## Abstract

We present a theoretical study of lineshape in polarization spectroscopy for the 6*S*_{1/2}-6*P*_{3/2}-7*S*_{1/2} transition line in cesium atoms. A circularly polarized pump beam is tuned either to the lower or the upper transition line, while a linearly polarized probe beam is tuned to the other transition line. The polarization rotation of the probe beam is accurately calculated using a semi-classical density-matrix formalism taking into account all relaxation processes.

© 2012 OSA

## 1. Introduction

Polarization spectroscopy (PS) [1] is one of the simplest schemes for sub-Doppler laser spectroscopy [2]. In PS, a circularly polarized pump beam generates optical anisotropy in a cell, this is then detected by a counterpropagating linearly polarized probe beam. Owing to the fact that the atoms belonging to a specific velocity class are able to experience the pump and probe beams simultaneously, sub-Doppler resolution in the polarization rotation signal can be obtained. This observed dispersive signal is usually used in laser frequency locking. There have been many papers on the theoretical and experimental aspects of PS [3–10]. As well as PS, several spectroscopic schemes have been used for laser frequency locking, such as saturated absorption spectroscopy (SAS) [2], dichroic atomic vapor laser lock (DAVLL) [11], sub-Doppler DAVLL [12], and modulation transfer spectroscopy (MTS) [13, 14].

In PS the pump and probe beams are usually derived from a single laser and tuned to the same transition line [1]. Recently there have been several reports for PS where the two beams are different in wavelength and are tuned to different transition lines. PS where the pump and probe beams were respectively tuned to the *D*_{1} and *D*_{2} transitions of Rb was reported in [15]. Akulshin *et al.* reported experimental results on PS for the transition 5*S*_{1/2}-5*P*_{3/2}-5*D*_{5/2} of ^{87}Rb atoms [16]. Carr *et al.* reported on PS for the transition 6*S*_{1/2}-6*P*_{3/2}-7*S*_{1/2} of Cs [17]. Very recently, Kulatunga *et al.* studied the dependence of a two-color PS signal on the pump beam detuning for the transition 5*S*_{1/2}-5*P*_{3/2}-5*D*_{5/2} of ^{85}Rb atoms [18]. In those reports, simple arguments using a two-level model system were presented, and no detailed study of lineshapes in two-color PS were included. Accurate theoretical studies on real atoms have been reported recently from the perspective of ladder-type electromagnetically induced transparency (EIT) [19] and double resonance optical pumping [20]. In this paper, we present an accurate theoretical calculation of two-color PS for the 6*S*_{1/2}-6*P*_{3/2}-7*S*_{1/2} transition line of cesium atoms, which was experimentally studied by Carr *et al.* [17]. In addition to PS spectra, we obtain transmission spectra, which exhibit sub-natural linewidths as recently discussed by Tanasittikosol *et al.* [21]. This paper is structured as follows: In Sec. 2, we present the theory for calculating the spectra. The calculated results are presented in Sec. 3. The final section briefly summaries the results.

## 2. Theory

The schematic diagrams and corresponding energy level diagrams are shown in Fig. 1(a). Here, we consider two different schemes: (i) Scheme A: The probe beam is tuned to 6*S*_{1/2}–6*P*_{3/2} (lower line) and the pump beam is scanned around 6*P*_{3/2}–7*S*_{1/2} (upper line) and (ii) Scheme B: *vice versa* as shown in Fig. 1. In scheme A, the pump beam is locked at the resonant transition line 6*P*_{3/2}–7*S*_{1/2}, whereas the probe beam is tuned and detected. In contrast, in Scheme B, the probe beam is locked at the resonant line 6*S*_{1/2}–6*P*_{3/2} and detected, whereas the pump beam is scanned around 6*P*_{3/2}–7*S*_{1/2}. Therefore, we can obtain a dispersive signal for frequency locking to the upper transition line 6*P*_{3/2}–7*S*_{1/2}. We explain the details of our calculation using Scheme A. Its application to Scheme B is straightforward. The energy level diagram under consideration is shown in Fig. 1(b). The lasers are tuned to 6*S*_{1/2}(*F* = 4)–6*P*_{3/2}(*F*′ = 5) and 6*P*_{3/2}(*F*′ = 5)–7*S*_{1/2}(*F*″ = 4), whose corresponding wavelengths are *λ*_{1} = 852 nm and *λ*_{2} = 1.47 *μ*m, respectively. For use later, we define the following values: the resonant angular frequency (*ω _{i}*

_{0}= 2

*πc*/

*λ*), angular frequency (

_{i}*ω*), Rabi frequency (Ω

_{i}*), and wave vector (*

_{i}*k*=

_{i}*ω*/

_{i}*c*) of the laser (

*i*= 1, 2) with

*c*being the speed of light. The dynamics of the populations and coherences between the magnetic sublevels belonging to these three hyperfine levels are described by the density matrix formalism, whereas all other relaxation phenomena are described by rate equations.

The electric fields of the coupling (*E*_{1}) and probe (*E*_{2}) beams in Scheme A are given by

*θ*is

*π*/4 in our scheme, and

*E*

_{i}_{0}is the amplitude of the laser beams (

*i*= 1, 2). The susceptibility of the

*σ*

^{±}component of the probe beam is then given by

*N*

_{at}is the atomic vapor density, ${{R}_{2}}_{{F}_{g},{m}_{g}}^{{F}_{e},{m}_{e}}$ is the normalized transition strength, which is presented in Ref. [20] and ${\sigma}_{{F}_{g},{m}_{g}}^{{F}_{e},{m}_{e}}$ is the slowly varying density matrix elements between |

*F*,

_{e}*m*〉 and |

_{e}*F*,

_{g}*m*〉.

_{g}The rotation angle is given by
${\eta}_{0}=\left({k}_{2}L/4\right)\left({\chi}_{-}^{r}-{\chi}_{+}^{r}\right)$, where *L* is the length of the cell and
${\chi}_{\pm}^{r}$ is the real part of the susceptibility for the *σ*^{±} component of the probe beam [22]. Because the refractive indices for a dilute vapor cell are given by
${n}_{\pm}\simeq 1+\left(1/2\right){\chi}_{\pm}^{r}$, the rotation angle can be written as *η*_{0} = (*k*_{2}*L*/2)(*n*_{−} − *n*_{+}) in terms of the refractive indices. Using Eq. (2), the rotation angle is explicitly given by

*v*, results from the Doppler shift as given below. Therefore, the rotation angle must be averaged over the Maxwell-Boltzmann velocity distribution, and various transit times as follows:

*d*while

*u*is the most probable velocity [20]. In a method analogous to obtaining the rotation angle, the absorption coefficient is given by

The density matrix elements in Eqs. (3) and (5) are obtained by numerically solving the density matrix equation. A detailed method of calculation of the density matrix elements was presented in Ref. [20]. The dependence of Eq. (3) on the velocity results from the relation: *ω*_{1} − *ω*_{10} = *δ*_{1} − *k*_{1}*v* and *ω*_{2} − *ω*_{20} = *δ*_{2} + *k*_{2}*v*. Thus *δ*_{1} is the frequency *ω*_{1} with respect to the transition line 6*S*_{1/2}(*F* = 4)–6*P*_{3/2}(*F*′ = 5), whereas *δ*_{2} is the frequency *ω*_{2} relative to the transition 6*P*_{3/2}(*F*′ = 5)–7*S*_{1/2}(*F*″ = 4).

## 3. Calculated results

The typical calculated results for Scheme A are shown in Fig. 2. In all the calculated results, the laser beam diameter was *d* = 2 mm. The rotation angles per unit length and the absorption coefficients are shown in Figs. 2(a) and 2(b), respectively. In Fig. 2, the Rabi frequency of the probe beam is Ω_{2} = 2*π* × 1.1 MHz, whereas those of the coupling beam (Ω_{1}) are 2*π* × 2.8 MHz (black curve) and 2*π* × 8.9 MHz (red curve). In Fig. 2(a) [2(b)], we can observe a dispersive [absorption] signal for a low coupling beam intensity. In the case of a strong coupling beam intensity, we can see Autler-Townes (AT) splitting of the energy level [23], thus two dispersive and absorptive signals are obtained. In particular, the full-width at half maximum (FWHM) value of the absorption signal for Ω_{2} = 2*π* × 2.8 MHz [Fig. 2(b)] is approximately 2*π* × 6.5 MHz, which is smaller than Γ_{1} + Γ_{2} = 2*π* × (5.234 + 3.3) MHz seen in [24, 25]. Thus we can observe sub-natural linewidths, as experimentally observed and theoretically explained in Ref. [21]. We note that the separation between the two peaks in Fig. 2(b) is approximately Ω_{1} as can be expected from the separation of the energy eigenvalues of the dressed states. A more detailed explanation of the separation is presented at the end of this section.

The results of the rotation angle (PS spectra) in Schemes A and B for various Rabi frequency of the coupling beam are shown in Figs. 3(a) and 3(b), respectively. In Fig. 3(a), the Rabi frequency of the probe beam is Ω_{2} = 2*π* × 1.1 MHz, whereas those of the coupling beam (Ω_{1}) are varied from 2*π* × 1.4 MHz to 2*π* × 89 MHz. In Fig. 3(b), the Rabi frequency of the probe beam is Ω_{1} = 2*π* × 1.4 MHz, whereas those of the coupling beam (Ω_{2}) are changed from 2*π* × 2.5 MHz to 2*π* × 80 MHz. In Fig. 3(a), we can clearly see that a single dispersive signal is changed into two separated dispersive signals resulting from the AT splitting of the energy level. We can also see that the separation of the two distinct dispersive signals in Fig. 3(a) is approximately Ω_{1}. In Fig. 3(b), the single dispersive signal becomes distorted as Ω_{2} increases. However, in contrast to Fig. 3(a), the AT splitting is not quite striking. This difference results from the wavelength difference between the two schemes.

In order to reveal the difference between Schemes A and B, we calculated the rotation angle and absorption coefficients by scanning *δ*_{1} while *δ*_{2} was set to zero, and *vice versa* as shown in Fig. 4. Figures 4(a) and 4(b) show the results for Schemes A and B, respectively. In Fig. 4, (i) and (ii) present the rotation angle and absorption coefficient, respectively, while (iii) and (iv) show the map for absorption coefficients for *δ*_{1} (blue curves) and *δ*_{2} scanning (red curves), respectively. The AT absorption resonance for the Scheme A is given by [26]

_{1}=

*δ*

_{1}−

*k*

_{1}

*v*and Δ

_{2}=

*δ*

_{2}+

*k*

_{2}

*v*. The two-photon resonance condition is given by Δ

_{1}+ Δ

_{2}= 0. The resonance condition in Eq. (6) can be found in the absorption coefficients map in (iii) and (iv) of Fig. 4. In Fig. 4(a), we can clearly see that the resonance peaks result from the absorption of the AT energy splitting. In the case of

*δ*

_{2}scanning, the velocities and corresponding detunings responsible for maximum absorption coefficient are given by

*s*is the sign of

*k*

_{1}− 2

*k*

_{2}. In Eq. (7), we can see that clear AT splitting is possible only when

*λ*

_{2}>

*λ*

_{1}. When

*λ*

_{2}≃ 2

*λ*

_{1}(in our case,

*λ*

_{2}= 1.72

*λ*

_{1}), the resonance detunings are given by Thus, the separation between the two AT splitted frequencies is approximately Ω

_{1}in the case of

*δ*

_{2}scanning. In the case of

*δ*

_{1}scanning, the velocities and detunings for resonance condition are given by

_{1}, but 1.7 Ω

_{1}in the case of

*δ*

_{1}scanning.

The AT absorption resonance condition for the Scheme B is given by [26]

where *κ*Ω_{2} ≃ Ω_{2}/3 is the effective Rabi frequency for the transition 6*P*_{3/2}(*F*′ = 5)–7*S*_{1/2}(*F*″ = 4). In the case of the scheme A, because the atoms are optically pumped to the state 6*P*_{3/2}(*F*″ = 5, *m*″ = 5), the effective Rabi frequency is almost equal to Ω_{1}. Therefore, it is legitimate to use Eq. (6). In contrast, the effective Rabi frequency (*κ*Ω_{2}) should be considered for the transition 6*P*_{3/2}(*F*′ = 5)–7*S*_{1/2}(*F*″ = 4). The resonance condition in Eq. (9) can be found in the absorption coefficients map in Fig. 4(b)(iii) and (iv). In contrast to the results in Fig. 4(a), it is not possible to observe well-separated AT energy splitting in Fig. 4(b). The reason for this is the short wavelength (*λ*_{1}) of the probe line compared to the wavelength (*λ*_{2}) of the coupling line. The results in Figs. 4(b)(i) and 4(b)(ii) can be easily explained by the results for map in Figs. 4(b)(iii) and 4(b)(iv). If the condition *λ*_{2} < *λ*_{1} is satisfied, it would be possible to observe AT energy splitting as in Fig. 4(a). If *λ*_{2} ≥ *λ*_{1}, we would observe a sharp EIT spectrum [27, 28].

## 4. Conclusions

In this paper we presented a theoretical study of two-color polarization spectroscopy for the transition 6*S*_{1/2}-6*P*_{3/2}-7*S*_{1/2} of Cs. We considered two different schemes where either the upper (Scheme A) or lower (Scheme B) transition line is used as a probe line. Since the accurate time-dependent density matrix equations are solved, and averaged over the velocity distribution, and no phenomenological constant is included, we can obtain accurate PS spectra in various conditions. From the calculations, we can see scheme A exhibits a larger rotation angle in typical experimental conditions. As the method of calculation is general, its application to other atomic species and other energy levels is straightforward. The calculation and experimental studies for other transition lines are currently under progress.

## Acknowledgments

This study was financially supported by Chonnam National University, 2011.

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