## Abstract

We demonstrate that the lin-par-lin Ramsey coherent population trapping ^{87}Rb clock using a dispersion detection technique has a promising performance. We theoretically and experimentally investigate the signal-to-noise ratio of the Ramsey spectrum signal by varying the relative angle of the polarizer and analyzer as well as the magnetic field. Based on the experimental results, the optimized relative angle and magnetic field are determined. This kind of atomic clock is attractive for the development of compact, high performance vapor clock based on CPT.

© 2016 Optical Society of America

## 1. Introduction

Since coherent population trapping (CPT) phenomenon was first discovered in 1976 [1], gas cell atomic clocks based on CPT have been widely studied [2–4]. Thanks to the very compact volume and low power consumption, chip-scale CPT clocks have been demonstrated based on microelectromechanical system (MEMS) fabrication techniques [5]. However, for the traditional circularly polarized CPT clock scheme in a cell, a large portion of the atoms is pumped into Zeeman end-state that does not contribute to the magnetic-insensitive 0-0 clock transition. This fact results in small signal amplitude, thus limiting the clock performance [6]. To overcome this problem, several methods have been proposed, such as polarized counterpropagating *σ*^{+} − *σ*^{−} optical field configuration [7], lin-per-lin configuration [8], push-pull optical pumping (PPOP) [9], four-wave-mixing [10]. Although these methods can enhance the signal’s contrast, many difficulties are encountered in miniaturization because relative complex optical systems are needed to generate specified polarization. Recently, the lin-par-lin CPT configuration has been demonstrated, by which one can obtain a high-contrast signal and keep the clock miniaturization simultaneously [11–13]. The well-known, Ramsey’s method of separated oscillation fields is also applied in the CPT clock to reduce the linewidth, referred to the Ramsey-CPT technique [8]. In order to suppress the strong background optical noise and obtain a high contrast, the dispersion detection by orthogonal polarizers has been proposed and demonstrated [12, 14–17]. However, the relevant techniques have been analyzed and demonstrated in those works, but the lin-par-lin Ramsey-CPT clock based on dispersion detection is seldom discussed thoroughly. Furthermore, we further study the factors limiting the signal-to-noise ratio (SNR) of the clock signal both theoretically and experimentally to demonstrate the promising performance of this kind of clock.

Compared to the traditional CPT clocks, the lin-par-lin Ramsey-CPT clock with dispersion detection has promising performances due to the high SNR and narrow linewidth. Although the techniques involved in this clock have been reported previously, how to choose optimal working parameters has not been studied thoroughly. In this paper, we report a detailed investigation, both theoretically and experimentally on the relationships between the clock’s SNR and the relative angle of the linear polarizers as well as the applied static magnetic field. The theoretical calculations agree with the experimental results very well, and the optimized working parameters of the relative angle and magnetic field are obtained.

This paper is organized as follows. In Sec. 2, we present the theoretical calculations of the dispersion detection in the lin-par-lin Ramsey-CPT spectrum and its relationship with magnetic field. In Sec. 3, the experimental setup is described. In Sec. 4, we present the measurement results and the calculations of the SNR by varying the relative angle between the two polarizers and the magnetic field. According to these results, the optimal relative angle and magnetic field to maximize the SNR are obtained. Finally, we discuss the expected short-term frequency stability of the lin-par-lin Ramsey-CPT clock and the summary is given in the last section.

## 2. Theory

#### 2.1. Dispersion detection in lin-par-lin CPT

In a lin-par-lin scheme, the two lineally polarized coherent optical field excited two pairs of ground-state hyperfine sublevels simultaneously as shown in Fig. 1. If an external magnetic field *B* is applied on the atom ensembles, the dispersions of *σ*^{+} and *σ*^{−} shift in opposite directions due to the Zeeman effect. When the external magnetic field is weak, the Zeeman shift can be calculated by

*B*is the magnetic field,

*ν*

_{0}the hyperfine splitting frequency of the ground states without magnetic field,

*g*the nuclear g-factor,

_{I}*g*the Lande g-factor,

_{J}*μ*the Bohr magneton, and

_{B}*h*the Planck constant.

Due to the difference of dispersions between *σ*^{+} and *σ*^{−} caused by atoms, the polarization of the transmitted light is rotated by a small angle *ϕ* in comparison to the input light, as indicated in Fig. 2. The relative angle of the transmission axes between the polarizer and analyzer is *θ*, which is zero when the polarizers are orthogonal to each other. The transmitted optical intensity can be written as [17]

*I*is the input optical intensity,

_{in}*l*the cell length,

*λ*wavelength of light,

*α*

_{+}and

*α*

_{−}the absorption coefficients of

*σ*

^{+}and

*σ*

^{−}.

*ϕ*can be further expressed as

*γ*is the relaxation between the ground state hyperfine splitting levels,

*χ*

_{0}the amplitude of linear susceptibility, Δ

_{±}is Raman detuning frequencies with Δ

_{±}=

*ν*

_{±}− (

*ω*

_{1}−

*ω*

_{2}),

*ξ*(Δ

_{±}) is the dispersion generated by CPT.

In Eq. (2), we assume that all incident light components contribute to CPT. In practice, the coherent light is usually generated by direct modulation of the laser’s drive current or phase modulation on the laser light, which can generate many unwanted sidebands simultaneously. Therefore, the total transmitted optical intensity is rewritten as

where*I*is the sum of light intensities contributing to CPT, and

_{c}*I*is the sum of background light intensities not contributing to CPT.

_{b}Typically, the traditional CPT contrast is no more than 5% [2], so that the DC level dominates in the output light. If the relative angle *θ* is larger compared to the rotation angle *ϕ*, then the DC level can be written as

And the signal can be expressed as [17]

From Eq. (5) and Eq. (6), we can see that *I _{DC}* increases with

*θ*quadratically, and

*I*increases with

_{s}*θ*lineally when

*θ*is close to zero. In the case of Ramsey-CPT, the DC level and the signal amplitude of the central Ramsey fringe have the same relationship with

*θ*. For the cross-polarizer detection, the intensity fluctuation of the incident light is the main source of the detection noise. Therefore, by varying the relative angle

*θ*, an optimal value can be obtained to maximize the SNR.

#### 2.2. Dependence of Ramsey-CPT spectrum’s dependence on B

The Ramsey-CPT spectrum of the Λ_{+} configuration as shown in Fig. 1 can be obtained by the quantum Liouville equations,

*b*=

_{ij}*b*

_{ji}^{*}(

*i*,

*j*= 1;2;3;

*i*≠

*j*),

*b*

_{12}=

*ρ*

_{12}

*e*

^{−}

^{i}^{(}

^{ω}^{1−}

^{ω}^{2)}

*,*

^{t}*b*

_{13}=

*ρ*

_{13}

*e*

^{−}

^{iω}^{1}

*,*

^{t}*b*

_{23}=

*ρ*

_{23}

*e*

^{−}

^{iω}^{2}

*, Ω*

^{t}_{1}and Ω

_{2}are Rabi frequencies (To simplify the calculation, we assume Ω

_{1}= Ω

_{2}), Γ is the decay rate from the excited state, and

*γ*

_{1}

*γ*

_{2}are the decay rates of the ground states.

*p*(

*t*) is the pulse modulation function of the light field.

In detection of the fluorescence or absorption of the transmitted light, the signal is represented by *ρ*_{33}. In dispersion detection, the signal depends on the dispersion shifts of *σ*^{+} and *σ*^{−}, which is given by the real part of *ρ*_{13}: Re(*ρ*_{13}) = −Re(*ρ*_{23}) ∝ *ξ*(Δ_{+}). The calculation is identical for configuration Λ_{−} (dashed line in Fig. 1) to obtain *ρ*_{43}: Re(*ρ*_{43}) = −Re(*ρ*_{53}) ∝ *ξ*(Δ_{−}). Some typical calculation results are shown in Fig. 3. The actual observable Ramsey fringes (as shown in Fig. 3(b), 3(d) and 3(f)) are the differences between two dispersion Ramsey spectra, *ξ*(Δ_{+}) and *ξ*(Δ_{−}) (Fig. 3(a), 3(c) and 3(e)). When *B* = 0, the two spectra of *ξ*(Δ_{+}) and *ξ*(Δ_{−}) are degenerate, and no signal can be observed. In the presence of external magnetic field, the two spectra shift in opposite directions, resulting in an observable Ramsey signal. By increasing the magnetic field, the peak to peak amplitude, Vpp, of the Ramsey signal reaches maximum when the peak of one spectrum coincides with the valley of the other one (see Fig. 3(c) and 3(d)). If the magnetic field continues to increase, Vpp begins to decrease because the adjacent peaks of *ξ*(Δ_{+}) and *ξ*(Δ_{−}) start to overlap (see Fig. 3(e) and 3(f)). The amplitude of Ramsey fringes varies with the magnetic field periodically, and the period is determined by the linewidth of the dispersion Ramsey spectrum.

Due to the maximum peak and valley of two dispersion Ramsey spectra coincide in the first period, Vpp should reach the maximum in the first period then decreases as the magnetic field increases. However, in the dispersion detection method, the detected CPT signal increases in the same time as the magnetic field increases [17]. As a result, the detected signal is determined by these two opposite mechanisms jointly, which we will discuss more with the experimental results in Sec. 4.2. It is worth noting that the central frequency of the Ramsey fringes is insensitive to the magnetic field to the first order, and the change of magnetic field mainly affects the amplitude of the detected signal.

## 3. Experimental setup

Figure 4 shows a schematic diagram of our experiment system. The 795 nm laser resonating with D1 lines of ^{87}Rb is generated by a distributed Bragg reflection (DBR) diode laser. The laser frequency is stabilized by saturated absorption spectrum (S.A.) technique to the transition line of 5*S*_{1}_{/}_{2}*F* = 2 *→* 5*P*_{1}_{/}_{2}*F*′ = 2. The laser is phase-modulated at 6.8 GHz by a fiber optical electronic-optical modulator (FEOM). Thus, the positive first sideband of the modulated laser is resonant with the 5*S*_{1}_{/}_{2}*F* = 1 *→* 5*P*_{1}_{/}_{2}*F*′ = 2 transition. Then the light beam passes through an acoustic-optic modulator (AOM), and the first order diffraction beam is directed to the Rb cell. The AOM also works as a fast optical switch by switching the radio frequency (RF) driving signal. In the cell, the diameter of the incident light is expanded to 8 mm. The quartz cell in this experiment is a 15mm cube containing natural-abundance ^{87}Rb, with 20 Torr Ne and 20 Torr N_{2} as buffer gases. The cell is installed in a four-layer magnetic shield and the temperature is stabilized at 71.8°C. A pair of Helmholtz coils provides a uniform longitudinal magnetic field. Meanwhile, a small part of the laser beam is sampled before the cell to monitor its intensity fluctuation, which can be used to normalize the final signals [18]. To implement the dispersion detection method, we place a Glan-Taylor polarizer with an extinction ratio of 60 dB in front of the cell, and another Glan-Taylor polarizer as an analyzer behind the cell. The analyzer is mounted in a rotation mount which enables the adjustment of two polarizer’s relative angle. A fast photo diode (PD) is placed after the analyzer to detect the transmitted optical signal, which is sampled by a data acquisition card.

In the Ramsey-CPT experiments, the laser in the cell is pulse modulated. In one optical pulse of duration *τ*, the dark state is created. After a period of free evolution time T, the atoms are probed in the following optical pulse at the moment *τ _{m}* by detecting the transmitted optical intensity after the analyzer (see Fig. 4 and 5(a)). When the microwave frequency is scanned over the two-photon resonance frequency, the Ramsey-CPT fringes can be obtained as shown in Fig. 5(b).

## 4. Experimental results and discussion

#### 4.1. SNR as a function of the relative angle θ

Because the SNR determines the clock performance, we study the Ramsey signal and noise as functions of the relative angle *θ*. The experimental results are shown in Fig. 6. The black solid squares are the measured amplitudes of the Ramsey signal, and the black solid line is the fitting curve based on Eq. (6). The shift of minimum of the Ramsey signal from 0°is caused by the magneto-optical rotation due to the interaction between light and atoms. The red solid triangles represent the noise, which are the standard deviations of the sampled signals while the Raman detuning is fixed. The measured noise includes the noise in the incident light and the electronic noise in the detection and sampling process. From Eq. (5), the total noise can be expressed as

*σ*is the noise of the incident light, and

_{in}*σ*the electronic noise. In Fig. 6, the red dashed line is a fit curve using Eq. (8). Those fitting curves match the experimental data very well. Because of the different relationships between the Ramsey signal (noise) and the relative angle, we are able to determine an optimal

_{e}*θ*to obtain the maximum SNR. In Fig. 7, the SNR calculated from the experimental data and the theoretical fitting are shown, and the optimal is obtained to be 5.5°.

#### 4.2. SNR as a function of magnetic field

As mentioned in Sec. 2, the magnetic field is a key factor to determine the amplitude of the Ramsey signal. In Fig. 8(a), the black solid squares are the experimental Ramsey signal data as a function of the magnetic field. The maximum of the measured magnetic field is limited by the current source of the coils. The Ramsey signal appears periodically as we predicted in Sec. 2.2. However, the experimental results show that the second peak is the highest among the four periods. In dispersion detection method, increasing magnetic field causes the magneto-optical rotation angle to increase, leading to larger CPT signal proportional to Ramsey signal (see the red triangles indicate in Fig. 8(a)). In weak magnetic field, the dependence of the CPT signal on the magnetic field can be simplified to a linear function, as showed in the red dashed line of Fig. 8(a). Simultaneously, increasing magnetic field leads to a larger shift of two Ramsey fringes, which results in the signal damping and suppresses the rise in CPT signal. Taking both of the two effects relevant to magnetic field into consideration, we numerically calculate the curve of the amplitude of Ramsey signal with respect to the magnetic field in Fig. 8(b). We can see that the theoretical calculation is consistent with the experimental data. Meanwhile, according to Eq. (3)–(5), the noise does not increase with the magnetic field, and which is also verified by experiments as shown in Fig. 9 (blue triangles). Thus, the varying tendency of SNR should be similar to that of the Ramsey signal (see fig. 8(b) and fig. 9). The calculated SNR based on the experimental data is also shown in Fig. 9 (black squares), and the optimal magnetic field is determined to be 31*μT* for our experimental setup.

## 5. Conclusion

The short-term frequency stability of a passive clock can be estimated according to

where*Q*is the quality factor of clock transition,

*SNR*the signal-to-noise ratio in 1 Hz bandwidth and

*τ*the sampling time. In the previous text, the SNR is calculated from the raw data sampled at 1 kHz. According to the measurement of noise spectra in the optimal condition, we estimate the short-term stability of our clock to be $6.6\times {10}^{-13}/\sqrt{\tau}$. As we can see, the lin-par-lin Ramsey-CPT clock based on dispersion detection has a promising high performance. Meanwhile, this kind of atomic clock has the potential of miniaturization for its simple setup [19–22]. It is worth noting that the generation of pulsed-modulated coherent Raman light (by EOM and AOM in our experimental setup) can be replaced by a microwave-modulated vertical-cavity surface-emitting laser (VCSEL), leading to a small package and low power consumption similar to currently available packaged CPT atomic clocks [23].

However, the above short-term stability of this kind of Ramsey-CPT clock is an estimation of the ultimate possible stability according to the SNR. Actually, the clock’s stability can still be affected by a couple of noise sources, such as the frequency and intensity noise of the laser, and the phase noise of the local oscillator due to the Dick effect. So, in order to reach the theoretical estimated stability, one has to deal with those noise sources. So far, the Rb cell in our experiments is filled with natural-abundance rubidium. If it is replaced with an ^{87}Rb isotope-enhanced cell, the clock signal can be enhanced at the same cell temperature. Besides, optimizing the buffer gases in the cell can help to decrease the temperature drift coefficient to improve the long-term frequency stability.

In conclusion, we investigate a lin-par-lin Ramsey-CPT clock prototype based on dispersion detection, and carefully discuss the relationships between the SNR of clock transition and the relative angle of polarizers as well as the applied magnetic field. The theoretical calculations agree with the experimental results very well. According to our calculation and experiment results, the optimal relative angle and magnetic field are determined for our Ramsey-CPT clock prototype. Therefore, it can be hoped that such an atomic clock can have a high-performance yet being compact.

## Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant No. 11304177, 11204154 and 11574016).

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