Steady-state mechanical squeezing in a double-cavity optomechanical system

We study the physical properties of double-cavity optomechanical system in which the mechanical resonator interacts with one of the coupled cavities and another cavity is used as an auxiliary cavity. The model can be expected to achieve the strong optomechanical coupling strength and overcome the optomechanical cavity decay, simultaneously. Through the coherent auxiliary cavity interferences, the steady-state squeezing of mechanical resonator can be generated in highly unresolved sideband regime. The validity of the scheme is assessed by numerical simulation and theoretical analysis of the steady-state variance of the mechanical displacement quadrature. The scheme provides a platform for the mechanical squeezing beyond the resolved sideband limit and solves the restricted experimental bounds at present.

Traditionally and generally, the decay rate of cavity, which is a dissipative factor in optomechanical systems, is considered to have negative effect on the performance of quantum manipulation and quantum information processing. The optomechanical coupling strength  ω ω = g L m ( / ) / c m (with the cavity frequency ω c , the mechanical resonator mass m, and the mechanical resonator frequency ω m ) is inverse relation to the cavity length L. While the cavity quality factor Q increases with increasing the cavity volume V. Thus it is difficult to achieve small decay rate and strong optomechanical coupling strength simultaneously. Here we propose a method to generate steady-state mechanical squeezing in a double-cavity optomechanical system with the highly dissipative cavity (κ 1 /ω m = 100). The scheme does not need to satisfy the conditions of the small cavity decay rate and the strong optomechanical coupling strength simultaneously. The coherently driving on the cavity mode is a monochromatic laser source which can generate strong optomechanical coupling between the mechanical and cavity modes. We show that, based on the mechanical nonlinearity and cavity cooling process in transformed frame, the steady-state mechanical squeezing can be successfully and effectively generated in the highly unresolved sideband regime via the coherent auxiliary cavity interfering. The result indicate that the squeezing can reach 4.4 dB, beyond the so-called 3 dB limit. Different from the hybrid atom-optomechanical systems 9,23,37 , the scheme does not have the challenge of putting a large number of atoms in the cavity. Unlike the dissipative coupling mechanism 32,43,49,54,55 , our scheme utilizes the destructive interference coming from the coherent auxiliary cavity to resist the influence of cavity decay.
The paper is organized as follows: In Sec. II, we describe the model of a double-cavity optomechanical system and derive the linearized Hamiltonian and the effective coupling between the auxiliary cavity and the mechanical resonator. In Sec. III, we engineer the mechanical squeezing and derive the analytical variance of the displacement quadrature of the mechanical resonator in the steady-state. In Sec. IV, we study the relationship between the variance of mechanical mode and the system parameters and obtain the steady-state mechanical squeezing in the highly unresolved sideband regime by numerical simulations method. A conclusion is given in Sec. V.

Results
Basic model. We consider a double-cavity optomechanical system, which is composed of a mechanical resonator and two coupled single-mode cavities, depicted in Fig. 1. The mechanical resonator couples to the first dissipative cavity which is driven by an external laser field, forming the standard optomechanical subsystem. The second high Q optical cavity is regarded as the auxiliary part, which couples to the first dissipative cavity with the coupling strength J. The total Hamiltonian H = H 0 + H I + H pump , which describes the double-cavity optomechanical system, consists of three parts, which reads (ħ = 1), respectively, The part H 0 accounts for the free Hamiltonian of the two cavity modes (with frequency ω 1 , ω 2 and decay rate κ 1 , κ 2 , respectively) and the mechanical resonator (with frequency ω m and damping rate γ m ). Here † a a ( ) 1 1 is the bosonic annihilation (creation) operator of the first optical cavity mode, † a a ( ) 2 2 is the bosonic annihilation (creation) operator of the second optical cavity mode, and b (b † ) is the bosonic annihilation (creation) operator of the mechanical mode. The last term of H 0 describes the Duffing nonlinearity 56,57 of the mechanical resonator with amplitude η. The intrinsic nonlinearity of the gigahertz mechanical resonator is usually very weak with nonlinear amplitude smaller than 10 −15 ω m . We can obtain a strong nonlinearity through coupling the mechanical mode to an auxiliary system 58 , such as the nonlinear amplitude of η = 10 −4 ω m can be obtained when we couple the mechanical resonator to an external qubit 44 . The resulting model is known as the Duffing oscillator and exhibits a bifurcation phenomenon as the strength of the mechanical driving is increased 59 . In our scheme, the bifurcation phenomenon 60,61 exists even in the joint influence of Duffing and optomechanical nonlinearities. While the driving power we need is far from reaching the bifurcation point, we will not discuss the bifurcation detailedly in here.
The part H I accounts for the interaction Hamiltonian consisting of the coupling interaction between two cavities and the optomechanical interaction derived from the radiation pressures. Where J represents the intercavity coupling strength between cavity mode a 1 and a 2 , and g is the single-photon optomechanical coupling strength.
The part H pump accounts for the external driving laser with frequency ω d used to coherently pump the cavity mode a 1 . The driving strength is related to the input laser power P, frequency ω d , and the decay rate of cavity 1 κ 1 .
In the rotating frame defined by the transformation operator 2 2 , the Hamiltonian of the system is given by where δ 1 = ω d − ω 1 and δ 2 = ω d − ω 2 are the detunings of the two cavity modes from the driving field, respectively. Considering the effect of the thermal environment, the quantum Heisenberg-Langevin equations for the system are written as where the corresponding noise operators a 1in , a 2in , and b in satisfy the following correlations: 1 is the mean thermal excitation number of bath of the mechanical resonator at temperature T, k B is the Boltzmann constant. And under the assumption of Markovian baths, the noise operators a 1in , a 2in , and b in have zero mean values.
Since the system is driven by a classical laser field, in the case of strong driving field, we can treat the field operators as the sum of their mean values and small quantum fluctuation. So we can apply a displacement transformation to linearize the equations, The steady-state amplitudes of the optical and mechanical modes are relative to the driving power P, and the relationship can be derived by solving the above equations under the condition of steady situation. One can see that when the driving power P is in the microwatt range, the amplitudes of the cavity and mechanical modes satisfy the relationships: α β  , 1 1 , as shown in Fig. 2. And the amplitudes of the cavity and mechanical modes increase with increasing the driving power. At the point of the driving power P = 0.53 mW, the result of α  390 1 and β  40 can be obtained, respectively.
Under the conditions of strong driving, the nonlinear terms are neglected. The quantum fluctuations satisfy the following linearized equations: in in m m m in The linearized Hamiltonian is given by When considering the system-reservoir interaction, which results in the dissipations of the system, the full dynamics of the system is described by the master equation is the standard Lindblad operators. κ 1 , κ 2 , and γ m are the decay rate of cavity mode a 1 , a 2 , and the damping rate of mechanical resonator, respectively. n th is the average phonon number in thermal equilibrium.
Effective coupling between the auxiliary cavity and the mechanical resonator. Since the decay rate of cavity 1 (κ 1 ) is much larger than the decay rate of cavity 2 (κ 2 ) and the damping rate of mechanical resonator (γ m ), the cavity mode a 1 can be eliminated adiabatically for the time scales longer than κ − 1 1 . The steady solution of the first equation in Eq. (6) about cavity mode a 1 can be written as Substituting Eq. (10) into the rear two equations of Eq. (6), we can obtain the effective coupling between the cavity mode a 2 and the mechanical mode b, which can be described by the following equations: Thus the effective Hamiltonian, describing the effective coupling between the auxiliary cavity mode and the mechanical resonator, is written as The effective Hamiltonian describes the effective interaction between the cavity 2 and the mechanical resonator. As we all know, if the Hamiltonian in the interaction picture has the form b 2 + b †2 , the corresponding evolution operator is a squeezed operator.

Engineering the mechanical squeezing. Applying the unitary transformation S(ζ)
which is the single-mode squeezing operator with the squeezing parameter m to the total system. Then the transformed effective Hamiltonian becomes where ω ′ m is the transformed effective mechanical frequency and G ′ is the transformed effective optomechanical coupling. The transformed Hamiltonian is a standard cavity cooling Hamiltonian and the best cooling in the transformed frame is at the optimal detuning ω ∆ = − ′ m eff . In the transformed frame, the master equation, which is used for describing the system-reservoir interaction, can be obtained via applying the squeezing transformation S(ζ) to the master equation Eq. (14) and the transformed density matrix ρ s = S † (ζ)ρS(ζ). The transformed master equation can achieve the cooling process, which can be seen from the Hamiltonian Eq. (16). Here, is the transformed thermal phonon number. The steady-state density matrix ρ (in the original frame) can be obtained by solving the master equation Eq. (14). Defining the displacement quadrature X = b + b † for the mechanical mode, the steady-state variance of X is given by 〈 δX 2 〉 = 〈 X 2 〉 − 〈 X〉 2 , which can be derived in the transformed frame as where ′ n eff is the steady-state phonon number coming from the cooling process in the transformed frame. When the best cooling in ideal situation ′ = n 0 eff is achieved by the cooling process, the steady-state variance of the mechanical resonator displacement quadrature is 〈 δX 2 〉 = e −2ζ .

Discussion
In this section, we solve the original master equation Eq. (9) numerically to calculate the steady-state variance of the mechanical displacement quadrature X. Firstly, we should provide the time evolution of variance 〈 δX 2 〉 about the mechanical displacement quadrature, which is shown in Fig. 3. It indicate that the variance 〈 δX 2 〉 gradually tends to be stable after a period of time. For simplicity, we have assumed that the system is initially prepared in its ground state and the system parameters are chosen to be the same as in Fig. 2.
The relationship between the steady-state variance and intercavity coupling strength is shown in Fig. 4. Before we study their relationship, we should recalculate the steady-state amplitudes of the optical and mechanical modes |α 1 |, |α 2 |, and |β| with the different intercavity coupling strengths. We can find that the steady-state mechanical squeezing can be achieved effectively when the intercavity coupling strength is appropriate, which reaches a balance between the enough large photons number in cavity 1 and the coherent auxiliary cavity interferences. However, when we remove the coherent auxiliary cavity interferences (J = 0), the mechanical steady-state squeezing can not be obtained effectively under the present condition.
The relationship between the steady-state variance and driving power is shown in Fig. 5. One can see from Fig. 5 that the steady-state squeezing of the mechanical resonator changes observably with the laser driving power. We can obtain the steady-state mechanical squeezing effectively when the driving power is in milliwatts level. At last, we consider the effect of the cavity 1 decay κ 1 . When calculate the relationship between the steady-state variance and cavity 1 decay, we consider a more variable J as shown in Fig. 6. The result show that the maximum value of squeezing can be reached with a appropriate J when the cavity 1 decay is certain. In Fig. 6, the minimum value of the steady-state variance is 0.36, corresponding to the 4.4 dB.
In the above, we study the steady-state squeezing of the mechanical resonator in a double-cavity optomechanical system and illustrate that the steady-state squeezing can be effectively generated in highly unresolved sideband regime with appropriate intercavity coupling strength and driving power. When the decay rate of cavity is known, the maximum value of the squeezing parameter ζ is achieved at the point of Δ a = κ 1 /2, which can be easily seen from Eq. (12). The experimental studies of the double-cavity optomechanical system with whispering-gallery microcavities have been reported [62][63][64][65] . Besides, in the latest experiment report 66 , the tunable nonlinearity of the mechanical resonator has been greatly improved by exploring the anharmonicity in chemical bonding interactions. And our method, utilizing the coherent auxiliary cavity 2 to resist the influence of decay coming from cavity 1, is also feasible with the cubic nonlinearity of mechanical resonator, which is easy to prove as refs 37, 44. We also notice anther approach beyond the resolved sideband limit and demonstrated experimentally in optomechanical system 67,68 .  Furthermore, the generated steady-state mechanical squeezing in the present scheme can be detected based on the method proposed in refs 22, 44. As illustrated in refs 22, 44, for detecting the mechanical resonator, we consider another auxiliary cavity mode a s (another mode of the cavity a 1 or adding another cavity on the right) with resonant frequency ω s , which is driven by a weak pump laser field of amplitude Ω p and frequency ω p . The presence of the cavity a s will affect the mirror dynamics, which is no more exactly described by Eqs (6 where δ s = ω p − ω s and g s is the strength of the single-photon optomechanical coupling. However, if the intracavity field is very weak under the weak driving field (the cavity mode steady-state amplitude α α  s 1 ), the cavity backaction on the mechanical mode can be neglected and the relevant dynamics is still well described by Eq. (6). Through homodyning detection of the output field of another auxiliary cavity mode with an appropriate phase, we can obtain the information of the position and the momentum quadratures of the mechanical resonator. Effective detection of the mechanical state requires that α  1 s while α κ  g s s s , where κ s is decay rate of the another auxiliary cavity. The experimental detection technology of the output field has also been realized.
In conclusion, we have proposed a scheme for generating the steady-state squeezing of the mechanical resonator in a double-cavity optomechanical system via the mechanical nonlinearity and cavity cooling process in transformed frame. The steady-state squeezing of the mechanical resonator can be obtained in the highly unresolve sideband regime through the coherent auxiliary cavity interferences. Since the auxiliary cavity mode is not directly coupled to the mechanical resonator, it can be a high Q optical cavity with big cavity volume V, while another cavity coupling with the mechanical resonator can have a short cavity length L to possess good mechanical properties. The effective coupling between the mechanical resonator and the auxiliary cavity can be obtained by reducing the cavity mode adiabatically. We simulate the steady-state variance of the mechanical displacement quadrature numerically at a determinate laser driving power and find that under an appropriate intercavity coupling strength the steady-state mechanical squeezing can be achieved effectively in highly unresolve sideband regime. Our scheme opens up the possibility for application of cavity quantum optomechanics beyond the resolved sideband regime, solving the restricted experimental bounds at present.  The effective interaction between the auxiliary cavity and the mechanical resonator. Here, we will introduce another way to derive the effective coupling between the auxiliary cavity and the mechanical resonator. The quantum Langevin equations Eq. (6) can be formally integrated as Since the decay rate of cavity 1 κ 1 is much larger than the decay rate of cavity 2 κ 2 and the damping rate of mechanical resonator γ m , the dynamics of mode b and a 2 are only slightly affected by mode a 1 . We obtain the approximated expressions