## Main

Conventional optomechanical systems rely on fixed frames to support mechanical elements, leading to unavoidable energy dissipation and thermal loading1. Levitated optomechanical (LOM) systems, that is, controlling motions of levitated objects with optical forces2, have provided unique advantages3, such as fundamental minima of damping and noises, the possibility for levitating complex objects, and a high degree of control over both conservative dynamics and coupling to the environment. These advantages are of significance for both fundamental studies of non-equilibrium physics and applications in metrology4,5,6,7,8,9,10,11. In recent years, remarkable achievements have been witnessed in LOM systems12,13,14, such as the realizations of motional ground-state cooling15,16, room-temperature strong coupling17 and ultrahigh-precision torque sensing18, to name a few. In a recent study19, a phonon laser or coherent amplification of phonons, the quanta of vibrations, was demonstrated for a levitated nanosphere, based on dispersive LOM coupling, in which the optical resonance frequency is modulated by mechanical motion. This study offers exciting opportunities for exploring the boundary of classical and quantum worlds with levitated macroscopic objects16,20,21, as well as making various levitated sensors22,23. Nevertheless, sophisticated feedback controls based on electronic loops19 are needed to provide both nonlinear cooling and linear heating, and only a single-mode phonon laser was observed, without any evidence of nonlinear mechanical harmonics.

Besides LOM systems, phonon lasers have also been built by using semiconductor superlattices24, nanomagnets25, ions26, and nanomechanical27 or electromechanical28 devices. These coherent sound sources, with a shorter wavelength of operation than that of a photon laser of the same frequency29, are indispensable in steering phonon chips30, improving the resolution of motional sensors31 and exploring various effects of phonons32,33,34. However, as far as we know, the ability to achieve multi-frequency phonon lasers with micro-size levitated objects has not been reported. This ability is promising for many important applications, such as multi-frequency motional sensors, exceptional point optomechanics32,33 and topological sound-wave control35.

In this Article, we develop a strategy to achieve nonlinear phonon lasers for a levitated object at microscales by utilizing an active LOM system. We show that, in such a system, dissipative LOM coupling36,37,38,39 can be significantly enhanced by introducing an optical gain, thus leading to not only efficient output of fundamental-mode phonon lasing but also spontaneous emergence of mechanical harmonics. The active gain plays a key role in our work as, for passive systems, only thermal phonons exist and no phonon lasing can appear. To steer this system from a chaotic regime into a phonon lasing regime, an optical gain is used to increase the photon lifetime and thus enhance the LOM coupling; as a result, a 3-order enhancement in the power spectrum of the fundamental-mode phonons is achieved, with a 40-fold narrowing in its linewidth. More importantly, above the lasing threshold, we observe nonlinear harmonics with double and triple mechanical frequencies, as clear evidence of gain-enhanced nonlinearity in an active LOM system.

We stress that we achieve phonon lasing with a micro-object, which is three to four orders larger in size or mass than a nanosphere19. This ability, which has not been achieved in previous studies24,25,26,27,28, is an important step towards applications based on coherent motional control of a wide range of typical micro-size objects. Also, our work demonstrates mechanical harmonics accompanying phonon lasing, with evidence of threshold features and high-order correlations. The key role of active gain in enhancing nonlinear LOM effects opens up the door to engineer and utilize more nonlinear phononic effects in LOM systems and to achieve exciting goals well beyond the reach of passive systems, for example, parity–time symmetric or gain-enhanced LOM sensing. Finally, our work shows dissipative optomechanical coupling in LOM architectures, and compared with dispersive architectures, dissipative LOM systems are expected to have unique advantages in acoustic metrology40.

Our experimental platform includes an active optical cavity and a dual-beam optical tweezer for trapping a single microsphere (Fig. 1a,b). The active cavity works for all three translational degrees of freedom, well tunable along both the longitudinal and the vertical directions. Its quality factor is 106 but can be enhanced to 109 by applying a Yb3+-doped gain fibre. The photon lifetime is thus increased to more than 1 μs, enhancing coherent vibrational amplification of the sphere. Technically, the optical gain is achieved simply by using a Yb3+-doped fibre with a pumping laser at 976 nm, and no other limitation exists. Compared with the cavity-free work19, there is no need to design and carefully control both nonlinear cooling and linear heating of the mechanical motion with complicated electronic loops or algorithms19 (see Supplementary Section 1.1 for details).

As shown in Fig. 1c, essentially different features can be observed in the power spectrums of phonons for the cases with or without the gain, that is, only thermal phonons can exist in the passive case, while a three-order-of-magnitude enhancement can be achieved in the active case for the fundamental mode with mechanical frequency Ω0 = 14.4 kHz. In particular, nonlinear mechanical harmonics with frequencies 2Ω0, 3Ω0 and so on emerge spontaneously in the spectrum, enabling the observation of tunable multi-frequency phonon lasers (Supplementary Section 1.4). Besides the role of gain in achieving phonon lasers, we expect that active LOM systems can also serve as an important tool for studying, for example, parity–time symmetric optomechanics32 or gain-enhanced sound sensing41.

To confirm the phonon lasing of the fundamental mode and its harmonics, we first measure the steady-state phonon population 〈N〉 = 0x2〉/ћ, where M is the mass of the levitated sphere, Ω0 is the oscillation frequency of the mode, x is the centre of mass displacement of the sphere and ћ is the reduced Planck’s constant. Explicit signatures of lasing threshold are observed for the fundamental mode, as shown in Fig. 2a, by increasing the intracavity mean power P. The threshold value is Pth = 0.39 mW, which agrees well with theoretical calculations (Methods). The insets of Fig. 2a showcase a linewidth narrowing, accompanying the transition from thermal to coherent oscillations. Below the threshold, the oscillator features thermal dynamics with mean phonon number 4.63 × 108, and the phonon probability distribution is well described by the Boltzmann distribution. By surpassing the lasing threshold, the phonon number is greatly enhanced towards its saturation value 2.3 × 1010 at P = 1.3 mW, also accompanied by a significant narrowing of linewidth. This is closely related to the fact that phonon lasing emerges in the system, resulting in the Gaussian distribution of the generated coherent phonons. By further increasing the power, the phonon population gradually approaches its saturation value.

Figure 2b further presents the experimental results of the dynamical behaviours of the microsphere. We find that significant motional amplifications of the microsphere emerge above the lasing threshold, with a 40-fold improvement in linewidth narrowing, compared with the case without any gain. Clearly, in the passive case, the interaction between the light and the levitated microsphere is rather weak, due to the large optical loss, and the linewidth remains at about 1.5 kHz. In contrast, in the active LOM system, the linewidth can approach as low as 0.03 kHz above the threshold.

Accompanying the giant enhancement of fundamental-mode phonon lasing, we also observe spontaneous emerging of nonlinear mechanical harmonics. We find that similar lasing features for the double-frequency mechanical mode can be achieved as shown in Fig. 2c, which, as far as we know, have not been achieved in LOM systems19. It is also different from the multi-mode phonon laser shown in a flat membrane trapped in a Fabry–Pérot cavity42, in which mode competitions make it only possible to stimulate a single-phonon mode into the lasing regime.

To further reveal the coherence of the phonon lasers, we study the kth-order phonon autocorrelation functions at zero-time delay $$g^{(k)}(0)$$ (Supplementary Section 2.2). For the phonon laser in fundamental mode, we find that the kth-order correlations of the phonons satisfy $$g_0^{(k)} = k!$$ below the threshold Pth, demonstrating their thermal statistics. As P exceeds Pth, $$g^{(k)}(0)$$ decreases and approaches 1, that is, the phonon dynamics changes from the thermal state to the coherent state. Moreover, $$g^{(k)}(0)$$ approaches 1 for the harmonics with 2Ω0 when operating in the lasing regime, as shown in the inset of Fig. 2d.

We remark that the origin of these nonlinear mechanical harmonics is the anharmonic optical potential produced by the optical-gain-enhanced nonlinearity. Without the gain, for a sphere with larger size than that used in ref. 19, stronger scattering losses lead to a smaller cavity quality factor and thus weaker light–motion coupling. Therefore, we find that the intracavity power P is independent of the x position of the oscillator (Supplementary Section 2.1), while the optical force Fopt, relying on both P and x, responds linearly to the position x (Fig. 3a). By measuring the phonon dynamics for spheres with different sizes (Fig. 3b), we also find that the linewidth of the fundamental mode is invariant for spheres smaller than 1.2 μm; the phonon lasing only emerges for spheres with larger sizes and becomes stronger when the size is increased to 2 μm. For the active case, the intracavity power can be modulated by the mechanical position due to strong light–motion coupling, and thus we find a strongly nonlinear optical force, as shown in Fig. 3a. As a result, the double-frequency component emerges in the phonon power spectrum (Fig. 3c), with also a giant enhancement compared with the passive case with the same intracavity power P = 1.3 mW. Similar lasing features are also observed for the harmonics with the triple-frequency 3Ω0 as shown in Fig. 3d, which is otherwise impossible in the absence of an optical gain.

We note that our results are essentially different from those in a very recent experiment on mechanical parametric amplification43 that originates from the nonparabolicity of the optical potential and typically dominates in the micrometre region for motional displacement (see also ref. 44). In contrast, in our work, the maximum displacement of the oscillator is 300 nm, which is small enough and thus far away from the region of observing parametric amplification. In fact, in our work, the nonlinear optical force coming from dissipative LOM coupling exists even for smaller displacements (Supplementary Section 1.2). Moreover, in the parametric amplification system, the nonparabolicity of the optical potential can lead to not only frequency multiplication but also frequency shifts. When the oscillator experiences larger displacements, the absolute value of the optical force is a convex function, the slope of which (and thus the natural frequency of the oscillator) decreases, while, in our work, the natural frequency increases above the lasing threshold. In addition, the origin of the nonlinearity of our work is fundamentally different from that in ref. 19 (see Supplementary Section 1.5 for details). In our future work, by further combining with other existing techniques used in previous studies19,43, it is possible to probe different nonlinear mechanisms, allowing studies of more nonlinear LOM effects and additional flexible control of LOM devices.

In summary, we have experimentally demonstrated a nonlinear phonon laser in an active LOM system. By introducing optical gain, we have realized a phonon laser on the fundamental mode with a three-order-of-magnitude enhancement in the power spectrum and a 40-fold improvement in linewidth narrowing, without the need of any complicated external feedback control techniques. We also present unequivocal evidence of lasing threshold behaviour and the phase transition from thermal to coherent phonons by measuring the phonon autocorrelations. More interestingly, we observe nonlinear phonon lasers with multiple frequencies, resulting from the optical-gain-enhanced nonlinearity. This observation of such nonlinear mechanical harmonics in LOM systems does not rely on the specific material or the shape of the oscillator45,46. We measured also correlations $$g^{(k)}(0)$$ of harmonic phonon lasing. These results push phonon lasers into the nonlinear regime and make many exciting applications more accessible, such as optomechanical combs47, high-precision metrology and non-classical state engineering. Our work opens up perspectives for achieving levitated phonon devices with active LOM systems and enables a wide range of applications, such as quantum phononics, multi-frequency mechanical sensors and high-precision acoustic frequency combs.

We stress that the purpose of our work is not to outperform dispersive phonon lasers; instead, it enriches the present toolbox of phonon lasing by confirming that, even for a micro-object containing as many as 1011 atoms, that is, four orders of magnitude higher than that used in ref. 19, it is still possible to achieve collective mechanical amplifications and observe the accompanying nonlinear effects. In fact, our work can be combined with existing techniques from previous studies, for future studies of more exciting projects, for example, switch phonon lasing from dissipative-coupling-governed regime to dispersive-coupling-governed regime, transient LOM effects with comparable couplings and the role of gain in operating nonlinear phonon lasers for sound-sensing applications.

Note added in proof: When submitting our revised manuscript, an experiment appeared40, and in that work, different features of dispersive and dissipative couplings in sound sensing are compared, by using a different system, that is, a suspended fibre coupled with an optical resonator.

## Methods

### Experimental details

A dual-beam optical tweezer and an active optical cavity are used in our experimental apparatus. These two set-ups are vertical to each other around the trapping region as shown in Extended Data Fig. 1. A silica sphere is trapped by the dual-beam optical tweezer inside a tiny chamber (Extended Data Fig. 1a, inset). Extended Data Fig. 1b shows a diagram of the experimental set-up. The origin of the cylindrical coordinate system is located at the lens’s focal point of the dual-beam optical tweezer. We install the free-space laser path of the active cavity to a three-dimensional translation stage. The relative position of the active cavity to the trapped sphere can be tuned. A balanced photodetector is installed to monitor the centre of mass displacement of the trapped sphere. There is no need for any external hardware or software to implement feedback control.

The trapping power of the dual-beam optical tweezer (>100 mW) is much stronger than that of the active cavity (1 mW), which can produce robust restoring force to trap the sphere. The wavelength of the low-noise laser (Laser Quantum, Axiom 532) is far away from the absorption spectrum of Yb3+, thus minimizing its influence on the active cavity. We equally split the trapping laser into s- and p-polarized parts by using a halfwave plate and a polarizing beam splitter (PBS1). Then, the s- and p-polarized light can be focused on the sphere through two high-numerical-aperture (NA) objectives L3 and L4 (Mitutoyo, M Plan Apo 100x, NA = 0.7). The portion of the s-polarized light scattered from the sphere is channelled by the PBS2 to a beam splitter whose outputs are then sent to a balanced photodetector (BPD). As the sphere alters the spatial distribution of the scattering light, the differential mode signal from the BPD reveals the sphere’s position48.

The active cavity comprises a continuous-wave ring-cavity fibre laser emitting along with the clockwise and anticlockwise directions. We use a single-mode Yb3+-doped fibre (nLIGHT, Yb1200-6/125, core diameter of 6 μm, cladding diameter of 125 μm) as a gain medium, pumped by a single-mode diode laser at 976 nm through wavelength division multiplexer (WDM1). The residual part of the pump laser is coupled out of the ring cavity through WDM2. The emitted lasers, centred at 1,030 nm, are expanded to free space by collimators C1 and C2 and then focused onto the trapped sphere using two collimating objective lenses L1 and L2 (NA = 0.25). The lenses are specially coated to enhance transmissivity and reduce their influence on the cavity loss. Collimators C1 and C2 are then used to couple the transmitted light through the trapping region back into the fibre loop. We tune the intracavity laser power (0–1.2 mW) by adjusting the power of the pump laser. Photodetectors PD1 and PD2 are installed to monitor the clockwise and anticlockwise laser powers.

We have confirmed that in our experiment, if there is no active medium, no phonon lasing can appear for micro-size levitated spheres, due to strong optical scattering losses. Thus, in our system, the active feedback plays a key role in compensating losses and achieving the nonlinear phonon laser. As already confirmed in our experiment, our active cavity indeed works for all three translational degrees of freedom, with well-tunable feedbacks along the longitudinal and vertical directions (by regulating the foci distance of the trapping beam). The optical gain is achieved by using a Yb3+-doped fibre with a pumping laser at 976 nm. As far as we know, no other limitation exists in our experiment. We expect that our active LOM system can also be an important tool for studying, for example, parity–time symmetric optomechanics and gain-enhanced metrology.

### Cavity alignment

We mount the pumping laser to one collimator (Thorlabs, ZC618FC-B) and a power meter to another. The alignment is evaluated by the coupling coefficient from one collimator to another. Through adjusting the lenses and mirrors, the loss of the free-space optical path can be regulated to its lowest value (usually lower than 0.31 dB).

### Microsphere trapping

The sphere was loaded into the trapping region by using an ultrasonic nebulizer, composed of an ultrasonic sheet metal with a great number of 5 μm holes distributed. It was trapped at atmospheric pressure. In most cases, a microsphere can be trapped within 30 s. Then, we reduced the pressure to the desired experimental level.

### Phonon population distribution

The phonon population is obtained according to the relationship $$\langle N\rangle = M\Omega _0\langle x^2\rangle /\hbar$$, where M is the mass of the levitated sphere, Ω0 is the oscillation frequency of the mechanical mode, $$\left\langle {x^2} \right\rangle$$ is the mean squared displacement of the sphere’s centre of mass and $$\hbar$$ is the reduced Planck’s constant. Given a series of trajectories xi with i = 1, …, L, the distribution $$\rho (n)$$ is constructed as follows. First, starting from i = 1, extract a finite dataset from xi with length k. Loop all datasets to calculate the phonon population as $$N_j = \frac{{M{{\varOmega }}_0}}{\hbar }\frac{1}{k}\mathop {\sum}\limits_{i = j}^k {\left( {x_i - \bar x} \right)^2}$$, where j = 1,...,i, and $$\bar{x}$$ = <x>. Second, determine the common maximum and minimum value of the phonon population variable, denoted by Nmax and Nmin, and discretize the interval $$\left[ {N_{\max },N_{\min }} \right]$$ into P bins as $$\begin{array}{*{20}{c}} {N_\alpha = N_{\min } + \frac{\alpha }{P}\left( {N_{\max } - N_{\min }} \right),} & {\alpha = 0,1, \ldots ,P - 1} \end{array}$$, where P is the total number of bins and α is a particular bin’s number. Then, use the ith time series to calculate the frequency of data points falling in each subinterval, obtaining the distribution for the experiments $$\rho (n)$$.

### Theoretical model

The dissipative optomechanical system, with the first-order dissipative optomechanical coupling $$\kappa (x) \approx \kappa + g_\kappa x$$, can be described via the Hamiltonian49

$$\hat H = \hbar \varDelta \hat a^{\dagger} \hat a + \frac{1}{2}\hbar \omega _{\mathrm{M}}\left( {\hat Q^2 + \hat P^2} \right) + {i}\hbar \sqrt {2\kappa } \left( {1 + \frac{{\sqrt 2 g}}{{2\kappa }}\hat Q} \right)\left[ {\hat a^{\dagger} \left( {\varepsilon _{\mathrm{l}} + \hat c_{\mathrm{l}}} \right) - \hat a\left( {\varepsilon _{\mathrm{l}} + \hat c_{\mathrm{l}}^{\dagger} } \right)} \right],$$
(1)

where $$\hat a$$ ($$\hat a^{\dagger}$$) are the annihilation (creation) operators of the cavity field satisfying the commutation relation $$[\hat a,\hat a^{\dagger} ] = 1$$; $$\varDelta = \omega _{\mathrm{C}} - \omega _{\mathrm{l}}$$ is the cavity detuning with respect to the frequency of the input laser; $$\hat Q = \hat x/(\sqrt 2 x_{{{{\mathrm{zpt}}}}})$$ and $$\hat P = \hat p/(\sqrt 2 p_{{{{\mathrm{zpt}}}}})$$ are the dimensionless position and momentum operators of the oscillator with zero-point motion $$x_{{{{\mathrm{zpt}}}}} = \sqrt {\hbar /2m\omega _{\mathrm{M}}}$$ and zero-point momentum $$p_{{{{\mathrm{zpt}}}}} = \sqrt {\hbar m\omega _{\mathrm{M}}/2}$$, respectively, satisfying the commutation relation $$[\hat Q,\hat P] = {i}$$; $$\omega _{\mathrm{M}}$$ and m are the frequency and mass of the mechanical oscillator; $$\varepsilon _{\mathrm{l}} = \sqrt {P_{{{{\mathrm{in}}}}}/(\hbar \omega _{\mathrm{l}})}$$ is the amplitude of the input laser related to the input power Pin and the input vacuum noise $$\hat c_{\mathrm{l}}$$; and the dissipative coupling constant is $$g = g_\kappa x_{{{{\mathrm{zpt}}}}}$$.

Using the Heisenberg equations of motion $$\dot{\hat{{\mathscr{O}}}}=-{\mathrm{i}}[\hat{{\mathscr{O}}},\hat{H}]/\hslash$$ and adding the corresponding damping and noise terms, the quantum Langevin equations of the system operators are given by

$$\begin{array}{c}\dot{\hat{Q}}={\omega }_{{\mathrm{M}}}\hat{P},\\ \dot{\hat{P}}=-{\omega }_{{\mathrm{M}}}\hat{Q}-i\frac{g}{\sqrt{\kappa }}[{\hat{a}}^{\dagger }({\varepsilon }_{{\mathrm{l}}}+{\hat{c}}_{{\mathrm{l}}})-\hat{a}({\varepsilon }_{{\mathrm{l}}}+{\hat{c}}_{{\mathrm{l}}}^{\dagger })-\sqrt{2\kappa }{\hat{a}}^{\dagger }\hat{a}]-{\gamma }_{{\mathrm{M}}}\hat{P}+\hat{\xi },\\ \dot{\hat{a}}=-[i\varDelta +(\kappa +\sqrt{2}g\hat{Q}-G)]\hat{a}+\sqrt{2\kappa }\left(1+\frac{\sqrt{2}g}{2\kappa }\hat{Q}\right)({\varepsilon }_{{\mathrm{l}}}+{\hat{c}}_{{\mathrm{l}}}),\end{array}$$
(2)

where the input vacuum noise operator $$\hat c_{\mathrm{l}}$$ has zero mean value and delta correlation $$\langle \hat c_{\mathrm{l}}(t)\hat c_{\mathrm{l}}^{\dagger} (t^\prime )\rangle = \delta (t - t^\prime )$$, where δ indicates the Dirac delta function; G is the optical gain of the active cavity; and ξ is the thermal noise with zero mean value and following correlation function $$\langle {\hat {\xi}} (t){\hat {\xi}} ^{\dagger} (t^\prime )\rangle = \frac{1}{{2\uppi }}\frac{{{{\varGamma }}_{\mathrm{M}}}}{{{\it{\omega }}_{\mathrm{M}}}}{\int} {\omega {\it{{\mathrm{e}}}}^{ - {\mathrm{i}}\omega (t - t^\prime )}\left[ {1 + \coth \left( {\frac{{\hbar \omega }}{{2k_{\mathrm{B}}T}}} \right)} \right]} {\mathrm{d}}\omega ,$$ where kB is the Boltzmann constant, t is time and T is temperature. By setting the time derivatives in above equations to 0, we find the steady state of the dynamical variables:

$$\begin{array}{*{20}{c}} {\bar Q = \displaystyle {i\frac{{g\varepsilon _{\mathrm{l}}}}{{\omega _{\mathrm{M}}\sqrt \kappa }}}\left( {\bar a - \bar a^ \ast } \right),} & {\bar a = \displaystyle{ \frac{{\left( {\sqrt {2\kappa } + \frac{g}{{\sqrt \kappa }}\bar Q} \right)\varepsilon _{\mathrm{l}}}}{{i\varDelta + \kappa + \sqrt 2 g\bar Q - G}}}.} \end{array}$$
(3)

Now we consider $$\hat c_{\mathrm{l}}$$ ($$\hat c_{\mathrm{l}}^{\dagger}$$) as the annihilation (creation) operator of the input field satisfying the commutation relation $$[\hat c_{\mathrm{l}},\hat c_{\mathrm{l}}^{\dagger} ] = 1$$. The Hamiltonian can be rewritten as

$$\begin{array}{l}\hat H = \hat H_0 + \hat H_{{{{\mathrm{int}}}}} + \hat H_{{{{\mathrm{dr}}}}},\\ \hat H_0 = \hbar \left( {\varDelta - i\kappa + iG} \right)\hat a^{\dagger} \hat a + \hbar \omega _{\mathrm{M}}\hat b^{\dagger} \hat b,\\ \hat H_{{{{\mathrm{int}}}}} = - {\mathrm{i}}\hbar g\left( {\hat b^{\dagger} + \hat b} \right)\hat a^{\dagger} \hat a + i\hbar \sqrt {2\kappa } \left( {\hat a^{\dagger} \hat c_{\mathrm{l}} - \hat a\hat c_{\mathrm{l}}^{\dagger} } \right),\\ \hat H_{{{{\mathrm{dr}}}}} = i\hbar \sqrt {2\kappa } \varepsilon _{\mathrm{l}}\left( {\hat a^{\dagger} - \hat a} \right),\end{array}$$
(4)

where $$\hat b$$ ($$\hat b^{\dagger}$$) are the annihilation (creation) operator of the mechanical mode. We introduce the supermode operators $$\hat a_ + = \left( {\hat a + i\hat c_{\mathrm{l}}} \right)/\sqrt 2$$ and $$\hat a_ - = \left( {\hat a - i\hat c_{\mathrm{l}}} \right)/\sqrt 2$$, which satisfy the commutation relations $$\left[ {\hat a_ + ,\hat a_ + ^{\dagger} } \right] = \left[ {\hat a_ - ,\hat a_ - ^{\dagger} } \right] = 1$$ and $$\left[ {\hat a_ + ,\hat a_ - } \right] = \left[ {\hat a_ + ,\hat a_ - ^{\dagger} } \right] = 0$$. Under the rotating-wave approximation, $$2\sqrt {2\kappa } + \omega _{\mathrm{M}}$$ and $$\omega _{\mathrm{M}}$$ are much larger than $$\left| {2\sqrt {2\kappa } - \omega _{\mathrm{M}}} \right|$$, and the equations of motion of the system are given by

$$\begin{array}{c}{\dot{\hat{a}}}_{+}=-(i{\omega }_{+}+\gamma ){\hat{a}}_{+}-\left(\frac{g}{2}\hat{b}+\gamma +i\frac{\varDelta }{2}\right){\hat{a}}_{-}+\sqrt{\kappa }{\varepsilon }_{{\mathrm{l}}},\\ {\dot{\hat{a}}}_{-}=-(i{\omega }_{-}+\gamma ){\hat{a}}_{-}-\left(\frac{g}{2}{\hat{b}}^{\dagger }+\gamma +i\frac{\varDelta }{2}\right){\hat{a}}_{+}+\sqrt{\kappa }{\varepsilon }_{{\mathrm{l}}},\\ \dot{\hat{p}}=-2\left(\gamma +i\sqrt{2\kappa }\right)\hat{p}+\left(\frac{g}{2}\hat{b}+\gamma +i\frac{\varDelta }{2}\right)\delta \hat{n}+\sqrt{\kappa }{\varepsilon }_{{\mathrm{l}}}({\hat{a}}_{-}^{\dagger }+{\hat{a}}_{+}),\\ \dot{\hat{b}}=-(i{\omega }_{{\mathrm{M}}}+{\gamma }_{{\mathrm{M}}})\hat{b}-\frac{g}{2}\hat{p},\end{array}$$
(5)

where $$\hat p = \hat a_ - ^{\dagger} \hat a_ +$$ and $$\delta \hat n = \hat a_ + ^{\dagger} \hat a_ + - \hat a_ - ^{\dagger} \hat a_ -$$ are the ladder operator and the population inversion operator, respectively; $$\omega _ \pm = \frac{\varDelta }{2} \pm \sqrt {2\kappa }$$; $$\gamma = \frac{{\kappa - G}}{2}$$; and $$\gamma _{\mathrm{M}}$$ denotes the damping of the oscillator. Since $$\gamma \gg \gamma _{\mathrm{M}}$$, we can adiabatically eliminate the degrees of freedom of the optical modes by setting the time derivatives of the optical components to 0. Then, we obtain the steady-state values:

$$\begin{array}{lll} {a_ + = \displaystyle{ \frac{{\sqrt \kappa \varepsilon _{\mathrm{l}}\left( { - gb - i2\sqrt {2\kappa } } \right)}}{{\beta - i\zeta }}},} \\ {a_ - = \displaystyle{ \frac{{\sqrt \kappa \varepsilon _{\mathrm{l}}\left( { - gb^ \ast + i2\sqrt {2\kappa } } \right)}}{{\beta - i\zeta }}},} \\ {p = \displaystyle{\frac{{2\sqrt \kappa \varepsilon _{\mathrm{l}}\left( {a_ - ^{\dagger} + a_ + } \right) + \left( {gb + 2\gamma + i\varDelta } \right)\varDelta n}}{{4\gamma + i2\left( {2\sqrt {2\kappa } - \omega _{\mathrm{M}}} \right)}}},} \end{array}$$
(6)

with $$\beta = \beta _0 - \left( {\kappa - G} \right)\frac{{gQ}}{{\sqrt 2 }}$$, $$\beta _0 = 4\kappa - \frac{{g^2}}{2}n_b$$, $$\zeta = \frac{{\varDelta gQ}}{{\sqrt 2 }}$$, and the mean phonon number $$n_b = b^{\dagger} b$$.

Then, the dynamics of the mechanical oscillator can be described by $$\dot b = \left( {G_{\mathrm{M}} - \gamma _{\mathrm{M}} - i\omega _{\mathrm{M}} - i\omega _{\mathrm{M}}^\prime } \right)b - D$$, where the parameters $$\omega _{\mathrm{M}}^\prime$$ and D can be obtained by comparing this motion equation with equation (7). The effective mechanical gain is given by

$$G_{\mathrm{M}} = \frac{{\left( {G - \kappa } \right)g^2/4}}{{\left( {G - \kappa } \right)^2 + \left( {2\sqrt {2\kappa } - \omega _{\mathrm{M}}} \right)^2}}\left( {\delta n - \frac{{4\kappa \beta \varepsilon _{\mathrm{l}}^2}}{{\beta ^2 + \zeta ^2}}} \right).$$
(7)

By setting $$G_{\mathrm{M}} = \gamma _{\mathrm{M}}$$ and assuming that the phonon laser satisfies the condition of complete inversion δnn+, we obtain the threshold of the phonon laser with $$P_{{{{\mathrm{th}}}}} = \hbar \omega _ + \left( {G - \kappa } \right)n_ +$$, that is

$$\begin{array}{l}P_{{{{\mathrm{th}}}}} = P_{{{{\mathrm{th}}}},0} + P_{{{{\mathrm{th}}}},1},\\ P_{{{{\mathrm{th}}}},0} = 2\frac{{\hbar \gamma _{\mathrm{M}}}}{{g^2}}\left( {\varDelta + 2\sqrt {2\kappa } } \right)\left[ {\left( {2\sqrt {2\kappa } - \omega _{\mathrm{M}}} \right)^2 + \left( {G - \kappa } \right)^2} \right]\\ +\displaystyle{ \frac{{2\hbar \kappa \beta \varepsilon _{\mathrm{l}}^2}}{{\beta ^2 + \zeta ^2}}}\left( {\varDelta + 2\sqrt {2\kappa } } \right)\left( {G - \kappa } \right),\\ P_{{{{\mathrm{th}}}},1} =\displaystyle{ \frac{{2\hbar \kappa \beta \varepsilon _{\mathrm{l}}^2\left( {\varDelta + 2\sqrt {2\kappa } } \right)\left( {G - \kappa } \right)}}{{\beta ^2 + \zeta ^2}}}.\end{array}$$
(8)

### Phonon correlations

The normalized equal-time kth-order phonon correlation is given by50

$$\begin{array}{c}g^{(k)}(0) =\displaystyle{ \frac{{\left\langle {\hat b^{{\dagger} k}\hat b^k} \right\rangle }}{{\left\langle {\hat b^{\dagger} \hat b} \right\rangle ^k}}} = \left\langle {\hat N} \right\rangle ^{ - k}\displaystyle{\mathop {\sum}\limits_{N = k}^\infty {N(N - 1) \cdots (N - k + 1)P(N)}} \\ = \displaystyle{\mathop {\sum}\limits_{N = k}^\infty {\frac{{N!}}{{(N - k)!}}\frac{{P(N)}}{{\left\langle {\hat N} \right\rangle ^k}}}} \end{array}$$
(9)

In particular, the second-order phonon correlation function is

$$g^{(2)}(0) = \frac{{\left\langle {\hat b^{{\dagger} 2}\hat b^2} \right\rangle }}{{\left\langle {\hat b^{\dagger} \hat b} \right\rangle ^2}} = \frac{{\langle \hat N(\hat N - 1)\rangle }}{{\langle \hat N\rangle ^2}} = \frac{{\langle \hat N\rangle ^2 - \langle \hat N\rangle }}{{\langle \hat N\rangle ^2}}$$
(10)

The third-order and fourth-order phonon correlation functions are $$g^{(3)}(0) = \left( {\left\langle {\hat N^3} \right\rangle - 3\left\langle {\hat N^2} \right\rangle + 2\langle \hat N\rangle } \right)/\langle \hat N\rangle ^3$$ and $$g^{(4)}(0) =$$$$\left( {\left\langle {\hat N^4} \right\rangle - 6\left\langle {\hat N^3} \right\rangle + 11\left\langle {\hat N^2} \right\rangle - 6\langle \hat N\rangle } \right)/\langle \hat N\rangle ^4$$, respectively.