Fig. 1. (a) Experimental setup of the MTS system, with an identical MTS system set up for heterodyne measurement. 780 nm IF ECDL, 780 nm interference filter configuration external cavity diode laser; HWP, half-wave plate; PBS, polarizing beam splitter; PD, photodetector; SIG, signal generator; PID, proportion-integral-derivative locking system. Inset: hyperfine levels of the ⁸⁷Rb 5²S_1/2 and 5²P_3/2 energy levels. (b) Physical layout of the MTS system, with optical dimensions of 365 mm × 205 mm × 76 mm. The laser measures 120 mm × 65 mm × 50 mm. The atomic vapor cell, filled with ⁸⁷Rb atoms, measures 6 mm × 6 mm × 6 mm.

Fig. 2. (a) SAS and MTS spectra of the ⁸⁷Rb 5²S_1/2 (F = 2) − 5²P_3/2 (F′ = 3) transition, the maximum amplitude of the MTS occurs at the cyclic transition. (b) MTS signal slope at zero crossing and amplitude measured at different cell temperatures. The optimal slope value is observed at approximately 62.5°C.

Fig. 3. The 1, 10, and 100 s Allan deviation of the system at various vapor cell temperatures. The figure illustrates that both the 1-s and 10-s stability metrics initially decrease and then increase as the vapor cell temperature rises, reaching optimal stability at 62.5°C.

Fig. 4. (a) Allan deviation of the beating frequency. In the short term, the stability of the beat signal over time follows the relationship 3.67 × 10⁻¹³/τ. After 1 s, the frequency stability shows an upward trend. The stability at 1 s for the beat frequency signal is 5.8 × 10⁻¹³, corresponding to a stability of 4.1 × 10⁻¹³ @1 s for each system. (b) 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.

Fig. 5. (a) Lorentzian linewidth of the beat frequency signals from 100 sets of freely running lasers. The most probable beating linewidth is approximately 65 kHz. (b) Typical beating data of two identical lasers with the resolution band width (RBW) set to 10 kHz. The Lorentz fitted linewidth is 64.2 kHz, corresponding to an individual laser linewidth of 45.4 kHz. (c) The beat frequency linewidth distribution of the two laser systems after locking. The most probable linewidth is around 10 kHz. (d) Lorentzian fit of the typical beat frequency signal after locking, with an RBW of 3 kHz and a linewidth of 9.8 kHz, corresponding to an individual system linewidth of 6.9 kHz.