1. Introduction

Metrology deals with high precision measurements of physical parameters, among which frequency can be determined with the highest degree of accuracy using atomic clocks [1,2]. Thus far, optical clocks have surpassed their microwave counterparts in both stability and accuracy by over two orders of magnitude [3–5], which paves the way towards an optical redefinition of the SI second. Most optical clocks are operated in a passive fashion, where the frequency of a pre-stabilized laser is referenced to a narrow-line optical transition of carefully engineered atoms [6]. Active optical clocks, which directly produce highly stable optical frequency standards without extra steps of stabilizing the laser frequency to an atomic transition, have also been proposed [7] and demonstrated [8]. Indeed, the underlying mechanism of such active operation is the substantial suppression of the cavity pulling effect on the lasing dynamics in the so-called bad-cavity limit. Frequency comparison is essential for evaluating the performance of an atomic clock and optical frequency combs bridge frequency standards from microwave to optical domain with orders of magnitude difference, facilitating the frequency comparison between different atomic clocks [9]. Yet, to our best knowledge, the direct frequency comparison between passive and active optical clocks through a frequency comb has not been demonstrated.

Engineering integrated metrology systems that include optical frequency comparison, synchronization, and synthesis benefits broad out-of-the-lab applications such as satellite-based geo-positioning and communication [10]. Its implementation demands a vast simplification of the metrology configuration and the subtle design of each component, for example, reducing the numbers of light sources and optical cavities and minimizing their physical volumes. Owing to ultrahigh $Q$ factors and small mode volumes, whispering-gallery-mode (WGM) microresonators enable nonlinear frequency conversion at low pump thresholds and large conversion efficiencies at room temperature [11–13]. In particular, microresonator-based frequency combs, i.e., microcombs, which make use of the nonlinear Kerr process and four-wave mixing, possess the advantages of miniaturized size, simple structure, and low power consumption, compared to conventional optical frequency combs. Recently, microcombs have been employed in diverse fields, including atomic and molecular spectroscopy [14], integrated photonics for optical communications and data processing [15], and chip-scale frequency metrology [16]. Various approaches of fully stabilizing microcombs have been exploited, for example, referencing two comb modes respectively to two atomic transitions through frequency doubling [17] and $f-2f$ self-referencing in an octave-spanning spectrum [18]. Nevertheless, the microcomb spectra mainly cover the infrared regime while the wavelengths of most optical clocks are in the visible band. Extending the microcomb spectra to the visible regime is still challenging [19,20].

In this paper, we propose a hybrid clock scheme, where the full stabilization of a Kerr microcomb is implemented using two optical atomic clocks that are operated in distinct, i.e., passive and active, modes. Both clock wavelengths, 1359 and 1377 nm, are in the infrared regime that is accessible by the microcomb spectrum. The whole system contains only one light source that also serves as a passive optical clock via indirectly locking its frequency to an atomic transition. Other optical modes at different wavelengths are generated through the third harmonic (TH) nonlinear optical process and the lasing action. We evaluate the performance of optical clocks and microcomb through the simulation analysis. The numerical results show that the frequency stabilities of passive and active optical clocks follow $9.3\times 10^{-14}\tau ^{-1/2}$ and $6.2\times 10^{-15}\tau ^{-1/2}$, respectively, with the averaging time $\tau$, and the mode spacing of the microcomb has an Allan deviation of $\sigma _{y}(\tau )=1.9\times 10^{-11}\tau ^{-1/2}$. Recent technologies in (micro)photonics and atomic physics ensure the successful implement of the proposed hybrid scheme.

2. Physical system

Figure 1(a) illustrates the hybrid clock scheme, in which there is only one light source at $\lambda _{\textrm {clock1}}=1377$ nm. This light source may be a commercial laser system. As we will see below, the light source can serve as a passive optical clock by indirectly locking its frequency to an atomic transition. The laser beam is split into two sub-beams. One sub-beam enters the microcavity I, where the TH optical wave at $\lambda _{\textrm {TH}}=459$ nm is generated. The output TH light excites the ensemble I of $^{133}$Cs atoms from the ground $|1\rangle =6s~^{2}S_{1/2}$ state to the excited $|4\rangle =7p~^{2}P_{1/2}$ state. Atoms in $|4\rangle$ emit fluorescent photons that are collected by a photodetector (PD). The electrical signal produced by the PD is fed back into the light source, indirectly stabilizing its central frequency $\omega _{\textrm {clock1}}=2\pi c/\lambda _{\textrm {clock1}}$ to the atomic $|1\rangle -|4\rangle$ transition. Here, $c$ is the speed of light in vacuum. Passing an optical amplifier (OA), the TH light further drives the $^{133}$Cs ensemble II that is placed inside a low-$Q$ (bad) optical cavity (frequency $\omega _{L}$). This cavity is resonantly coupled with the $|2\rangle =6p~^{2}P_{1/2}-|3\rangle =7s~^{2}S_{1/2}$ transition in $^{133}$Cs. A sufficient driving strength creates the population inversion between two atomic states and, as a result, the lasing action occurs. Due to the substantial suppression of the cavity pulling effect, this bad-cavity laser is operated as an active optical clock at $\lambda _{\textrm {clock2}}=1359$ nm [7]. The other sub-beam of the laser source (i.e., passive optical clock) is coupled into the microcavity II, where an optical frequency comb (microcomb) is created via the nonlinear Kerr process. The beat note between microcomb and active clock laser is used to control the light power launched into the microcavity II, thereby stabilizing the mode spacing of the microcomb [21].

 

Fig. 1. Hybrid clock system. (a) Schematic diagram. One sub-beam from a light source at $\lambda _{\textrm {clock1}}=1377$ nm undergoes the TH nonlinear process and drives the fluorescence of $^{133}$Cs atoms. The light source serves as a passive optical clock when its frequency is stabilized using the fluorescence signal. The TH light further pumps an active clock laser at $\lambda _{\textrm {clock2}}=1359$ nm. The other sub-beam from the light source pumps a Kerr microcomb. (b) Energy-level structure of $^{133}$Cs. The $6s~^{2}S_{1/2}-7p~^{2}P_{1/2}$ transition serves as the frequency reference line for the passive optical clock and the pump line for the active optical clock. The $6p~^{2}P_{1/2}-7s~^{2}S_{1/2}$ transition plays the role of the active clock laser transition. (c) Allan deviation of the free-running light source (open circles), the passive optical clock (filled circles), and the active optical clock (filled squares). Solid curves: curve fitting. Dashed line: quantum noise limit.

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In our hybrid scheme, two atomic ensembles play different roles, where the ensemble I is used to stabilize the central frequency of the passive optical clock while the ensemble II acts as the optical gain medium for the active clock laser. Figure 1(b) depicts the energy level structure of $^{133}$Cs. For simplicity, here we do not consider the hyperfine structure of $^{133}$Cs. The atomic $|1\rangle -|4\rangle$ transition at 459 nm is used as the indirect frequency reference for the passive optical clock and the pump line for the active optical clock. The atomic $|2\rangle -|3\rangle$ transition at 1359 nm works as the laser transition that couples to the bad cavity. We use $\gamma _{ij}$ with ($i,j=1,2,3,4$) to denote the (effective) decay rate of the atom from $|i\rangle$ to $|j\rangle$ with $\gamma _{21}=2\pi \times 4.6$ MHz, $\gamma _{31}=2\pi \times 1.8$ MHz, $\gamma _{32}=2\pi \times 1.0$ MHz, $\gamma _{41}=2\pi \times 0.5$ MHz, $\gamma _{42}=2\pi \times 0.02$ MHz, and $\gamma _{43}=2\pi \times 0.6$ MHz [22]. It should be noted that although, for example, the $|1\rangle -|3\rangle$ transition is electric-dipole forbidden, atoms in $|3\rangle$ can decay to $|1\rangle$ through the spontaneous emission from $|3\rangle$ to $6p~^{2}P_{3/2}$ and further to $|1\rangle$, leading to the effective decay rate $\gamma _{31}$. In this work, we assume that two atomic ensembles have been carefully engineered such that Doppler effect and interatomic collisions are negligible.

In what follows, we discuss the passive and active optical clocks and the microcomb in detail.

3. Third harmonic generation and passive optical clock

The light at $\lambda _{\textrm {TH}}=459$ nm can be generated through the TH nonlinear optical process in the silica WGM microcavity I (for example, toroid with a major radius of 13 $\mu$m and a minor radius of 3 $\mu$m [12]), where the sub-beam (power $P$ and frequency $\omega _{\textrm {clock1}}$) from the light source excites a WGM that we label with "$p$" (frequency $\omega _{p}$ and quality factor $Q_{p}$) at 1377 nm via a tapered optical fiber and the TH optical wave is resonant to a WGM that we label with "$s$" (frequency $\omega _{s}$ and quality factor $Q_{s}$), as shown in Fig. 1(a). We write the intracavity fields of the two WGMs as $E_{j=p,s}(\textbf {r},t)=\frac {1}{2}a_{j}(t)\psi _{j}(\textbf {r})e^{-i\omega _{j}t}$ with amplitudes $a_{j}(t)$ and normalized WGMs $\psi _{j}(\textbf {r})$. It is convenient to introduce the amplitudes $b_{j}(t)=[\frac {\epsilon _{0}}{2}\int n_{j}^{2}|\psi _{j}(\textbf {r})|^{2}d\textbf {r}]a_{j}(t)$ with the vacuum permittivity $\epsilon _{0}$ and refractive indices $n_{j}$, and $|b_{j}(t)|^{2}$ correspond to the intracavity energies of two fields. The coupling dynamics between $b_{p}(t)$ and $b_{s}(t)$ is described by [23]

In the steady state ($ss$), one obtains $(1+xy^{4})y=1$ with $x=3(\frac {2Q_{s}}{\omega _{s}})(\frac {2Q_{p}}{\omega _{p}})^{5}(|\gamma _{\textrm {TH}}|\kappa _{p}P)^{2}$ and $y=(\frac {\omega _{p}}{2Q_{p}})\frac {|b_{p,ss}|}{\sqrt {\kappa _{p}P}}$ for the resonant pump $\omega _{\textrm {clock1}}=\omega _{p}$. The TH conversion efficiency is then given by $\eta _{\textrm {TH}}=\kappa _{s}\frac {|b_{s,ss}|^{2}}{3P}=(\frac {\kappa _{s}Q_{s}}{\omega _{s}})(\frac {\kappa _{p}Q_{p}}{\omega _{p}})4xy^{6}$. Here, we have defined $b_{j,ss}=b_{j}(t\rightarrow \infty )$ with $j=p,s$ and $\kappa _{s}$ ($<\frac {\omega _{s}}{Q_{s}}$) accounts for the coupling rate of the TH signal out of the microcavity I. The efficiency $\eta _{\textrm {TH}}$ reaches its maximum $\textrm {max}(\eta _{\textrm {TH}})=(\frac {\kappa _{s}Q_{s}}{\omega _{s}})(\frac {\kappa _{p}Q_{p}}{\omega _{p}})$ when $x=16$ and $y=1/2$ (Supplement 1). Generally, the pump mode is a fundamental WGM and $\kappa _{p}$ is equal to half of $\frac {\omega _{p}}{Q_{p}}$ under the critical coupling point [25]. In contrast, the TH coupling efficiency is relatively low because of the high-order transverse mode. Substituting typical values $Q_{s,p}=10^{7}$, $\frac {\kappa _{s}Q_{s}}{\omega _{s}}=0.5~\%$, and $\gamma _{\textrm {TH}}=7\times 10^{18}$ J$^{-1}$ s$^{-1}$, we have $\textrm {max}(\eta _{\textrm {TH}})=0.25~\%$ with the corresponding pump power $P=2.8$ mW. The resultant output power of the TH light reaches $\kappa _{s}|b_{s,ss}|^{2}=7.0$ $\mu$W.

In practice, environmental noise perturbs the frequencies of light source and WGMs, $\omega _{\textrm {clock1}}=\omega _{\textrm {clock1},0}+\delta \omega _{\textrm {clock1}}(t)$, $\omega _{p}=\omega _{p,0}+\delta \omega _{p}(t)$, and $\omega _{s}=\omega _{s,0}+\delta \omega _{s}(t)$ with constant values $(\omega _{\textrm {clock1},0},~\omega _{p,0},~\omega _{s,0})$ and fluctuations $(\delta \omega _{\textrm {clock1}}(t),~\delta \omega _{p}(t),~\delta \omega _{s}(t))$. The TH generation requires $\omega _{s,0}=3\omega _{p,0}$. Generally, the frequency noise spectrum of the free-running light source $S_{\textrm {clock1}}(f)=\omega _{\textrm {clock1},0}^{-2}\int \langle \delta \omega _{\textrm {clock1}}(t)\delta \omega _{\textrm {clock1}}(t+\tau )\rangle e^{-i2\pi f\tau }d\tau$ takes the form $S_{\textrm {clock1}}(f)=h_{\textrm {clock1},0}f^{0}+h_{\textrm {clock1},-2}f^{-2}$. The white frequency component ($\propto f^{0}$) determines the laser linewidth $\Delta \omega _{\textrm {clock1}}=\omega _{\textrm {clock1},0}^{2}h_{\textrm {clock1},0}/2$ while the random walk component ($\propto f^{-2}$) leads to the long-term drift of the laser frequency. The typical $\Delta \omega _{\textrm {clock1}}=2\pi \times 1$ MHz gives $h_{\textrm {clock1},0}=6.7\times 10^{-24}$ s and the corresponding short-term Allan deviation is $\sigma _{y}(\tau )=\sqrt {h_{\textrm {clock1},0}/2\tau }=1.8\times 10^{-12}\tau ^{-1/2}$. Similarly, we have the frequency noise spectra of two WGMs [26], $S_{p}(f)=\omega _{p,0}^{-2}\int \langle \delta \omega _{p}(t)\delta \omega _{p}(t+\tau )\rangle e^{-i2\pi f\tau }d\tau =h_{p,0}f^{0}+h_{p,-2}f^{-2}$ and $S_{s}(f)=\omega _{s,0}^{-2}\int \langle \delta \omega _{s}(t)\delta \omega _{s}(t+\tau )\rangle e^{-i2\pi f\tau }d\tau =h_{s,0}f^{0}+h_{s,-2}f^{-2}$ with $h_{p,0}=\frac {2}{Q_{p}\omega _{p,0}}=1.5\times 10^{-22}$ s and $h_{s,0}=\frac {2}{Q_{s}\omega _{s,0}}=5.0\times 10^{-23}$ s. For simplicity, we assume $h_{-2}=h_{\textrm {clock1},-2}=h_{p,-2}= h_{s,-2}=1.0\times 10^{-20}$ s$^{-1}$ with the corresponding long-term Allan deviation $\sigma _{y}(\tau )=\sqrt {2\pi ^{2}h_{-2}\tau /3}=2.5\times 10^{-10}\tau ^{1/2}$.

The absolute frequency $\omega _{\textrm {clock1}}$ of the light source may be indirectly stabilized to an atomic transition. We let the output TH light $\sqrt {\kappa _{s}}b_{s}$ (frequency $\omega _{\textrm {TH}}=2\pi c/\lambda _{\textrm {TH}}=3\omega _{\textrm {clock1}}$) further derive the $|1\rangle -|4\rangle$ transition (frequency $\omega _{41}$) of the $^{133}$Cs ensemble I (atom number $N_{at}$). The dynamics of atomic variables is governed by the following $c$-number Langevin equations [27] (Supplement 1)

Dropping fluctuation terms $\delta \omega _{\textrm {clock1},p,s}(t)$ and $F_{\alpha }(t)$ with $\alpha =N_{1,2,3,4}$ and $M_{14}$ in Eqs. (1)–(7), one may compute the dependence of the steady-state population $N_{4,ss}=N_{4}(t\rightarrow \infty )$ of atoms in $|4\rangle$ on the pump detuning $\Delta _{p}=\omega _{\textrm {clock1},0}-\omega _{p,0}$. As shown in Fig. 2, the atomic excitation exhibits a Lorentzian lineshape with a full width at half maximum (FWHM) $\Gamma _{p}=2\pi \times 1.4$ MHz, close to the natural linewidth of the $|1\rangle -|4\rangle$ transition. Atoms in $|4\rangle$ emit fluorescent photons at 459 nm, which are measured by a photodetector (PD). Detecting fluorescence allows for locking the frequency $\omega _{\textrm {clock1}}(t)$ to the atomic $|1\rangle -|4\rangle$ transition in an indirect manner. We set the locking point at $\Delta _{p}=-\Gamma _{p}/2$, where the gradient of the frequency discrimination curve is $k=dN_{4,ss}/d\Delta _{p}=0.44$ s.

 

Fig. 2. Dependence of the fluorescence response on the pump detuning $\Delta _{p}=\omega _{\textrm {clock1},0}-\omega _{p,0}$. The frequency $\omega _{\textrm {clock1}}(t)$ of the light source is locked to the side of the fluorescence curve. The dashed line corresponds to the gradient $k=dN_{4,ss}/d\Delta _{p}$ at the locking point. Inset: Frequency fluctuations $\delta \omega _{\textrm {clock1}}(t)$ of the free-running/frequency-locked light source. The stabilized light source is operated as a passive optical clock.

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According to the frequency noise spectra $(S_{\textrm {clock1}}(f),~S_{p}(f),~S_{s}(f))$, one may numerically produce the frequency fluctuations $(\delta \omega _{\textrm {clock1}}(t),~\delta \omega _{p}(t),~\delta \omega _{s}(t))$ by digitally filtering a stochastic white field [28]. Langevin forces $F_{\alpha }(t)$ with $\alpha =N_{1,2,3,4}$ and $M_{14}$ can be generated following the method in [29,30] (Supplement 1). Thus, we simulate the frequency stabilization of the light source based on Eqs. (1)–(7). The integration time of the detection electronics is set to be $T_{i}=0.1$ ms. Within the time period from $nT_{i}$ to $(n+1)T_{i}$ with $n\in \mathbb {Z}$, the number of fluorescent photons collected by the photodetector is $N_{ph}=\gamma _{41}\int N_{4}(t)dt+\Delta N_{ph}$. The extra term $\Delta N_{ph}$ is a random number that originates from the shot noise whose mean is zero and standard deviation is $N_{ph}^{1/2}$. Here, we have assumed 100-percent efficiency of the photon detection. The average value of $\delta \omega _{\textrm {clock1}}(t)$ within this integration period is then derived as $\langle \delta \omega _{\textrm {clock1}}\rangle _{T_{i}}=(N_{ph}-N_{ph,bia})/(k\gamma _{41}T_{i})$, where $N_{ph,bia}$ denotes the steady-state number of fluorescent photons at the locking point. As a result, the light source frequency $\omega _{\textrm {clock1}}$ is corrected accordingly in the next integration period.

The Allan deviation $\sigma _{y}(\tau )$ of the simulated $\omega _{\textrm {clock1}}$ is summarized in Fig. 1(c). The free-running $\omega _{\textrm {clock1}}$ follows the short-term $\sigma _{y}(\tau )=1.8\times 10^{-12}\tau ^{-1/2}$ and the long-term $\sigma _{y}(\tau )=2.5\times 10^{-10}\tau ^{1/2}$. In contrast, the stabilized $\omega _{\textrm {clock1}}$ obeys the short-term $\sigma _{y}(\tau )=9.3\times 10^{-14}\tau ^{-1/2}$, over one order of magnitude better than that of the free-running laser, and the long-term $\sigma _{y}(\tau )=5.2\times 10^{-13}\tau ^{1/2}$, corresponding to an improvement factor of about 500. Consequently, the light source at $\lambda _{\textrm {clock1}}$ is operated as an optical frequency standard whose frequency is passively stabilized to the atomic $|1\rangle -|4\rangle$ transition in an indirect manner. The short-term stability is better than the recently demonstrated compact optical atomic clock, where a 1556-nm laser beam is frequency doubled (via the second harmonic generation) and further stabilized to the 778-nm two-photon transition in rubidium [31]. We compute the stability limit set by atomic fluctuations, $\sigma _{y}(\tau )=\frac {\Gamma _{p}}{\omega _{\textrm {clock1},0}}\frac {1}{\textrm {SNR}}\sqrt {\frac {T_{i}}{\tau }}=9.1\times 10^{-14}\tau ^{-1/2}$ with the signal-to-noise ratio $\textrm {SNR}=\frac {N_{4,ss}}{2}\sqrt {\frac {\Gamma _{41}}{2D(N_{4},N_{4})}}$ and the diffusion coefficient $2D(N_{4},N_{4})$ associated with the population $N_{4}$ of atoms in $|4\rangle$ (Supplement 1). In addition, the stability limited by the photon shot noise is evaluated as $\sigma _{y}(\tau )=\frac {\Gamma _{p}}{\omega _{\textrm {clock1},0}}\frac {1}{\sqrt {N_{ph}}}\sqrt {\frac {T_{i}}{\tau }}=3.1\times 10^{-15}\tau ^{-1/2}$. Therefore, the stability of the passive optical clock is primarily limited by atomic fluctuations.

4. Active optical clock

As shown in Fig. 1(a), after passing an optical amplifier (OA), the TH light $\sqrt {\kappa _{s}}b_{s}$ interacts with the $^{133}$Cs ensemble II. For simplicity, we still use the symbols $N_{i=1,2,3,4}$, $M_{14}$, $N_{at}$, and $\Omega$ to denote the population of atoms in $|i\rangle$, the macroscopic polarization corresponding to the $|1\rangle -|4\rangle$ transition, the atom number, and the driving strength. Here, we assume that the frequency $\omega _{\textrm {TH}}$ of the TH light has been shifted to $\omega _{41}$. Atoms in $|1\rangle$ are excited to $|4\rangle$ and then decay to $|3\rangle$, creating the population inversion between $|2\rangle$ and $|3\rangle$. A low-$Q$ cavity (frequency $\omega _{L}$ and loss rate $\kappa _{L}$) is coupled with the $|2\rangle -|3\rangle$ transition (frequency $\omega _{32}$ and transition dipole moment $d_{32}=1.6~ea_{0}$) with the single-photon coupling strength $g=-\sqrt {\frac {\omega _{L}}{2\hbar \epsilon _{0}V_{L}}}d_{32}=-2\pi \times 10$ kHz and the cavity mode volume $V_{L}=\pi (0.5~\textrm {mm})^{2}(5~\textrm {cm})$. The lasing action (wavelength $\lambda _{\textrm {clock2}}=1359$ nm) occurs once the optical gain overcomes optical losses. Following the Heisenberg–Langevin approach [27], the equations of motion of intracavity field $A$ and atomic variables are derived as (Supplement 1)

Let us first consider the steady state of the laser system in the resonant coupling situation, $\omega _{L}=\omega _{32}$ and $\omega _{\textrm {TH}}=\omega _{41}$. The corresponding analytical solutions are given in Supplement 1. Figures 3(a) and (b) illustrate the dependence of the output power $P_{\textrm {clock2}}=\kappa _{L}\hbar \omega _{L}|A_{ss}|^{2}$ with $A_{ss}=A(t\rightarrow \infty )$ and the laser linewidth $\Delta \omega _{\textrm {clock2}}$ on the cavity loss rate $\kappa _{L}$ and the atom number $N_{at}$. In the good-cavity regime ($\kappa _{L}<\Gamma _{32}$), the required minimal $N_{at}$ for the lasing action can be small, for example, $N_{at}\sim 10^{5}$ with $\kappa _{L}/\Gamma _{32}=0.1$, corresponding to $Q_{L}=\omega _{L}/\kappa _{L}=5\times 10^{8}$. The linewidth $\Delta \omega _{\textrm {clock2}}$ reaches the Hz level (or even less than 1 Hz). However, in this regime the central frequency of the clock laser suffers huge cavity frequency fluctuations through the cavity pulling effect, consequently deteriorating the laser frequency stability [32]. As $\kappa _{L}$ grows, the increased cavity loss strongly elevates the threshold and $\Delta \omega _{\textrm {clock2}}$ also goes up. Nevertheless, in the bad-cavity regime ($\kappa >\Gamma _{32}$), $P_{\textrm {clock2}}$ can exceed 1 $\mu$W, which is high enough for locking the phase of an optical local oscillator. Interestingly, once the system enters the bad-cavity regime, $\Delta \omega _{\textrm {clock2}}$ starts declining as $\kappa _{L}$ is increased. According to [33], the quantum-limited linewidth of a laser takes the form $\Delta \omega _{\textrm {clock2}}=\Delta \omega _{\textrm {ST}}(1+\kappa _{L}/2\Gamma _{32})^{-2}$ with the usual Schawlow–Townes linewidth $\Delta \omega _{\textrm {ST}}$. When $\kappa _{L}\gg \Gamma _{32}$, the factor $(1+\kappa _{L}/2\Gamma _{32})^{-2}$ decreases rapidly, thereby suppressing $\Delta \omega _{\textrm {clock2}}$. The underlying mechanism may be ascribed to the fact that the phase diffusion process gets slowed down due to the memory effect of the relatively long-lifetime polarization of atoms [34]. We choose $N_{at}=2.5\times 10^{8}$, i.e., the two atomic ensembles have the same atom number. As shown in Fig. 3(c), in the bad-cavity regime $\Delta \omega _{\textrm {clock2}}$ reaches its minimum $\textrm {min}(\Delta \omega _{\textrm {clock2}})=2\pi \times 2.8$ Hz at $\kappa /\Gamma _{32}=90$, where, in what follows, we set the operating point of the active clock laser. The corresponding cavity quality factor is $Q_{L}=6\times 10^{5}$ and the output power $P_{\textrm {clock2}}$ approximates 20 $\mu$W.

 

Fig. 3. Steady state of the active optical clock. (a) Dependence of the output power $P_{\textrm {clock2}}$ on the cavity loss rate $\kappa _{L}$ and the atom number $N_{at}$. The dashed line denotes the laser threshold. The dash-dotted line corresponds to the boundary between good- and bad-cavity regimes. (b) Laser linewidth $\Delta \omega _{\textrm {clock2}}$ vs. $(N_{at},\kappa _{L})$. (c) Dependence of $P_{\textrm {clock2}}$ and $\Delta \omega _{\textrm {clock2}}$ on $\kappa _{L}$ with $N_{at}=2.5\times 10^{8}$. The five-pointed star symbol: The operating point of the active optical clock is set at $\kappa _{L}/\gamma _{32}=90$, where $P_{\textrm {clock2}}=20$ $\mu$W and $\Delta \omega _{\textrm {clock2}}=2\pi \times 2.8$ Hz (five-pointed star symbol in (a) and (b)). (d) Power density spectrum of the active clock laser at the operating point. The curve fitting gives the FWHM of $2\pi \times 3$ Hz. For all plots, the pump strength is $|\Omega |/\Gamma _{41}=5$.

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We now consider the frequency stability of the active clock laser. In the adiabatic approximation, $\kappa _{L}\gg \Gamma _{41,32}$, one has $A(t)=[\kappa _{L}/2+i(\omega _{L}-\omega _{32})]^{-1}gM_{23}(t)$. It is seen that two factors, noise in $\omega _{L}(t)=\omega _{L,0}+\delta \omega _{L}(t)$ and fluctuations of the polarization $M_{23}$, influence $A(t)$. The resonant atom–cavity coupling gives $\omega _{L,0}=\omega _{32}$. Generally, the power spectral density of the cavity frequency noise $S_{L}(f)=\omega _{L,0}^{-2}\int \langle \delta \omega _{L}(t)\delta \omega _{L}(t+\tau )\rangle e^{-i2\pi f\tau }d\tau$ takes the form $S_{L}(f)=h_{L,0}f^{0}+h_{L,-1}f^{-1}+h_{L,-2}f^{-2}$ with $h_{L,0}=2\kappa _{L}/\omega _{L,0}^{2}=2.3\times 10^{-21}$ s. Here, we ignore the flicker frequency noise component ($\propto f^{-1}$), which originates from Brownian thermal-mechanical fluctuations of cavity mirrors, due to the low $Q_{L}$ and assume $h_{L,-2}=h_{-2}$. One may simulate the dynamics of the active optical clock by numerically generating the frequency noise $\delta \omega _{L}(t)$ and Langevin fluctuations $F_{\alpha }(t)$ with $\alpha =N_{1,2,3,4}$ and $M_{14,23}$ according to $S_{L}(f)$ and diffusion coefficients $2D(\alpha,\beta )$ of atomic variables (Supplement 1). Figure 3(d) depicts the laser spectrum $\mathcal {S}_{\textrm {clock2}}(\omega )=\int \langle A(t)A(t+\tau )\rangle e^{-i\omega \tau }d\tau$ in the absence of fluctuations in $\omega _{L}$ and $b_{s}$. The corresponding spectral broadening completely arises from the fluctuations associated with atomic variables. The curve fitting gives the FWHM of $2\pi \times 3$ Hz, well matching the analytical result of $2\pi \times 2.8$ Hz and verifying the validity of the simulation. Since this $\Delta \omega _{\textrm {clock2}}$ is entirely induced by atomic fluctuations, the corresponding Allan deviation $\sigma _{y}(\tau )=\omega _{\textrm {clock2},0}^{-1}(\Delta \omega _{\textrm {clock2}}/\tau )^{1/2}=3.0\times 10^{-15}\tau ^{-1/2}$ is referred to as the quantum-noise-limited stability.

Considering the fluctuations in $\omega _{L}$ and $b_{s}$, we perform the long-term simulation of the active clock dynamics, compute the clock frequency $\omega _{\textrm {clock2}}$ (see Supplement 1), and evaluate its Allan deviation. The simulation result is $\sigma _{y}(\tau )=6.2\times 10^{-15}\tau ^{-1/2}$ (see Fig. 1(c)), which is higher than the quantum-noise-limited stability due to the residual cavity-pulling effect. As we will see below, this active optical clock can be used to stabilize a Kerr microcomb.

5. Microcomb

Thus far, we have only considered one sub-beam from the light source (i.e., passive optical clock). As shown in Fig. 1(a), the other sub-beam whose power is enhanced using a semiconductor optical amplifier (SOA) drives a certain mode (i.e., pump mode) in the silica microcavity II that may be a toroid microresonator (major radius of 109 $\mu$m and minor radius of 3 $\mu$m). A sufficient input power excites multiple WGMs with stable phases around the pump mode through the nonlinear Kerr process and four-wave mixing [37], forming an optical frequency microcomb [38]. For the sake of simplicity, we still use the symbols $\omega _{p}$, $Q_{p}$, and $\Delta _{p}=\omega _{\textrm {clock1}}-\omega _{p}$ to denote the pump mode frequency, the corresponding quality factor, and the detuning between the passive clock laser and the pump mode, respectively.

The dynamics of the microcomb field $E_{\textrm {comb}}(t,\theta )$ (in units of W$^{1/2}$) is governed by the Lugiato–Lefever equation [39]

 

Fig. 4. Kerr microcomb. (a) Dependence of the group refractive index $n_{g}$ on the wavelength change $\delta \lambda =\lambda -\lambda _{\textrm {clock1}}$ and temperature $T$ around ($\lambda _{\textrm {clock1}}=1377$ nm, $T_{0}=293$ K). (b) Single-soliton microcomb. The 10th comb mode on the blue-detuned side of the pump mode is near resonant to the active optical clock. Inset: single soliton. (c) Allan deviation $\sigma _{y}(\tau )$ of the mode spacing $f_{\textrm {rep}}(t)$ for the microcomb in the free-running/stabilized state. Inset: fluctuations $\delta f_{\textrm {rep}}(t)$ of the mode spacing when the feedback loop is switched off/on.

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Table 1. Physical parameters of the microresonator-based optical frequency comb at $\lambda _{\textrm {clock1}}$ and room temperature $T_{0}=293$ K.

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Figure 4(b) shows the microcomb spectrum that is numerically derived from Eq. (15) with the input power $P=1.5$ mW and the detuning $\Delta _{p}=-3(\omega _{p}/Q_{p})$. Multiple sidebands are located around the pump mode (corresponding to the highest peak). The entire microcomb is characterized by two degrees of freedom, namely the offset frequency of the carrier envelope and the mode spacing (repetition frequency) $f_{\textrm {rep}}$. The microcomb is in the single-soliton state, leading to $f_{\textrm {rep}}=\textrm {FSR}$. The first degree of freedom has already been controlled since the passive optical clock directly drives one comb mode. Next we consider the stabilization of $f_{\textrm {rep}}$.

In practice, the temperature $T$ fluctuates around $T_{0}$. The small mode volume of the microcavity makes the mode spacing $f_{\textrm {rep}}$ sensitive to the fluctuation $\delta T(t)=T-T_{0}$ that is averaged over the whole mode volume. In the linear approximation, the change of the group index $n_{g}$ caused by $\delta T$ is expressed as $\delta n_{g}=n_{g}(T)-n_{g}(T_{0})=\eta _{n}\delta T$ with $\eta _{n}=\partial n_{g}(T_{0})/\partial T=1.6\times 10^{-5}$ K$^{-1}$. Here, we neglect the thermo-mechanical effects that change the circulating length $L$. The induced fluctuation of $f_{\textrm {rep}}$ is then written as $\delta f_{\textrm {rep}}(t)=-[\eta _{n}f_{\textrm {rep},0}/n_{g}(T_{0})]\delta T(t)$ with $f_{\textrm {rep},0}=c/Ln_{g}(T_{0})$. According to the experimental results in [18,20], the typical Allan deviation of $y(t)=\delta f_{\textrm {rep}}(t)/f_{\textrm {rep},0}$ follows $\sigma _{y}(\tau >5\times 10^{-2}~\textrm {s})=5\times 10^{-8}\tau ^{1/2}$. Thus, the long-term drift of $\delta T(t)$ has $\sigma _{y}(\tau )=\sqrt {2\pi ^{2}h_{T,-2}\tau /3}$ with $y(t)=\delta T(t)/T_{0}$ and $h_{T,-2}=3.7\times 10^{-11}$ s$^{-1}$. To verify this, we numerically generate the temperature fluctuations $\delta T(t)$ and compute the mode spacing $f_{\textrm {rep}}(t)$ by solving Eq. (15), where we have used $\textrm {GVD}(T)=\textrm {GVD}(T_{0})+\eta _{\textrm {GVD}}\delta T$ with $\eta _{\textrm {GVD}}=-10.1$ fs$^{2}$ mm$^{-1}$ K$^{-1}$ and $\tau _{R}(T)=\tau _{R}(T_{0})+\eta _{\tau }\delta T$ with $\eta _{\tau }=3.7\times 10^{-5}$ ps K$^{-1}$. In addition, the pump mode frequency can be approximated by $\omega _{p}(T)=\omega _{p}(T_{0})+\eta _{\omega }\delta T$ with $\eta _{\omega }=-2\pi \times 2.4$ GHz K$^{-1}$ and the stabilized clock frequency $\omega _{\textrm {clock1}}$ follows the Allan deviation shown in Fig. 1(c). The numerical $\sigma _{y}(\tau )$ of $\delta f_{\textrm {rep}}(t)$ is presented in Fig. 4(c), well matching the analytical expression.

In experiment, the change of the mode spacing can be derived by measuring the beat note $i_{\textrm {beat}}(t)=\textrm {Re}[A^{\ast }(t)E_{m}(t)]$ between the active optical clock $A(t)$ and its nearest mode $E_{m}(t)$, whose frequency is $\omega _{p}+mf_{\textrm {rep}}$ with $m=10$, in the microcomb $E_{\textrm {comb}}(t,\theta )$. The beat frequency is set at $\omega _{\textrm {beat}}=2\pi \times 50$ MHz. We compare the beat note with a local RF signal at $\omega _{\textrm {beat}}$ and extract the phase difference $\Delta \phi$ over the integration time $T_{i}$ of the detection electronics. Consequently, the average value $\langle \delta f_{\textrm {rep}}\rangle _{T_{i}}$ of $\delta f_{\textrm {rep}}(t)$ within $T_{i}$ is given by $\langle \delta f_{\textrm {rep}}\rangle _{T_{i}}=\frac {\Delta \phi /T_{i}}{2\pi m}+\frac {\omega _{\textrm {beat}}}{2\pi m}\frac {\Delta N_{ph}}{\sqrt {\xi N_{1}N_{2}}}$. The extra noise $\Delta N_{ph}$ represents the shot noise that has a zero mean and a standard deviation $\sqrt {N_{1}+N_{2}}$ with $N_{1}=(\frac {2Q_{p}}{\omega _{p}\tau _{R}})^{2}[1+(\frac {2\Delta _{p}Q_{p}}{\omega _{p}})^{2}]^{-1}\frac {PT_{i}}{2\hbar \omega _{\textrm {clock1}}}$ and $N_{2}=\frac {P_{\textrm {clock2}}T_{i}}{\hbar \omega _{\textrm {clock2}}}$. Here, $\xi =6.6\times 10^{-3}$ accounts for the energy percentage of $E_{m}(t)$ in $E_{\textrm {comb}}(t,\theta )$ and we have assumed the critical coupling between the fiber and the microcavity II.

The dynamics of temperature fluctuations $\delta T(t)$ follows the equation [42]

One may numerically simulate the stabilization of the mode spacing $f_{\textrm {rep}}(t)$. We firstly calculate the average value $\langle \delta f_{\textrm {rep}}\rangle _{T_{i}}$ of $\delta f_{\textrm {rep}}(t)$ within the integration time $T_{i}=10^{-2}$ s. The temperature change averaged over $T_{i}$ is then computed as $\langle \delta T\rangle _{T_{i}}=\langle \delta f_{\textrm {rep}}\rangle _{T_{i}}/(\eta _{\tau }f_{\textrm {rep},0}^{2})$. The corresponding change of the light energy absorbed by the microcavity II reads $\langle \delta \mathcal {E}\rangle _{T_{i}}=(\Gamma _{T}/\eta _{T})\langle \delta T\rangle _{T_{i}}$. Consequently, the input power $P$ is corrected by $(\frac {\omega _{p}\tau _{R}}{2Q_{p}})^{2}[1+(\frac {2\Delta _{p}Q_{p}}{\omega _{p}})^{2}]\langle \delta \mathcal {E}\rangle _{T_{i}}/T_{i}$ for the next integration period. Numerical results are presented in Fig. 4(c). The Allan deviation of the stabilized $f_{\textrm {rep}}(t)$ follows $\sigma _{y}(\tau )=1.9\times 10^{-11}\tau ^{-1/2}$, matching the shot-noise limit $\sigma _{y}(\tau )=\frac {\omega _{\textrm {beat}}}{2\pi mf_{\textrm {rep},0}}\sqrt {\frac {N_{1}+N_{2}}{\xi N_{1}N_{2}}}\sqrt {\frac {T_{i}}{\tau }}=1.8\times 10^{-11}\tau ^{-1/2}$.

6. Discussion

In summary, we have proposed a hybrid clock scheme, where a passive optical clock pumps an active optical clock and a Kerr microcomb whose mode spacing is stabilized using the active optical clock. The passive optical clock is locked to one side of the fluorescence spectrum of the $6s~^{2}S_{1/2}-7p~^{2}P_{1/2}$ transition in $^{133}$Cs via the TH generation. The frequency stabilization may be improved by locking the clock frequency to the zero crossing point in the modulation transfer spectroscopy for $^{133}$Cs [43]. The cancellation of Doppler broadening allows the use of thermal atoms, simplifying the preparation of the atomic ensemble. In particular, the passive optical clock can be miniaturized by confining $^{133}$Cs atoms in a chip-scale vapor cell [16].

The active optical clock, i.e., bad-cavity laser, produces an infrared frequency standard that is located in the microcomb spectrum. The pump energy is supplied by the passive optical clock through the TH generation. Although in this work we focus on the atomic $6p~^{2}P_{1/2}-7s~^{2}S_{1/2}$ (clock) and $6s~^{2}S_{1/2}-7p~^{2}P_{1/2}$ (pump) transitions in $^{133}$Cs, one may also choose the $6p~^{2}P_{3/2}-7s~^{2}S_{1/2}$ clock transition at 1470 nm. In addition, the active optical clock scheme is applicable to rubidium atoms with clock $5p~^{2}P_{1/2,3/2}-6s~^{2}S_{1/2}$ (wavelengths of 1323 and 1366 nm) and pump $5s~^{2}S_{1/2}-6p~^{2}P_{1/2}$ (wavelength of 421 nm) transitions. Despite the absence of the Allan deviation measurement, active optical clocks have been demonstrated in [44,45] with thermal atoms. Employing cold atoms as the gain medium potentially improves the clock stability [46].

Two degrees of freedom of the microcomb are stabilized using passive and active optical clocks. Alternatively, one may re-design the microcavity II to reduce the FSR down to tens of GHz. Thus, the mode spacing $f_{\textrm {rep}}$ can be directly measured using a fast photodiode and the pump power launched into the microcavity II is controlled accordingly. In this case, the active clock is separated from the stabilization loop of the microcomb, allowing for performing the frequency comparison between two optical clocks with distinct (i.e., passive vs. active) operation modes through the microcomb.