## Abstract

A new source of two diode laser beams, spatially separated but optically phase-locked with each other, is used to study the modulation transfer spectroscopy of coherent population trapping resonance (CPT). The spectrum for the ^{87}Rb D2 line is obtained with narrow linewidth and high signal-to-noise ratio, and analyzed with different experimental parameters. A theoretical analysis of the CPT modulation transfer spectra is deduced from the density matrix equation of motion, and found to be in good agreement with the experimental results.

© 2009 Optical Society of America

## 1. Introduction

Atomic clocks based on coherent population trapping (CPT) effect have been of great concern for several years [1, 2]. Since this is a promising scheme for micro atomic clocks, people’s interests are focused on the compactness and low power consumption. And due to all the efforts, the first commercial CPT atomic clock (SA3Xm, Symmetricom Inc.) came out recently with the size as small as 51mm×51mm×18mm, while providing a stability of 3×10^{-11} and 8×10^{-12} for an averaging time of 1s and 100s respectively. On the other hand, some people are drawing their attention to the best performance now, with secondary concerns about the size and power consumption of the clock [3].

The remarkable advantages of conventional MTS come from the four-wave mixing (FWM) process [4]. Due to the nonlinear process of FWM, the detected signal is insensitive to the linear absorption of the background. Hence the MTS technique can readily generate dispersion-like lineshapes which sit on a flat, zero background and is especially suitable for detection of weak transitions and could obtain Doppler-free and high signal-to-noise-ratio (SNR) spectra [5].

In this paper, we deduce a simple line-shape formula for the modulation transfer spectra of the Λ-type three-level atomic system from the matrix equation of motion, in the case that the frequency difference between two ground states is much higher than Doppler linewidth of the upper energy level, and verify the correctness of the formula experimentally. To observe the MTS signal in Λ configuration of the three-level atomic system, two spatially separated and optically phase-locked laser beams are necessary. We have constructed an optical phase-locking system to generate these coherent Raman beams with one of them modulated at angular frequency *δ* [6]. Using this source of coherent beams, we obtained the MTS signal at different modulation frequencies and different demodulating phases. We calculated the lineshape formula for Λ configuration of the three-level ^{87}Rb D2 atomic system, and find that the experimental results are in good agreement with the theoretical predictions. We also give theoretical simulations which, by changing the modulation frequency and modulation index, gain the largest signal gradient to optimize the parameters which will be used for microwave frequency locking in the atomic clock.

## 2. Theoretical analysis

Consider a Λ-configuration of the three-level quantum systems showed in Fig. 1. Two coherent Raman beams with angular frequency of *ω* and *ω* + Ω, named probe and pump, are resonant with energy levels ∣*c*〈 ↔ ∣*b*〉 and ∣*a*〉 ↔ ∣*b*〉, respectively. The pump beam is frequency modulated with a sinusoidal wave at frequency *δ*, leaving the sidebands at frequencies *ω* + Ω + *2δ*, where *n* is the order of the sidebands. The modulated pump and the co-propagating unmodulated probe beams are aligned collinearly through a vapor cell. If the interaction of the pump and probe beams with the atomic vapor are sufficiently nonlinear, the modulation sidebands appear on the unmodulated probe beam. The optical heterodyne beating between the weak new sidebands and the probe beam can be demodulated by the original modulation signal at frequency *δ*.

The fields of the frequency-modulated pump beam are represented in terms of the carrier frequency *ω* + Ω and the sidebands separated by the modulation frequency *δ*:

where *β* is the modulation index and *J _{n}*(

*β*) is the Bessel function of order

*n*. The unmodulated probe beam is expressed as

All the possible combinations we are concerned about are listed in Table 1. These combinations regenerate the fields of *E _{r}* with the frequencies

*ω*=

_{r}*ω*

_{1}−

*ω*

_{2}+

*ω*

_{3}=

*ω*±

*δ*, according to the request of four-wave mixing, when up to the

*s*order of the pump beam sidebands are considered. In Table 1,

^{th}*n*is the order of the sidebands (

*n*= −

*s*, −

*s*+ 1,…,

*s*− 1,

*s*). Using perturbation theory and the rotating-wave approximation, the third-order elements of the density matrix are given by Eq. (12) – (17) in [7]. Considering the experimental quantum system of gas-buffered

^{87}Rb, the relaxation rates

*γ*,

_{a}*γ*and

_{c}*γ*are of the order of several hundred hertz, while

_{ac}*γ*,

_{b}*γ*and

_{ab}*γ*are approximately several hundred mega-hertz. They are all much less than the hyper-fine splitting,

_{cb}*ω*~ 6.8GHz, of the ground state. The modulation frequency

_{ac}*δ*is also ignorable compared with

*ω*. Using all the approximations above, we obtain the third-order elements of the density matrix in our system:

_{ac}$$\times \left\{\frac{1}{{\gamma}_{ac}-i\left({\Delta}_{ac}\mp n\delta \right)}\right[\frac{1}{{\gamma}_{ab}-i\left({\Delta}_{ab}\mp n\delta -\mathrm{k\upsilon}\right)}$$

$$+\frac{1}{{\gamma}_{cb}+i\left({\Delta}_{cb}-k\upsilon \right)}]-\frac{1}{{\gamma}_{ac}-i\left[{\Delta}_{ac}\mp \left(n-1\right)\delta \right]}$$

$$\times [\frac{1}{{\gamma}_{ab}-i\left[{\Delta}_{ab}\pm \left(n-1\right)\delta -k\upsilon \right]}+\frac{1}{{\gamma}_{cb}+i\left({\Delta}_{cb}-k\upsilon \right)}\left]\right\}$$

where *δ _{ab}* =

*ω*+ Ω −

*ω*,

_{ab}*δ*=

_{cb}*ω*−

*ω*,

_{cb}*δ*= Ω −

_{ac}*ω*,

_{ac}*ν*is the velocity component along

*z*axis, and

*s*is the maximum order of sidebands considered.

*N*is the total number of atoms, and we assume that

*n*=

_{ab}*n*=

_{cb}*N*/2. The reemitted field is given by [7]

where an optically thin absorption cell is assumed. The integration is somewhat different from that discussed in [5] because the Doppler limitation *ku* ≫ *γ _{ab}*,

*γ*is unavailable here. When the light field is in near resonance with the atomic system, the detuning ∆

_{cb}*, ∆*

_{ab}*and the modulation frequency*

_{cb}*δ*are far less than the relaxation

*γ*and

_{ab}*γ*. Therefore, the integrations over velocity

_{cb}*ν*become a constant factor

Numerical calculation indicates that *C* is real. Since the beating current at the photodiode is proportional to

The modulation transfer signal in case of the Λ configuration of the three-level ^{87}Rb quantum system is expressed by

$$\times \{\frac{1}{{\gamma}_{ac}-i\left[{\Delta}_{ac}+\left(n-1\right)\delta \right]}-\frac{1}{{\gamma}_{cb}+i[{\Delta}_{ac}-\left(n-1\right)\delta ]}$$

$$+\frac{1}{{\gamma}_{ac}+i\left({\Delta}_{ac}+n\delta \right)}-\frac{1}{{\gamma}_{ac}-i\left({\Delta}_{ac}-n\delta \right)}\}$$

where *γ _{ac}* is the linewidth of the CPT resonance, and ∆

*is the detuning from the resonance center. The real part*

_{ac}*ℜ*[

*S*(

*δ*)] represents the quadrature component of the signal and the imaginary part

*ℑ*[

*S*(

*δ*)] represents the in-phase component of the signal. The real demodulated signal is a combination of these two parts with a form of

where *ϕ* is the detector phase with respect to the modulation field applied to the pump laser. The maximum signal amplitude is obtained at $\mathrm{tan}\left(\varphi \right)=\frac{\Im \left[S\left(\delta \right)\right]}{\Re \left[S\left(\delta \right)\right]}$. The quadrature component, in-phase component and maximum-amplitude signal are shown in Fig. 2.

## 3. Experiment

The experimental setup is shown in Fig. 3. To obtain the spatially separated and phase coherent Raman beams, we use a microwave frequency-modulated vertical-cavity surface-emitted laser (VCSEL) to phase connect two diode lasers by a two-step injection locking [6]. As a result, the master laser, a narrow-linewidth external-cavity diode laser (ECDL), and the slave laser, a Fabry-Perot diode laser, are phase locked with the frequency difference equal to the modulation frequency, 6.8GHz, of the VCSEL, which is controlled by the frequency synthesizer (E8257D, Agilent). The slave laser could be further modulated at a low frequency, e.g., 400 Hz, by modulating the high modulation frequency of VCSEL. The modulated slave laser playing the role of the pump beam, and the unmodulated master laser, as the probe beam, are combined with orthogonal polarizations by a polarization beam splitter (PBS). The combined beams are expanded to 12.7 mm in diameter and then incidented into the cell with the light power 50 *μ*W of each beam. Two quarter plates at the entrance and the exit of the cell change the polarization of the beams to be circular to ensure the maximum CPT signal. The photodiode (PD) detects the probe beam and the reemitted sidebands by placing a PBS in front of it, filtering the pump beam out.

With the 400-Hz modulation turned off, a CPT resonant absorption signal is achieved with direct observation of the PD output signal (Fig.4(a)). With the modulation turned on, the modulation transfer signal can be obtained by demodulating the PD output signal using a lock-in amplifier (Fig.4(b)). The linewidth of the CPT absorption spectrum is measured to be as narrow as 400Hz with the intensity of the superposed Raman lasers less than 100*μ*W/cm^{2}, and is mainly broadened by saturation broadening and collision broadening.

Fig. 5 shows the MTS spectra with the modulation index of 1 and modulation frequencies of 400 Hz, 1 kHz and 3 kHz respectively. The solid curves are the theoretical predictions with the experimental parameters. And the “o” curves are the experimental results. It is clear that the experimental results are in good agreement with the theoretical predictions.

The most concerning thing in using such dispersion-like spectrum in frequency stabilization is to maximize the gradient of the MTS signal at the center point which determines the frequency discrimination sensitivity. To optimize the gradient of the MTS signal, we theoretically predicted the dependence of signal gradient on the modulation frequency and the modulation index in Fig.6. It indicates that signal gradients can be optimized by using the proper parameter of the modulation frequency and modulation index. For the rubidium atom used in our experiment, the optimized modulation frequency is about 320 Hz. However, heterodyne detection with this low modulation frequency will be contaminated by acoustic noise. Since the shot noise limitation occurs in detection with high modulation frequency, it is necessary to carefully weigh the benefits of both factors.

## 4. Conclusion

We have demonstrated a method for obtaining the MTS spectroscopy of CPT resonance by the example of the 5*S*
_{1/2} ↔ 5*P*
_{3/2} three-level system of the ^{87}Rb atom. We recorded the MTS signal of CPT resonances in this system and studied the dependence of the MTS signal gradient on the modulation frequency and modulation index. The high-phase-coherence laser beams allows us to obtain the MTS signal of narrow linewidth and high signal-to-noise ratio. A theoretical analysis of the CPT modulation transfer spectra is deduced from the density matrix equation of motion, and found to be in good agreement with the experimental results.

This work is partially supported by the state Key Development Program for Basic Research of China (No. 2005CB724503, 2006CB921401 and 2006CB921402), and NSFC (No. 10874008).

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