In this study, we investigate a hybrid system consisting of an atomic ensemble trapped inside a dissipative optomechanical cavity assisted with perturbative oscillator-qubit coupling. Such a system is generally very suitable for generating stationary squeezing of the mirror motion in the long-time limit under the unresolved sideband regime. Based on the master equation and covariance matrix approaches, we discuss in detail the respective squeezing effects. We also determine that in both approaches, simplifying the system dynamicswith adiabatic elimination of the highly dissipative cavity mode is very effective. In the master equation approach, we find that the squeezing is a resulting effect of the cooling process and is robust against thermal fluctuations of the mechanical mode. In the covariance matrix approach, we can approximately obtain the analytical result of the steady-state mechanical position variance from the reduced dynamical equation. Finally, we compare the two approaches and observe that they are completely equivalent for the stationary dynamics. Moreover, the scheme may be useful for possible ultraprecise quantum measurement that involves mechanical squeezing.
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61822114, 11465020, and 61465013), the Project of Jilin Science and Technology Development for Leading Talent of Science, and the Technology Innovation in Middle and Young and Team Project (Grant No. 20160519022JH).
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Figure 1
(Color online) Schematic diagram of the considered system. A cloud of identical two-level atoms is trapped in a dissipative optomechanical cavity, which is driven by an external laser field. The qubit (within the black dashed elliptical ring) which is denoted by a yellow dot inside the movable mirror has the levels $|\uparrow\rangle$ and $|\downarrow\rangle$.
Figure 2
(Color online) Time evolution of the real and imaginary parts of the cavity mode mean value $\langle~a(t)\rangle$ and the mechanical mode mean value $\langle~b(t)\rangle$. The system parameters are chosen as: $\omega_\text{m}=\pi\times10^6~\mathrm{Hz}$, $\gamma_\text{m}=10^{-6}\omega_\text{m}$, $g_0^{\prime}=10^{-3}\omega_\text{m}$, $\omega_\text{c}=10^8\omega_\text{m}$, $\delta_\text{c}=-250\omega_\text{m}$, $\kappa=3\omega_\text{m}$, $\Delta_\text{a}=1.1\omega_\text{m}$, $\gamma_\text{a}=0.1\omega_\text{m}$, $G=8\omega_\text{m}$, $\eta=0.2\omega_\text{m}$, $P=20~\mathrm{mW}$, and $E=\sqrt{2P\kappa/(\hbar\omega_\text{l})}$.
Figure 3
(Color online) Time evolution of the variance $\langle\delta~X^2\rangle$ for the qua- drature operator $X$. The red solid line and blue dashed line denote, respectively, the corresponding numerical result with original linearized Hamiltonian $H_{\mathrm{lin}}$ and effective Hamiltonian $H_{\mathrm{eff}}$. The system parameters are presented in Figure
Figure 4
(Color online) Steady-state variance $\langle\delta~X^2\rangle$ for the quadrature operator $X$ versus the cavity decay rate $\kappa$ and atom-cavity coupling strength $G$ with mean thermal phonon number $n_\text{m}=0$. Here the parameters are the same as those in Figure
Figure 5
(Color online) The dependence of steady-state variance $\langle\delta~X^2\rangle$ for the quadrature operator $X$ on the effective detuning $\Delta_{\mathrm{eff}}$ in the case of different mean thermal phonon numbers. The other parameters are fixed as in Figure
Figure 6
(Color online) Energy-level diagram of the transformed Hamiltonian in eq. (
Figure 7
(Color online) Time evolution of the mean phonon number $\langle~b^{\dag}b\rangle$ corresponding to the optimal effective detuning $\Delta_{\mathrm{eff}}=1.4\omega_\text{m}$ when the initial state of the mechanical oscillator is a thermal state with certain mean thermal phonon number $n_\text{m}$. The other system parameters are the same as those in Figure
Figure 8
(Color online) Time evolution of the real and imaginary parts of the mirror position mean value $\langle~q(t)\rangle$ and the cavity mode mean value $\langle~a(t)\rangle$. The parameters are the same as those in Figure
Figure 9
(Color online) Time evolution of the variance $\langle\delta~q^2\rangle$ for the mirror position. The red solid line and blue dashed line denote, respectively, the corresponding numerical result with full covariance matrix $\bf~V$ and reduced covariance matrix $~\bf{V}^{\prime}$. The system parameters are the same as those in Figure
Figure 10
(Color online) Steady-state variance $\langle\delta~q^2\rangle$ for the mirror position versus $\eta$. The blue line, red line, and yellow dots indicate, respectively, the numerical result with full covariance matrix $\textbf{V}$, numerical result with reduced covariance matrix $\textbf{V}^{\prime}$, and analytical result with reduced covariance matrix $\textbf{V}^{\prime}$. The other parameters are the same as those in Figure
Figure 11
(Color online) Time evolution of variance $\langle\delta~X^2\rangle$ for the quadrature operator $X$ (variance $\langle\delta~q^2\rangle$ for the mirror position) obtained from the master equation (dynamical equation for covariance matrix) in the both cases of without and with approximations. The system parameters are the same as those in Figure
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