**2. Description of saturated absorption spectroscopy**

Saturated absorption spectroscopy is a simple technique to measure the narrow-line atomic spectral feature limited only by the natural linewidth, that is typically 6 MHz or less (Milonni, 2010). A strong laser beam called the pump beam is directed through an optical cell that contains a vapour as shown in Fig.1. A small part of the pump beam used as a probe beam is sent through the cell in the counter-propagating direction and detected by a simple photodiode.

Fig. 1. Basic setup for saturation absorption spectroscopy.

Cold Atoms Experiments: Influence of Laser Intensity Imbalance on Cloud Formation 159

In the case of a three level system with two closely spaced upper levels and one ground level the spectral features presents two ordinary Lamb dips at the resonance frequencies of the associated transitions and one cross over peak situated just between these two dips at the average of these frequencies as shown in Fig.4. When the laser is tuned at the cross over frequency it is absorbed by one transition from atoms moving toward the laser and by the other transition of the same atom by the laser beam oriented in the opposite direction. The increase of the population of the upper level caused by the strongest laser (pump beam) produces an increase of the transmission of the probe beam at the cross over frequency.



relative frequency (GHz)

When the system has two closely spaced ground levels and one single upper level the cross over is still half between the ordinary Lamb dips but it exhibits a reduction of transmission due a process named "optical pumping" (Fig.5). Here the laser is absorbed by one optical transition from atoms moving toward the laser. The atoms decay to the second ground level producing an increase of absorption of the probe laser beam driven in the opposite

relative frequency (GHz)

**2.1 Multilevel atoms** 

0.4

Fig. 4. Positive cross over.

0.4

Fig. 5. Negative cross over.

0.6

transmission (a.u.)

0.8

1

direction.

0.6

transmission (a.u.)

0.8

1

The probe beam can be disposed at a small angle or collinear with respect to the pump beam. The laser frequency is scanned close to the atomic resonance. In the case of a twolevel atom system the spectral feature looks like Fig.2. The upper feature is the detected absorption feature when the pump beam is blocked. It shows a Doppler-broadened line which is much broader than the natural linewidth. In the case of weak absorption the feature is a Gaussian profile. The atoms in the vapour move with different velocities in different directions following the Boltzmann velocity distribution. Considering the velocity component of the atoms along the probe beam we have that some atoms move with velocity component in the same direction as the probe beam propagation and other in the opposite direction. The lower feature is the detected intensity with pump laser (Fig.3). It shows a spike just at the atomic resonance frequency <sup>0</sup> . This spike is known as Lamb dip. When the laser is tuned at <sup>0</sup> , it will be absorbed only by atoms moving toward the probe laser with longitudinal velocity / <sup>0</sup> *c* . The beam will not be absorbed by atoms with different longitudinal velocities because they are not in resonance so they don't contribute to absorption. Atoms with zero velocity absorb light from the pump laser and become saturated. The probe laser moves through a saturated transparent group of atoms reducing the absorption and producing the Lamb dip.

Fig. 2. Absorption line.

Fig. 3. Doppler free saturated absorption line.

#### **2.1 Multilevel atoms**

158 Quantum Optics and Laser Experiments

The probe beam can be disposed at a small angle or collinear with respect to the pump beam. The laser frequency is scanned close to the atomic resonance. In the case of a twolevel atom system the spectral feature looks like Fig.2. The upper feature is the detected absorption feature when the pump beam is blocked. It shows a Doppler-broadened line which is much broader than the natural linewidth. In the case of weak absorption the feature is a Gaussian profile. The atoms in the vapour move with different velocities in different directions following the Boltzmann velocity distribution. Considering the velocity component of the atoms along the probe beam we have that some atoms move with velocity component in the same direction as the probe beam propagation and other in the opposite direction. The lower feature is the detected intensity with pump laser (Fig.3). It shows a

different longitudinal velocities because they are not in resonance so they don't contribute to absorption. Atoms with zero velocity absorb light from the pump laser and become saturated. The probe laser moves through a saturated transparent group of atoms reducing


relative frequency (GHz)


relative frequency (GHz)

<sup>0</sup> . This spike is known as Lamb dip. When the

. The beam will not be absorbed by atoms with

, it will be absorbed only by atoms moving toward the probe laser

spike just at the atomic resonance frequency

the absorption and producing the Lamb dip.

 *c* 

 <sup>0</sup> 

0.4

0.4

Fig. 3. Doppler free saturated absorption line.

0.6

transmission (a.u.)

0.8

1

0.6

transmission (a.u.)

Fig. 2. Absorption line.

0.8

1

with longitudinal velocity / <sup>0</sup>

laser is tuned at

In the case of a three level system with two closely spaced upper levels and one ground level the spectral features presents two ordinary Lamb dips at the resonance frequencies of the associated transitions and one cross over peak situated just between these two dips at the average of these frequencies as shown in Fig.4. When the laser is tuned at the cross over frequency it is absorbed by one transition from atoms moving toward the laser and by the other transition of the same atom by the laser beam oriented in the opposite direction. The increase of the population of the upper level caused by the strongest laser (pump beam) produces an increase of the transmission of the probe beam at the cross over frequency.

Fig. 4. Positive cross over.

When the system has two closely spaced ground levels and one single upper level the cross over is still half between the ordinary Lamb dips but it exhibits a reduction of transmission due a process named "optical pumping" (Fig.5). Here the laser is absorbed by one optical transition from atoms moving toward the laser. The atoms decay to the second ground level producing an increase of absorption of the probe laser beam driven in the opposite direction.

Fig. 5. Negative cross over.

Cold Atoms Experiments: Influence of Laser Intensity Imbalance on Cloud Formation 161

states when the system is illuminated by the strong pump laser it is neccesary to write the

1 2 12 1 21 2 <sup>1</sup> ( ) *P P IS B P B P <sup>p</sup> <sup>c</sup>*

> 3 21 3 <sup>0</sup> 8

the absorption coeficient, 1 *g* and 2 *g* are the degeneracy's of the ground and excited states

2 2 /2 ( ) / 4

pump laser is in the counterpropagating direction in relation to the probe laser. In stationary

2 2 2 / 2 1 4/

intensity. To plot a profile with one single Lamb dip we used the calculated excited population from Ec.9. For example, Fig. 2 was obtained integrating numerically the

The energy level diagram (Fig. 7) contains two ground hyperfine levels separated by nearly 3 GHz and four excited levels separated by less than the Doppler broadened line. As the atoms pumped by the cooling laser from the F = 3 level into the F' = 4 level decay into the F = 2 level it is necessary to optically pump the atoms from this level back to the F = 3 level

*S*

Solving Ec. 4 for *P*2 in stationary state and assuming that 1 2 *g g* we have that

where / *sat sII* , *I* the intensity of the pump laser and 2 3 *I hc sat* 2 /

*<sup>s</sup> <sup>P</sup> s* 

(4)

*<sup>h</sup>* (5)

12 1 2 12 *B g gB* (/) (6)

(7)

12 2 *PP P* 1 2 (8)

(9)

780 nm, 0

is the saturation

1 , 300 *T* K and

( /) *c* . The minus sign is explained because the

where corresponds to the excited lifetime, *pI* is the intensity of the pump laser and

*<sup>c</sup> <sup>B</sup>* 

rate equation for a two level system as

is the stimulated emission coefficient,

the atom lineshape, with 0 0

 

transmission coefficient for rubidium atoms with 0

through the F' = 3 level. This is done by the repumping laser.

6 MHz and considering the pump laser.

**2.4 Energy level diagram** 

state we have 1 2 *P P* <sup>0</sup> and as 1 2 *P P* 1 we have

respectively, and

#### **2.2 The saturated absorption spectrometer**

The optical setup of the saturated absorption spectrometer is depicted in Fig.6. The signal obtained by the photodiode PD1 can be used as a reference for the Doppler limited spectra.

Fig. 6. Saturated absorption spectrometer. The pump beam is indicated with a broader line. The signal obtained by the photodiode PD2 contains the Doppler free feature.

#### **2.3 Semiquantitative ideas at two level atoms**

The saturated absorption spectra can be calculated with a simplified model for two level atoms. The differential contribution to the absorption coefficient by atoms with velocity between and *d*can be written as

$$d\pi(\nu,\nu) = \tau\_0 \mathbb{X}\_0 \left(P\_1 - P\_2\right) F d\mathfrak{n} \tag{1}$$

where 0 is the optical depth at the centre of the resonance, *P*1 and *P*2 are the relative populations of the ground and excited state respectively,

$$F(\nu, \nu) = \frac{\Gamma / 2\pi}{(\nu - \nu\_0 + \nu\_0 \nu / c) + \Gamma^2 / 4} \tag{2}$$

is the normalized Lorentzian absorption profile for atoms with natural linewidth including the Doppler shift and

$$d\mathfrak{m} \propto e^{-m\upsilon^2/kT} d\upsilon\tag{3}$$

the Boltzman distribution for velocities along the beam axis. The transmission of the probe beam through the cell is ( ) *e* . In the case that the pump laser is turned off and the probe laser beam intensity is low we have that few atoms will be excited and most of the atoms will remain in its ground state. In this case 2*P* 0 and 1*P* 1 . For example in the case of rubidium when 0 1 , T = 300ºK, and 6 MHz we obtained by numerical integration of Ec. 2 the profile shown in Fig.1. To obtain the relative populations of the ground and excited states when the system is illuminated by the strong pump laser it is neccesary to write the rate equation for a two level system as

$$\dot{P}\_1 = \Gamma P\_2 - \frac{1}{c} I\_p S(\nu) \left( B\_{12} P\_1 - B\_{21} P\_2 \right) \tag{4}$$

where corresponds to the excited lifetime, *pI* is the intensity of the pump laser and

$$B\_{21} = \frac{c^3}{8\pi\hbar\nu\_0^3} \Gamma$$

is the stimulated emission coefficient,

160 Quantum Optics and Laser Experiments

The optical setup of the saturated absorption spectrometer is depicted in Fig.6. The signal obtained by the photodiode PD1 can be used as a reference for the Doppler limited spectra.

Fig. 6. Saturated absorption spectrometer. The pump beam is indicated with a broader line.

The saturated absorption spectra can be calculated with a simplified model for two level atoms. The differential contribution to the absorption coefficient by atoms with velocity

> 0 0 / 2 (,) ( / ) /4

is the normalized Lorentzian absorption profile for atoms with natural linewidth

<sup>2</sup> *mv kT* / *dn e d*

the Boltzman distribution for velocities along the beam axis. The transmission of the probe

laser beam intensity is low we have that few atoms will be excited and most of the atoms will remain in its ground state. In this case 2*P* 0 and 1*P* 1 . For example in the case of

Ec. 2 the profile shown in Fig.1. To obtain the relative populations of the ground and excited

is the optical depth at the centre of the resonance, *P*1 and *P*2 are the relative

*c* 

1 , T = 300ºK, and 6 MHz we obtained by numerical integration of

2

. In the case that the pump laser is turned off and the probe

(,) 00 1 2 *P P Fdn* (1)

(2)

(3)

 

 

The signal obtained by the photodiode PD2 contains the Doppler free feature.

**2.2 The saturated absorption spectrometer** 

**2.3 Semiquantitative ideas at two level atoms** 

PD1

PD2

can be written as

populations of the ground and excited state respectively,

  *F*

 

*d*

between

where 0 

 and *d*

including the Doppler shift and

beam through the cell is ( ) *e*

rubidium when 0

$$B\_{12} = (\mathcal{g}\_1 \; / \; \mathcal{g}\_2) B\_{12} \tag{6}$$

the absorption coeficient, 1 *g* and 2 *g* are the degeneracy's of the ground and excited states respectively, and

$$S(\nu) = \frac{\Gamma \, / \, 2\pi}{\delta^2 + \Gamma^2 \, / \, 4} \tag{7}$$

the atom lineshape, with 0 0 ( /) *c* . The minus sign is explained because the pump laser is in the counterpropagating direction in relation to the probe laser. In stationary state we have 1 2 *P P* <sup>0</sup> and as 1 2 *P P* 1 we have

$$P\_1 - P\_2 = 1 - 2P\_2 \tag{8}$$

Solving Ec. 4 for *P*2 in stationary state and assuming that 1 2 *g g* we have that

$$P\_2 = \frac{s/2}{1 + s + 4\delta^2/\Gamma^2} \tag{9}$$

where / *sat sII* , *I* the intensity of the pump laser and 2 3 *I hc sat* 2 / is the saturation intensity. To plot a profile with one single Lamb dip we used the calculated excited population from Ec.9. For example, Fig. 2 was obtained integrating numerically the transmission coefficient for rubidium atoms with 0 780 nm, 0 1 , 300 *T* K and 6 MHz and considering the pump laser.

#### **2.4 Energy level diagram**

The energy level diagram (Fig. 7) contains two ground hyperfine levels separated by nearly 3 GHz and four excited levels separated by less than the Doppler broadened line. As the atoms pumped by the cooling laser from the F = 3 level into the F' = 4 level decay into the F = 2 level it is necessary to optically pump the atoms from this level back to the F = 3 level through the F' = 3 level. This is done by the repumping laser.

Cold Atoms Experiments: Influence of Laser Intensity Imbalance on Cloud Formation 163

2

66 of the 85Rb D2 line are given by

 

 

33 44 55 11

44 55 66 22

3

 

 

 

 

1

4 4

1 2

5 5

1 2

*<sup>T</sup>* /*d* is the transit time broadening with *d* the diameter of the beam and

 

25 7 9 9 12 *<sup>T</sup>*

 

*<sup>g</sup> <sup>W</sup> g*

*g g W W g g*

*g g W W g g*

 

 

velocity of the atoms along the beam diameter, ' *<sup>T</sup>*

*if*

*dI h n W*

 

 

 

 

> 74 5 9 9 12 *<sup>T</sup>*

 

is the square of the reduced matrix element, <sup>9</sup> *Ke* 8.99 10 Vm/C the Coulomb constant,

the lifetime of the excited atoms and 1 *Lf* . The optical Bloch equations for the relative

11 13 33 11 14 44 11 15 55 11

22 24 44 22 25 55 22 26 66 22

33 13 11 33 33

44 14 11 44 24 22 44 44

55 15 11 55 25 22 55 55

11 22 33 44 55 66 1

 

where 12345 6 *ggggg g* 5, 7, 3, 5, 7 and 9 , the levels labeled with *N* = 1 and 2 corresponds to the ground states and the levels labeled with *N* = 3 to 6 are the excited states,

 33 44 55 66 is the total population of the various excited states. In stationary state the time derivatives of the relative popuations become zero. The absorption of the laser

' *if if ff ii if e*

The angle brackets indicate the average over the velocity distribution for vapor at

 

*g gg W WW*

  *g gg W WW*

*D*

populations

 

 *<sup>e</sup>* 

temperature *T* , given by

11 to 

 

light in a vapor with density *n* and length *dx*

3

*c L*

 *K* 

3 1 3 (2 1) 4

*f e*

3 45

( )

 

 

( ) *<sup>T</sup>*

( ) *<sup>T</sup>*

(17)

 , / 2 *<sup>T</sup>* 

the average

and

 

 

1 11

 

4 56

2 22

 

 

  ( ) *<sup>T</sup>*

 

 

> 

 *dx h n dx* (18)

(16)

(15)

(14)

 

*g gg*

 

*g gg*

(11)

(12)

(13)

Fig. 7. Energy level diagram. The transitions for cooling and optical repumping are indicated with arrows.

#### **3. Detailed saturated absorption using density matrix elements**

The transition rate is given by

$$\mathcal{W}\_{jk} = \frac{\frac{1}{2} \cdot \left(\frac{8\pi K\_{\varepsilon}}{c\hbar^2} \|D\|^2\right) \cdot \Gamma}{\left((o\rho - o\_0) - (o\rho\_j - o\_k) - k\nu\right)^2 + \Gamma^2} \cdot \mathcal{N}\_{jk} \cdot I \tag{10}$$

with 13 *N* 1/9 , 14 *N* 7 /81 , 15 *N* 4 /81 , 24 *N* 2 /81 , 25 *N* 5 /81 and 26 *N* 1/9 , *I*  the laser intensity,

162 Quantum Optics and Laser Experiments

100.2 MHz

*F'* = 4, *N* = 6

*F'* = 1, *N* = 3

*F'* = 2, *N* = 4

*F'* = 3, *N* = 5

20.4 MHz

63.4 MHz

29.4 MHz

cooling laser

Fig. 7. Energy level diagram. The transitions for cooling and optical repumping are

 

( )( )

2

*e jk jk j k*

 

*<sup>K</sup> <sup>D</sup> <sup>c</sup> <sup>W</sup> N I*

with 13 *N* 1/9 , 14 *N* 7 /81 , 15 *N* 4 /81 , 24 *N* 2 /81 , 25 *N* 5 /81 and 26 *N* 1/9 , *I* 

2

*k*

 

1.77 GHz

1.26 GHz

<sup>2</sup> <sup>2</sup>

(10)

*F* = 2, *N* = 1

*F* = 3, *N* = 2

**3. Detailed saturated absorption using density matrix elements** 

0

1 8 2

indicated with arrows.

the laser intensity,

The transition rate is given by

2 1/2 5 S

2 3/2 5 P

85Rb, D2

780.241 nm

repumping laser

$$\left\|\left\|\boldsymbol{D}\right\|\right\|^{2} = \frac{1}{\tau} \cdot \frac{3\hbar c^{3} \left(2L\_{f} + 1\right)}{4\alpha^{3} K\_{e}} \tag{11}$$

is the square of the reduced matrix element, <sup>9</sup> *Ke* 8.99 10 Vm/C the Coulomb constant, the lifetime of the excited atoms and 1 *Lf* . The optical Bloch equations for the relative populations 11 to 66 of the 85Rb D2 line are given by

$$\begin{split} \dot{\rho}\_{11} &= \mathcal{W}\_{13} \Big( \rho\_{33} - \frac{\mathcal{g}\_3}{\mathcal{g}\_1} \cdot \rho\_{11} \Big) + \mathcal{W}\_{14} \Big( \rho\_{44} - \frac{\mathcal{g}\_4}{\mathcal{g}\_1} \rho\_{11} \Big) + \mathcal{W}\_{15} (\rho\_{55} - \frac{\mathcal{g}\_5}{\mathcal{g}\_1} \rho\_{11}) \\ &+ \gamma \Big( \rho\_{33} + \frac{7}{9} \rho\_{44} + \frac{4}{9} \rho\_{55} \Big) + \gamma\_T \left( \frac{5}{12} - \rho\_{11} \right) \end{split} \tag{12}$$

$$\begin{split} \dot{\rho}\_{22} &= W\_{24} \Big( \rho\_{44} - \frac{\mathcal{g}\_4}{\mathcal{g}\_2} \cdot \rho\_{22} \Big) + W\_{25} \Big( \rho\_{55} - \frac{\mathcal{g}\_5}{\mathcal{g}\_2} \rho\_{22} \Big) + W\_{26} \Big( \rho\_{66} - \frac{\mathcal{g}\_6}{\mathcal{g}\_2} \rho\_{22} \Big) \\ &+ \gamma \Big( \frac{2}{9} \rho\_{44} + \frac{5}{9} \rho\_{55} + \rho\_{66} \Big) + \gamma\_T \Big( \frac{7}{12} - \rho\_{22} \Big) \end{split} \tag{13}$$

$$
\dot{\rho}\_{33} = \mathcal{W}\_{13} \left( \frac{\mathcal{g}\_3}{\mathcal{g}\_1} \rho\_{11} - \rho\_{33} \right) - (\mathcal{\gamma} + \mathcal{\gamma}\_T) \rho\_{33} \tag{14}
$$

$$
\dot{\rho}\_{44} = \mathcal{W}\_{14} \left( \frac{\mathcal{g}\_4}{\mathcal{g}\_1} \rho\_{11} - \rho\_{44} \right) + \mathcal{W}\_{24} \left( \frac{\mathcal{g}\_4}{\mathcal{g}\_2} \rho\_{22} - \rho\_{44} \right) - (\mathcal{y} + \mathcal{y}\_T)\rho\_{44} \tag{15}
$$

$$
\dot{\rho}\_{55} = \mathcal{W}\_{15} \left( \frac{g\_5}{g\_1} \rho\_{11} - \rho\_{55} \right) + \mathcal{W}\_{25} \left( \frac{g\_5}{g\_2} \rho\_{22} - \rho\_{55} \right) - (\chi + \chi\_T) \rho\_{55} \tag{16}
$$

$$1 = \rho\_{11} + \rho\_{22} + \rho\_{33} + \rho\_{44} + \rho\_{55} + \rho\_{66} \tag{17}$$

where 12345 6 *ggggg g* 5, 7, 3, 5, 7 and 9 , the levels labeled with *N* = 1 and 2 corresponds to the ground states and the levels labeled with *N* = 3 to 6 are the excited states, *<sup>T</sup>* /*d* is the transit time broadening with *d* the diameter of the beam and the average velocity of the atoms along the beam diameter, ' *<sup>T</sup>* , / 2 *<sup>T</sup>* and *<sup>e</sup>* 33 44 55 66 is the total population of the various excited states. In stationary state the time derivatives of the relative popuations become zero. The absorption of the laser light in a vapor with density *n* and length *dx*

$$\text{rad}\,I = \text{h}\,\nu\_{\text{if}}\,\text{tr}\left\langle \sum\_{\text{if}} \text{V}\mathbb{V}\_{\text{if}}\left(\rho\_{\text{ff}} - \rho\_{\text{ii}}\right) \right\rangle \text{d}\,\text{x} = -\text{h}\,\nu\_{\text{if}}\,\text{tr}\,\gamma'\{\rho\_e\} \,\text{d}\,\text{x} \tag{18}$$

The angle brackets indicate the average over the velocity distribution for vapor at temperature *T* , given by

Cold Atoms Experiments: Influence of Laser Intensity Imbalance on Cloud Formation 165

The scanning confocal Fabry-Perot interferometer (Fig.9) is a nice tool to check if the laser is running in single mode operation specially. One of the main features of the Fabry-Perot interferometer is that it can measure with high resolution the spectral content of the laser. A basic Fabry-Perot consists of two identical spherical mirrors with radius *R* separated by a distance *L*. The use of two curved mirrors is convenient as they permit a good match to the

Two parameters defines the properties of a Fabry-Perot, the free spectral range and the

*L* = *R*

4 *<sup>c</sup> FSR*

where *n* is the index of refraction of the air between the mirrors, *c* the speed of light, *L* the distance between mirrors. Near the centre of the mirrors we have that every time the

spectrum will be reproduced. The mirrors used in our interferometer 3 have a radius of 75

\* <sup>1</sup> *FSR R <sup>F</sup>*

the mirrors. The *finesse* depends on the mirror reflectivity, the losses due to imperfections on the mirror surfaces or dust, and the alignment of the mirrors. In our interferometer the highest *finesse* reported was larger than *F*\* = 450. A cylindrical piezoelectric transducer (PZT) is attached to one mirror and can move it in small displacements. To displace the mirror a high voltage is applied between the inner and the outer side of the PZT. The interferometer can be used in scanning mode when the laser wavelength is fixed and the piezo transducer is displaced continuously with a ramp function. In this case it is possible to observe the detailed spectra of the laser. Another option is to scan the laser wavelength with a ramp function and the distance between mirrors remains constant. In this case one can observe the laser spectra and change its absolute position in the oscilloscope by applying a

*R* 

is the full with at half maximum of the interference maxima and *R* reflectivity of

*nL* (21)

the same part of the

(22)

**4.2 Scanning confocal interferometer** 

Gaussian beam coming from the laser.

where

3 Toptica Photonics, Model FPI100

Fig. 9. Confocal scanning Fabry-Perot interferometer.

*finesse* or resolution. The free spectral range (FSR) is defined by

distance between mirrors is changed by a quarter wavelength ( /4)

mm and a *FSR* = 1GHz. The resolution of the interferometer is given by its *finesse* 

$$F(\nu) = \left(\frac{m}{2\,\pi\,k\_BT}\right)^{1/2} \exp\left(-\frac{m\nu^2}{2\,k\_BT}\right) \tag{19}$$

Extending the absorption equations to te Doppler-free saturation spectroscopy we have

$$dI = \hbar \left. \nu\_{if} \, n \left\langle \sum\_{\vec{\mathcal{Y}}} \mathcal{W}\_{\vec{\mathcal{Y}}} \, ^+ \left( \boldsymbol{\rho}\_{\vec{\mathcal{Y}}} - \boldsymbol{\rho}\_{\vec{\mathcal{U}}} \, ^- \right) \right\rangle d\mathbf{x} \tag{20}$$

where the population depends on the transition rate *W WI <sup>d</sup>* , determined by the probe beam with intensity *dI* and the transition probability *W WI <sup>p</sup>* , due to the pump beam with intensity *pI* propagating in the opposite direction, and *ii iiW W* .

## **4. Experimental details**

The experiment was installed in a 6x12 feet optical top 1 that was passively damped. The experiment included two tuneable diode lasers, two saturated absorption spectrometers, two scanning interferometers, a complete vacuum system, beam expanders, polarizing optics, infrared camera, optics and mechanics components, a rubidium cells, and photodiodes.

#### **4.1 The saturated absorption spectrometer**

The saturated absorption spectrometer is shown in Fig.8. The laser beam was lifted 15 cm above the optical top level by the mirrors M1 and M2, and directed to the first optical glass beam divider. A small part of the beam was directed to the second optical glass divider, the strongest beam went to the trap. The second beam divider drives the strongest beam to the interferometer and the small beam act as a pump laser in the rubidium cell. The beam reflected off the mirror M3 acts as a test weak beam that was measured by a photodiode 2.

Fig. 8. Saturated absorption spectrometer. Pump and probe laser are collinear. PD = photodiode, OGD = optical glass divider, M1, M2, M3 = mirrors. Distance between closest optical components are given in inches.

<sup>1</sup> Thorlabs, Model PTR12114-PTH503

<sup>2</sup> Thorlabs, Model DET10
