Active optomechanics

Cavity optomechanics explores the coupling between optical and mechanical modes mediated by the radiation pressure force. Unlike the passive scheme, the active optomechanics with optical gain directly imposes the mechanical motion upon the lasing dynamics, unveiling the intrinsic properties determined by the system itself. Here we numerically explore the general characteristics of the active optomechanics. The effects of the mechanical oscillation on the macroscopic laser include introducing multiple unstable regimes in the lasing phase, shifting the laser central frequency, broadening the laser spectrum, and degrading the laser frequency stability. Reducing the optical gain down to one active atom highlights the quantum nature of atom–cavity and photon–phonon interactions. The one-atom optomechanical microlaser does not only emit nonclassical photons but also generate nonclassical photon–phonon pairs. Our work extends the cavity optomechanics to the active fashion, paving the way towards optomechanical light sources for photonic integrated circuits, on-chip quantum communication, and biosensing. Active optomechanics offers a route to precise microscopic sensing by measuring the intrinsic laser dynamics. Here, the principles of active optomechanics are studied numerically at the limit of a single active atom, opening a path to quantum applications.

T he parametric coupling between optical and mechanical degrees of freedom originates from the light momentum transfer to the mechanical objects that are located inside the optical cavities 1 and is usually modelled as the spring-mass oscillators. Various motivations are driving the vastly growing interest in cavity optomechanics 2,3 and a great variety of optomechanical devices have been developed, including suspended macroscopic/microscopic mirrors 4-6 , micropillars 7 , membranes 8 , whispering-gallery-mode microcavities [9][10][11][12] , and nanorods 13 . The retarded nature of the radiation pressure force allows for suppressing 11 /pumping 14 the mechanical vibration through controlling the incident light. In particular, the optomechanical coupling in the resolved-sideband limit (i.e. the mechanical oscillation frequency greatly exceeds the cavity loss rate) enables cooling of mechanical vibration to the quantum-mechanical ground state 10,15 , leading to the ultrasensitive mass/displacement/ force transduction [16][17][18] . In single-molecule detection, employing the optomechanical effect boosts the sensing resolution by the quality factor of the mechanical mode compared to the conventional methods 19 . In nonequilibrium thermodynamics, an optomechanical array mediated by cavity field allows for monitoring the heat transport in real time 20 . Additionally, hybridizing optomechanical systems with lattice-confined ultracold atoms enables the spatial modulation of atomic density 21 and the manipulation of the motion of atoms 22 . Quantum states can be also transferred between optical and mechanical modes in a coherent fashion (i.e. the optomechanical-coupling strength well exceeds both optical and mechanical decoherence rates 9 ) by using modern quantum optical techniques 9 . Furthermore, the optomechanical coupling may lead to a stationary entanglement between an optical cavity mode and a macroscopic vibrating mirror 23 .
Numerous studies have been focused on the cavity optomechanical systems that are operated in a passive manner 3 . In such setups, an optical cavity is pumped by an external laser beam whose central frequency is detuned from the resonance frequency of the cavity mode. The optomechanical coupling is simply tuned by changing the intensity and frequency of the input laser. The mechanical vibration is mapped onto the optical phase of the intracavity field and the transmission spectrum of the cavity is monitored. The noise spectrum of the mechanical displacement can be attained by using conventional methods, such as balanced homodyne detection 24 , Hänsch-Coulliaud polarization spectroscopy 25 , and Pound-Drever-Hall technique 26 . Nevertheless, the optical phase measurement suffers from unavoidable technical noise sources that are associated with intensity and phase fluctuations in probe laser and local oscillator. Addressing these issues requires extra frequency and polarization stabilization steps and associated optical and electrical components 27 , complicating the design of optomechanical systems. In addition, the limited tuning speed and range of the probe laser frequency impede the detection of high-frequency mechanical vibration.
Recently, the active cavity optomechanics 28 , also known as optomechanical lasing, has drawn much attention. In this new configuration, a spring-mass oscillator is mounted onto one end mirror of a laser cavity. The gain particle-photon and photon-phonon interactions interfere with each other, transcribing the mechanical oscillation directly onto the lasing dynamics 29 . Up to now, the active optomechanical scheme has been implemented only on the vertical-external-cavity surfaceemitting laser with an ultralight-weight mirror [29][30][31][32] . The results manifest a substantially enhanced photon-phonon interaction 32 and a high-speed wide-range wavelength sweep 31 . In comparison to the passive operation fashion, the active optomechanics holds the advantages of simple system structures (i.e. no need for extra frequency and intensity stabilization procedures or associated optical and electrical components), revealing the intrinsic optical properties (i.e. frequency shift, spectral broadening, temporal coherence, and photon statistics) that are entirely determined by an optomechanical system itself, and the potential role of wavelength-tunable light sources in nanophotonic integrated circuits 30 . Nevertheless, the previous demonstrations 29- 32 have been less focused on the effects of the mechanical oscillation on the optical properties of the lasing dynamics.
In addition, thus far, research has been carried out mainly on the optomechanical coupling between a classical optical mode and a mechanical oscillation mode that may be modelled as either a classical spring-mass or a quantum harmonic oscillator with particle-like excitations, known as phonons. By contrast, less attention has been given to the cavity optomechanical systems with small amounts of light quanta (photons) and phonons. In this limit, the quantum nature of light plays a key role in its interaction with the mechanical oscillator and the full quantum treatment should be applied to analyze the cavity optomechanics 33 . The resultant optomechanical microlasers potentially act as nonclassical light sources that are of particular importance to quantum communication and computing.
In this paper, we numerically investigate the active optomechanics where a macroscopic/microscopic lasing dynamics is coupled to a mechanical oscillator. For the macroscopic system composed of an optical cavity interacting with an ensemble of active atoms, the mechanical oscillation induces multiple unstable steady-state regimes in the lasing phase, broadening the power spectral density compared to the corresponding conventional laser system, and degrading the laser frequency stability. In the microscopic limit, the one-atom optomechanical microlaser emits nonclassical photons, the dependence of which on the atom-cavity detuning exhibits multiple submaxima because of the strong photon-phonon interface. The spectrum and photon statistics of the optomechanical microlaser also differ much from that of the common one-atom microlaser. Interestingly, the crosscorrelation function between photons and phonons shows a nonclassical behaviour of the photon-phonon pair generation. To our best knowledge, such a one-atom optomechanical microlaser has not been studied yet and may operate as a nonclassical light source, whose optical properties can be tuned by controlling the mechanical oscillator, for quantum networking. The recent microcavity and nanophotonic technologies allow for testing the predictions obtained in this study.

Results
Physical model. We restrict our attention to the simplest cavity optomechanics, where one optical mode interacts with one mechanical mode. Figure 1a illustrates a generic scheme of the active optomechanical system. An ensemble of atoms is located inside an optical cavity that is composed of a fixed and movable mirror. The movable mirror undergoes a one-dimensional harmonic oscillation x t ð Þ at a radio frequency Ω. The damping rate of the mechanical vibration is Γ with the corresponding quality factor Q m ¼ Ω=Γ. The optical cavity has a resonance frequency ω C when x ¼ 0. The cavity loss rate is κ with the corresponding quality factor Q ¼ ω C =κ. To the linear approximation, the nonzero mechanical displacement x ≠ 0 introduces a frequency shift δ t ð Þ ¼ ξx t ð Þ to the optical cavity mode. The frequency pull parameter ξ takes the form ξ ¼ Àω C =L for a Fabry-Pérot-type cavity with a length L and ξ ¼ Àω C =R WGM for the fundamental mode in a whispering-gallery-mode microcavity with a radius R WGM . The zero-point fluctuation of the mechanical oscillator is defined as , where the effective mass m eff depends on the specific mode under consideration and the choice of the normalized displacement 34 . The atoms play the role of the source of the intracavity field EðtÞ ¼ aðtÞ with the reduced Planck's constant _, the vacuum electric permittivity ε 0 , the effective volume V eff of the cavity mode, and the dimensionless amplitude aðtÞ. Each atom is modelled as a two-level system, i.e. upper |e〉 and lower |g〉 states, with a transition frequency ω A . The cavity mode frequency ω C is close to ω A with a detuning Δ ¼ ω C À ω A . The total decay rate of |e〉 is γ 0 e , wherein the amount corresponding to the |g〉-|e〉 branch is γ e . The decay rate γ g of |g〉 should be larger than γ e for achieving the population inversion. The decay rate of the atomic polarization is then given by γ eg ¼ ðγ e þ γ g Þ=2. The coupling strength between the optical cavity and the atomic |g〉-|e〉 transition is μ ¼ d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω C =2ε 0 _V eff p with the associated electric dipole moment d of the atom. The atoms are continuously pumped into the upper |e〉 state at a rate R. The lasing action occurs for a large enough R. The radiation pressure force F rad ðtÞ ¼ À_ξN photon ðtÞ exerted on the movable mirror leads to the parametric coupling between optical and mechanical degrees of freedom of the cavity. Here, N photon ðtÞ ¼ jaðtÞj 2 accounts for the intracavity photon number.
The generic scheme described in Fig. 1a is actually a four-level optomechanical laser, which is feasible by means of current atomic and photonic technologies, for instances, active optical clocks with gaseous alkali-metal atoms 35,36 and whisperinggallery-mode microresonators doped by rare-earth ions 37,38 and organic molecules 39,40 . In what follows, we take neutral caesium atoms, whose external motion and internal states can be Fig. 1 Active optomechanics in the macroscopic limit. a Schematic of a generic cavity optomechanical system that operates in an active fashion. An ensemble of two-level (upper jei and lower jgi) atoms is coupled to an optical cavity with a single-photon coupling strength μ. A pump light excites the atoms onto jei at a rate R. The total decay rate of jei is γ 0 e , wherein the decay rate of the atom from jgi to jei is γ e . The fast decay rate γ g of jgi ensures the population inversion between jei and jgi. The active atoms emit photons into the optical cavity. The position of one cavity mirror is fixed while the other mirror is movable. The motion of the movable mirror is modelled as a mechanical oscillator with a displacement xðtÞ, oscillation frequency Ω, and damping rate Γ. The radiation pressure of the intracavity field drives the mechanical oscillator. b Steady-state photon number N photon;ss (in units of the saturation photon number N ðsatÞ photon ) as a function of the detuning Δ ¼ ω C À ω A between the cavity mode frequency ω C and the atomic transition frequency ω A . The pump rate is set at R ¼ 10 3 R 0 with the minimum pump threshold R 0 and the cavity quality factor is Q ¼ 10 7 . The cavity loss rate κ ¼ ω C =Q is chosen as the frequency unit in the plot. The solid curves denote the stable steady-state solutions while the dashed lines correspond to the unstable steady-state solutions. c Time evolutions of the mechanical-displacement-induced detuning δðtÞ and the intracavity photon number N photon ðtÞ for the optomechanical system operating at an unstable steady state. The dashed line corresponds to the average value of δðtÞ. d, e Dependences of the steady-state N photon;ss on the detuning Δ and the pump rate R for Q ¼ 10 5 and Q ¼ 10 7 , respectively. The boundaries of unstable steady-state regions have been plotted. The red shading denotes the stable steady states while the blue shading corresponds to the unstable steady states. f, g Photon differences ΔN photon;ss between the active optomechanical system and the corresponding conventional laser for Q ¼ 10 5 and Q ¼ 10 7 , respectively. manipulated and controlled at a high degree of precision by using laser beams in experiments 41 , as the intracavity gain medium with the laser transition between |e〉 (5p 6 7s) 2 S 1/2 and |g〉 (5p 6 6p) 2 P 3/2 (Supplementary Note 1). The transition frequency is ω A ¼ 2π 204 THz (wavelength of 1470 nm) and the relevant decay rates 42 are γ e ¼ 2π 1:8 MHz, γ ' e ¼ 2π 2:8 MHz, γ g ¼ 2π 5:2 MHz, and γ eg ¼ 2π 4:0 MHz. A pump laser at 455 nm continuously drives the atoms from the ground (5p 6 6s) 2 S 1/2 to the excited (5p 6 7p) 2 P 3/2 state. The atoms are accumulated in |e〉 via the rapid decay from (5p 6 7p) 2 P 3/2 to |e〉 with the corresponding spontaneous emission rate of 2π 0:7 MHz. The pump rate R can be as high as 2π 13 GHz for 2 10 4 atoms located inside the cavity. We further assume the cavity mode volume V eff =ð2πc=ω A Þ 3 ¼ 2:5 10 3 , which is experimentally feasible for various microcavities [43][44][45] . Here, c denotes the speed of light in vacuum. The atom-cavity coupling strength is then computed as μ ¼ 20γ eg .
The cavity Q factor depends on the specific cavity structure. Fabry-Pérot 46 and whispering-gallery-mode 47 (micro)cavities may have an ultrahigh quality factor of the order of 10 10 . The corresponding photon loss rate κ $ 2π 20:4 kHz is much smaller than the decay rate γ eg of the medium polarization. As a result, the laser system accesses the good-cavity regime with κ ( γ eg . In contrast, the attainable Q factor of micropillar 48 and photonic crystal 49 microcavities is relatively low, ranging from 10 3 to 10 6 , and thus the optomechanical laser operates in the badcavity limit with κ ) γ eg . Without losing the generality, we restrict ourselves to the range of 10 5 ≤ Q ≤ 10 7 in this work. On the other hand, the vibrational frequency Ω and quality factor Q m of the mechanical oscillator also depend on the specific structure of the optical cavity, for examples, Ω=2π of 10 $ 10 2 MHz and Q m of 10 3 for whispering-gallery-mode microcavities 11,27 , Ω=2π of 10 À2 $ 10 2 MHz and Q m of 10 $ 10 4 for cantilevers [50][51][52] , and Ω=2π of 10 À2 $ 1 MHz and Q m of 10 3 for nanowires 53,54 . In this work, we study both macroscopic (i.e. atomic ensemble) and microscopic (i.e. one atom) active optomechanical systems. For the macroscopic optomechanics, without losing the generality, we choose Ω ¼ 2π 10 MHz and Q m ¼ 10 2 and hence the system stands outside the so-called sideband-resolved regime, i.e. κ > Ω. We further assume the entire system is operated at the cryogenic temperature 55 T ¼ 1 K. The average number of thermal photons approximates zero due to k B T=_ω C % 0 with the Boltzmann constant k B . In contrast, the mechanical degree of freedom has a thermal occupation of n thrm ¼ 1=ðe _Ω=k B T À 1Þ % 2083, whose influence on the lasing dynamics is non-negligible. The assumption of cryogenic temperature facilitates the performance of the numerical simulation. For a higher T, for example, room temperature, n thrm goes up strongly and the analysis presented below may be extended accordingly. For the microscopic optomechanics, in order to carry out the numerical simulation, we set Ω ¼ 2π 100 MHz, Q m ¼ 10 2 and the cryogenic temperature 56,57 T ¼ 10 mK with n thrm % 1:6. Vastly reducing n thrm is highly desirable because it highlights the quantum nature of phonons. In addition, to reveal the quantum behaviour of the photon-phonon interface, we set the cavity Q factor at 10 7 and as a result, the microscopic system accesses the sideband-resolved regime, i.e. κ < Ω.
Macroscopic optomechanics. We first consider the macroscopic system with an optical cavity simultaneously interacting with a large number of active atoms and a classical mechanical oscillator. The light-matter interaction inside the cavity is described by a set of Heisenberg-Langevin equations while the mechanical displacement xðtÞ follows a second-order differential equation (i.e. spring-mass oscillator) driven by the radiation pressure force F rad ðtÞ (see Methods). It is worth noting that the macroscopic system may be also studied by using the density matrix method 58 . However, the huge dimension of the Hilbert space makes it challenging to perform the investigation, especially on the power spectral densities of the intracavity field and mechanical oscillation. In contrast, the Heisenberg-Langevin method strongly simplifies the mathematical modelling of the macroscopic system, allowing one to numerically simulate the lasing dynamics in an optomechanical cavity. In active optomechanics, the mechanical displacement xðtÞ introduces an extra detuning δðtÞ to the atom-cavity interaction, thereby affecting the intracavity photon number N photon ðtÞ. This further changes the force F rad ðtÞ exerted on the movable mirror and in return affects xðtÞ. Thus, the optomechanical dynamics can be solved through a self-consistent method.
The system reaches the steady-state (denoted by the subscript ss) when the optical pump R is balanced by the total energy loss experienced by the photons and atoms inside the cavity and the mechanical oscillator. The steady-state intracavity photon number is derived as (Supplementary Note 2) with the saturation photon number N ðsatÞ Here, h i accounts for the expected value operation and φ ss denotes the relative phase difference of the intracavity field with respect to the medium polarization. We have with the steady-state mechanical-displacement-induced detuning δ ss hδðt ! 1Þi ¼ ξx ss ¼ Àδ 0 N photon;ss ≤ 0, δ 0 2δ 2 ZPF =Ω, and the zero-point-fluctuation-induced detuning δ ZPF ξx ZPF . It is seen that the steady-state mechanical displacement x ss hxðt ! 1Þi ¼ Àðδ 0 =ξÞN photon;ss is positive and proportional to the photon number N photon;ss . Equation (1) is actually a quadratic equation with the variable N photon;ss and only one solution is physically acceptable for the system parameters chosen in this work. Setting δ ss ¼ 0 in Eq. (2), one obtains a conventional laser with fixed cavity mirrors (ξ ¼ 0) and the corresponding steadystate photon number is given by N ðconvlÞ photon;ss N photon;ss j δ ss ¼0 . Without losing the generality, in what follows, we choose δ 0 ¼ 2π 40 MHz, leading to δ ZPF ¼ À2π 14:1 MHz. Such large δ ZPF is accessible in an experiment by reducing the cavity length and the effective mass of the mechanical oscillator 53 . It should be also noted that even reducing δ 0 (or δ ZPF ) by a factor of 10 does not affect the main conclusions derived in this work.
Due to the nonzero δ ss , N photon;ss lacks the symmetry between red-detuned Δ < 0 and blue-detuned Δ > 0 regimes (Fig. 1b). The maximum of N photon;ss occurs in the blue-detuned regime. Unlike the conventional lasers whose steady-state solutions are always stable, the steady-state solution of the optomechanical laser may be unstable to a small perturbation. One may perform the linear stability analysis to examine the stability of the steady-state solution in the lasing phase with N photon;ss > 0 (Supplementary Note 2). At an unstable steady state, δðtÞ undergoes a nearly harmonic oscillation around a nonzero average value (Fig. 1c). Whenever δðtÞ passes across zero, N photon ðtÞ increases abruptly and then descends to zero in an oscillatory manner because the cavity loss rate κ greatly exceeds the damping rate Γ of the mechanical oscillation. Figure 1d and e shows the photon number N photon;ss in the R À Δ plane for different Q factors. There exist several unstable zones, whose areas depend closely on the cavity Q factor, in the lasing phase (Supplementary Movie 1). A large Q increases the intracavity photon number and thus boosts the radiation pressure force. The resulting enhanced mechanical oscillation strongly affects the laser dynamics in return. Consequently, a steady-state that is stable for a low Q may become unstable for a high Q and the unstable zones are expanded. We consider the difference ΔN photon;ss N photon;ss À N ðconvlÞ photon;ss between the steady-state photon numbers of the optomechanical laser N photon;ss and the corresponding conventional laser N ðconvlÞ photon;ss with fixed mirrors (ξ ¼ 0). As illustrated in Fig. 1f, g, ΔN photon;ss is mainly positive (negative) in the red-detuned Δ < 0 (bluedetuned Δ > 0) regime, indicating that the optomechanical coupling facilitates (supresses) the lasing action. It is also found that the ratio of ΔN photon;ss to N ðconvlÞ photon;ss is less than 1 percent for a low-Q cavity and thus, the influence of the mechanical oscillation on the lasing dynamics may be viewed as a perturbation. In contrast, ΔN photon;ss =N ðconvlÞ photon;ss can be over 20% for a high-Q cavity due to the strong optomechanical coupling, and the mechanical oscillation cannot be treated as a perturbation to the lasing dynamics.
Besides the stability of steady-state solutions, the effects of the optomechanical coupling can be also revealed by the power spectral density S photon ðωÞ of the laser light. According to the Wiener-Khinchin theorem, when the system undergoes a stationary random process, one has S photon ðωÞ / hjF½aðtÞj 2 i with the Fourier transform F½aðtÞ R þ1 À1 aðtÞe Àiωt dt and the normalization N photon;ss ¼ R þ1 À1 S photon ðωÞdω. For the conventional laser with fixed mirrors, the fluctuations of the macroscopic polarization of active atoms primarily contribute to the spectral broadening of the laser light 59 . For the optomechanical laser, the mechanical oscillation of the movable mirror may strongly affect S photon ðωÞ. We numerically simulated the amplitude aðtÞ of the intracavity field (see Methods) and computed S photon ðωÞ accordingly. As shown in the panel of Q ¼ 10 5 in Fig. 2a, for a low-Q cavity, S photon ðωÞ consists of the main peak surrounded by equally separated sidebands. Indeed, these sideband spikes directly result from the mechanical oscillator modulating the laser dynamics and the separation between two adjacent sidebands approximates Ω. The main spectral peak is characterized by the central oscillation frequency ω L of the laser light and a full width at half maximum Δω L . The cavity pulling effect 60 shifts ω L away from the atomic transition frequency ω A . However, unlike conventional laser whose central frequency ω L is independent of the photon number N photon;ss , the frequency difference jω L À ω A j of the optomechanical laser goes up linearly as N photon;ss is enhanced because of the extra mechanicaldisplacement-induced detuning δ ss . In the low-Q limit, the perturbation approximation leads to whose validity can be confirmed by the panel of Q ¼ 10 5 in Fig. 2b.
The above equation allows one to evaluate δ ss (or the steady-state mechanical displacement x ss ) by measuring the laser frequency ω L . The sensitivity is given by ∂x ss =∂ω L ¼ ξ À1 ð1 þ κ=2γ eg Þ, which, interestingly, can be enhanced by increasing the cavity loss rate κ.
The spectral broadening Δω L of the laser light may be predicted by using the Schawlow-Townes formula 61 with the laser output power P out =_ω L ¼ κN photon;ss and the steady-state population N u;ss of atoms in the ju ¼ e; gi state. The panel of Q ¼ 10 5 in Fig. 2b confirms the match between Eq. (4) and numerical results. When jΔj ) jδ ss j, increasing R (or N photon;ss ) reduces Δω L . We also computed the power spectral density of the mechanical-displacement-induced detuning, S δ ðωÞ ¼ hjF½δðtÞj 2 i. Actually, S δ ðωÞ=ξ 2 corresponds to the spectral density of the mechanical displacement xðtÞ. For a low Q, the spectrum S δ ðωÞ is single peaked at Ω (the panel of Q ¼ 10 5 in Fig. 2c).
As the cavity Q factor is increased, the main spectral peak in S photon ðωÞ is strongly broadened, i.e. Δω L exceeds Ω, and consequently the optomechanical-coupling-induced sidebands vanish (the panel of Q ¼ 10 6 in Fig. 2a). This is attributed to the fact that a high Q boosts the intracavity field and the resulting enhanced optomechanical-coupling maps more thermal fluctuations of the mechanical oscillator onto the lasing dynamics, thereby broadening the laser spectrum. Both Eqs. (3) and (4) fail in estimating ω L and Δω L since the perturbation approximation becomes invalid (the panel of Q ¼ 10 6 in Fig. 2b). Deriving the analytical expressions for ω L and Δω L is impractical. The mechanical spectrum S δ ðωÞ also becomes broadened and, interestingly, the sideband peaks emerge at ω ¼ mΩ with m 2 Z (the panel of Q ¼ 10 6 in Fig. 2c). For a high enough Q, the spectrum S photon ðωÞ apparently violates the Lorentzian line shape as shown in the panel of Q ¼ 10 7 in Fig. 2a. It is seen that S photon ðωÞ consists of a narrow high peak, corresponding to the lasing dynamics, and a wide background that arises from the thermal mechanical fluctuations. The linewidth Δω L grows as R is increased (the panel of Q ¼ 10 7 in Fig. 2b). The sideband peaks in the mechanical spectrum S δ ðωÞ become more distinct (the panel of Q ¼ 10 7 in Fig. 2c).
We are interested in the efficiency of the optical energy transfer to the spring-mass oscillator. Let us assume that the positions of both cavity mirrors are initially fixed, i.e. xðt ¼ 0Þ ¼ 0 and none energy is stored in the spring-mass oscillator, and the intracavity field reaches the steady state at a certain pump rate R (Fig. 3a). Then, we let the mechanical oscillator (i.e. the movable mirror) free to move. Under the radiation pressure F rad ðtÞ, the mechanical displacement xðtÞ goes towards the steady-state value x ss . The work done by F rad ðtÞ during this process is given by W ¼ R t¼1 t¼0 F rad ðtÞdxðtÞ, which relies on the specific route of the movable mirror moving from xðt ¼ 0Þ ¼ 0 to xðt ¼ 1Þ ¼ x ss . A portion of W is dissipated through the damping of the spring-mass oscillator while the rest is converted to the elastic potential energy U ¼ m eff Ω 2 x 2 ss =2 ¼ _δ 2 ss =2δ 0 of the spring. One may define the efficiency of the optomechanical coupling as η ¼ U=W. Figure 3b illustrates the dependence of η on the pump rate R for different cavity Q factors. It is seen that η grows strongly in the low-R regime (still above the threshold) and then gradually declines after reaching a maximum that is close to unity. The efficiency η is eventually saturated in the high-R regime, denoting W scales as x 2 ss for a large R. The saturation value of η is about 0.5 for a low Q while it approximates unity for a high Q.
Let us further examine the influence of optomechanical coupling on laser frequency stability. We assume that in addition to the radiation pressure F rad ðtÞ an extra environmental noise force F xtra ðtÞ that arises from, for example, the ground vibration is exerted on the mechanical oscillator. We focus on the dimensionless signal yðtÞ ¼ ω L ðtÞ=ω A with the laser frequency ω L ðtÞ perturbed by the mechanical vibration. The power spectral density of laser frequency fluctuations is given by S y ðωÞ ¼ hjF½yðtÞj 2 i, which can be measured by means of standard optical techniques in experiments 27 . The stability of yðtÞ is commonly quantified by the Allan deviation σ y ðτÞ with a sampling time τ. We numerically simulated the optomechanical laser dynamics and extracted yðtÞ from the trajectories of the laser amplitude aðtÞ.
For a low cavity Q factor, the spectrum S y ðωÞ scales as ω 0 (Fig. 4a), that is, the white frequency noise plays the predominant role, and the corresponding Allan deviation follows the scaling law σ y ðτÞ / 1= ffiffi ffi τ p (Fig. 4b). When F xtra ðtÞ ¼ 0, the atom-cavity interface induces a wide peak located at ω ¼ 2μ in S y ðωÞ and the optomechanical coupling gives rise to the other narrow peak at ω ¼ Ω (Fig. 4a). In comparison, we also computed the spectral density of frequency fluctuations of the corresponding conventional laser with fixed cavity mirrors (ξ ¼ 0). Due to the weak optomechanical coupling in the low-Q limit, optomechanical and conventional lasers have a similar spectral density of frequency fluctuations, except the spectral peak at ω ¼ Ω (Fig. 4a). As a result, their Allan deviations are close to each other, making it hardly to analyze mechanical fluctuations from the measurement of the laser frequency stability (Fig. 4b). Additionally, in the practical measurement the photon shot noise whose spectral density of frequency fluctuations is given by ðκ 2 þ 4Ω 2 Þ=4ω 2 A κN photon;ss sets the standard quantum limit to the photodetector's sensitivity 2 . As shown in Fig. 4a, only the spectral peaks at ω ¼ Ω and ω ¼ 2μ in S y ðωÞ Power spectral densities of laser field and mechanical oscillation. a Numerically simulated spectrum S photon ðωÞ of the intracavity light field for different cavity Q factors and pump rates R. The atom-cavity detuning is set at Δ ¼ Àκ with the cavity loss rate κ, and R 0 denotes the minimum pump threshold. At Δ ¼ Àκ, the optomechanical laser has stable steady-state solutions for different Q and R. The laser frequency ω L is shifted from the atomic transition frequency ω A and the laser linewidth (i.e. full width half maximum) is Δω L . The mechanical oscillation frequency Ω is chosen as the frequency unit. b Dependence of laser frequency shift ðω L À ω A Þ and linewidth Δω L on the pump rate R for different Q factors with Δ ¼ Àκ. Symbols corresponds to the numerical results. The curves derived from the cavity pulling effect and Schawlow-Townes formula are also plotted for comparison. c Numerically simulated spectrum S δ ðωÞ of the mechanical-displacement-induced detuning δðtÞ for different Q and R with Δ ¼ Àκ.
exceed the shot-noise limit. The Allan deviation of the low-Q optomechanical laser is below the shot-noise-limited Allan deviation (Fig. 4b), impeding the detection of mechanical motion. When F xtra ðtÞ ≠ 0, the extra environmental noise source strongly elevates S y ðωÞ above the shot noise (Fig. 4a). The spectral peak at ω ¼ 2μ in S y ðωÞ disappears while the peak at ω ¼ Ω is retained. The Allan deviation σ y ðτÞ with F xtra ðtÞ ≠ 0 may exceed the shot-noise-limited σ y ðτÞ, providing a way to study the optomechanical coupling from the measurement of the laser frequency stability. Due to the frequency band-pass filtering feature of the mechanical oscillator, the noise force F xtra ðtÞ in a certain color does not lead to the corresponding scaling of σ y ðτÞ with respect to τ. For instance, the pink-colored F xtra ðtÞ with a power spectrum scaling as ω À1 may not lead to the scaling σ y ðτÞ / τ 0 (Fig. 4b).
For a high cavity Q factor, the enhanced intracavity field suppresses the shot noise and both S y ðωÞ and σ y ðτÞ with F xtra ðtÞ ¼ 0 exceed the shot-noise limit (Fig. 4c, d), allowing for detecting the mechanical vibration. A spectral peak is presented at ω ¼ Ω in S y ðωÞ of the optomechanical laser. The strong optomechanical coupling lifts the spectral density of frequency fluctuations of the optomechanical laser well above that of the corresponding conventional laser. That is, the laser frequency stability degrades since more mechanical fluctuations are mapped onto the lasing dynamics. This paves a way to evaluate the mechanical vibration from the measurement of the Allan deviation.
One-atom active optomechanics. In the above, we have discussed the active optomechanics in the macroscopic limit, where a large number of active atoms contribute to the intracavity field. We now consider the optomechanical microlaser with only one atom placed inside the optical cavity. Such a one-atom microlaser is feasible by means of current atom optical techniques [62][63][64][65] . The maximum pump rate R is limited by half of the spontaneous emission rate of the atom from (5p 6 7p) 2 P 3/2 to jei, i.e. R $ 2π 0:35 MHz. In the one-atom active optomechanics, the quantum nature of light plays the main role in the atom-cavity interaction since the intracavity field is significantly reduced. In addition, the quantum behaviour of the mechanical oscillator is highlighted at an extremely low temperature. Thus, the Heisenberg-Langevin method becomes invalid and the full quantum-mechanical treatment (based on the density matrix method) is necessarily employed. The complex field amplitude aðtÞ is replaced with the photon annihilation operatorâ whose conjugate transpose gives the photon creation operatorâ y . The harmonic mechanical oscillation is modelled as a vibrational mode with creationb ðtÞ, respectively, with the trace operation Trð Þ. We set the cavity Q factor at 10 7 and the phonon frequency Ω is chosen as 2π 100 MHz. That is, the optomechanical system accesses the sideband-resolved regime with κ < Ω, highlighting the quantum features of the photon-phonon interface. We numerically study this one-atom optomechanical microlaser.
The dependences of the steady-state photon N photon;ss and phonon N phonon;ss numbers on the atom-cavity detuning Δ are shown in Fig. 5a and b, respectively. It is seen that N photon;ss differs much from that of the corresponding common one-atom microlaser with fixed mirrors (i.e. ξ ¼ 0). Due to κ < Ω, the transitions between the product states jn photon i jei jn phonon i and jn photon þ 1i jgi jn 0 phonon i are well resolved. Here, jn photon i and jn phonon i denote the Fock (number) states of photons and phonons, respectively. Consequently, several submaxima of N photon;ss are presented in the N photon;ss À Δ curve and the separation between two adjacent submaxima approximates Ω. For a large Δ, N phonon;ss decreases strongly because of the weakened atom-cavity interaction. The N phonon;ss À Δ curve also exhibits several submaxima/subminima. Comparing N phonon;ss with the average number n thrm of thermal quanta, one obtains the heating regime with N phonon;ss > n thrm and the cooling regime with N phonon;ss < n thrm (Fig. 5b). The boundary between two regimes is located on the blue-detuned side of the atomic transition. Tuning  Fig. 3 Efficiency of the optomechanical coupling. a Energy transfer from the laser light to the spring-mass oscillator. Initially, the positions of two mirrors of the optical cavity are fixed and the intracavity field reaches the steady state. Then, the moveable mirror is free to move. The radiation pressure F rad ðtÞ pushes the movable mirror away from the initial position xðt ¼ 0Þ ¼ 0. As the time goes infinity, t ! 1, the position of the movable mirror approaches the steady-state displacement xðt ¼ 1Þ ¼ x ss . During this process, the radiation pressure does the work W ¼ R t¼1 t¼0 F rad ðtÞdxðtÞ. When the moveable mirror is displaced at x ss , the potential energy stored in the spring-mass oscillator is U. The energy conversion efficiency from W to U is given by η ¼ U=W. b Efficiency η vs. the pump rate R for different cavity Q factors. Here, R 0 is the minimum pump threshold. The atom-cavity detuning is set at Δ ¼ Àκ with the cavity loss rate κ.
Δ allows for exciting/suppressing the mechanical oscillation. Due to the huge difference between the lifetimes of photons and phonons, κ ) Γ, the intracavity photons are mainly distributed in the vacuum state jn photon ¼ 0i while the phonons can be populated on the high number states (Fig. 5c). The phonon distribution, especially the probability in jn phonon ¼ 0i, strongly depends on the detuning Δ (Supplementary Note 4).
We employ the second-order correlation function g ð2Þ photon ðτÞ ¼ â y ðtÞâ y ðt þ τÞâðt þ τÞâðtÞ = â y ðtÞâðtÞ 2 at the zero-time delay τ ¼ 0 to characterize the photon statistics. Here,â y ðtÞ andâðtÞ are the photon operators in the Heisenberg picture. As shown in Fig. 5d, g ð2Þ photon ðτ ¼ 0Þ is well below unity, indicating the antibunching of photon emission, and appears an oscillatory behaviour vs. the atom-cavity detuning Δ due to the well-resolved phonon states (i.e. κ < Ω). In comparison to the common one-atom microlaser with fixed mirrors (i.e. ξ ¼ 0), one finds that g ð2Þ photon ðτ ¼ 0Þ is lower (higher) than that of the common one-atom microlaser in the red-detuned (blue-detuned) regime with Δ < 0 (Δ > 0). We also  Fig. 4 Frequency stability of the optomechanical laser. a Numerically simulated spectrum S y ðωÞ of the signal yðtÞ ¼ ω L ðtÞ=ω A with the perturbed laser frequency ω L ðtÞ and the atomic transition frequency ω A . The optical cavity has a low Q factor of 10 5 . For the optomechanical laser, the optical cavity has a movable mirror. For the corresponding conventional laser, the positions of both cavity mirrors are fixed. In addition to the photon radiation pressure, an extra environmental noise force F xtra ðtÞ is exerted upon the mechanical oscillator whose oscillation frequency Ω is chosen as the frequency unit. The pinkcolored noise force F xtra ðtÞ is chosen as F xtra ðtÞ ¼ 200_ξ Á FðtÞ with the frequency pull parameter ξ. The spectral density of the noise function FðtÞ is given by 1=ω. The pump rate R is set at 10 3 R 0 with the minimum pump threshold R 0 and the atom-cavity detuning is chosen as Δ ¼ Àκ with the cavity loss rate κ. The spectra S y ðωÞ under different conditions are plotted. Two spectral peaks are presented in S y ðωÞ at around ω ¼ 2μ and ω ¼ Ω with the atom-cavity coupling strength μ and the mechanical oscillation frequency Ω. The corresponding Allan deviations σ y ðτÞ are shown in b. c, d Spectra S y ðωÞ and Allan deviations σ y ðτÞ with a high Q factor of 10 7 under different conditions. For comparison, the shot-noise-limited spectra and Allan deviations have been inserted.  ¼ 0) is also plotted. The mechanical oscillation frequency Ω is chosen as the frequency unit. The corresponding steady-state number N phonon;ss of phonons is shown in b. The blue shade denotes the heating regime with N phonon;ss > n thrm while the pink shade corresponds to the cooling regime with N phonon;ss < n thrm . Here, n thrm denotes the average number of thermal quanta. c Distributions of photon and phonon numbers with Δ ¼ ± Ω. d Dependence of the second-order correlation function g ð2Þ photon ðτÞ of photons at the zero-time delay τ ¼ 0 on the detuning Δ. The second-order correlation function of the common one-atom microlaser with fixed cavity mirrors (ξ ¼ 0) is also plotted for comparison. e Second-order correlation function g ð2Þ phonon ðτ ¼ 0Þ of phonons relative to the second-order correlation function g ð2Þ thrm ðτ ¼ 0Þ of the thermal state. f Ratio χ ½g ð2Þ photonÀphonon ð0Þ 2 =g ð1Þ photon ð0Þg ð1Þ phonon ð0Þ vs. the atom-cavity detuning Δ. Here, g ð2Þ photonÀphonon ð0Þ denotes the cross-correlation function between photons and phonons. For all curves, the cavity Q factor is 10 7 and the pump rate is set at R $ 2π 0:35 MHz.
compute the second-order correlation function g ð2Þ phonon ðτÞ ¼ b y ðtÞb y ðt þ τÞbðt þ τÞbðtÞ = b y ðtÞbðtÞ 2 of phonons. It is found that unlike g ð2Þ photon ðτ ¼ 0Þ, g ð2Þ phonon ðτ ¼ 0Þ is always above unity, i.e. the bunching of phonons. As illustrated in Fig. 5e, g ð2Þ phonon ðτ ¼ 0Þ exhibits a dispersion behaviour around Δ ¼ 0. For an ensemble of thermal quanta, the corresponding secondorder correlation function g ð2Þ thrm ðτ ¼ 0Þ is equal to two 66 . The function g ð2Þ phonon ðτ ¼ 0Þ is lower (higher) than g ð2Þ thrm ðτ ¼ 0Þ mainly in the red-detuned (blue-detuned) regime with Δ < 0 (Δ > 0). We further examine the Cauchy-Schwarz inequality 67 χ ½g ð2Þ photonÀphonon ð0Þ 2 =g ð2Þ photon ð0Þg ð2Þ phonon ð0Þ < 1 with the crosscorrelation function g ð2Þ photonÀphonon ðτÞ ¼ â y ðtÞb y ðt þ τÞbðt þ τÞâðtÞ = â y ðtÞâðtÞ b y ðtÞbðtÞ between photons and phonons. Figure 5f shows that χ is larger than unity within the detuning regime of interest. This violation of the Cauchy-Schwarz inequality is a concrete measure for the nonclassical photon-phonon pair generation 68,69 . The minimum of χ approaches unity and occurs at the heating-cooling boundary. Compared to nonclassical correlations for photon pairs 70 , generating nonclassical correlations between single photons and phonons with six orders of magnitude difference in frequency is more challenging, and here we provide a potential way. The nonclassical photon-phonon pairs can be used to remotely control mechanical states and allow for mechanicsbased long-distance quantum communication 71 .
The first-order correlation functions of photons and phonons are defined as g ð1Þ photon ðτÞ ¼ â y ðtÞâðt þ τÞ = â y ðtÞâðtÞ and g ð1Þ phonon ðτÞ ¼ b y ðtÞbðt þ τÞ = b y ðtÞbðtÞ , respectively. It is seen from Fig. 6a that g ð1Þ photon ðτÞ rapidly decays from unity to zero while g ð1Þ phonon ðτÞ experiences a long-term damped oscillation process. That is, the phonons possess a lifetime much longer than that of the photons. The Fourier transform of g ð1Þ photon ðτÞ gives the spectrum of photons, S photon ðωÞ / F½g ð1Þ photon ðτÞ. We find that the spectrum of the one-atom optomechanical microlaser is more complex than that of the corresponding one-atom microlaser with fixed mirrors (Fig. 6b). Multiple peaks exhibit in S photon ðωÞ of the one-atom optomechanical microlaser. Two central peaks result from the atom-cavity coupling while others are located at ω ¼ ±mΩ with a non-negative integer m and correspond to the photon-phonon interaction. The appearance of photon-phononinteraction-induced spectral peaks is a direct consequence of the system operating in the resolved-sideband regime κ < Ω. We also calculated the spectrum S phonon ðωÞ / F½g ð1Þ phonon ðτÞ of phonons and found S phonon ðωÞ is always single peaked (Fig. 6c). The linewidth and central frequency of S phonon ðωÞ depends less on the atom-cavity detuning Δ (Fig. 6d), that is, S phonon ðωÞ is mainly determined by the thermal fluctuations.

Discussion
Besides the frequency shift and optical phase measurements that are usually employed in passive optomechanics, the mechanical motion may be also detected by measuring the laser intensity, spectral broadening, and laser frequency stability in the active optomechanics. Generally, the intracavity field in the typical passive optomechanics comes from an external driving beam and can be adjusted freely by changing the driving power and the detuning between the optical cavity and the driving beam. In contrast, the energy source in active optomechanics is the intracavity active medium. The lasing action occurs when the optical gain exceeds the optical loss. Thus, an active optomechanical system may act as a light source, whose output can be tuned by controlling the mechanical motion, in a photonic integrated circuit 30 . In addition, the passive optomechanical system may exhibit a bistable behaviour, i.e. two stable steady-state solutions and one unstable steady-state solution 2 . However, the active optomechanical system has at most one stable steady-state. This may be ascribed to the extra condition imposed by the lasing threshold. Moreover, in passive optomechanics, when the cavity is pumped in a detuned manner, the radiation pressure force exerted on the mechanical oscillator becomes viscous, i.e. dynamic backaction. This effect has been applied to suppress the mechanical motion of the optical cavity, i.e. optomechanical cooling 11 . However, the situation becomes more complicated in macroscopic active optomechanics. Since the intracavity field is completely determined by the active optomechanical system itself, it is challenging to fulfil the optomechanical-cooling condition 72 N photon;ss Δω L ( κ. Furthermore, unlike the passive optomechanics, the active operation manner reveals the intrinsic properties (such as stability and temporal coherence) of the optomechanical systems.
Up to now, the active optomechanics has been only demonstrated on the vertical-cavity surface-emitting lasers [30][31][32] . Other laser platforms that are potentially utilized to implement the active optomechanics include Fabry-Pérot resonators with neutral atom gases 36,73 , whispering-gallery-mode microcavities 74-76 /droplets 77,78 doped with ions and molecules, and fibre microcavities 79,80 . The one-atom microlaser has been demonstrated based on the schemes of an atom/ion trapped in a high-finesse optical cavity 62,63 , a quantum dot embedded in a nanocavity 64 , and a nitrogen-vacancy centre in diamond 65 . The optomechanical coupling may be induced by inserting a nanorod/nanowire into the optical cavity 53 or bringing a silicon nitride nanomechanical string close to a silicon microdisk/microtoroid 81,82 . These nano-scaled objects have an effective mass as small as a few pictograms or even a few femtograms 54 and a mechanical oscillation frequency of the order of megahertz. The single-photon optomechanical-coupling strength can reach $ 2π 1 MHz. Another possible platform is soap bubbles 83 , which may be further extended to optomechanical single-molecule detection 19 . Although here we only discussed the coupling between one optical cavity mode and one mechanical mode, the analysis can be extended to the active optomechanics with multiple mechanical modes 27 . The inter-mechanical-mode coupling mediated by light may lead to nonlinear phenomena.
In order to suppress the influence of thermal fluctuations on the active optomechanical dynamics, the average number n thrm of thermal quanta should be sufficiently reduced by means of, for examples, raising the mechanical oscillation frequency and operating at cryogenic temperature. An integrated optical and nanomechanical resonator formed on a microchip can have a mechanical oscillation frequency Ω of a few gigahertzs 15 . Despite the difficulty, a cryogenic temperature below 10 mK is feasible 84,85 . When Ω $ 2π 1 GHz, n thrm is much less than unity 68 , highlighting the quantum nature of phonons and enabling quantum control of single phonons through single optical photons. Besides optical modes, mechanical oscillators also coherently interact with other physical quantum systems, such as spins 86 and superconducting circuits 87 , allowing for the mechanics-based quantum information transfer and remote quantum state control.

Methods
Classical optomechanical system. The dynamics of a macroscopic laser system is governed by a set of c-number Heisenberg-Langevin equations 69 _ N e ðtÞ ¼ R À γ 0 e N e ðtÞ À iμ½M * ðtÞaðtÞ À a * ðtÞMðtÞ; ð5aÞ _ N g ðtÞ ¼ Àγ g N g ðtÞ þ γ e N e ðtÞ þ iμ½M * ðtÞaðtÞ À a * ðtÞMðtÞ; ð5bÞ _ MðtÞ ¼ Àγ eg MðtÞ þ iμ½N e ðtÞ À N g ðtÞaðtÞ þ F M ðtÞ; ð5cÞ Here, N u¼e;g ðtÞ denotes the population of atoms in the ju ¼ e; gi state, MðtÞ accounts for the macroscopic polarization of atoms, and δðtÞ corresponds to the mechanical-displacement-induced detuning. In above equations, we have only included the Langevin force F M ðtÞ associated with MðtÞ and omitted the Langevin forces acting on N u¼e;g ðtÞ since the laser spectral broadening arises predominantly from the fluctuations in MðtÞ. The noise F M ðtÞ has the correlation functions hF M ðtÞi ¼ 0, hF M ðtÞF M ðt 0 Þi ¼ 2iμhMðtÞaðtÞiδðt À t 0 Þ, and hF * M ðtÞF M ðt 0 Þi ¼ ½ð2γ eg À γ 0 e ÞhN e ðtÞi þ Rδðt À t 0 Þ. Further, the Langevin force F a ðtÞ acting on the optical cavity is negligible under the assumption of low temperature T.
The mechanical displacement xðtÞ is driven by the radiation pressure force F rad ðtÞ ¼ À_ξN photon ðtÞ with the intracavity photon number N photon ðtÞ ¼ jaðtÞj 2 . The equation of motion of the mechanical-displacement-induced detuning δðtÞ ¼ ξxðtÞ is given by € δðtÞ þ Γ _ δðtÞ þ Ω 2 δðtÞ ¼ ξ½F rad ðtÞ þ F thrm ðtÞ þ F xtra ðtÞ=m eff : In above equation, the thermal force noise F thrm ðtÞ ¼ ð_ ffiffi ffi Γ p =x ZPF Þf thrm ðtÞ acting on the movable mirror has been included. The white noise f thrm ðtÞ has the correlation functions f thrm ðtÞ ¼ 0 and f thrm ðtÞf thrm ðt0Þ ¼ n thrm δðt À t0Þ with the average number n thrm of thermal quanta. The extra environmental force F xtra ðtÞ is introduced to include all other noise sources such as ground vibrations. For the macroscopic optomechanics, we choose the frequency Ω ¼ 2π 10 MHz and Q m ¼ 10 2 of the mechanical oscillator and the environmental temperature T ¼ 1 K with n thrm % 2083. The active optomechanics can be investigated by selfconsistently solving Eqs. (5) and (6). Setting _ N u¼e;g ðtÞ ¼ _ MðtÞ ¼ _ AðtÞ ¼ € δðtÞ ¼ _ δðtÞ ¼ 0 and dropping all Langevin noises, one obtains the steady-state solutions N photon;ss hN photon ðt ! 1Þi and δ ss hδðt ! 1Þi from Eqs. (5) and (6). The stability of the derived steady-state solutions can be examined by performing the linear analysis. We write each variable as a sum of its steady-state value and a small fluctuation, e.g. N e ðtÞ ¼ N e;ss þ ϵ N e ðtÞ. Then, Eqs. (5) and (6) are reduced to a matrix linear differential equation _ ϵ ¼ Lϵ with ϵ ¼ ðϵ N e ϵ δ Þ T (Supplementary Note 2). A stable steady-state solution requires that the real parts of all eigenvalues of the 7 7 matrix L are all negative.
One-atom optomechanical microlaser. The active optomechanical system with only one atom placed inside the optical cavity can be described by the Hamiltonian H=_ ¼ ω Câ yâ þ ω A 2 ðσ ee Àσ gg Þ þ Ωb yb þ μðσ egâ þâ yσ ge Þ þ δ ZPFâ yâ ðb þb y Þ; with the photon creation (â y ) and annihilation (â) operators and the phonon creation (b y ) and annihilation (b) operators of the quantum-mechanical oscillator. The population operators of the atom are given byσ ee ¼ jeihej andσ gg ¼ jgihgj. The atomic lowering operator is defined asσ ge ¼ jgihej and the raising operator takes the formσ eg ¼ jeihgj. The first three terms inĤ denote the free energies of photons, atom, and phonons, respectively. The fourth term inĤ corresponds to the atom-cavity interaction while the last them inĤ gives the optomechanical coupling between photons and phonons. The Hilbert space is spanned by fjn photon i jui jn phonon i; n photon ; n phonon 2 Z þ 0 ; u ¼ e; g; sg with the photon number state jn photon i, the atomic state jui, and the phonon number state jn phonon i. Here, an extra atomic state jsi is introduced to model the pumping process of the atom 88 . The symbol Z þ 0 denotes the non-negative integers. The dynamics of the one-atom optomechanical system follows the Lindblad master equation whereρ is the system's density operator and the dissipative term, for example, D½âρ ¼ 2âρâ y Àâ yâρ Àρâ yâ denotes the photons escaping from the optical cavity. The photon and phonon numbers are given by N photon ðtÞ ¼ â y ðtÞâðtÞ ¼ Tr½â yâρ ðtÞ and N phonon ðtÞ ¼ by ðtÞbðtÞ ¼ Tr½b ybρ ðtÞ, respectively. When the pump laser is strong enough, the pump rate R approximates the half of the decay rate of the atom from (5p 6 7p) 2 P 3/2 to jei, i.e. R $ 2π 0:35 MHz. For the microscopic optomechanics, in order to carry out the numerical simulation we choose the mechanical frequency Ω ¼ 2π 100 MHz and Q m ¼ 10 2 of the mechanical oscillator and the environmental temperature 56,57 T ¼ 10 mK with the average number n thrm % 1:6 of thermal quanta. The master Eq. (8) may be solved by using the diagonalization method 89 . The two-time correlation function Â ðtÞBðt þ τÞ between two arbitrary operatorsÂ andB can be calculated by using the quantum regression theorem 66 .

Data availability
All data supporting the findings of this study are available from the corresponding authors upon reasonable request.

Code availability
The computer code to simulate the dynamics is available from the corresponding authors upon reasonable request.