimated X-ray beam can be obtained. The divergence angle of the X-ray beam is given by ∼ 1/*γ*. Here, *γ* is the Lorentz factor of an electron energy. The divergence of an X-ray beam is estimated to be approximately 10 mrad from the observed peak energy of 50 MeV. The observed divergence angle was close to the predicted value from the electron energy. The maximum photon energy

Fig. 16. (a) Image of X-rays produced by laser Compton scattering and (b) the vertical profile of the image. A well-collimated X-ray beam with a divergence angle of 5 mrad at the FWHM in the vertical direction is observed.

of the X-rays was estimated to be 60 keV from the peak energy of the QME beam and the interaction angle between the electron and laser pulses. The X-ray yield was also estimated to be approximately 105 photons/pulse from the charge of the QME beam and the irradiation conditions of the colliding laser pulse by including the dependence of the differential cross section of scattering on the scattered angle of X-rays.

The allowance range of the delay between the main and colliding laser pulses for X-ray generation was investigated. The allowance range was approximately 100 fs, which was close to the duration of the colliding laser pulse. This result suggests that a pulse duration of a QME beam is nearly equal to or less than 100 fs. The generation of an ultrashort electron pulse has been demonstrated at the same time.

#### **5. Recent progress toward next step**

The present status of LPA research has been presented as an example of works conducted at the AIST. In addition to our works, various works on improvement of the performance of electron beams have been so far conducted, and characterization and applications of electron beams have been also demonstrated. In this section, these woks are briefly reviewed.

Nishimura & Daido, 2009). For an argon gas jet, a preformed plasma suitable for guiding a main pulse is produced by ASE accompanying a main pulse due to the lower ionization

Electron Acceleration Using an Ultrashort Ultraintense Laser Pulse 45

The increase in a charge and the decrease in a beam divergence have been achieved using a gas jet composed helium and controlled amounts of various high-Z gases (McGuffey et al., 2010; Pak et al., 2010). Optical field ionization of inner shell electrons of the high-Z gas plays an important role in increasing the number of electrons injected into a plasma wave. This is

As described in Sec. 5.1, a capillary discharge plasma has been used for producing a long plasma and guiding an intense laser pulse. With a steady-state-flow gas cell using a hollow capillary without discharge, stable generation of QME beams has been achieved (Osterhoff et al., 2008). The steady-state gas flow forms a reproducible, homogeneous gas distribution along the laser propagation direction, which brings about the stable QME beam generation. An increase in a charge and a decrease in a divergence of electron beams have been observed by applying a static external magnetic field along the laser pulse propagation axis, although the electron energy spectra are Maxwell-like distributions (Hosokai et al., 2006). The shape of a preformed plasma suitable for guiding a main pulse is formed by applying a static magnetic

A femtosecond electron pulse can be produced in LPA. To prove the generation of a femtosecond electron pulse, a temporal characterization of an electron pulse has been conducted. In the temporal characterization, coherent transition radiation (CTR) emitted at the plasma-vacuum boundary or through a thin metallic foil is used. An electron pulse duration is measured with an electro-optical sampling technique using CTR in the THz region, and the temporal resolution of several tens of femtoseconds has been achieved. (Debus et al., 2010; van Tilborg et al., 2006) An electron pulse duration is also estimated from the spectrum of CTR (Glinec et al., 2007; Ohkubo et al., 2007). Recently, generation of a few femtosecond electron pulse has been demonstrated using an optical injection scheme (Lundh et al., 2011). Such ultrashort electron pulse is a useful tool to investigate physical and chemical kinetics of ultrafast phenomena initiated by ionization radiation referred as pulsed radiolysis. Applications of a laser-accelerated electron beam to pulsed radiolysis have been

Using a femtosecond electron pulse obtained in LPA, ultrashort short-wavelength radiation sources from extreme ultraviolet to X-ray can be obtained as the secondary source. Such ultrashort short-wavelength sources are attractive for applications to observe ultrafast phenomena such as time-resolved X-ray diffraction. As shown in Sec. 4, X-ray generation by laser Compton scattering using a laser-accelerated electron beam has been

The generation of synchrotron radiation has been demonstrated by using a laser-accelerated QME beam through an undulator (Fuchs et al., 2009; Schlenvoigt et al., 2008). The wavelength of the radiation still remains visible to extreme ultraviolet due to the low electron energy. In the near future, the wavelength range will be extended to X-rays, as the electron energy

Generation of keV X-rays by betatron radiation in a plasma has been demonstrated (Rousse et al., 2004). Laser-accelerated electrons undergo a transverse electric field in an ion channel,

potential of argon than that of helium.

referred as ionization induced trapping.

demonstrated (Brozek-Pluska et al., 2005).

demonstrated (Miura et al., 2011; Schwoerer et al., 2006).

**5.4 Ultrashort X-ray radiation**

increases.

**5.3 Demonstration of femtosecond electron pulse generation**

field.

#### **5.1 Toward more energetic electron acceleration**

LPA is expected as technologies obtaining extremely high energy particles using the extremely high accelerating field. A longer plasma and a longer acceleration length are required for obtaining more energetic electrons. One of the approaches is to form a plasma waveguide for guiding an intense laser pulse using a preformed plasma produced by a discharge, which has a transversely hollow density profile. A 1-GeV QME beam has been produced from a 3-cm-long capillary discharge plasma (Leemans et al., 2006). After that, several groups have reported the generation of GeV-class QME beams from a centimeter-scale capillary discharge plasma (Kameshima et al., 2008; Karsch et al., 2007; Rowlands-Rees et al., 2008). On the other hand, GeV-class electron beams have been also produced using a centimeter-scale gas jet based on a self-guiding of an intense laser pulse (Clayton et al., 2010; Hafz et al., 2008).

Experiments to accelerate extremely high energy electrons have been also conducted using a PW-class laser system of single shot operation. Energetic electron beams have been produced from a hollow glass capillary attached with a gold cone irradiated by an intense laser pulse (Kitagawa et al., 2004; Mori, Sentoku, Kondo, Tsuji, Nakanii, Fukumochi, Kashihara, Kimura, Takeda, Tanaka, Norimatsu, Tanimoto, Nakamura, Tampo, Kodama, Miura, Mima & Kitagawa, 2009). In this experiment, the gold cone attached at the entrance of a laser pulse plays an important role for the electron injection into a plasma wave driven inside the hollow capillary. Energetic electrons have been produced by the interaction of an intense laser pulse with a plasma preformed from a hollow plastic cylinder via laser-driven implosion (Nakanii et al., 2008). Electrons with energies of more than 600 MeV are observed from a 3-mm-long plasma tube.

## **5.2 Improvement, stabilization, and control of electron beam quality**

#### **Optical injection scheme**

In a self-injection scheme, the injection of electrons into a plasma wave is based on the wave-breaking. The wave-breaking is a nonlinear phenomenon which is substantially unstable, and it is difficult to actively control it. Characteristics of electron beams are strongly affected by the shot-to-shot fluctuations of conditions of a gas jet, a laser pulse, and so on. Stabilization of electron beam qualities will be achieved by controlling the injection of electrons into a plasma wave. It is possible to control the injection of electrons into a plasma wave using multiple laser pulses. One laser pulse drives a plasma wave with an amplitude, in which wave-breaking does not occur, and other laser pulses control the injection of electrons into a plasma wave. This scheme is called optical injection scheme, or colliding pulse scheme. Although there have been several theoretical proposals (Esarey et al., 1997; Kotaki et al., 2004; Umstadter, Kim & Dodd, 1996), it took time for the experimental demonstration due to the requirement for using multiple laser pulses. Recently, experimental demonstrations of an optical injection scheme have been reported (Faure et al., 2006; Kotaki et al., 2009). When two counter-propagating laser pulses collide, a beat wave is formed and preaccelerates electrons in a plasma. The preaccelerated electrons are trapped into a plasma wave and accelerated. The stability of QME beam qualities has been improved, as compared with a self-injection scheme.

#### **Plasma control in self-injection scheme**

In a self-injection scheme, stabilization and improvement of electron beam qualities have been achieved by controlling the characteristics of a plasma.

In most experiments using a gas jet, a helium gas jet has been used. The pointing stability and divergence of QME beams have been improved by using an argon gas jet (Mori, Kondo, Mizuta, Kando, Kotaki, Nishiuchi, Kado, Pirozhkov, Ogura, Sugiyama, Bulanov, Tanaka, 22 Will-be-set-by-IN-TECH

LPA is expected as technologies obtaining extremely high energy particles using the extremely high accelerating field. A longer plasma and a longer acceleration length are required for obtaining more energetic electrons. One of the approaches is to form a plasma waveguide for guiding an intense laser pulse using a preformed plasma produced by a discharge, which has a transversely hollow density profile. A 1-GeV QME beam has been produced from a 3-cm-long capillary discharge plasma (Leemans et al., 2006). After that, several groups have reported the generation of GeV-class QME beams from a centimeter-scale capillary discharge plasma (Kameshima et al., 2008; Karsch et al., 2007; Rowlands-Rees et al., 2008). On the other hand, GeV-class electron beams have been also produced using a centimeter-scale gas jet based on a self-guiding of an intense laser pulse (Clayton et al., 2010; Hafz et al., 2008). Experiments to accelerate extremely high energy electrons have been also conducted using a PW-class laser system of single shot operation. Energetic electron beams have been produced from a hollow glass capillary attached with a gold cone irradiated by an intense laser pulse (Kitagawa et al., 2004; Mori, Sentoku, Kondo, Tsuji, Nakanii, Fukumochi, Kashihara, Kimura, Takeda, Tanaka, Norimatsu, Tanimoto, Nakamura, Tampo, Kodama, Miura, Mima & Kitagawa, 2009). In this experiment, the gold cone attached at the entrance of a laser pulse plays an important role for the electron injection into a plasma wave driven inside the hollow capillary. Energetic electrons have been produced by the interaction of an intense laser pulse with a plasma preformed from a hollow plastic cylinder via laser-driven implosion (Nakanii et al., 2008). Electrons with energies of more than 600 MeV are observed from a 3-mm-long

**5.1 Toward more energetic electron acceleration**

**5.2 Improvement, stabilization, and control of electron beam quality**

In a self-injection scheme, the injection of electrons into a plasma wave is based on the wave-breaking. The wave-breaking is a nonlinear phenomenon which is substantially unstable, and it is difficult to actively control it. Characteristics of electron beams are strongly affected by the shot-to-shot fluctuations of conditions of a gas jet, a laser pulse, and so on. Stabilization of electron beam qualities will be achieved by controlling the injection of electrons into a plasma wave. It is possible to control the injection of electrons into a plasma wave using multiple laser pulses. One laser pulse drives a plasma wave with an amplitude, in which wave-breaking does not occur, and other laser pulses control the injection of electrons into a plasma wave. This scheme is called optical injection scheme, or colliding pulse scheme. Although there have been several theoretical proposals (Esarey et al., 1997; Kotaki et al., 2004; Umstadter, Kim & Dodd, 1996), it took time for the experimental demonstration due to the requirement for using multiple laser pulses. Recently, experimental demonstrations of an optical injection scheme have been reported (Faure et al., 2006; Kotaki et al., 2009). When two counter-propagating laser pulses collide, a beat wave is formed and preaccelerates electrons in a plasma. The preaccelerated electrons are trapped into a plasma wave and accelerated. The stability of QME beam qualities has been improved, as compared with a self-injection

In a self-injection scheme, stabilization and improvement of electron beam qualities have been

In most experiments using a gas jet, a helium gas jet has been used. The pointing stability and divergence of QME beams have been improved by using an argon gas jet (Mori, Kondo, Mizuta, Kando, Kotaki, Nishiuchi, Kado, Pirozhkov, Ogura, Sugiyama, Bulanov, Tanaka,

plasma tube.

scheme.

**Plasma control in self-injection scheme**

achieved by controlling the characteristics of a plasma.

**Optical injection scheme**

Nishimura & Daido, 2009). For an argon gas jet, a preformed plasma suitable for guiding a main pulse is produced by ASE accompanying a main pulse due to the lower ionization potential of argon than that of helium.

The increase in a charge and the decrease in a beam divergence have been achieved using a gas jet composed helium and controlled amounts of various high-Z gases (McGuffey et al., 2010; Pak et al., 2010). Optical field ionization of inner shell electrons of the high-Z gas plays an important role in increasing the number of electrons injected into a plasma wave. This is referred as ionization induced trapping.

As described in Sec. 5.1, a capillary discharge plasma has been used for producing a long plasma and guiding an intense laser pulse. With a steady-state-flow gas cell using a hollow capillary without discharge, stable generation of QME beams has been achieved (Osterhoff et al., 2008). The steady-state gas flow forms a reproducible, homogeneous gas distribution along the laser propagation direction, which brings about the stable QME beam generation.

An increase in a charge and a decrease in a divergence of electron beams have been observed by applying a static external magnetic field along the laser pulse propagation axis, although the electron energy spectra are Maxwell-like distributions (Hosokai et al., 2006). The shape of a preformed plasma suitable for guiding a main pulse is formed by applying a static magnetic field.

#### **5.3 Demonstration of femtosecond electron pulse generation**

A femtosecond electron pulse can be produced in LPA. To prove the generation of a femtosecond electron pulse, a temporal characterization of an electron pulse has been conducted. In the temporal characterization, coherent transition radiation (CTR) emitted at the plasma-vacuum boundary or through a thin metallic foil is used. An electron pulse duration is measured with an electro-optical sampling technique using CTR in the THz region, and the temporal resolution of several tens of femtoseconds has been achieved. (Debus et al., 2010; van Tilborg et al., 2006) An electron pulse duration is also estimated from the spectrum of CTR (Glinec et al., 2007; Ohkubo et al., 2007). Recently, generation of a few femtosecond electron pulse has been demonstrated using an optical injection scheme (Lundh et al., 2011). Such ultrashort electron pulse is a useful tool to investigate physical and chemical kinetics of ultrafast phenomena initiated by ionization radiation referred as pulsed radiolysis. Applications of a laser-accelerated electron beam to pulsed radiolysis have been demonstrated (Brozek-Pluska et al., 2005).

#### **5.4 Ultrashort X-ray radiation**

Using a femtosecond electron pulse obtained in LPA, ultrashort short-wavelength radiation sources from extreme ultraviolet to X-ray can be obtained as the secondary source. Such ultrashort short-wavelength sources are attractive for applications to observe ultrafast phenomena such as time-resolved X-ray diffraction. As shown in Sec. 4, X-ray generation by laser Compton scattering using a laser-accelerated electron beam has been demonstrated (Miura et al., 2011; Schwoerer et al., 2006).

The generation of synchrotron radiation has been demonstrated by using a laser-accelerated QME beam through an undulator (Fuchs et al., 2009; Schlenvoigt et al., 2008). The wavelength of the radiation still remains visible to extreme ultraviolet due to the low electron energy. In the near future, the wavelength range will be extended to X-rays, as the electron energy increases.

Generation of keV X-rays by betatron radiation in a plasma has been demonstrated (Rousse et al., 2004). Laser-accelerated electrons undergo a transverse electric field in an ion channel,

**8. References**

21: 414–416.

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beams, *Nature* 431: 541–544.

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which is formed inside a low electron density region of a plasma wave. They undergo betatron oscillation and emitted a collimated X-ray beam. Generation of a short X-ray pulse of less than 1 ps has been proven by measuring the temporal variation of X-ray reflectivity of a crystal pumped by a femtosecond laser pulse (Phuoc et al., 2007).

Extreme ultraviolet radiations have been demonstrated by the laser light reflection from a plasma wave driven by an intense laser pulse (Kando et al., 2007; 2009). The plasma wave moving almost at the speed of light acts as a relativistic flying mirror, and brings about frequency upshift and compression of the incident laser pulse due to the double Doppler effect. This regime is one of notable methods for generation of ultrashort short-wavelength radiations using an intense laser pulse, although laser-accelerated electron beams are not used.

## **6. Summary and future prospects**

In this chapter, we provide the overview of the present status of research on LPA. Remarkable progress such as QME beam generation has been made especially in the last several years. Although the quality and stabilization of electron beams have been improved, there is still room for improvement of beam qualities for practical applications. For further improvement of electron beam qualities, theoretical and experimental studies should be continued to answer various questions in physics of laser-plasma interactions and to explore new acceleration regime.

Compactness of a plasma accelerating electrons, corresponding to an electron gun and a series of rf cavities in rf accelerators, has been proven. However, an ultrashort ultraintense laser system for producing the plasma is still large, although the laser system is called "table-top" system. More compact laser system is necessary. In addition, the repetition rate of present laser systems is still 10 Hz. For increasing the average flux and luminosity, the development of an efficient laser system with much higher repetition rate is also dispensable. It is essential to conduct the research on plasma physics and the development of laser technologies in parallel. There are several prominent features in LPA such as femtosecond electron pulse generation. It is important to prove promising applications that make the best use of the features of LPA. Several promising applications have been already demonstrated. It becomes a stage when the design of laser electron accelerators for practical applications should be conducted by including a laser system. The recent progress of ultrashort ultraintense laser technologies is rapid and remarkable. In addition, the recent progress of high performance computer system is also rapid and remarkable. It will play a major role to investigate physics in laser-plasma interactions by numerical simulations. The set of rapid progress of laser technologies and investigation of physics will dramatically improve the performance of laser-accelerated electron beams. In the near future, laser electron accelerators useful for fundamental physics, and industrial and societal applications will be realized.

## **7. Acknowledgments**

I thank all the coworkers for their supports and fruitful discussion, especially S. Masuda, S. Ishii, K. Tanaka, T. Ooyama, R. Kuroda, and H. Toyokawa, who have contributed to the works presented here. A part of our works presented here was financially supported by the Budget for Nuclear Research of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, based on the screening and counseling by the Atomic Energy Commission, the Advanced Compact Accelerator Development of the MEXT, and the Matsuo Foundation. I also wish to acknowledge the support of the Advanced Compact Accelerator Development Office of the National Institute of Radiological Sciences, Japan.

#### **8. References**

24 Will-be-set-by-IN-TECH

which is formed inside a low electron density region of a plasma wave. They undergo betatron oscillation and emitted a collimated X-ray beam. Generation of a short X-ray pulse of less than 1 ps has been proven by measuring the temporal variation of X-ray reflectivity of a crystal

Extreme ultraviolet radiations have been demonstrated by the laser light reflection from a plasma wave driven by an intense laser pulse (Kando et al., 2007; 2009). The plasma wave moving almost at the speed of light acts as a relativistic flying mirror, and brings about frequency upshift and compression of the incident laser pulse due to the double Doppler effect. This regime is one of notable methods for generation of ultrashort short-wavelength radiations using an intense laser pulse, although laser-accelerated electron beams are not used.

In this chapter, we provide the overview of the present status of research on LPA. Remarkable progress such as QME beam generation has been made especially in the last several years. Although the quality and stabilization of electron beams have been improved, there is still room for improvement of beam qualities for practical applications. For further improvement of electron beam qualities, theoretical and experimental studies should be continued to answer various questions in physics of laser-plasma interactions and to explore new acceleration

Compactness of a plasma accelerating electrons, corresponding to an electron gun and a series of rf cavities in rf accelerators, has been proven. However, an ultrashort ultraintense laser system for producing the plasma is still large, although the laser system is called "table-top" system. More compact laser system is necessary. In addition, the repetition rate of present laser systems is still 10 Hz. For increasing the average flux and luminosity, the development of an efficient laser system with much higher repetition rate is also dispensable. It is essential to conduct the research on plasma physics and the development of laser technologies in parallel. There are several prominent features in LPA such as femtosecond electron pulse generation. It is important to prove promising applications that make the best use of the features of LPA. Several promising applications have been already demonstrated. It becomes a stage when the design of laser electron accelerators for practical applications should be conducted by including a laser system. The recent progress of ultrashort ultraintense laser technologies is rapid and remarkable. In addition, the recent progress of high performance computer system is also rapid and remarkable. It will play a major role to investigate physics in laser-plasma interactions by numerical simulations. The set of rapid progress of laser technologies and investigation of physics will dramatically improve the performance of laser-accelerated electron beams. In the near future, laser electron accelerators useful for fundamental physics,

I thank all the coworkers for their supports and fruitful discussion, especially S. Masuda, S. Ishii, K. Tanaka, T. Ooyama, R. Kuroda, and H. Toyokawa, who have contributed to the works presented here. A part of our works presented here was financially supported by the Budget for Nuclear Research of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, based on the screening and counseling by the Atomic Energy Commission, the Advanced Compact Accelerator Development of the MEXT, and the Matsuo Foundation. I also wish to acknowledge the support of the Advanced Compact Accelerator Development

pumped by a femtosecond laser pulse (Phuoc et al., 2007).

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Office of the National Institute of Radiological Sciences, Japan.

**7. Acknowledgments**

**6. Summary and future prospects**

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**0**

**3**

*United States*

**Coherent Laser Manipulation**

Elena Kuznetsova1,2, Robin Côté1 and S. F. Yelin1,2

<sup>1</sup>*Department of Physics, University of Connecticut, Storrs, Connecticut*

<sup>2</sup>*ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts*

The realization of rovibrationally stable dense samples of ultracold diatomic molecules remains one of the main stepping stones to achieve the next slate of major goals in the field of atomic and molecular physics. Though obtaining diatomic alkali molecules was seen as a logical next step following the optical cooling of atoms, many of the possible applications currently under investigation extend beyond atomic and molecular physics. For example, spectroscopy of ultracold molecules can help in testing extensions of the Standard Model via the search for a permanent electric dipole moment of the electron (1; 2), or the energy difference between enantiomers of chiral molecules (3). Various molecular transitions can be utilized to track the time dependence of fundamental constants, including the fine structure constant and the proton to electron mass ratio (4). They also open the way for cold and ultracold chemistry, where the interacting species and products are in a coherent quantum superposition state (5) and reactions can happen via quantum tunneling. Dipolar ultracold quantum gases promise to show a plethora of new phenomena due to anisotropic long-range dipole-dipole interactions (6). Dipolar molecules in optical lattices can be employed as a quantum simulator of condensed matter systems, and they are predicted to demonstrate new quantum phases such as a dipolar crystal, supersolid, checkerboard and collapse phases (7; 8). Ultracold polar molecules also represent an attractive platform for quantum computation (9). They offer a variety of long-lived states for qubit encoding, including rotational, spin and hyperfine (if electronic and nuclear spins are non-zero), Λ and Ω-doublet states (10) and scalability to a large number of qubits. Polar molecules can be easily controlled by DC electric and magnetic fields, as well as by microwave and optical fields, allowing the design of various traps (11; 12). The main appeal of polar molecules for quantum information processing, however, comes from their permanent electric dipole moment, permitting them to interact via a long-range dipole-dipole interaction. The dipole-dipole interaction offers a tool to construct

two-qubit gates, required for universal quantum computation (9; 13).

Ultracold molecules in their ground vibrational state *v* = 0 (and even in specific rotational, hyperfine or Zeeman states) are required for many of these applications since they have a large permanent electric dipole moment and are stable with respect to collisions and spontaneous emission. Currently translationally ultracold (100 nK - 1 mK) molecules are produced by magneto- (14) and photo-association (15) techniques. In a typical photoassociation scheme,

**1. Introduction**

**of Ultracold Molecules**


## **Coherent Laser Manipulation of Ultracold Molecules**

Elena Kuznetsova1,2, Robin Côté1 and S. F. Yelin1,2 <sup>1</sup>*Department of Physics, University of Connecticut, Storrs, Connecticut* <sup>2</sup>*ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts United States*

#### **1. Introduction**

30 Will-be-set-by-IN-TECH

52 Femtosecond-Scale Optics

Schwoerer, H., Liesfeld, B., Schlenvoigt, H.-P., Amthor, K.-U. & Sauerbrey, R. (2006).

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96: 014802.

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*Commun.* 56: 219–221.

*Phys. Rev. Lett.* 96: 014801.

*Phys. B* 74: 327–331.

spectrometer, *Rev. Sci. Instrum.* 76: 013507.

wakefield plasma waves, *Phys. Rev. Lett.* 76: 2073–2076.

filter: pulse compression and shaping, *Opt. Lett.* 25: 575–577.

with very low density plasmas, *Phys. Plasmas* 12: 093101.

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optics in relativistic plasmas and laser wake field acceleration of electrons, *Science*

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control of ultrashort pulses by use of an acousto-optic programmable dispersive

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Fluctuation of femtosecond x-ray pulses generated by a laser-compton scheme, *Appl.*

The realization of rovibrationally stable dense samples of ultracold diatomic molecules remains one of the main stepping stones to achieve the next slate of major goals in the field of atomic and molecular physics. Though obtaining diatomic alkali molecules was seen as a logical next step following the optical cooling of atoms, many of the possible applications currently under investigation extend beyond atomic and molecular physics. For example, spectroscopy of ultracold molecules can help in testing extensions of the Standard Model via the search for a permanent electric dipole moment of the electron (1; 2), or the energy difference between enantiomers of chiral molecules (3). Various molecular transitions can be utilized to track the time dependence of fundamental constants, including the fine structure constant and the proton to electron mass ratio (4). They also open the way for cold and ultracold chemistry, where the interacting species and products are in a coherent quantum superposition state (5) and reactions can happen via quantum tunneling. Dipolar ultracold quantum gases promise to show a plethora of new phenomena due to anisotropic long-range dipole-dipole interactions (6). Dipolar molecules in optical lattices can be employed as a quantum simulator of condensed matter systems, and they are predicted to demonstrate new quantum phases such as a dipolar crystal, supersolid, checkerboard and collapse phases (7; 8). Ultracold polar molecules also represent an attractive platform for quantum computation (9). They offer a variety of long-lived states for qubit encoding, including rotational, spin and hyperfine (if electronic and nuclear spins are non-zero), Λ and Ω-doublet states (10) and scalability to a large number of qubits. Polar molecules can be easily controlled by DC electric and magnetic fields, as well as by microwave and optical fields, allowing the design of various traps (11; 12). The main appeal of polar molecules for quantum information processing, however, comes from their permanent electric dipole moment, permitting them to interact via a long-range dipole-dipole interaction. The dipole-dipole interaction offers a tool to construct two-qubit gates, required for universal quantum computation (9; 13).

Ultracold molecules in their ground vibrational state *v* = 0 (and even in specific rotational, hyperfine or Zeeman states) are required for many of these applications since they have a large permanent electric dipole moment and are stable with respect to collisions and spontaneous emission. Currently translationally ultracold (100 nK - 1 mK) molecules are produced by magneto- (14) and photo-association (15) techniques. In a typical photoassociation scheme,

deeply bound molecules, starting from Feshbach molecules (23; 24). It allows to realize high transfer efficiency and preserve the high phase-space density of an initial atomic gas. In STIRAP, the laser pulses, coupling an initial and a final state to an intermediate excited state, are applied in a counter-intuitive sequence where a pump pulse is preceeded by a Stokes pulse. During the transfer, the system stays in a "dark" state, *i.e.*, a coherent superposition of initial and final states, preventing any losses that would otherwise occur from the excited state. By adiabatically changing amplitudes of the laser pulses, the "dark" state evolves from the initial to the final state, resulting in nearly 100% transfer efficiency. A STIRAP transfer from a Feshbach to a next lower vibrational state of a ground electronic potential has been demonstrated in 87Rb2 molecules held in an optical lattice to avoid inelastic collisions (25). Recently, heteronuclear 40K87Rb molecules have been transferred from a Feshbach to a deeply bound (> 10 GHz binding energy) vibrational state using STIRAP (26). Prospects of STIRAP-based photoassociation of a thermal ensemble of 85Rb atoms have recently been analyzed theoretically in (27). Finally, in breakthrough experiments at JILA and Innsbruck ultracold weakly bound KRb (23) and Cs2 (24) Feshbach molecules have been successfully brought to their ground rovibrational state via two- and multi-photon STIRAP, respectively. In principle, STIRAP allows lossless transfer of the population from an initial to the target state with 100 % efficiency. The main difficulty with a two-pulse STIRAP in molecules is to find an intermediate vibrational state of the excited electronic potential which has a good Franck-Condon overlap with both highly delocalized initial high vibrational state and a tightly localized ground vibrational state (28). It is particularly difficult for homonuclear atoms having the ground electronic potential scaling as 1/*R*<sup>6</sup> and the first excited potentials as 1/*R*<sup>3</sup> with interatomic distance *R*, resulting in a non-favorable potential curves' overlap. It is less an issue for heteronuclear molecules having both potentials falling off with distance according to the 1/*R*<sup>6</sup> law, making the overlap better and enabling a one-step STIRAP (23). It was therefore proposed in (29) to transfer population in several steps down the ladder of vibrational states using a sequence of stimulated optical Raman transitions. In this case the initial and final vibrational levels of each step do not differ significantly, and it is easier to find a suitable intermediate vibrational level in the excited electronic state. In this step-wise approach, however, the population is transfered through a number of vibrational levels in the ground electronic state subject to vibrational relaxation due to inelastic collisions with a background atom or another molecule. The released kinetic energy greatly exceeds the trap depth resulting in loss of both molecules and atoms from the trap. The step-wise transfer therefore has to be faster than the vibrational relaxation time. In the next section we describe a multistate chainwise STIRAP in which molecules are brought to the ground rovibrational state through a series of intermediate states in one run, which allows to minimize collisional losses in intermediate states. In Section III we discuss direct STIRAP conversion of ultracold atoms from a scattering continuum into deeply bound molecules in the presence of a Feshbach resonance. Direct conversion without first forming Feshbach molecules allows to reduce

Coherent Laser Manipulation of Ultracold Molecules 55

collisional losses during formation of molecules.

**2. Theory of multistate chainwise adiabatic passage in the presence of decay**

the population loss due to inelastic collisions during the transfer process (30).

In this section, we present a multistate chainwise STIRAP technique allowing an efficient transfer of a molecule from a high-lying state to the ground vibrational state which minimizes

a pair of colliding atoms is photoassociated into a bound electronically excited molecular state that spontaneously decays, forming molecules in the electronic ground state. In magnetoassociation, a magnetic field is adiabatically swept across a Feshbach resonance, converting two atoms from a scattering state into a bound molecular state. These techniques have most successfully been applied to form alkali dimers from ultracold alkali metal atoms (14–16). In both techniques the molecules are translationally cold, but vibrationally hot, since they are formed in high vibrational states near a dissociation limit of the electronic ground state. Therefore, once created, molecules have to be rapidly transfered to the ground rovibrational state.

Basically all successfull methods for cooling vibrational (and rotational) degrees of freedom require refined laser pulse techniques from simple STIRAP pulses to optical control. In the recent past, several methods have been proposed to reach this goal: they can be divided into non-coherent and coherent techniques. Non-coherent methods include the pump-dump technique and radiative vibrational cascade. In the pump-dump technique a pump pulse transfers population from an initial state to some intermediate vibrational level of an electronically excited state, followed by a dump pulse which brings the population from the intermediate to the ground vibrational level (17). Ultracold molecules in the ground vibrational state have been produced using this technique in a photoassociation experiment (16) with a 6% efficiency. The pump-dump technique requires pulses shorter than the excited state lifetime (large intensity is therefore needed to achieve reasonable transfer efficiency). The transfer efficiency is low, unless pulses of specific area (e.g. *π* pulses) are used. To increase the transfer efficiency and avoid losses due to spontaneous emission from the excited electronic state, a sequence of alternating short pump and dump pulses can be applied, each transferring a small fraction of the population to the target state (18). Pulses in this case can be weak, since each has to transfer only a small fraction of the population. Additional spectral shaping of the pulses can provide population transfer to a desirable target state, where population is coherently accumulated. In this case molecules formed in the ground vibrational state have to be removed from the interaction volume to avoid excitation by subsequent pulses, leading to increase in the duration of the transfer process. In the second non-coherent method molecules in the ground electronic state are allowed to radiatively decay from the initial high vibrational level reaching the ground *v* = 0 state after several decay steps (19). In this case many intermediate vibrational states are populated which would result in loss of molecules from a trap due to vibrationally inelastic collisions with background atoms and formed molecules. Another incoherent technique, named molecular optical pumping (20), allows to transfer molecules from high vibrational states to the ground state using a shaped laser pulse. Molecules are excited to vibrational levels of higher-energy electronic states and spontaneously decay back to the ground one. The excited state vibrational levels are chosen to have a good Franck-Condon factors with the *v* = 0 vibrational level in the ground electronic state. The laser pulse is spectrally shaped so that all frequencies allowing molecules to be excited back from the *v* = 0 state are removed, and a significant fraction of molecules accumulates in the ground vibrational state after a few excitation- spontaneous emission cycles.

The major coherent methods are adiabatic passage and coherent control techniques. The latter one utilizes spectrally shaped broadband optical pulses to transfer the molecules from an initial to the ground vibrational state with high efficiency (21). Stimulated Raman Adiabatic Passage (STIRAP) (22) has recently attracted significant interest as an efficient way to produce 2 Will-be-set-by-IN-TECH

a pair of colliding atoms is photoassociated into a bound electronically excited molecular state that spontaneously decays, forming molecules in the electronic ground state. In magnetoassociation, a magnetic field is adiabatically swept across a Feshbach resonance, converting two atoms from a scattering state into a bound molecular state. These techniques have most successfully been applied to form alkali dimers from ultracold alkali metal atoms (14–16). In both techniques the molecules are translationally cold, but vibrationally hot, since they are formed in high vibrational states near a dissociation limit of the electronic ground state. Therefore, once created, molecules have to be rapidly transfered to the ground

Basically all successfull methods for cooling vibrational (and rotational) degrees of freedom require refined laser pulse techniques from simple STIRAP pulses to optical control. In the recent past, several methods have been proposed to reach this goal: they can be divided into non-coherent and coherent techniques. Non-coherent methods include the pump-dump technique and radiative vibrational cascade. In the pump-dump technique a pump pulse transfers population from an initial state to some intermediate vibrational level of an electronically excited state, followed by a dump pulse which brings the population from the intermediate to the ground vibrational level (17). Ultracold molecules in the ground vibrational state have been produced using this technique in a photoassociation experiment (16) with a 6% efficiency. The pump-dump technique requires pulses shorter than the excited state lifetime (large intensity is therefore needed to achieve reasonable transfer efficiency). The transfer efficiency is low, unless pulses of specific area (e.g. *π* pulses) are used. To increase the transfer efficiency and avoid losses due to spontaneous emission from the excited electronic state, a sequence of alternating short pump and dump pulses can be applied, each transferring a small fraction of the population to the target state (18). Pulses in this case can be weak, since each has to transfer only a small fraction of the population. Additional spectral shaping of the pulses can provide population transfer to a desirable target state, where population is coherently accumulated. In this case molecules formed in the ground vibrational state have to be removed from the interaction volume to avoid excitation by subsequent pulses, leading to increase in the duration of the transfer process. In the second non-coherent method molecules in the ground electronic state are allowed to radiatively decay from the initial high vibrational level reaching the ground *v* = 0 state after several decay steps (19). In this case many intermediate vibrational states are populated which would result in loss of molecules from a trap due to vibrationally inelastic collisions with background atoms and formed molecules. Another incoherent technique, named molecular optical pumping (20), allows to transfer molecules from high vibrational states to the ground state using a shaped laser pulse. Molecules are excited to vibrational levels of higher-energy electronic states and spontaneously decay back to the ground one. The excited state vibrational levels are chosen to have a good Franck-Condon factors with the *v* = 0 vibrational level in the ground electronic state. The laser pulse is spectrally shaped so that all frequencies allowing molecules to be excited back from the *v* = 0 state are removed, and a significant fraction of molecules accumulates in the ground vibrational state after a few excitation- spontaneous

The major coherent methods are adiabatic passage and coherent control techniques. The latter one utilizes spectrally shaped broadband optical pulses to transfer the molecules from an initial to the ground vibrational state with high efficiency (21). Stimulated Raman Adiabatic Passage (STIRAP) (22) has recently attracted significant interest as an efficient way to produce

rovibrational state.

emission cycles.

deeply bound molecules, starting from Feshbach molecules (23; 24). It allows to realize high transfer efficiency and preserve the high phase-space density of an initial atomic gas. In STIRAP, the laser pulses, coupling an initial and a final state to an intermediate excited state, are applied in a counter-intuitive sequence where a pump pulse is preceeded by a Stokes pulse. During the transfer, the system stays in a "dark" state, *i.e.*, a coherent superposition of initial and final states, preventing any losses that would otherwise occur from the excited state. By adiabatically changing amplitudes of the laser pulses, the "dark" state evolves from the initial to the final state, resulting in nearly 100% transfer efficiency. A STIRAP transfer from a Feshbach to a next lower vibrational state of a ground electronic potential has been demonstrated in 87Rb2 molecules held in an optical lattice to avoid inelastic collisions (25). Recently, heteronuclear 40K87Rb molecules have been transferred from a Feshbach to a deeply bound (> 10 GHz binding energy) vibrational state using STIRAP (26). Prospects of STIRAP-based photoassociation of a thermal ensemble of 85Rb atoms have recently been analyzed theoretically in (27). Finally, in breakthrough experiments at JILA and Innsbruck ultracold weakly bound KRb (23) and Cs2 (24) Feshbach molecules have been successfully brought to their ground rovibrational state via two- and multi-photon STIRAP, respectively. In principle, STIRAP allows lossless transfer of the population from an initial to the target state with 100 % efficiency. The main difficulty with a two-pulse STIRAP in molecules is to find an intermediate vibrational state of the excited electronic potential which has a good Franck-Condon overlap with both highly delocalized initial high vibrational state and a tightly localized ground vibrational state (28). It is particularly difficult for homonuclear atoms having the ground electronic potential scaling as 1/*R*<sup>6</sup> and the first excited potentials as 1/*R*<sup>3</sup> with interatomic distance *R*, resulting in a non-favorable potential curves' overlap. It is less an issue for heteronuclear molecules having both potentials falling off with distance according to the 1/*R*<sup>6</sup> law, making the overlap better and enabling a one-step STIRAP (23). It was therefore proposed in (29) to transfer population in several steps down the ladder of vibrational states using a sequence of stimulated optical Raman transitions. In this case the initial and final vibrational levels of each step do not differ significantly, and it is easier to find a suitable intermediate vibrational level in the excited electronic state. In this step-wise approach, however, the population is transfered through a number of vibrational levels in the ground electronic state subject to vibrational relaxation due to inelastic collisions with a background atom or another molecule. The released kinetic energy greatly exceeds the trap depth resulting in loss of both molecules and atoms from the trap. The step-wise transfer therefore has to be faster than the vibrational relaxation time. In the next section we describe a multistate chainwise STIRAP in which molecules are brought to the ground rovibrational state through a series of intermediate states in one run, which allows to minimize collisional losses in intermediate states. In Section III we discuss direct STIRAP conversion of ultracold atoms from a scattering continuum into deeply bound molecules in the presence of a Feshbach resonance. Direct conversion without first forming Feshbach molecules allows to reduce collisional losses during formation of molecules.

#### **2. Theory of multistate chainwise adiabatic passage in the presence of decay**

In this section, we present a multistate chainwise STIRAP technique allowing an efficient transfer of a molecule from a high-lying state to the ground vibrational state which minimizes the population loss due to inelastic collisions during the transfer process (30).

*i*.

by the expression

states decay.

**2.1 c-STIRAP with two pulses**

define an effective Rabi frequency <sup>Ω</sup>(*t*) =

taking into account decay in the bare state basis is

A), where only population decays (∝ *T*−<sup>1</sup>

 Φ0 

where Ω1(*t*) = *μ*1E1(*t*)/2¯*h*, Ω2(*t*) = *μ*2E2(*t*)/2¯*h*, Ω3(*t*) = *μ*3E3(*t*)/2¯*h* and Ω4(*t*) = *μ*4E4(*t*)/2¯*h* are the Rabi frequencies of optical fields; E*<sup>i</sup>* is the amplitude of *i*th optical field, *μ<sup>i</sup>* is the dipole matrix element along the respective transition, Δ<sup>1</sup> = *ω*<sup>1</sup> − *ν*<sup>1</sup> and Δ<sup>2</sup> = *ω*<sup>4</sup> − *ν*<sup>4</sup> are one-photon detunings of the fields, and the *ω<sup>i</sup>* are the molecular frequencies along transition

Coherent Laser Manipulation of Ultracold Molecules 57

We assumed in Eq. (2) that pairs of fields coupling two neighboring ground state vibrational levels are in a two-photon (Raman) resonance; in this case, the system has a dark state, given

In c-STIRAP (as in classical STIRAP) the optical fields are applied in a counterintuitive way, *i.e.* at *t* = −∞ only a combination of the Ω4, Ω3, Ω<sup>2</sup> fields, and at *t* = +∞ only of Ω3, Ω<sup>2</sup> and Ω<sup>1</sup> is present. As a result the dark state is initially associated with the |*g*1� and finally with the |*g*3� states. Adiabatically changing the Rabi frequencies of the optical fields so that the system stays in the dark state during its evolution, one can transfer the system from the initial high-lying |*g*1� to the ground vibrational |*g*3� state with unit efficiency, defined as the population of the |*g*3� state at *t* = +∞. The dark state does not have contributions from the |*e*1� and |*e*2� excited states, which means that they are not populated during the transfer process. As a result, the decay from these states does not affect the transfer efficiency. Decay from the |*g*1�, |*g*2�, |*g*3� states will, however, degrade the coherent superposition (3) and result in population loss from the dark state and reduction of the transfer efficiency. In the next two subsections we consider two c-STIRAP schemes which can be used for efficient population transfer to the ground vibrational state with minimal population loss due to intermediate

Two regimes can provide efficient population transfer to the ground state using multiple intermediate states. In the first one, called c-STIRAP, as also proposed in (31), the Stokes pulses Ω<sup>2</sup> and Ω<sup>4</sup> are applied simultaneously followed with a delay by pump pulses Ω<sup>1</sup> and Ω3, applied at the same time as well. It means that, ideally, only two pulses can be used so that Ω2(*t*) = Ω4(*t*) = Ω*s*(*t*) are Stokes pulses and Ω1(*t*) = Ω3(*t*) = Ω*p*(*t*) are pump pulses. To simplify the analysis, we set the one-photon detunings to zero Δ<sup>1</sup> = Δ<sup>2</sup> = 0, and

tan *θ* = Ω*p*/Ω*s*. In this case the Hamiltonian (2) has a zero eigenvalue describing the dark state (3). Four other eigenvalues correspond to bright states and are given in Appendix A along with a rotation matrix *W* converting adiabatic eigenstates into the bare ones. To study the effect of the decay from the intermediate state |*g*2� and the initial state |*g*1� on the dark state evolution, we turn to a density matrix description (32). The density matrix equation

where the Liouville operator L consists of the usual decays (see exact form in the Appendix

*ih*¯ *dρ*

considered. Initially, all population is assumed to be in state |*g*1�.

 Ω<sup>2</sup> 4Ω<sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>Ω</sup><sup>2</sup> 1Ω<sup>2</sup> <sup>3</sup> <sup>+</sup> <sup>Ω</sup><sup>2</sup> 2Ω<sup>2</sup> 4

<sup>=</sup> <sup>Ω</sup>2Ω<sup>4</sup> <sup>|</sup>*g*1� <sup>−</sup> <sup>Ω</sup>4Ω<sup>1</sup> <sup>|</sup>*g*2� <sup>+</sup> <sup>Ω</sup>1Ω<sup>3</sup> <sup>|</sup>*g*3�

. (3)

Ω*p*(*t*)<sup>2</sup> + Ω*s*(*t*)<sup>2</sup> and a rotation angle *θ*(*t*) by

*dt* <sup>=</sup> [*H*, *<sup>ρ</sup>*] − L*ρ*, (4)

<sup>1</sup> ) into other vibrational states or the continuum are

Fig. 1. Schematic showing the multistate chainwise STIRAP transfer of population from the Feshbach |*g*1� to the ground |*g*3� vibrational state.

We analyze a simple five-level model molecular system with states chainwise coupled by optical fields as illustrated in Fig. 1. The states |*g*1�, |*g*2� and |*g*3� are vibrational levels of a ground electronic molecular state, while |*e*1� and |*e*2� are vibrational states of an excited electronic molecular state. Molecules are formed in a high energy state |*g*1�, which in the following we assume to be a molecular Feshbach state. The state |*g*3� is the deepest bound vibrational state *v* = 0, and |*g*2� is an intermediate vibrational state. The goal is to efficiently transfer the population from the state |*g*1� to state |*g*3�. At least two vibrational levels |*e*1� and |*e*2� in an excited electronic state are required, one which has a good Franck-Condon overlap with |*g*3�, and the other having a good overlap with the initial Feshbach molecular state |*g*1�. In the states |*e*1� and |*e*2�, molecules decay due to spontaneous emission and collisions, and in the states |*g*1� (for bosonic molecules) and |*g*2� they experience fast inelastic collisions with background atoms leading to loss of molecules from a trap. It means that populating the states |*e*1�, |*e*2� and |*g*2� has to be avoided when a background atomic gas is present, or the transfer process has to be faster than the collisional relaxation time.

First, we analyze the system neglecting all decays. The wave function of the system is |Ψ� = ∑*<sup>i</sup> Ci* exp (−*iφi*(*t*))|*i*�, where *i* = *g*1,*e*1, *g*2,*e*2, *g*3; *φg*<sup>1</sup> = 0, *φe*<sup>1</sup> = *ν*1*t*, *φg*<sup>2</sup> = (*ν*<sup>2</sup> − *ν*1)*t*, *φe*<sup>2</sup> = (*ν*<sup>3</sup> + *ν*<sup>2</sup> − *ν*1)*t*, *φg*<sup>3</sup> = (*ν*<sup>4</sup> − *ν*<sup>3</sup> + *ν*<sup>2</sup> − *ν*1)*t*; *ν<sup>i</sup>* is the frequency of the *i*th optical field. The evolution is then governed by the Schrödinger equation

$$i\hbar \frac{\partial \left| \Psi \right>}{\partial t} = H(t) \left| \Psi \right>\,. \tag{1}$$

The time-dependent Hamiltonian of the system in the rotating wave approximation is given by

$$H(t) = \begin{pmatrix} 0 & -\Omega\_4(t) & 0 & 0 & 0 \\ -\Omega\_4(t) & \Delta\_2 & -\Omega\_3(t) & 0 & 0 \\ 0 & -\Omega\_3(t) & 0 & -\Omega\_2(t) & 0 \\ 0 & 0 & -\Omega\_2(t) & \Delta\_1 & -\Omega\_1(t) \\ 0 & 0 & 0 & -\Omega\_1(t) & 0 \end{pmatrix},\tag{2}$$

4 Will-be-set-by-IN-TECH

Δ1 Ω1


Γ1


<sup>Δ</sup> <sup>γ</sup> <sup>2</sup> <sup>2</sup>


γ1

<sup>Ω</sup> <sup>Ω</sup> <sup>3</sup> <sup>4</sup>


Ω2

Γ2


>=|v=0>

Energy

Feshbach |*g*1� to the ground |*g*3� vibrational state.

process has to be faster than the collisional relaxation time.

evolution is then governed by the Schrödinger equation

⎛

⎜⎜⎜⎜⎝

*H*(*t*) =

by

*ih*¯ *∂* |Ψ�

Separation

Fig. 1. Schematic showing the multistate chainwise STIRAP transfer of population from the

We analyze a simple five-level model molecular system with states chainwise coupled by optical fields as illustrated in Fig. 1. The states |*g*1�, |*g*2� and |*g*3� are vibrational levels of a ground electronic molecular state, while |*e*1� and |*e*2� are vibrational states of an excited electronic molecular state. Molecules are formed in a high energy state |*g*1�, which in the following we assume to be a molecular Feshbach state. The state |*g*3� is the deepest bound vibrational state *v* = 0, and |*g*2� is an intermediate vibrational state. The goal is to efficiently transfer the population from the state |*g*1� to state |*g*3�. At least two vibrational levels |*e*1� and |*e*2� in an excited electronic state are required, one which has a good Franck-Condon overlap with |*g*3�, and the other having a good overlap with the initial Feshbach molecular state |*g*1�. In the states |*e*1� and |*e*2�, molecules decay due to spontaneous emission and collisions, and in the states |*g*1� (for bosonic molecules) and |*g*2� they experience fast inelastic collisions with background atoms leading to loss of molecules from a trap. It means that populating the states |*e*1�, |*e*2� and |*g*2� has to be avoided when a background atomic gas is present, or the transfer

First, we analyze the system neglecting all decays. The wave function of the system is |Ψ� = ∑*<sup>i</sup> Ci* exp (−*iφi*(*t*))|*i*�, where *i* = *g*1,*e*1, *g*2,*e*2, *g*3; *φg*<sup>1</sup> = 0, *φe*<sup>1</sup> = *ν*1*t*, *φg*<sup>2</sup> = (*ν*<sup>2</sup> − *ν*1)*t*, *φe*<sup>2</sup> = (*ν*<sup>3</sup> + *ν*<sup>2</sup> − *ν*1)*t*, *φg*<sup>3</sup> = (*ν*<sup>4</sup> − *ν*<sup>3</sup> + *ν*<sup>2</sup> − *ν*1)*t*; *ν<sup>i</sup>* is the frequency of the *i*th optical field. The

The time-dependent Hamiltonian of the system in the rotating wave approximation is given

0 −Ω4(*t*) 000 −Ω4(*t*) Δ<sup>2</sup> −Ω3(*t*) 0 0 0 −Ω3(*t*) 0 −Ω2(*t*) 0 0 0 −Ω2(*t*) Δ<sup>1</sup> −Ω1(*t*) 000 −Ω1(*t*) 0

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>H</sup>*(*t*)|Ψ�. (1)

⎞

⎟⎟⎟⎟⎠ , (2)

where Ω1(*t*) = *μ*1E1(*t*)/2¯*h*, Ω2(*t*) = *μ*2E2(*t*)/2¯*h*, Ω3(*t*) = *μ*3E3(*t*)/2¯*h* and Ω4(*t*) = *μ*4E4(*t*)/2¯*h* are the Rabi frequencies of optical fields; E*<sup>i</sup>* is the amplitude of *i*th optical field, *μ<sup>i</sup>* is the dipole matrix element along the respective transition, Δ<sup>1</sup> = *ω*<sup>1</sup> − *ν*<sup>1</sup> and Δ<sup>2</sup> = *ω*<sup>4</sup> − *ν*<sup>4</sup> are one-photon detunings of the fields, and the *ω<sup>i</sup>* are the molecular frequencies along transition *i*.

We assumed in Eq. (2) that pairs of fields coupling two neighboring ground state vibrational levels are in a two-photon (Raman) resonance; in this case, the system has a dark state, given by the expression

$$
\left| \Phi^0 \right> = \frac{\Omega\_2 \Omega\_4 \left| g\_1 \right> - \Omega\_4 \Omega\_1 \left| g\_2 \right> + \Omega\_1 \Omega\_3 \left| g\_3 \right>}{\sqrt{\Omega\_4^2 \Omega\_1^2 + \Omega\_1^2 \Omega\_3^2 + \Omega\_2^2 \Omega\_4^2}}. \tag{3}
$$

In c-STIRAP (as in classical STIRAP) the optical fields are applied in a counterintuitive way, *i.e.* at *t* = −∞ only a combination of the Ω4, Ω3, Ω<sup>2</sup> fields, and at *t* = +∞ only of Ω3, Ω<sup>2</sup> and Ω<sup>1</sup> is present. As a result the dark state is initially associated with the |*g*1� and finally with the |*g*3� states. Adiabatically changing the Rabi frequencies of the optical fields so that the system stays in the dark state during its evolution, one can transfer the system from the initial high-lying |*g*1� to the ground vibrational |*g*3� state with unit efficiency, defined as the population of the |*g*3� state at *t* = +∞. The dark state does not have contributions from the |*e*1� and |*e*2� excited states, which means that they are not populated during the transfer process. As a result, the decay from these states does not affect the transfer efficiency. Decay from the |*g*1�, |*g*2�, |*g*3� states will, however, degrade the coherent superposition (3) and result in population loss from the dark state and reduction of the transfer efficiency. In the next two subsections we consider two c-STIRAP schemes which can be used for efficient population transfer to the ground vibrational state with minimal population loss due to intermediate states decay.

#### **2.1 c-STIRAP with two pulses**

Two regimes can provide efficient population transfer to the ground state using multiple intermediate states. In the first one, called c-STIRAP, as also proposed in (31), the Stokes pulses Ω<sup>2</sup> and Ω<sup>4</sup> are applied simultaneously followed with a delay by pump pulses Ω<sup>1</sup> and Ω3, applied at the same time as well. It means that, ideally, only two pulses can be used so that Ω2(*t*) = Ω4(*t*) = Ω*s*(*t*) are Stokes pulses and Ω1(*t*) = Ω3(*t*) = Ω*p*(*t*) are pump pulses. To simplify the analysis, we set the one-photon detunings to zero Δ<sup>1</sup> = Δ<sup>2</sup> = 0, and define an effective Rabi frequency <sup>Ω</sup>(*t*) = Ω*p*(*t*)<sup>2</sup> + Ω*s*(*t*)<sup>2</sup> and a rotation angle *θ*(*t*) by tan *θ* = Ω*p*/Ω*s*. In this case the Hamiltonian (2) has a zero eigenvalue describing the dark state (3). Four other eigenvalues correspond to bright states and are given in Appendix A along with a rotation matrix *W* converting adiabatic eigenstates into the bare ones. To study the effect of the decay from the intermediate state |*g*2� and the initial state |*g*1� on the dark state evolution, we turn to a density matrix description (32). The density matrix equation taking into account decay in the bare state basis is

$$i\hbar\frac{d\rho}{dt} = [H,\rho] - \mathcal{L}\rho.\tag{4}$$

where the Liouville operator L consists of the usual decays (see exact form in the Appendix A), where only population decays (∝ *T*−<sup>1</sup> <sup>1</sup> ) into other vibrational states or the continuum are considered. Initially, all population is assumed to be in state |*g*1�.

As expected, the dark state is not affected by the decay from the excited states |*e*1,2�, but only by the decay from the states |*g*1,2�, which destroys the coherent superposition the dark state

Coherent Laser Manipulation of Ultracold Molecules 59

Taking Gaussian laser pulses with the pump and Stokes pulse Rabi frequencies Ω*<sup>p</sup>* = <sup>Ω</sup><sup>0</sup> exp (−(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*/2)/*T*2) and <sup>Ω</sup>*<sup>S</sup>* <sup>=</sup> <sup>Ω</sup><sup>0</sup> exp (−(*<sup>t</sup>* <sup>+</sup> *<sup>τ</sup>*/2)/*T*2) with tan *<sup>θ</sup>* <sup>=</sup> exp (2*tτ*/*T*2), one can see that at the moment *t* = 0 of maximal interaction the rotation angle is *θ* = *π*/4. The

Equation (9) shows that in this regime the decay of both the initial and intermediate vibrational states is not suppressed, *i.e.* directly influences the dark state evolution. This scheme therefore can be applied only if Γ1,2*T*tr � 1. It implies that c-STIRAP pulses have to be shorter than the collisional relaxation time. This requirement combined with the adiabaticity condition might result in large Rabi frequencies needed for high transfer effciency. However, as will be shown in the next section, using reasonable values of Rabi frequencies (∼ 10 MHz) and pulse durations, rather high transfer efficiencies of the order of 90% and 85% for fermionic and bosonic alkali dimers, respectively, can be realized in this scheme in the presence of collisions. High transfer efficiency with weaker and longer pulses is possible in this scheme if the remaining atoms are removed after the molecules are formed. It can be done by applying a "blast" laser pulse resonant with an atomic, but not a molecular electronic transition. In this case Γ1,2 are determined by vibrationally inelastic molecule-molecule collisions, which result in slower decay compared to atom-molecule collisions due to a typically smaller density of molecules. Although, it is worth noting that in this case a collision results in two molecules lost from a trap. Another situation when this regime gives high transfer efficiency using weaker pulses can be used is when molecules are formed in an optical lattice with initially two atoms per site. In this case Γ1,2 will be determined by off-resonant Raman scattering of lattice and

As we discussed in the previous subsection, if the remaining atoms are not removed from the trap, the molecules in the |*g*1,2� states are subject to vibrationally inelastic atom-molecule collisions. To maximaize the number of molecules transferred to the ground |*g*3� state the population of the decaying intermediate state |*g*2� has to be minimized. This can be achieved using an extension of the STIRAP technique to multiple chainwise-coupled states, called a "straddling" STIRAP (33). Namely, if Ω2, Ω<sup>3</sup> � Ω1, Ω<sup>4</sup> and Ω2, Ω<sup>3</sup> temporally overlap both the Ω<sup>1</sup> and Ω<sup>4</sup> pulses, populations in all intermediate states are minimized. To simplify the analysis, we assume Ω<sup>2</sup> = Ω<sup>3</sup> = Ω0; Ω<sup>0</sup> is independent of time (in practice the corresponding pulses just have to be much longer than Ω1(*t*), Ω4(*t*) and overlap both of them), and Ω<sup>0</sup> � |Ω1|, |Ω4|. As in the previous subsection, we set one-photon detunings to zero <sup>Δ</sup><sup>1</sup> <sup>=</sup> <sup>Δ</sup><sup>2</sup> <sup>=</sup> 0, and define the effective Rabi frequency <sup>Ω</sup>(*t*) = <sup>Ω</sup>1(*t*)<sup>2</sup> <sup>+</sup> <sup>Ω</sup>4(*t*)<sup>2</sup> and a rotation angle by tan *θ*(*t*) = Ω1(*t*)/Ω4(*t*). The eigenvalues of the system (2) are

states. Adiabatic eigenstates <sup>|</sup>Φ� <sup>=</sup> {|Φ*n*�}, *<sup>n</sup>* <sup>=</sup> 0, ...4 and the bare states <sup>|</sup>Ψ� <sup>=</sup>

*l* = *g*1,*e*1, *g*2,*e*2, *g*<sup>3</sup> are transformed via a rotation matrix (A2), given in the Appendix A.

√

2, and *ε*3,4 = ±

<sup>√</sup>2Ω<sup>0</sup> to bright

 Ψ*l* ,

<sup>00</sup>(Γ<sup>1</sup> + Γ2)/3. (9)

is based on.

density matrix equation (8) then takes a form

*ρ*˙ *a* <sup>00</sup> <sup>=</sup> <sup>−</sup>*ρ<sup>a</sup>*

STIRAP laser fields (25), which can be sufficiently suppressed.

*ε*<sup>0</sup> = 0, corresponding to the dark state, and *ε*1,2 = ±Ω/

**2.2 Chainwise "straddling" STIRAP**

It is convenient to use the adiabatic basis states to study the effect of decay. In the adiabatic basis the density matrix equation (4) transforms into

$$i\hbar \frac{d\rho^a}{dt} = [H^a, \rho^a] - i\hbar \left[\mathcal{W}^T \dot{\mathcal{W}}\_\prime \rho^a\right] - \mathcal{L}^a \rho^a \tag{5}$$

where the density matrix and the Liouville operator in this basis are given by *ρ<sup>a</sup>* = *WTρW* and <sup>L</sup>*aρ<sup>a</sup>* <sup>=</sup> *<sup>W</sup>T*L*ρW*, and the Hamiltonian *<sup>H</sup><sup>a</sup>* is diagonal; *<sup>W</sup>* is the rotation matrix, given by (A1) in the Appendix A. Initial conditions for Eq. (5) read as *ρ<sup>a</sup>* <sup>00</sup> <sup>=</sup> 1, *<sup>ρ</sup><sup>a</sup> nm* = 0 for *n*, *m* �= 0, where *ρa* <sup>00</sup> denotes the dark state population. The second term on the right hand side of Eq. (5) is responsible for non-adiabatic couplings. This term results in excitation of nondiagonal density matrix elements due to coupling with the dark state. Since the nondiagonal elements *ρ<sup>a</sup>* 0*i* , *i* = 0, ..4, are excited first, the second term in the r.h.s. of Eq. (5), which is ∝ ˙ *θρ<sup>a</sup>* <sup>00</sup>, has to be much smaller than the first term ∼ |*ε*<sup>0</sup> <sup>−</sup> *<sup>ε</sup>i*|*ρ<sup>a</sup>* <sup>0</sup>*<sup>i</sup>* to keep these non-diagonal elements negligible. Thus, in order to maintain adiabaticity for the system to stay in the dark state during the transfer process, we must have

$$
\dot{\theta} \ll \left| \varepsilon\_0 - \varepsilon\_{1,2} \right| \ll \left| \varepsilon\_0 - \varepsilon\_{3,4} \right|. \tag{6}
$$

This gives a standard STIRAP adiabaticity requirement Ω*T*tr � 1, where *T*tr is the c-STIRAP transfer time.

From Eq. (5), the density matrix equation for the dark state population in terms of density matrix elements in the bare state basis can be written as

$$\begin{split} \rho\_{00}^{d}/\xi &= -\Gamma\_{1}\cos^{4}\theta\rho\_{\mathcal{S}1\mathcal{S}1} - \Gamma\_{2}\sin^{2}\theta\cos^{2}\theta\rho\_{\mathcal{S}2\mathcal{S}2} + \\ &+ \frac{1}{2} \Big( (\Gamma\_{2} + \Gamma\_{1})\sin\theta\cos^{3}\theta\rho\_{\mathcal{S}1\mathcal{S}2} + \Gamma\_{2}\cos\theta\sin^{3}\theta\rho\_{\mathcal{S}2\mathcal{S}3} - \Gamma\_{1}\sin^{2}\theta\cos^{2}\theta\rho\_{\mathcal{S}1\mathcal{S}3} + c.c.\Big) (7) \end{split}$$

with *<sup>ξ</sup>* = (<sup>1</sup> <sup>−</sup> <sup>1</sup> <sup>4</sup> sin2 <sup>2</sup>*θ*)−1. Provided that the transfer is adiabatic, so that the system stays in the dark state during the evolution: *ρ<sup>a</sup>* <sup>00</sup> ≈ 1, the density matrix elements in the bare state basis appearing in (7) are expressed via the dark state population in the following way:

$$\rho\_{\mathcal{S}1\mathcal{S}1} \approx \rho\_{00}^{a} \xi \cos^{4} \theta\_{\prime}$$

$$\rho\_{\mathcal{S}2\mathcal{S}2} \approx \rho\_{00}^{a} \xi \sin^{2} \theta \cos^{2} \theta\_{\prime}$$

$$\text{Re}\left(\rho\_{\mathcal{S}1\mathcal{S}2}\right) = \text{Re}\left(\rho\_{\mathcal{S}2\mathcal{S}1}\right) \approx -\rho\_{00}^{a} \xi \sin \theta \cos^{3} \theta\_{\prime}$$

$$\text{Re}\left(\rho\_{\mathcal{S}2\mathcal{S}3}\right) = \text{Re}\left(\rho\_{\mathcal{S}3\mathcal{S}2}\right) \approx -\rho\_{00}^{a} \xi \cos \theta \sin^{3} \theta\_{\prime}$$

$$\text{Re}\left(\rho\_{\mathcal{S}1\mathcal{S}3}\right) = \text{Re}\left(\rho\_{\mathcal{S}3\mathcal{S}1}\right) \approx \rho\_{00}^{a} \xi \sin^{2} \theta \cos^{2} \theta.$$

Inserting these back into Eq. (7), one obtains the equation describing the dark state population decay

$$
\rho\_{00}^a \approx -\left(\Gamma\_1 \cos^4 \theta + \Gamma\_2 \sin^2 \theta \cos^2 \theta\right) \xi \rho\_{00}^a. \tag{8}
$$

6 Will-be-set-by-IN-TECH

It is convenient to use the adiabatic basis states to study the effect of decay. In the adiabatic

where the density matrix and the Liouville operator in this basis are given by *ρ<sup>a</sup>* = *WTρW* and <sup>L</sup>*aρ<sup>a</sup>* <sup>=</sup> *<sup>W</sup>T*L*ρW*, and the Hamiltonian *<sup>H</sup><sup>a</sup>* is diagonal; *<sup>W</sup>* is the rotation matrix, given by (A1)

<sup>00</sup> denotes the dark state population. The second term on the right hand side of Eq. (5) is responsible for non-adiabatic couplings. This term results in excitation of nondiagonal density matrix elements due to coupling with the dark state. Since the nondiagonal elements *ρ<sup>a</sup>*

Thus, in order to maintain adiabaticity for the system to stay in the dark state during the

This gives a standard STIRAP adiabaticity requirement Ω*T*tr � 1, where *T*tr is the c-STIRAP

From Eq. (5), the density matrix equation for the dark state population in terms of density

basis appearing in (7) are expressed via the dark state population in the following way:

*ρg*<sup>2</sup> *<sup>g</sup>*<sup>1</sup>

*ρg*<sup>3</sup> *<sup>g</sup>*<sup>2</sup>

*ρg*<sup>3</sup> *<sup>g</sup>*<sup>1</sup>

*<sup>ρ</sup>g*<sup>1</sup> *<sup>g</sup>*<sup>1</sup> <sup>≈</sup> *<sup>ρ</sup><sup>a</sup>*

*<sup>ρ</sup>g*<sup>2</sup> *<sup>g</sup>*<sup>2</sup> <sup>≈</sup> *<sup>ρ</sup><sup>a</sup>*

≈ −*ρ<sup>a</sup>*

≈ −*ρ<sup>a</sup>*

<sup>≈</sup> *<sup>ρ</sup><sup>a</sup>*

Inserting these back into Eq. (7), one obtains the equation describing the dark state population

Γ<sup>1</sup> cos4 *θ* + Γ<sup>2</sup> sin2 *θ* cos<sup>2</sup> *θ*

(Γ<sup>2</sup> <sup>+</sup> <sup>Γ</sup>1) sin *<sup>θ</sup>* cos<sup>3</sup> *θρg*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> <sup>+</sup> <sup>Γ</sup><sup>2</sup> cos *<sup>θ</sup>* sin3 *θρg*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> <sup>−</sup> <sup>Γ</sup><sup>1</sup> sin2 *<sup>θ</sup>* cos<sup>2</sup> *θρg*<sup>1</sup> *<sup>g</sup>*<sup>3</sup> <sup>+</sup> *<sup>c</sup>*.*c*.

<sup>4</sup> sin2 <sup>2</sup>*θ*)−1. Provided that the transfer is adiabatic, so that the system stays

<sup>00</sup> *<sup>ξ</sup>* cos<sup>4</sup> *<sup>θ</sup>*,

<sup>00</sup> *<sup>ξ</sup>* sin2 *<sup>θ</sup>* cos<sup>2</sup> *<sup>θ</sup>*,

<sup>00</sup> *<sup>ξ</sup>* sin *<sup>θ</sup>* cos3 *<sup>θ</sup>*,

<sup>00</sup> *<sup>ξ</sup>* cos *<sup>θ</sup>* sin3 *<sup>θ</sup>*,

 *ξρ<sup>a</sup>*

<sup>00</sup>. (8)

<sup>00</sup> *<sup>ξ</sup>* sin2 *<sup>θ</sup>* cos<sup>2</sup> *<sup>θ</sup>*.

*WTW*˙ , *ρ<sup>a</sup>*

<sup>00</sup> <sup>=</sup> 1, *<sup>ρ</sup><sup>a</sup>*

*θ* � |*ε*<sup>0</sup> − *ε*1,2| , |*ε*<sup>0</sup> − *ε*3,4| . (6)

<sup>00</sup> ≈ 1, the density matrix elements in the bare state

<sup>0</sup>*<sup>i</sup>* to keep these non-diagonal elements negligible.

− L*aρ<sup>a</sup>* (5)

*nm* = 0 for *n*, *m* �= 0, where

*θρ<sup>a</sup>*

0*i* ,

<sup>00</sup>, has to be

 (7)

, *ρ<sup>a</sup>* ] − *ih*¯ 

*i* = 0, ..4, are excited first, the second term in the r.h.s. of Eq. (5), which is ∝ ˙

˙

matrix elements in the bare state basis can be written as

<sup>00</sup>/*<sup>ξ</sup>* <sup>=</sup> <sup>−</sup>Γ<sup>1</sup> cos4 *θρg*<sup>1</sup> *<sup>g</sup>*<sup>1</sup> <sup>−</sup> <sup>Γ</sup><sup>2</sup> sin2 *<sup>θ</sup>* cos2 *θρg*<sup>2</sup> *<sup>g</sup>*<sup>2</sup> <sup>+</sup>

Re *ρg*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> = Re

Re *ρg*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> = Re

Re *ρg*<sup>1</sup> *<sup>g</sup>*<sup>3</sup> = Re

*ρ*˙ *a* 00 ≈ − in the dark state during the evolution: *ρ<sup>a</sup>*

basis the density matrix equation (4) transforms into

*ih*¯ *dρ<sup>a</sup> dt* <sup>=</sup> [*H<sup>a</sup>*

in the Appendix A. Initial conditions for Eq. (5) read as *ρ<sup>a</sup>*

much smaller than the first term ∼ |*ε*<sup>0</sup> <sup>−</sup> *<sup>ε</sup>i*|*ρ<sup>a</sup>*

transfer process, we must have

transfer time.

+ 1 2 

with *<sup>ξ</sup>* = (<sup>1</sup> <sup>−</sup> <sup>1</sup>

*ρ*˙ *a*

decay

*ρa*

As expected, the dark state is not affected by the decay from the excited states |*e*1,2�, but only by the decay from the states |*g*1,2�, which destroys the coherent superposition the dark state is based on.

Taking Gaussian laser pulses with the pump and Stokes pulse Rabi frequencies Ω*<sup>p</sup>* = <sup>Ω</sup><sup>0</sup> exp (−(*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*/2)/*T*2) and <sup>Ω</sup>*<sup>S</sup>* <sup>=</sup> <sup>Ω</sup><sup>0</sup> exp (−(*<sup>t</sup>* <sup>+</sup> *<sup>τ</sup>*/2)/*T*2) with tan *<sup>θ</sup>* <sup>=</sup> exp (2*tτ*/*T*2), one can see that at the moment *t* = 0 of maximal interaction the rotation angle is *θ* = *π*/4. The density matrix equation (8) then takes a form

$$
\dot{\rho}\_{00}^{a} = -\rho\_{00}^{a} (\Gamma\_1 + \Gamma\_2) / 3. \tag{9}
$$

Equation (9) shows that in this regime the decay of both the initial and intermediate vibrational states is not suppressed, *i.e.* directly influences the dark state evolution. This scheme therefore can be applied only if Γ1,2*T*tr � 1. It implies that c-STIRAP pulses have to be shorter than the collisional relaxation time. This requirement combined with the adiabaticity condition might result in large Rabi frequencies needed for high transfer effciency. However, as will be shown in the next section, using reasonable values of Rabi frequencies (∼ 10 MHz) and pulse durations, rather high transfer efficiencies of the order of 90% and 85% for fermionic and bosonic alkali dimers, respectively, can be realized in this scheme in the presence of collisions. High transfer efficiency with weaker and longer pulses is possible in this scheme if the remaining atoms are removed after the molecules are formed. It can be done by applying a "blast" laser pulse resonant with an atomic, but not a molecular electronic transition. In this case Γ1,2 are determined by vibrationally inelastic molecule-molecule collisions, which result in slower decay compared to atom-molecule collisions due to a typically smaller density of molecules. Although, it is worth noting that in this case a collision results in two molecules lost from a trap. Another situation when this regime gives high transfer efficiency using weaker pulses can be used is when molecules are formed in an optical lattice with initially two atoms per site. In this case Γ1,2 will be determined by off-resonant Raman scattering of lattice and STIRAP laser fields (25), which can be sufficiently suppressed.

#### **2.2 Chainwise "straddling" STIRAP**

As we discussed in the previous subsection, if the remaining atoms are not removed from the trap, the molecules in the |*g*1,2� states are subject to vibrationally inelastic atom-molecule collisions. To maximaize the number of molecules transferred to the ground |*g*3� state the population of the decaying intermediate state |*g*2� has to be minimized. This can be achieved using an extension of the STIRAP technique to multiple chainwise-coupled states, called a "straddling" STIRAP (33). Namely, if Ω2, Ω<sup>3</sup> � Ω1, Ω<sup>4</sup> and Ω2, Ω<sup>3</sup> temporally overlap both the Ω<sup>1</sup> and Ω<sup>4</sup> pulses, populations in all intermediate states are minimized. To simplify the analysis, we assume Ω<sup>2</sup> = Ω<sup>3</sup> = Ω0; Ω<sup>0</sup> is independent of time (in practice the corresponding pulses just have to be much longer than Ω1(*t*), Ω4(*t*) and overlap both of them), and Ω<sup>0</sup> � |Ω1|, |Ω4|. As in the previous subsection, we set one-photon detunings to zero <sup>Δ</sup><sup>1</sup> <sup>=</sup> <sup>Δ</sup><sup>2</sup> <sup>=</sup> 0, and define the effective Rabi frequency <sup>Ω</sup>(*t*) = <sup>Ω</sup>1(*t*)<sup>2</sup> <sup>+</sup> <sup>Ω</sup>4(*t*)<sup>2</sup> and a rotation angle by tan *θ*(*t*) = Ω1(*t*)/Ω4(*t*). The eigenvalues of the system (2) are *ε*<sup>0</sup> = 0, corresponding to the dark state, and *ε*1,2 = ±Ω/ √ 2, and *ε*3,4 = ± <sup>√</sup>2Ω<sup>0</sup> to bright states. Adiabatic eigenstates <sup>|</sup>Φ� <sup>=</sup> {|Φ*n*�}, *<sup>n</sup>* <sup>=</sup> 0, ...4 and the bare states <sup>|</sup>Ψ� <sup>=</sup> Ψ*l* , *l* = *g*1,*e*1, *g*2,*e*2, *g*<sup>3</sup> are transformed via a rotation matrix (A2), given in the Appendix A.

second regime analyzed in this subsection the population of all intermediate states is strongly

Coherent Laser Manipulation of Ultracold Molecules 61

The major prerequisite for high transfer efficiency in STIRAP is the two-photon resonance between fields coupling vibrational levels in the ground electronic state via Raman transitions. It requires all fields to be phase coherent. In a general case the frequency difference between any fields in the chain can be in the THz range. To maintain phase coherence at these large frequency differences the fields can then be phase locked to an optical frequency comb (35).

Magneto- and photo-association techniques produce molecules mostly from ultracold Bose, two-spin component Fermi and mixture alkali metal atomic gases. Weakly bound Feshbach molecules rapidly decay due to vibrationally inelastic atom-molecule collisions, which were found to be the major molecule lifetime limiting factor in atomic traps. Depending on the quantum statistics due to the nuclear spin of the constituent atoms, the alkali dimers show different behavior with respect to inelastic atom-molecule and molecule-molecule collisions. Fermionic alkali dimers in the Feshbach state are very stable with respect to collisions. They are particularly stable close to the resonance, where the scattering length is large. The stability of the fermionic molecules has been explained based on the Pauli exclusion principle combined with significantly different length scales associated with the initial and final vibrational states (36). Lifetimes of the Feshbach molecular states of the order of 1 s have been observed experimentally for 6Li2 fermionic molecules (37; 38), giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 1 s−1, and of the order of 100 ms for 40K2 molecules (39), giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 10 s−1. More deeply bound Feshbach molecules have larger decay rates, with the corresponding collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>10</sup>−<sup>11</sup> cm3s−1. With typical densities of atoms in traps *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−3, it gives <sup>Γ</sup><sup>1</sup> <sup>=</sup> *<sup>k</sup>*inel*n*at <sup>∼</sup> (<sup>1</sup> <sup>−</sup> 103) <sup>s</sup>−<sup>1</sup> for these molecules. To calculate the intermediate state decay rate Γ<sup>2</sup> we use results of a theoretical analysis of collisional stability of low-lying vibrational states of fermionic and bosonic Li2 molecules (40). In low vibrational states, as was shown, fermionic molecules experience fast vibrational quenching due to collisions with surrounding atoms, leading to loss of both molecules and atoms from a trap. The inelastic atom-molecule collision coefficient for fermionic molecules in these low vibrational states is of the order of *<sup>k</sup>*inel <sup>∼</sup> <sup>3</sup> · <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> (calculated in Ref.(40) for fermionic 6Li2 in *<sup>v</sup>* <sup>=</sup> 1). The vibrational relaxation rate <sup>Γ</sup><sup>2</sup> can then be estimated using *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−3, giving

In contrast to fermionic alkali dimers, bosonic dimers experience fast vibrational quenching due to inelastic atom-molecule collisions, even in their Feshbach state. As was observed experimentally for 23Na (41) and 133Cs (42), the inelastic collision coefficient for bosonic molecules due to atom-molecule collisions is of the order of *<sup>k</sup>*inel <sup>∼</sup> <sup>5</sup> · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> for the Feshbach state. An inelastic atom-molecule collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> and an elastic collision coefficient of the same order have been theoretically predicted for 87Rb2 molecules for magetic fields below the Feshbach 1007.4 G resonance (43). Fast vibrationally inelastic atom-molecule collisions thus limit the lifetime of the molecules in the trap to 100 *μ*s - 1 ms. They also limit the atom-to-molecule conversion efficiency during the magnetic field ramp across the resonance. The lifetime of the bosonic molecules in the trap can be significantly extended if at the end of the magnetic field ramp a "blast" laser pulse is applied, selectively removing atoms from the trap (41). In this case, the main loss mechanism is vibrationally inelastic molecule-molecule collisions. The corresponding collision coefficient

suppressed, and the transfer efficiency close to unity can be realized.

**2.3 Collisional relaxation rates for fermionic and bosonic alkali dimers**

<sup>Γ</sup><sup>2</sup> <sup>=</sup> *<sup>k</sup>*inel*n*at <sup>∼</sup> <sup>3</sup> · (101 <sup>−</sup> 104) <sup>s</sup>−1.

The adiabaticity condition (6) in this case becomes ˙ *<sup>θ</sup>* � <sup>Ω</sup>, <sup>Ω</sup>0. If the condition ˙ *θ* � Ω is satisfied, the dark state will not couple to the <sup>Φ</sup>1,2 states. Coupling to the <sup>Φ</sup>3,4 states will be suppressed even more strongly, since Ω � Ω0.

The density matrix equation for the dark state population in terms of density matrix elements in the bare state basis in this case is given by

$$
\rho\_{00}^a = \Gamma\_1 \cos^2 \theta \rho\_{\mathcal{S}1\mathcal{S}1} - \Gamma\_2 \left(\frac{\Omega \sin 2\theta}{2\Omega\_0}\right)^2 \rho\_{\mathcal{S}2\mathcal{S}2} +
$$

$$
+ \left(\Gamma\_2 \frac{\Omega}{4\Omega\_0} \sin 2\theta \sin \theta \rho\_{\mathcal{S}2\mathcal{S}3} + \frac{\Gamma\_1}{2} \sin \theta \cos \theta \rho\_{\mathcal{S}1\mathcal{S}3} +
$$

$$
+ (\Gamma\_1 + \Gamma\_2) \frac{\Omega}{2\Omega\_0} \sin 2\theta \cos \theta \rho\_{\mathcal{S}1\mathcal{S}2} + c.c.\right) \tag{10}
$$

For adiabatic evolution *ρ<sup>a</sup>* <sup>00</sup> ≈ 1, and the density matrix elements in the bare state basis are expressed via the dark state population as:

$$\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^2} \approx \sin^2 2\theta \frac{\Omega^2}{4\Omega\_0^2} \rho\_{00'}^a$$

$$\text{Re}\left(\rho\_{\mathcal{S}1\mathcal{S}^2}\right) = \text{Re}\left(\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^1}\right) \approx -\frac{\Omega}{2\Omega\_0} \sin 2\theta \cos \theta \rho\_{00'}^a$$

$$\text{Re}\left(\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^1}\right) = \text{Re}\left(\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^1}\right) \approx -\frac{\Omega}{2\Omega\_0} \sin 2\theta \sin \theta \rho\_{00'}^a$$

$$\text{Re}\left(\rho\_{\mathcal{S}1\mathcal{S}^1}\right) = \text{Re}\left(\rho\_{\mathcal{S}\mathcal{S}1}\right) \approx \sin 2\theta \rho\_{00'}^a / 2.$$

The decay of the dark state due to the population loss from the |*g*1� and |*g*2� states is then described by the equation (keeping only terms up to the Ω2/Ω<sup>2</sup> <sup>0</sup> order)

$$\dot{\rho}\_{00}^{a} \approx -\left( (\Gamma\_2 + \Gamma\_1 \cos^2 \theta) \left( \frac{\Omega}{2\Omega\_0} \sin 2\theta \right)^2 + \Gamma\_1 \cos^2 \theta \right) \rho\_{00}^{a} \,. \tag{11}$$

Equation (11) shows that intermediate state decay can be neglected during the transfer time *T*tr if (Γ<sup>1</sup> + Γ2)*T*tr (sin 2*θ*Ω/2Ω0) <sup>2</sup> � 1. From this expression one can see that the intermediate state decay rate is reduced by a factor (Ω/Ω0)<sup>2</sup> � 1 in this regime. It also follows from Eq. (11) that decay from the initial state |*g*1� is not suppressed, so that the transfer process has to be faster than this decay.

Both schemes can be readily extended to a general N-state chainwise-linked system with odd number of states having (*N* + 1)/2 ground and (*N* − 1)/2 excited levels (33; 34). In the two-pulse STIRAP scheme, considered in the previous subsection, all Stokes pulses are applied simultaneously followed with a delay by pump pulses applied at the same time as well. In the second regime counterintuitively ordered pump and Stokes pulses drive the first and the last transitions in the chain, while intermediate states are coupled by strong CW (or pulsed with durations longer than that of the pump and Stokes pulses) fields. In the 8 Will-be-set-by-IN-TECH

The density matrix equation for the dark state population in terms of density matrix elements

sin 2*θ* sin *θρg*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> +

*ρg*<sup>2</sup> *<sup>g</sup>*<sup>1</sup>

*ρg*<sup>3</sup> *<sup>g</sup>*<sup>1</sup>

*ρg*<sup>3</sup> *<sup>g</sup>*<sup>1</sup>

2Ω<sup>0</sup>

<sup>Φ</sup>1,2

 Ω sin 2*θ* 2Ω<sup>0</sup>

Γ1

sin 2*θ* cos *θρg*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> + *c*.*c*.

*<sup>ρ</sup>g*<sup>2</sup> *<sup>g</sup>*<sup>2</sup> <sup>≈</sup> sin2 <sup>2</sup>*<sup>θ</sup>* <sup>Ω</sup><sup>2</sup>

 ≈ − <sup>Ω</sup> 2Ω<sup>0</sup>

 ≈ − <sup>Ω</sup> 2Ω<sup>0</sup>

<sup>≈</sup> sin 2*θρ<sup>a</sup>*

sin 2*θ* <sup>2</sup>

The decay of the dark state due to the population loss from the |*g*1� and |*g*2� states is then

 Ω 2Ω<sup>0</sup>

Equation (11) shows that intermediate state decay can be neglected during the transfer time

state decay rate is reduced by a factor (Ω/Ω0)<sup>2</sup> � 1 in this regime. It also follows from Eq. (11) that decay from the initial state |*g*1� is not suppressed, so that the transfer process has to be

Both schemes can be readily extended to a general N-state chainwise-linked system with odd number of states having (*N* + 1)/2 ground and (*N* − 1)/2 excited levels (33; 34). In the two-pulse STIRAP scheme, considered in the previous subsection, all Stokes pulses are applied simultaneously followed with a delay by pump pulses applied at the same time as well. In the second regime counterintuitively ordered pump and Stokes pulses drive the first and the last transitions in the chain, while intermediate states are coupled by strong CW (or pulsed with durations longer than that of the pump and Stokes pulses) fields. In the

<sup>2</sup>

*<sup>θ</sup>* � <sup>Ω</sup>, <sup>Ω</sup>0. If the condition ˙

states. Coupling to the

*ρg*<sup>2</sup> *<sup>g</sup>*2+

<sup>2</sup> sin *<sup>θ</sup>* cos *θρg*<sup>1</sup> *<sup>g</sup>*3<sup>+</sup>

<sup>00</sup> ≈ 1, and the density matrix elements in the bare state basis are

4Ω<sup>2</sup> 0 *ρa* 00,

<sup>00</sup>/2.

sin 2*θ* cos *θρ<sup>a</sup>*

sin 2*θ* sin *θρ<sup>a</sup>*

00,

00,

<sup>0</sup> order)

 *ρa*

<sup>00</sup> . (11)

+ Γ<sup>1</sup> cos<sup>2</sup> *θ*

<sup>2</sup> � 1. From this expression one can see that the intermediate

*θ* � Ω is

states will

<sup>Φ</sup>3,4

. (10)

The adiabaticity condition (6) in this case becomes ˙

<sup>00</sup> <sup>=</sup> <sup>Γ</sup><sup>1</sup> cos2 *θρg*<sup>1</sup> *<sup>g</sup>*<sup>1</sup> <sup>−</sup> <sup>Γ</sup><sup>2</sup>

+(Γ<sup>1</sup> <sup>+</sup> <sup>Γ</sup>2) <sup>Ω</sup>

satisfied, the dark state will not couple to the

in the bare state basis in this case is given by

*ρ*˙ *a*

expressed via the dark state population as:

Re *ρg*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> = Re

Re *ρg*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> = Re

Re *ρg*<sup>1</sup> *<sup>g</sup>*<sup>3</sup> = Re

*ρ*˙ *a* 00 ≈ −

*T*tr if (Γ<sup>1</sup> + Γ2)*T*tr (sin 2*θ*Ω/2Ω0)

faster than this decay.

described by the equation (keeping only terms up to the Ω2/Ω<sup>2</sup>

(Γ<sup>2</sup> + Γ<sup>1</sup> cos2 *θ*)

For adiabatic evolution *ρ<sup>a</sup>*

+ Γ2 Ω 4Ω<sup>0</sup>

be suppressed even more strongly, since Ω � Ω0.

second regime analyzed in this subsection the population of all intermediate states is strongly suppressed, and the transfer efficiency close to unity can be realized.

The major prerequisite for high transfer efficiency in STIRAP is the two-photon resonance between fields coupling vibrational levels in the ground electronic state via Raman transitions. It requires all fields to be phase coherent. In a general case the frequency difference between any fields in the chain can be in the THz range. To maintain phase coherence at these large frequency differences the fields can then be phase locked to an optical frequency comb (35).

#### **2.3 Collisional relaxation rates for fermionic and bosonic alkali dimers**

Magneto- and photo-association techniques produce molecules mostly from ultracold Bose, two-spin component Fermi and mixture alkali metal atomic gases. Weakly bound Feshbach molecules rapidly decay due to vibrationally inelastic atom-molecule collisions, which were found to be the major molecule lifetime limiting factor in atomic traps. Depending on the quantum statistics due to the nuclear spin of the constituent atoms, the alkali dimers show different behavior with respect to inelastic atom-molecule and molecule-molecule collisions. Fermionic alkali dimers in the Feshbach state are very stable with respect to collisions. They are particularly stable close to the resonance, where the scattering length is large. The stability of the fermionic molecules has been explained based on the Pauli exclusion principle combined with significantly different length scales associated with the initial and final vibrational states (36). Lifetimes of the Feshbach molecular states of the order of 1 s have been observed experimentally for 6Li2 fermionic molecules (37; 38), giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 1 s−1, and of the order of 100 ms for 40K2 molecules (39), giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 10 s−1. More deeply bound Feshbach molecules have larger decay rates, with the corresponding collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>10</sup>−<sup>11</sup> cm3s−1. With typical densities of atoms in traps *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−3, it gives <sup>Γ</sup><sup>1</sup> <sup>=</sup> *<sup>k</sup>*inel*n*at <sup>∼</sup> (<sup>1</sup> <sup>−</sup> 103) <sup>s</sup>−<sup>1</sup> for these molecules. To calculate the intermediate state decay rate Γ<sup>2</sup> we use results of a theoretical analysis of collisional stability of low-lying vibrational states of fermionic and bosonic Li2 molecules (40). In low vibrational states, as was shown, fermionic molecules experience fast vibrational quenching due to collisions with surrounding atoms, leading to loss of both molecules and atoms from a trap. The inelastic atom-molecule collision coefficient for fermionic molecules in these low vibrational states is of the order of *<sup>k</sup>*inel <sup>∼</sup> <sup>3</sup> · <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> (calculated in Ref.(40) for fermionic 6Li2 in *<sup>v</sup>* <sup>=</sup> 1). The vibrational relaxation rate <sup>Γ</sup><sup>2</sup> can then be estimated using *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−3, giving <sup>Γ</sup><sup>2</sup> <sup>=</sup> *<sup>k</sup>*inel*n*at <sup>∼</sup> <sup>3</sup> · (101 <sup>−</sup> 104) <sup>s</sup>−1.

In contrast to fermionic alkali dimers, bosonic dimers experience fast vibrational quenching due to inelastic atom-molecule collisions, even in their Feshbach state. As was observed experimentally for 23Na (41) and 133Cs (42), the inelastic collision coefficient for bosonic molecules due to atom-molecule collisions is of the order of *<sup>k</sup>*inel <sup>∼</sup> <sup>5</sup> · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> for the Feshbach state. An inelastic atom-molecule collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> and an elastic collision coefficient of the same order have been theoretically predicted for 87Rb2 molecules for magetic fields below the Feshbach 1007.4 G resonance (43). Fast vibrationally inelastic atom-molecule collisions thus limit the lifetime of the molecules in the trap to 100 *μ*s - 1 ms. They also limit the atom-to-molecule conversion efficiency during the magnetic field ramp across the resonance. The lifetime of the bosonic molecules in the trap can be significantly extended if at the end of the magnetic field ramp a "blast" laser pulse is applied, selectively removing atoms from the trap (41). In this case, the main loss mechanism is vibrationally inelastic molecule-molecule collisions. The corresponding collision coefficient

The set of requirements in the table 1 allows one to obtain a general range of Rabi frequencies and pulse durations, providing optimal population transfer. Since the goal is the production of dense molecular gases with a large number of molecules, we consider a high initial atomic density *<sup>n</sup>*at <sup>∼</sup> 1014 cm−<sup>3</sup> in a trap in our estimates. For bosonic molecules the inelastic atom-molecule collision coefficient differs at resonance and for deeper bound Feshbach molecules by about a factor of two, giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> <sup>5</sup> · 103 <sup>−</sup> 104 <sup>s</sup>−<sup>1</sup> at this density. One can see that the c-STIRAP transfer time has be *<sup>T</sup>*tr � <sup>10</sup>−<sup>4</sup> s. Adiabaticity typically requires <sup>Ω</sup>*T*tr <sup>∼</sup> 102, giving a lower limit on the Rabi frequencies of the c-STIRAP pulses <sup>Ω</sup> � 106 s−1. As our analysis shows transfer efficiencies > 90% can be achieved with pulse durations of several *μ*s and Rabi frequencies of 5 − 10 MHz. It means that to minimize dissociation of molecules due to the back transfer deeper bound Feshbach molecules with *E*Fesh > 1 MHz are prefered. For deeper bound fermionic molecules the decay rate <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 103 <sup>s</sup>−<sup>1</sup> at this high atomic density, resulting in the same pulse durations and Rabi frequencies, maximizing the transfer efficiency, as for bosonic molecules. At resonance, fermionic molecules have significantly smaller decay rates <sup>Γ</sup><sup>1</sup> <sup>∼</sup> <sup>1</sup> <sup>−</sup> 10 s−1. In this case high transfer efficiencies <sup>&</sup>gt; 90% can be realized with longer and weaker pulses of hundred *μ*s duration and Rabi frequencies

Coherent Laser Manipulation of Ultracold Molecules 63

Next we illustrate the technique with numerical simulations of a transfer process based on the system (4) for model seven-state bosonic and fermionic molecular systems. A seven-state system shown in Fig. 2a is easier to realize experimentally, *i.e.* to find transitions with good Franck-Condon factors, than the five-level scheme, analyzed in the previous subsection. For example, a seven-state chainwise path from the Feshbach to the ground vibrational state was found in 87Rb2. The Feshbach state is experimentally formed after a magnetic field crosses the Feshbach resonance at 1007.4 G. Far from the resonance at 973 G the Feshbach state binding energy is 24 MHz (25). In the first step it can be coupled to an electronically

 0−

asymptote. The corresponding Rabi frequency scales as Ω<sup>2</sup> = 2*π* × 6

*<sup>g</sup>* , *v* = 31, *J* = 0

*<sup>g</sup>* , *v*, *J* = 0

dissociation asymptote (45). For example, *v* = 31 vibrational level can be chosen located 6.87 cm−<sup>1</sup> (<sup>≈</sup> 206 GHz) below the asymptote. The corresponding Rabi frequency scales with the field intensity as <sup>Ω</sup><sup>1</sup> <sup>=</sup> <sup>2</sup>*<sup>π</sup>* <sup>×</sup> 2.9*<sup>I</sup>*(*W*/*cm*2) <sup>s</sup>−1, giving the transition dipole moment *μ*<sup>1</sup> ∼ 0.3 D. The second transition of the STIRAP scheme in (25) was to the second-to-last bound vibrational state, located 637 MHz below the ground electronic state dissociation

the transition dipole moment *μ*<sup>2</sup> ∼ 0.6 D. The authors mention that the Franck-Condon

*<sup>g</sup>* (*v* = 119) from where the ground vibrational state can be reached in five steps (29):

The results of the numerical solution of the density matrix equation (4) for a fermionic molecular system are given in Fig. 2. The left column presents the maximal transfer efficiency

*<sup>g</sup>* (*v* = 116) are similar to the second-to-last vibrational state. This includes the

*<sup>g</sup>* (*<sup>v</sup>* <sup>=</sup> 119, *<sup>J</sup>* <sup>=</sup> <sup>0</sup>) <sup>→</sup> *<sup>A</sup>*1Σ<sup>+</sup>

*<sup>u</sup>* (*v*� <sup>=</sup> 185, *<sup>J</sup>* <sup>=</sup> <sup>1</sup>) <sup>→</sup> *<sup>X</sup>*1Σ<sup>+</sup>

*<sup>g</sup>* (*<sup>v</sup>* <sup>=</sup> 52, *<sup>J</sup>* <sup>=</sup> <sup>0</sup>) <sup>→</sup> *<sup>A</sup>*1Σ<sup>+</sup>

*<sup>u</sup>* (*v*� <sup>=</sup> 24, *<sup>J</sup>* <sup>=</sup> <sup>1</sup>) <sup>→</sup> *<sup>X</sup>*1Σ<sup>+</sup>

, located close to the 5*S*1/2 + 5*P*3/2

state to the ground state vibrational levels down

*<sup>u</sup>* (*v*� = 185, *J* = 1),

*<sup>g</sup>* (*v* = 52, *J* = 0),

*<sup>u</sup>* (*v*� = 24, *J* = 1),

*<sup>u</sup>* (*v* = 0, *J* = 0).

*I*(*W*/*cm*2) s−1, giving

∼ 1 MHz.

excited pure long range molecular state

 0−

*X* <sup>1</sup>Σ<sup>+</sup>

*A* <sup>1</sup>Σ<sup>+</sup>

*X* <sup>1</sup>Σ<sup>+</sup>

*A* <sup>1</sup>Σ<sup>+</sup>

factors from the excited

to the *X* <sup>1</sup>Σ<sup>+</sup>

*X* <sup>1</sup>Σ<sup>+</sup>

Two-pulse c-STIRAP "Straddling" STIRAP <sup>1</sup> � <sup>Ω</sup>*T*tr <sup>1</sup> � <sup>Ω</sup>*T*tr � <sup>Ω</sup><sup>2</sup> 0 ΩΓ<sup>2</sup> Ω ∼ *E*Fesh Ω ∼ *E*Fesh 1 � Γ1,2*T*tr 1 � Γ1*T*tr

Table 1

in the Feshbach state was measured for a bosonic 23Na2 molecule as *<sup>k</sup>*inel <sup>∼</sup> 5.1 · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> (41). In this experiment, the initial atomic density and the atom-to-molecule conversion efficiency were *<sup>n</sup>*at <sup>∼</sup> 1.7 · 1014 cm−<sup>3</sup> and 4%, respectively, giving the molecular density of *<sup>n</sup>*mol <sup>∼</sup> <sup>6</sup> · 1012 cm−<sup>3</sup> and therefore the decay rate <sup>Γ</sup><sup>1</sup> <sup>=</sup> *<sup>k</sup>*inel*n*mol <sup>∼</sup> 300 s−1. The vibrational relaxation rate Γ<sup>2</sup> of intermediate vibrational states for bosonic molecules can be estimated from the inelastic atom-molecule collision coefficient in low vibrational states *<sup>k</sup>*inel <sup>∼</sup> <sup>6</sup> · <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> (calculated in Ref.(40) for bosonic 7Li in *v* = 1). For a typical density of atoms in a trap, *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−<sup>3</sup> and the resulting relaxation rate is <sup>Γ</sup><sup>2</sup> <sup>∼</sup> <sup>6</sup> · (101 <sup>−</sup> 104) <sup>s</sup>−1. The heteronuclear molecules are formed from a mixture of Bose and Fermi atomic gases, and their collisional properties are expected to differ from pure fermionic and bosonic molecules discussed above. Stability of the KRb molecules with respect to collisions with initial fermionic and bosonic atoms has been recently studied in (44). Vibrationally inelastic

relaxation was found to be dominated by atom-molecule collisions, and the corresponding collision coefficients to strongly depend on the quantum statistics of the atoms. Close to the heteronuclear Feshbach resonance the collision coefficient for collisions with indistinguishable fermions (40K in the same hyperfine state) was found to be *k*inel < 10−<sup>11</sup> cm3s−1; for collisions with indistinguishable bosons (87Rb in the same hyperfine state) *<sup>k</sup>*inel <sup>∼</sup> (<sup>2</sup> <sup>−</sup> <sup>3</sup>)· <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> close to the resonance. Finally, for collisions with distinduishable atoms (40K in a different hyperfine state) the collision coeffcient *kinel* <sup>∼</sup> (<sup>3</sup> <sup>−</sup> <sup>5</sup>) · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> was measured. These results are therefore consistent with coefficients of vibrationally inelastic collisions with pure fermionic and bosonic atoms, considered in previous paragraphs.

Let us now estimate the parameters of the optical pulses providing maximal transfer efficiency. As follows from Eqs. (3) and Eq. (11), in both STIRAP schemes the decay of the Feshbach state strongly affects the transfer efficiency, and the condition Γ1*T*tr � 1 has to be satisfied to minimize molecular loss. At the same time the adiabaticity condition requires Ω*T*tr � 1. Weakly bound Feshbach states are very close to a dissociation threshold, a typical binding energy *E*Fesh being tens kHz -tens MHz (∼ 1 *μ*K - 1 mK). At these small binding energies the first pulse in the chain (Ω*<sup>p</sup>* or Ω<sup>1</sup> in the two STIRAP schemes), coupling the Feshbach state |*g*1� to the first excited state |*e*1� can lead to back-action, *i.e.* back transfer of the molecule to the scattering contunuum, thus molecular dissociation, via a stimulated Raman process. This effect is minimized if the binding energy of a molecule *E*Fesh is much larger than the effective Rabi frequency corresponding to the coupling between the |*e*1� state and the scattering continuum. Typically, the dipole moment of the bound-bound transition greatly exceeds the dipole moment of the bound-continuum transition. Therefore, the Rabi frequency of the E<sup>1</sup> field between the bound-bound transition |*g*�<sup>1</sup> − |*e*�<sup>1</sup> will be much larger than the effective Rabi frequency for the same field corresponding to coupling between the |*e*1� and the scattering continuum. It means that choosing Ω ∼ *E*Fesh one can make the back transfer process negligible. Finally, we have the following requirements for the Rabi frequencies and durations of the STIRAP pulses (amplitudes and durations are assumed the same for the Stokes and the pump pulse to maximize the transfer efficiency).

10 Will-be-set-by-IN-TECH

Two-pulse c-STIRAP "Straddling" STIRAP <sup>1</sup> � <sup>Ω</sup>*T*tr <sup>1</sup> � <sup>Ω</sup>*T*tr � <sup>Ω</sup><sup>2</sup>

in the Feshbach state was measured for a bosonic 23Na2 molecule as *<sup>k</sup>*inel <sup>∼</sup> 5.1 · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> (41). In this experiment, the initial atomic density and the atom-to-molecule conversion efficiency were *<sup>n</sup>*at <sup>∼</sup> 1.7 · 1014 cm−<sup>3</sup> and 4%, respectively, giving the molecular density of *<sup>n</sup>*mol <sup>∼</sup> <sup>6</sup> · 1012 cm−<sup>3</sup> and therefore the decay rate <sup>Γ</sup><sup>1</sup> <sup>=</sup> *<sup>k</sup>*inel*n*mol <sup>∼</sup> 300 s−1. The vibrational relaxation rate Γ<sup>2</sup> of intermediate vibrational states for bosonic molecules can be estimated from the inelastic atom-molecule collision coefficient in low vibrational states *<sup>k</sup>*inel <sup>∼</sup> <sup>6</sup> · <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> (calculated in Ref.(40) for bosonic 7Li in *v* = 1). For a typical density of atoms in a trap, *<sup>n</sup>*at <sup>∼</sup> 1011 <sup>−</sup> 1014 cm−<sup>3</sup> and the resulting relaxation rate is <sup>Γ</sup><sup>2</sup> <sup>∼</sup> <sup>6</sup> · (101 <sup>−</sup> 104) <sup>s</sup>−1. The heteronuclear molecules are formed from a mixture of Bose and Fermi atomic gases, and their collisional properties are expected to differ from pure fermionic and bosonic molecules discussed above. Stability of the KRb molecules with respect to collisions with initial fermionic and bosonic atoms has been recently studied in (44). Vibrationally inelastic relaxation was found to be dominated by atom-molecule collisions, and the corresponding collision coefficients to strongly depend on the quantum statistics of the atoms. Close to the heteronuclear Feshbach resonance the collision coefficient for collisions with indistinguishable fermions (40K in the same hyperfine state) was found to be *k*inel < 10−<sup>11</sup> cm3s−1; for collisions with indistinguishable bosons (87Rb in the same hyperfine state) *<sup>k</sup>*inel <sup>∼</sup> (<sup>2</sup> <sup>−</sup> <sup>3</sup>)· <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> close to the resonance. Finally, for collisions with distinduishable atoms (40K in a different hyperfine state) the collision coeffcient *kinel* <sup>∼</sup> (<sup>3</sup> <sup>−</sup> <sup>5</sup>) · <sup>10</sup>−<sup>11</sup> cm3s−<sup>1</sup> was measured. These results are therefore consistent with coefficients of vibrationally inelastic collisions with pure

Let us now estimate the parameters of the optical pulses providing maximal transfer efficiency. As follows from Eqs. (3) and Eq. (11), in both STIRAP schemes the decay of the Feshbach state strongly affects the transfer efficiency, and the condition Γ1*T*tr � 1 has to be satisfied to minimize molecular loss. At the same time the adiabaticity condition requires Ω*T*tr � 1. Weakly bound Feshbach states are very close to a dissociation threshold, a typical binding energy *E*Fesh being tens kHz -tens MHz (∼ 1 *μ*K - 1 mK). At these small binding energies the first pulse in the chain (Ω*<sup>p</sup>* or Ω<sup>1</sup> in the two STIRAP schemes), coupling the Feshbach state |*g*1� to the first excited state |*e*1� can lead to back-action, *i.e.* back transfer of the molecule to the scattering contunuum, thus molecular dissociation, via a stimulated Raman process. This effect is minimized if the binding energy of a molecule *E*Fesh is much larger than the effective Rabi frequency corresponding to the coupling between the |*e*1� state and the scattering continuum. Typically, the dipole moment of the bound-bound transition greatly exceeds the dipole moment of the bound-continuum transition. Therefore, the Rabi frequency of the E<sup>1</sup> field between the bound-bound transition |*g*�<sup>1</sup> − |*e*�<sup>1</sup> will be much larger than the effective Rabi frequency for the same field corresponding to coupling between the |*e*1� and the scattering continuum. It means that choosing Ω ∼ *E*Fesh one can make the back transfer process negligible. Finally, we have the following requirements for the Rabi frequencies and durations of the STIRAP pulses (amplitudes and durations are assumed the same for the

Ω ∼ *E*Fesh Ω ∼ *E*Fesh 1 � Γ1,2*T*tr 1 � Γ1*T*tr

fermionic and bosonic atoms, considered in previous paragraphs.

Stokes and the pump pulse to maximize the transfer efficiency).

Table 1

0 ΩΓ<sup>2</sup> The set of requirements in the table 1 allows one to obtain a general range of Rabi frequencies and pulse durations, providing optimal population transfer. Since the goal is the production of dense molecular gases with a large number of molecules, we consider a high initial atomic density *<sup>n</sup>*at <sup>∼</sup> 1014 cm−<sup>3</sup> in a trap in our estimates. For bosonic molecules the inelastic atom-molecule collision coefficient differs at resonance and for deeper bound Feshbach molecules by about a factor of two, giving <sup>Γ</sup><sup>1</sup> <sup>∼</sup> <sup>5</sup> · 103 <sup>−</sup> 104 <sup>s</sup>−<sup>1</sup> at this density. One can see that the c-STIRAP transfer time has be *<sup>T</sup>*tr � <sup>10</sup>−<sup>4</sup> s. Adiabaticity typically requires <sup>Ω</sup>*T*tr <sup>∼</sup> 102, giving a lower limit on the Rabi frequencies of the c-STIRAP pulses <sup>Ω</sup> � 106 s−1. As our analysis shows transfer efficiencies > 90% can be achieved with pulse durations of several *μ*s and Rabi frequencies of 5 − 10 MHz. It means that to minimize dissociation of molecules due to the back transfer deeper bound Feshbach molecules with *E*Fesh > 1 MHz are prefered. For deeper bound fermionic molecules the decay rate <sup>Γ</sup><sup>1</sup> <sup>∼</sup> 103 <sup>s</sup>−<sup>1</sup> at this high atomic density, resulting in the same pulse durations and Rabi frequencies, maximizing the transfer efficiency, as for bosonic molecules. At resonance, fermionic molecules have significantly smaller decay rates <sup>Γ</sup><sup>1</sup> <sup>∼</sup> <sup>1</sup> <sup>−</sup> 10 s−1. In this case high transfer efficiencies <sup>&</sup>gt; 90% can be realized with longer and weaker pulses of hundred *μ*s duration and Rabi frequencies ∼ 1 MHz.

Next we illustrate the technique with numerical simulations of a transfer process based on the system (4) for model seven-state bosonic and fermionic molecular systems. A seven-state system shown in Fig. 2a is easier to realize experimentally, *i.e.* to find transitions with good Franck-Condon factors, than the five-level scheme, analyzed in the previous subsection. For example, a seven-state chainwise path from the Feshbach to the ground vibrational state was found in 87Rb2. The Feshbach state is experimentally formed after a magnetic field crosses the Feshbach resonance at 1007.4 G. Far from the resonance at 973 G the Feshbach state binding energy is 24 MHz (25). In the first step it can be coupled to an electronically excited pure long range molecular state 0− *<sup>g</sup>* , *v*, *J* = 0 , located close to the 5*S*1/2 + 5*P*3/2 dissociation asymptote (45). For example, *v* = 31 vibrational level can be chosen located 6.87 cm−<sup>1</sup> (<sup>≈</sup> 206 GHz) below the asymptote. The corresponding Rabi frequency scales with the field intensity as <sup>Ω</sup><sup>1</sup> <sup>=</sup> <sup>2</sup>*<sup>π</sup>* <sup>×</sup> 2.9*<sup>I</sup>*(*W*/*cm*2) <sup>s</sup>−1, giving the transition dipole moment *μ*<sup>1</sup> ∼ 0.3 D. The second transition of the STIRAP scheme in (25) was to the second-to-last bound vibrational state, located 637 MHz below the ground electronic state dissociation asymptote. The corresponding Rabi frequency scales as Ω<sup>2</sup> = 2*π* × 6 *I*(*W*/*cm*2) s−1, giving the transition dipole moment *μ*<sup>2</sup> ∼ 0.6 D. The authors mention that the Franck-Condon factors from the excited 0− *<sup>g</sup>* , *v* = 31, *J* = 0 state to the ground state vibrational levels down to the *X* <sup>1</sup>Σ<sup>+</sup> *<sup>g</sup>* (*v* = 116) are similar to the second-to-last vibrational state. This includes the *X* <sup>1</sup>Σ<sup>+</sup> *<sup>g</sup>* (*v* = 119) from where the ground vibrational state can be reached in five steps (29):

$$\begin{split} &X^{\ 1}\Sigma\_{\mathcal{S}}^{+}(v=119,\mathcal{J}=0) \to A^{1}\Sigma\_{\mathcal{U}}^{+}(v'=185,\mathcal{J}=1), \\ &A^{\ 1}\Sigma\_{\mathcal{U}}^{+}(v'=185,\mathcal{J}=1) \to X^{1}\Sigma\_{\mathcal{S}}^{+}(v=52,\mathcal{J}=0), \\ &X^{\ 1}\Sigma\_{\mathcal{S}}^{+}(v=52,\mathcal{J}=0) \to A^{1}\Sigma\_{\mathcal{U}}^{+}(v'=24,\mathcal{J}=1), \\ &A^{\ 1}\Sigma\_{\mathcal{U}}^{+}(v'=24,\mathcal{J}=1) \to X^{1}\Sigma\_{\mathcal{U}}^{+}(v=0,\mathcal{J}=0). \end{split}$$

The results of the numerical solution of the density matrix equation (4) for a fermionic molecular system are given in Fig. 2. The left column presents the maximal transfer efficiency

**0.0**

**3x107 s -1**

**0**

*<sup>p</sup>* = Ωmax

STIRAP Ωmax

<sup>1</sup> <sup>=</sup> <sup>Ω</sup>max

Ωmax

**1x107**

**2x107**

**e1 ,e1 , e2 ,e2 , e3 ,e3**

**-4 -2 0 2 4 6**

**t, s**


<sup>4</sup> <sup>=</sup> <sup>6</sup> · 106 <sup>s</sup>−1, <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>3</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>μ</sup>*s.

**<sup>S</sup> <sup>p</sup>**

**g3 ,g3** **(c)**

**0.0 0.2 0.4 0.6 0.8 1.0**

**0.06**

**0.00**

**3x107 s -1**

**0**


*<sup>S</sup>* <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>2</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>μ</sup>*s; for "straddling" STIRAP


**1**

**(a)**



**<sup>3</sup> <sup>2</sup>**

**<sup>6</sup> <sup>3</sup> <sup>2</sup> <sup>1</sup>**


Fig. 2. Results of numerical solution of Eq. (4) for a seven-state model fermionic molecular system, shown in (a). Left colunm (figures (b),(c),(d)) and right column (figures ((e),(f),(g))) present results for a two-pulse and "straddling" STIRAP schemes, respectively. Parameters used: <sup>Γ</sup><sup>1</sup> <sup>=</sup> 103 <sup>s</sup>−1, <sup>Γ</sup><sup>2</sup> <sup>=</sup> <sup>Γ</sup><sup>3</sup> <sup>=</sup> <sup>3</sup> · 104 <sup>s</sup>−1, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*<sup>3</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for a two-pulse

**<sup>5</sup> <sup>4</sup>**


**1x107**

**2x107**

**0.02**

**0.04**

**g4 ,g4**

**g3 ,g3**

**<sup>4</sup> <sup>1</sup>**

**e1 ,e1 , e2 ,e2 , e3 ,e3**

**-5 0 5 10**

**t, s**

**(e)**

**(f)**

**(g)**

**g1 ,g1**

**g2 ,g2**

**(d)**

Coherent Laser Manipulation of Ultracold Molecules 65

**g4 ,g4**

**(b)**

**0.1**

**0.2**

**g2 ,g2**

**g1 ,g1**

**0.3**

**0.0 0.2 0.4 0.6 0.8 1.0**

case for the the two-pulse c-STIRAP scheme, and the right column for the "straddling" STIRAP scheme. In the two-pulse c-STIRAP scheme, the states |*g*1� − |*e*1�; |*g*2� − |*e*2�; |*g*3� − |*e*3� are coupled by the pump field Ω*<sup>p</sup>* = Ωmax *<sup>p</sup>* (1 + tanh (*t* − *τ*/2)/*T*)/2, while the states |*e*1� − |*g*2�; <sup>|</sup>*e*2� <sup>−</sup> <sup>|</sup>*g*3�; <sup>|</sup>*e*3� <sup>−</sup> <sup>|</sup>*g*4� are coupled by the Stokes field <sup>Ω</sup>*<sup>S</sup>* <sup>=</sup> <sup>Ω</sup>max *<sup>S</sup>* (1 − tanh (*t* + *τ*/2)/*T*)/2. In the "straddling" STIRAP scheme the states |*e*1� − |*g*2�; |*g*2� − |*e*2�, |*e*2� − |*g*3�; and |*g*3� − |*e*3� are coupled by CW laser fields with a Rabi frequency Ω0, the first transition |*g*1� − |*e*1� and the last transition <sup>|</sup>*e*3� <sup>−</sup> <sup>|</sup>*g*4� in the chain are coupled by the fields <sup>Ω</sup><sup>1</sup> <sup>=</sup> <sup>Ω</sup>max <sup>1</sup> (1 + tanh (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*/2)/*T*)/2 and <sup>Ω</sup><sup>4</sup> <sup>=</sup> <sup>Ω</sup>max <sup>4</sup> (1 − tanh (*t* + *τ*/2)/*T*)/2, respectively. The transfer efficiency does not strongly depend on the form of optical pulses in this case, the same transfer efficiency was obtained using Gaussian pulses. In the numerical analysis a deeper bound Feshbach state with *<sup>E</sup>*Fesh <sup>∼</sup> tens MHz was assumed, which was the case in the 87Rb2 STIRAP experiment (25). The Rabi frequencies of STIRAP fields were chosen to satisfy the second condition of the 1, and the pulse duration *T* and delay *τ* were varied to find the maximal transfer efficiency. To estimate the decay rate of intermediate vibrational states, the atomic density *<sup>n</sup>*at <sup>∼</sup> 1014 cm−<sup>3</sup> was used along with the inelastic collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>3</sup> · <sup>10</sup>−<sup>10</sup> cm3s−1, giving <sup>Γ</sup>2,3 <sup>=</sup> <sup>3</sup> · 104 <sup>s</sup>−1. Lifetimes of vibrational states of an excited electronic state of the order of 30 ns (*γ*1,2,3 <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1) were also assumed.

The numerical analysis demonstrates that > 90% of population in the case of two-pulse c-STIRAP (see Fig. 2d) and > 96% in the case of "straddling" STIRAP (see Fig. 2e) can be transfered from the Feshbach to the ground vibrational state for the chosen Rabi frequencies using the two-pulse and "straddling" STIRAP schemes, respectively, even in the presence of fast collisional decay of the Feshbach state. This transfer efficiency is realized using STIRAP pulses much shorter than the Feshbach state lifetime. Thus the influence of the decay of this state is significantly reduced.

Results of a similar analysis for a model seven-level bosonic molecular system are shown in Fig. 3. Transfer efficiency of the order of 85% and 92% can be realized with the two-pulse and "straddling" STIRAP schemes, respectively. In this case the form of the STIRAP pulses plays an important role due to the fast decay of the Feshbach state. Using Gaussian pulses instead of tanh pulses results in significantly smaller transfer efficiency since by the time the Stokes pulse arrives the Feshbach state experiences noticable decay. With tanh pulses it is, however, possible to make the delay time between the moment of molecule formation and the start of the transfer process reasonably small to minimize the Feshbach state decay.

We can now estimate intensities of CW and pulsed fields corresponding to Rabi frequencies used in our calculations. Typical dipole moments of electric dipole-allowed transitions between molecular electronic states are of the order of 1 D (Debye) and larger. Assuming that the chosen transitions have reasonably large Franck-Condon factors, we use an estimate of transition dipole moments between vibrational levels in the ground and excited electronic state *Dv*,*v*� <sup>∼</sup> 1 D. Taking the peak Rabi frequency of the pump and Stokes fields <sup>Ω</sup>max *max* 1,4 <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1, the corresponding intensity is *<sup>I</sup>* peak 1,4 <sup>=</sup> *<sup>c</sup>*E<sup>2</sup> 1,4/4*<sup>π</sup>* <sup>=</sup> *<sup>c</sup>*(Ωmax 1,4 *<sup>h</sup>*¯ /*Dv*,*v*�)2/4*<sup>π</sup>* <sup>∼</sup> 0.2 W/cm2; for CW fields with a Rabi frequency <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−<sup>1</sup> the corresponding intensity is *<sup>I</sup>*2,3 <sup>∼</sup> 0.9 W/cm2.

To conclude this section, we analyzed a method to coherently transfer ultracold molecules formed in high-lying vibrational states to the ground vibrational state, based on the multilevel chainwise STIRAP technique. Molecules are transfered from a high vibrational state into a ground rovibrational state *v* = 0, *J* = 0 using Raman transitions via several intermediate vibrational states in the ground electronic state. The former one has lower transfer efficiency 12 Will-be-set-by-IN-TECH

case for the the two-pulse c-STIRAP scheme, and the right column for the "straddling" STIRAP scheme. In the two-pulse c-STIRAP scheme, the states |*g*1� − |*e*1�; |*g*2� − |*e*2�; |*g*3� − |*e*3� are

the "straddling" STIRAP scheme the states |*e*1� − |*g*2�; |*g*2� − |*e*2�, |*e*2� − |*g*3�; and |*g*3� − |*e*3� are coupled by CW laser fields with a Rabi frequency Ω0, the first transition |*g*1� − |*e*1� and the last transition <sup>|</sup>*e*3� <sup>−</sup> <sup>|</sup>*g*4� in the chain are coupled by the fields <sup>Ω</sup><sup>1</sup> <sup>=</sup> <sup>Ω</sup>max

efficiency does not strongly depend on the form of optical pulses in this case, the same transfer efficiency was obtained using Gaussian pulses. In the numerical analysis a deeper bound Feshbach state with *<sup>E</sup>*Fesh <sup>∼</sup> tens MHz was assumed, which was the case in the 87Rb2 STIRAP experiment (25). The Rabi frequencies of STIRAP fields were chosen to satisfy the second condition of the 1, and the pulse duration *T* and delay *τ* were varied to find the maximal transfer efficiency. To estimate the decay rate of intermediate vibrational states, the atomic density *<sup>n</sup>*at <sup>∼</sup> 1014 cm−<sup>3</sup> was used along with the inelastic collision coefficient *<sup>k</sup>*inel <sup>∼</sup> <sup>3</sup> · <sup>10</sup>−<sup>10</sup> cm3s−1, giving <sup>Γ</sup>2,3 <sup>=</sup> <sup>3</sup> · 104 <sup>s</sup>−1. Lifetimes of vibrational states of an excited electronic state

The numerical analysis demonstrates that > 90% of population in the case of two-pulse c-STIRAP (see Fig. 2d) and > 96% in the case of "straddling" STIRAP (see Fig. 2e) can be transfered from the Feshbach to the ground vibrational state for the chosen Rabi frequencies using the two-pulse and "straddling" STIRAP schemes, respectively, even in the presence of fast collisional decay of the Feshbach state. This transfer efficiency is realized using STIRAP pulses much shorter than the Feshbach state lifetime. Thus the influence of the decay of this

Results of a similar analysis for a model seven-level bosonic molecular system are shown in Fig. 3. Transfer efficiency of the order of 85% and 92% can be realized with the two-pulse and "straddling" STIRAP schemes, respectively. In this case the form of the STIRAP pulses plays an important role due to the fast decay of the Feshbach state. Using Gaussian pulses instead of tanh pulses results in significantly smaller transfer efficiency since by the time the Stokes pulse arrives the Feshbach state experiences noticable decay. With tanh pulses it is, however, possible to make the delay time between the moment of molecule formation and the start of

We can now estimate intensities of CW and pulsed fields corresponding to Rabi frequencies used in our calculations. Typical dipole moments of electric dipole-allowed transitions between molecular electronic states are of the order of 1 D (Debye) and larger. Assuming that the chosen transitions have reasonably large Franck-Condon factors, we use an estimate of transition dipole moments between vibrational levels in the ground and excited electronic state *Dv*,*v*� <sup>∼</sup> 1 D. Taking the peak Rabi frequency of the pump and Stokes fields <sup>Ω</sup>max *max* 1,4 <sup>=</sup>

> peak 1,4 <sup>=</sup> *<sup>c</sup>*E<sup>2</sup>

W/cm2; for CW fields with a Rabi frequency <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−<sup>1</sup> the corresponding intensity is

To conclude this section, we analyzed a method to coherently transfer ultracold molecules formed in high-lying vibrational states to the ground vibrational state, based on the multilevel chainwise STIRAP technique. Molecules are transfered from a high vibrational state into a ground rovibrational state *v* = 0, *J* = 0 using Raman transitions via several intermediate vibrational states in the ground electronic state. The former one has lower transfer efficiency

1,4/4*<sup>π</sup>* <sup>=</sup> *<sup>c</sup>*(Ωmax

1,4 *<sup>h</sup>*¯ /*Dv*,*v*�)2/4*<sup>π</sup>* <sup>∼</sup> 0.2

the transfer process reasonably small to minimize the Feshbach state decay.

*<sup>p</sup>* (1 + tanh (*t* − *τ*/2)/*T*)/2, while the states |*e*1� − |*g*2�;

<sup>4</sup> (1 − tanh (*t* + *τ*/2)/*T*)/2, respectively. The transfer

*<sup>S</sup>* (1 − tanh (*t* + *τ*/2)/*T*)/2. In

<sup>1</sup> (1 +

coupled by the pump field Ω*<sup>p</sup>* = Ωmax

tanh (*<sup>t</sup>* <sup>−</sup> *<sup>τ</sup>*/2)/*T*)/2 and <sup>Ω</sup><sup>4</sup> <sup>=</sup> <sup>Ω</sup>max

state is significantly reduced.

<sup>3</sup> · 107 <sup>s</sup>−1, the corresponding intensity is *<sup>I</sup>*

*<sup>I</sup>*2,3 <sup>∼</sup> 0.9 W/cm2.

<sup>|</sup>*e*2� <sup>−</sup> <sup>|</sup>*g*3�; <sup>|</sup>*e*3� <sup>−</sup> <sup>|</sup>*g*4� are coupled by the Stokes field <sup>Ω</sup>*<sup>S</sup>* <sup>=</sup> <sup>Ω</sup>max

of the order of 30 ns (*γ*1,2,3 <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1) were also assumed.

Fig. 2. Results of numerical solution of Eq. (4) for a seven-state model fermionic molecular system, shown in (a). Left colunm (figures (b),(c),(d)) and right column (figures ((e),(f),(g))) present results for a two-pulse and "straddling" STIRAP schemes, respectively. Parameters used: <sup>Γ</sup><sup>1</sup> <sup>=</sup> 103 <sup>s</sup>−1, <sup>Γ</sup><sup>2</sup> <sup>=</sup> <sup>Γ</sup><sup>3</sup> <sup>=</sup> <sup>3</sup> · 104 <sup>s</sup>−1, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*<sup>3</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for a two-pulse STIRAP Ωmax *<sup>p</sup>* = Ωmax *<sup>S</sup>* <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>2</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>μ</sup>*s; for "straddling" STIRAP Ωmax <sup>1</sup> <sup>=</sup> <sup>Ω</sup>max <sup>4</sup> <sup>=</sup> <sup>6</sup> · 106 <sup>s</sup>−1, <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>3</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>3</sup> *<sup>μ</sup>*s.

a molecule using a magnetic field modulated at a frequency close to a binding frequency of a Feshbach molecule (46). Another possibility, which we discuss in the next section, is to use the magnetic field dependent DC interchannel coupling between an entrance and a closed channel states as a first transition in the STIRAP chain (47) followed by optical transitions to the ground vibrational state. The first transition in these cases will directly couple the continuum states of colliding atoms and either a Feshbach molecular state or a bound state in the closed channel. As a result the overall atom-to-molecule conversion efficiency is expected to be higher compared to the two-step conversion sequence, when first a Feshbach molecular state is created, from where a molecule is transferred to the ground vibrational state. The chainwise STIRAP can be applied to resonant photoassociation as well, then the first transition in the STIRAP chain will couple the continuum states to a high energy vibrational state in the

Coherent Laser Manipulation of Ultracold Molecules 67

**3. Efficient formation of ground state ultracold molecules via STIRAP from the**

from the atomic scattering continuum, avoiding formation of Feshbach molecules.

not only those atoms that were first transferred to a Feshbach molecular state.

In this section we describe photoassociative Stimulated Raman Adiabatic Passage (STIRAP) near a Feshbach resonance in a thermal atomic gas (49). We show that it is possible to use *low intensity* laser pulses to directly excite pairs of atoms in the continuum near a Feshbach resonance and to transfer nearly the entire atomic population to the lowest rovibrational level of the molecular ground state. This differs from the STIRAP techniques used in creation of ground state KRb (23) and Cs2 (24) molecules in that the formation process starts directly

Feshbach molecules, used in the STIRAP transfer of KRb and Cs2 to the ground rovibrational state (23; 24) are usually short-lived because of inelastic collisions with background atoms or other molecules. This is especially true for those produced from bosonic or mixed bosonic/fermionic atoms, for which collisions are not suppressed by the Fermi statistics at ultralow temperatures. In a dense atomic gas of density *nat* <sup>∼</sup> 1013 <sup>−</sup> 1014 cm−3, the collisional decay rate can be up to <sup>∼</sup> 104 <sup>s</sup>−<sup>1</sup> (with inelastic rate coefficient *<sup>K</sup>*inel <sup>∼</sup> 0.5 <sup>−</sup> 1.0 <sup>×</sup> <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> for Feshbach molecules (50; 51)). The STIRAP transfer must therefore be fast enough to avoid loosing molecules by inelastic decay. To alleviate this problem, we propose to start the STIRAP process directly from the scattering continuum without first forming Feshbach molecules. Using this approach with many STIRAP pulses and a fast repetition rate would also allow the conversion of nearly an entire atomic ensemble into ground state molecules,

Efficient adiabatic passage from the continuum requires laser pulses shorter than the coherence time of the continuum (27; 52; 53). The adiabaticity condition of STIRAP, Ω*T* � 1, where *T* is the transfer time, therefore implies a large effective Rabi frequency Ω for the pulses. In addition, dipole matrix elements between the continuum and the bound state are usually small, and so the pump pulse that couples the continuum and the excited state would require a very high intensity, which proves impractical. Thus the previous STIRAP experiments (23), being restricted by the very short coherence time of the continuum, used a Feshbach molecular

The small continuum-bound dipole matrix elements can be dramatically increased by photoassociating atoms in the vicinity of a Feshbach resonance. It has been shown, both theoretically and experimentally, that the photoassociation rate increases in the presence

ground electronic state (48).

state as an initial state.

**continuum at a Feshbach resonance**

Fig. 3. Results of numerical solution of Eq. (4) for a model seven-state bosonic molecular system. Parameters used: <sup>Γ</sup><sup>1</sup> <sup>=</sup> 104 <sup>s</sup>−1, <sup>Γ</sup><sup>2</sup> <sup>=</sup> <sup>Γ</sup><sup>3</sup> <sup>=</sup> <sup>6</sup> · 104 <sup>s</sup>−1, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*<sup>3</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for a two-pulse STIRAP Ωmax *<sup>p</sup>* = Ωmax *<sup>S</sup>* <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for "straddling" STIRAP Ωmax <sup>1</sup> <sup>=</sup> <sup>Ω</sup>max <sup>4</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1, <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>1</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>2</sup> *<sup>μ</sup>*s for both schemes.

for both fermionic and bosonic molecules compared to the "straddling" scheme, but is experimentally simpler, since it can be realized with two pulses, while the "straddling" scheme requires at least three optical fields (two pulsed and one CW). Numerical analysis of the transfer process for a typical bosonic and fermionic molecular system in a trap with an atomic density *nat* <sup>∼</sup> 1014 cm−<sup>3</sup> shows that transfer efficiencies <sup>∼</sup> 92% and <sup>∼</sup> 96% respectively are possible even in the presence of fast collisional relaxation of the Feshbach molecular state. Multistate chainwise STIRAP, as described in subsection 2.2, has been recently used in the Innsbruck experiment to transfer Cs2 molecules from a Feshbach to the ground rovibrational state. The transfer efficiency of 55% has been achieved, limited by insufficient laser field intensities resulting in imperfect adiabaticity of the transfer and finite laser linewidth.

The multistate chainwise STIRAP technique allows one to use various transitions, coupled by *e.g.* rf fields and DC interactions. It can therefore be used in combination with the recently demonstrated resonant association method, where a pair of atoms is converted into 14 Will-be-set-by-IN-TECH

**(c)**

**(b)**

**(a)**

Fig. 3. Results of numerical solution of Eq. (4) for a model seven-state bosonic molecular system. Parameters used: <sup>Γ</sup><sup>1</sup> <sup>=</sup> 104 <sup>s</sup>−1, <sup>Γ</sup><sup>2</sup> <sup>=</sup> <sup>Γ</sup><sup>3</sup> <sup>=</sup> <sup>6</sup> · 104 <sup>s</sup>−1, *<sup>γ</sup>*<sup>1</sup> <sup>=</sup> *<sup>γ</sup>*<sup>2</sup> <sup>=</sup> *<sup>γ</sup>*<sup>3</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for

<sup>4</sup> <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1, <sup>Ω</sup><sup>0</sup> <sup>=</sup> <sup>6</sup> · 107 <sup>s</sup>−1, *<sup>T</sup>* <sup>=</sup> <sup>1</sup> *<sup>μ</sup>*s, *<sup>τ</sup>* <sup>=</sup> <sup>−</sup><sup>2</sup> *<sup>μ</sup>*s for both schemes.

for both fermionic and bosonic molecules compared to the "straddling" scheme, but is experimentally simpler, since it can be realized with two pulses, while the "straddling" scheme requires at least three optical fields (two pulsed and one CW). Numerical analysis of the transfer process for a typical bosonic and fermionic molecular system in a trap with an atomic density *nat* <sup>∼</sup> 1014 cm−<sup>3</sup> shows that transfer efficiencies <sup>∼</sup> 92% and <sup>∼</sup> 96% respectively are possible even in the presence of fast collisional relaxation of the Feshbach molecular state. Multistate chainwise STIRAP, as described in subsection 2.2, has been recently used in the Innsbruck experiment to transfer Cs2 molecules from a Feshbach to the ground rovibrational state. The transfer efficiency of 55% has been achieved, limited by insufficient laser field

intensities resulting in imperfect adiabaticity of the transfer and finite laser linewidth.

The multistate chainwise STIRAP technique allows one to use various transitions, coupled by *e.g.* rf fields and DC interactions. It can therefore be used in combination with the recently demonstrated resonant association method, where a pair of atoms is converted into

**0**

*<sup>S</sup>* <sup>=</sup> <sup>3</sup> · 107 <sup>s</sup>−1; for "straddling" STIRAP

**1x107**

**2x107**

**3x107 s -1**

**0.00**

**0.02**

**0.04**

**0.06**

**0.08**

**0.0 0.2 0.4 0.6 0.8 1.0**

**-2 -1 0 1 2 3**

**t, s**

**<sup>1</sup> <sup>4</sup>**

**e1 ,e1 , e2 ,e2 , e3 ,e3**

**g2 ,g2**

**g1 ,g1**

> **g3 ,g3**

**g4 ,g4**

**(f)**

**(e)**

**(d)**

**g4 ,g4**

> **g3 ,g3**

**-2 -1 0 1 2 3**

*<sup>p</sup>* = Ωmax

**t, s**

**<sup>S</sup> <sup>p</sup>**

**0.0 0.2 0.4 0.6 0.8 1.0**

**0.3**

**0.0**

**3x107 s -1**

**0**

a two-pulse STIRAP Ωmax

Ωmax

<sup>1</sup> <sup>=</sup> <sup>Ω</sup>max

**1x107**

**2x107**

**0.1**

**0.2**

**g1 ,g1**

> **e1 ,e1 , e2 ,e2 , e3 ,e3**

**g2 ,g2** a molecule using a magnetic field modulated at a frequency close to a binding frequency of a Feshbach molecule (46). Another possibility, which we discuss in the next section, is to use the magnetic field dependent DC interchannel coupling between an entrance and a closed channel states as a first transition in the STIRAP chain (47) followed by optical transitions to the ground vibrational state. The first transition in these cases will directly couple the continuum states of colliding atoms and either a Feshbach molecular state or a bound state in the closed channel. As a result the overall atom-to-molecule conversion efficiency is expected to be higher compared to the two-step conversion sequence, when first a Feshbach molecular state is created, from where a molecule is transferred to the ground vibrational state. The chainwise STIRAP can be applied to resonant photoassociation as well, then the first transition in the STIRAP chain will couple the continuum states to a high energy vibrational state in the ground electronic state (48).

## **3. Efficient formation of ground state ultracold molecules via STIRAP from the continuum at a Feshbach resonance**

In this section we describe photoassociative Stimulated Raman Adiabatic Passage (STIRAP) near a Feshbach resonance in a thermal atomic gas (49). We show that it is possible to use *low intensity* laser pulses to directly excite pairs of atoms in the continuum near a Feshbach resonance and to transfer nearly the entire atomic population to the lowest rovibrational level of the molecular ground state. This differs from the STIRAP techniques used in creation of ground state KRb (23) and Cs2 (24) molecules in that the formation process starts directly from the atomic scattering continuum, avoiding formation of Feshbach molecules.

Feshbach molecules, used in the STIRAP transfer of KRb and Cs2 to the ground rovibrational state (23; 24) are usually short-lived because of inelastic collisions with background atoms or other molecules. This is especially true for those produced from bosonic or mixed bosonic/fermionic atoms, for which collisions are not suppressed by the Fermi statistics at ultralow temperatures. In a dense atomic gas of density *nat* <sup>∼</sup> 1013 <sup>−</sup> 1014 cm−3, the collisional decay rate can be up to <sup>∼</sup> 104 <sup>s</sup>−<sup>1</sup> (with inelastic rate coefficient *<sup>K</sup>*inel <sup>∼</sup> 0.5 <sup>−</sup> 1.0 <sup>×</sup> <sup>10</sup>−<sup>10</sup> cm3s−<sup>1</sup> for Feshbach molecules (50; 51)). The STIRAP transfer must therefore be fast enough to avoid loosing molecules by inelastic decay. To alleviate this problem, we propose to start the STIRAP process directly from the scattering continuum without first forming Feshbach molecules. Using this approach with many STIRAP pulses and a fast repetition rate would also allow the conversion of nearly an entire atomic ensemble into ground state molecules, not only those atoms that were first transferred to a Feshbach molecular state.

Efficient adiabatic passage from the continuum requires laser pulses shorter than the coherence time of the continuum (27; 52; 53). The adiabaticity condition of STIRAP, Ω*T* � 1, where *T* is the transfer time, therefore implies a large effective Rabi frequency Ω for the pulses. In addition, dipole matrix elements between the continuum and the bound state are usually small, and so the pump pulse that couples the continuum and the excited state would require a very high intensity, which proves impractical. Thus the previous STIRAP experiments (23), being restricted by the very short coherence time of the continuum, used a Feshbach molecular state as an initial state.

The small continuum-bound dipole matrix elements can be dramatically increased by photoassociating atoms in the vicinity of a Feshbach resonance. It has been shown, both theoretically and experimentally, that the photoassociation rate increases in the presence

Fig. 4. Schematics: population from the initial state |Ψ*�*� is transferred to a final target state |1� via an intermediate state |2�. Both |Ψ*�*� and |1� are coupled to |2� by a pump and a Stokes pulse, respectively labeled Ω*<sup>P</sup>* and Ω*S*. A bound level |*b*� corresponding to a closed channel

Coherent Laser Manipulation of Ultracold Molecules 69

We assume that the levels associated with states |1� and |2� are well-isolated, and that there are no off-resonant laser couplings to other levels: this ensures the sufficient accuracy of the

No restriction applies to the definition of the continuum state |Ψ*�*� as it can be associated to either a single-channel or a multi-channel scattering state. In this work, we consider the multi-channel case in which a bound level |*b*� associated to a closed channel is embedded in

with that of |*b*�, a Feshbach resonance (14) occurs. These are common in binary collisions of alkali atoms due to hyperfine mixing and the tuning of the Zeeman interaction by an external magnetic field, hence the possibility to control interatomic interactions with a magnetic field. Following the Fano theory presented in Ref.(60), the scattering state |Ψ*�*� can be expressed as:

2

sinΔ

scattering state |*�*� of the open channel. We assume Δ ∈ [−*π*/2,*π*/2]. The width of the

interaction strength between the open and closed channels. The position of the resonance,

*d�*� *b*(*�*, *�*�

*�* <sup>−</sup> *�*� <sup>−</sup> cos <sup>Δ</sup> *<sup>δ</sup>*(*�* <sup>−</sup> *�*�

*�*−*�*� , includes an interaction induced shift from the energy of the bound

<sup>2</sup>(*�*−*�F*)) is the phase shift due to the interaction between <sup>|</sup>*b*� and the

2, is weakly dependent on the energy, while *V*(*�*) is the


 Γ(*�*� ) Γ(*�*)

*b*(*�*, *�*�

) = <sup>1</sup> *π*

*<sup>a</sup>*(*�*) =

� of an open channel. When the energy of |*�*�

) *�*� � coincides

, (13)

) . (15)

*<sup>π</sup>*Γ(*�*) sin<sup>Δ</sup> , (14)

can be imbedded in the continuum.

three-state model (see e.g. (23; 24; 59)).

the continuum of scattering states |*�*�

with

and

Here, <sup>Δ</sup> <sup>=</sup> <sup>−</sup> arctan( <sup>Γ</sup>

*�<sup>F</sup>* = *Eb* + *P*

state *Eb*.

Feshbach resonance, Γ = 2*π*|*V*(*�*)|

 |*V*(*�*� )| <sup>2</sup>*d�*�

of a Feshbach resonance by several orders of magnitude (54–57). This can be explained by considering that delocalized scattering states acquire some bound-state character due to admixture of a bound level associated with a closed channel, resulting in a large increase of the Franck-Condon factor between the initial scattering state and the final excited state. The recently proposed Feshbach Optimized Photoassociation (FOPA) technique (57) relies on this enhancement to directly reach deeply bound ground state vibrational levels from the scattering continuum. Consequently, photoassociation in the vicinity of a Feshbach resonance is expected to increase molecular formation rate up to 106 molecules/s (57).

Here we show that the approach used in FOPA can be combined with STIRAP for reducing the required pulse intensity. We predict highly efficient transfer of an entire atomic ensemble into the lowest rovibrational level in the molecular ground state. We note that a proposal, where the admixing of a short-range potential to a longer-range excited electronic potential, was recently suggested for improving a two-color pump-dump photoassociation scheme (58). The scattering continuum states have good overlap with the long-range potential, while the admixed short-range potential provides a good overlap with tightly bound vibrational levels of the ground electronic state, greatly improving the efficiency of photoassociation.

This section is organized as follows. In the next subsection, we derive a theoretical model of a combined atomic and molecular system. Fano theory is used to describe the interaction of a bound molecular state with the scattering continuum, represented as closed and open channel, respectively. The resulting continuum states are coupled by two laser fields to the vibrational target state in the ground state via the intermediate excited molecular electronic vibrational state. Next, we describe STIRAP-assisted conversion of a pair of colliding atoms into a deeply bound molecule. In subsection B, we present the results of numerical solutions of the model described in subsection A and in the Appendix B, using typical parameters of alkali dimers. We find optimal Rabi frequencies and profiles of STIRAP pulses. In subection C we average the pair-of-atoms STIRAP transfer efficiency over a thermal atomic ensemble, calculate a fraction of atoms that can be converted into molecules by one STIRAP sequence, and the number of pulses and overall time required to convert an entire atomic ensemble into molecules.

#### **3.1 Model**

We consider a three level system plus a continuum as shown in Figure 4, representing scattering states of two colliding atoms and bound states of a molecule. The ground level labeled |1� is the final product state to which a maximun of population must be transfered. Typically, this level will be the lowest vibrational level (*v*�� = 0, *J*�� = 0) of a ground molecular potential. This ground level is coupled to an excited bound level |2� of an excited molecular potential via a "Stokes" pulse depicted by the blue down-arrow in Figure 4. This level |2� is itself coupled via a pump pulse (red up-arrow) to an initial continuum of unbound scattering states |Ψ� of energies (shaded area in Figure 4). If we denote *C*1(*t*), *C*2(*t*) and *C*(,*t*) the time dependent amplitudes associated to the final, intermediate, and initial states |1�, |2�, and |Ψ�, respectively, then the total wave function |Φ� of the system is given by:

$$\left|\left|\Phi\right> = \mathbb{C}\_1(t)\left|1\right> + \mathbb{C}\_2(t)\left|2\right> + \int d\varepsilon \, \mathbb{C}(\varepsilon, t) \left|\Psi\_{\varepsilon}\right>. \tag{12}$$

Fig. 4. Schematics: population from the initial state |Ψ*�*� is transferred to a final target state |1� via an intermediate state |2�. Both |Ψ*�*� and |1� are coupled to |2� by a pump and a Stokes pulse, respectively labeled Ω*<sup>P</sup>* and Ω*S*. A bound level |*b*� corresponding to a closed channel can be imbedded in the continuum.

We assume that the levels associated with states |1� and |2� are well-isolated, and that there are no off-resonant laser couplings to other levels: this ensures the sufficient accuracy of the three-state model (see e.g. (23; 24; 59)).

No restriction applies to the definition of the continuum state |Ψ*�*� as it can be associated to either a single-channel or a multi-channel scattering state. In this work, we consider the multi-channel case in which a bound level |*b*� associated to a closed channel is embedded in the continuum of scattering states |*�*� � of an open channel. When the energy of |*�*� � coincides with that of |*b*�, a Feshbach resonance (14) occurs. These are common in binary collisions of alkali atoms due to hyperfine mixing and the tuning of the Zeeman interaction by an external magnetic field, hence the possibility to control interatomic interactions with a magnetic field. Following the Fano theory presented in Ref.(60), the scattering state |Ψ*�*� can be expressed as:

$$\left| \left| \Psi\_{\varepsilon} \right> \right| = a(\varepsilon) \left| b \right> + \int d\varepsilon' \, b(\varepsilon, \varepsilon') \left| \varepsilon' \right> \, , \tag{13}$$

with

16 Will-be-set-by-IN-TECH

of a Feshbach resonance by several orders of magnitude (54–57). This can be explained by considering that delocalized scattering states acquire some bound-state character due to admixture of a bound level associated with a closed channel, resulting in a large increase of the Franck-Condon factor between the initial scattering state and the final excited state. The recently proposed Feshbach Optimized Photoassociation (FOPA) technique (57) relies on this enhancement to directly reach deeply bound ground state vibrational levels from the scattering continuum. Consequently, photoassociation in the vicinity of a Feshbach resonance

Here we show that the approach used in FOPA can be combined with STIRAP for reducing the required pulse intensity. We predict highly efficient transfer of an entire atomic ensemble into the lowest rovibrational level in the molecular ground state. We note that a proposal, where the admixing of a short-range potential to a longer-range excited electronic potential, was recently suggested for improving a two-color pump-dump photoassociation scheme (58). The scattering continuum states have good overlap with the long-range potential, while the admixed short-range potential provides a good overlap with tightly bound vibrational levels

This section is organized as follows. In the next subsection, we derive a theoretical model of a combined atomic and molecular system. Fano theory is used to describe the interaction of a bound molecular state with the scattering continuum, represented as closed and open channel, respectively. The resulting continuum states are coupled by two laser fields to the vibrational target state in the ground state via the intermediate excited molecular electronic vibrational state. Next, we describe STIRAP-assisted conversion of a pair of colliding atoms into a deeply bound molecule. In subsection B, we present the results of numerical solutions of the model described in subsection A and in the Appendix B, using typical parameters of alkali dimers. We find optimal Rabi frequencies and profiles of STIRAP pulses. In subection C we average the pair-of-atoms STIRAP transfer efficiency over a thermal atomic ensemble, calculate a fraction of atoms that can be converted into molecules by one STIRAP sequence, and the number of pulses and overall time required to convert an entire atomic ensemble into

We consider a three level system plus a continuum as shown in Figure 4, representing scattering states of two colliding atoms and bound states of a molecule. The ground level labeled |1� is the final product state to which a maximun of population must be transfered. Typically, this level will be the lowest vibrational level (*v*�� = 0, *J*�� = 0) of a ground molecular potential. This ground level is coupled to an excited bound level |2� of an excited molecular potential via a "Stokes" pulse depicted by the blue down-arrow in Figure 4. This level |2� is itself coupled via a pump pulse (red up-arrow) to an initial continuum of unbound scattering states |Ψ� of energies (shaded area in Figure 4). If we denote *C*1(*t*), *C*2(*t*) and *C*(,*t*) the time dependent amplitudes associated to the final, intermediate, and initial states |1�, |2�, and

*d C*(,*t*)|Ψ�. (12)



is expected to increase molecular formation rate up to 106 molecules/s (57).

of the ground electronic state, greatly improving the efficiency of photoassociation.

molecules.

**3.1 Model**

$$a(\varepsilon) = \sqrt{\frac{2}{\pi \Gamma(\varepsilon)}} \sin \Delta \,, \tag{14}$$

and

$$b(\mathfrak{e}, \mathfrak{e}') = \frac{1}{\pi} \sqrt{\frac{\Gamma(\mathfrak{e}')}{\Gamma(\mathfrak{e})}} \frac{\sin \Delta}{\mathfrak{e} - \mathfrak{e}'} - \cos \Delta \,\delta(\mathfrak{e} - \mathfrak{e}') \,. \tag{15}$$

Here, <sup>Δ</sup> <sup>=</sup> <sup>−</sup> arctan( <sup>Γ</sup> <sup>2</sup>(*�*−*�F*)) is the phase shift due to the interaction between <sup>|</sup>*b*� and the scattering state |*�*� of the open channel. We assume Δ ∈ [−*π*/2,*π*/2]. The width of the Feshbach resonance, Γ = 2*π*|*V*(*�*)| 2, is weakly dependent on the energy, while *V*(*�*) is the interaction strength between the open and closed channels. The position of the resonance, *�<sup>F</sup>* = *Eb* + *P* |*V*(*�*� )| <sup>2</sup>*d�*� *�*−*�*� , includes an interaction induced shift from the energy of the bound state *Eb*.

where *t* = 0 is some moment before the collision of the two atoms. The resulting continuum

Coherent Laser Manipulation of Ultracold Molecules 71

Inserting this result into Eq. (22), we obtain a final system of equations for the amplitudes of

 ∞ *�*th

*dt*� Ω*�*(*t* � )∗*c*2(*t* � )*e i*Δ*�*(*t* � −*t*)

The third term of Eq.(27), labelled *S*, corresponds to the source function, wheareas the last term, labelled *T*, corresponds to the "back-stimulation" term (or back-conversion) which accounts for the transfer of the bound molecules back into the continuum. The initial amplitude of the continuum wave function *s*(*�*, *t* = 0) appearing in the source term has been discussed in various contributions (27; 52; 53). A Gaussian wavepacket provides the most classical description of a two-atom collision characterized by a minimal uncertainty relation between the energy bandwidth *δ�* of the wavepacket and the duration of the collision:

> <sup>−</sup> (*�*−*�*0)<sup>2</sup> 2*δ*2 *�* + *<sup>i</sup> <sup>h</sup>*¯ (*�*−*�*0)*t*<sup>0</sup>

 *t* 0

<sup>−</sup>*t*) + *s*(*�*, *t* = 0)*e*

*d�* Ω*�*(*t*)*s*(*�*, *t* = 0)*e*

≡ (*δ* − *iγ*)*c*<sup>2</sup> − Ω*Sc*<sup>1</sup> − *S* + *T* , (27)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*<sup>2</sup> , (26)

−*i*Δ*�t*

−*i*Δ*�t*

, (28)

(Γ/2)<sup>2</sup> + (*�* <sup>−</sup> *�F*)<sup>2</sup> sgn(*�* <sup>−</sup> *�F*), (29)

*<sup>π</sup>V*∗(*�*)(*<sup>μ</sup>*2*�* · *<sup>e</sup>*ˆ*p*) , (30)

. (25)

amplitude is

the bound states:

given by (60)

*c*(*�*, *t*) = *i*

*i ∂c*<sup>1</sup>

*i ∂c*<sup>2</sup>  *t* 0

*dt*� Ω∗ *�* (*t* � )*c*2(*t* � )*ei*Δ*�*(*<sup>t</sup>* �

*<sup>∂</sup><sup>t</sup>* = (*<sup>δ</sup>* <sup>−</sup> *<sup>i</sup>γ*)*c*<sup>2</sup> <sup>−</sup> <sup>Ω</sup>*Sc*<sup>1</sup> <sup>−</sup>

where we introduced a spontaneous decay term *γc*<sup>2</sup> in Eq.(27).

*<sup>s</sup>*(*�*, *<sup>t</sup>* <sup>=</sup> <sup>0</sup>) = <sup>1</sup>

*h*¯

*<sup>q</sup>* <sup>=</sup> (*<sup>μ</sup>*2*<sup>b</sup>* · *<sup>e</sup>*ˆ*p*) + *<sup>P</sup>*

<sup>Ω</sup>*�* <sup>=</sup> *<sup>μ</sup>*2*�* · *<sup>e</sup>*ˆ*p*E*<sup>p</sup>*

state |2�, and *q* is the Fano parameter, expressed as:

(*πδ*<sup>2</sup> *�* )1/4 *<sup>e</sup>*

where *t*<sup>0</sup> is the moment of the collision and *�*<sup>0</sup> is the central energy of the wavepacket.

Futhermore, the Rabi frequency of the field coupling continuum states |Ψ*�*� to the state |2� is

where *μ*2*�* is the dipole matrix element between an unperturbed scattering state |*�*� and the

where *e*ˆ*<sup>p</sup>* is the polarization vector of the pump field, and *μ*2*<sup>b</sup>* is the dipole matrix element between bound states |2� and |*b*�. The *q* factor is essentially the ratio of the dipole matrix elements from the state |2� to the bound state |*b*� (modified by the continuum) and to an unperturbed continuum state |*�*�. This factor can be made much larger than unity, and as will be shown below, the total dipole matrix element from the continuum can be enhanced by this factor in the presence of the resonance. The magnitude of *q* can be controlled by the choice of the vibrational state |2�. Selecting a tightly bound excited vibrational state will increase the

*q*Γ/2 + *�* − *�<sup>F</sup>*

*<sup>V</sup>*(*�*�

)(*μ*2*�*�·*e*ˆ*p*)*d�*� *�*−*�*�

*d�* Ω*�*(*t*)

+*i* ∞ *�*th

If we label *Ei* the energy of the state |*i*�, the total Hamiltonian *H* is given by:

$$H = \sum\_{i=1,2} E\_i |i\rangle\langle i| + \int d\varepsilon \,\varepsilon |\Psi\_{\varepsilon}\rangle\langle \Psi\_{\varepsilon}| + V\_{\text{light}}\,. \tag{16}$$

The light-matter interaction Hamiltonian *V*light takes the form:

$$\begin{split} V\_{\text{light}} &= -\left[ \vec{\mu}\_{21} | \2 \rangle \langle 1 | + \text{H.c.} \right] \cdot \left( \vec{\mathcal{E}}\_{p} + \vec{\mathcal{E}}\_{S} + \text{c.c.} \right) \\ &- \int d\varepsilon \, \left[ \vec{\mu}\_{2\Psi\_{\varepsilon}} | 2 \rangle \left< \Psi\_{\varepsilon} | + \text{H.c.} \right] \cdot \left( \vec{\mathcal{E}}\_{p} + \vec{\mathcal{E}}\_{S} + \text{c.c.} \right) \,, \tag{17} \end{split} \tag{17}$$

where <sup>E</sup>*<sup>p</sup>*,*<sup>S</sup>* <sup>=</sup> *<sup>e</sup>*ˆ*p*,*S*E*p*,*<sup>S</sup>* exp(−*iωp*,*St*) are the pump and Stokes laser fields of polarization *<sup>e</sup>*ˆ*p*,*S*, respectively, while *μ*<sup>21</sup> and *μ*2Ψ*�* are the dipole transition moments between the states |2� and |1�, and |2� and |Ψ*�*�, respectively. In this form, the Hamiltonian already takes into account the mixing between the bound state of the closed channel and the scattering states of the open channel. The Schrödinger equation describing the STIRAP conversion of two atoms into a molecule gives:

$$i\hbar \frac{\partial \mathbb{C}\_1}{\partial t} = E\_1 \mathbb{C}\_1 - \vec{\mu}\_{21}^\* \cdot \vec{\mathcal{E}}\_S^\* \, \mathbb{C}\_2 \tag{18}$$

$$i\hbar \frac{\partial \mathbb{C}\_2}{\partial t} = E\_2 \mathbb{C}\_2 - \vec{\mu}\_{21} \cdot \vec{\mathcal{E}}\_S \, \mathbb{C}\_1 \tag{19}$$

$$-\int\_{\varepsilon\_{th}}^{\infty} d\varepsilon \,\vec{\mu}\_{2\mathbf{V}\_{\varepsilon}} \cdot \vec{\mathcal{E}}\_{p}^{\*} \, \mathsf{C}(\varepsilon, t),$$

$$i\hbar \frac{\partial \mathsf{C}(\varepsilon, t)}{\partial t} = \varepsilon \,\mathsf{C}(\varepsilon, t) - \vec{\mu}\_{2\mathbf{V}\_{\varepsilon}}^{\*} \cdot \vec{\mathcal{E}}\_{p}^{\*} \, \mathsf{C}\_{2}.\tag{20}$$

For simplicity, we set the origin of the energy to be the position of the ground state |1�, and use the rotating wave approximation with *C*<sup>1</sup> = *c*1, *C*<sup>2</sup> = *c*2*e*−*iωSt* , and *C*(*�*,*t*) = *c*(*�*, *t*)*e*−*i*(*ωS*−*ωP*)*<sup>t</sup>* . Eqs.(18)-(20) then become:

$$i\frac{\partial c\_1}{\partial t} = -\Omega\_S c\_{2'} \tag{21}$$

$$i\frac{\partial c\_2}{\partial t} = \delta c\_2 - \Omega\_S c\_1 - \int\_{\varepsilon\_{\rm th}}^{\infty} d\varepsilon \, \Omega\_{\rm \varepsilon} c(\varepsilon, t), \tag{22}$$

$$i\frac{\partial c(\varepsilon,t)}{\partial t} = \Delta\_{\varepsilon}c(\varepsilon,t) - \Omega\_{\varepsilon}^{\*}c\_{2\prime} \tag{23}$$

where *δ* = *E*2/¯*h* − *ωS*, Δ*�* = *�*/¯*h* − (*ω<sup>S</sup>* − *ωp*), and *�*th is the dissociation energy of the ground electronic potential with respect to the state |1�. The Rabi frequencies of the fields are Ω*<sup>S</sup>* = *μ*<sup>21</sup> · *e*ˆ*S*E*S*/¯*h* (assumed real), Ω*�* = *μ*2Ψ*�* · *e*ˆ*p*E*p*/¯*h*.

The previous system of three equations can be reduced into a two-equation system by eliminating the continuum amplitude *c*(*�*, *t*) in Eq.(23). Introducing a solution in the form of *c*(*�*, *t*) = *s*(*�*,*t*) exp (−*i*Δ*�t*) into Eq.(23), we get

$$s = \mathrm{i} \int\_0^t dt' \, \Omega\_\epsilon^\*(t') c\_2(t') e^{i\Delta\_\epsilon t'} + s(\epsilon, t = 0), \tag{24}$$

18 Will-be-set-by-IN-TECH

*d�* [*μ*2Ψ*�* |2��Ψ*�*| + H.c.] ·

*ih*¯ *∂C*<sup>1</sup>

*ih*¯ *∂C*<sup>2</sup>

*∂C*(*�*,*t*)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>δ</sup>c*<sup>2</sup> <sup>−</sup> <sup>Ω</sup>*Sc*<sup>1</sup> <sup>−</sup>

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>Δ</sup>*�c*(*�*, *<sup>t</sup>*) <sup>−</sup> <sup>Ω</sup><sup>∗</sup>

*dt*� Ω∗ *�* (*t* � )*c*2(*t* � )*ei*Δ*�<sup>t</sup>* �

For simplicity, we set the origin of the energy to be the position of the ground state |1�, and use

where *δ* = *E*2/¯*h* − *ωS*, Δ*�* = *�*/¯*h* − (*ω<sup>S</sup>* − *ωp*), and *�*th is the dissociation energy of the ground electronic potential with respect to the state |1�. The Rabi frequencies of the fields are Ω*<sup>S</sup>* =

The previous system of three equations can be reduced into a two-equation system by eliminating the continuum amplitude *c*(*�*, *t*) in Eq.(23). Introducing a solution in the form

*<sup>d</sup>� <sup>μ</sup>*2Ψ*�* · <sup>E</sup>*<sup>p</sup> <sup>C</sup>*(*�*,*t*),

*ih*¯

the rotating wave approximation with *C*<sup>1</sup> = *c*1, *C*<sup>2</sup> = *c*2*e*−*iωSt*

*i ∂c*<sup>1</sup>

*i ∂c*<sup>2</sup>

*i ∂c*(*�*, *t*)

*μ*<sup>21</sup> · *e*ˆ*S*E*S*/¯*h* (assumed real), Ω*�* = *μ*2Ψ*�* · *e*ˆ*p*E*p*/¯*h*.

of *c*(*�*, *t*) = *s*(*�*,*t*) exp (−*i*Δ*�t*) into Eq.(23), we get

*s* = *i t* 0

where <sup>E</sup>*<sup>p</sup>*,*<sup>S</sup>* <sup>=</sup> *<sup>e</sup>*ˆ*p*,*S*E*p*,*<sup>S</sup>* exp(−*iωp*,*St*) are the pump and Stokes laser fields of polarization *<sup>e</sup>*ˆ*p*,*S*, respectively, while *μ*<sup>21</sup> and *μ*2Ψ*�* are the dipole transition moments between the states |2� and |1�, and |2� and |Ψ*�*�, respectively. In this form, the Hamiltonian already takes into account the mixing between the bound state of the closed channel and the scattering states of the open channel. The Schrödinger equation describing the STIRAP conversion of two atoms into a

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>E</sup>*<sup>1</sup> *<sup>C</sup>*<sup>1</sup> <sup>−</sup>*<sup>μ</sup>*<sup>∗</sup>

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *� <sup>C</sup>*(*�*,*t*) <sup>−</sup>*<sup>μ</sup>*<sup>∗</sup>

 ∞ *�*th

<sup>E</sup>*<sup>p</sup>* <sup>+</sup> <sup>E</sup>*<sup>S</sup>* <sup>+</sup> c.c.

<sup>21</sup> · <sup>E</sup> <sup>∗</sup>

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>E</sup>*<sup>2</sup> *<sup>C</sup>*<sup>2</sup> <sup>−</sup>*<sup>μ</sup>*<sup>21</sup> · <sup>E</sup>*<sup>S</sup> <sup>C</sup>*<sup>1</sup> (19)

<sup>2</sup>Ψ*�* · <sup>E</sup> <sup>∗</sup>

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*2, (21)

<sup>E</sup>*<sup>p</sup>* <sup>+</sup> <sup>E</sup>*<sup>S</sup>* <sup>+</sup> c.c.

*d� �*|Ψ*�*��Ψ*�*| + *V*light . (16)

, (17)

*<sup>S</sup> C*2, (18)

*<sup>p</sup> C*2. (20)

.

, and *C*(*�*,*t*) = *c*(*�*, *t*)*e*−*i*(*ωS*−*ωP*)*<sup>t</sup>*

*d�* Ω*�c*(*�*, *t*), (22)

+ *s*(*�*, *t* = 0), (24)

*� c*2, (23)

If we label *Ei* the energy of the state |*i*�, the total Hamiltonian *H* is given by:

*Ei*|*i*��*i*| +

*H* = ∑ *i*=1,2

> −

− ∞ *�th*

Eqs.(18)-(20) then become:

molecule gives:

The light-matter interaction Hamiltonian *V*light takes the form:

*V*light = − [*μ*21|2��1| + H.c.] ·

where *t* = 0 is some moment before the collision of the two atoms. The resulting continuum amplitude is

$$\mathcal{L}(\mathfrak{e},t) = i \int\_0^t dt' \, \Omega\_{\mathfrak{e}}^\*(t') c\_2(t') e^{i\Delta\_{\mathfrak{e}}(t'-t)} + s(\mathfrak{e}, t=0) e^{-i\Delta\_{\mathfrak{e}}t}. \tag{25}$$

Inserting this result into Eq. (22), we obtain a final system of equations for the amplitudes of the bound states:

$$\begin{split} i\frac{\partial c\_{1}}{\partial t} &= -\Omega\_{S}c\_{2} \\ i\frac{\partial c\_{2}}{\partial t} &= (\delta - i\gamma)c\_{2} - \Omega\_{S}c\_{1} - \int\_{\varepsilon\_{\text{th}}}^{\infty} d\varepsilon \, \Omega\_{\varepsilon}(t)s(\varepsilon, t = 0)e^{-i\Delta\_{\text{f}}t} \\ &+ i\int\_{\varepsilon\_{\text{th}}}^{\infty} d\varepsilon \, \Omega\_{\varepsilon}(t) \int\_{0}^{t} dt' \, \Omega\_{\varepsilon}(t')^{\*}c\_{2}(t')e^{i\Delta\_{\varepsilon}(t' - t)} \\ &\equiv (\delta - i\gamma)c\_{2} - \Omega\_{S}c\_{1} - S + T \end{split} \tag{27}$$

where we introduced a spontaneous decay term *γc*<sup>2</sup> in Eq.(27).

The third term of Eq.(27), labelled *S*, corresponds to the source function, wheareas the last term, labelled *T*, corresponds to the "back-stimulation" term (or back-conversion) which accounts for the transfer of the bound molecules back into the continuum. The initial amplitude of the continuum wave function *s*(*�*, *t* = 0) appearing in the source term has been discussed in various contributions (27; 52; 53). A Gaussian wavepacket provides the most classical description of a two-atom collision characterized by a minimal uncertainty relation between the energy bandwidth *δ�* of the wavepacket and the duration of the collision:

$$s(\varepsilon, t = 0) = \frac{1}{(\pi \delta\_{\varepsilon}^{2})^{1/4}} e^{-\frac{(\varepsilon - \varepsilon\_{0})^{2}}{2\delta\_{\varepsilon}^{2}} + \frac{i}{\hbar}(\varepsilon - \varepsilon\_{0})t\_{0}}\tag{28}$$

where *t*<sup>0</sup> is the moment of the collision and *�*<sup>0</sup> is the central energy of the wavepacket. Futhermore, the Rabi frequency of the field coupling continuum states |Ψ*�*� to the state |2� is given by (60)

$$
\Omega\_{\varepsilon} = \frac{\vec{\mu}\_{2\varepsilon} \cdot \theta\_p \mathcal{E}\_p}{\hbar} \frac{q \Gamma/2 + \varepsilon - \varepsilon\_F}{\sqrt{(\Gamma/2)^2 + (\varepsilon - \varepsilon\_F)^2}} \text{sgn}(\varepsilon - \varepsilon\_F), \tag{29}
$$

where *μ*2*�* is the dipole matrix element between an unperturbed scattering state |*�*� and the state |2�, and *q* is the Fano parameter, expressed as:

$$q = \frac{(\vec{\mu}\_{2b} \cdot \pounds\_p) + P \int \frac{V(\epsilon')(\vec{\mu}\_{2\epsilon'} \cdot \pounds\_p) d\epsilon'}{\varepsilon - \varepsilon'}}{\pi V^\*(\varepsilon)(\vec{\mu}\_{2\epsilon} \cdot \pounds\_p)}\tag{30}$$

where *e*ˆ*<sup>p</sup>* is the polarization vector of the pump field, and *μ*2*<sup>b</sup>* is the dipole matrix element between bound states |2� and |*b*�. The *q* factor is essentially the ratio of the dipole matrix elements from the state |2� to the bound state |*b*� (modified by the continuum) and to an unperturbed continuum state |*�*�. This factor can be made much larger than unity, and as will be shown below, the total dipole matrix element from the continuum can be enhanced by this factor in the presence of the resonance. The magnitude of *q* can be controlled by the choice of the vibrational state |2�. Selecting a tightly bound excited vibrational state will increase the

Reso- *δ�* Γ Ω<sup>0</sup>

*μ*2*<sup>b</sup>* = *μ*<sup>21</sup> = 0.1 D. Rabi frequencies are modeled by Gaussians

which can lead to a slight translational heating of the sample.

*<sup>S</sup>*,*<sup>p</sup>* exp (−(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*<sup>0</sup> <sup>±</sup> *<sup>τ</sup>S*,*p*))/*T*<sup>2</sup>

Ω*S*,*<sup>p</sup>* = Ω<sup>0</sup>

*c*(Ω<sup>0</sup>

*Sh*¯)2/8*πμ*<sup>2</sup>

Eq. (38) in Appendix B):

in *Ip* = *q*2*c*(Ω<sup>0</sup>

<sup>21</sup> and *Ip* <sup>=</sup> *<sup>c</sup>*E<sup>2</sup>

*p*)2*δε*Γ/64√*πμ*<sup>2</sup>

respectively.

broad and narrow resonances.

*<sup>S</sup> IS Ip TS Tp τ<sup>S</sup> τ<sup>p</sup>*

*<sup>S</sup>*,*p*, where ± refers to the Stokes and pump pulse,

*<sup>S</sup>*/8*π* =

<sup>√</sup>*π*Γ, resulting

<sup>2</sup>*�*, where we use Eq.(30) to estimate

. (35)

nance *μ*K *μ*K 108 s−<sup>1</sup> W/cm<sup>2</sup> W/cm<sup>2</sup> *μ*s *μ*s *μ*s *μ*s

Coherent Laser Manipulation of Ultracold Molecules 73

None 10 — 0.72 62 4 <sup>×</sup> 105 1.5 3 0.75 1.0 Broad 10 1000 0.74 65 4000 1.4 3.4 0.65 1.0 Narrow 100 1 2.24 600 400 0.157 0.3 0.1 0.207

resonances, more practical expressions for both the source term *S* and the back stimulation term *T* can be found. The derivation of the final system of equations used in numerical solutions is given in Appendix B. Here, we describe the solutions of these systems for both

We note that during the transfer an initial incoherent mixture of atomic scattering states is converted into a pure internal state, which seems to decrease the entropy of the system. However, the entropy is transferred to the center-of-mass motion of the created molecules,

Using Eqs.(A3)-(A4) and (A7)-(A8) with the parameters listed in Table 2 for a broad (Γ = 1 mK) and a narrow (Γ = 1 *μ*K) Feshbach resonance, we obtain the results for the STIRAP transfer of an atom pair, depicted in Fig. 5. Here the left column corresponds to the broad resonance, and the right column to the narrow resonance. The top row shows the variation of the Rabi frequencies over the time period required for the population transfer along with

population in the intermediate state |2� (middle row) and final state |1� (bottom row). For the broad case, we considered a Feshbach resonance with a width Γ = 1 mK (typical for broad resonances), and a thermal atomic ensemble with an energy bandwidth *δ�* = 10 *μ*K. We see that the transfer efficiency can reach ∼ 97% of the continuum state into the target state |1� (see Fig. 5 c). The parameters of the Gaussian laser pulses used (optimized Rabi frequencies, durations and delays of laser pulses) are given in Table 2: the peak intensities

of the Stokes and pump fields were calculated from Rabi frequencies as *IS* <sup>=</sup> *<sup>c</sup>*E<sup>2</sup>

*p*)2*δ�*/32*π*3/2*μ*<sup>2</sup>

<sup>Γ</sup> (*�*<sup>0</sup> − *�F*)

<sup>Γ</sup><sup>2</sup> (*�*<sup>0</sup> − *�F*)<sup>2</sup>

When comparing the results for a broad resonance to that of the unperturbed continuum (*i.e.*, far from the resonance), we find that the source term *S* is enhanced by the factor *g*(*q*, *�*0) (see

This factor has a maximum at 2(*�*<sup>0</sup> − *�F*)/Γ = 1/*q*, with the corresponding maximum value <sup>1</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup> <sup>≈</sup> *<sup>q</sup>* for *<sup>q</sup>* � 1: hence, the source amplitude is enhanced *<sup>q</sup>* times. In this limit, all populated continuum states experience the same transition dipole matrix element enhancement factor to the state |2�, so that the system essentially reduces to the case of a flat

*<sup>p</sup>*/8*π* = *c*(Ω<sup>0</sup>

2*b*.

the continuum-bound dipole matrix element *<sup>μ</sup>*2*�* <sup>≈</sup> *<sup>μ</sup>*2*b*/*qπV*(*�*) = <sup>√</sup>2*μ*2*b*/*<sup>q</sup>*

*<sup>g</sup>*(*q*, *�*0) = *<sup>q</sup>* <sup>+</sup> <sup>2</sup>

 1 + <sup>4</sup>

Table 2. Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer for a pair of atoms shown in Fig.5. We use *q* = 10, *γ* = 108 s−1, and

bound-bound and decrease the continuum-bound dipole matrix elements, resulting in larger *q*, whereas choosing a highly excited state close to a dissociation threshold will decrease *q*. Using the expressions given in Eqs.(28), (29), and (30) for the initial amplitude of the continuum wave function, the Rabi frequency between the continuum state |Ψ*�*� and the excited bound state |2�, and the Fano parameter, respectively, we obtain the following complete expression for the source term:

$$S = S\_0 \int\_{\varepsilon\_{\rm th}}^{\infty} d\varepsilon \, g(q\_\prime \varepsilon) \, \text{sgn}(\varepsilon - \varepsilon\_F) e^{-\frac{(\varepsilon - \varepsilon\_0)^2}{2\delta\_\varepsilon^2} + \frac{i(\varepsilon - \varepsilon\_0)t\_0}{\hbar}} e^{-i\Delta\_i t} \,\tag{31}$$

with *<sup>S</sup>*<sup>0</sup> <sup>=</sup> �*μ*2*�* · *<sup>e</sup>*ˆ*p*E*p*/¯*h*(*πδ�*)1/4, and where the function *<sup>g</sup>*(*q*, *�*) is defined as

$$\lg(q,\varepsilon) \equiv \frac{q + \frac{2}{\Gamma}(\varepsilon - \varepsilon\_F)}{\sqrt{1 + \frac{4}{\Gamma^2}(\varepsilon - \varepsilon\_F)^2}}.\tag{32}$$

We assume that the unperturbed continuum is structureless and that the corresponding Rabi frequency �*μ*2*�* · *e*ˆ*p*E*p*/¯*h* depends only weakly on the energy. We also extend *�*th to −∞ to have the initial continuum wavefunction normalized to unity1: ∞ <sup>−</sup><sup>∞</sup> *<sup>d</sup>�*|*C*(*�*)<sup>|</sup> <sup>2</sup> = 1.

We can as well obtain a complete expression for the back-stimulation term *T*. We have:

$$T = \left| \frac{\vec{\mu}\_{2\epsilon} \hat{e}\_p}{\hbar} \right|^2 \mathcal{E}\_p(t) \int\_{\varepsilon\_{\rm th}}^{\infty} d\epsilon \,\, g^2(\eta, \epsilon) \int\_0^t dt' \, c\_2(t') \mathcal{E}\_p(t') e^{i\Delta\_i(t'-t)}.\tag{33}$$

Extending the lower integration limit2 allows for an analytical solution for the integrals over energy and time, leading to the following expression for the back-stimulation term:

$$\begin{split} T &= \left| \frac{\vec{\mu}\_{2\varepsilon} \mathfrak{E}\_p}{\hbar} \right|^2 \left[ \pi \hbar \mathcal{E}\_p^2(t) c\_2(t) + \frac{\pi \Gamma}{2} (q - i)^2 \mathcal{E}\_p(t) \\ &\times \int\_0^t dt' \, c\_2(t') \mathcal{E}\_p(t') e^{\left[\Gamma/2\hbar + i(\varepsilon\_\mathbb{F}/\hbar - \omega\_\mathbb{S} + \omega\_p)\right](t' - t)} \right]. \tag{34}$$

#### **3.2 Results of STIRAP transfer for a pair of atoms**

In this subsection, we consider two different cases: first, when Γ � *δ�*, *i.e.*, when the width Γ of the Feshbach resonance is much larger than the thermal energy spread *δ�* of the colliding atoms, and second when Γ � *δ�*. By considering these two limiting cases of broad and narrow

<sup>1</sup> Extension of *�*th to <sup>−</sup><sup>∞</sup> in the source term (31) can be justified by the sharp reduction of the Gaussian term exp(−(*�*th <sup>−</sup> *�*0)2/2*δ*<sup>2</sup> *�* ) for *�*<sup>0</sup> − *�th* > *δ�*. For *�*<sup>0</sup> close to *�*th, this approximation is less accurate, but as will be shown in Section IV, these low energies give negligible contribution to transfer efficiency averaged over an atomic ensemble due to their small weight in the Maxwell-Boltzmann distribution.

<sup>2</sup> The extension of the lower integration limit to <sup>−</sup><sup>∞</sup> in the back-stimulation term (33) can be explained by the following argument. For an optimal transfer the duration of laser pulses has to be of the order of the coherence time of the populated continuum, given by 1/*δ�*. The period of the exponent exp(*i*Δ*�*(*t* � − *t*)) on the other hand is given by 2*π*/Δ*�*. The integral over time therefore quickly goes to zero if this period is smaller then the duration of pulses. As a result, the time integral is non-zero only for energies |Δ*�*| = |*�*/¯*h* − (*ω<sup>S</sup>* − *ωp*)| < *δ�*. In the case of the pump laser resonant with the center of the thermal distribution |*�*th − (*ω<sup>S</sup>* − *ωp*)| ∼ *δ�*, and the extension of the energy integration to −∞ is well-justified.

20 Will-be-set-by-IN-TECH

bound-bound and decrease the continuum-bound dipole matrix elements, resulting in larger *q*, whereas choosing a highly excited state close to a dissociation threshold will decrease *q*. Using the expressions given in Eqs.(28), (29), and (30) for the initial amplitude of the continuum wave function, the Rabi frequency between the continuum state |Ψ*�*� and the excited bound state |2�, and the Fano parameter, respectively, we obtain the following

> <sup>−</sup> (*�*−*�*0)<sup>2</sup> 2*δ*2 *�*

<sup>Γ</sup> (*�* − *�F*)

 *t* 0

<sup>Γ</sup><sup>2</sup> (*�* − *�F*)<sup>2</sup>

∞

<sup>2</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>i</sup>*)2E*p*(*t*)

*�* ) for *�*<sup>0</sup> − *�th* > *δ�*. For *�*<sup>0</sup> close to *�*th, this approximation is less accurate,

[Γ/2¯*h*+*i*(*�F*/¯*h*−*ωS*+*ωp*)](*t*�

*dt*� *c*2(*t* � )E*p*(*t* � )*e i*Δ*�*(*t*� −*t*)

<sup>−</sup><sup>∞</sup> *<sup>d</sup>�*|*C*(*�*)<sup>|</sup>

−*t*) 

<sup>+</sup> *<sup>i</sup>*(*�*−*�*0)*t*<sup>0</sup> *<sup>h</sup>*¯ *<sup>e</sup>*−*i*Δ*�<sup>t</sup>*

, (31)

. (33)

. (34)

� − *t*))

. (32)

<sup>2</sup> = 1.

*d� g*(*q*, *�*) sgn(*�* − *�F*)*e*

*<sup>g</sup>*(*q*, *�*) <sup>≡</sup> *<sup>q</sup>* <sup>+</sup> <sup>2</sup>

 1 + <sup>4</sup>

We can as well obtain a complete expression for the back-stimulation term *T*. We have:

*d� g*2(*q*, *�*)

energy and time, leading to the following expression for the back-stimulation term:

Extending the lower integration limit2 allows for an analytical solution for the integrals over

*<sup>p</sup>* (*t*)*c*2(*t*) + *<sup>π</sup>*<sup>Γ</sup>

In this subsection, we consider two different cases: first, when Γ � *δ�*, *i.e.*, when the width Γ of the Feshbach resonance is much larger than the thermal energy spread *δ�* of the colliding atoms, and second when Γ � *δ�*. By considering these two limiting cases of broad and narrow

<sup>1</sup> Extension of *�*th to <sup>−</sup><sup>∞</sup> in the source term (31) can be justified by the sharp reduction of the Gaussian

but as will be shown in Section IV, these low energies give negligible contribution to transfer efficiency averaged over an atomic ensemble due to their small weight in the Maxwell-Boltzmann distribution. <sup>2</sup> The extension of the lower integration limit to <sup>−</sup><sup>∞</sup> in the back-stimulation term (33) can be explained by the following argument. For an optimal transfer the duration of laser pulses has to be of the order of the coherence time of the populated continuum, given by 1/*δ�*. The period of the exponent exp(*i*Δ*�*(*t*

on the other hand is given by 2*π*/Δ*�*. The integral over time therefore quickly goes to zero if this period is smaller then the duration of pulses. As a result, the time integral is non-zero only for energies |Δ*�*| = |*�*/¯*h* − (*ω<sup>S</sup>* − *ωp*)| < *δ�*. In the case of the pump laser resonant with the center of the thermal distribution |*�*th − (*ω<sup>S</sup>* − *ωp*)| ∼ *δ�*, and the extension of the energy integration to −∞ is well-justified.

We assume that the unperturbed continuum is structureless and that the corresponding Rabi frequency �*μ*2*�* · *e*ˆ*p*E*p*/¯*h* depends only weakly on the energy. We also extend *�*th to −∞ to have

with *<sup>S</sup>*<sup>0</sup> <sup>=</sup> �*μ*2*�* · *<sup>e</sup>*ˆ*p*E*p*/¯*h*(*πδ�*)1/4, and where the function *<sup>g</sup>*(*q*, *�*) is defined as

 ∞ *�th*

complete expression for the source term:

*T* = 

term exp(−(*�*th <sup>−</sup> *�*0)2/2*δ*<sup>2</sup>

*S* = *S*<sup>0</sup>

 ∞ *�*th

the initial continuum wavefunction normalized to unity1:

  2 E*p*(*t*)

�*μ*2*�e*ˆ*<sup>p</sup> h*¯

× *t* 0

**3.2 Results of STIRAP transfer for a pair of atoms**

 2 *<sup>π</sup>h*¯ <sup>E</sup><sup>2</sup>

*dt*� *c*2(*t* � )E*p*(*t* � )*e*

�*μ*2*�e*ˆ*<sup>p</sup> h*¯

*T* = 


Table 2. Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer for a pair of atoms shown in Fig.5. We use *q* = 10, *γ* = 108 s−1, and *μ*2*<sup>b</sup>* = *μ*<sup>21</sup> = 0.1 D. Rabi frequencies are modeled by Gaussians

Ω*S*,*<sup>p</sup>* = Ω<sup>0</sup> *<sup>S</sup>*,*<sup>p</sup>* exp (−(*<sup>t</sup>* <sup>−</sup> *<sup>t</sup>*<sup>0</sup> <sup>±</sup> *<sup>τ</sup>S*,*p*))/*T*<sup>2</sup> *<sup>S</sup>*,*p*, where ± refers to the Stokes and pump pulse, respectively.

resonances, more practical expressions for both the source term *S* and the back stimulation term *T* can be found. The derivation of the final system of equations used in numerical solutions is given in Appendix B. Here, we describe the solutions of these systems for both broad and narrow resonances.

We note that during the transfer an initial incoherent mixture of atomic scattering states is converted into a pure internal state, which seems to decrease the entropy of the system. However, the entropy is transferred to the center-of-mass motion of the created molecules, which can lead to a slight translational heating of the sample.

Using Eqs.(A3)-(A4) and (A7)-(A8) with the parameters listed in Table 2 for a broad (Γ = 1 mK) and a narrow (Γ = 1 *μ*K) Feshbach resonance, we obtain the results for the STIRAP transfer of an atom pair, depicted in Fig. 5. Here the left column corresponds to the broad resonance, and the right column to the narrow resonance. The top row shows the variation of the Rabi frequencies over the time period required for the population transfer along with population in the intermediate state |2� (middle row) and final state |1� (bottom row).

For the broad case, we considered a Feshbach resonance with a width Γ = 1 mK (typical for broad resonances), and a thermal atomic ensemble with an energy bandwidth *δ�* = 10 *μ*K. We see that the transfer efficiency can reach ∼ 97% of the continuum state into the target state |1� (see Fig. 5 c). The parameters of the Gaussian laser pulses used (optimized Rabi frequencies, durations and delays of laser pulses) are given in Table 2: the peak intensities of the Stokes and pump fields were calculated from Rabi frequencies as *IS* <sup>=</sup> *<sup>c</sup>*E<sup>2</sup> *<sup>S</sup>*/8*π* = *c*(Ω<sup>0</sup> *Sh*¯)2/8*πμ*<sup>2</sup> <sup>21</sup> and *Ip* <sup>=</sup> *<sup>c</sup>*E<sup>2</sup> *<sup>p</sup>*/8*π* = *c*(Ω<sup>0</sup> *p*)2*δ�*/32*π*3/2*μ*<sup>2</sup> <sup>2</sup>*�*, where we use Eq.(30) to estimate the continuum-bound dipole matrix element *<sup>μ</sup>*2*�* <sup>≈</sup> *<sup>μ</sup>*2*b*/*qπV*(*�*) = <sup>√</sup>2*μ*2*b*/*<sup>q</sup>* <sup>√</sup>*π*Γ, resulting in *Ip* = *q*2*c*(Ω<sup>0</sup> *p*)2*δε*Γ/64√*πμ*<sup>2</sup> 2*b*.

When comparing the results for a broad resonance to that of the unperturbed continuum (*i.e.*, far from the resonance), we find that the source term *S* is enhanced by the factor *g*(*q*, *�*0) (see Eq. (38) in Appendix B):

$$\log(q\_\prime \varepsilon\_0) = \frac{q + \frac{2}{\Gamma}(\varepsilon\_0 - \varepsilon\_F)}{\sqrt{1 + \frac{4}{\Gamma^2}(\varepsilon\_0 - \varepsilon\_F)^2}} \,. \tag{35}$$

This factor has a maximum at 2(*�*<sup>0</sup> − *�F*)/Γ = 1/*q*, with the corresponding maximum value <sup>1</sup> <sup>+</sup> *<sup>q</sup>*<sup>2</sup> <sup>≈</sup> *<sup>q</sup>* for *<sup>q</sup>* � 1: hence, the source amplitude is enhanced *<sup>q</sup>* times. In this limit, all populated continuum states experience the same transition dipole matrix element enhancement factor to the state |2�, so that the system essentially reduces to the case of a flat

**|1**

states |±�. We have the Rabi frequency <sup>Ω</sup>2*<sup>c</sup>* <sup>=</sup> <sup>Ω</sup><sup>2</sup>

and Δ<sup>+</sup> = −Δ−.

*μ*2*<sup>c</sup>* ∼ *μ*2*b*/*q*


gives ∼ 50% transfer efficiency.

reduced by a factor of <sup>∼</sup> 1/*q*2.

**|2**

**E Ep <sup>S</sup>**

**(a)**

**|b |c**

**|1**

Fig. 6. Illustration of the reduction of STIRAP transfer efficiency due to destructive quantum interference for a narrow resonance: (a) a simplified level scheme where the scattering continuum is modeled by a single state |*c*� and the interaction between the continuum and the Feshbach state |*b*� is neglected; (b) an equivalent scheme, where the strong coupling between the Feshbach state |*b*� and the excited state |*c*� by the pump field forms "dressed"

prediction that in the presence of a wide resonance the required pump laser intensity is

Results of adiabatic passage for a pair of atoms in a narrow resonance limit are shown in Fig. 5 (right column). We considered typical values for a narrow resonance width Γ = 1 *μ*K and the ensemble energy bandwidth *δ�* = 100 *μ*K. Again, we give the parameters providing the optimal transfer in Table 2. In this limit, the transfer efficiency is lower: in the specific case analyzed here, it does not exceed 47%. The reason for this lower efficiency compared to a wide resonance is the destructive quantum interference which leads to electromagnetically induced transparency (61) in the transition from the continuum to the excited state. It can be explained using the following argument (see Fig. 6). The limit of a narrow Feshbach resonance corresponds to a weak coupling between the bound Feshbach state and the scattering continuum, and thus can be neglected in this simplified explanation. The system then can be viewed as consisting of bound and continuum states |*b*� and |*c*� having the same energy, which are coupled by the pump field to a molecular state |2�, itself coupled to the state |1� by the Stokes field. Assuming that initially all the population is in the state |*c*�, due to the small interaction strength between |*b*� and |*c*�, we can eliminate the state |*b*�, taking into account its coupling to |2� by the pump laser as the formation of "dressed" states |±� = (|2� ± |*b*�)/

the dipole matrix element of the |*b*� → |2� transition is much larger than that of the |*c*� → |2�

the one-photon coupling of |*c*� to the excited state, as well as two-photon coupling to |1� vanishes, preventing the adiabatic transfer. This mechanism is similar to the Fano interference effect, the difference is that the continuum is initially populated. One can therefore view it as an inverse Fano effect. The effective dipole matrix element of the |*c*� → |2� transition is

The transfer efficiency increases if the Feshbach state is far detuned from the populated continuum. Our calculations show that for a Feshbach state detuning *�F*/¯*h* − (*ω<sup>S</sup>* − *ωp*) �

2/*γ*, the transfer efficiency reaches 70% using the laser pulse parameters in Table 2. We note that the smaller intensity of the pump pulse used for the narrow resonance, as compared to the broad resonance, is due to the fact that we used the same *q* = 10 and assumed *μ*2*<sup>b</sup>* = 0.1

√

transition, the detuning of the "dressed" states satisfies <sup>|</sup>Δ±| <sup>=</sup> <sup>Ω</sup>2*<sup>b</sup>*

<sup>√</sup>*ξ*. In the case we analyzed, *<sup>q</sup>* <sup>=</sup> 10, *<sup>ξ</sup>* <sup>=</sup> <sup>Γ</sup>/

**+ -**

Coherent Laser Manipulation of Ultracold Molecules 75

**E Ep <sup>S</sup>**

**(b)**

<sup>+</sup>*c*/Δ<sup>+</sup> + <sup>Ω</sup><sup>2</sup>

**|c**

(| <sup>2</sup> <sup>|</sup> ) <sup>2</sup> <sup>1</sup> <sup>|</sup> *<sup>b</sup>*

(| <sup>2</sup> <sup>|</sup> ) <sup>2</sup> <sup>1</sup> <sup>|</sup> *<sup>b</sup>*

<sup>−</sup>*c*/Δ<sup>−</sup> <sup>=</sup> 0, since <sup>Ω</sup>+*<sup>c</sup>* <sup>=</sup> <sup>Ω</sup>−*<sup>c</sup>*

*<sup>p</sup>* � <sup>Ω</sup>2*<sup>c</sup>*

2*δ�* = 0.01, and *μ*2*<sup>c</sup>* ≈ *μ*2*b*, which

<sup>√</sup>2. If

*<sup>p</sup>* , Ω*S*. As a result,

Fig. 5. Time-dependence of the Stokes and pump pulses (top row) and population in state |2� (middle row) and target state |1� (bottom row) for the STIRAP transfer of a pair of atoms within the center of the thermal distribution. The left column is for a broad Feshbach resonance, while the right column is for a narrow resonance (see Table 2 for values of parameters used). The dashed blue lines in the left column are the results obtained without resonance, when the parameters are adjusted to obtain the same overall transfer efficiency as for the broad resonance. The Stokes Rabi frequency is in units of 106 s−1, while the pump Rabi frequency is in dimensionless units (16*π*/*δ�*) 1/4 *<sup>μ</sup>*2*�e*ˆ*p*E*<sup>p</sup>* in the broad resonance limit and (2*π*/Γ) 1/2 *<sup>μ</sup>*2*�e*ˆ*p*E*<sup>p</sup>* in the narrow resonance limit. Note that the scale for the Rabi frequencies in the narrow resonance case is 40 times the scale for the broad resonance, and the magnitude of the pump Rabi frequency is enlarged 10 times for better visibility.

continuum with an uniformly enhanced transition dipole matrix element. One thus expects that in this limit, the adiabatic passage should be efficient, requiring less pump laser intensity when compared to the unperturbed (*i.e.* without resonance) scattering continuum. This is clearly demonstrated in Fig. 5 (left column, dashed lines): to reach the same ∼ 97% transfer efficiency achieved with the broad resonance, a very large pump laser intensity (∼ 100 times larger) is required if there is no resonance in the continuum (Fig. 5 a), while the Stokes laser intensity is basically the same. Considering the intensity used in this particular example, this would lead to intensities in the range of 5 <sup>×</sup> 105 W/cm2, making STIRAP from the continuum technically impossible to achieve without a resonance. This is consistent with the analysis of photoassociative adiabatic passage from an unstructured continuum (27), and the above 22 Will-be-set-by-IN-TECH

(a)

Ωp no-res

(b)

(c)

Fig. 5. Time-dependence of the Stokes and pump pulses (top row) and population in state |2� (middle row) and target state |1� (bottom row) for the STIRAP transfer of a pair of atoms within the center of the thermal distribution. The left column is for a broad Feshbach resonance, while the right column is for a narrow resonance (see Table 2 for values of parameters used). The dashed blue lines in the left column are the results obtained without resonance, when the parameters are adjusted to obtain the same overall transfer efficiency as for the broad resonance. The Stokes Rabi frequency is in units of 106 s−1, while the pump

1/2 *<sup>μ</sup>*2*�e*ˆ*p*E*<sup>p</sup>* in the narrow resonance limit. Note that the scale for the Rabi frequencies in the narrow resonance case is 40 times the scale for the broad resonance, and the magnitude of the pump Rabi frequency is enlarged 10 times for better visibility.

continuum with an uniformly enhanced transition dipole matrix element. One thus expects that in this limit, the adiabatic passage should be efficient, requiring less pump laser intensity when compared to the unperturbed (*i.e.* without resonance) scattering continuum. This is clearly demonstrated in Fig. 5 (left column, dashed lines): to reach the same ∼ 97% transfer efficiency achieved with the broad resonance, a very large pump laser intensity (∼ 100 times larger) is required if there is no resonance in the continuum (Fig. 5 a), while the Stokes laser intensity is basically the same. Considering the intensity used in this particular example, this would lead to intensities in the range of 5 <sup>×</sup> 105 W/cm2, making STIRAP from the continuum technically impossible to achieve without a resonance. This is consistent with the analysis of photoassociative adiabatic passage from an unstructured continuum (27), and the above

Broad resonance Narrow resonance


1/4 *<sup>μ</sup>*2*�e*ˆ*p*E*<sup>p</sup>* in the broad resonance limit

(d)

(e)

(f)

0.00000

0

0.5


and (2*π*/Γ)


2

1


Rabi frequency is in dimensionless units (16*π*/*δ�*)

Ωp broad-res

no-res

Ωs Ω broad-res <sup>s</sup>

0.00025



2

0.00050

Rabi frequency

Fig. 6. Illustration of the reduction of STIRAP transfer efficiency due to destructive quantum interference for a narrow resonance: (a) a simplified level scheme where the scattering continuum is modeled by a single state |*c*� and the interaction between the continuum and the Feshbach state |*b*� is neglected; (b) an equivalent scheme, where the strong coupling between the Feshbach state |*b*� and the excited state |*c*� by the pump field forms "dressed" states |±�. We have the Rabi frequency <sup>Ω</sup>2*<sup>c</sup>* <sup>=</sup> <sup>Ω</sup><sup>2</sup> <sup>+</sup>*c*/Δ<sup>+</sup> + <sup>Ω</sup><sup>2</sup> <sup>−</sup>*c*/Δ<sup>−</sup> <sup>=</sup> 0, since <sup>Ω</sup>+*<sup>c</sup>* <sup>=</sup> <sup>Ω</sup>−*<sup>c</sup>* and Δ<sup>+</sup> = −Δ−.

prediction that in the presence of a wide resonance the required pump laser intensity is reduced by a factor of <sup>∼</sup> 1/*q*2.

Results of adiabatic passage for a pair of atoms in a narrow resonance limit are shown in Fig. 5 (right column). We considered typical values for a narrow resonance width Γ = 1 *μ*K and the ensemble energy bandwidth *δ�* = 100 *μ*K. Again, we give the parameters providing the optimal transfer in Table 2. In this limit, the transfer efficiency is lower: in the specific case analyzed here, it does not exceed 47%. The reason for this lower efficiency compared to a wide resonance is the destructive quantum interference which leads to electromagnetically induced transparency (61) in the transition from the continuum to the excited state. It can be explained using the following argument (see Fig. 6). The limit of a narrow Feshbach resonance corresponds to a weak coupling between the bound Feshbach state and the scattering continuum, and thus can be neglected in this simplified explanation. The system then can be viewed as consisting of bound and continuum states |*b*� and |*c*� having the same energy, which are coupled by the pump field to a molecular state |2�, itself coupled to the state |1� by the Stokes field. Assuming that initially all the population is in the state |*c*�, due to the small interaction strength between |*b*� and |*c*�, we can eliminate the state |*b*�, taking into account its coupling to |2� by the pump laser as the formation of "dressed" states |±� = (|2� ± |*b*�)/ <sup>√</sup>2. If the dipole matrix element of the |*b*� → |2� transition is much larger than that of the |*c*� → |2� transition, the detuning of the "dressed" states satisfies <sup>|</sup>Δ±| <sup>=</sup> <sup>Ω</sup>2*<sup>b</sup> <sup>p</sup>* � <sup>Ω</sup>2*<sup>c</sup> <sup>p</sup>* , Ω*S*. As a result, the one-photon coupling of |*c*� to the excited state, as well as two-photon coupling to |1� vanishes, preventing the adiabatic transfer. This mechanism is similar to the Fano interference effect, the difference is that the continuum is initially populated. One can therefore view it as an inverse Fano effect. The effective dipole matrix element of the |*c*� → |2� transition is *μ*2*<sup>c</sup>* ∼ *μ*2*b*/*q* <sup>√</sup>*ξ*. In the case we analyzed, *<sup>q</sup>* <sup>=</sup> 10, *<sup>ξ</sup>* <sup>=</sup> <sup>Γ</sup>/ √ 2*δ�* = 0.01, and *μ*2*<sup>c</sup>* ≈ *μ*2*b*, which gives ∼ 50% transfer efficiency.

The transfer efficiency increases if the Feshbach state is far detuned from the populated continuum. Our calculations show that for a Feshbach state detuning *�F*/¯*h* − (*ω<sup>S</sup>* − *ωp*) � |Ω2*b*| 2/*γ*, the transfer efficiency reaches 70% using the laser pulse parameters in Table 2. We note that the smaller intensity of the pump pulse used for the narrow resonance, as compared to the broad resonance, is due to the fact that we used the same *q* = 10 and assumed *μ*2*<sup>b</sup>* = 0.1

0

0.00000



2

Table 3.

*P*avg*ρ*

0 0.2 0.4 0.6 0.8 1

by the two pulses is *f*(*�*) = *P*(*�*)*N*(*�*), or


*f*(*�*) =

0.00020

0.00040



2

0.00060

25

50

Rabi frequency

75

Ωp no-res

Ωp broad-res

100

Broad resonance, averaged Narrow resonance, averaged

(a)

Ωp no-res

Coherent Laser Manipulation of Ultracold Molecules 77

(b)

Ωp broad-res

(c)

Fig. 7. Same as Fig. 5, but for the energy averaged transfer. The parameters are listed in

<sup>√</sup>2*πh*¯ <sup>2</sup>

The total fraction of atoms photoassociated by a pair of pulses is *f* = <sup>∞</sup>

of pulses to convert an entire atomic ensemble into deeply bound molecular levels.

As was shown in (62), in the limit of a narrow resonance longer pulses with durations *TS*, *Tp* ∼ 1/Γ can be used. The reason is that population gets "trapped" in the bound state |*b*� for a time ∼ 1/Γ (it can be seen from the expression for the narrow resonance source function A.6), and as a result coherent transfer is still possible. The fraction of atoms associated per pulse pair in this case is comparable to the case of a wide resonance, since the smaller transfer efficiency *P*(*�*) is compensated by a larger pulse overlap *τ*. The long pulse duration results in

<sup>√</sup>2*πτh*¯ 2/4*μ*3/2√*kBT*, where we assumed that *<sup>P</sup>*(*�*) does not significantly vary within the ensemble, and approximated it by the averaged value *P*avg. Considering as an example 6Li atoms at *<sup>T</sup>* <sup>=</sup> <sup>100</sup> *<sup>μ</sup>*K with an atomic density *<sup>ρ</sup>* <sup>=</sup> 1012 cm−3, an overlap time *<sup>τ</sup>* <sup>∼</sup> <sup>1</sup> *<sup>μ</sup>*s, and assuming *P*avg = 0.7, the fraction of atoms photoassociated by the Stokes and pump pulses is *<sup>f</sup>* <sup>∼</sup> 2.5 <sup>×</sup> <sup>10</sup>−4: for heavier atoms *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>6</sup> <sup>−</sup> <sup>10</sup>−5. It will therefore require <sup>∼</sup> 104 <sup>−</sup> 106 pairs


<sup>4</sup>(*μkBT*)3/2 *τρP*(*�*) exp (−*�*/*kBT*). (37)

(d)

(e)

(f)

<sup>0</sup> *d� f*(*�*) ≈


Table 3. Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer shown in Fig.7 for averaging over a Maxwell-Boltzmann distribution of energies. We use *q* = 10, *γ* = 108 s−1, and *μ*2*<sup>b</sup>* = *μ*<sup>21</sup> = 0.1 D (1 D=10−<sup>18</sup> esu cm = 0.3934 *ea*0).

D for both resonances. From the definition of *q*, it means that the continuum-bound dipole matrix element *μ*2*<sup>ε</sup>* is higher in the narrow than in the broad resonance we considered. This explains the smaller resulting pump pulse intensity. The overall conclusion for a narrow resonance is that, as opposed to a broad resonance, the presence of the Feshbach resonance prevents one from realizing high transfer efficiencies. It should be noted, however, that the destructive quantum interference effect is based on negligible interaction between the Feshbach and continuum states during the transfer time, since *<sup>T</sup>* <sup>&</sup>lt; *<sup>δ</sup>*−<sup>1</sup> *<sup>ε</sup>* � <sup>Γ</sup>−1. This argument shows that already for Γ ≥ *δ�*, there is enough interaction to neutralize the effect of destructive interference. Therefore, we expect that the broad resonance limit can be extended down to Γ ∼ *δ�*, making it applicable to a wide variety of atomic species.

#### **3.3 Conversion of atomic ensembles into ground state molecules**

The results of Fig. 5 were obtained for a pair of atoms having a specific mean collision energy *�*<sup>0</sup> = *h*¯(*ω<sup>S</sup>* − *ωp*). Such a situation could be realized in very tight traps, *e.g.*, in tight optical lattices. For a system with a wider energy distribution, one would like to find an ensemble averaged transfer efficiency, and thus one needs to calculate the transfer probability *P*(*�*0) = |*c*1| <sup>2</sup> for all central wavepacket energies *�*<sup>0</sup> within the thermal spread of energies, and perform the averaging as

$$P\_{\rm avg} = \frac{2}{\sqrt{\pi} (k\_B T)^{3/2}} \int\_0^\infty e^{-\epsilon\_0/k\_B T} \sqrt{\epsilon\_0} P(\epsilon\_0) d\epsilon\_0 \tag{36}$$

where we assume a Maxwell-Boltzmann energy distribution, the pump laser resonant with the center of the distribution at �*�*� = 3/2*kBT*, and set the bandwidth of the distribution at *δ�* <sup>=</sup> �(Δ*�*)2� <sup>=</sup> <sup>√</sup>3/2*kBT*. The results are shown in Fig.7: while the maximum transfer efficiency in the broad resonance case is ∼ 70%, it can be achieved with lower laser intensities than in the case of a pair of atoms of Fig.5.

Given the adiabatic photoassociation probability *P*(*�*) for two colliding atoms with relative energy *�*, we can calculate the number of atoms photoassociated during the time overlap *τ* of the Stokes and pump pulses. During this time, the atom with the energy *�* = *μv*2/2, where *μ* is the reduced mass, will collide with atoms in the volume *πb*2*vτ*, where *πb*<sup>2</sup> is the collision cross-section. The impact parameter for the collision corresponding to a partial wave with angular momentum � is *b* = (� + 1/2)*h*¯ /*p* = (� + 1/2)*h*¯ / 2*μ�*. The number of collisions that atoms with a relative energy in the interval (*�*, *�* + *d�*) will experience during the transfer time is therefore *<sup>N</sup>*(*�*)*d�* <sup>=</sup> *<sup>π</sup>b*2*vτρ*(*�*)*d�*, where *<sup>ρ</sup>*(*�*) = <sup>2</sup>*<sup>ρ</sup>* exp (−*�*/*kBT*) <sup>√</sup>*�*/ <sup>√</sup>*π*(*kBT*)3/2 is the spectral density of the atoms (*ρ* is the density of the sample). Finally, � = 0 for ultracold *s*-wave collisions, and the fraction of atoms in the energy interval (*�*, *�* + *d�*) photoassociated 24 Will-be-set-by-IN-TECH

nance *μ*K *μ*K 108 s−<sup>1</sup> W/cm<sup>2</sup> W/cm<sup>2</sup> *μ*s *μ*s *μ*s *μ*s

None 10 — 0.50 30 1.7 <sup>×</sup> 105 1.5 3.3 0.75 1.3 Broad 10 1000 0.60 40 2500 1.3 3.2 0.7 1.25 Narrow 100 1 2.24 600 400 0.157 0.3 0.1 0.207

D for both resonances. From the definition of *q*, it means that the continuum-bound dipole matrix element *μ*2*<sup>ε</sup>* is higher in the narrow than in the broad resonance we considered. This explains the smaller resulting pump pulse intensity. The overall conclusion for a narrow resonance is that, as opposed to a broad resonance, the presence of the Feshbach resonance prevents one from realizing high transfer efficiencies. It should be noted, however, that the destructive quantum interference effect is based on negligible interaction between the Feshbach and continuum states during the transfer time, since *<sup>T</sup>* <sup>&</sup>lt; *<sup>δ</sup>*−<sup>1</sup> *<sup>ε</sup>* � <sup>Γ</sup>−1. This argument shows that already for Γ ≥ *δ�*, there is enough interaction to neutralize the effect of destructive interference. Therefore, we expect that the broad resonance limit can be extended down to

The results of Fig. 5 were obtained for a pair of atoms having a specific mean collision energy *�*<sup>0</sup> = *h*¯(*ω<sup>S</sup>* − *ωp*). Such a situation could be realized in very tight traps, *e.g.*, in tight optical lattices. For a system with a wider energy distribution, one would like to find an ensemble averaged transfer efficiency, and thus one needs to calculate the transfer probability *P*(*�*0) =

<sup>2</sup> for all central wavepacket energies *�*<sup>0</sup> within the thermal spread of energies, and perform

<sup>−</sup>*�*0/*kBT*√*�*0*P*(*�*0)*d�*0, (36)

2*μ�*. The number of collisions

<sup>√</sup>*�*/

<sup>√</sup>*π*(*kBT*)3/2 is

 ∞ 0 *e*

where we assume a Maxwell-Boltzmann energy distribution, the pump laser resonant with the center of the distribution at �*�*� = 3/2*kBT*, and set the bandwidth of the distribution at *δ�* <sup>=</sup> �(Δ*�*)2� <sup>=</sup> <sup>√</sup>3/2*kBT*. The results are shown in Fig.7: while the maximum transfer efficiency in the broad resonance case is ∼ 70%, it can be achieved with lower laser intensities

Given the adiabatic photoassociation probability *P*(*�*) for two colliding atoms with relative energy *�*, we can calculate the number of atoms photoassociated during the time overlap *τ* of the Stokes and pump pulses. During this time, the atom with the energy *�* = *μv*2/2, where *μ* is the reduced mass, will collide with atoms in the volume *πb*2*vτ*, where *πb*<sup>2</sup> is the collision cross-section. The impact parameter for the collision corresponding to a partial wave with

that atoms with a relative energy in the interval (*�*, *�* + *d�*) will experience during the transfer

the spectral density of the atoms (*ρ* is the density of the sample). Finally, � = 0 for ultracold *s*-wave collisions, and the fraction of atoms in the energy interval (*�*, *�* + *d�*) photoassociated

Table 3. Parameters of the Stokes and pump photoassociating pulses providing optimal population transfer shown in Fig.7 for averaging over a Maxwell-Boltzmann distribution of energies. We use *q* = 10, *γ* = 108 s−1, and *μ*2*<sup>b</sup>* = *μ*<sup>21</sup> = 0.1 D (1 D=10−<sup>18</sup> esu cm = 0.3934 *ea*0).

*<sup>S</sup> IS Ip TS Tp τ<sup>S</sup> τ<sup>p</sup>*

Reso- *δ�* Γ Ω<sup>0</sup>

Γ ∼ *δ�*, making it applicable to a wide variety of atomic species.

*<sup>P</sup>*avg <sup>=</sup> <sup>2</sup>

angular momentum � is *b* = (� + 1/2)*h*¯ /*p* = (� + 1/2)*h*¯ /

than in the case of a pair of atoms of Fig.5.

<sup>√</sup>*π*(*kBT*)3/2

time is therefore *<sup>N</sup>*(*�*)*d�* <sup>=</sup> *<sup>π</sup>b*2*vτρ*(*�*)*d�*, where *<sup>ρ</sup>*(*�*) = <sup>2</sup>*<sup>ρ</sup>* exp (−*�*/*kBT*)


the averaging as

**3.3 Conversion of atomic ensembles into ground state molecules**

Fig. 7. Same as Fig. 5, but for the energy averaged transfer. The parameters are listed in Table 3.

by the two pulses is *f*(*�*) = *P*(*�*)*N*(*�*), or

$$f(\epsilon) = \frac{\sqrt{2\pi}\hbar^2}{4(\mu k\_B T)^{3/2}} \text{tr}\rho P(\epsilon) \exp\left(-\epsilon/k\_B T\right). \tag{37}$$

The total fraction of atoms photoassociated by a pair of pulses is *f* = <sup>∞</sup> <sup>0</sup> *d� f*(*�*) ≈ *P*avg*ρ* <sup>√</sup>2*πτh*¯ 2/4*μ*3/2√*kBT*, where we assumed that *<sup>P</sup>*(*�*) does not significantly vary within the ensemble, and approximated it by the averaged value *P*avg. Considering as an example 6Li atoms at *<sup>T</sup>* <sup>=</sup> <sup>100</sup> *<sup>μ</sup>*K with an atomic density *<sup>ρ</sup>* <sup>=</sup> 1012 cm−3, an overlap time *<sup>τ</sup>* <sup>∼</sup> <sup>1</sup> *<sup>μ</sup>*s, and assuming *P*avg = 0.7, the fraction of atoms photoassociated by the Stokes and pump pulses is *<sup>f</sup>* <sup>∼</sup> 2.5 <sup>×</sup> <sup>10</sup>−4: for heavier atoms *<sup>f</sup>* <sup>∼</sup> <sup>10</sup>−<sup>6</sup> <sup>−</sup> <sup>10</sup>−5. It will therefore require <sup>∼</sup> 104 <sup>−</sup> 106 pairs of pulses to convert an entire atomic ensemble into deeply bound molecular levels.

As was shown in (62), in the limit of a narrow resonance longer pulses with durations *TS*, *Tp* ∼ 1/Γ can be used. The reason is that population gets "trapped" in the bound state |*b*� for a time ∼ 1/Γ (it can be seen from the expression for the narrow resonance source function A.6), and as a result coherent transfer is still possible. The fraction of atoms associated per pulse pair in this case is comparable to the case of a wide resonance, since the smaller transfer efficiency *P*(*�*) is compensated by a larger pulse overlap *τ*. The long pulse duration results in

atomic gas can be greatly facilitated by a Feshbach resonance. The presence of a bound state imbedded in and resonant with scattering continuum states strongly enhances the continuum-bound transition dipole matrix element to an excited electronic state, thus requiring less laser intensity for efficient transfer. In the limit of a wide resonance, when compared to the thermal spread of collision energies, the dipole matrix element is enhanced by the Fano parameter *q*. By choosing a tightly bound excited vibrational state, *q* can be made much larger than unity, resulting in the intensity of the pump pulse required for efficient adiabatic passage to be <sup>∼</sup> 1/*q*<sup>2</sup> times smaller than in the absence of the resonance. Numeical modeling of the adiabatic passage using typical parameters of alkali dimers shows that intensities of the pump pulse, coupling the continuum to an excited state, of kW/cm<sup>2</sup> are sufficient for optimal transfer, which is ∼ 100 times smaller than without resonance. Optimal pulse durations are several *μ*s, resulting in energies per pulse ∼ 10 *μ*J for a focus area of 1

Coherent Laser Manipulation of Ultracold Molecules 79

If the Feshbach resonance is narrow compared to the thermal energy spread of colliding atoms, adiabatic passage is hindered by destructive quantum interference. The reason is that electromagnetically induced transparency significantly reduces the transition dipole matrix element from the scattering continuum to an excited state in the presence of the bound Feshbach state. In the narrow resonance limit, photoassociative adiabatic passage is therefore

Due to low atomic collision rates at ultracold temperatures, only a small fraction of atoms can be converted into molecules by a pair of photoassociative pulses. To convert an entire atomic ensemble, a train of pulse pairs can be applied. We estimate that 104 <sup>−</sup> 106 pulse pairs will associate an atomic gas of alkali dimers with a density 1012 cm−<sup>3</sup> in an illuminated volume of 10−<sup>2</sup> <sup>−</sup> <sup>10</sup>−<sup>3</sup> mm<sup>3</sup> in 0.1 <sup>−</sup> 10 s, resulting in extremely high production rates of 105 <sup>−</sup> 108 molecules/s. High transfer efficiencies combined with low intensities of adiabatic photoassociative pulses also make the broad resonance limit attractive for quantum computation. For example, a scheme proposed in (63) can be realized, where qubit states are encoded into a scattering and a bound molecular states of polar molecules. To perform one and two-qubit operations, this scheme requires a high degree of control over the system,

Finally, marrying FOPA and STIRAP is a very promising avenue to produce large amounts of molecules, for a variety of molecular species. In fact, although we described here examples based on magnetically induced Feshbach resonances, such resonances are extremely common, and can be induced by several interactions, such as external electric fields or optical fields. Even in the absence of hyperfine interactions, other interactions can provide the necessary coupling, such as in the case of the magnetic dipole-dipole interaction in 52Cr (64; 65).

Precise control over internal and external degrees of freedom of molecules will open the way for new fundamental studies and applications in physics and chemistry. As has been clearly seen with atoms in the recent decades, well-controlled laser fields offer an exquisite control tool over atomic internal and external states, including laser cooling and trapping, coherent manipulation of atomic quantum states and in particular various techniques used for quantum information applications, atomic spectroscopy. Recent years have witnessed mastering of single atom manipulation in individual traps, including optical dipole traps and

mm2.

more efficient if the resonance is far-detuned.

which our model readily offers.

**4. Conclusions**

its narrow bandwidth ∼ Γ, much smaller than the thermal ensemble energy *δ�*. Conversion efficiency per pulse pair in this case might be increased by simultaneously chirping the Feshbach resonance energy and the pump pulse frequency, i.e. by tuning *�<sup>F</sup>* and *ω<sup>P</sup>* in time keeping the two-photon resonance condition *�<sup>F</sup>* = *ω<sup>S</sup>* − *ω<sup>p</sup>* satisfied.

While only a small fraction of atoms can be transferred to |1� by a pair of STIRAP pulses, a train of pulse pairs can be applied to photoassociate the entire atomic ensemble. To prevent excitation of molecules in |1� back to the continuum by subsequent pulses, they have to be removed before the next pair of pulses is applied. This could be realized by applying, after each pair of Stokes and pump pulses, a relatively long pulse resonant to a transition from |1� to some other vibrational level in the excited electronic potential which decays spontaneously to a deep vibrational state in the ground electronic potential. This long pulse would optically pump molecules out of the state |1� to deeper vibrational states in the ground electronic potential. It therefore has to be longer than the spontaneous decay time of the excited state. The excited state has to be chosen carefully so that it does not decay back into the scattering continuum. This would empty the |1� state and deposit molecules into ground potential vibrational states according to Franck-Condon factors before the next pair of pulses arrives. Finally, after all atoms have been converted into molecules the recently demonstrated optical pumping for molecules method (20) can be applied, which would transfer molecules from all populated vibrational states into the ground level *v* = 0.

The optimal strategy is to actually choose an excited state that decays mostly to the *v* = 0 level. This would allow one to avoid storing molecules in unstable vibrational states and using the optical pumping method. If such a state cannot be directly reached from |1�, a four-photon STIRAP transfer can be applied (30), which provides efficient transfer to deeply bound molecular states. It allows one to choose the final state |1�, from which the excited state decaying predominantly to *v* = 0 can be easily reached. In this case rotational selectivity can also be preserved, since only *v* = 0, *J* = 0 and *v* = 0, *J* = 2 states will be populated.

The total time required to photoassociate the whole atomic ensemble and transfer it to the *v* = 0 level can be estimated as follows. As the numerical results show, the adiabatic passage requires ∼ 5 *μ*s, the follow-up pulse emptying state |1� can have a ∼ 100 ns duration, if the excited state lifetime is tens of ns, resulting in the whole sequence ∼ 6 *μ*s. Then the train of 104 <sup>−</sup> 106 pulse pairs will take <sup>∼</sup> 0.1 <sup>−</sup> 10 s. The final step, optical pumping to the *<sup>v</sup>* <sup>=</sup> 0 level, requires ∼ hundred *μ*s, so the overall formation time is ∼ 0.1 − 10 s. Given an illuminated volume <sup>∼</sup> <sup>10</sup>−<sup>2</sup> <sup>−</sup> <sup>10</sup>−<sup>3</sup> mm<sup>3</sup> and an atomic density *<sup>ρ</sup>* <sup>∼</sup> 1012 cm−3, the resulting production rate is expected to be 105 <sup>−</sup> 108 molecules/s. This compares well with recent experiments on STIRAP production of ground state KRb molecules starting from the Feshbach bound state, where <sup>∼</sup> <sup>3</sup> <sup>×</sup> 104 ground state molecules are produced during the entire cycle, including creation of Feshbach molecules, taking ∼ 10 − 30 s (23).

Alternatively, back-stimulation of formed molecules into the continuum by subsequent STIRAP pulses can be avoided by placing them in a moving optical lattice, holding molecules but not atoms (27). Another way to avoid back-stimulation, applicable to polar molecules, is to overlap the atomic trap with a gradient of a DC electric field. It will leave dipoleless atoms unaffected, while shifting molecules out of STIRAP laser beams.

To summarize, combining both photoassociation and coherent optical transfer to rovibrational levels of the ground electronic molecular potential can allow one to convert an entire atomic ensemble into deeply bound molecular states, and to produce an ultracold molecular gas with high phase-space density. Photoassociative adiabatic passage in a thermal ultracold 26 Will-be-set-by-IN-TECH

its narrow bandwidth ∼ Γ, much smaller than the thermal ensemble energy *δ�*. Conversion efficiency per pulse pair in this case might be increased by simultaneously chirping the Feshbach resonance energy and the pump pulse frequency, i.e. by tuning *�<sup>F</sup>* and *ω<sup>P</sup>* in time

While only a small fraction of atoms can be transferred to |1� by a pair of STIRAP pulses, a train of pulse pairs can be applied to photoassociate the entire atomic ensemble. To prevent excitation of molecules in |1� back to the continuum by subsequent pulses, they have to be removed before the next pair of pulses is applied. This could be realized by applying, after each pair of Stokes and pump pulses, a relatively long pulse resonant to a transition from |1� to some other vibrational level in the excited electronic potential which decays spontaneously to a deep vibrational state in the ground electronic potential. This long pulse would optically pump molecules out of the state |1� to deeper vibrational states in the ground electronic potential. It therefore has to be longer than the spontaneous decay time of the excited state. The excited state has to be chosen carefully so that it does not decay back into the scattering continuum. This would empty the |1� state and deposit molecules into ground potential vibrational states according to Franck-Condon factors before the next pair of pulses arrives. Finally, after all atoms have been converted into molecules the recently demonstrated optical pumping for molecules method (20) can be applied, which would transfer molecules from all

The optimal strategy is to actually choose an excited state that decays mostly to the *v* = 0 level. This would allow one to avoid storing molecules in unstable vibrational states and using the optical pumping method. If such a state cannot be directly reached from |1�, a four-photon STIRAP transfer can be applied (30), which provides efficient transfer to deeply bound molecular states. It allows one to choose the final state |1�, from which the excited state decaying predominantly to *v* = 0 can be easily reached. In this case rotational selectivity can

The total time required to photoassociate the whole atomic ensemble and transfer it to the *v* = 0 level can be estimated as follows. As the numerical results show, the adiabatic passage requires ∼ 5 *μ*s, the follow-up pulse emptying state |1� can have a ∼ 100 ns duration, if the excited state lifetime is tens of ns, resulting in the whole sequence ∼ 6 *μ*s. Then the train of 104 <sup>−</sup> 106 pulse pairs will take <sup>∼</sup> 0.1 <sup>−</sup> 10 s. The final step, optical pumping to the *<sup>v</sup>* <sup>=</sup> 0 level, requires ∼ hundred *μ*s, so the overall formation time is ∼ 0.1 − 10 s. Given an illuminated volume <sup>∼</sup> <sup>10</sup>−<sup>2</sup> <sup>−</sup> <sup>10</sup>−<sup>3</sup> mm<sup>3</sup> and an atomic density *<sup>ρ</sup>* <sup>∼</sup> 1012 cm−3, the resulting production rate is expected to be 105 <sup>−</sup> 108 molecules/s. This compares well with recent experiments on STIRAP production of ground state KRb molecules starting from the Feshbach bound state, where <sup>∼</sup> <sup>3</sup> <sup>×</sup> 104 ground state molecules are produced during the entire cycle, including

Alternatively, back-stimulation of formed molecules into the continuum by subsequent STIRAP pulses can be avoided by placing them in a moving optical lattice, holding molecules but not atoms (27). Another way to avoid back-stimulation, applicable to polar molecules, is to overlap the atomic trap with a gradient of a DC electric field. It will leave dipoleless atoms

To summarize, combining both photoassociation and coherent optical transfer to rovibrational levels of the ground electronic molecular potential can allow one to convert an entire atomic ensemble into deeply bound molecular states, and to produce an ultracold molecular gas with high phase-space density. Photoassociative adiabatic passage in a thermal ultracold

also be preserved, since only *v* = 0, *J* = 0 and *v* = 0, *J* = 2 states will be populated.

keeping the two-photon resonance condition *�<sup>F</sup>* = *ω<sup>S</sup>* − *ω<sup>p</sup>* satisfied.

populated vibrational states into the ground level *v* = 0.

creation of Feshbach molecules, taking ∼ 10 − 30 s (23).

unaffected, while shifting molecules out of STIRAP laser beams.

atomic gas can be greatly facilitated by a Feshbach resonance. The presence of a bound state imbedded in and resonant with scattering continuum states strongly enhances the continuum-bound transition dipole matrix element to an excited electronic state, thus requiring less laser intensity for efficient transfer. In the limit of a wide resonance, when compared to the thermal spread of collision energies, the dipole matrix element is enhanced by the Fano parameter *q*. By choosing a tightly bound excited vibrational state, *q* can be made much larger than unity, resulting in the intensity of the pump pulse required for efficient adiabatic passage to be <sup>∼</sup> 1/*q*<sup>2</sup> times smaller than in the absence of the resonance. Numeical modeling of the adiabatic passage using typical parameters of alkali dimers shows that intensities of the pump pulse, coupling the continuum to an excited state, of kW/cm<sup>2</sup> are sufficient for optimal transfer, which is ∼ 100 times smaller than without resonance. Optimal pulse durations are several *μ*s, resulting in energies per pulse ∼ 10 *μ*J for a focus area of 1 mm2.

If the Feshbach resonance is narrow compared to the thermal energy spread of colliding atoms, adiabatic passage is hindered by destructive quantum interference. The reason is that electromagnetically induced transparency significantly reduces the transition dipole matrix element from the scattering continuum to an excited state in the presence of the bound Feshbach state. In the narrow resonance limit, photoassociative adiabatic passage is therefore more efficient if the resonance is far-detuned.

Due to low atomic collision rates at ultracold temperatures, only a small fraction of atoms can be converted into molecules by a pair of photoassociative pulses. To convert an entire atomic ensemble, a train of pulse pairs can be applied. We estimate that 104 <sup>−</sup> 106 pulse pairs will associate an atomic gas of alkali dimers with a density 1012 cm−<sup>3</sup> in an illuminated volume of 10−<sup>2</sup> <sup>−</sup> <sup>10</sup>−<sup>3</sup> mm<sup>3</sup> in 0.1 <sup>−</sup> 10 s, resulting in extremely high production rates of 105 <sup>−</sup> 108 molecules/s. High transfer efficiencies combined with low intensities of adiabatic photoassociative pulses also make the broad resonance limit attractive for quantum computation. For example, a scheme proposed in (63) can be realized, where qubit states are encoded into a scattering and a bound molecular states of polar molecules. To perform one and two-qubit operations, this scheme requires a high degree of control over the system, which our model readily offers.

Finally, marrying FOPA and STIRAP is a very promising avenue to produce large amounts of molecules, for a variety of molecular species. In fact, although we described here examples based on magnetically induced Feshbach resonances, such resonances are extremely common, and can be induced by several interactions, such as external electric fields or optical fields. Even in the absence of hyperfine interactions, other interactions can provide the necessary coupling, such as in the case of the magnetic dipole-dipole interaction in 52Cr (64; 65).

#### **4. Conclusions**

Precise control over internal and external degrees of freedom of molecules will open the way for new fundamental studies and applications in physics and chemistry. As has been clearly seen with atoms in the recent decades, well-controlled laser fields offer an exquisite control tool over atomic internal and external states, including laser cooling and trapping, coherent manipulation of atomic quantum states and in particular various techniques used for quantum information applications, atomic spectroscopy. Recent years have witnessed mastering of single atom manipulation in individual traps, including optical dipole traps and

where *c*± = cos *θ*/

<sup>L</sup>*<sup>ρ</sup>* <sup>=</sup> <sup>1</sup> 2 <sup>√</sup><sup>1</sup> <sup>±</sup> sin 2*θ*/2, *<sup>s</sup>*<sup>±</sup> <sup>=</sup> <sup>±</sup> sin *<sup>θ</sup>*/

<sup>−</sup>2 cos *<sup>θ</sup>* 0 sin 2*<sup>θ</sup>* <sup>Ω</sup>

Coherent Laser Manipulation of Ultracold Molecules 81

<sup>√</sup>2 sin *<sup>θ</sup>* <sup>2</sup> <sup>−</sup> cos 2*<sup>θ</sup>* <sup>√</sup>

<sup>√</sup>2 sin *<sup>θ</sup>* <sup>−</sup><sup>1</sup> <sup>−</sup> cos 2*<sup>θ</sup>* <sup>√</sup>

In the "straddling" STIRAP scheme the rotation matrix reads as:

⎛

⎜⎜⎜⎜⎜⎜⎝

The Liouville operator in the bare state basis has a form

−

−

sin *θ* <sup>Ω</sup>

sin *θ* <sup>Ω</sup>

assume that molecules in the ground vibrational state |*g*1� do not decay.

*<sup>W</sup>* <sup>=</sup> <sup>1</sup> 2

where terms of the order of *O*(Ω2/Ω<sup>2</sup>

**B.1 Limit of a broad Feshbach resonance**

*Sw* = *S*<sup>0</sup>

√

⎛

⎜⎜⎜⎜⎝

<sup>√</sup><sup>1</sup> <sup>±</sup> sin 2*θ*/2.

Ω 2Ω<sup>0</sup> −1

Ω 2Ω<sup>0</sup>

<sup>Ω</sup><sup>0</sup> <sup>−</sup><sup>1</sup> <sup>√</sup><sup>2</sup> <sup>−</sup>1 cos *<sup>θ</sup>* <sup>Ω</sup>

<sup>Ω</sup><sup>0</sup> <sup>1</sup> <sup>√</sup>2 1 cos *<sup>θ</sup>* <sup>Ω</sup>

<sup>0</sup>) and higher are neglected.

2Γ1*ρg*<sup>1</sup> *<sup>g</sup>*<sup>1</sup> (*γ*<sup>1</sup> + Γ1)*ρg*<sup>1</sup> *<sup>e</sup>*<sup>1</sup> (Γ<sup>2</sup> + Γ1)*ρg*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> (*γ*<sup>2</sup> + Γ1)*ρg*<sup>1</sup> *<sup>e</sup>*<sup>2</sup> Γ1*ρg*<sup>1</sup> *<sup>g</sup>*<sup>3</sup> (*γ*<sup>1</sup> + Γ1)*ρe*<sup>1</sup> *<sup>g</sup>*<sup>1</sup> 2*γ*1*ρe*1*e*<sup>1</sup> (*γ*<sup>1</sup> + Γ2)*ρe*<sup>1</sup> *<sup>g</sup>*<sup>2</sup> (*γ*<sup>1</sup> + *γ*2)*ρe*1*e*<sup>2</sup> *γ*1*ρe*<sup>1</sup> *<sup>g</sup>*<sup>3</sup> (Γ<sup>2</sup> + Γ1)*ρg*<sup>2</sup> *<sup>g</sup>*<sup>1</sup> (*γ*<sup>1</sup> + Γ2)*ρg*<sup>2</sup> *<sup>e</sup>*<sup>1</sup> 2Γ2*ρg*<sup>2</sup> *<sup>g</sup>*<sup>2</sup> (*γ*<sup>2</sup> + Γ2)*ρg*<sup>2</sup> *<sup>e</sup>*<sup>2</sup> Γ2*ρg*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> (*γ*<sup>2</sup> + Γ1)*ρe*<sup>2</sup> *<sup>g</sup>*<sup>1</sup> (*γ*<sup>1</sup> + *γ*2)*ρe*2*e*<sup>1</sup> (*γ*<sup>2</sup> + Γ2)*ρe*<sup>2</sup> *<sup>g</sup>*<sup>2</sup> 2*γ*2*ρe*2*e*<sup>2</sup> *γ*2*ρe*<sup>2</sup> *<sup>g</sup>*<sup>3</sup> Γ1*ρg*<sup>3</sup> *<sup>g</sup>*<sup>1</sup> *γ*1*ρg*<sup>3</sup> *<sup>e</sup>*<sup>1</sup> Γ2*ρg*<sup>3</sup> *<sup>g</sup>*<sup>2</sup> *γ*2*ρg*<sup>3</sup> *<sup>e</sup>*<sup>2</sup> 0

We include the decay from the Feshbach and the intermediate state using a rate Γ<sup>1</sup> and Γ2, respectively, and from the excited states |*e*1,2�, given by *γ*1,2, and assume that all decay is due to population loss out of the system, e.g. to other vibrational levels or continuum. We also

**B. Adiabatic passage in the limits of broad and narrow Feshbach resonances**

In this appendix, we discuss Eqs.(26) and (27) for various relative widths of the Feshbach resonance Γ with respect to the thermal energy spread *δ�* of the colliding atoms. We first describe the case of a broad resonance, *i.e.* when the width of the Feshbach resonance greatly exceeds the thermal energy spread (Γ � *δ�*), and second consider the opposite situation of a narrow resonance (Γ � *δ�*). Finally, we briefly present the case where there is no resonance.

The typical thermal energy spread for colliding atoms in photoassociation experiments with non-degenerate gases is *δ�* ∼ 10 − 100 *μ*K. The broad resonance case occurs for resonances having a width of several Gauss (∼ 1 mK), for which we have Γ/*δ�* ∼ 10 − 100. A wide variety of systems exhibit broad resonances. For instance, they can be found in collision of 6Li atoms at 834 G for the <sup>|</sup> *<sup>f</sup>* <sup>=</sup> 1/2, *mf* <sup>=</sup> 1/2�⊗| *<sup>f</sup>* <sup>=</sup> 1/2, *mf* <sup>=</sup> <sup>−</sup>1/2� entrance channel (<sup>Γ</sup> <sup>=</sup> 302 G= 40 mK) and in 7Li at 736 G for the <sup>|</sup> *<sup>f</sup>* <sup>=</sup> 1, *mf* <sup>=</sup> <sup>1</sup>�⊗| *<sup>f</sup>* <sup>=</sup> 1, *mf* <sup>=</sup> <sup>1</sup>� entrance channel (Γ = 145 G = 19 mK). We note here that these values of Γ are slightly different than the "magnetic" width Δ*B* usually given and based on the modelling of the scattering length. The source function can be readily calculated from Eq.(31) by noticing that the Rabi frequency term can be set at *�* = *�*<sup>0</sup> corresponding to the maximum of the Gaussian function in the

<sup>−</sup>(*t*−*t*0)<sup>2</sup>*δ*<sup>2</sup>

= *S*no−res*g*(*q*, *�*0)sgn(*�*<sup>0</sup> − *�F*), (38)

*�*/2¯*h*<sup>2</sup>

−*i*(*�*0/¯*h*−(*ωS*−*ω<sup>p</sup>* ))*t*

integrand. Using the function *g*(*q*, *�*) defined in Eq.(32), the result takes the form

2*πδ�g*(*q*, *�*0)sgn(*�*<sup>0</sup> − *�F*)*e*

<sup>Ω</sup><sup>0</sup> 0 −2 sin *θ*

<sup>√</sup>2 cos *<sup>θ</sup>*

Ω<sup>0</sup>

⎞

⎟⎟⎟⎟⎟⎟⎠ ,

⎞

⎟⎟⎟⎟⎠ .

Ω<sup>0</sup>

<sup>1</sup> <sup>√</sup>2 cos *<sup>θ</sup>*

atom chips, and optical lattices, with most manipulation techniques relying on laser fields. There is a great incentive in the atomic and molecular optics community to extend the precise control techniques developed for atoms to molecules. We have outlined in this chapter some experimentally relatively simple laser pulse techniques that can accomplish this task.

A prerequisite for many of the new studies is a high phase space density molecular sample in a stable internal state, specifically in the ground rovibrational state and preferably in the lowest hyperfine sublevel. We have in particular discussed two examples of coherent laser control of molecular states, multistate chainwise STIRAP and photoassociatice adiabatic passage near Feshbach resonance, which provide efficient transfer of molecules to the ground rovibrational state. In chainwise STIRAP the transfer is based on a generalized dark state, which is a superposition of all ground vibrational levels involved in the process. Selecting a special time order of the laser pulses coupling vibrational states and optimizing durations and intensities transfer efficiencies > 90% are predicted even in the presence of fast collisional decay of intermediate vibrational states. This technique has recently been applied to transfer Cs2 Feshbach molecules to the ground rovibrational state with 55% efficiency, limited by technical issues. Additionally, we outlined how the step from the atomic scattering continuum to the ground rovibrational molecular state can be done in one coordinated step. In the presence of a Feshbach resonance delocalized scattering states acquire some bound-state character due to admixture of a bound level associated with a closed channel. It strongly enhances the Franck-Condon factor between the initial scattering state and a bound intermediate excited molecular state, a technique named Feshbach Optimized Photoassociation. We analyzed the transfer efficiency and intensities of the laser pulses required for optimal transfer both with and without the resonance and found that > 70% efficiencies are possible with relatively low intensity pulses of several W/cm<sup>2</sup> in the presence of the resonance.

#### **5. Acknowledgments**

We gratefully acknowledge finantial support from NSF and AFOSR under the MURI award FA9550-09-1-0588.

#### **6. Appendix**

#### **A. Rotation and dephasing matrices**

The Hamiltonian (2) in the case of the two-pulse STIRAP scheme, discussed in Section 2.1 has a zero eigenvalue *ε*<sup>0</sup> = 0, describing the dark state, and four eigenvalues, *ε*1,2 = <sup>±</sup>Ω√<sup>1</sup> <sup>−</sup> sin 2*θ*/2 and *<sup>ε</sup>*3,4 <sup>=</sup> <sup>±</sup>Ω√<sup>1</sup> <sup>+</sup> sin 2*θ*/2, corresponding to bright states. Adiabatic eigenstates <sup>|</sup>Φ� <sup>=</sup> {|Φ*n*�}, *<sup>n</sup>* <sup>=</sup> 0, ...4 and the bare states <sup>|</sup>Ψ� <sup>=</sup> �� � � Ψ*l* ��, *<sup>l</sup>* <sup>=</sup> *<sup>g</sup>*1,*e*1, *<sup>g</sup>*2,*e*2, *<sup>g</sup>*<sup>3</sup> are transformed as � � � Ψ*l* � <sup>=</sup> <sup>∑</sup>*<sup>n</sup> Wln* <sup>|</sup>Φ*n*�, <sup>|</sup>Φ*n*� <sup>=</sup> <sup>∑</sup>*<sup>l</sup> Wln* � � � Ψ*l* � via an orthogonal (*W*−<sup>1</sup> = *WT*) rotation matrix, given by the expression

$$
\mathcal{W} = \frac{1}{2} \begin{pmatrix}
2\mathcal{c}^+\mathcal{c}^- & \mathcal{s}^- & \mathcal{s}^- & \mathcal{s}^+ & \mathcal{s}^+ \\
0 & 1 & -1 & -1 & 1 \\
2\mathcal{s}^-\mathcal{c}^+ & -(\mathcal{s}^- + \mathcal{c}^-) - (\mathcal{s}^- + \mathcal{c}^-) \ (\mathcal{s}^+ + \mathcal{c}^+) \ (\mathcal{s}^+ + \mathcal{c}^+) \\
0 & -1 & 1 & -1 & 1 \\
\end{pmatrix},
$$

28 Will-be-set-by-IN-TECH

atom chips, and optical lattices, with most manipulation techniques relying on laser fields. There is a great incentive in the atomic and molecular optics community to extend the precise control techniques developed for atoms to molecules. We have outlined in this chapter some

A prerequisite for many of the new studies is a high phase space density molecular sample in a stable internal state, specifically in the ground rovibrational state and preferably in the lowest hyperfine sublevel. We have in particular discussed two examples of coherent laser control of molecular states, multistate chainwise STIRAP and photoassociatice adiabatic passage near Feshbach resonance, which provide efficient transfer of molecules to the ground rovibrational state. In chainwise STIRAP the transfer is based on a generalized dark state, which is a superposition of all ground vibrational levels involved in the process. Selecting a special time order of the laser pulses coupling vibrational states and optimizing durations and intensities transfer efficiencies > 90% are predicted even in the presence of fast collisional decay of intermediate vibrational states. This technique has recently been applied to transfer Cs2 Feshbach molecules to the ground rovibrational state with 55% efficiency, limited by technical issues. Additionally, we outlined how the step from the atomic scattering continuum to the ground rovibrational molecular state can be done in one coordinated step. In the presence of a Feshbach resonance delocalized scattering states acquire some bound-state character due to admixture of a bound level associated with a closed channel. It strongly enhances the Franck-Condon factor between the initial scattering state and a bound intermediate excited molecular state, a technique named Feshbach Optimized Photoassociation. We analyzed the transfer efficiency and intensities of the laser pulses required for optimal transfer both with and without the resonance and found that > 70% efficiencies are possible with relatively low

We gratefully acknowledge finantial support from NSF and AFOSR under the MURI award

The Hamiltonian (2) in the case of the two-pulse STIRAP scheme, discussed in Section 2.1 has a zero eigenvalue *ε*<sup>0</sup> = 0, describing the dark state, and four eigenvalues, *ε*1,2 = <sup>±</sup>Ω√<sup>1</sup> <sup>−</sup> sin 2*θ*/2 and *<sup>ε</sup>*3,4 <sup>=</sup> <sup>±</sup>Ω√<sup>1</sup> <sup>+</sup> sin 2*θ*/2, corresponding to bright states. Adiabatic

> 2*c*+*c*<sup>−</sup> *s*<sup>−</sup> *s*<sup>−</sup> *s*<sup>+</sup> *s*<sup>+</sup> 0 1 −1 −1 1 <sup>2</sup>*s*−*c*<sup>+</sup> <sup>−</sup>(*s*<sup>−</sup> <sup>+</sup> *<sup>c</sup>*−) <sup>−</sup>(*s*<sup>−</sup> <sup>+</sup> *<sup>c</sup>*−) (*s*<sup>+</sup> <sup>+</sup> *<sup>c</sup>*+) (*s*<sup>+</sup> <sup>+</sup> *<sup>c</sup>*+) 0 −1 1 −1 1 <sup>−</sup>2*s*+*s*<sup>−</sup> *<sup>c</sup>*<sup>−</sup> *<sup>c</sup>*<sup>−</sup> *<sup>c</sup>*<sup>+</sup> (*s*<sup>+</sup> <sup>−</sup> *<sup>c</sup>*+)

<sup>=</sup> <sup>∑</sup>*<sup>n</sup> Wln* <sup>|</sup>Φ*n*�, <sup>|</sup>Φ*n*� <sup>=</sup> <sup>∑</sup>*<sup>l</sup> Wln*

�� � � Ψ*l* ��

� � � Ψ*l* � , *l* = *g*1,*e*1, *g*2,*e*2, *g*<sup>3</sup>

via an orthogonal (*W*−<sup>1</sup> = *WT*)

⎞

⎟⎟⎟⎟⎠ ,

experimentally relatively simple laser pulse techniques that can accomplish this task.

intensity pulses of several W/cm<sup>2</sup> in the presence of the resonance.

eigenstates <sup>|</sup>Φ� <sup>=</sup> {|Φ*n*�}, *<sup>n</sup>* <sup>=</sup> 0, ...4 and the bare states <sup>|</sup>Ψ� <sup>=</sup>

**5. Acknowledgments**

**A. Rotation and dephasing matrices**

� � � Ψ*l* �

rotation matrix, given by the expression

⎛

⎜⎜⎜⎜⎝

*<sup>W</sup>* <sup>=</sup> <sup>1</sup> 2

FA9550-09-1-0588.

are transformed as

**6. Appendix**

where *c*± = cos *θ*/ <sup>√</sup><sup>1</sup> <sup>±</sup> sin 2*θ*/2, *<sup>s</sup>*<sup>±</sup> <sup>=</sup> <sup>±</sup> sin *<sup>θ</sup>*/ <sup>√</sup><sup>1</sup> <sup>±</sup> sin 2*θ*/2. In the "straddling" STIRAP scheme the rotation matrix reads as:

$$\begin{array}{cccc} W = \frac{1}{2} \begin{pmatrix} -2\cos\theta & 0 & \sin 2\theta \frac{\Omega}{\Omega\_{0}} & 0 & -2\sin\theta \\ -\sqrt{2}\sin\theta & 2 & -\cos 2\theta \frac{\Omega}{\sqrt{2}\Omega\_{0}} & -1 & \sqrt{2}\cos\theta \\ -\sqrt{2}\sin\theta & -1 & \cos 2\theta \frac{\Omega}{\sqrt{2}\Omega\_{0}} & 1 & \sqrt{2}\cos\theta \\ \sin\theta \frac{\Omega}{\Omega\_{0}} & -1 & \sqrt{2} & -1 \cos \theta \frac{\Omega}{\Omega\_{0}} \\ \sin\theta \frac{\Omega}{\Omega\_{0}} & 1 & \sqrt{2} & 1 & \cos \theta \frac{\Omega}{\Omega\_{0}} \end{pmatrix} \end{array}$$

where terms of the order of *O*(Ω2/Ω<sup>2</sup> <sup>0</sup>) and higher are neglected. The Liouville operator in the bare state basis has a form

$$
\mathcal{L}\rho = \frac{1}{2} \begin{pmatrix}
2\Gamma\_1\rho\_{\mathcal{S}\cup\mathcal{S}\_1} & (\gamma\_1+\Gamma\_1)\rho\_{\mathcal{S}\cup\mathcal{E}\_1} \ (\Gamma\_2+\Gamma\_1)\rho\_{\mathcal{S}\cup\mathcal{S}\_2} \ (\gamma\_2+\Gamma\_1)\rho\_{\mathcal{S}\cup\mathcal{E}\_2} \ \Gamma\_1\rho\_{\mathcal{S}\cup\mathcal{S}\_3} \\
(\gamma\_1+\Gamma\_1)\rho\_{\mathcal{e}\_1\mathcal{S}\_1} & 2\gamma\_1\rho\_{\mathcal{e}\_1\mathcal{E}\_1} \ (\gamma\_1+\Gamma\_2)\rho\_{\mathcal{e}\_1\mathcal{S}\_2} \ (\gamma\_1+\gamma\_2)\rho\_{\mathcal{E}\_1\mathcal{E}\_2} \ \gamma\_1\rho\_{\mathcal{E}\_3\mathcal{S}\_3} \\
(\Gamma\_2+\Gamma\_1)\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^1} \ (\gamma\_1+\Gamma\_2)\rho\_{\mathcal{S}\mathcal{S}^2\mathcal{E}\_1} & 2\Gamma\_2\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^2\mathcal{E}} \ (\gamma\_2+\Gamma\_2)\rho\_{\mathcal{S}\mathcal{S}^2\mathcal{E}} \ \Gamma\_2\rho\_{\mathcal{S}\mathcal{S}^3\mathcal{E}} \\
(\gamma\_2+\Gamma\_1)\rho\_{\mathcal{e}\_2\mathcal{S}\_1} & (\gamma\_1+\gamma\_2)\rho\_{\mathcal{e}\_2\mathcal{E}\_1} \ (\gamma\_2+\Gamma\_2)\rho\_{\mathcal{e}\_2\mathcal{S}\_2} & 2\gamma\_2\rho\_{\mathcal{e}\_2\mathcal{E}\_2} & 2\gamma\_2\rho\_{\mathcal{e}\_2\mathcal{E}\_3} \\
\Gamma\_1\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}\mathcal{S}^1} & \gamma\_1\rho\_{\mathcal{S}\mathcal{S}^3\mathcal{E}} & \Gamma\_2\rho\_{\mathcal{S}\mathcal{S}\mathcal{S}^2} & 0
\end{pmatrix}.
$$

We include the decay from the Feshbach and the intermediate state using a rate Γ<sup>1</sup> and Γ2, respectively, and from the excited states |*e*1,2�, given by *γ*1,2, and assume that all decay is due to population loss out of the system, e.g. to other vibrational levels or continuum. We also assume that molecules in the ground vibrational state |*g*1� do not decay.

#### **B. Adiabatic passage in the limits of broad and narrow Feshbach resonances**

In this appendix, we discuss Eqs.(26) and (27) for various relative widths of the Feshbach resonance Γ with respect to the thermal energy spread *δ�* of the colliding atoms. We first describe the case of a broad resonance, *i.e.* when the width of the Feshbach resonance greatly exceeds the thermal energy spread (Γ � *δ�*), and second consider the opposite situation of a narrow resonance (Γ � *δ�*). Finally, we briefly present the case where there is no resonance.

#### **B.1 Limit of a broad Feshbach resonance**

The typical thermal energy spread for colliding atoms in photoassociation experiments with non-degenerate gases is *δ�* ∼ 10 − 100 *μ*K. The broad resonance case occurs for resonances having a width of several Gauss (∼ 1 mK), for which we have Γ/*δ�* ∼ 10 − 100. A wide variety of systems exhibit broad resonances. For instance, they can be found in collision of 6Li atoms at 834 G for the <sup>|</sup> *<sup>f</sup>* <sup>=</sup> 1/2, *mf* <sup>=</sup> 1/2�⊗| *<sup>f</sup>* <sup>=</sup> 1/2, *mf* <sup>=</sup> <sup>−</sup>1/2� entrance channel (<sup>Γ</sup> <sup>=</sup> 302 G= 40 mK) and in 7Li at 736 G for the <sup>|</sup> *<sup>f</sup>* <sup>=</sup> 1, *mf* <sup>=</sup> <sup>1</sup>�⊗| *<sup>f</sup>* <sup>=</sup> 1, *mf* <sup>=</sup> <sup>1</sup>� entrance channel (Γ = 145 G = 19 mK). We note here that these values of Γ are slightly different than the "magnetic" width Δ*B* usually given and based on the modelling of the scattering length. The source function can be readily calculated from Eq.(31) by noticing that the Rabi frequency term can be set at *�* = *�*<sup>0</sup> corresponding to the maximum of the Gaussian function in the integrand. Using the function *g*(*q*, *�*) defined in Eq.(32), the result takes the form

*Sw* = *S*<sup>0</sup> √ 2*πδ�g*(*q*, *�*0)sgn(*�*<sup>0</sup> − *�F*)*e* <sup>−</sup>(*t*−*t*0)<sup>2</sup>*δ*<sup>2</sup> *�*/2¯*h*<sup>2</sup> −*i*(*�*0/¯*h*−(*ωS*−*ω<sup>p</sup>* ))*t* = *S*no−res*g*(*q*, *�*0)sgn(*�*<sup>0</sup> − *�F*), (38)

width to the width of the thermal energy spread, gives the ratio of contributions from the

Coherent Laser Manipulation of Ultracold Molecules 83

It is then easier to notice that in the limit of a narrow resonance, the Gaussian function in the integrand of Eq.(42) is much narrower than the Bessel and Struve functions, which change on

Since *ξ* � 1, the real part of the source function is given by the first term in the square brackets, which is a pure continuum source function, while the imaginary part is due to the admixed bound state and its magnitude depends on the product *ξq*. Using asymptotic expansions of modified Bessel and Struve functions *<sup>I</sup>*0(*x*) <sup>−</sup> *<sup>L</sup>*0(*x*) → −2/*πx*, *<sup>I</sup>*1(*x*) <sup>−</sup> *<sup>L</sup>*−1(*x*) → −2/*πx*2, it is seen from Eq.(43) that the contribution to the source function from the bound state decays on the time scale |*τ* − *τ*0| ∼ 1/*ξ*, while the contribution from the unperturbed continuum

[*e*−(*τ*−*τ*0)<sup>2</sup>

−*iq*(*I*0(*ξ*|*τ* − *τ*0|) − *L*0(*ξ*|*τ* − *τ*0|))sgn(*τ* − *τ*0))]. (43)

(*I*1(*ξ*|*<sup>τ</sup>* − *<sup>τ</sup>*0|) − *<sup>L</sup>*−1(*ξ*|*<sup>τ</sup>* − *<sup>τ</sup>*0|)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*2, (44)

<sup>2</sup>E*p*(*t*)

−*t*)/2¯*h*+*i*(*�F*/¯*h*−(*ωS*−*ωp*))(*t*�

Ω*�* = Ωno−res = *μ*2*�* · *e*ˆ*<sup>p</sup>* E*p*/¯*h*, (46)

−*i*(*�*0/¯*h*−(*ωS*−*ωp*))*t*

−*t*) 

. (45)

. (47)

<sup>2</sup> *<sup>c</sup>*2, (48)

*π*Γ <sup>2</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>i</sup>*)

Finally, let us consider the case of a continuum without resonance. In this case the

<sup>2</sup>*πδ�e*−(*t*−*t*0)<sup>2</sup>*δ*<sup>2</sup>

*�*/2¯*h*<sup>2</sup>

*<sup>p</sup> c*<sup>2</sup> = *πh*¯ |Ωno−res(*t*)|

bound state and the unperturbed continuum, respectively.

√

+*ξ* <sup>√</sup>*π<sup>e</sup>*

*Sn* = *S*<sup>0</sup>

decays on the time scale |*τ* − *τ*0| ∼ 1 � 1/*ξ*.

−*i* 

× *t* 0

*S*no−res = *S*<sup>0</sup>

*μ*2*�* · *e*ˆ*p*/¯*h*

continuum-bound Rabi frequency Eq.(29) is:

The back-stimulation term (34) reduces to

*i ∂c*<sup>1</sup>

*i ∂c*<sup>2</sup>

**B.3 Continuum without resonance**

and the source function is

In the limit of a narrow resonance the system (26)-(27) becomes:

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*<sup>1</sup> <sup>−</sup> *Sn* + (*<sup>δ</sup>* <sup>−</sup> *<sup>i</sup>γ*)*c*<sup>2</sup>

 2 *<sup>π</sup>h*¯ <sup>E</sup><sup>2</sup> *<sup>p</sup> c*<sup>2</sup> +

*dt*� *c*2(*t* � )E*p*(*t* � )*e* Γ(*t* �

√

 2 *<sup>π</sup>h*¯ <sup>E</sup><sup>2</sup>

*μ*2*�e*ˆ*<sup>p</sup> h*¯

the time scale ∼ 1/*ξ*. Therefore the source term can be aproximated as:

<sup>2</sup>*iD*−*D*<sup>2</sup>

2*πδ�e*−*i*(*�*0/¯*h*−(*ωS*−*ωp*))*<sup>t</sup>*

where *S*no−res is the source function without a resonance given below in Eq.(47). Strictly speaking, this expression is valid for |*�<sup>F</sup>* − *�*0| ≥ *δ�*, but since Γ � *δ�* Eq.(38) is a good approximation for a wide range of detunings *�<sup>F</sup>* − *�*0.

The back-stimulation term (34) can be further simplified in the limit of a broad resonance. In this case, both *c*2(*t*) and E*p*(*t*) change on a time scale ∼ 1/*δ�*, *i.e.* slowly compared to the decay time ∼ *h*¯ /Γ of the exponent. Therefore, we can rewrite (34) as:

$$\left|\frac{\vec{\mu}\_{2\epsilon}\ell\_p}{\hbar}\right|^2 \pi\hbar \left[1 + \frac{(q-i)^2}{1 + 2i(\epsilon\_F - \hbar(\omega\_S - \omega\_p))/\Gamma}\right] c\_2(t)\mathcal{E}\_p^2(t). \tag{39}$$

The system (26)-(27) in the case of a broad resonance becomes:

$$i\hbar\frac{\partial c\_1}{\partial t} = -\Omega\_S c\_{2\prime} \tag{40}$$

$$i\frac{\partial c\_2}{\partial t} = -\Omega\_S c\_1 - S\_w + (\delta - i\gamma)c\_2$$

$$-i\pi\hbar|\Omega\_{\rm no-res}(t)|^2 \left[1 + \frac{(q-i)^2}{1 + 2i(\varepsilon\_F - \hbar(\omega\_S - \omega\_p))/\Gamma}\right]c\_{2\prime} \tag{41}$$

where Ωno−res = *μ*2*�e*ˆ*p*E*p*/¯*h* is the continuum-bound Rabi frequency in the absence of resonance. We also added a spontaneous decay term *γc*2, assuming that the excited molecules dissociate into high energy continuum states and the resulting atoms leave a trap. From Eq.(38), one can see that in a broad resonance case, the source amplitude is enhanced by the factor *g*(*q*, *�*0)=(*q* + 2(*�*<sup>0</sup> − *�F*)/Γ)/ <sup>1</sup> <sup>+</sup> <sup>4</sup>(*�*<sup>0</sup> <sup>−</sup> *�F*)2/Γ<sup>2</sup> when compared to the unperturbed continuum case. This factor has a maximum at 2(*�*<sup>0</sup> − *�F*)/Γ = 1/*q*, with the corresponding maximum value *g*max ∼ *q* for *q* � 1.

#### **B.2 Limit of a narrow Feshbach resonance**

This situation occurs when the width of the resonance is of the order of a few micro-Gauss or less. Examples of narrow resonances include 6Li23Na at 746 G for the <sup>|</sup> *<sup>f</sup>*<sup>1</sup> <sup>=</sup> 1/2, *mf* <sup>1</sup> <sup>=</sup> 1/2�| *<sup>f</sup>*<sup>2</sup> <sup>=</sup> 1, *mf* <sup>2</sup> <sup>=</sup> <sup>1</sup>� channel (<sup>Γ</sup> <sup>=</sup> 7.8 mG = 1 *<sup>μ</sup>*K) (66), or 6Li87Rb at 882 G for the <sup>|</sup> *<sup>f</sup>*<sup>1</sup> <sup>=</sup> 1/2, *mf* <sup>1</sup> = 1/2�| *f*<sup>2</sup> = 1, *mf* <sup>2</sup> = 1� channel (*p*-wave, Γ = 10 mG = 1.3 *μ*K).

We note that the source term expressed in Eq.(31) can be rewritten in a time representation:

$$\begin{split} S &= S\_0 \sqrt{2\pi} \delta\_\ell e^{-i(\epsilon\_0/\hbar - (\omega\_\delta - \omega\_p))t} \\ &\times \left[ e^{-(\tau - \tau\_0)^2} + \xi e^{2iD - D^2} \int\_{-\infty}^{\infty} e^{-(\tau' - iD)^2} (I\_1(\xi | \tau - \tau\_0 - \tau'|) \\ &- L\_{-1}(\xi | \tau - \tau\_0 - \tau'|) - iq (I\_0(\xi | \tau - \tau\_0 - \tau'|) \\ &- L\_0(\xi | \tau - \tau\_0 - \tau'|)) \text{sgn}(\tau - \tau\_0 - \tau')) d\tau' \right], \end{split} \tag{42}$$

where we introduced the dimensionless variables *τ* = *tδ�*/ <sup>√</sup>2¯*h*, *<sup>D</sup>* = (*�<sup>F</sup>* <sup>−</sup> *�*0)/ <sup>√</sup>2*δ�*, *ξ* = Γ/ <sup>√</sup>2*δ�*; *<sup>I</sup>*0,1 and *<sup>L</sup>*0,−<sup>1</sup> are modified Bessel and Struve functions. One can see from this expression that the source function is a sum of the pure source function of the unperturbed continuum, given by the first term in square brackets, and of the admixed bound state, given by the integral. The coefficient *ξ* = Γ/ <sup>√</sup>2*δ�*, which is the ratio of the Feshbach resonance 30 Will-be-set-by-IN-TECH

where *S*no−res is the source function without a resonance given below in Eq.(47). Strictly speaking, this expression is valid for |*�<sup>F</sup>* − *�*0| ≥ *δ�*, but since Γ � *δ�* Eq.(38) is a good

The back-stimulation term (34) can be further simplified in the limit of a broad resonance. In this case, both *c*2(*t*) and E*p*(*t*) change on a time scale ∼ 1/*δ�*, *i.e.* slowly compared to the

1 + 2*i*(*�<sup>F</sup>* − *h*¯(*ω<sup>S</sup>* − *ωp*))/Γ

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*2, (40)

<sup>1</sup> <sup>+</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>i</sup>*)<sup>2</sup>

1 + 2*i*(*�<sup>F</sup>* − *h*¯(*ω<sup>S</sup>* − *ωp*))/Γ

*<sup>c</sup>*2(*t*)E<sup>2</sup>

<sup>1</sup> <sup>+</sup> <sup>4</sup>(*�*<sup>0</sup> <sup>−</sup> *�F*)2/Γ<sup>2</sup> when compared to the

*<sup>p</sup>* (*t*). (39)

*c*2, (41)

<sup>1</sup> <sup>+</sup> (*<sup>q</sup>* <sup>−</sup> *<sup>i</sup>*)<sup>2</sup>

2 

where Ωno−res = *μ*2*�e*ˆ*p*E*p*/¯*h* is the continuum-bound Rabi frequency in the absence of resonance. We also added a spontaneous decay term *γc*2, assuming that the excited molecules dissociate into high energy continuum states and the resulting atoms leave a trap. From Eq.(38), one can see that in a broad resonance case, the source amplitude is enhanced

unperturbed continuum case. This factor has a maximum at 2(*�*<sup>0</sup> − *�F*)/Γ = 1/*q*, with the

This situation occurs when the width of the resonance is of the order of a few micro-Gauss or less. Examples of narrow resonances include 6Li23Na at 746 G for the <sup>|</sup> *<sup>f</sup>*<sup>1</sup> <sup>=</sup> 1/2, *mf* <sup>1</sup> <sup>=</sup> 1/2�| *<sup>f</sup>*<sup>2</sup> <sup>=</sup> 1, *mf* <sup>2</sup> <sup>=</sup> <sup>1</sup>� channel (<sup>Γ</sup> <sup>=</sup> 7.8 mG = 1 *<sup>μ</sup>*K) (66), or 6Li87Rb at 882 G for the <sup>|</sup> *<sup>f</sup>*<sup>1</sup> <sup>=</sup>

We note that the source term expressed in Eq.(31) can be rewritten in a time representation:

−∞

*e*−(*τ*�


expression that the source function is a sum of the pure source function of the unperturbed continuum, given by the first term in square brackets, and of the admixed bound state, given


2*δ�*; *I*0,1 and *L*0,−<sup>1</sup> are modified Bessel and Struve functions. One can see from this

<sup>−</sup>*iD*)<sup>2</sup>

(*I*1(*ξ*|*τ* − *τ*<sup>0</sup> − *τ*�

<sup>√</sup>2*δ�*, which is the ratio of the Feshbach resonance


))*dτ*�  |)

<sup>√</sup>2¯*h*, *<sup>D</sup>* = (*�<sup>F</sup>* <sup>−</sup> *�*0)/

, (42)

<sup>√</sup>2*δ�*,

1/2, *mf* <sup>1</sup> = 1/2�| *f*<sup>2</sup> = 1, *mf* <sup>2</sup> = 1� channel (*p*-wave, Γ = 10 mG = 1.3 *μ*K).

2*πδ�e*−*i*(*�*0/¯*h*−(*ωS*−*ωp*))*<sup>t</sup>*

<sup>2</sup>*iD*−*D*<sup>2</sup> <sup>∞</sup>

+ *ξe*

−*L*−1(*ξ*|*<sup>τ</sup>* − *<sup>τ</sup>*<sup>0</sup> − *<sup>τ</sup>*�

−*L*0(*ξ*|*τ* − *τ*<sup>0</sup> − *τ*�

where we introduced the dimensionless variables *τ* = *tδ�*/

approximation for a wide range of detunings *�<sup>F</sup>* − *�*0.

 

*i ∂c*<sup>1</sup>

*i ∂c*<sup>2</sup> *μ*2*�e*ˆ*<sup>p</sup> h*¯

 2 *πh*¯ 

by the factor *g*(*q*, *�*0)=(*q* + 2(*�*<sup>0</sup> − *�F*)/Γ)/

**B.2 Limit of a narrow Feshbach resonance**

*S* = *S*<sup>0</sup>

× *e* <sup>−</sup>(*τ*−*τ*0)<sup>2</sup>

by the integral. The coefficient *ξ* = Γ/

*ξ* = Γ/

√

√

corresponding maximum value *g*max ∼ *q* for *q* � 1.

decay time ∼ *h*¯ /Γ of the exponent. Therefore, we can rewrite (34) as:

The system (26)-(27) in the case of a broad resonance becomes:

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>Ω*Sc*<sup>1</sup> <sup>−</sup> *Sw* + (*<sup>δ</sup>* <sup>−</sup> *<sup>i</sup>γ*)*c*<sup>2</sup>

−*iπh*¯|Ωno−res(*t*)|

width to the width of the thermal energy spread, gives the ratio of contributions from the bound state and the unperturbed continuum, respectively.

It is then easier to notice that in the limit of a narrow resonance, the Gaussian function in the integrand of Eq.(42) is much narrower than the Bessel and Struve functions, which change on the time scale ∼ 1/*ξ*. Therefore the source term can be aproximated as:

$$S\_{\rm II} = S\_0 \sqrt{2\pi} \delta\_{\rm ef} e^{-i(\varepsilon\_0/\hbar - (\omega\_S - \omega\_p))t} [e^{-(\tau - \tau\_0)^2} $$

$$+ \xi \sqrt{\pi} e^{2iD - D^2} (I\_1(\xi |\tau - \tau\_0|) - L\_{-1}(\xi |\tau - \tau\_0|) $$

$$- iq (I\_0(\xi |\tau - \tau\_0|) - L\_0(\xi |\tau - \tau\_0|)) \text{sgn}(\tau - \tau\_0))). \tag{43}$$

Since *ξ* � 1, the real part of the source function is given by the first term in the square brackets, which is a pure continuum source function, while the imaginary part is due to the admixed bound state and its magnitude depends on the product *ξq*. Using asymptotic expansions of modified Bessel and Struve functions *<sup>I</sup>*0(*x*) <sup>−</sup> *<sup>L</sup>*0(*x*) → −2/*πx*, *<sup>I</sup>*1(*x*) <sup>−</sup> *<sup>L</sup>*−1(*x*) → −2/*πx*2, it is seen from Eq.(43) that the contribution to the source function from the bound state decays on the time scale |*τ* − *τ*0| ∼ 1/*ξ*, while the contribution from the unperturbed continuum decays on the time scale |*τ* − *τ*0| ∼ 1 � 1/*ξ*.

In the limit of a narrow resonance the system (26)-(27) becomes:

$$\begin{split} \dot{q}\frac{\partial c\_{1}}{\partial t} &= -\Omega\_{S}c\_{2}, \\ \dot{q}\frac{\partial c\_{2}}{\partial t} &= -\Omega\_{S}c\_{1} - S\_{\text{ll}} + (\delta - i\gamma)c\_{2} \\ &- i\left|\frac{\vec{\mu}\_{2\varepsilon}\mathcal{E}\_{p}}{\hbar}\right|^{2} \left[\pi\hbar \mathcal{E}\_{p}^{2}c\_{2} + \frac{\pi\Gamma}{2}(q-i)^{2}\mathcal{E}\_{p}(t) \\ &\times \int\_{0}^{t} dt' \, c\_{2}(t')\mathcal{E}\_{p}(t')e^{\Gamma(t'-t)/2\hbar + i(\varepsilon\_{\rm{f}}/\hbar - (\omega\_{\rm{S}}-\omega\_{\rm{p}}))(t'-t)}\right]. \end{split} \tag{45}$$

#### **B.3 Continuum without resonance**

Finally, let us consider the case of a continuum without resonance. In this case the continuum-bound Rabi frequency Eq.(29) is:

$$
\Omega\_{\text{ef}} = \Omega\_{\text{no-res}} = \vec{\mu}\_{2\varepsilon} \cdot \pounds\_p \mathcal{E}\_p / \hbar \,\tag{46}
$$

and the source function is

$$S\_{\rm no-res} = S\_0 \sqrt{2\pi} \delta\_\varepsilon e^{-\left(t - t\_0\right)^2 \delta\_\iota^2 / 2\hbar^2 - i\left(\varepsilon\_0/\hbar - (\omega\_S - \omega\_p)\right)t}.\tag{47}$$

The back-stimulation term (34) reduces to

$$\left|\vec{\mu}\_{2\varepsilon} \cdot \hat{\mathbf{e}}\_p / \hbar\right|^2 \pi \hbar \mathcal{E}\_p^2 c\_2 = \pi \hbar \left|\Omega\_{\rm no-res}(t)\right|^2 c\_{2\prime} \tag{48}$$

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and the system (26)-(27) takes the simple form:

$$\begin{split} i\frac{\partial c\_1}{\partial t} &= -\Omega\_S c\_2, \\ i\frac{\partial c\_2}{\partial t} &= -\Omega\_S c\_1 + (\delta - i\gamma)c\_2 - i\pi\hbar |\Omega\_{\rm no-res}(t)|^2 c\_2 - S\_{\rm no-res}. \end{split} \tag{50}$$

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*i ∂c*<sup>1</sup>

*i ∂c*<sup>2</sup>

**C. References**


**4** 

*Russian Federation* 

**Fast Charged Particles and Super-**

**Strong Magnetic Fields Generated** 

**by Intense Laser Target Interaction** 

The development of a new generation of solid-state lasers has resulted in unique conditions for irradiating laser targets by light pulses, with radiation intensity ranging from 1017 to 1021

At such intensities, the laser pulse produces superstrong electric fields which could not be obtained earlier and considerably exceed the atomic electric field of strength *Ea* = 5.14109 V/cm. In these conditions, there arises a new physical picture of laser pulse interaction with plasma produced when the pulse leading edge or a pre-pulse affects solid targets. Laser radiation is rather efficiently transformed into fluxes of fast charged particles such as electrons and atomic ions. The latter interact with the ambient material of the target, which leads to the generation of hard X-rays, when inner atomic shells are ionized, and to various

One important area in investigating the interaction of sub-picosecond laser pulses with solid targets is related to the important role which arising superstrong quasistatic magnetic fields and electronic structures play in laser plasma dynamics. This area of research became most attractive after carrying out the direct measurements of quasistatic magnetic fields on the Vulcan laser system (Great Britain) (Tatarakis et al., 2002), in particular, after the pinch effect

The relativistic character of laser radiation with intensity *I* is realized at the magnitude of a dimensionless parameter *a* > 1. This parameter represents the dimensionless momentum of the electron oscillating in the electric field of linearly polarized laser radiation and can be

*eE I*

V/cm W/cm *E I*

18 2 0.85 μm 10 W/cm

<sup>2</sup> 27.7

1 2

1 2 (1)

(2)

has been found experimentally in laser plasma (Beg et al., 2004).

*a mc*

**1. Introduction** 

expressed as

W/cm2 and a duration of 20 - 1000 fs.

nuclear and photonuclear reactions.

Vadim Belyaev and Anatoly Matafonov *Central Research Institute of Machine Building* 

