Proposal for a hybrid clock system consisting of passive and active optical clocks and a fully stabilized microcomb

: Optical atomic clocks produce highly stable frequency standards and frequency combs bridge clock frequencies with hundreds of terahertz difference. In this paper, we propose a hybrid clock scheme, where a light source pumps an active optical clock through a microresonator-based nonlinear third harmonic process, serves as a passive optical clock via indirectly locking its frequency to an atomic transition, and drives a chip-scale microcomb whose mode spacing is stabilized using the active optical clock. The operation of the whole hybrid system is investigated through simulation analysis. The numerical results show: (i) The short-term frequency stability of the passive optical clock follows an Allan deviation of σ y ( τ ) = 9.3 × 10 − 14 τ − 1 / 2 with the averaging time τ , limited by the population fluctuations of interrogated atoms. (ii) The frequency stability of the active optical clock reaches σ y ( τ ) = 6.2 × 10 − 15 τ − 1 / 2 , which is close to the quantum noise limit. (iii) The mode spacing of the stabilized microcomb has a shot-noise-limited Allan deviation of σ y ( τ ) = 1.9 × 10 − 11 τ − 1 / 2 . Our hybrid scheme may be realized using recently developed technologies in (micro)photonics and atomic physics, paving the way towards on-chip optical frequency comparison, synthesis, and synchronization.


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][4][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][12][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 × 10 −14 τ −1/2 and 6.2 × 10 −15 τ −1/2 , respectively, with the averaging time τ, and the mode spacing of the microcomb has an Allan deviation of σ y (τ) = 1.9 × 10 −11 τ −1/2 .Recent technologies in (micro)photonics and atomic physics ensure the successful implement of the proposed hybrid scheme.

Physical system
Figure 1(a) illustrates the hybrid clock scheme, in which there is only one light source at λ 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 λ TH = 459 nm is generated.The output TH light excites the ensemble I of 133 Cs atoms from the ground |1⟩ = 6s 2 S 1/2 state to the excited |4⟩ = 7p 2 P 1/2 state.Atoms in |4⟩ 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 ω clock1 = 2πc/λ clock1 to the atomic |1⟩ − |4⟩ 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 ω L ).This cavity is resonantly coupled with the |2⟩ = 6p 2 P 1/2 − |3⟩ = 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 λ 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].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⟩ − |4⟩ 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⟩ − |3⟩ transition at 1359 nm works as the laser transition that couples to the bad cavity.We use γ ij with (i, j = 1, 2, 3, 4) to denote the (effective) decay rate of the atom from |i⟩ to |j⟩ with γ 21 = 2π × 4.6 MHz, γ 31 = 2π × 1.8 MHz, γ 32 = 2π × 1.0 MHz, γ 41 = 2π × 0.5 MHz, γ 42 = 2π × 0.02 MHz, and γ 43 = 2π × 0.6 MHz [22].It should be noted that although, for example, the |1⟩ − |3⟩ transition is electric-dipole forbidden, atoms in |3⟩ can decay to |1⟩ through the spontaneous emission from |3⟩ to 6p 2 P 3/2 and further to |1⟩, leading to the effective decay rate γ 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.

Third harmonic generation and passive optical clock
The light at λ 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 µm and a minor radius of 3 µm [12]), where the sub-beam (power P and frequency ω clock1 ) from the light source excites a WGM that we label with "p" (frequency ω 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 ω 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 (r, t) = 1 2 a j (t)ψ j (r)e −iω j t with amplitudes a j (t) and normalized WGMs with the vacuum permittivity ϵ 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] where the nonlinear TH coefficient takes the form dr with the TH susceptibility χ TH ∼ 10 −22 m 2 V −2 for fused silica [24] and κ p (< is the coupling rate of the pump beam into the microcavity I.The condition of energy conservation requires ω s = 3ω p .The WGM dispersion caused by different high-order transverse modes compensates the material dispersion, ensuring the phase matching condition [12].
In the steady state (ss), one obtains (1 + xy 4 )y = 1 with x = 3( 2Q s ω s )( Here, we have defined b j,ss = b j (t → ∞) with j = p, s and κ s (< ω s Q s ) accounts for the coupling rate of the TH signal out of the microcavity I.The efficiency η TH reaches its maximum max(η TH ) = ( κ s Q s ω s )( κ p Q p ω p ) when x = 16 and y = 1/2 (Supplement 1).Generally, the pump mode is a fundamental WGM and κ p is equal to half of ω 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 , κ s Q s ω s = 0.5 %, and γ TH = 7 × 10 18 J −1 s −1 , we have max(η TH ) = 0.25 % with the corresponding pump power P = 2.8 mW.The resultant output power of the TH light reaches κ s |b s,ss | 2 = 7.0 µW.
According to the frequency noise spectra (S clock1 (f ), S p (f ), S s (f )), one may numerically produce the frequency fluctuations (δω clock1 (t), δω p (t), δω s (t)) by digitally filtering a stochastic white field [28].Langevin forces F α (t) with α = 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 ∈ Z, the number of fluorescent photons collected by the photodetector is The extra term ∆N ph is a random number that originates from the shot noise whose mean is zero and standard deviation is N 1/2 ph .Here, we have assumed 100-percent efficiency of the photon detection.The average value of δω clock1 (t) within this integration period is then derived as ⟨δω clock1 ⟩ T i = (N ph − N ph,bia )/(kγ 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 ω clock1 is corrected accordingly in the next integration period.The Allan deviation σ y (τ) of the simulated ω clock1 is summarized in Fig. 1(c).The free-running ω clock1 follows the short-term σ y (τ) = 1.8×10 −12 τ −1/2 and the long-term σ y (τ) = 2.5×10 −10 τ 1/2 .In contrast, the stabilized ω clock1 obeys the short-term σ y (τ) = 9.3 × 10 −14 τ −1/2 , over one order of magnitude better than that of the free-running laser, and the long-term σ y (τ) = 5.2 × 10 −13 τ 1/2 , corresponding to an improvement factor of about 500.Consequently, the light source at λ clock1 is operated as an optical frequency standard whose frequency is passively stabilized to the atomic |1⟩ − |4⟩ 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, and the diffusion coefficient 2D(N 4 , N 4 ) associated with the population N 4 of atoms in |4⟩ (Supplement 1).In addition, the stability limited by the photon shot noise is evaluated as . Therefore, the stability of the passive optical clock is primarily limited by atomic fluctuations.

Active optical clock
As shown in Fig. 1 2ℏϵ 0 V L d 32 = −2π × 10 kHz and the cavity mode volume V L = π(0.5 mm) 2 (5 cm).The lasing action (wavelength λ 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) where All nonvanishing diffusion coefficients 2D(α, β) are listed in Supplement 1.As we will see below, this laser has an outstanding frequency stability in the bad-cavity limit (κ L ≫ Γ 32 ) due to the substantial suppression of the cavity pulling effect [7].Thus, the laser directly serves as an optical frequency standard, i.e., an active optical clock.Let us first consider the steady state of the laser system in the resonant coupling situation, ω L = ω 32 and ω TH = ω 41 .The corresponding analytical solutions are given in Supplement 1. Figures 3(a) and (b) illustrate the dependence of the output power P clock2 = κ L ℏω L |A ss | 2 with A ss = A(t → ∞) and the laser linewidth ∆ω clock2 on the cavity loss rate κ L and the atom number N at .In the good-cavity regime (κ L <Γ 32 ), the required minimal N at for the lasing action can be small, for example, N at ∼ 10 5 with κ L /Γ 32 = 0.1, corresponding to Q L = ω L /κ L = 5 × 10 8 .The linewidth ∆ω 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 κ L grows, the increased cavity loss strongly elevates the threshold and ∆ω clock2 also goes up.Nevertheless, in the bad-cavity regime (κ>Γ 32 ), P clock2 can exceed 1 µW, which is high enough for locking the phase of an optical local oscillator.Interestingly, once the system enters the bad-cavity regime, ∆ω clock2 starts declining as κ L is increased.According to [33], the quantum-limited linewidth of a laser takes the form ∆ω clock2 = ∆ω ST (1 + κ L /2Γ 32 ) −2 with the usual Schawlow-Townes linewidth ∆ω ST .When κ L ≫ Γ 32 , the factor (1+ κ L /2Γ 32 ) −2 decreases rapidly, thereby suppressing ∆ω 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 × 10 8 , i.e., the two atomic ensembles have the same atom number.As shown in Fig. 3(c), in the bad-cavity regime ∆ω clock2 reaches its minimum min(∆ω clock2 ) = 2π × 2.8 Hz at κ/Γ 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 × 10 5 and the output power P clock2 approximates 20 µW.
We now consider the frequency stability of the active clock laser.In the adiabatic approximation, κ L ≫ Γ 41,32 , one has and fluctuations of the polarization M 23 , influence A(t).The resonant atom-cavity coupling gives ω L,0 = ω 32 .Generally, the power spectral density of the cavity frequency noise Here, we ignore the flicker frequency noise component (∝ 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 δω L (t) and Langevin fluctuations F α (t) with α = N 1,2,3,4 and M 14,23 according to S L (f ) and diffusion coefficients 2D(α, β) of atomic variables (Supplement 1). Figure 3(d the validity of the simulation.Since this ∆ω clock2 is entirely induced by atomic fluctuations, the corresponding Allan deviation σ y (τ) = ω −1 clock2,0 (∆ω clock2 /τ) 1/2 = 3.0 × 10 −15 τ −1/2 is referred to as the quantum-noise-limited stability.
Considering the fluctuations in ω L and b s , we perform the long-term simulation of the active clock dynamics, compute the clock frequency ω clock2 (see Supplement 1), and evaluate its Allan deviation.The simulation result is σ y (τ) = 6.2 × 10 −15 τ −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.

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 µm and minor radius of 3 µm).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 ω p , Q p , and ∆ p = ω clock1 − ω 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 comb (t, θ) (in units of W 1/2 ) is governed by the Lugiato-Lefever equation [39] ∂ ∂t with the slow time t, the fast phase 0<θ<2π, the dispersion coefficient β = (2π/τ R ) 2 GVD, the round-trip duration τ R = L/v g , the microcavity circumference L, the group velocity v g , and the group dispersion GVD.The nonlinear Kerr coefficient is given by γ Kerr = 8πn 0 n 2 3A eff λ clock1 with the refractive index n 0 , the second-order nonlinear refractive index n 2 , and the effective cross-section area A eff of the pump mode.Varying SOA tunes the input pump power P launched into the microcavity II.Indeed, both v g and GVD are linked to the group refractive index n g (ω, T) = n(ω, T) + ω ∂n(ω,T) ∂ω , i.e., v g = c/n g and GVD = ∂ ∂ω 1 v g .The common refractive index n(ω, T) of the medium depends on the optical frequency ω and the temperature T. We compute n g according to [40,41] (see Fig. 4(a)).Table 1 lists the relevant physical parameters of the microcomb at λ clock1 and room temperature T 0 = 293 K.  shows the microcomb spectrum that is numerically derived from Eq. ( 15) with the input power P = 1.5 mW and the detuning ∆ p = −3(ω 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 In practice, the temperature T fluctuates around T 0 .The small mode volume of the microcavity makes the mode spacing f rep sensitive to the fluctuation δT(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 δT is expressed as δn g = n g (T) − n g (T 0 ) = η n δT with η n = ∂n g (T 0 )/∂T = 1.6 × 10 −5 K −1 .Here, we neglect the thermo-mechanical effects that change the circulating length L. The induced fluctuation of f rep is then written as δf rep (t) = −[η n f rep,0 /n g (T 0 )]δT(t) with f rep,0 = c/Ln g (T 0 ).According to the experimental results in [18,20], the typical Allan deviation of y(t) = δf rep (t)/f rep,0 follows σ y (τ>5 × 10 −2 s) = 5 × 10 −8 τ 1/2 .Thus, the long-term drift of δT(t) has σ y (τ) = √︁ 2π 2 h T,−2 τ/3 with y(t) = δT(t)/T 0 and h T,−2 = 3.7 × 10 −11 s −1 .To verify this, we numerically generate the temperature fluctuations δT(t) and compute the mode spacing f rep (t) by solving Eq. ( 15), where we have used GVD(T) = GVD(T 0 ) + η GVD δT with η GVD = −10.1 fs 2 mm −1 K −1 and τ R (T) = τ R (T 0 ) + η τ δT with η τ = 3.7 × 10 −5 ps K −1 .In addition, the pump mode frequency can be approximated by ω p (T) = ω p (T 0 ) + η ω δT with η ω = −2π × 2.4 GHz K −1 and the stabilized clock frequency ω clock1 follows the Allan deviation shown in Fig. 1(c).The numerical σ y (τ) of δf 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 beat (t) = Re[A * (t)E m (t)] between the active optical clock A(t) and its nearest mode E m (t), whose frequency is ω p + mf rep with m = 10, in the microcomb E comb (t, θ).The beat frequency is set at ω beat = 2π × 50 MHz.We compare the beat note with a local RF signal at ω beat and extract the phase difference ∆ϕ over the integration time T i of the detection electronics.Consequently, the average value ⟨δf rep ⟩ T i of δf rep (t) within T i is given by ⟨δf . The extra noise ∆N ph represents the shot noise that has a zero mean and a standard deviation Here, ξ = 6.6 × 10 −3 accounts for the energy percentage of E m (t) in E comb (t, θ) and we have assumed the critical coupling between the fiber and the microcavity II.
The dynamics of temperature fluctuations δT(t) follows the equation [42] δ where Γ T = D/V th denotes the thermal relaxation rate and η T = 2 × 10 14 K J −1 s −1 represents the photothermal heating coefficient.The fluctuations of the intracavity light energy δE , where the component associated with δE(t) has been omitted.Thus, the thermal-noise-limited stability of the mode spacing f rep (t), i.e., Allan deviation σ y (τ) of y(t) = δf rep (t)/f rep,0 , is given by σ . Generally, this thermal limit is well below the shot-noise-limited stability (see below).In addition, the time scale of interest in this work is much larger than the thermal dissipation scale 1/Γ T .Thus, we simplify Eq. ( 16) as δT(t) = (η T /Γ T )δE(t), that is, the temperature fluctuations is mainly caused by the pump power fluctuations.Controlling SOA (so as to control the pump power launched into the microcavity II) allows for stabilizing δT, thereby stabilizing δf rep .
One may numerically simulate the stabilization of the mode spacing f rep (t).We firstly calculate the average value ⟨δf rep ⟩ T i of δf rep (t) within the integration time T i = 10 −2 s.The temperature change averaged over T i is then computed as ⟨δT⟩ T i = ⟨δf rep ⟩ T i /(η τ f 2 rep,0 ).The corresponding change of the light energy absorbed by the microcavity II reads ⟨δE⟩ T i = (Γ T /η T )⟨δT⟩ T i .Consequently, the input power P is corrected by ( ⟨δE⟩ T i /T i for the next integration period.Numerical results are presented in Fig. 4(c).The Allan deviation of the stabilized f rep (t) follows σ y (τ) = 1.9 × 10 −11 τ −1/2 , matching the shot-noise limit σ y (τ) =

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 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.

Fig. 1 .
Fig. 1.Hybrid clock system.(a) Schematic diagram.One sub-beam from a light source at λ 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 λ 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.

Fig. 2 .
Fig. 2. Dependence of the fluorescence response on the pump detuning ∆ p = ω clock1,0 −ω p,0 .The frequency ω 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∆ p at the locking point.Inset: Frequency fluctuations δω clock1 (t) of the free-running/frequency-locked light source.The stabilized light source is operated as a passive optical clock.

τ = 9 . 1 ×N 4
10 −14 τ −1/2 with the signal-to-noise ratio SNR = (a), after passing an optical amplifier (OA), the TH light √ κ 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 Ω to denote the population of atoms in |i⟩, the macroscopic polarization corresponding to the |1⟩ − |4⟩ transition, the atom number, and the driving strength.Here, we assume that the frequency ω TH of the TH light has been shifted to ω 41 .Atoms in |1⟩ are excited to |4⟩ and then decay to |3⟩, creating the population inversion between |2⟩ and |3⟩.A low-Q cavity (frequency ω L and loss rate κ L ) is coupled with the |2⟩ − |3⟩ transition (frequency ω 32 and transition dipole moment d 32 = 1.6 ea 0 ) with the single-photon coupling strength g = − √︂ ω L

Fig. 3 .
Fig. 3. Steady state of the active optical clock.(a) Dependence of the output power P clock2 on the cavity loss rate κ 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 ∆ω clock2 vs. (N at , κ L ).(c) Dependence of P clock2 and ∆ω clock2 on κ L with N at = 2.5 × 10 8 .The five-pointed star symbol: The operating point of the active optical clock is set at κ L /γ 32 = 90, where P clock2 = 20 µW and ∆ω clock2 = 2π × 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 2π × 3 Hz.For all plots, the pump strength is |Ω|/Γ 41 = 5.

Fig. 4 .
Fig. 4. Kerr microcomb.(a) Dependence of the group refractive index n g on the wavelength change δλ = λ − λ clock1 and temperature T around (λ 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 σ y (τ) of the mode spacing f rep (t) for the microcomb in the free-running/stabilized state.Inset: fluctuations δf rep (t) of the mode spacing when the feedback loop is switched off/on.

Figure 4 (
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 ∆ p = −3(ω 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 originate from the fluctuations δP(t) of the input power.Here, D = 9.5 × 10 −7 m 2 s −1 corresponds to the thermal diffusivity and V 2×10 −14 m 3 is the effective thermal volume of the microcavity, leading to Γ T = 2π ×4 MHz.The correlation function of thermal fluctuations F T (t) is given by ⟨FT (t)F T (t ′ )⟩ = 2Γ T k B T 2 0 ρCV th δ(t − t ′ )with the Boltzmann constant k B , the material density ρ = 2200 kg m −3 , and the specific heat capacity C = 670 J kg −1 K −1 .From Eq. (16), one may derive the noise spectrum of temperature fluctuations S T

Table 1 . Physical parameters of the microresonator-based optical frequency comb at λ clock1 and room temperature T 0 = 293 K.
and the mode spacing (repetition frequency) f rep .The microcomb is in the single-soliton state, leading to f rep = 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 rep .