Optical frequency standards referenced to thermal atoms, employing sub-Doppler spectroscopy techniques, including saturation absorption spectroscopy (SAS) [1–3], polarization spectroscopy [4–6], two-photon spectroscopy [7–10], dual-frequency sub-Doppler spectroscopy [11–13], frequency modulation spectroscopy [14,15], and modulation transfer spectroscopy (MTS) [16–20], have facilitated advancements in precision measurement [10,21,22], optical clocks [10,23–25], cold atom physics [26,27], and navigational systems [28,29]. Benefiting from enhanced sensitivity and noise rejection, which yield a high signal-to-noise ratio and superior frequency instability, MTS and frequency modulation spectroscopy are essential for the stabilization of compact laser systems [24]. They have been effectively applied to iodine [30], acetylene [31], and alkali atoms [32–34], achieving the best short-term instability in thermal atomic ensembles to date [24,30,35]. Utilizing the quantum transitions of iodine and acetylene molecules facilitates optical frequency standards at diverse wavelength regions, crucial for the realization of meter precision metrology [36] and optical telecommunication [37,38]. In various application scenarios, such as constructing optical frequency networks [39,40] and cold atom physics, the establishment of multi-wavelength frequency standards necessitates multiple independent quantum ensembles, increasing system complexity. Nevertheless, the potential for realizing multiple wavelength standards within a single quantum ensemble remains uninvestigated. Here, we implement modulation transfer spectroscopy between different quantum transitions to establish the first dual-wavelength optical frequency standard based on a single quantum ensemble.

Research on modulation transfer spectroscopy has primarily focused on single quantum transitions [41–44], where both pump and probe lights originate from the same laser beam and modulation is transferred through the same transition, thereby limiting its application to single-quantum transition laser stabilization. Modulation transfer between different quantum transitions was preliminarily realized using Ne transitions [45], with the initial theoretical analysis in 1986 [46]. After years of stagnation, in 2015, progress was made by transferring modulation from a prelocked 780 nm laser in a separate atomic vapor cell to a 1529 nm laser, resulting in the locking of a second atomic vapor cell at 1529 nm with a frequency instability of 3.3×10−12τ/s, derived from the error signal [47]. In 2022, following the work made in 2015, a four-level modulation transfer scheme involving 770 nm, 1529 nm, and 780 nm was locked with three independent atomic vapor cells independently for 780 nm, 1529 nm, and 770 nm [48]. In this paper, we demonstrate dual-optical-transition modulation transfer spectroscopy (DOT-MTS) using solely a single atomic cell, where the modulation frequency sidebands of the pump beam, resonant with the Rb87 D2 line (780 nm), are simultaneously transferred to both the 780 nm (resonant with the Rb87 D2 line) and 795 nm (resonant with the Rb87 D1 line) probe beams. 780 and 795 nm lasers are simultaneously stabilized on a thermal atomic ensemble through the modulation transfer. The resultant frequency instabilities reach the low 10−14 level at 1 s of averaging, as estimated from the residual error signal. The approach enables the realization of a dual-wavelength or even a multi-wavelength frequency standard. And a theoretical model based on the V-type atomic level structure is developed to illustrate the dual-wavelength spectroscopy. To our knowledge, this work represents the first demonstration of dual-wavelength laser stabilization in a single atomic gas cell, enabling the realization of multiple quantum standards within the same system to enhance coherence, necessitating further investigation into the underlying fundamental physics and complex correlations.

The schematic of the experimental setup and the energy level structure of rubidium atoms are shown in Fig. 1. The 780 nm and 795 nm laser beams are generated from two external-cavity diode lasers (ECDL 780 and ECDL 795, respectively) followed by an isolator to avoid the optical feedback. First we employ the common MTS to lock the 780 nm laser to the rubidium D2 line, where the 780 nm laser is divided into two (strong pump and weak probe) beams by a half-wave plate and a polarization beam splitter (PBS). The pump light (0.5 mW) passes through a Rb87 vapor cell and is modulated by an EOM at the frequency of 6.5 MHz and the probe light (0.1 mW) counter-propagates with the pump. The ratio of pump to probe power can be adjusted precisely through two half-wavelength plates. The direct current (DC) signal of the first photoelectric detector (PD1) is used to derive the SAS of 780 nm, via scanning the 780 nm laser. As shown in Fig. 2(a), the SAS of the Rb87 D2 line contains three resonance peaks and three crossover peaks. The alternating current (AC) signal after amplification is mixed with the demodulation signal by a laser servo to derive the 780 nm MTS [see Fig. 2(a)]. Afterwards the laser servo provides the first proportional integral differential feedback to stabilize the 780 nm laser to the 52S1/2F=2→52P3/2F=3 transition, which shows the steepest slope in MTS, in Rb87 atoms.

The 795 nm laser is also divided into two beams. One is used to derive the SAS of 795 nm, which features two resonance peaks and one crossover peak [see Fig. 2(b)] for calibration. The other passes through the M1 mirror and is coincident with the 780 nm probe light. M1 and M2 are dichroic mirrors to separate 780 and 795 nm lasers. Since the 780 nm laser has been stabilized to the 52S1/2F=2→52P3/2F=3 transition, it couples with the 795 nm probe light through atoms with zero-velocity in the ground state F=2. As a result, the modulation on the 780 nm pump light is transferred to the 795 nm probe light. After passing through the M2 mirror, the 795 nm probe light is incident on PD2. The 780–795 nm SAS reveals velocity transfer peaks that are induced by the 780 nm pump light and the 780 nm probe light [49], alongside the standard saturated absorption peaks [see Fig. 2(b) red circle].

The amplified AC signal is mixed with the demodulation signal to derive the 780–795 nm DOT-MTS, which is used to lock the 795 nm laser to the 52S1/2F=2→52P1/2F=1 transition in Rb87 atoms. Thus, the initial stabilization of the 780 nm laser via MTS is essential to the overall stabilization process. As shown in Fig. 2(b), the DOT-MTS includes two dispersion curves corresponding to the transitions 52S1/2F=2→52P1/2F=1 and F=2. Under identical conditions including the atomic ensemble, modulation frequency δ, and modulation depth, the amplitudes of the 780 nm MTS and the DOT-MTS are essentially the same. However, the linewidth of the DOT-MTS is twice as broad as that of the single-frequency 780 nm MTS. This results in a twofold difference in their signal gradients, consistent with the theoretical explanations provided (see below). Figure 2(c) illustrates the signal gradients of the 780 nm MTS and DOT-MTS across various δ, both exhibiting maxima at 6.5 MHz. The corresponding spectral linewidth for the 52S1/2F=2→52P3/2F=3 transition is approximately 25 MHz.

We model the DOT-MTS using the four-wave mixing in a V-type atomic system. As illustrated in Fig. 3(a), the ground state and the two excited states of Rb87 are labeled as |1⟩, |2⟩, and |3⟩, corresponding to states 52S1/2F=2, 52P3/2F=2, 52P1/2F=1, respectively. The Lindblad master equation describes the dynamics of the V-system: ρ^˙=1iℏ[H^0+V^,ρ^]+γ2D[c^2]ρ^+γ3D[c^3]ρ^,(1)where ρ^ is the density matrix operator, V^ is the interaction Hamiltonian, c^2=|1⟩⟨2|, c^3=|1⟩⟨3|, D[c^]ρ^=c^ρ^c^†−12(c^†c^ρ^+ρ^c^†c^) is the Lindblad superoperator, and γ2 and γ3 are relaxation rates of excited levels |2⟩ and |3⟩. The Hamiltonian of the unperturbed atomic system is given by H0=ℏω21c^2†c^2+ℏω31c^3†c^3, where ωij is the transition frequency between |i⟩ and |j⟩.

In the DOT-MTS, the pump beam for the transition |1⟩→|2⟩ is modulated, while the counter-propagating probe beam is detected over the transition |1⟩→|3⟩. Here we consider only the carrier and the first-order sidebands of the pump beam due to the small modulation index. Consequently, the pump beam is described by Epump=E02[J0(κ)eikcz−iωct+J1(κ)eikcz−i(ωc+δ)t−J1(κ)eikcz−i(ωc−δ)t+c.c], where E0 is the amplitude, κ is the modulation index, Jn(κ) is the nth-order Bessel function, ωc is the frequency of the carrier, kc=ωc/c is the wave vector of the carrier, and δ is the modulation frequency. Assuming δ is small compared to ωc, the wave vectors of the sidebands are approximately equal to that of the carrier. Similarly, the counter-propagating probe beam can be described by Eprobe=Ep2[e−i(ωpt+kpz)+c.c], where Ep, kp, and ωp are the amplitude, wave vector, and frequency of the probe beam, respectively. Therefore, the interaction Hamiltonian can be given by $$\begin{gathered} V^=−ℏ2[(Ωceikcz−iωct+Ωseikcz−i(ωc+δ)t \\ −Ωseikcz−i(ωc−δ)t)⁢(c^2+c^2†) \\ −Ωpe−ikpz−iωpt(c^3+c^3†)]+h.c. \end{gathered}$$(2)Here Ωk=μijEk/ℏ is the Rabi frequency for k=c (carrier), s (sidebands), and p (probe). μij is the matrix element of the dipole operator related to |i⟩ and |j⟩ and is assumed to be real.

Then, we use the third-order perturbation theory to describe the four-wave mixing and modulation transfer process. In four-wave mixing, new photons carrying modulation information are re-emitted along the same direction as the probe beam, and their generation is proportional to the induced nonlinear macroscopic polarization of the system. By applying the rotating wave approximation and solving for the system’s steady state, we obtain the third-order perturbation density matrix element ρ13(3) related to the macroscopic polarization, which in the laboratory frame can be written as P(3)=μ31∫∞∞ρ13(3)(v)f(v)dv+c.c, where f(v)=e−v2/u2/uπ is the Maxwell velocity distribution and u is the most probable speed of atoms.

For an optically thin sample, the re-emitted light fields in the four-wave mixing process can be described as Ere=−ikpL2ε0P(3), where kp is the wave vector of the probe beam, L is the length of the vapor cell, and ε0 is the vacuum permittivity. The frequency and wave vector of the new laser generated by four-wave mixing satisfy the momentum and energy conservation conditions: ωl−ωm+ωn=ωr,kl−km+kn=kr, where the subscript symbols are shown in Fig. 3(a). Table 1 summarizes the four combinations of frequency and wave vector conditions, as well as the detuning of the three-photon resonance peaks relative to the one-photon resonance frequency ω0 of the probe transition, similar to the situation in MTS described in the literature [41,42].Table 1.

Frequency Combinations Generating Sidebands in the Probe Light during Four-Wave Mixing

Following the theoretical framework, the demodulated spectrum of the DOT-MTS is shown in Fig. 3(b), featuring three distinct spectra: in-phase, mixed components, and quadrature phases. The red line represents the DOT-MTS line shape, while the blue curve shows the normal 780 nm MTS signal for comparison. Phase changes in MTS and DOT-MTS patterns experimentally match theoretical results, as shown in Fig. 3(c). We find that the DOT-MTS linewidth is approximately twice that of the MTS, despite nearly identical signal amplitudes, which is consistent with the phenomenon observed in hole burning [50] and our experiment. The theoretical δ is 4.2 MHz (equivalent to 0.7γ2/2π), differing from the experimental 6.5 MHz because saturation effects are not considered in the theoretical simulations.

Moreover, numerical calculations indicate that DOT-MTS is primarily constrained by probe relaxation and the ratio of pump to probe wavelengths. Corresponding energy level diagrams are also provided as shown in Fig. 4(a). The normalized zero crossing gradients of DOT-MTS for various transitions are depicted as a function of δ in Figs. 4(b) and 4(c). Here, the gradient of the 780 nm MTS is used for normalization purposes. For all depicted curves, the pump laser intensity is consistently set at one-tenth of the saturation intensity. For transitions with similar wavelengths and emission rates, such as 780–795 nm or 420–421 nm, the DOT-MTS gradient is approximately half that of the single-quantum-transition MTS, with twice the linewidth. When the spontaneous emission rates of two optical transitions differ significantly, modulation transfer efficiency at a constant pump Rabi frequency is limited by the probe transition’s emission rate. In our study, the 780–420 nm DOT-MTS is limited by the 420 nm transition’s radiation rate. With significant wavelength differences between two optical transitions, the smaller wavelength ratio between the pump and probe transitions in a four-wave mixing process causes the two external three-photon resonance peaks (see Table 1) to converge inward. This convergence interferes with the error signal, reducing the DOT-MTS gradient. Our study on 420–780 nm DOT-MTS exemplifies this effect, primarily due to the Doppler shift in thermal optical transitions.

We also investigate the lineshape, considering saturation effects, using numerical methods for the 780–795 nm DOT-MTS [51]. The effects of spectral broadening and increased amplitude due to saturation are depicted in Fig. 4(d), with each case normalized to when the saturation parameter equals one. The results illustrate notable changes in the lineshape: as the saturation parameter increases, both the broadening of the frequency and the amplitude of the line shape are enhanced. Figure 4(e) illustrates the slope gradient, taking into account saturation effects, and reveals a maximum at a saturation parameter of approximately 15, which is selected as the experimental operating point. The corresponding DOT-MTS experimental and theoretical results, depicted in Fig. 4(f), emphasize this peak gradient and further confirm the validity of the theoretical model.

Through theoretical analysis, DOT-MTS can be employed for multi-wavelength locking, thereby establishing a multi-wavelength frequency standard within the same atomic ensemble. Experimentally validating this, we applied the 780 nm MTS and the DOT-MTS to stabilize the 780 nm and 795 nm lasers, respectively. We investigated the instability of both lasers using the residual error signal, with the Allan deviation results presented in Fig. 5. Figure 5(a) displays the 1 s and 10 s instability for both lasers at varying temperatures of the atomic cell. At the optimal temperature of 40°C, the instability of the 780 nm laser is 1.13×10−14 at 1 s, and the instability of the 795 nm laser is 2.2×10−14 at 1 s. The long-term instability is depicted in Fig. 5(b), where the instability of the 795 nm laser is found to be twice as large as that of the 780 nm laser. This discrepancy aligns with the difference in the slope of their error signals. Consequently, the 780 and 795 nm lasers are both stabilized to the same atomic ensemble through the utilization of the DOT-MTS, demonstrating excellent in-loop locking precision. The self-estimated instability confirms the dual-wavelength optical frequency standard’s ability to “closely track” the atomic frequency and verifies its coherence. However, it typically degrades by an order of magnitude compared to that measured via heterodyne methods between two identical systems [24]. The instability of our system, constrained by the inherent frequency fluctuations of the ensemble, can be optimized by extending the approach to atomic beam or cold atom systems, thereby establishing a more precise multi-wavelength standard in the future.

This study successfully demonstrates DOT-MTS on the Rb87 D1 and D2 lines, with both 780 and 795 nm lasers frequency-stabilized to a single atomic cell via DOT-MTS, achieving high locking precision. We also developed a general expression for DOT-MTS based on a V-type energy level configuration. This theoretical model not only supports our experimental results but also enhances the understanding of the modulation transfer mechanisms. The thermal atomic ensemble may experience frequency fluctuations and long-term drift due to atomic collisions and motion, yet the atomic resonance frequency shifts due to external factors such as temperature and magnetic fields, and the locking points of both wavelengths shift in tandem, ensuring high coherence between the dual-wavelength frequency standards. Extending this approach to other atomic systems, such as the K39 D1 (770 nm) and D2 (766 nm) lines, enables the realization of an integrated optical-THz frequency reference. Similarly, for the Rb87 D1 line (795 nm) and ground-state microwave transition, it is possible to achieve an integrated optical-microwave frequency reference.

Looking ahead, future work could focus on extending the DOT-MTS to a multi-frequency scheme, incorporating V-type, Λ-type, and ladder-type configurations in various atomic structures. Next, we will focus on the iodine molecule, with its rich spectral features, which holds great potential for developing multi-wavelength standards that can enhance precision measurements. These advancements would expand applications beyond optical clocks and length metrology, establishing a new frontier in multi-wavelength optical frequency standards realized with a single quantum atomic ensemble and significantly advancing laser stabilization, with far-reaching implications for quantum metrology.