Steady-state mechanical squeezing in a hybrid atom-optomechanical system with a highly dissipative cavity

Quantum squeezing of mechanical resonator is important for studying the macroscopic quantum effects and the precision metrology of weak forces. Here we give a theoretical study of a hybrid atom-optomechanical system in which the steady-state squeezing of the mechanical resonator can be generated via the mechanical nonlinearity and cavity cooling process. The validity of the scheme is assessed by simulating the steady-state variance of the mechanical displacement quadrature numerically. The scheme is robust against dissipation of the optical cavity, and the steady-state squeezing can be effectively generated in a highly dissipative cavity.

Scientific RepoRts | 6:24421 | DOI: 10.1038/srep24421 Lü et al. 23 proposed a scheme to generate steady-state mechanical squeezing via mechanical nonlinearity, which showed that squeezing could be achieved by the joint effect of nonlinearity-induced parametric amplification and cavity cooling process.
Traditionally and generally, the decay rate of cavity field, which is a dissipative factor in optomechanical system, is considered to have negative effect on the performance of quantum manipulation of mechanical modes. With the development of the hybrid system 42 , here we propose a method to generate steady-state mechanical squeezing in a hybrid atom-optomechanical system where the atomic ensemble is trapped in the optical cavity consisting of a fixed mirror and a movable mirror. 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, via the mechanical nonlinearity and cavity cooling process in transformed frame, the steady-state mechanical squeezing can be successfully and effectively generated in a highly dissipative cavity. Different with ref. 23, our scheme is feasible for Low Q cavity via the coherent auxiliary atomic ensemble interfering.
The paper is organized as follows: In Section II, we describe the model of a hybrid atom-optomechanical system and derive the effective coupling between the atomic ensemble and the mechanical resonator. In Section III, we engineer the mechanical squeezing and derive the analytical variance of the displacement quadrature of the movable mirror in the steady-state. In Section IV, we study the variance of mechanical mode with the large decay rate of cavity by numerical simulations method and discuss the validity of the scheme in the highly and lowly dissipative cavities. A conclusion is given in Section V.

Results
Basic model. We consider a hybrid atom-optomechanical system depicted in Fig. 1, in which N identical two-level atoms are trapped in the optical cavity consisting of a fixed mirror and a movable mirror. The total Hamiltonian H = H 0 + H pump , which describes the hybrid system, consists of three parts, which reads ( = 1), respectively, The part H 0 accounts for the free Hamiltonian of the cavity mode (with frequency ω a and decay rate κ), the atoms (with transition frequency ω c and linewidth γ c ), and the mechanical resonator (with frequency ω m and damping rate γ m ). Here a(a † ) is the bosonic annihilation (creation) operator of the optical cavity mode, b(b † ) is the bosonic annihilation (creation) operator of the mechanical mode, and is the collective z-spin operator of the atoms. The last term of H 0 describes the cubic nonlinearity of the mechanical resonator with amplitude η. For mechanical resonator in the gigahertz range, the intrinsic nonlinearity 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 ancillary system 43,44 , such as the nonlinear amplitude of η = 10 −4 ω m can be obtained when we couple the mechanical resonator to an external qubit 23 .
The part H I accounts for the interaction Hamiltonian consisting of the atom-field interaction and the optomechanical interaction derived from the radiation pressures.
represents the averaged atom-field coupling strength with g i 0 being the coupling strength between the ith atom and single-photon, 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. The driving strength is related to the input laser power P, the mechanical resonator frequency ω d , and the decay rate of cavity κ.
The spin operators S − (S + ) of the atomic ensemble can be transformed to a collective bosonic operator c(c † ) in the Holstein-Primakoff representation 10 where operators c and c † obey the standard boson commutator [c, c † ] = 1. Under the conditions of sufficiently large atom number N and weak atom-photon coupling g 0 , the total Hamiltonian in the frame rotating at input laser frequency ω d is written as . Applying a displacement transformation to linearize the where α, β, and ξ are c numbers denoting the steady-state amplitude of the cavity, mechanical, and collective atomic modes, which are derived by solving the following equations: 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, as shown in Fig. 2. And the amplitudes of the cavity and mechanical modes increase with increasing the driving power. For example, at the point of the driving power P = 49 mW, |α| ≃ 670 and β ≃ 330 can be obtained, respectively.
After the standard linearization procedure, the linearized Hamiltonian is given by And the Hamiltonian of the nonlinear terms, which come from the radiation-pressure interaction and the cubic nonlinearity, is written as Under the conditions of g, η ≪ Λ , G, G 0 , the nonlinear terms in H NL can be neglected because they are much weaker than the linear terms in H L .
Considering the effect of the thermal environment and basing on the linearized Hamiltonian H L , the quantum Langevin equations for the system are written as where the corresponding noise operators a in , b in , and c in satisfy correlations is the mean thermal excitation number of bath of the movable mirror at temperature T, k B is the Boltzmann constant, and one recovers a Markovian process. Since the decay rate of cavity, κ, is much larger than the linewidth of the atoms, and under the conditions a a a i n 2 0 2 where A in (t) denotes the modified noise term. The detailed derivation is displayed in methods part. Neglecting the fast decaying term which contains exp(− κt/2) and substituting Eq. (9) into Eq. (8), we can obtain the effective coupling between the mechanical mode b and collective atoms mode c, which can be written as  Thus the effective Hamiltonian is rewritten as m eff e ff eff 2 2 When considering the system-reservoir interaction, which results in the dissipations of the system, the full dynamics of the effective system is described by the master equation ) /2 is the standard Lindblad operators, γ eff is the effective damping rate of the mode c, and n th is the average phonon number in thermal equilibrium.

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 system is at the optimal detuning ω ∆ = − ′ m eff . In the transformed frame, the master equation ρ′ = S † (ζ)ρS(ζ) of system-reservoir interaction can be approximatively written as 23 , is the transformed thermal phonon number. The steady-state density matrix ρ can be obtained by solving the master equation Eq. (13) numerically. 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 as

Discussion
In this section, we solve the master equation numerically to calculate the steady-state variance of the mechanical displacement quadrature X. Firstly, we show the numerical results for the time evolution of the variance with the Hamiltonian H L and H eff as shown in Fig. 3. We can find that the variance will be stable after time evolution. The relationship between the steady-state variance and effective detuning is shown in Fig. 4. One can see from Fig. 4 that the minimum value of variance can be achieved at the optimal detuning point of ω ∆ = − ′ m eff , which comes from the standard cavity cooling Hamiltonian in Eq. (15) under the transformed frame. The change rate of variance on the effective detuning increases with increasing the average phonon number n th . In the process of numerical simulation, the parameters are set to be ω m /(2π) = 5 MHz, ω a /(2π) = 500 THz, δ a = 400 ω m , Δ c = − 0.9 ω m , G 0 = 6.3 ω m , g = 10 −3 ω m , 1, 10, 50 th respectively, which satisfy the conditions , the steady-state variance of the displacement quadrature is 〈 δX 2 〉 = e −2ζ = 0.65. However, one can see from Fig. 4 that we need a more precise control for Δ eff to achieve the optimal steady-state squeezing of the mechanical resonator with the temperature rising constantly.
In addition, considering the situation of the smaller cavity decay rate κ = 0.1 ω m . The relationship between the steady-state amplitudes (|α|, |β|) and driving power P is shown in Fig. 5 and the relationship between the steady-state variance and effective detuning is shown in Fig. 6 (here we calculate the steady-state variance of the mechanical displacement quadrature X numerically by setting P = 5 mW, |α| = 680, and β = 330), respectively. At the optimal detuning point ω ω ω ∆ = − ′ = − + Λ′ 1 4 / m m m eff , the steady-state variance of the displacement quadrature is 〈 δX 2 〉 = e −2ζ = 0.65.
In the above, we study the steady-state squeezing of the mechanical resonator in a hybrid atom-optomechanical system and illustrate that it can be effectively generated in both the highly and lowly dissipative cavities. The steady-state squeezing can be generated at the optimal detuning point by adjusting the parameters appropriately. Furthermore, the generated steady-state mechanical squeezing in the present scheme can be detected based on the method proposed in refs 23,45. As illustrated in refs 23,45, for detecting the steady-state mechanical squeezing, we can measure the position and the momentum quadratures of the mechanical resonator via homodyning detection of the output field of another auxiliary cavity mode with an appropriate phase, and the auxiliary cavity is driven by another pump laser field under a much weaker intracavity field so that its backaction on the mechanical mode can be neglected. Furthermore, to achieve the steady-state mechanical squeezing, the number of atoms N should be a large number and approximate million. This might be a high challenge in present experiment of the macroscopic objects.
In conclusion, we have proposed a scheme for generating the steady-state squeezing of the mechanical resonator in a hybrid atom-optomechanical system via the mechanical nonlinearity and cavity cooling process in transformed frame. The atomic ensemble is trapped in the optomechanical cavity, which is driven by an external monochrome laser. The effective coupling between the mechanical resonator and the atomic ensemble can be obtained by reducing the cavity mode in the case of large detuning. We simulate the steady-state variance of the mechanical displacement quadrature numerically at a determinate laser driving power and find that the Here Δ eff can be tuned individually by varying Δ c , the average phonon number n th is set to be 1, 10, and 50 respectively, and the other parameters are chosen to be the same as in Fig. 2.  steady-state variance has the minimum value at the optimal detuning point, where the effective detuning is in resonance with the effective transformed mechanical frequency.

Methods
The effective atom-mechanical interaction. To derive the effective coupling between the mechanical resonator and atomic ensemble, the quantum Langevin equations Eq. (8) can be formally integrated as  Since the decay rate of cavity κ is much larger than the linewidth of atoms and the damping rate of resonator, and under the case of ω Λ 2 m , the dynamics of mode b and c are only slightly affected by mode a. We obtain the approximated expressions