Optical frequency standards are renowned for their superior stability and accuracy (below 10−18) compared to their microwave counterparts, primarily due to the higher quality factors of optical transitions^[1–4]. In the domain of precision timekeeping and frequency standards, the quest for miniaturization without compromising accuracy has become a focal point of current research. Compact optical frequency standards, which leverage the unique properties of atomic transitions, are emerging as pivotal components in various advanced technological applications, from global positioning systems (GPS) to telecommunications and beyond^[5–9]. By utilizing thermal atomic vapor cells, compact optical frequency standards can achieve competitive frequency precision^[10–22].

To further reduce system size, adopting miniaturized devices, including miniaturized atomic vapor cells, is essential. These miniature cells, typically filled with atomic vapors such as rubidium, are the core components of compact optical frequency standards. However, smaller atomic vapor cells inherently produce a lower signal-to-noise ratio, potentially limiting the system’s achievable frequency precision. Consequently, the development of compact optical frequency standards using miniaturized atomic vapor cells has become a significant area of research. Key spectroscopic methods explored in this context include saturation absorption spectroscopy (SAS)^[10,11], two-photon spectroscopy^[12–14], and dual-frequency sub-Doppler spectroscopy^[23–25]. These studies have shown that miniature vapor cells can achieve high-frequency precision through appropriate spectroscopic techniques. Nevertheless, to the best of our knowledge, optical frequency standards based on miniature vapor cells and modulation transfer spectroscopy (MTS) have not yet been validated, even though MTS has emerged as a powerful method for enhancing the signal-to-noise ratio (SNR), making it an ideal technique for thermal atomic cell-based frequency standards^[6,18–21].

References [18–21] have demonstrated MTS using alkali-metal thermal vapor cells, achieving short-term frequency stability on the order of 10−13 or better in optical frequency standards. Typically, vapor cells with lengths of 4–10 cm are employed to ensure a sufficient number of interacting atoms and a high SNR. However, for micro-atomic frequency standards, the vapor cell volume is generally required to be less than 1 cm^3[26–29]. Therefore, to advance the development of miniaturized optical frequency standards based on MTS, it is crucial to explore whether such standards can be realized in vapor cells smaller than 1cm3 while maintaining high SNRs and excellent frequency stability.

This research aims to explore the potential of a compact optical frequency standard based on the MTS technique using a miniature Rb87 glass cell, with dimensions of merely 6mm×6mm×6mm. Through meticulous experimentation and analysis, this study will demonstrate the feasibility and performance of such a system, highlighting its linewidth and short-term stability to be 6.9 kHz and 4.1×10−13@1s. By providing a detailed blueprint for developing small-scale optical frequency standards, this study lays the groundwork for subsequent innovations in chip-level optical frequency standard technologies.

The experimental setup for the miniature rubidium atomic optical frequency standard is shown in Fig. 1(a). Post-isolation, the laser beam is split into two paths using a half-wave plate (HWP) and a polarizing beam splitter (PBS). One beam, with an approximate power of 11 mW, is directed toward the optical frequency standard output. The second beam, with a power of 1 mW, is employed for MTS stabilization. This 1 mW beam is further divided into pump and probe beams using an additional HWP and PBS, and the intensities of both beams can be finely adjusted. The pump beam undergoes phase modulation using an electro-optic modulator (EOM) before entering the miniature atomic vapor cell through a third HWP and PBS, where it interacts with the rubidium atoms. The third HWP is optimized to maximize the light intensity reflected into the rubidium vapor cell. Simultaneously, the probe beam, reflected by a mirror, enters the vapor cell and interacts with the modulated pump beam and rubidium atoms. The vapor cell is heated by a TEC temperature controller (THORLABS TC300) to obtain more gaseous atoms. The temperature setting point resolution is 0.1°C, and the temperature stability is ±0.1°C. A high-speed photodetector captures the post-interaction probe beam. The detected signal is mixed with the EOM drive signal to demodulate the MTS. The resulting signal is then filtered, amplified, and fed into a proportional-integral-derivative (PID) circuit (Vescent D2-125). The PID circuit outputs are connected to the piezoelectric transducer (PZT) terminal, the current terminal, and the laser head of the laser driver power supply. These terminals exhibit distinct feedback response bandwidths: approximately 1 kHz for the PZT terminal, around 10 kHz for the current terminal, and up to 1 MHz for the laser head. The feedback gains and frequencies for these terminals are independently adjustable. By synchronizing these components, the system loop bandwidth is maximized.

Figure 1(b) presents the physical layout of the MTS system, with optical dimensions of 365mm×205mm×76mm. The atomic vapor cell, filled with Rb87 atoms, measures 6mm×6mm×6mm. The device employs a homemade narrow linewidth interference filter laser as the clock laser, emitting light with a wavelength of 780.46 nm. The laser diode is packaged in a TO9 housing, and the external cavity is based on an interference filter structure. The overall dimensions of the laser are approximately 120mm×65mm×50mm, with a typical Gaussian beam profile and a beam waist radius of 840μm×850μm.

By scanning the laser frequencies, we obtain the modulation transfer spectra, as illustrated in Fig. 2(a). In this figure, the blue line corresponds to the saturated absorption spectrum (SAS) of the rubidium atoms, while the black line represents the MTS. It is evident that the maximum amplitude of the MTS occurs at the cyclic transition of the rubidium atoms from 52S1/2 (F=2) to 52P3/2 (F'=3). Consequently, we can effectively lock the laser to this specific cyclic transition.

We evaluated the peak-to-peak intensity and slope of the modulation transfer spectrum at various atomic vapor cell temperatures, as illustrated in Fig. 2(b). As the temperature of the atomic vapor cell increases, the atomic number density also rises, leading to an increase in the peak-to-peak value of the modulation transfer spectrum. However, when the atomic number density becomes significantly high, the saturation absorption effect on the pump light diminishes, resulting in a decrease in both the saturated absorption spectrum and the peak-to-peak value of the modulation transfer spectrum. Notably, at 70°C, the peak-to-peak value of the modulation transfer spectrum reaches its maximum. Concurrently, with rising temperatures, the collisional broadening of the atoms causes an increase in the linewidth of the modulation transfer spectrum. The slope of the modulation transfer spectrum is influenced by both the peak value and the linewidth. As shown in Fig. 2(b), the optimal slope value is observed at approximately 62.5°C.

By adjusting the error signal via PID control, laser servo feedback enables closed-loop control of an optical frequency standard. In this closed-loop system, frequency stability within the loop is a critical performance indicator and is closely related to the slope of the error signal. We assessed the in-loop frequency stability of the system, also known as self-assessment frequency stability, at various temperatures. The results, depicted in Fig. 3, illustrate the variation of self-assessment frequency stability over time at different temperatures. It is evident that at 62.5°C, the self-assessment stability reaches its minimum, achieving a stability level of 5.5 × 10⁻¹⁵@1 s. This indicates that the system demonstrates optimal stability and accuracy at this temperature. At this point, the SNR within the loop can exceed a level of 4.5×106.

To evaluate the system’s stability outside the loop, we established two systems for beat frequency comparison. The frequency beat signal is sent to a frequency counter (Keysight 53230 A) with a reference signal from an Rb microwave clock, and the result is shown in Fig. 4(a). In the short term, the stability of the beat signal over time follows the relationship 3.67×10−13/τ. Significantly, the frequency stability shows an upward trend when the time constant is greater than 1 s. The stability at 1 s for the beat frequency signal is 5.8×10−13, corresponding to a stability of 4.1×10−13@1s for each system when divided by 2. Figure 4(b) illustrates the primary factors affecting frequency stability. For averaging time in the 1–20 s range, temperature-induced frequency shifts in the atomic vapor cell, and the EOM have a significant impact. After 20 s, the residual amplitude modulation effect caused by the EOM becomes the dominant factor limiting frequency stability. In future work, precise temperature control of the EOM will be implemented to optimize mid- and long-term frequency stability.

Compared to optical frequency standards based on large-volume cells, our system exhibits comparable stability in both the inner and outer loops^[18–22]. Achieving high-precision locking with smaller cells suggests that for quantum references, small-volume cells are adequate to attain high SNRs, thereby supporting high-precision frequency stabilization.

During the locking process, the laser linewidth can be significantly narrowed. In Fig. 5(a), we present the Lorentzian linewidth of the beat frequency signals from 100 sets of our custom-built freely running lasers. The most probable beating linewidth is approximately 65 kHz. Figure 5(b) shows a typical beat frequency signal from a freely running laser, fitted with a Lorentzian function, resulting in a linewidth of 64.2 kHz, which corresponds to an individual laser linewidth of 45.4 kHz. In Fig. 5(c), we illustrate the beat frequency linewidth distribution of the two laser systems after locking, with the most probable linewidth being 10 kHz. Figure 5(d) displays the Lorentzian fit of the beat frequency signal after locking, yielding a linewidth of 9.8 kHz, corresponding to an individual system linewidth of 6.9 kHz. This demonstrates the effectiveness of our locking mechanism in narrowing the laser linewidth.

To the best of our knowledge, this study presents the first demonstration of a compact rubidium atomic optical frequency standard using the MTS with a miniature Rb87 glass cell with a size of 6mm×6mm×6mm. By employing the transition spectrum from 52S1/2(F=2) to 52P3/2(F′=3), we achieve a 780 nm optical frequency standard with a linewidth of 6.9 kHz and an out-of-loop short-term stability of 4.1×10−13@1s. These results were validated through beat frequency analysis, demonstrating the system’s ability to achieve high-precision locking with smaller cells. This research provides a practical blueprint for developing small-scale optical atomic frequency standards and lays essential groundwork for future advancements in chip-level optical frequency standard technologies. The study’s outcomes suggest that further enhancements in temperature control can lead to greater long-term stability.