## Abstract

We realize a high-stability laser by modulation transfer spectroscopy and apply it to implement a high-performance compact optically pumped cesium beam atomic clock. Evaluated by the optical heterodyne method with two identical frequency-stabilized lasers, the frequency instability of the 852 nm laser directly referenced on thermal atoms is 2.6×10^{−13} at the averaging time of 5 s. Factors degrading the frequency stability of the laser are analyzed, and we will further control it to reduce the frequency noise of the laser. By comparing with a Hydrogen maser, the measured Allan deviation of the high-stability-laser-based cesium beam atomic clock is 2$\times 10^{-12}/\sqrt {\tau }$, dropping to 1×10^{−14} in less than half a day of averaging time. To our knowledge, the Allan deviation of our cesium clock is better than that of any reported compact cesium beam atomic clocks at the averaging time of half-day. The high-performance atomic clock can promote the fields in metrology and timekeeping, and the high-stability laser additionally possesses great potential to be a compact optical frequency standard.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Atomic clocks play a crucial role in fundamental research and practical applications. They are essential not only in the conventional fields, including navigation, calibration, and communications [1,2], but also in the state-of-the-art precision measurement areas, such as gravitational wave detection [3] and geodesy [4,5]. The improvement of the clock performance will continue to promote the physics probing and technological advance. Optical atomic clocks have better performance than microwave atomic clocks. However, they still face the challenge of working out of the laboratory because of complicated systems and huge size [5,6], even if researches are engaged in realizing a transportable optical lattice clock (overall size is still 15 m$^{3}$) [7] or a miniaturized thermal beam system (overall size is 1 m$^{3}$) [8]. To meet with the further application requirements, we necessitate improving the performance of microwave atomic clocks that have been maturely industrialized and transportable. Among all atomic clocks, cesium clocks stand out because the SI (International System of Units) second is defined on the microwave hyperfine quantum transition of the cesium ground state.

Based on the optically pumped scheme [9], cesium beam atomic clocks with a meter-level-long free drift region have developed as primary frequency standards, which are restricted to the laboratory because of large and sensitive systems. Resulting from long Ramsey evolution time, they can acquire narrow linewidth Ramsey fringes [10–13], which are beneficial to clock performance. Makdissi *et al.* [10] reported a cesium beam primary frequency standard with a frequency instability of $3.5 \times 10^{-13}/\sqrt {\tau }$, but at the cost of massive consumption of atoms. Otherwise, frequency instabilities of optically pumped cesium beam primary frequency standards were generally around $9 \times 10^{-13}/\sqrt {\tau }$ [11–13]. Different from primary frequency standards that are large in overall size (generally have a thermal beam vacuum tube longer than two meters) [10–13], the compact optically pumped cesium beam atomic clock [14–19] enclosed in a transportable mechanical box of dozens of liters typically employs a small vacuum tube and integrated electronics. Thus the compact clock can support the flexibility of the operation site. Due to the reduction of the size of the vacuum tube, a shorter free drift region leading to broader Ramsey fringe linewidth is not conducive to the frequency stability of the compact optically pumped cesium beam atomic clock. In spite of this, the compact system can still obtain satisfactory frequency stability by raising the signal-to-noise ratio of the Ramsey fringes [16,18]. Bousset *et al.* realized a compact clock with a frequency instability of $5 \times 10^{-12}/\sqrt {\tau }$, and it reached the flicker floor of $2 \times 10^{-14}$ at the averaging time of one day [14]. Investigating different pump-probe configuration, Sallot *et al.* achieved a frequency instability of $3 \times 10^{-12}/\sqrt {\tau }$ [16]. The space-borne project group showed notable short-term stability, but the frequency stability drifted at the averaging time of a few thousand seconds [18,19].

When employing the cycling transition detection scheme, the frequency stability of the compact cesium beam atomic clock is mainly limited by the laser frequency noise and atomic shot noise. By increasing the atomic flux, the frequency stability can be significantly improved, unless the laser frequency noise surpasses the atomic shot noise. At specific atomic flux, with the laser frequency noise reduced, the signal-to-noise ratio of the Ramsey fringes will be enhanced [20–22]. Dimarcq *et al.* theoretically and experimentally revealed that the signal-to-noise ratio could be improved by almost one order when the laser linewidth was reduced from 5 MHz to 100 kHz at high atomic flux [20]. Whereas, without the availability of compact and durable low-frequency noise laser system, previous optically pumped cesium beam atomic clocks [14–19] generally adopted commercial lasers with broad linewidth ($\geq$ 1MHz). Therefore, in addition to introducing less frequency noise, a frequency-stabilized laser with high spectral purity provides a feasible route to further improve the frequency stability of the optically pumped cesium beam atomic clock.

Locking the frequency by Doppler-free spectroscopy is a typical way to acquire lasers with high stability and spectral purity. Abundant latest works on frequency-stabilized 852 nm laser corresponding to the cesium atomic D$_{2}$ transition line have been done [23–30]. Roverae *et al.* locked an external cavity diode laser by saturation absorption spectroscopy, and obtained an instability of $5 \times 10^{-13}$ at the averaging time of $\sim$ 1 s, but the frequency stability immediately reached to the flicker floor [23]. Based on heterodyne detection, modulation transfer spectroscopy (MTS) without Doppler background has revealed its potential to achieve a high-stability laser source theoretically and experimentally [31–38]. In a combination of frequency modulation spectroscopy, Zi *et al.* reported an MTS-stabilized 852 nm laser with a frequency instability of $5.5 \times 10^{-12}$ at the averaging time of 1 s [30].

In this paper, we implement a high-stability narrow-linewidth 852 nm laser by MTS and demonstrate its application into the compact optically pumped cesium beam atomic clock. To evaluate the performance of the laser, we build two identical frequency-stabilized laser configurations (whose sizes are 430 mm $\times$ 200 mm $\times$ 60 mm) to heterodyne. The beating results give a 3.2 kHz Lorentz linewidth for each laser. Besides, factors adverse to laser frequency stabilization are analyzed. By comparing with a Hydrogen maser, the Allan deviation of the compact optically pumped cesium beam atomic clock based on the high-stability 852 nm laser is measured to be 2$\times 10^{-12}/\sqrt {\tau }$. Then our cesium clock is contrasted with a distributed feedback (DFB) laser-based compact optically pumped cesium beam atomic clock.

## 2. Experimental apparatus and methods

The overall configuration of the compact optically pumped cesium beam atomic clock based on the high-stability laser is depicted in Fig. 1. The integrated laser source, whose pedestal and optical elements’ mounts are machined from a whole piece of aviation aluminum, is frequency stabilized by the MTS. After power-splitting and frequency-shifting, the laser interacts with the cesium atomic beam which jets from the heated oven. The size of the optical part is 430 $\times$ 200 $\times$ 60 mm$^{3}$ (whose photograph see Fig. 1), and the size of the whole compact optically pumped cesium beam atomic clock is 446 $\times$ 550 $\times$ 177 mm$^{3}$ [39]. Detailed description of the experimental set-up is as below.

#### 2.1 High stability laser source

The laser source mainly consists of a home-made narrow-linewidth external cavity diode laser (ECDL) [40], cesium atomic vapor cell, and electronic servo circuits. The frequency of the laser is tuned near to the transition of 6$s$ $^{2}S_{1/2}$ $|F=4\rangle - 6p$ $^{2}P_{3/2}$ (852 nm, relevant energy level see Fig. 2(a)). We use a 40 dB optical isolator (Thorlabs, IO-5-850-VLP) to eliminate the retro-reflected light. Then we divide the laser into two paths by a polarization beam splitter (PBS), with a half-wave plate (HWP) added to adjust the laser power for different paths. One path is utilized for the frequency stabilization of the laser source, and the other acts as the pump laser and probe laser for the cesium beam atomic clock.

To stabilize the frequency of the diode laser on the cycling transition (see Fig. 2(a)), we adopt the MTS with inherent advantages of high-resolution, high-sensitivity, and free-of-Doppler background [31,32], and it was ever used to realize rubidium optical frequency standard [35,37]. A 50 mm-length pure cesium atomic vapor cell as the frequency reference is magnetically shielded by the double-layer mu-metal to reduce undesirable effects of the external magnetic field. The controlled-temperature of the vapor cell is set as 29 $^{\circ }$C, slightly higher than room temperature. The polarization-orthogonal probe laser and pump laser interact with cesium atoms in counter-propagation direction. The intensity of the probe laser and pump laser are 0.7 mW/cm$^{2}$ and 3.3 mW/cm$^{2}$ (size of the oval laser spot is $\sim$ 3 mm $\times$ 4.5 mm, saturation parameter is $\sim$ 3.6), respectively. To phase modulate the pump laser, a sinusoidal signal generated by a direct digital synthesizer is applied to the EOM (electro-optic modulator), and the modulation depth is about 1 rad. After four-wave mixing, the transferred modulation on the probe laser is detected by the high-speed photo-detector. The MTS signal (shown in Fig. 2(b)) corresponding to the transition of 6$s$ $^{2}S_{1/2}$ $|F=4\rangle - 6p$ $^{2}P_{3/2}$ is acquired after demodulation. We notice that the signal-to-noise ratio of the spectroscopy is significantly related to the modulation frequency. We experimentally measure the peak-to-peak frequency interval [41] and slope of the MTS signal corresponding to the transition of 6$s$ $^{2}S_{1/2}$ $|F=4\rangle - 6p$ $^{2}P_{3/2}$ $|F'=5\rangle$ against the modulation frequency. As shown in Fig. 2(c), it exhibits that the maximum slope is obtained at the modulation frequency of 4.5 MHz.

The dispersive MTS signal is fed back into the controller of the ECDL by servo circuits, including the slow- and high-speed ports. Before entering the servo circuits, a fraction of the error signal is split out by a dual-way radio-frequency (RF) power splitter and sent into a spectrum analyzer. The frequency noise spectral density is monitored to indicate noise suppression and feedback bandwidth. The whole laser source is enclosed in an integrated aluminum box with a relatively thick base to improve the robustness. Besides, the ECDL is surrounded by thermally insulating foam to minimize the influence of the environmental temperature fluctuations on the slow frequency drift of the laser, since only the pedestal of the laser diode is temperature-controlled.

#### 2.2 High performance cesium beam atomic clock

Before interacting with the cesium atomic beam, the frequency-stabilized laser source is divided into two paths by two sets of HWP and PBS, and power adjustment for each path is also available. One path performs as probe laser, and the other acts as pump laser after frequency-shifting by 251 MHz via an acousto-optic modulator (AOM). The oven of the fully-sealed cesium beam vacuum tube is temperature-controlled at 105 $^{\circ }$C, to form a collimated atomic beam with a height of 5 mm (in the direction vertical to the paper plane of Fig. 1). The pump laser, whose spot size is expanded by a combination of plano-concave and plano-convex lenses, perpendicularly excites the atoms from the state of $|F=4\rangle$ to $|F=3\rangle$. The pump laser is retroreflected after passing through the atomic beam to make sure the pumping efficiency is near to 1. Under the excitation of Ramsey separated oscillating fields, atoms populated on the state of $|F=4\rangle$ are subsequently detected by the probe laser. Similar to the pump laser, the probe laser spot size is also expanded to cover the atomic beam. However, to avoid laser scattering due to the finite size of the cesium tube optical window, the dimensions of the laser beams are only enlarged by nearly two times, namely $\sim$ 5 mm $\times$ 8 mm. When interacting with the atomic beam, the laser beams have been collimated.

We tune the frequency of the microwave signal which comes from the frequency synthesis of a 10 MHz oven-controlled crystal oscillator (OCXO), and apply it to the microwave cavity (shown in Fig. 1). The Ramsey fringes are detected by the fluorescence-collection photodetector and depicted in Fig. 2(d). The scanning range is 2 kHz with a central frequency of 9.192 631 770 GHz corresponding to the transition of ground state $|F=3,m_{F}=0\rangle$ - $|F=4, m_{F}=0\rangle$. The linewidth of the central fringe is about 600 Hz when the temperature of the oven is at 105 $^{\circ }$C. After modulating and demodulating the Ramsey fringes, an error signal is sent into the electronic control module to servo the frequency of OCXO. More detailed experimental design of the atomic clock, including the electronics and microwave source, see [39].

## 3. Results and analysis

#### 3.1 Properties of frequency-stabilized laser

To evaluate the linewidth denoting frequency noise of the ECDL, we build two identical laser sources whose frequencies are both stabilized on the transition of 6$s$ $^{2}S_{1/2}$ $|F=4\rangle - 6p$ $^{2}P_{3/2}$ $|F'=5\rangle$. The two laser sources interrogate different atomic vapor cells; besides, the optics and electronics for each construction are independent. One of them is frequency shifted by 150 MHz with an AOM to generate a beating frequency interval. The beating signal is sent into a spectrum analyzer (Keysight N9935A, RBW is 750 Hz), and the with-locking linewidth is measured when two laser sources are both frequency stabilized. According to Fig. 3, the Lorentz linewidth (-3 dB) of the frequency-stabilized laser is fitted to be 3.2 kHz considering that the two identical lasers make the same contributions to the frequency noise. This narrow linewidth benefits from a relatively long external invar cavity jointed with an integrated design of the whole optical and mechanical configuration of the laser sources. The blue-solid line in the inset figure of Fig. 3 is the beating signal when one laser source is free-running. The free-running linewidth is fitted to be 28.6 kHz, which is approximately consistent with the results in [40]. There is an extremely narrow signal at 150 MHz, which is induced by the residual amplitude modulation of the 150 MHz AOM. The influence of the residual amplitude modulation signal on the measurements of the two laser sources is negligible due to narrower linewidth and smaller intensity in contrast to the beating signal of the two lasers. The measured feedback bandwidth of the locking loop is 210 kHz, which is much larger than the linewidth of the laser, and this will further reduce the laser frequency noise when the technical noises discussed in next subsection are sufficiently suppressed. The beating signal of the two identical frequency-stabilized laser sources is sent into a frequency counter (Keysight 53230A, gate time is set as 10 ms), then we calculate the standard Allan deviation of the frequency-stabilized ECDL, as depicted in Fig. 4. Frequency instability of the laser is 4.8 $\times$ 10$^{-13}$/$\tau ^{1/2}$ at short term, dropping to 2.6 $\times$ 10$^{-13}$ at the averaging time of 5 s. However, the Allan deviation deteriorates at a few seconds with an approximate slope of $\tau$, corresponding to linear frequency drift of 3.5 $\times 10^{-14}$/s. Notwithstanding, longer-term stability recovers a bit because the frequency drift is periodic. With the relatively large linear frequency drift removed, the frequency instability would be comparable with the reported compact rubidium optical atomic clock based on two-photon transition [42]. Except for the optical heterodyne method, we also evaluate the frequency instability of the lasers by the self-estimation method [35]. According to the results (see Fig. 4), the self-estimation instability is around 1$\times 10^{-14}$. Although heterodyne instability is authentic, self-estimation results can reflect the in-loop locking performance.

#### 3.2 Factors adverse to laser locking

The factors that might result in frequency drift and the difference between the two frequency stability evaluation methods will be discussed in this section. Since our goals are to analyze the principal limitations rather than to provide uncertainty budgets, we do not undertake precise measurements of the magnitude of each frequency variations. For a complete locking loop, the relationship between the reference frequency $\nu _{R}$ and the controlled laser frequency $\nu _{L}$ can be represented by $\nu _{L}-\nu _{R}=\delta \nu$, where $\delta \nu$ is the frequency difference between the reference and the laser. The equation still holds under Allan deviation calculation. Namely, ${\sigma _{\nu _{L}}^{2}}={\sigma _{\nu _{R}}^{2}}+{\sigma _{\delta \nu }^{2}}$, the Allan deviation of the locked-laser $\sigma _{\nu _{L}}$ can be directly measured by the optical heterodyne method and $\sigma _{\delta \nu }$ is evaluated by the self-estimation method. Obviously, $\sigma _{\nu _{L}} \simeq \sigma _{\nu _{R}}$ when $\sigma _{\nu _{R}}$ is significantly larger than $\sigma _{\delta \nu }$ which is around 1$\times 10^{-14}$ in our systems. It indicates that even though the servo-loop is in good locking, the factors leading to variations of the reference frequency would limit the frequency stability of the lasers.

### 3.2.1 Collisional shift

Collisions between atoms will disturb the wave functions of ground- and excited-state, giving rise to an atomic density-dependent frequency shift. It should be estimated, especially in vapor cell-based frequency locking, because interactions of thermal atoms enriched in a limited size vapor cell are relatively intense. In a heated vapor cell, the atomic density fluctuates with the temperature of the vapor cell cold finger, leading to variations of the collisional shift. In our current frequency-stabilized laser configuration, the vapor cell is heated by a twisted copper heating wire driven by an analog circuit, and it is enclosed by finite-layer thermal insulation materials. Thus the temperature of the cold finger might be sensitive to variations of the ambient temperature. To estimate the effects, we continuously monitor the temperature of the cold finger over a 24 h period. As shown in Fig. 5(a), the variation of the beating frequency is inconsistent with the fluctuations of the cold finger temperature. After recording a longer time, although the temperature of the cold finger (see Fig. 5(b)) oscillates with a period of four hours, it does not affect the beating frequency until the long term. The drift of the monitored temperature is about 0.4 mK/s, resulting in an atomic density variation of 2$\times 10^{7}$cm$^{-3}$ at 1 s. According to the experimentally measured collisional shift coefficient (for the transition of 6$s$ $^{2}S_{1/2}$ $|F=4\rangle - 6p$ $^{2}P_{3/2}$ $|F'=5\rangle$), which is $(-4\pm 6)$ $\times$ $10^{-9}$ Hz$\cdot$cm$^{3}$ for pure cesium [43], we obtain a maximum calculated frequency drift of 6 $\times 10^{-16}$/s. Based on the data in Fig. 5(b), we calculate the cold finger temperature-limited Allan deviation, as shown in Fig. 4 (Black dash-dot line). Although it might affect the long term (after the averaging time of 1000 s) performance, the collisional shift is not the principal factor degrading the frequency stability of the lasers currently.

### 3.2.2 External electromagnetic fields

Quantum absorbers inevitably suffer from frequency shifts induced by external electromagnetic fields, which mainly causes Stark effect and Zeeman effect [42]. In our systems, the dc Stark shift is negligible since there are no strong dc electric fields applied. Ac Stark effect, namely light shift, arises once the electric fields of light interact with atoms. The frequency shift induced by the ac Stark effect can be expressed with $\frac {\Delta }{2\pi }\cdot I/\lbrack 4I_{s}(\Delta /\Gamma )^{2}+I+I_{s}\rbrack$, where $\frac {\Delta }{2\pi }$ and $I$ are frequency detuning and laser intensity, respectively. Spontaneous decay rate $\Gamma =2\pi \cdot 5.2$ MHz, saturation intensity $I_{s}\simeq 1.1$ mW/cm$^{2}$. When the servo loop is locked, $\vert \Delta \vert \ll \Gamma$, the frequency shift is nearly $\frac {\Delta }{2\pi }\cdot I/(I+I_{s})$. The measured intensity variation is $4 \times 10^{-4}$ at 1 s, even if the frequency detuning is a few kHz (actually smaller), the frequency instability induced by the intensity variations is at the level of 10$^{-16}$ or less. We find that the frequency shift is nearly equal to the frequency detuning in value in our system. Although the frequency detuning is tricky to measure, the frequency instability induced by the variations of the frequency detuning is small because the laser is tightly locked [5,7,42].

Since the measured magnetic field in the double-layer mu-metal is smaller than 1 mG, we only consider the second-order effect of the Zeeman shift in weak magnetic fields. The quadratic shift of ground state $|F=4\rangle$ can be calculated based on the Breit-Rabi formula [44], which is 213.7 Hz/G$^{2}$. The excited state shift can barely be simply-analyzed. Generally, one needs to diagonalize the Hamiltonian numerically [42,44]. However, the quadratic shift would not exceed the adjacent sub-level magnetic splitting of $|F'=5\rangle$, so we estimate the quadratic shift to be less than 560 kHz/G$^{2}$. Considering that the variation of magnetic fields is assumed to be less than 10$\%$, frequency instability induced by the quadratic Zeeman effect is less than 4$\times 10^{-16}$.

### 3.2.3 Residual amplitude modulation

Residual amplitude modulation (RAM) occurs associated with phase modulation in frequency modulation technique, mainly arising from the variable birefringence of EOM crystal and the etalon effects [34,45–47]. Active approaches to servo the temperature and the bias voltage applied to the EOM crystal can control the RAM on the parts-per-million level. Typically, several hundred kHz servo-bandwidth is necessary to suppress the in-phase components [46], inferring that high-speed variations are written into the frequency discrimination signal. In combination with the quadrature components, the RAM would simultaneously deteriorate the short- and long-term signal-to-noise ratio of the discrimination signal.

To suppress the RAM, we now place a half-wave plate before the EOM to align the polarization between the light and the principal axis of the crystal [48], and relevant optical surfaces are tilted slightly off of normal incidence for the laser beam. To monitor the amplitude of the RAM, we split a fraction of light after the EOM and send it to a photodetector whose signal is received by a spectrum analyzer (RBW 100 Hz). The measured RAM is $\sim$ 40 dB higher than the shot-noise limit [46], which additionally fluctuates with the periodically varied ambient temperature. The measured fluctuation of ambient temperature at 1 s is about 20 mK. The temperature-related frequency shift is typically a few kHz per $^{\circ }$C [34], therefore in our system, we estimate the frequency instability induced by RAM to be at the level of 10$^{-13}$, which is near to the heterodyne results (see Fig. 4). After assessing relevant factors, we judge that RAM might be the principal source degrading the short- and long-term frequency stability of lasers, further works will focus on the cancellation of the RAM by a combination of active and passive methods [46,47].

#### 3.3 Performance evaluation of high-stability-laser-based atomic clock

After locking the 10 MHz OCXO, we compare the frequency instability of it with a Hydrogen maser whose Allan deviation is shown in Fig. 6. To find the best frequency stability of the atomic clock, we vary the experimental conditions, including atomic flux and laser intensity, which significantly influence the signal-to-noise ratio of the Ramsey fringes when utilizing narrow-linewidth laser source [20]. Allan deviation at the averaging time of 100 s is observed to find out the optimal experimental conditions. At the oven temperature of 100 $^{\circ }$C, we notice that the 100 s Allan deviation is 5.8$\times 10^{-13}$ when the probe laser power is 1.0 mW, while drops to 3.7$\times 10^{-13}$ when the probe laser power is 1.9 mW. However, larger probe laser intensity will degrade the Allan deviation, because the simultaneously increasing background signal will deteriorate the signal-to-noise ratio of the Ramsey fringes. The pump laser intensity is synchronously adjusted with the probe laser intensity to ensure that the pump efficiency keeps being $\sim$ 1. The frequency instability at 100 s is reduced to 2.3$\times 10^{-13}$ after the temperature of the oven is raised from 100 $^{\circ }$C to 105 $^{\circ }$C. Considering the lifetime of the vacuum tube, we do not increase the temperature of the oven anymore. After optimizing the servo parameters, we continuously measure the Allan deviation of the compact optically pumped cesium beam atomic clock, as depicted in Fig. 6. The Allan deviation of our optically pumped cesium beam atomic clock is 2$\times 10^{-12}/\sqrt {\tau }$, dropping to 1$\times 10^{-14}$ in less than half a day of averaging time. Our results are better than the ever best compact cesium beam atomic clock [16] by 1.5 times (see Fig. 6), benefiting from the use of the low-frequency noise 852 nm laser. To verify the effect of laser frequency noise on clock performance, we only replace our high-stability laser with a frequency-stabilized DFB laser reported in [39]. Results show that the measured Allan deviation of the DFB laser-based cesium beam atomic clock is consistent with [39], which is three times worse than ours. The daily frequency drift calculated from long term assembled data is 3.2$\times 10^{-15}$. According to the asymptotes of white-frequency-noise and daily frequency drift, we predict that the flicker floor will be around 5$\times 10^{-15}$ at the averaging time of 3 days once the performance is not limited by other parts, such as the physical part and electronic control module. In further works, the laser will be automatically locked and re-locked to prolong the measurement time. Additionally, the clock performance can be further improved by employing a better frequency-stabilized laser, and subsequently by increasing atomic flux.

## 4. Conclusions

In summary, we implement a high-stability narrow-linewidth 852 nm laser, which is frequency stabilized by MTS, and demonstrate its application into the compact optically pumped cesium beam atomic clock. The fitted Lorentz linewidth of the laser is 3.2 kHz, and the frequency instability is 2.6$\times 10^{-13}$ at the averaging time of 5 s. We have analyzed the factors adverse to the frequency stability of the lasers, including collisional shifts, external electromagnetic fields, and residual amplitude modulation that might be the principal limitation currently. Further works will focus on controlling the residual amplitude modulation to reduce the frequency noise of the laser. Based on this high-stability laser, the Allan deviation of the compact optically pumped cesium beam atomic clock is measured to be 2$\times 10^{-12}/\sqrt {\tau }$ by comparing with a Hydrogen maser. The results indicate that the frequency instability can reach 1$\times 10^{-14}$ in less than half a day of averaging time, which is better than any reported compact cesium beam atomic clock to our knowledge. Compared with previous works [14–19,39], better frequency stability of our atomic clock benefits from the low-frequency noise 852 nm laser. According to the measured daily frequency drift, we predict that the flicker floor of the high-stability laser-based atomic clock will be around 5$\times 10^{-15}$ at the averaging time of three days once other parts do not limit the clock performance. The short-term performance of the atomic clock can be improved by further reducing the laser frequency noise and enhancing the atomic flux. This high-performance compact optically pumped cesium beam atomic clock can promote the fields where it has been used, such as metrology and timekeeping. The high-stability laser also has excellent potential to be a compact optical frequency standard, broadening the applications in geophysical measurements and space optical clocks. Jointed with photonic integration optical frequency comb [49], our apparatus with a highly stable optical oscillator and a microwave atomic clock can provide a relatively small and accurate frequency standard that has good frequency stability both at short- and long-term.

## Funding

National Natural Science Foundation of China (91436210).

## Acknowledgments

The authors would like to thank Dongxu Li and Yuao Li for technical discussions of optically pumped cesium beam atomic clock.

## Disclosures

The authors declare no conflicts of interest.

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