Enhancement of steady-state bosonic squeezing and entanglement in a dissipative optomechanical system

We systematically study the influence of amplitude modulation on the steady-state bosonic squeezing and entanglement in a dissipative three-mode optomechanical system, where a vibrational mode of the membrane is coupled to the left and right cavity modes via the radiation pressure. Numerical simulation results show that the steady-state bosonic squeezing and entanglement can be significantly enhanced by periodically modulated external laser driving either or both ends of the cavity. Remarkably, the fact that as long as one periodically modulated external laser driving either end of the cavities is sufficient to enhance the squeezing and entanglement is convenient for actual experiment, whose cost is that required modulation period number for achieving system stability is more. In addition, we numerically confirm the analytical prediction for optimal modulation frequency and discuss the corresponding physical mechanism.


I. INTRODUCTION
Driven by a variety of different goals and promising prospects, cavity optomechanics, a field at the intersection of nanophysics and quantum optics, has developed over the past few years [1][2][3]. It has been known that nonclassical states of macroscopic mechanical resonators, especially the squeezed and entangled states, play a key role in test of the fundamental principles of quantum mechanics, quantum information processing, and ultrahigh-precision measurements. Many researches have been investigated on quantum squeezing and entanglement generation in cavity optomechanical interfaces. Normally, one can simply use radiation pressure forces or combine continuous quantum measurements and feedback to obtain stationary squeezing [4][5][6] and stationary entanglement [7,8] in a two-mode optomechanical system. In order to increase the richness of the research, the three-mode optomechanical setting was introduced and has been realized experimentally recently [9][10][11]. Several theoretical schemes for generating quantum squeezing and entanglement in the three-mode optomechanical system have been proposed based on the basic idea that the auxiliary mode mediates an effective two-mode squeezing interaction between the two target modes [12][13][14][15][16].
However, the schemes are generally restricted to the requirement of stability so that they yield at best a relatively small amount of squeezing and entanglement.
Resent studies show that large degrees of squeezing and entanglement can be achieved by mildly modulating the amplitude of the driving field [17][18][19][20][21][22] or combining with dissipation mechanism [23,24], where no feedback is needed. Moreover, the modulation-assisted driving can give rise to interesting and rich quantum dynamics [25,26]. Farace and Giovannetti [27] further investigated this modulation regime and showed that simultaneous modulations of the mechanical frequency and input laser intensity can either enhance or weaken the desired quantum effects. Newly, the robust entanglement is generated by modulating the coupling strength between two mechanical oscillators [28,29]. Besides, the modulation-induced mechanical parametric amplification effectively enhances the resonant optomechanical interaction and leads to single-photon strong-coupling [30]. Remarkably, several works [31][32][33][34][35][36][37] reveal that optimizing relative ratio of optomechanical couplings, rather than simply increasing their magnitudes, is essential for achieving strong steady-state squeezing and entanglement via dissipation mechanism. These schemes exploit the Bogoliubov-mode-based method [38] instead of the Sφrensen-Mφlmer approach [39]. Another promising means for generating strong entanglement or squeezing is the phonon-mediated four-wave mixing process [40]. Although the physical explanations for these schemes are not quite the same, a common feature is to induce an effective engineered reservoir by driving the optomechanical systems with proper blue and red detuned lasers [38][39][40][41][42].
In this work, combination of the modulation and the dissipation is considered. We expand the optomechanical model in [17] to three-mode optomechanical system, which is similar to that in [43] and [44] except being driven by periodic modulation field. A single-cavity optomechanical system usually requires an external laser to drive the mechanical resonator out of its zero steady state at equilibrium position. For the system considered here, an external laser being applied to either end of the cavity is sufficient to drive the vibrating membrane. Numerical simulation results show that the squeezing and entanglement can be enhanced with one-end or two-end periodically modulated external laser. The time required for the two-end modulation when the system achieves a stable state is shorter than that for the one-end modulation, but the one-end modulation reduces the difficulty of the experiment. What is more, with the help of the third mode acted as an engineered reservoir, dissipation mechanism is explored. Compared to the previous studies of threemode modulated optomechanics [21,22,24], more general modulations of quantum dynamics are discussed here.
In what follows, we give a detailed description of our model and obtain the linearized dynamical equations for the system in Sec. II. In Sec. III, analytical solutions for mean values in the cases of symmetric and asymmetric modulation are obtained in a perturbative way. Then We analyze in detail the characters of the mean values, where the numerical results agree well with the analytical results. In Sec. IV, the mechanisms of squeezing and entanglement via combinations of the periodic amplitude modulation and the dissipation regime are discussed by assuming a simple but justifiable form of the effective coupling.
Finally, conclusions are presented in Sec. V.

II. THEORETICAL MODEL
The considered system is depicted in Fig. 1. A dielectric membrane as a mechanical oscillator separates an optical cavity into two cavities and constructs a "membrane-in-themiddle" configuration, which has been theoretically studied [20,[45][46][47][48][49][50][51][52][53] and experimentally implemented [54][55][56][57][58][59][60][61][62]. The mechanical oscillator with frequency ω m is simultaneously coupled to the left and right cavity modes via the radiation pressure difference between the two cavities, where tunneling of photons through the membrane is allowed. The two cavity modes with frequency ω cL and ω cR are respectively driven by external lasers with periodically modulated amplitudes E L (t) and E R (t). In the rotating frame with respect to laser frequencies ω L and ω R , the corresponding Hamiltonian reads ( = 1) Here, ∆ j = ω cj − ω j denotes the jth cavity mode detuning, A † j and A j represent the creation and annihilation operators of the jth cavity mode, Q and P are the dimensionless position and momentum operators of the mechanical mode with the standard canonical commutation relation [Q, P ] = i, J expresses the cavity-cavity coupling strength which is in the regime J ω cL , ω cR , and g signifies the phonon-photon coupling coefficient. The time-dependent amplitude E j (t) is a period function with the period τ , i.e., E j (t + τ ) = E j (t). Taken into account the cavity leakage and membrane damping, the dissipative dynamics of the system is described by the following nonlinear quantum Langevin equations (QLEs) where κ and γ m are severally the leakage rate of the cavities and the mechanical damping rate. The zero-mean fluctuation terms a in j (t) obey the correlation relations [63] a in j (t)a in † j (t ) = (n a + 1)δ(t − t ), where n a = [exp( ω cj /k B T ) − 1] −1 is the mean bath photon number at the environmental temperature T . The correlation function of zero-mean Brownian motion noise operator ξ(t) in the case of the large mechanical quality factor Q = ω m /γ m 1 can be approximately described by the Markovian process and satisfies where n m = [exp( ω m /k B T ) − 1] −1 is the mean thermal phonon number at the environmental temperature T .
In the presence of strong external driving fields, we can rewrite each Heisenberg operator the linearized QLEs for the quantum fluctuationṡ q =ω m p, (6a) and the corresponding linearized system Hamiltonian

III. THE CHARACTERS OF THE MEAN VALUES
It is difficult to find exact solutions of the mean values in Eq. (5) in general. But when the system is far away from optomechanical instabilities and multistabilities [64], the optomechanical coupling can be treated in a perturbative way. More specifically, approximately analytical solutions of the mean values can be found by expanding them in power series of the coupling costant g. Besides, it is justifiable that stable solution has the same periodicity τ as the implemented modulation field E j (t). Hence, we can perform double expansions for the mean values O(t) in power series of g and Fourier series, i.e., where Ω = 2π/τ is the fundamental modulation frequency. Similarly, Fourier series for the periodic driving amplitudes can be written as After directly substituting Eqs. (8) and (9) into Eq. (5), the coefficients O n,l are completely determined by the following relations corresponding to the 0-order perturbation with respect to g, and corresponding to the l-order coefficients in a recursive way.
In the case of identical cavity detuning (∆ = ∆ L = ∆ R ) and symmetric modulation of , it is reasonable to expect that the mean values A L and A R have the same stable solutions. Thus, Eqs. (10) and (11) can be further simplified as follows: To gain more insights about the dynamics, we respectively plot the phase space trajectories of the mean values for symmetric and asymmetric modulations in Fig. 3. As shown in Fig. 3(a), when the system is stable after dozens of modulation periods, the numerical phase space trajectories of A L (t) (or A R (t) ) finally converge to a limit cycle in the case of symmetric modulation, which agrees well with analytical prediction. In the cases of single cavity driving and single cavity modulation, the numerical results in Fig. 3(b) display that the phase space trajectories of the cavity mode mean values A L (t) , A R (t) , and the dimensionless mechanical position and momentum mean values almost converge to a limit cycle after hundreds of modulation periods.

IV. STATIONARY BOSONIC SQUEEZING AND ENTANGLEMENT
Since the asymptotic evolution period of the system is τ , without loss of generality, we assume the asymptotic form for time-dependent mean values of the cavity modes as follows: where A j0 and A j1 are positive real number and related to the driving amplitude components E L n and E R n in Eq. (9). When t → ∞ and ω m γ m > 0, the corresponding mechanical mean values and the driving amplitude can be readily derived from Eq. (5) via Laplace transformation and inverse transformation with the driving amplitude components (15h) In the long time limit, when driving amplitudes E L (t) and E R (t) with forms as Eqs. (14) and (15)  0.02/ √ 2g. In fact, the above four parameters can be arbitrary assigned when the requirement is met, which ensures stability. Thus, one can always design the corresponding modulation driving laser to realize mean values of the cavity modes with any periodic form (the specific form is dependent on what effect we want to achieve).
In the following, based on the assumption of Eq. (13) we analyze how to enhance squeezing and entanglement via the symmetrically and asymmetrically periodic modulation. By introducing the position and momentum quadratures for the two cavity modes and their input noises and the column vectors of all quadratures and noises Eq. (6) can be rewritten asU with where the effective time-modulated detuning G jr (t) and G ji (t) are respectively real and imaginary parts of the effective coupling coefficient When the system is stable, it converges to a time-dependent Gaussian state [65], which is independently from the initial condition. Thus, the asymptotic state of the fluctuation is fully described by the covariance matrix (CM) σ(t) of the pairwise correlation among the quadratures, where the entries of the CM are defined as From Eqs. (18) and (22), it can be deduceḋ where D is a diffusion matrix whose components are associated with the noise correlation functions and defined as It can be gained from Eqs. (3) and (4) D = diag(0, γ m (2n m + 1), κ(2n a + 1), κ(2n a + 1), κ(2n a + 1), κ(2n a + 1)). (25) In the long time limit, based on Floquet's theorem [17,18,20,66] the periodicity of the entries of R(t) implies that asymptotic solution of the linear differential Eq. (23) will have the same period τ , i.e., The CM σ(t) can be written as a block matrix where each block represents a 2 × 2 matrix. The diagonal blocks represent the variance within each subsystem (for example, resonator M, the left cavity mode L, and the right cavity mode R), while the off-diagonal blocks denote covariance across different subsystems.
Since the asymptotic state of the system is Gaussian, it is convenient to measure the pairwise entanglement E N with the logarithmic negativity [67,68], which can be readily computed from the reduced 4 × 4 CM σ r (t) for two subsystems The logarithmic negativity E N is then given by with where the Routh-Hurwitz criterion [69]. Obviously, the squeezing of the mechanical mode and the cavity-cavity entanglement are indeed τ period when the system finally tends to be stable in the long time limit. Noticeably, the squeezing and the entanglement can be significantly enhanced compared with the parametric interaction, which are limited by a factor of 1/2 below the zero-point level, i.e., 0.25 (the so-called 3dB limit) [4-6, 17, 19], and 0.69 [7,13,17,19], respectively.
In order to better understanding the physical reality, we introduce the creation and annihilation operators of the mechanical fluctuations and the nonlocal bosonic modes Thus, the linearized system Hamiltonian in Eq. (7) can be rewritten as where ∆ 3 = ∆ 1 (t) + J, ∆ 4 = ∆ 2 (t) − J. In the interaction picture with respect to the free if the relationship between the effective coupling G j (t) and effective mean value of the cavity modes A j (t) is taken as Eq. (21), Eq. (34) is transformed Here we focus on the range gA j0 , gA j1 ω m , Ω, and set J = 2ω m , ∆ L = ∆ R = 3ω m without loss of generality. Based on Eqs. (14) and (20), we have where the fast oscillating terms e ±iΩt have been neglected. When the modulation frequency is chosen to match with the resonance frequency of the nonlocal cavity and mechanical modes, i.e., all rapid oscillating terms in Eq. (35) can be neglected and the Hamiltonian can be rewritten Due to the fact that ig 2 ( the simplicity in the following analyses.
Introducing two Bogoliubov-mode annihilation operators where the squeezing parameter r is defined as tanh r = (A L1 + A R1 )/(A L0 + A R0 ). Assuming , which ensures stability of the system, the Hamiltonian of Eq. (38) becomes with the coupling This is a beam-splitter-like Hamiltonian, which is well known from optomechanical sideband cooling [70,71]. Obviously, the ground state of β 1 or β 2 is the single-mode squeezed state of the mechanical mode b or two-mode squeezed state of the cavity modes a L and a R , respectively. When the mechanical decay rate γ m is small, which ensures that the mechanical mode b only weakly couples to the mechanical thermal baths with relatively large mean thermal occupancies, the dynamics of mechanical mode b, i.e., the Bogoliubov mode β 1 , is dominated by the interaction with the nonlocal bosonic modes c 2 , namely, the cavity modes a L and a R . Therefore, the Bogoliubov mode β 1 can be cooled to near ground state via the beam-splitter-like interaction [Eq. (41)] with the nonlocal bosonic modes c 2 , which strongly interacts with optical thermal baths with neglectable small mean thermal occupancies. In other words, the dissipative dynamics of the cavity modes can be used to cool the Bogoliubov mode β 1 , generating single-mode squeezing of the mechanical mode. In contrary, if the cavity decay rate κ is smaller compared with the mechanical decay rate γ m and the mechanical mode b has been precooled by a cold reservoir, as discussed in [31], the beam-splitter-like interaction between the mechanical mode b and the Bogoliubov mode β 2 [Eq. (42)] can be exploited to cool the cavities, obtaining the stationary two-mode squeezing state of two cavities. The system dynamics behaviors numerically shown in Figs. 5 and 6 can be explained very well by the above analyses. Notably, all of the above analyses are based on the assumption that the system is stable and does not enter the chaotic regime [72][73][74]. Under the circumstance, the amount of stationary squeezing or entanglement is a and 6 may be not the optimal, which is not our focus of concern. Here, we only verify the enhancement of the squeezing and entanglement via symmetrically and asymmetrically pe-riodically modulated lasers. As shown in Fig. 2 Eq. (37). Besides, compared to the squeezing of the mechanical oscillator position operator, the cavity-cavity entanglement has a larger scale of modulation frequency Ω, which implies that the squeezing is more sensitive to the variation of the modulation frequency.

V. CONCLUSIONS
In summary, we have explored the mechanism of periodic driving laser modulation in a dissipative three-mode optomechanical system. Our studies show that combinations of the modulation and the dissipation can significantly enhance the mechanical squeezing and cavity-cavity entanglement. What is more, both symmetric and asymmetric modulations of the external driving laser are effective when we carefully balance the two opposing effects by varying the ratio of the effective mean values of cavity modes or effective coupling.
The numerical simulation results signify that it is sufficient to enhance the squeezing and entanglement effects as long as one periodically modulated laser is applied to either end of the cavities, which is convenient for actual experiment. However, the cost is more modulation periods required for achieving system stability. In order to achieve large squeezing and entanglement, apart from selecting appropriate ratio of the effective mean values of cavity modes or effective coupling, the modulation frequency should also be chosen carefully.