Optomechanical squeezing with strong harmonic mechanical driving

Xin-Yu Lin; Guang-Zheng Ye; Ye Liu; Yun-Kun Jiang; Huaizhi Wu

doi:10.1364/OE.516529

1. Introduction

Quantum squeezing of macroscopic objects, refers to that the fluctuation in its position or momentum is suppressed below the vacuum level [1–12]. Generation of squeezed states for a macroscopic mechanical oscillator is important not only for fundamental test of quantum mechanical theory, but also has wide applications in high-precision measurements [13–18] , even for detection of the mechanical displacement induced by the gravitational wave [19,20]. Cavity optomechanics [21], where the radiation pressure force of light allows to cool the mechanical resonators at macroscopic scales down close to the quantum ground state, provides an ideal platform for studying quantum squeezing of mechanical fluctuations. In comparison with the parametric approach [22], where the mechanical spring constant is periodically modulated with twice the mechanical frequency, optomechanical setups are promising to generate a steady-state squeezing of mechanical motion below one half of the zero-point level (i.e. the 3-dB limit) [10,11], beyond the restriction of the stability condition for the parametrically modulated system.

Generating mechanical squeezing over the 3-dB limit can be realized with cavity optomechanical setups under different cavity driving schemes [23–42] , e.g. by driving cavity with a broadband squeezed light [25,26], short optical pulses [5], two-tone control lasers of different amplitudes [10,27], or with a periodically amplitude-modulated driving laser [23]. In particular, the reservoir engineering approach based on the two-tone driving [10,27,43] has been experimentally demonstrated with the steady-state mechanical squeezing stronger than the 3-dB limit [11], where classical feedback and squeezed-light input are not involved [44]. Alternatively, there are mechanical-driving-based proposals, which show that mechanical squeezing can be generated again by combining parametrical mechanical modulation [45]. While mechanical instability may occur for a strong parametric driving, the optomechanically induced cooling can help to stabilize the mechanical oscillator for better squeezing. The parametric method can also be applied for mechanical cooling [46], generation of phase-sensitive amplification [47] and squeezing of optical signals reflected from the cavity [48,49]. Moreover, mechanical resonators subjected to linear mechanical driving has been discussed widely in the classical regime, which enables to observe optomechanically induced transparency or amplification [50–52] and four-wave-mixing [53], but it has been rarely explored for the quantum regime. Indeed, the intracavity optical pressure can inherently generate weak harmonic mechanical driving, which is normally neglected for an appropriate laser driving strength.

In this paper, we propose to generate strong mechanical squeezing in an optomechanical setup by simply applying a constant red-detuned cavity driving and a harmonic mechanical force. The force periodically shakes the mechanical oscillator at twice the mechanical eigenfrequency, and has a large driving amplitude modulating the cavity frequency shift on the order of the mechanical eigenfrequency. The effect of optomechanically induced heating may occur when the mechanical mirror is pushed over the cavity resonance. By avoiding heating-induced unstable regime, it ensures that the mechanical mirror is cooled down to a dynamically squeezed steady state with the maximum degree of squeezing over 8 dB. The underlying physics can again be interpreted by the dissipative mechanism with the optomechanical cavity acting as an engineered reservoir [27]. Remarkably, the sideband driving tones here are effectively introduced by the Landau-Zener analogical dynamics [54], which is induced by the strong harmonic driving, instead of the mechanical self-sustained oscillation under a blue-detuned cavity driving [54]. Moreover, we show that the harmonic mechanical force can be implemented by coupling the mirror to an auxiliary cavity under strong cavity driving, where the mechanical oscillator is driven by the strong radiation pressure force approximately described by a harmonic function and the force noise associated with the cavity input noise is included. The scheme provides a simple way for generating strong squeezing of a macroscopic mirror, with potential applications in quantum sensing and in addressing multi-mode quantum entanglement.

The paper is organized as follows. In Sec. 2, we introduce the standard optomechanical setup with the mechanical resonator driven by a harmonic force. In Sec. 3, we derive the effective Hamiltonian for the periodical asymptotic dynamics, and study the system stability by considering the Routh-Hurwitz criterion. In Sec. 4, we discuss the dissipative mechanism for generating mechanical squeezing, and then compare it with the numerical results shown in Sec. 5 by considering the experimental feasibility.

2. Model and Hamiltonian

We consider a standard optomechanical cavity with the length $L$ and the finesse $\mathcal {F},$ as schematically shown in Fig. 1. The cavity mode of the resonant frequency $\omega _{c}$ is driven by an external laser of the carrier frequency $\omega _{l}$ (along the cavity axis) and the amplitude $E=\sqrt {2\kappa P/\hbar \omega _{l}}$, with $P$ and $\kappa =\pi c/2\mathcal {F}L$ being the laser power and the cavity decay rate, respectively. The movable mirror, which is treated as a quantum-mechanical harmonic oscillator with effective mass $m$, frequency $\omega _{m}$, and damping rate $\gamma _{m}$, is driven by a harmonic force of the amplitude $\epsilon$ and the frequency $\Omega$ with an initial phase $\phi$. Here, we allow for any periodic driving in principle, but we set $\Omega =2\omega _{m}$ for generating multiple driving sidebands around cavity resonance, which are then modulated by a large driving amplitude. The total Hamiltonian of the system in the frame rotating at the laser frequency $\omega _{l}$ can be written as ($\hbar =1$)

(1)$$H ={-}\Delta_{0}a^{{\dagger}}a+\frac{\omega_{m}}{2}(p^{2}+q^{2})-ga^{{\dagger}}aq +\epsilon\text{sin}(\Omega t+\phi)q+iE(a^{{\dagger}}-a),$$

where $a$ and $a^{\dagger }$ are annihilation and creation operators of the cavity mode, $q$ and $p$ are the dimensionless position and momentum operators for the movable mirror satisfying the standard canonical commutation relation $[q,\:p]=i$. $\Delta _{0}=\omega _{l}-\omega _{c}$ is the laser detuning from the cavity resonance. $g$=$x_{\text {ZPF}}\omega _{c}/L$ is the single-photon coupling strength between light and mechanical mirror arising from the radiation pressure force, with $x_{\text {ZPF}}=\sqrt {\hbar /2m\omega _{m}}$ being the zero point motion of the mechanical mode. The driven-dissipative dynamics of the light-mechanical system can be described by the quantum Langevin equations (QLEs)

(2)$$\begin{aligned} \dot{q} & = \omega_{m}p,\\ \dot{p} & = -\omega_{m}q-\gamma_{m}p-\epsilon\text{sin}(\Omega t+\phi)+ga^{{\dagger}}a+\xi,\\ \dot{a} & = -(\kappa-i\Delta_{0})a+igaq+E+\sqrt{2\kappa}a_{in}.\end{aligned}$$

where the optical ($a_{in}$) and mechanical ($\xi$) noise operators have zero mean values, and are characterized by the nonzero correlation functions $\langle a_{in}(t)a_{in}^{\dagger }(t')\rangle =\delta (t-t')$ and $\langle \xi (t)\xi (t')+\xi (t')\xi (t)\rangle /2=\gamma _{m}(2n_{m}+1)\delta (t-t')$ (with $n_{m}=[\exp (\hbar \omega _{m}/k_{B}T)-1]^{-1}$ being the mean thermal phonon number) under the Markovian approximation [23,55]. In addition, we first neglect the noise from the harmonic driving force, which is considered later with the harmonic driving force being implemented with strong radiation pressure force via an auxiliary cavity.

Fig. 1. Sketch of the optomechanical setup. The optomechanical cavity is driven by an external laser of the strength $E$. The movable mirror acts as the mechanical resonator and is driven by a harmonic force with the strength $\epsilon$, and the frequency $\Omega$ being twice the mechanical eigen-frequency $\omega _{m}$. See text for the details.

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For a sufficiently strong laser drive and a large intracavity photon number $\langle a^{\dagger }(t)a(t)\rangle \gg 1$, we can linearize the dynamics by rewriting each Heisenberg operator as $O=\langle O(t)\rangle +\delta O$ ($O=q,\,p,\,a$), where $\delta O$ are quantum fluctuation operators with zero-mean value, and justify that the mean-field calculations $\langle a^{\dagger }(t)a(t)\rangle \simeq \left |\langle a(t)\rangle \right |^{2}$ and $\langle a(t)q(t)\rangle \simeq \langle a(t)\rangle \langle q(t)\rangle$ are good approximations. We thus obtain the equation of motion for the first moments of the optical and mechanical modes

(3)$$\begin{aligned} \left\langle \dot{q}(t)\right\rangle & = \omega_{m}\left\langle p(t)\right\rangle ,\\ \left\langle \dot{p}(t)\right\rangle & = -\omega_{m}\left\langle q(t)\right\rangle -\gamma_{m}\left\langle p(t)\right\rangle -\epsilon\text{sin}(\Omega t+\phi) +g\left|\left\langle a(t)\right\rangle \right|^{2},\\ \left\langle \dot{a}(t)\right\rangle & ={-} (\kappa-i\Delta_{0})\left\langle a(t)\right\rangle +ig\left\langle a(t)\right\rangle \left\langle q(t)\right\rangle +E,\end{aligned}$$

and the linearized QLEs for the corresponding quantum fluctuation operators

(4)$$\begin{aligned} \delta\dot{q} & = \omega_{m}\delta p,\\ \delta\dot{p} & = -\omega_{m}\delta q-\gamma_{m}\delta p+g[\left\langle a(t)\right\rangle \delta a^{{\dagger}}+\left\langle a(t)\right\rangle ^{*}\delta a]+\xi,\\ \delta\dot{a} & = -(\kappa-i\Delta)\delta a+ig\left\langle a(t)\right\rangle \delta q+\sqrt{2\kappa}a_{in},\end{aligned}$$

where the effective laser detuning $\Delta (t)=\Delta _{0}+\delta _{\text {d}}(t)$ is modulated by the mechanical oscillation with the frequency drift $\delta _{\text {d}}(t)=g\left \langle q(t)\right \rangle$. Without mechanical driving $\epsilon =0$, $\delta _{\text {d}}(t)$ can normally be treated as a stationary frequency drift for weak laser driving, and the laser driving on the red-sideband driving tone $\Delta (t)\approx -\omega _{m}$ can induce optomechanical cooling [56]. Here, alternatively, we consider the strong mechanical driving regime, where the frequency drift can be large compared with the mechanical frequency, i.e. $\text {max}(\delta _{\text {d}})/\omega _{m}\sim 1$. This results in two effects. First, the time dependent $\delta _{\text {d}}(t)$ together with the nonlinear optomechanical interaction strongly modulates the effective multiple sideband couplings [23]. Second, $\Delta (t)$ may change its sign in the dynamical evolution, leading to transient heating of the mechanical mirror by the radiation pressure and even the system instability [54,57,58].

3. Time-periodic linearized Hamiltonian and quadrature squeezing

Assume that the system keeps far away from the optomechanical instabilities and multistabilities for the strong mechanical driving [54,58], which is numerically checked throughout the paper with the Routh-Hurwitz criterion (see the later discussion), and the semiclassical dynamics will undergo an initial transient and then gradually evolves toward a dynamical steady state with the oscillation period same to the periodicity of the harmonic driving force $\tau =2\pi /\Omega$. The asymptotic solutions of $\langle O(t)\rangle$ ($O=q,p,a$) for the nonlinear Eq. (3) can then be expanded perturbatively in the powers of $g$ via the Fourier expansion [23,59]

(5)$$\left\langle O(t)\right\rangle = \sum_{j=0}^{\infty}\sum_{n={-}\infty}^{\infty}O_{n,\,j}e^{in\Omega t}g^{j},$$

where the optomechanical coupling strength is assumed to be $g\ll \omega _{m}$ concerning a weak single-photon nonlinearity. Substituting this into Eq. (3), we find the following recursive formulas for the time-independent coefficients $O_{n,j}$ [23], where for $j=0$ the nonvanishing terms are

(6)$$q_{\pm1,0}=\frac{\mp\epsilon\omega_{m}e^{{\pm} i\phi}}{2i(\omega_{m}^{2}\pm i\Omega\gamma_{m}-\Omega^{2})},\text{ } p_{\pm1,0}={\pm}\frac{i\Omega}{\omega_{m}}q_{\pm1,0},\text{ }a_{0,0}=\frac{E}{\kappa-i\Delta_{0}},$$

corresponding to the light-mechanics decoupling case, and for $j>0$,

q_{n,j} = \sum_{k=0}^{j-1}\sum_{m={-}\infty}^{\infty}\frac{\omega_{m}a_{m,k}^{*}a_{n+m,j-k-1}}{\omega_{m}^{2}-(n\Omega)^{2}+i\gamma_{m}n\Omega},\text{ } p_{n,j}=\frac{in\Omega}{\omega_{m}}q_{n,j},

(7)$$a_{n,j}=i\sum_{k=0}^{j-1}\sum_{m={-}\infty}^{\infty}\frac{a_{m,k}q_{n-m,j-k-1}}{\kappa+i(n\Omega-\Delta_{0})}.$$

Then, we obtain the linearized Hamiltonian

(8)$$H_{\text{Lin}} ={-}\Delta(\epsilon,t)\delta a^{{\dagger}}\delta a+\frac{\omega_{m}}{2}(\delta p^{2}+\delta q^{2})-\sqrt{2}\left[G(t)\delta a^{{\dagger}}+G^{*}(t)\delta a\right]\delta q,$$

where

(9)$$\Delta(\epsilon,t)=\Delta_{0}+\sum_{j=0}^{+\infty}\sum_{n={-}\infty}^{+\infty}q_{n,j}(\epsilon)e^{in\Omega t}g^{j+1}$$

strongly depends on the mechanical driving amplitude $\epsilon$ and its sign may be changed for $g|q_{\pm n,0}(\epsilon )|\sim |\Delta _{0}|$, and

(10)$$G(t)\equiv\frac{g}{\sqrt{2}}\langle a(t)\rangle =\sum_{n={-}\infty}^{\infty}g_{n}e^{{-}in\Omega t}$$

is the cavity enhanced optomechanical coupling with $g_{n}=\frac {1}{\sqrt {2}}\sum _{j=0}^{\infty }a_{-n,j}g^{j+1}$. Equation (10) can be equivalently regarded as multiple classical drive tones driving the optomechanical cavity at the mechanical sideband frequencies $n\Omega$ with imbalanced amplitudes $g_{n}$.

Now let us turn to the stability condition and the measures of the mechanical squeezing. For the purpose, we now introduce the amplitude and phase quadratures of the cavity mode as $\delta x=(\delta a+\delta a^{\dagger })/\sqrt {2}$, $\delta y=(\delta a-\delta a^{\dagger })/i\sqrt {2}$ and the analogous input quantum noise quadratures as $\delta x_{in}=(\delta a_{in}+\delta a_{in}^{\dagger })/\sqrt {2}$, $\delta y_{in}=(\delta a_{in}-\delta a_{in}^{\dagger })/i\sqrt {2}$. Then the time-dependent equations of motion for the quadrature operators $u(t)=[\delta q,\delta p,\delta x,\delta y]^{T}$ read

(11)$$\dot{u}(t) = M(t)u(t)+N(t),$$

with the drift matrix

(12)$$\begin{array}{ccc} M(t) & = \left[\begin{array}{cccc} 0 & \omega_{m} & 0 & 0\\ -\omega_{m} & -\gamma_{m} & 2G_{x}(t) & 2G_{y}(t)\\ -2G_{y}(t) & 0 & -\kappa & -\Delta(\epsilon,t)\\ 2G_{x}(t) & 0 & \Delta(\epsilon,t) & -\kappa \end{array}\right]\end{array}$$

and the diffusion $N(t)=[0,\xi (t),\sqrt {2\kappa }\delta x_{in},\sqrt {2\kappa }\delta y_{in}]^{T}$ being the noise sources. Here $G_{x}(t)$ and $G_{y}(t)$ are real part and imaginary parts of the effective optomechanical coupling $G(t)$. Since Eq. (12) is time-dependent, there does not exist a general rule for determining the stability of the system. We thus consider the sufficient but not necessary condition (referring to the Routh-Hurwitz criterion [60]) that all the eigenvalues of the matrix $M(t)$ have negative real parts at any time, which gives rise to stability conditions,

(13)$$\begin{aligned} S_{1}=2\Delta^{2}\kappa+2\kappa^{3}+4\kappa^{2}\gamma_{m}+\omega_{m}^{2}\gamma_{m}+2\kappa\gamma_{m}^{2} & > & 0,\\ S_{2}=4\Delta G^{2}\omega_{m}+\Delta^{2}\omega_{m}^{2}+\kappa^{2}\omega_{m}^{2} & > & 0,\\ S_{3}=(2\kappa\omega_{m}^{2}+\Delta^{2}\gamma_{m}+\kappa^{2}\gamma_{m})S_{1}-S_{2}(2\kappa+\gamma_{m})^{2} & > & 0, \end{aligned}$$

and is a conservative choice. In the following, we numerically confirm the system stability via Eq. (13) for all the plots.

On the other hand, since the linearized system under the stable regime will evolve onto an asymptotic Gaussian state for the Gaussian-typed white noise [61], we can then characterize the second moments of the quadratures of the asymptotic state through the covariance matrix (CM) $V(t)$, with the matrix elements being $V_{k,l}(t) = \langle u_{k}(t)u_{l}^{\dagger }(t)+u_{l}^{\dagger }(t)u_{k}(t)\rangle /2.$ Here, the diagonal elements $V_{11}(t)=\left \langle \delta q(t)^{2}\right \rangle$, $V_{22}(t)=\left \langle \delta p(t)^{2}\right \rangle$ of $V(t)$ represent the quantum fluctuations in the mechanical position and momentum, and the other two diagonal terms $V_{33}(t)=\left \langle \delta x(t)^{2}\right \rangle$, $V_{44}(t)=\left \langle \delta y(t)^{2}\right \rangle$ represent the variances in the amplitude and phase quadratures of the cavity mode. From Eq. (11), we can readily obtain the linear differential equation governing the evolution of the CM $V(t)$

(14)$$\dot{V}(t) = M(t)V(t)+V(t)M^{T}(t)+D,$$

where $\begin {array}{ccc} D & = diag[0,\gamma _{m}(2n_{m}+1),\kappa,\kappa ]\end {array}$ is a diagonal matrix of noise correlations, derived from $\delta (t-t')D_{k,l}=\langle n_{k}(t)n_{l}^{\dagger }(t')+n_{l}^{\dagger }(t')n_{k}(t)\rangle /2$. By solving Eq. (14) in the long time limit, we can examine whether the mechanical oscillator is position- or momentum-squeezed corresponding to either $\left \langle \delta q(t)^{2}\right \rangle <1/2$ or $\left \langle \delta p(t)^{2}\right \rangle <1/2$ for the dynamical steady state. For convenience, the degree of the squeezing can be further quantified in the dB unit, which is calculated by

(15)$$\xi_{O}(t)={-}10\log_{10}\frac{\left\langle \delta O(t)^{2}\right\rangle }{\left\langle \delta O(t)^{2}\right\rangle _{vac}},\text{ }\{O=p,\,q\},$$

with $\left \langle \delta O(t)^{2}\right \rangle _{vac}=1/2$ being the zero-point fluctuations.

4. Mechanical squeezing stabilized by dissipation

According to the linearized Hamiltonian Eq. (8), the periodical mechanical oscillation with large amplitude (approximately proportional to $\epsilon$) modulates the optomechanical system in two aspects: first, it effectively introduces the sideband driving tones (including the blue-sideband driving tone) with imbalanced amplitudes via the nonlinear photon-phonon interaction; second, the effective detuning $\Delta (\epsilon,t)$ centering around the red-sideband driving tone periodically sweeps the mechanics from center frequency to near cavity resonance, leading to Landau-Zener analogous dynamics, which further modulates the amplitudes of the effective driving tones. The physical insight can be well studied by considering the steady-state behavior with the ansatz

(16)$$\Delta(\epsilon,t)=\Delta_{0}+g[q_{0}+2\left|q_{1}\right|\text{cos}(\Omega t+\varphi)],$$

where $q_{n}\equiv e^{in\Omega t}\sum _{j}q_{n,j}(\epsilon )g^{j}$ ($n=0,1$). We then rewrite the linearized Hamiltonian (8) in terms of the mechanical annihilation and creation operators $\delta b=(\delta q+i\delta p)/\sqrt {2}$, $\delta b^{\dagger }=(\delta q-i\delta p)/\sqrt {2}i$,

(17)$$H_{\text{Lin}}^{\prime} = H_{0}-\Delta^{\prime}\delta a^{{\dagger}}\delta a-\left[G(t)\delta a^{{\dagger}}(\delta b+\delta b^{{\dagger}})+\text{H.c.}\right]$$

with

(18)$$H_{0}=\left[\omega_{m}-2g\left|q_{1}\right|\text{cos}(\Omega t)\right]\delta a^{{\dagger}}\delta a+\omega_{m}\delta b^{{\dagger}}\delta b,$$

and $\Delta ^{\prime }=\Delta _{0}+gq_{0}+\omega _{m}$. Since $\varphi$ only induces translational invariant behavior, it is set to zero for convenience. Furthermore, we move to the frame rotating at the mechanical frequency $\omega _{m}$, and transform the Hamiltonian onto the interaction picture with the unitary operator

(19)$$\begin{aligned}U(t) & = \text{exp}[i\int_{0}^{t}H_{0}(t)dt]\\ & = \text{exp}\left[i\left(\omega_{m}t-\frac{2g|q_{1}|}{\Omega}\text{sin}\left(\Omega t\right)\right)\delta a^{{\dagger}}\delta a\right] \otimes\text{ }\text{exp}\left(i\omega_{m}t\delta b^{{\dagger}}\delta b\right), \end{aligned}$$

leading to the effective Hamiltonian

(20)$$\begin{aligned}H_{\text{eff}} & = U(t)H_{\text{Lin}}^{\prime}U^{{\dagger}}(t)-iU(t)\dot{U}^{{\dagger}}(t)\\ & = -\Delta^{\prime}\delta a^{{\dagger}}\delta a-G(t)e^{{-}ix\text{sin}(\Omega t)}\delta a^{{\dagger}}\delta b -G(t)e^{{-}ix\text{sin}(\Omega t)}\delta a^{{\dagger}}\delta b^{{\dagger}}e^{i\Omega t}+\text{H.c.}\\ & = -\Delta^{\prime}\delta a^{{\dagger}}\delta a-\sum_{n,n^{\prime}={-}\infty}^{\infty}g_{n^{\prime}}J_{n}(x) \delta a^{{\dagger}}\left[ e^{{-}i(n+n^{\prime})\Omega t}\delta b +e^{{-}i(n+n^{\prime}-1)\Omega t}\delta b^{{\dagger}}\right]+\text{H.c.}, \end{aligned}$$

where $x=2g|q_{1}|/\Omega$, and we have used the Jacobi-Anger expansion

(21)$$e^{{-}ix\text{sin}\Omega t}=\sum_{n={-}\infty}^{\infty}J_{n}(x)e^{{-}in\Omega t},$$

with $J_{n}(x)$ being the $n$th-order Bessel function of the first kind. Considering the resolved sideband limit and the weak coupling condition, i.e. $\omega _{m}\gg \kappa$, $|g_{n^{\prime }}|$, we can then make the rotating wave approximation (RWA) and safely ignore the fast oscillating terms. It follows that

(22)$$\begin{aligned} H_{\text{eff}} & \approx & -\Delta^{\prime}\delta a^{{\dagger}}\delta a-\delta a^{{\dagger}}\left[G_{0}\left(x\right)\delta b+G_{1}\left(x\right)\delta b^{{\dagger}}\right]+\text{H.c}.,\\ \end{aligned}$$

where

(23)$$\begin{aligned} G_{0}\left(x\right) = \left|G_{0}\left(x\right)\right|e^{i\phi_{0}}=\sum_{n={-}\infty}^{+\infty}g_{n}J_{{-}n}(x),\\ G_{1}\left(x\right) = \left|G_{1}\left(x\right)\right|e^{i\phi_{1}}=\sum_{n={-}\infty}^{+\infty}g_{n}J_{1-n}(x),\end{aligned}$$

are effective coupling strengths for the optomechanical beam-splitter and two-mode squeezing interactions. Note that $G_{0}$ and $G_{1}$ are jointly contributed by all effective sideband driving tones oscillating at frequencies “$n\Omega$” with weights given by the Bessel functions $J_{n}$, which is referred to as the Landau-Zener analogous effect. Moreover, the effective Hamiltonian (22) shows that the optomechanical cavity acts as the engineered reservoir [27], which can in principle cool the mechanical Bogoliubov-mode

(24)$$\delta\beta=\text{cosh}R\delta b+e^{i(\phi_{1}-\phi_{0})}\text{sinh}R\delta b^{{\dagger}}$$

[with $R=\text {atanh}(|G_{1}\left (x\right )/G_{0}(x)|)$ being the squeezing parameter] down towards the ground state. If $\delta b$ is initially in the vacuum, the mechanical mirror will be dissipatively stabilized to the squeezed vacuum state with the position or momentum fluctuation being squeezed by $e^{-2R}$.

In Fig. 2(a), we show the temporal dynamics of the mechanical momentum fluctuation $\left \langle \delta p(t)^{2}\right \rangle$ by simulating Eq. (14) for $(E,\Delta _{0},\epsilon,\Omega,\kappa,\gamma _m,g)/\omega _{m}=(2\times 10^{4},-1,1.1\times 10^{6},2,0.1,10^{-3},4\times 10^{-6})$. Squeezing of $\left \langle \delta p(t)^{2}\right \rangle$ over the 3 dB limit is found periodically with the time period $\tau =\text {2}{\pi }/ \Omega$, and due to the Gaussian nature, it can be further examined by the Wigner density function [62]

(25)$$\mathcal{W}(D) = \frac{1}{2\pi\sqrt{\text{Det}[V_{m}]}}\text{exp}[-\frac{1}{2}D^{T}V_{m}^{{-}1}D],$$

where $D^{T}=[\delta q,\ \delta p]$, and $V_{m}$ is the subblock matrix of $V$ for the mechanical mode. As shown in Fig. 2(b)-(e), the squeezing direction with respect to the minimum of the mechanical momentum fluctuation [or the ridge of $\mathcal {W}(D)$] continuously rotates in phase space in accordance with the modulation period $\text {2}{\pi }/\Omega$. In comparison, we also show $\left \langle \delta p^{2}\right \rangle$ and $\mathcal {W}(D)$ under the RWA by using the effective Hamiltonian (22) with $\phi _{1}-\phi _{0}=\pi$. We find that the degree of momentum squeezing under the phase matching condition [as indicated by the green horizontal line in Fig. 2(a)] agrees well with the numerical counterpart corresponding to the periodic minimum [denoted by $\langle \delta p(t)^{2}\rangle _{\text {min}}$ from now on] of the dynamical steady state. The same conclusion can be drawn by comparing the time-independent $\mathcal {W_{\text {RWA}}}(D)$ shown in Fig. 2(g) with that at the dynamical minimum as well.

Fig. 2. (a) Variance of the mechanical momentum $\left \langle \delta p(t)^{2}\right \rangle$ versus the rescaled time $t/\tau$ with $(E,\Delta _{0},\epsilon,\Omega,\kappa,\gamma _m,g)/\omega _{m}=(2\times 10^{4},-1,1.1\times 10^{6},2,0.1,10^{-3},4\times 10^{-6})$ and $\phi =0$, $n_{m}=0$. The green horizontal line indicates the steady-state $\left \langle \delta {p}^{2}\right \rangle$ [Eq. (27)] calculated under the RWA. (b)-(f) Temporal dynamics of the mechanical Wigner function $\mathcal {W}(D)$ at the time period highlighted by the shaded region in (a). (g) Mechanical Wigner function $\mathcal {W}_{\text {RWA}}(D)$ under the RWA, corresponding to the green horizontal line in (a).

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To examine the cooperative effect between the harmonic driving and the radiation pressure force, we show the phase diagram of $\langle \delta p(t)^{2}\rangle _{\text {min}}$ versus the cavity driving $E$ and the mechanical driving strength $\epsilon$ in Fig. 3(a), and find that the momentum squeezing surpasses the 3 dB limit (indicated by the region I) only when the harmonic force ($\sim \epsilon$) is strong enough such that $\Delta _{0}+2g|q_{1}(\epsilon )|>0$. In this case, mechanical oscillations periodically sweep the system through the cavity resonance, leading to transient heating for the mirror, but the system remains stable due to the positive net damping rate, which allows $\delta \beta$ subject to cavity cooling.

Fig. 3. (a) $\left \langle \delta p^{2}\right \rangle _{\text {min}}$ versus $E$ and $\epsilon$ by simulating (4). The right-hand side of the black dashed line denotes that the mechanics will sweep through the cavity resonance for large mechanical driving strengths. The yellow dashed line sets the boundary for the regions I and II, corresponding to the momentum squeezing larger and smaller than the 3 dB limit, respectively. The unstable parameter region is indicated by the white area according to Eq. (13). (b) Density plot of $\left \langle \delta {p}^{2}\right \rangle$ with Eq. (27), which excepts for a slight deformation, agrees well that under the fully numerical simulation in (a). Other parameters are the same as that in Fig. 2.

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For a generic case, if $\delta \beta$ is cooled down to the ground state, then it follows that the squeezed position or momentum fluctuation is given by [27]

(26)$$\left\langle \delta {O}^{2}\right\rangle = \frac{1}{2}\frac{1-\text{tanh}R}{1+\text{tanh}R},$$

with ${O}={q}$ and ${p}$ for $\phi _{0}-\phi _{1}=0$ and $\pi$, respectively. However, we note that the mechanical oscillator is subject to non-zero temperature thermal bath, which heats the mechanics away from the true ground state. Now starting from the effective Hamiltonian (22) and setting $\phi _{0}-\phi _{1}=\pi$, $\Delta ^{\prime }=0$, we can derive the steady-state momentum fluctuation by solving the equation of motion for the CM. It leads to

(27)$$\left\langle \delta{p}^{2}\right\rangle \approx \frac{\frac{1}{4}(1-\text{tanh}R)^{2}\mathcal{C}+n_{m}+\frac{1}{2}}{\frac{1}{2}(1-\text{tanh}^{2}R)\mathcal{C}+1},$$

where we have introduced the cooperativity parameter $\mathcal {C}=4|G_{0}|^{2}/(\kappa \gamma _{m})$, and assumed $\kappa \gg \gamma _{m}$. In Fig. 3(b), we show the density plot of $\left \langle \delta {p}^{2}\right \rangle$ based on Eq. (27), and compare it with the numerical result $\left \langle \delta p^{2}\right \rangle _{\text {min}}$ [i.e. Figure 3(a)] simulated via the original Hamiltonian (4). In general, the results based on Eq. (27) (i.e. the steady-state assumption) and Eq. (4) (corresponding to the original Hamiltonian) are in good agreement with each other except for a slight deformation, which arises from the cavity driving $E$-dependent finite detuning $\Delta ^{\prime }=g\langle q_{0}\rangle \neq 0$ for $\Delta _{0}=-\omega _{m}$. For zero optomechanical coupling, i.e. $\mathcal {C}=0$, Eq. (27) reduces to $\left \langle \delta {p}^{2}\right \rangle \thickapprox n_{m}+\frac {1}{2}$, implying that the mechanics is in equilibrium with the thermal bath. While for $(1-\text {tanh}^{2}R)\mathcal {C}\gg 1$ and $n_{m}=0$, we recover the Eq. (26), where the Bogoliubov-mode $\delta \beta$ is cooled down towards its ground state.

Furthermore, we consider the strong driving with $E/\omega _{m}=6\times 10^{4}$ and the low damping rate $\gamma _{m}/\omega _{m}=10^{-6}$, we find that the system becomes unstable if the mechanical oscillations slightly pass through the cavity resonance. Thus, the amplitude of the harmonic driving force should be restrained to avoid mechanically self-sustained amplifications. As a result, the optimal squeezing parameter $\text {tanh}R=\left |G_{1}/G_{0}\right |$ (obtained by $\langle \delta p(t)^{2}\rangle _{\text {min}}$) is cut off at the critical mechanical driving amplitude $\epsilon _{c}$, corresponding to $\Delta _{0}+g\langle q_{0}\rangle +2g|q_{1}(\epsilon _{c})|=0$, see Fig. 4. To get an intuitive understanding, we simply truncate $G_{0}$ and $G_{1}$ in Eq. (23) to $n=0,\text { }1$, and obtain

(28)$$\text{tanh}R \thickapprox \left|\frac{g_{1}J_{0}(x)+g_{0}J_{1}(x)}{g_{0}J_{0}(x)-g_{1}J_{1}(x)}\right| = \left|\frac{\text{tanh}r-J(x)}{J(x)\text{tanh}r+1}\right|,$$

where $\text {tanh}r=\left |g_{1}/g_{0}\right |$ is associated with the squeezing parameter obtained by driving an optomechanical cavity with two control lasers of different amplitudes [27], and the Bessel factor $\text { }J(x)=J_{1}(x)/J_{0}(x)$ is explicitly introduced by the harmonic driving force. Thus, Eq. (28) intuitively shows the interplay between the Landau-Zener dynamics [$\sim J(x)$] and the sideband drive tones ($\sim \text {tanh}r$). Indeed, both $g_{1}/g_{0}\propto q_{-1,0}$ (for $j=1$) and $J(x)$ are modulated by $\epsilon$ as $q_{-1,0}\propto \epsilon$ and $x\propto \epsilon$, and therefore, the mechanical squeezing ($\text {tanh}R$) is approximately proportional to the driving strength $\epsilon$. On the other hand, when the driving strength $\epsilon$ is fixed, by shifting the laser driving frequency weakly red-detuned from $\Delta _{0}=-\omega _{m}$, we find that the degree of squeezing can be further improved. This is due to the fact that the effective driving component near the blue sideband $\Delta _{0}+2\omega _{m}$ is shifted closer to the cavity resonance correspondingly. As a result, $g_{1}/g_{0}$ is increased, while $J(x)$ is unaffected, giving rise to the growth of the squeezing parameter. Moreover, we show that Eq. (28) with $g_{0/1,}{}_{j}$ being truncated to $j\leq 1$ or $j\leq 3$ can be used to estimate the degree of mechanical squeezing, which may nicely agree with the numerical counterpart depending on the laser detuning. As an example, in the inset of the Fig. 4, we show the temporal dynamics of $\left \langle \delta p(t)^{2}\right \rangle$ with the parameter regime $\left (E,\epsilon,\Delta _{0},\gamma _{m}\right )/\omega _{m}$ $=\left (6\times 10^{4},10^{6},-1.4,10^{-6}\right )$ as indicated by the red spot in Fig. 4. The minimum of $\left \langle \delta p(t)^{2}\right \rangle$ is precisely given by the analytical result (27) with $n_{m}=0$ and is equal to $\left \langle \delta p(t)^{2}\right \rangle _{\text {min}}\approx 0.08$ ($\sim$8 dB) indicated by the green horizontal line. Finally, it should be noted that the system may become unstable if we further increase the laser detuning $\left |\Delta _{0}\right |$. In this case, the stability condition (13) is violated and the ansatz about periodical mechanical oscillations does not hold.

Fig. 4. Squeezing parameter dependent $\text {tanh}R={\left |G_{1}/G_{0}\right |}$ versus $\epsilon$ for laser detuning being $\Delta _{0}/\omega _{m}=-1,-1.2,-1.4$, respectively. The blue solid lines are numerical results with the original Hamiltonian (4). The red dotted and green dashed lines are simulated with Eq. (28), where $g_{0/1,}{}_{j}$ are truncated to $j \leq 1$ and $j\leq 3$, respectively. The curves are cut off at the critical driving strengths $\epsilon _{c}$, where $\Delta _{0}+g\langle q_{0}\rangle +\Delta _{1}(\epsilon _{c})\rightarrow 0$ and the system reaches the unstable regime. The inset shows the steady-state dynamics of $\langle \delta p(t)^{2}\rangle$ for the parameter regime indicated by the red spot, and the green line indicates the stabilized squeezing in Eq. (27). Parameters are $E/\omega _{m}=6\times 10^{4}$ and $\gamma _{m}/\omega _{m}=10^{-6}.$ Other parameters are the same as that in Fig. 2.

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5. Experimental feasibility and conclusion

Considering the experimental feasibility, we envision a setup as illustrated in Fig. 5. A movable mirror is coupled to a high-finesse optical cavity (with resonant frequency $\omega _{c1}$, length $L_{1}$ and decay rate $\kappa _{1}$) and a fast dissipative cavity (with resonant frequency $\omega _{c2},$ length $L_{2}$ and decay rate $\kappa _{2}$). The cavity mode $a_{1}$ is driven by a weak driving $E_{1}$ from the left port, while the (auxiliary) cavity mode $a_{2}$ is driven by a two-tone field with large amplitudes $E_{2}(t)=E_{20}-iE_{21}e^{-i(\Omega t+\phi )}$ and $\Omega =2\omega _{m}$, where the periodic radiation pressure force exerted on the mirror by $a_{2}$ acts as the strong harmonic mechanical driving. The Hamiltonian of the cascaded optomechanical system can be written by ($\hbar =1$):

(29)$$\begin{aligned} H_{\text{com}}= & \left(\omega_{c1}-g_{a}q\right)a_{1}^{{\dagger}}a_{1}+\left(\omega_{c2}+g_{b}q\right)a_{2}^{{\dagger}}a_{2}+\frac{\omega_{m}}{2}(p^{2}+q{}^{2})\\ & +\left[iE_{1}a_{1}^{{\dagger}}e^{{-}i\omega_{1}t}+iE_{2}(t)a_{2}^{{\dagger}}e^{{-}i\omega_{2}t}+\text{H.c.}\right].\end{aligned}$$

We consider the experimentally feasible parameters for the mechanical oscillator $\omega _{m}/2\pi =100$ kHz, $\gamma _{m}/\omega _{m}=10^{-6}$, $m=150$ ng, and those for the cavity modes, the cavity lengths $L_{1}=1$ mm and $L_{2}=20$ mm, the cavity finesses $\mathcal {F}_{1}=1.5\times 10^{6}$ and $\mathcal {F}_{2}=375$, and the wavelength of the driving laser $\lambda =1064$ nm [25]. As such, the single-photon optomechanical coupling strengths are $g_{a}/\omega _{m}=10^{-4}$ and $g_{b}/\omega _{m}=5\times 10^{-6}$ for the left and right cavity modes, and the adiabatic conditions $\kappa _{2}\gg \left \{ g_{b}\left \langle q(t)\right \rangle,\Omega,\kappa _{1}\right \}$ can be well satisfied under appropriate cavity drivings. Thus, the radiation pressure force exerted on the mirror by $a_{2}$ can be regarded as the strong harmonic force after a transient process, leading to the effective Hamiltonian (in the rotating frame) given by (see Appendix)

(30)$$\begin{aligned} H_{\text{com}}^{\prime} & = -\Delta_{1}a_{1}^{{\dagger}}a_{1}+\frac{\omega_{m}}{2}(p^{2}+q^{2})-g_{a}a_{1}^{{\dagger}}a_{1}q\\ & +\frac{2g_{b}E_{20}E_{21}}{\kappa_{2}^{2}}\text{sin}(\Omega t+\phi)q+iE_{1}(a_{1}^{{\dagger}}-a_{1}),\end{aligned}$$

which is equivalent to the mechanic-driven Hamiltonian (1) with $\epsilon \rightarrow 2g_{b}E_{20}E_{21}/\kappa _{2}^{2}$ and the equilibrium position of the mechanical oscillator is simply shifted by $q_{0}=g_{b}(E_{20}^{2}+E_{21}^{2})/\kappa _{2}^{2}\omega _{m}$, i.e. $q+q_{0}\rightarrow q$. Note that here we have treated the macroscopic mirror as a single-mode harmonic oscillator. Instead of a macroscopic end mirror, state-of-the-art optomechanical experiments use cantilevers as an end mirror in an optical cavity and couple light with the fundamental mode of oscillation. For a clamped cantilever beam of length $l$, width $w$, and thickness $t$, the frequency of the fundamental mode is $\omega _m/2\pi =3.516(t/l^{2})\sqrt {E/12\rho }$, with $E$ the Young’s modulus and $\rho$ the density; and the second mode is 6.3 times higher in frequency than the fundamental mode [63]. While a large driving can lead to the generation of high harmonic sidebands due to cavity nonlinearity, Eq. (28) [and also Fig. 4] shows that the truncated squeezing parameter can be effective with $n\leq 1$, i.e., the effective sideband driving tones at "$n\Omega$" ($n\geq 3$ ) can be neglected. Thus, the effect of the second and higher modes of the clamped beam can be ignored.

Fig. 5. Optomechanical setup consists of two Fabry-Perot cavities and a movable mirror with the cavity lengths satisfying $L_{1}{\ll }L_{2}$. The left cavity is driven by an external laser of the strength $E_{1}$. The right cavity is driven by a periodically amplitude-modulated laser field $E_{2}(t)$, which effectively acts as a harmonic force on the mirror. See text for the details.

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Now we can examine mechanical squeezing by following the same recipe as Eqs (11)-(14), where the mechanical Brownian noise and the input vacuum noises by the cavity modes are included. In Fig. 6, we show variance of the mechanical momentum $\left \langle \delta p(t)^{2}\right \rangle$ versus the rescaled time with laser detuning $\Delta _{1}/\omega _{m}=(-1.4,-1.2,-1.0)$ and the driving power $P_{1}=\hbar \omega _{1}E_{1}^{2}/2\kappa _{1}=2.84$ $\mu$W and $P_{2} = \hbar \omega _{2}E_{20}^{2}/2\kappa _{2} = 113.9$ mW. For comparison, we have also shown the results given by Eq. (14), as indicated by the blue dashed lines, which agree well with that from the effective model (30), and have the minimum $\left \langle \delta p(t)^{2}\right \rangle _{\text {min}}\thickapprox 0.0975,0.2201,0.3383$ for $\Delta _{1}/\omega _{m}=-1.4,-1.2,-1.0$. Moreover, we show the optimal steady-state squeezing $\left \langle \delta p(t)^{2}\right \rangle _{\text {min}}$ as a function of mechanical thermal occupation for $\gamma _{m}/\omega _{m}=10^{-6}$, $10^{-5}$, $10^{-4}$ and $10^{-3}$, respectively. The result in Fig. 6(d) shows that the mechanical squeezing becomes more robust to thermal fluctuations as $\gamma _{m}$ decreases, and for $\gamma _{m}/\omega _{m}=10^{-6}$, the momentum squeezing can overcome the 3dB limit for $n_{m}\approx 2.4\times 10^{3}$, namely, for a mechanical frequency $\omega _{m}/2\pi \sim 1$ MHz, one can in principle reach the 3 dB limit at a few hundred mK.

Fig. 6. $\left \langle \delta p(t)^{2}\right \rangle$ versus the rescaled time $t/\tau$ with (a) $(\Delta _{1},E_{20})/\omega _{m}=(-1.4,1.4\times 10^{7})$, (b) $(\Delta _{1},E_{20})/\omega _{m}=(-1.2,1.28\times 10^{7})$, and (c) $(\Delta _{1},E_{20})/\omega _{m}=(-1.0,1.16\times 10^{7})$, respectively. The green dashed lines and blue solid lines are simulated with the Hamiltonians Eq. (29) and Eq. (30), respectively. Other parameters are $g_{a}/\omega _{m}=10^{-4}$, $\kappa _{1}/\omega _{m}=0.1$, $\kappa _{2}/\omega _{m}=200$, $g_{a}/g_{b}=20$, $E_{1}/\omega _{m}=2.2\times 10^{3}$, $\Delta _{2}/\omega _{m}=-1$ and $E_{21}=0.8E_{20}$. (d) $\left \langle \delta p(t)^{2}\right \rangle _{\text {min}}$ in units of dB versus the thermal phonon occupation number $n_{m}$ for $\gamma _{m}/\omega _{m}=10^{-6}$, $10^{-5}$, $10^{-4}$ and $10^{-3}$, respectively. The shaded region indicates the mechanical squeezing below the 3 dB limit, and other parameters are the same as (a).

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Finally, we briefly comment on the detection of mechanical squeezing. For the effective Hamiltonian Eq. (22) within the RWA, the system dynamics are governed by a Langevin equation without explicit time dependence, the stationary spectrum for squeezing quadrature can be readily obtained, and can be detected simply by following the quadrature and making measurements in the special frame rotating with the mechanical frequency $\omega _m$ [see Fig. 2(g)] [27]. For the general time-periodic squeezing, the time-averaged power spectrum of the squeezed quadrature can be calculated by the Floquet approach [64], and can be observed through a weakly coupled (readout) cavity mode, driven bichromatically close to the red and blue sidebands [17,65]. Moreover, for the readout drivings with a balanced weight, the QND measurement is in principle applicable to the effective model (22) [64], but is sensitive to variance in laser detunings and phases.

In conclusion, optomechanically induced mechanical squeezing can be realized by shaking the mechanical mirror with a strong harmonic force at twice the mechanical frequency. When the optical radiation pressure force induced damping rate keeps the system away from the self-sustained oscillation regime, the mechanical mirror can be stabilized into a deeply squeezed steady state by using the dissipatively optomechanical cavity as the engineered reservoir. The squeezing strength is cooperatively determined by the effective sideband drive tones induced by the nonlinear optomechanical interaction, and the harmonic driving amplitude via the Landau-Zener analogous dynamics. The mechanical squeezing reaches 8 dB for the thermal bath of $T=0$ $\text {mK}$. The scheme shows an example for cooperative addressing of macroscopic objects with both optical radiation pressure force and mechanical driving force, which can potentially be applied for addressing multi-mode optomechanical systems and for mechanical sensing.

Appendix

The total Hamiltonian of the system in the frame rotating at the laser frequency $\omega _{1,2}$ can be written as

(31)$$\begin{aligned}H= & -(\Delta_{1}+g_{a}q)a_{1}^{{\dagger}}a_{1}-(\Delta_{2}-g_{b}q)a_{2}^{{\dagger}}a_{2}+\frac{\omega_{m}}{2}(p^{2}+q^{2})\\ & +i(E_{1}a_{1}^{{\dagger}}-E_{1}^{*}a_{1})+i\left[E_{2}(t)a_{2}^{{\dagger}}-E_{2}^{*}(t)a_{2}\right], \end{aligned}$$

where $\Delta _{1}=\omega _{1}-\omega _{c1}$ and $\Delta _{2}=\omega _{2}-\omega _{c2}$. By including the mechanical damping and the cavity decay, the dissipative dynamics of the open system can be described by the quantum Langevin equations (QLEs)

(32)$$\begin{aligned} \dot{q} & = \omega_{m}p,\\ \dot{p} & =-\omega_{m}q-\gamma_{m}p+g_{a}a_{1}^{{\dagger}}a_{1}-g_{b}a_{2}^{{\dagger}}a_{2}+\xi(t),\\ \dot{a}_{1} & = -\left[\kappa_{1}-i(\Delta_{1}+g_{a}q)\right]a_{1}+E_{1}+\sqrt{2\kappa_{1}}a_{1in}(t),\\ \dot{a}_{2} & = -\left[\kappa_{2}-i(\Delta_{2}-g_{b}q)\right]a_{2}+E_{2}(t)+\sqrt{2\kappa_{2}}a_{2in}(t). \end{aligned}$$

For $\{\langle a_{1}^{\dagger }a_{1}\rangle, \langle a_{2}^{\dagger }a_{2}\rangle \}\gg 1$, we can again linearize the system, obtaining the equations of motion for the first moments of the optical and mechanical modes

(33)$$\begin{aligned} \left\langle \dot{q}(t)\right\rangle &= \omega_{m}\left\langle p(t)\right\rangle,\\ \left\langle \dot{p}(t)\right\rangle &= -\omega_{m}\left\langle q(t)\right\rangle -\gamma_{m}\left\langle p(t)\right\rangle +g_{a}\left|\left\langle a_{1}(t)\right\rangle \right|^{2}-g_{b}\left|\left\langle a_{2}(t)\right\rangle \right|^{2},\\ \left\langle \dot{a_{1}}(t)\right\rangle &= -\left[\kappa_{1}-i(\Delta_{1}+g_{a}\left\langle q(t)\right\rangle )\right]\left\langle a_{1}(t)\right\rangle +E_{1},\\ \left\langle \dot{a_{2}}(t)\right\rangle &= -\left[\kappa_{2}-i(\Delta_{2}-g_{b}\left\langle q(t)\right\rangle )\right]\left\langle a_{2}(t)\right\rangle +E_{2}(t), \end{aligned}$$

and for the corresponding quantum fluctuation operators

(34)$$\begin{aligned} \delta\dot{q} & = \omega_{m}\delta p,\\ \delta\dot{p} & = -\omega_{m}\delta q-\gamma_{m}\delta p+\xi(t) +g_{a}[\left\langle a_{1}(t)\right\rangle \delta a_{1}^{{\dagger}}+\left\langle a_{1}(t)\right\rangle ^{*}\delta a_{1}]\\ &-g_{b}[\left\langle a_{2}(t)\right\rangle \delta a_{2}^{{\dagger}}+\left\langle a_{2}(t)\right\rangle ^{*}\delta a_{2}],\\ \delta\dot{a_{1}} & = -[\kappa_{1}-i\Delta_{1}^{\prime}(t)]\delta a_{1}+ig_{a}\left\langle a_{1}(t)\right\rangle \delta q+\sqrt{2\kappa_{1}}a_{1in}(t),\\ \delta\dot{a_{2}} & = -[\kappa_{2}-i\Delta_{2}^{\prime}(t)]\delta a_{2}-ig_{b}\left\langle a_{2}(t)\right\rangle \delta q+\sqrt{2\kappa_{2}}a_{2in}(t), \end{aligned}$$

where the effective laser detuning $\Delta _{1}^{\prime }(t)=\Delta _{1}+\delta _{\text {d1}}(t)$, $\Delta _{2}^{\prime }(t)=\Delta _{2}-\delta _{\text {d2}}(t)$ are modulated by the mechanical oscillation with the frequency drifts $\delta _{\text {d1(2)}}(t)=g_{a(b)}\left \langle q(t)\right \rangle$. The equation of motion for $\left \langle a_{2}(t)\right \rangle$ can be solved in the weak coupling regime $g_{b}\ll g_{a}$ and $g_{b}\left \langle q(t)\right \rangle \ll \{\omega _{m},\kappa _{2},\kappa _{1}\}$, and the back action of the mechanical oscillation on the cavity intensity $|\left \langle a_{2}(t)\right \rangle |^2$ can be neglected, giving rise to

(35)$$\left\langle a_{2}(t)\right\rangle = \frac{iE_{20}}{\Delta_{2}+i\kappa_{2}}\left[1-e^{-\kappa t+i\Delta_{2}t}\right]-\frac{E_{21}e^{{-}i(\Omega t+\phi)}}{\Delta_{2}+i\kappa_{2}+\Omega}\left[1-e^{-\kappa t+i\left(\Delta_{2}t+\Omega t\right)}\right].$$

Considering the adiabatic approximation with $\kappa _{2}\gg \left \{ \Delta _{2},\Omega,\kappa _{1}\right \}$, the radiation pressure force $F_{\text {rad2}}=g_{b}\left |\left \langle a_{2}(t)\right \rangle \right |^{2}$ exerted by the cavity mode $a_{2}$, after a short transient dynamics, can then be given by :

(36)$$F_{\text{rad2}} \simeq \frac{g_{b}}{\kappa_{2}^{2}}\left[2E_{20}E_{21}\text{sin}(\Omega t+\phi)+E_{20}^{2}+E_{21}^{2}\right],$$

based on which we can write down the effective Hamiltonian (29), with $E_{21}/E_{20}<1$ for stablizing the system. We then set $E_{21}=0.8E_{20}$, $\kappa _{2}/\omega _{m}=200$ and $\Delta _{2}/\omega _{m}=-1$ to keep away from unstable region [defined by Eq. (13)], and numerically confirm that $F_{\text {rad2}}$ can be described by the form Eq. (36).

Funding

National Natural Science Foundation of China (11774058, 11874114, 12174058).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Optomechanical squeezing with strong harmonic mechanical driving

Abstract

1. Introduction

2. Model and Hamiltonian

3. Time-periodic linearized Hamiltonian and quadrature squeezing

4. Mechanical squeezing stabilized by dissipation

5. Experimental feasibility and conclusion

Appendix

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (37)

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