Since the 1970s, high-accuracy measurements with cold atom sensors have been rapidly developed with the introduction of laser cooling technology. The high intrinsic sensitivity of the quantum states of cold atoms toward the external perturbations facilitates exceptional measurement precision of a myriad of physical quantities such as time, acceleration, angular velocity, magnetic field, and gravity, surpassing the detection limit of classical technologies and thus offering a wide application prospect in the fields of metrology, medicine, geology, positioning, navigation, and timing (PNT), and fundamental physics^[1–4]. However, cold atom sensing devices have yet to make their way into the mainstream due to the great size ($\sim {\mathrm{m}}^3$), weight ($>100\mathrm{kg}$), complexity, and cost of the necessary laser, vacuum, and control systems. As the demand for practical cold atom sensors has become increasingly urgent in recent years, miniaturization of the key components of cold atom sensing has received extensive attention. Among the instrumental requirements, the robust, portable, and frequency-stable lasers are especially highlighted for their complicated and sophisticated characteristics, inspiring significant research on integrated laser sources and compact frequency discriminators^[5–7].

In a vacuum chamber of a magneto-optical trap, the neutral atoms [e.g.,  rubidium (Rb)] are trapped, cooled, and manipulated by lasers of multiple specific wavelengths through stimulating the respective transition processes. The laser frequencies, therefore, are required to be precisely stabilized at the exact atomic spectral lines in the long run. As the frequency of free-running laser is highly vulnerable to environmental fluctuations, such as current, temperature, air pressure, and vibration, the emission frequency can easily drift in a range that is far larger than the spectral linewidth, making the active laser frequency stabilization according to the frequency standards indispensable for cold atom preparation and detection. So far, numerous frequency stabilization techniques, including the frequency-modulation (FM) spectroscopy method, the Pound–Drever–Hall (PDH) technique^[8], the saturation absorption spectroscopy^[9], the modulation transfer spectroscopy (MTS) method^[10], and the dual-color atomic absorption (DAVLL) method^[11], have been developed to suppress the frequency drift to $\sim 10\mathrm{kHz}$ via feedback control, but they are generally based on bulky optical components and frequency references that are heavy and nearly unwieldy, causing great inconvenience of utilization. Although there has been remarkable progress in the integrated external cavity semiconductor lasers and fiber lasers with small form factor and low-frequency noise, most of them still rely on the conventional vapor cells as the absolute standard to ensure long-term frequency stability. Hence the miniaturization of the frequency stabilization system remains to be settled^[12–15].

In this study, a miniature laser frequency stabilization module for Rb atom cooling was demonstrated to address the current dilemma of excessive size and complexity through a compact optoelectronic co-package design. In this module, a simplified absorption optical setup and related control circuits are encapsulated on a $12{\mathrm{cm}}^2$ substrate, significantly reducing the overall volume and weight of the whole stabilization system, which can be compatible with any lasers through a modular design. Beyond a small form factor, the module provides stable saturated absorption lines as frequency references and succeeds in achieving frequency stabilization control of a commercial 780 nm laser. Finally, the magneto-optical trap experiment based on the module was conducted in the laboratory, demonstrating its potential application in portable cold atom sensing systems.

The three-dimensional schematic of the miniature frequency-stabilization module, consisting of three parts—a copper substrate, an absorption optical path, and circuits, is shown in Fig. 1(a). The substrate served as the package for the entire module, supporting both the optical and electrical components. The optical path includes a collimating lens, beam splitter, miniature Rb vapor cell, attenuators, and mirrors. All the components were fixed onto the corresponding slots of the substrate using ultraviolet-curable adhesive. The detector and temperature controller circuits were assembled on the side of the optical path and integrated into the single printed circuit board (PCB). The detector is aligned with the beam splitter in the optical path, receiving and amplifying the saturated absorption signal. The temperature control circuit is responsible for controlling the temperature of the Rb vapor cell to ensure the stability of the module operation. Figure 1(b) shows the miniature frequency-stabilization module after the final installation and calibration. The overall dimensions of the module are 40 mm × 15 mm × 30 mm, and the mass is 180 g. This represents a 57% reduction in size compared to the existing work^[14].

In nature, Rb exists in two isotopes: ${}^{85}\mathrm{Rb}$ (72%) and ${}^{87}\mathrm{Rb}$ (28%) with slightly different electronic energy levels. The absorption transitions of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ D2 intertwine around the 780 nm wavelength^[16], as shown in Fig. 2. In atom cooling experiments, we lock the laser frequency onto the saturated absorption peak of the ${\mathit{F}}_{\mathrm{g}}=2\to {\mathit{F}}_{\mathrm{e}}=1$ transition of ${}^{87}\mathrm{Rb}$$5^2{\mathrm{S}}_{1/2}\to 5^2{\mathrm{P}}_{3/2}$.

The frequency-stabilization process of the laser, as shown in Fig. 3, is based on the FM spectroscopy method using the D2 saturated absorption line as the frequency reference. When working, the output light of the laser is phase-modulated, emitted, collimated to the Rb cell, attenuated, and retroreflected to pass the Rb vapor again with weaker intensity. Due to the overlapping of the two counterpropagating beams and their power difference, the absorption of the reflected beam is saturated by the incident (pump) beam, producing a series of sharp peaks on the spectrum (shown in Fig. 4), of which positions correspond to the different transitions of the Rb atoms. Then the reflected beam is deflected by the beam splitter to the detector on the side, where the light is converted into an electrical signal with the information of frequency difference between laser and saturated absorption carried by the modulation sideband. Then the received signal is transmitted to the digital control circuit, amplified, demodulated, and filtered to obtain the frequency error signal, whose voltage is proportional to the frequency difference within a limited range. The error signal is processed by the proportional-integral-derivative (PID) controller and driver to generate the current signal, which will be finally injected into the laser diode to correct the frequency offset. As the core of the module, the Rb vapor cell is a glass cube with a length of 10 mm and sidewall thickness of 1 mm, filled with a mixture of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$ atoms, which can endure a working temperature of less than 200°C. Owing to the short length of the miniature vapor cell compared with conventional centimeter-sized cells, the effective light absorption path is shorter^[17], thus making it difficult to observe saturated absorption signals with a sufficiently high signal-to-noise ratio (SNR) at room temperature, which will severely degrade the performance of the subsequent frequency-stabilization feedback operations. Therefore, a thermoelectric cooler (TEC) and a thermistor, both of which are connected with the temperature-control circuit of the module, are attached to the top of the vapor cell to heat it and increase the concentration of the Rb vapor, with the temperature accuracy of $\pm 0.1°\mathrm{C}$ mainly limited by the thermistor. When powered, the heating element can control the temperature of the top side of the cell to be higher than that of the bottom side, thus preventing the condensation of Rb atoms in the optical window^[18].

The performance of feedback control circuits plays a crucial role in the frequency-stabilized lasers. Traditional feedback control circuits, primarily composed of analog circuits, include devices such as sinusoidal wave generators, multipliers (mixers), phase shifters, low-pass filters, and PID controllers. However, these circuits suffer from significant drawbacks such as large zero drift, high levels of noise interference, low signal filtering accuracy, and poor flexibility, consequently struggling to meet the demands of high-precision laser frequency stabilization. To address this issue, this study proposes fully digital signal processing and feedback control circuits and algorithm based on a field-programmable gate array (FPGA). This approach involves implementing all the aforementioned devices using digital circuits, which significantly enhances signal accuracy and flexibility. The modulation signal required by the modulator was generated using direct digital synthesis (DDS), modulation frequency 5 MHz, and modulation depth $\sim 2$. The modulated laser then enters the saturated absorption optical path and outputs optical signals, which are converted into electrical signals by the detector, amplified, and then fed into a fully digital signal processing circuit based on the FPGA. The frequency reference signal required for frequency stabilization was obtained through demodulation and filtering processes. Finally, the PID controller with carefully-picked parameters was employed to achieve frequency stabilization of the laser. In the specific implementation, the core FPGA adopts a first-generation SoC architecture Zynq-7000 series chip from Xilinx. This chip integrates a rich functionality processing system based on dual- or single-core ARM Cortex-A9 and 28 nm Xilinx programmable logic controllers into a single device. The ARM Cortex-A9 CPU serves as the core of the processing system, running real-time operating systems, coordinating hardware resource configurations, and communicating with the host computer. The programmable logic controller is responsible for signal acquisition, waveform generation, filtering operation, and other logical operations. The low-pass filter adopts a 4th-order Butterworth low-pass filter with a 3 dB cutoff bandwidth of 10 kHz, which can be flexibly adjusted according to actual needs. Digital filters have been used to replace traditional analog filter circuits. The characteristics of traditional analog filters are determined by the properties of their hardware components such as resistors, capacitors, and inductors, whereas those of digital filters are primarily determined by their coefficient properties. Digital filters are not affected by hardware, thus improving the overall signal accuracy and significantly increasing the flexibility of the entire system^[19]. With the development of digital chip technology, in the future, the feedback circuit function can be achieved through the design of specialized chips to reduce the size of the digital feedback circuit, the digital feedback part of the circuit, and the miniature frequency-stabilization module integrated package to further reduce the size of the overall frequency-stabilization system.

The absorption length of the Rb atomic vapor cell affects the amplitude of the saturated absorption spectroscopy and error signals, thereby influencing the SNR of the frequency reference signal according to the Beer–Lambert law^[15]. Experiments were conducted at room temperature using two types of vapor cells with the same pressure and optical aperture of 10 mm but lengths of 10 and 50 mm. The saturated absorption spectra and frequency reference signals obtained under the same experimental conditions are shown in Fig. 4. The amplitude of the saturated absorption signal obtained from the 50 mm Rb vapor cell is greater than that obtained from the 10 mm Rb vapor cell. The saturated absorption intensity obtained from the 10 mm cell is weaker with the absorption spectral line SNR of 40 dB, and its corresponding frequency reference signal for ${}^{87}\mathrm{Rb}$${\mathit{F}}_{\mathrm{g}}=2\to {\mathit{F}}_{\mathrm{e}}=1$, has a smaller amplitude with significant quantization noise, thus not satisfying the requirements for frequency-stabilization operations. If a 10 mm Rb atomic vapor cell is used to build the saturated absorption optical path for the miniature frequency-stabilization module, the Rb vapor cell must be heated to increase the atom density inside the cell, which in turn increases the intensity of both the saturated absorption signal and the error signal, thereby improving the SNR.

Utilizing miniature Rb vapor cells can significantly enhance the integration and reliability of frequency-stabilization modules. However, owing to the limited optical path length of the vapor cell, the resulting saturated absorption signal exhibits a relatively poor SNR, failing to meet the requirements for frequency-stabilization operations. Therefore, heating and temperature-control mechanisms for vapor cells are essential. This approach aims to amplify the intensity of the saturated absorption signal, thereby locking the laser frequency to the absorption peak of ${}^{87}\mathrm{Rb}$. A temperature scan of the vapor cell was conducted within the range of 40°C–100°C at intervals of 10°C. The resulting saturated absorption spectra of the miniature Rb atomic vapor cells at different temperatures are shown in Fig. 5. Notably, as the temperature increased, the intensity of the saturated absorption signal increased progressively. As shown in Fig. 5, when the vapor cell was heated to 90°C, the signal intensity of the saturated absorption spectrum reached its maximum value.

Theoretically, the intensity of the transmitted light passing through the vapor cell can be expressed using Eq. (1), $$I_T=I_0\exp [-n\sigma (v)L],$$(1)where $I_0$ indicates the incident light intensity, $n$ is the atomic number density of Rb atoms in the Rb gas chamber, $L$ is the length of the Rb gas chamber, and $\sigma (v)$ is the photon absorption cross section, as obtained from Eq. (2), $${\int }_0^{+\infty }\sigma (v)\mathrm{d}v=\pi c{r}_{\mathrm{e}}{f}_{\text{res}},$$(2)where $r_{\mathrm{e}}$ is the electron radius, $c$ is the speed of light in vacuum, and $f_{\text{res}}$ is the transition oscillator strength, corresponding to the proportion of a given resonance in the total cross section. For Rb atoms, the oscillator strength of the D2 line is approximated by $f_{{\mathrm{D}}_2}\approx \frac{2}{3}$^[20]. The number density of Rb atoms increases with the temperature of the vapor cell, as inferred from Eq. (1), which indicates a decrease in the intensity of light transmitted through the atomic vapor cell, leading to an increase in the absorption rate of Rb atoms within the cell. Consequently, the amplitude of the absorption peaks in the saturated absorption spectrum increased continuously with increasing temperature. Furthermore, when the temperature reaches 100°C, the signal tends to be smoothed, and some portions of the saturated absorption signal diminish. This phenomenon arises because of the exponential growth trend of the total atomic number density within the Rb cell with increasing temperature, significantly suppressing the burn-hole effect and weakening of the saturated absorption intensity^[17,21].

A commercial 780 nm laser (Precilasers FL-SF-780) was frequency-stabilized using this miniature module, with the variations of the laser frequency over 1000 s measured by a HighFinesse wavelength meter for both the free-running and stabilized conditions. The results are shown in Fig. 6 and are compared with the commercial stabilization setup. The peak-to-peak value of the frequency drift of the free-running laser within 1000 s was 22 MHz, whereas the frequency change after locking using the frequency-stabilization module was only 1 MHz, comparable to the commercial stabilization setup. Moreover, the performance of the module could be improved by optimizing the structure design, broadening feedback loop bandwidth, and reducing the noise of electronic devices. Integrating a powerful digital controller into the module may be able to solve the above issues simultaneously, which is difficult currently but promising in the future with the development of microelectronics.

Finally, the laser light stabilized by the frequency-stabilization module is shifted by phase lock loop and an acoustic-optic modulator to obtain the cooling and pumping light required for the cold atom experiment, which realizes the laser cooling of ${}^{87}\mathrm{Rb}$ atoms. Figure 7(a) shows the magneto-optical trap used to obtain cold atoms, while Fig. 7(b) shows a photo of the cold atom cloud obtained by a CCD camera after using the frequency-stabilized module proposed in this paper to stabilize the frequency of the laser. The diameter of the cold atom cluster is 3 mm, and the number of cold atoms obtained is $1\times {10}^8$, indicating that the cooling operation of the atoms can be realized after applying the frequency-stabilization module to stabilize the frequency-aligned laser, thus meeting the application requirements of miniaturized and portable equipment.

We designed and implemented a miniaturized laser frequency-stabilization module for cold atom sensing that provides the means for inherent rigidity and path stability. In addition, the optical components and its closely packaged control electronics could be contained in a single ruggedized module enabling deployment on small space platforms. The overall size of the designed miniaturized frequency-stabilization module was 40 mm × 15 mm × 30 mm, with temperature control of the Rb cell using a temperature control circuit to meet the requirements of portable devices. The laser frequency fluctuation after being locked by the module was 1 MHz, satisfying the requirements of the cold atom experiments. Such a stabilization module obviously has a very simple and compact setup, which is attractive for applications where simplicity, robustness, and low cost are of interest.