**Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field**

Sergei P. Roshchupkin, Alexandr A. Lebed', Elena A. Padusenko and Alexey I. Voroshilo *Institute of Applied Physics, NASU Ukraine*

### **1. Introduction**

106 Quantum Optics and Laser Experiments

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Studying of various aspects of laser field influence on physical processes is one of the most topical problems of modern applied and fundamental physics. Scientific interest is due to numerous unknown before phenomena, which are caused by laser radiation application and make enable coming to the main point of atomic and molecular structure of matter. These phenomena are of great importance over such fields of physics as holography, fiberglass optics, telecommunications, material authority, biophysics, plasma physics, nuclear fusion and so on. The lasers which generate radiation within the range from deep infrared to ultraviolet one and even the soft X-rays region with intensities up to 1022 W/cm<sup>2</sup> inclusive are made accessible at present. The sources of laser radiation had been put into practice of modern experiment widespread owing to its unique properties. The laser physics progress is generally concentrated on ever shorter and more powerful laser pulses production and on application of the lasts into various fields of scientific studies. New experimental conditions require continual improvements in computations and development of model of external field description.

Influence of laser field on kinematics and cross-sections of various quantum electrodynamics (QED) processes of the both first and second orders in the fine structure constant has been an object of study over a long period of time already. The characteristic feature of electrodynamics processes of the second order in the fine-structure constant in a laser field is associated with the possibility of their nonresonant and resonant modes. At this rate resonant cross-sections of scattering of particles may exceed the corresponding ones in external field absence in several orders of magnitude. Resonant character relates to the fact that lower-order processes, such as spontaneous emission or one-photon production and annihilation of electron-positron pairs, are allowed in the field of a light wave. Therefore, within a certain range of energy and momentum values a particle in an intermediate state may fall within the mass shell. Then the considered higher-order process effectively decomposes into two consecutive lower-order processes. Occurrence of resonances in a laser field is one of the fundamental problems of QED in strong fields.

Theoretical study of QED processes in an external laser field basis on solutions of the Dirac's equation for an electron in the field of a plane electromagnetic wave namely the Volkov functions (Volkov (1935)). Also one has to use the Green function of an intermediate particle in

number of the external field oscillations in an electromagnetic pulse *N* is large:

is satisfied.

*<sup>F</sup>*<sup>0</sup> <sup>∼</sup>

<sup>10</sup><sup>5</sup> <sup>÷</sup> 106

Fofanov (1996))

*A* (*ϕ*) = *g*

meeting the standard conditions: *e*<sup>2</sup>

 *ϕ ωτ* · *cF*<sup>0</sup> *ω* 

circular polarization case); *ex* = (0, **e***x*), *ey* =

*<sup>N</sup>* <sup>=</sup> *ωτ* 2*π*

where *ω* is the characteristic frequency of wave field oscillation, *τ* is the characteristic pulse duration. Quantity *τ* can approach a value of even tens of femtoseconds for fields within the optical range of frequency, thus the condition (1) is satisfied for the majority of modern intense pulsed lasers. Fields are named the quasi-monochromatic ones when the condition (1)

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 109

Hereinafter we consider the external electromagnetic pulse as a plane electromagnetic elliptically polarized wave propagating along *z*-axis with the four-potential (Narozhniy &

*ex* cos *ϕ* + *δey* sin *ϕ*

where *ϕ* is the wave phase; *c* is the velocity of light in vacuum, *F*<sup>0</sup> is the strength of a wave electric field in a pulse peak, *k* = (*ω*/*c*, **k**) is the wave four-vector; *δ* is the wave ellipticity parameter (*δ* = 0 corresponds to the linear polarization case, *δ* = ±1 corresponds to the

is the envelope function of the external wave four-potential that allows to take into account the pulsed character of a laser field. Generally it is chosen to be equal to unity in the center of a pulse and to decrease exponentially when |*ϕ*| � *ωτ*. Thus, in this case it is possible to

Nonlinear effects in the processes of interaction of particles with the field of wave are

*<sup>η</sup>*<sup>0</sup> <sup>=</sup> *eF*0*λ*¯

Its value equals to the ratio of work done by the field at the wavelength to the particle rest energy. The parameter (3) is one of the most important characteristics of the external field and means the velocity of particle oscillation in the field of a wave in the case *η*<sup>0</sup> � 1. Multiphoton processes occurring when particles interact in a light field are characterized also

*γ*<sup>0</sup> = *η*<sup>0</sup>

moderately strong field when the considered parameters obey the following conditions:

In the Eqs. (3), (4) *e* and *m* are the electron charge and mass, respectively; ¯*λ* = *c*/*ω* is the characteristic wavelength, *v* is the particle velocity. The multiphoton parameters (3), (4) vary considerably with external field intensity. Thus, within the range of optical frequencies (*ω* ∼ 1015 <sup>s</sup>−1) the classical parameter *<sup>η</sup>*<sup>0</sup> <sup>∼</sup> 1 when *<sup>F</sup>*<sup>0</sup> <sup>∼</sup> 1010 <sup>÷</sup> 1011 V/cm, and *<sup>γ</sup>*<sup>0</sup> <sup>∼</sup> 1 when

*mvc*

(*c*/*v*) V/cm. Consequently, we study all the processes within the range of

*<sup>x</sup>* = *e*<sup>2</sup>

consider the parameter *τ* as the laser pulse characteristic duration.

by the Bunkin–Fedorov quantum parameter (Bunkin & Fedorov (1966))

governed by the classical relativistic-invariant parameter

0, **e***<sup>y</sup>* 

*<sup>y</sup>* <sup>=</sup> <sup>−</sup>1, (*exk*) <sup>=</sup>

*eyk*

� 1, (1)

, *ϕ* = (*kx*) = *ωt* − **kx**, (2)

are the wave polarization four-vectors

*mc*<sup>2</sup> . (3)

*<sup>h</sup>*¯ *<sup>ω</sup>* . (4)

*η*<sup>0</sup> � 1, *γ*<sup>0</sup> 1. (5)

= 0. The function *g* (*ϕ*/*ωτ*)

a plane wave field when studying processes of the second order in the fine structure constant. The analytical expression of the Green function was obtained (Schwinger (1951); Brown & Kibble (1964)).

Several of significant reviews are already devoted to studying of QED processes in the field of a plane monochromatic wave. The review Nikishov & Ritus (1979) is to be mentioned as one of the earliest works, in which first order processes in the field of a plane electromagnetic wave are studied generally. Processes of an electron scattered by an atom and a multiphoton ionization were considered in Ehlotzky et al. (1998). Theoretical studies of resonant and coherent effects of QED in light field were systematized in the monograph Roshchupkin & Voroshilo (2008) and several QED processes in strong field were reviewed by Ehlotzky et al. (2009).

Detailed consideration of resonant processes in the field of a plane monochromatic wave was fulfilled by Roshchupkin (1996). It is necessary to emphasize, that, when the resonance conditions are satisfied, the amplitude of process of particles scattering in the field of a plane monochromatic wave becomes infinite nominally. The infinity is eliminated by introducing of radiative corrections into Green's function of an intermediate particle according to the Breit–Wigner prescription under consideration as usual. The resonant peak altitude is determined by the lifetime of a particle in the intermediate state (Oleinik (1967)).

Since 1996 experiments of verification of QED in strong fields had been started at SLAC National Accelerator Laboratory (Bula et al. (1996); Burke et al. (1997)) along with theoretical study. The earliest results were related to studying of photon emission by an electron in a collision with laser pulse and photoproduction of electron–positron pairs by a gamma-quantum in the field of a laser. Verification of QED in strong pulsed fields is also expected in the frame of the wide-ranging FAIR project (Darmstadt, Germany). Within the FAIR project the laser facility PHELIX was developed and constructed. It enables to get laser pulses with power up to petawatt range. The earliest experiments at this laser facility have been put into practice (Bagnoud et al. (2009)).

The present paper reviews studies of a number of resonant processes in the field of an intense pulsed laser. The earliest studies on spontaneous bremsstrahlung of an electron in a collision with a laser pulse and photoproduction of electron–positron pairs by a high–energy photon in the pulsed field were performed by Narozhniy & Fofanov (1996). Second order processes in the pulsed fields which allow resonances were analytically studied for the case of moderately strong field (Lebed' & Roshchupkin (2010); Padusenko & Roshchupkin (2010); Lebed' & Roshchupkin (2011); Voroshilo et al. (2011)). These works were performed in recent several years therefore the systematization and generalization of them is definitely significant. It is important to underline that resonant divergences in amplitudes of studied processes are eliminated in a consistent manner due to account of a pulsed character of the external field in mentioned works.

Amplitude of a field strength of intense ultra short laser pulses changes greatly in space and time. Thus, description of the external field as a plane monochromatic wave becomes the problematic one. The external field is modeled usually as a plane quasi-monochromatic wave for description of interaction of particles with a pulsed laser field when the characteristic 2 Will-be-set-by-IN-TECH

a plane wave field when studying processes of the second order in the fine structure constant. The analytical expression of the Green function was obtained (Schwinger (1951); Brown &

Several of significant reviews are already devoted to studying of QED processes in the field of a plane monochromatic wave. The review Nikishov & Ritus (1979) is to be mentioned as one of the earliest works, in which first order processes in the field of a plane electromagnetic wave are studied generally. Processes of an electron scattered by an atom and a multiphoton ionization were considered in Ehlotzky et al. (1998). Theoretical studies of resonant and coherent effects of QED in light field were systematized in the monograph Roshchupkin & Voroshilo (2008) and several QED processes in strong field were reviewed by Ehlotzky et al.

Detailed consideration of resonant processes in the field of a plane monochromatic wave was fulfilled by Roshchupkin (1996). It is necessary to emphasize, that, when the resonance conditions are satisfied, the amplitude of process of particles scattering in the field of a plane monochromatic wave becomes infinite nominally. The infinity is eliminated by introducing of radiative corrections into Green's function of an intermediate particle according to the Breit–Wigner prescription under consideration as usual. The resonant peak altitude is

Since 1996 experiments of verification of QED in strong fields had been started at SLAC National Accelerator Laboratory (Bula et al. (1996); Burke et al. (1997)) along with theoretical study. The earliest results were related to studying of photon emission by an electron in a collision with laser pulse and photoproduction of electron–positron pairs by a gamma-quantum in the field of a laser. Verification of QED in strong pulsed fields is also expected in the frame of the wide-ranging FAIR project (Darmstadt, Germany). Within the FAIR project the laser facility PHELIX was developed and constructed. It enables to get laser pulses with power up to petawatt range. The earliest experiments at this laser facility have

The present paper reviews studies of a number of resonant processes in the field of an intense pulsed laser. The earliest studies on spontaneous bremsstrahlung of an electron in a collision with a laser pulse and photoproduction of electron–positron pairs by a high–energy photon in the pulsed field were performed by Narozhniy & Fofanov (1996). Second order processes in the pulsed fields which allow resonances were analytically studied for the case of moderately strong field (Lebed' & Roshchupkin (2010); Padusenko & Roshchupkin (2010); Lebed' & Roshchupkin (2011); Voroshilo et al. (2011)). These works were performed in recent several years therefore the systematization and generalization of them is definitely significant. It is important to underline that resonant divergences in amplitudes of studied processes are eliminated in a consistent manner due to account of a pulsed character of the external field in

Amplitude of a field strength of intense ultra short laser pulses changes greatly in space and time. Thus, description of the external field as a plane monochromatic wave becomes the problematic one. The external field is modeled usually as a plane quasi-monochromatic wave for description of interaction of particles with a pulsed laser field when the characteristic

determined by the lifetime of a particle in the intermediate state (Oleinik (1967)).

been put into practice (Bagnoud et al. (2009)).

mentioned works.

Kibble (1964)).

(2009).

number of the external field oscillations in an electromagnetic pulse *N* is large:

$$N = \frac{\omega \tau}{2\pi} \gg 1,\tag{1}$$

where *ω* is the characteristic frequency of wave field oscillation, *τ* is the characteristic pulse duration. Quantity *τ* can approach a value of even tens of femtoseconds for fields within the optical range of frequency, thus the condition (1) is satisfied for the majority of modern intense pulsed lasers. Fields are named the quasi-monochromatic ones when the condition (1) is satisfied.

Hereinafter we consider the external electromagnetic pulse as a plane electromagnetic elliptically polarized wave propagating along *z*-axis with the four-potential (Narozhniy & Fofanov (1996))

$$A\left(\varphi\right) = \lg\left(\frac{\varphi}{\omega\tau}\right) \cdot \frac{cF\_0}{\omega} \left(e\_{\mathbf{x}}\cos\varphi + \delta e\_{\mathbf{y}}\sin\varphi\right), \quad \varphi = \left(k\mathbf{x}\right) = \omega t - \mathbf{k}\mathbf{x},\tag{2}$$

where *ϕ* is the wave phase; *c* is the velocity of light in vacuum, *F*<sup>0</sup> is the strength of a wave electric field in a pulse peak, *k* = (*ω*/*c*, **k**) is the wave four-vector; *δ* is the wave ellipticity parameter (*δ* = 0 corresponds to the linear polarization case, *δ* = ±1 corresponds to the circular polarization case); *ex* = (0, **e***x*), *ey* = 0, **e***<sup>y</sup>* are the wave polarization four-vectors meeting the standard conditions: *e*<sup>2</sup> *<sup>x</sup>* = *e*<sup>2</sup> *<sup>y</sup>* <sup>=</sup> <sup>−</sup>1, (*exk*) <sup>=</sup> *eyk* = 0. The function *g* (*ϕ*/*ωτ*) is the envelope function of the external wave four-potential that allows to take into account the pulsed character of a laser field. Generally it is chosen to be equal to unity in the center of a pulse and to decrease exponentially when |*ϕ*| � *ωτ*. Thus, in this case it is possible to consider the parameter *τ* as the laser pulse characteristic duration.

Nonlinear effects in the processes of interaction of particles with the field of wave are governed by the classical relativistic-invariant parameter

$$
\eta\_0 = \frac{eF\_0 \hbar}{mc^2}.\tag{3}
$$

Its value equals to the ratio of work done by the field at the wavelength to the particle rest energy. The parameter (3) is one of the most important characteristics of the external field and means the velocity of particle oscillation in the field of a wave in the case *η*<sup>0</sup> � 1. Multiphoton processes occurring when particles interact in a light field are characterized also by the Bunkin–Fedorov quantum parameter (Bunkin & Fedorov (1966))

$$
\gamma\_0 = \eta\_0 \frac{m\upsilon c}{\hbar \omega}.\tag{4}
$$

In the Eqs. (3), (4) *e* and *m* are the electron charge and mass, respectively; ¯*λ* = *c*/*ω* is the characteristic wavelength, *v* is the particle velocity. The multiphoton parameters (3), (4) vary considerably with external field intensity. Thus, within the range of optical frequencies (*ω* ∼ 1015 <sup>s</sup>−1) the classical parameter *<sup>η</sup>*<sup>0</sup> <sup>∼</sup> 1 when *<sup>F</sup>*<sup>0</sup> <sup>∼</sup> 1010 <sup>÷</sup> 1011 V/cm, and *<sup>γ</sup>*<sup>0</sup> <sup>∼</sup> 1 when *<sup>F</sup>*<sup>0</sup> <sup>∼</sup> <sup>10</sup><sup>5</sup> <sup>÷</sup> 106 (*c*/*v*) V/cm. Consequently, we study all the processes within the range of moderately strong field when the considered parameters obey the following conditions:

$$
\eta\_0 \ll 1, \quad \gamma\_0 \gtrsim 1. \tag{5}
$$

The *S*-matrix element is given by

+*A*ˆ� (*x*2, *k*�

*<sup>μ</sup>* and *k*� = (*ω*�

spontaneous photon, *k*�

photons, that is

Here, the functions *Bli* (*γ*˜0,*ε*ˆ

*Bli* (*γ*˜0,*ε*ˆ

Let us consider the diagram (a):

SB in Fig. 1; *ui*, *u*¯*<sup>f</sup>* are the Dirac bispinors.

∗) =

∞ ∑ *r*=−∞

of an intermediate electron for the diagram (a) (Fig. 1)

Here, *ε*∗

the Coulomb potential of a nucleus, and *A*�

radiated photon. They have the following forms

*A*0 �� � � **x***j* � � � � <sup>=</sup> *Ze* � � � **x***j* � � �

*A*� *μ* � *xj*, *k*� � =

, **k**�

*xj* = *ω*�

*tj* − **k**� **x***j*.

(2009)). This amplitude may be presented in the following form

*Sl* <sup>=</sup> <sup>−</sup>*<sup>i</sup> Ze*3√*<sup>π</sup>* �

2*ω*�*Ef Ei*

2*ωτ*<sup>2</sup> **<sup>q</sup>**<sup>2</sup> + *<sup>q</sup>*<sup>0</sup> (*q*<sup>0</sup> − <sup>2</sup>*qz*)

> ⎧ ⎨ ⎩

<sup>∗</sup>) and *Bl f* (*ε*ˆ

*Sf i* <sup>=</sup> <sup>−</sup>*ie*<sup>2</sup> � *<sup>d</sup>*4*x*1*d*4*x*2*ψ*¯*<sup>f</sup>* (*x*<sup>2</sup> <sup>|</sup>*<sup>A</sup>* )

) *G* (*x*2*x*<sup>1</sup> |*A* ) *γ*˜0*A*<sup>0</sup> (|**x1**|)

�

Resonant Effects of Quantum Electrodynamics in the Pulsed Light Field 111

�

Here, *<sup>ψ</sup><sup>i</sup>* (*x*1|*A*) and *<sup>ψ</sup>*¯*<sup>f</sup>* (*x*2|*A*) are wave functions of an electron in initial and final states in the field (2), and *G* (*x*2*x*<sup>1</sup> |*A* ) is the Green function of an intermediate electron in the field of a pulsed light wave (2). Hereafter, expressions with hats above mean scalar products

of corresponding four-vectors with the Dirac *γ*˜-matrices. In the amplitude (6) *A*<sup>0</sup>

�2*π ω*� *ε* ∗ *<sup>μ</sup>* exp � *ik*� *xj* �

The SB amplitude of an electron scattered by a nucleus in the field of a moderately strong pulsed wave (6) in the general relativistic case was derived early (Lebed' & Roshchupkin

> ∞ ∑ *l*=−∞

where *Sl* is the process partial amplitude with emission or absorption of |*l*| laser-wave

*Bli* (*γ*˜0,*ε*ˆ

�∞

*dξ*

−∞

where the four-vector *q* = (*q*0, **q**) is the transferred four-momentum, *qi* is the four-momentum

*q* = *pf* − *pi* + *k*� + *lk*, *qi* = *pi* − *k*� + *rk*, *q <sup>f</sup>* = *pf* + *k*� + (*l* + *r*) *k*;

<sup>∗</sup>) + *Bl f* (*ε*ˆ

Λ*l*+*<sup>r</sup>* (*ξ*)

*q*2

�

∗, *γ*˜0) �

∗, *γ*˜0) correspond to the diagrams of electron-nucleus

*<sup>q</sup>*ˆ*<sup>i</sup>* + *<sup>m</sup>* + *<sup>ξ</sup>* <sup>ˆ</sup>

*k* �

*<sup>i</sup>* <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> <sup>2</sup>*<sup>ξ</sup>* (*kqi*) <sup>+</sup> *<sup>i</sup>*<sup>0</sup> , (11)

*Sf i* =

*u*¯*f* �

*μ* � *xj*, *k*� �

*<sup>γ</sup>*˜0*A*<sup>0</sup> (|**x2**|) *<sup>G</sup>* (*x*2*x*<sup>1</sup> <sup>|</sup>*<sup>A</sup>* ) *<sup>A</sup>*ˆ� (*x*1, *<sup>k</sup>*�

) +

�� � � **x***j* � � � � is

*<sup>ψ</sup><sup>i</sup>* (*x*<sup>1</sup> <sup>|</sup>*<sup>A</sup>* ). (6)

is the four-potential of a spontaneously

, *j* = 1, 2. (8)

*Sl*, (9)

*ui*. (10)

Λ−*<sup>r</sup>* (*ξ*)

(12)

, (7)

) are the polarization four-vector and the four-momentum of a

The relativistic system of units, where ¯*h* = *c* = 1 and the standard metric for 4-vectors (*ab*) = *<sup>a</sup>*0*b*<sup>0</sup> <sup>−</sup> **ab** will be used throughout this paper.
