Lineshapes in two-color polarization spectroscopy for cesium

Heung-Ryoul Noh

doi:10.1364/OE.20.021784

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 D₁ and D₂ transitions of Rb was reported in [15]. Akulshin et al. reported experimental results on PS for the transition 5S_1/2-5P_3/2-5D_5/2 of ⁸⁷Rb atoms [16]. Carr et al. reported on PS for the transition 6S_1/2-6P_3/2-7S_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 5S_1/2-5P_3/2-5D_5/2 of ⁸⁵Rb 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 6S_1/2-6P_3/2-7S_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 6S_1/2–6P_3/2 (lower line) and the pump beam is scanned around 6P_3/2–7S_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 6P_3/2–7S_1/2, whereas the probe beam is tuned and detected. In contrast, in Scheme B, the probe beam is locked at the resonant line 6S_1/2–6P_3/2 and detected, whereas the pump beam is scanned around 6P_3/2–7S_1/2. Therefore, we can obtain a dispersive signal for frequency locking to the upper transition line 6P_3/2–7S_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 6S_1/2(F = 4)–6P_3/2(F′ = 5) and 6P_3/2(F′ = 5)–7S_1/2(F″ = 4), whose corresponding wavelengths are λ₁ = 852 nm and λ₂ = 1.47 μm, respectively. For use later, we define the following values: the resonant angular frequency (ω_i₀ = 2πc/λ_i), angular frequency (ω_i), Rabi frequency (Ω_i), and wave vector (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.

Fig. 1 (a) Two schematic diagrams and (b) an energy level diagram for the PS of the transition 6S_1/2-6P_3/2-7S_1/2 for cesium.

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The electric fields of the coupling (E₁) and probe (E₂) beams in Scheme A are given by

E_{1} = E_{10} {\hat{ɛ}}_{+} e^{- i ω_{1} t}, E_{2} = E_{20} [a_{+} {\hat{ɛ}}_{+} + a_{-} {\hat{ɛ}}_{-}] e^{- i ω_{2} t},

where

a_{\pm} = \mp e^{\mp i θ} / \sqrt{2}

, θ is π/4 in our scheme, and E_i₀ is the amplitude of the laser beams (i = 1, 2). The susceptibility of the σ^± component of the probe beam is then given by

χ_{\pm} = - N_{at} \frac{3 λ_{2}^{3}}{4 π^{2}} \frac{Γ_{2}}{a_{\pm} Ω_{2}} \sum_{m = - 5}^{5} {R_{2}}_{F^{'} = 5, m}^{F^{″} = 4, m \pm 1} σ_{F^{'} = 5, m}^{F^{″} = 4 \pm 1},

where 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

σ_{F_{g}, m_{g}}^{F_{e}, m_{e}}

is the slowly varying density matrix elements between |F_e, m_e〉 and |F_g, m_g〉.

The rotation angle is given by $η_{0} = (k_{2} L / 4) (χ_{-}^{r} - χ_{+}^{r})$ , where L is the length of the cell and $χ_{\pm}^{r}$ 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 $n_{\pm} ≃ 1 + (1 / 2) χ_{\pm}^{r}$ , the rotation angle can be written as η₀ = (k₂L/2)(n₋ − n₊) in terms of the refractive indices. Using Eq. (2), the rotation angle is explicitly given by

η_{0} (v, t) = N_{at} \frac{3 λ_{2}^{2}}{8 π} \frac{Γ_{2}}{Ω_{2}} L \sum_{m = - 5}^{5} Re (\frac{1}{a_{+}} {R_{2}}_{F^{'} = 5, m}^{F^{″} = 4, m + 1} σ_{F^{'} = 5, m}^{F^{″} = 4, m + 1} - \frac{1}{a_{-}} {R_{2}}_{F^{'} = 5, m}^{F^{″} = 4, m - 1} σ_{F^{'} = 5, m}^{F^{″} = 4, m - 1})

The dependence of the rotation angle, in Eq. (3), on the atomic velocity, 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:

η = \frac{1}{t_{av}} \int_{0}^{t_{av}} d t \int_{0}^{\infty} d v \frac{1}{π^{1 / 2} u} e^{- {(v / u)}^{2}} η_{0} (v, t),

where

t_{av} (= \sqrt{π} d / (2 u))

is the average transit time for traversing a laser beam with a diameter of d while u is the most probable velocity [20]. In a method analogous to obtaining the rotation angle, the absorption coefficient is given by

α_{0} (v, t) = - N_{at} \frac{3 λ_{2}^{2}}{2 π} \frac{Γ_{2}}{Ω_{2}} \sum_{q = \pm 1} \sum_{m = - 5}^{5} Im (a_{q}^{*} {R_{2}}_{F^{'} = 5, m}^{F^{″} = 4, m + q} σ_{F^{'} = 5, m}^{F^{″} = 4, m + q}),

this too should be averaged as shown in Eq. (4).

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: ω₁ − ω₁₀ = δ₁ − k₁v and ω₂ − ω₂₀ = δ₂ + k₂v. Thus δ₁ is the frequency ω₁ with respect to the transition line 6S_1/2(F = 4)–6P_3/2(F′ = 5), whereas δ₂ is the frequency ω₂ relative to the transition 6P_3/2(F′ = 5)–7S_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 Ω₂ = 2π × 1.1 MHz, whereas those of the coupling beam (Ω₁) are 2π × 2.8 MHz (black curve) and 2π × 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 Ω₂ = 2π × 2.8 MHz [Fig. 2(b)] is approximately 2π × 6.5 MHz, which is smaller than Γ₁ + Γ₂ = 2π × (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 Ω₁ 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.

Fig. 2 (a) Calculated rotation angles and (b) absorption coefficients of Scheme A for two different coupling beam intensities.

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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 Ω₂ = 2π × 1.1 MHz, whereas those of the coupling beam (Ω₁) are varied from 2π × 1.4 MHz to 2π × 89 MHz. In Fig. 3(b), the Rabi frequency of the probe beam is Ω₁ = 2π × 1.4 MHz, whereas those of the coupling beam (Ω₂) are changed from 2π × 2.5 MHz to 2π × 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 Ω₁. In Fig. 3(b), the single dispersive signal becomes distorted as Ω₂ 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.

Fig. 3 The calculated rotation angles of (a) Scheme A and (b) Scheme B for various values of the coupling beam intensities. The traces are displaced for clear view.

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In order to reveal the difference between Schemes A and B, we calculated the rotation angle and absorption coefficients by scanning δ₁ while δ₂ 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 δ₁ (blue curves) and δ₂ scanning (red curves), respectively. The AT absorption resonance for the Scheme A is given by [26]

Δ_{2} + \frac{1}{2} (Δ_{1} \pm \sqrt{Ω_{1}^{2} + Δ_{1}^{2}}) = 0,

where Δ₁ = δ₁ − k₁v and Δ₂ = δ₂ + k₂v. The two-photon resonance condition is given by Δ₁ + Δ₂ = 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 δ₂ scanning, the velocities and corresponding detunings responsible for maximum absorption coefficient are given by

v = \pm \frac{| k_{1} - 2 k_{2} | Ω_{1}}{2 k_{1} \sqrt{k_{2} (k_{1} - k_{2})}}, δ_{2} = \frac{\pm k_{1}^{2} + s {(k_{1} - 2 k_{2})}^{2}}{4 k_{1} \sqrt{k_{2} (k_{1} - k_{2})}},

respectively, where s is the sign of k₁ − 2k₂. In Eq. (7), we can see that clear AT splitting is possible only when λ₂ > λ₁. When λ₂ ≃ 2λ₁ (in our case, λ₂ = 1.72λ₁), the resonance detunings are given by

δ_{2} ≃ \pm \frac{Ω_{1}}{2} .

Thus, the separation between the two AT splitted frequencies is approximately Ω₁ in the case of δ₂ scanning. In the case of δ₁ scanning, the velocities and detunings for resonance condition are given by

v = \pm \frac{Ω_{1}}{2 \sqrt{k_{2} (k_{1} - k_{2})}}, δ_{1} = \pm \frac{\sqrt{k_{2} (k_{1} - k_{2})}}{k_{2}} Ω_{1} = \pm 0.85 Ω_{1},

which can be easily validated by looking at Fig. 4(a). We can see that the separation between the AT splitted frequencies is not Ω₁, but 1.7 Ω₁ in the case of δ₁ scanning.

Fig. 4 (i) The calculated rotation angles, (ii) absorption coefficients, (iii) velocity-detuning map for δ₁ scanning, and (iv) δ₂ scanning for (a) Scheme A and (b) Scheme B.

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

Δ_{1} + \frac{1}{2} (Δ_{2} \pm \sqrt{κ^{2} Ω_{2}^{2} + Δ_{2}^{2}}) = 0,

where κΩ₂ ≃ Ω₂/3 is the effective Rabi frequency for the transition 6P_3/2(F′ = 5)–7S_1/2(F″ = 4). In the case of the scheme A, because the atoms are optically pumped to the state 6P_3/2(F″ = 5, m″ = 5), the effective Rabi frequency is almost equal to Ω₁. Therefore, it is legitimate to use Eq. (6). In contrast, the effective Rabi frequency (κΩ₂) should be considered for the transition 6P_3/2(F′ = 5)–7S_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 (λ₁) of the probe line compared to the wavelength (λ₂) 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 λ₂ < λ₁ is satisfied, it would be possible to observe AT energy splitting as in Fig. 4(a). If λ₂ ≥ λ₁, 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 6S_1/2-6P_3/2-7S_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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Lineshapes in two-color polarization spectroscopy for cesium

Abstract

1. Introduction

2. Theory

3. Calculated results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express