Generation of squeezed states and single-phonon states via homodyne detection and photon subtraction on the filtered output of an optomechanical cavity

In this paper, we consider the generation of mechanical squeezed states and single-phonon Fock states, respectively, by homodyne detection and photon subtraction in a dissipative or dispersive optomechanical system under the situation that the cavity linewidth ${\kappa }_c$ is much larger than the mechanical frequency ${\omega }_m$. We at first show that strong mechanical squeezing beyond the 3-dB limit can be achieved by homodyning the filtered cavity output field in a purely dispersive or purely dissipative optomechanical system for ${\kappa }_c\gg {\omega }_m$, while the combination of the two kinds of coupling can also effectively enhance the mechanical squeezing. We next show that in these optomechanical systems, mechanical single-phonon states with high fidelity can be generated via photon subtraction on the filtered cavity output field in the regimes of weak optomechanical entanglement and ${\kappa }_c\gg {\omega }_m$. It is found that the single-phonon Fock state via purely dissipative optomechanical coupling is attainable in the blue-detuned regime, whereas it is present in the red-detuned regime by purely dispersive optomechanical coupling. In addition, the effects of thermal fluctuations on the mechanical squeezing and the negativity of the Wigner function of the mechanical single-phonon states are also studied. It is shown that the Gaussian squeezing is much more robust against decohering thermal fluctuations than the non-Gaussian nonclassicality.

Figure 1

The schematic plot of a dissipatively (dispersively) coupled optomechanical system in which the cavity dissipation rate ${\kappa }_c$ (resonant frequency ${\omega }_c$) is modulated by the position $x_m$ of the mechanical oscillator. The cavity output is subject to filtering and detection to generate mechanical squeezing [via homodyne detection in (a)] or single-phonon Fock states [via photon subtraction detection in (b), see the text].

Figure 2

The contour plot of the mechanical squeezing $V_m$ and OM entanglement $E_{\mathrm{ab}}$ versus the OM coupling $G_{\kappa ,\omega }$ and the detuning $\mathrm{\Delta }$, for purely dissipative coupling in (a) and (b) and for purely dispersive coupling in (c) and (d), with the parameters ${\kappa }_c=10{\omega }_m, {\mathrm{\Omega }}_f=-{\omega }_m, \tau ={\omega }_m^{-1}, {\gamma }_m={10}^{-5}{\omega }_m$, and ${\overline{n}}_{\mathrm{th}}=0$. The green shaded areas denote instability regimes.

Figure 3

The contour plot of the mechanical squeezing $V_m$ and OM entanglement $E_{\mathrm{ab}}$ versus the cavity dissipation rate ${\kappa }_c$ and the detuning $\mathrm{\Delta }$, for purely dissipative coupling in (a) and (b) with $G_{\kappa }=3{\omega }_m$ and for purely dispersive coupling in (c) and (d) with $G_{\omega }=1.5{\omega }_m$. The others parameters are the same as in Fig. 2, and the green shaded areas denote instability regimes.

Figure 4

The dependence of the mechanical squeezing $V_m$ and OM entanglement $E_{\mathrm{ab}}$ on the detuning $\mathrm{\Delta }$ for different values of the couplings $G_{\kappa }$ and $G_{\omega }$. The other parameters are same as in Fig. 2.

Figure 5

The contour plot of the mechanical squeezing $V_m$ and OM entanglement $E_{\mathrm{ab}}$ versus the Fourier frequency ${\mathrm{\Omega }}_f$ and the detuning $\mathrm{\Delta }$, for purely dissipative coupling in (a) and (b) with $G_{\kappa }=3{\omega }_m$ and for purely dispersive coupling in (c) and (d) with $G_{\omega }=1.5{\omega }_m$. The others parameters are the same as in Fig. 2.

Figure 6

The dependence of the mechanical squeezing on the mean thermal phonon number ${\overline{n}}_{\mathrm{th}}$ for purely dissipative coupling with $G_{\kappa }=4{\omega }_m$ and $\mathrm{\Delta }=-3.5{\omega }_m$ (red solid line) and purely dispersive coupling with $G_{\omega }=1.5{\omega }_m$ and $\mathrm{\Delta }=4.5{\omega }_m$ (green dashed line). The other parameters are the same as in Fig. 2.

Figure 7

The dependence of the OM entanglement $E_{\mathrm{ab}}$ in (a), the mechanical Wigner function at the phase-space origin $W(0,0)$ in (b), and the fidelity $F_1$ of the mechanical states with respect to the single-phonon state $|1\rangle$ in (c) on the detuning $\mathrm{\Delta }$ via purely dissipative coupling, for ${\kappa }_c=10{\omega }_m, G_{\kappa }=0.1{\omega }_m, {\mathrm{\Omega }}_f=-{\omega }_m, \tau =10{\omega }_m^{-1}, {\gamma }_m={10}^{-5}{\omega }_m$, and ${\overline{n}}_{\mathrm{th}}=0$. (d) Plots the Wigner function of the mechanical states for the detuning $\mathrm{\Delta }=-0.45{\omega }_m$.

Figure 8

The dependence of the OM entanglement $E_{\mathrm{ab}}$ in (a), the mechanical Wigner function at the phase-space origin $W(0,0)$ in (b), and the fidelity $F_1$ of the mechanical states with respect to the single-phonon state $|1\rangle$ in (c) on the detuning $\mathrm{\Delta }$ via purely dispersive coupling, for ${\kappa }_c=10{\omega }_m, G_{\omega }=0.001{\omega }_m, {\mathrm{\Omega }}_f=-{\omega }_m, {\tau }^{-1}=10{\omega }_m, {\gamma }_m={10}^{-5}{\omega }_m$, and ${\overline{n}}_{\mathrm{th}}=0$. (d) Plots the Wigner function of the mechanical states for the detuning $\mathrm{\Delta }=5{\omega }_m$.

Figure 9

The dependence of the OM entanglement $E_{\mathrm{ab}}$ in (a), the mechanical Wigner function at the phase-space origin $W(0,0)$ in (b), and the fidelity $F_1$ of the mechanical states with respect to the single-phonon state $|1\rangle$ in (c) on the detuning $\mathrm{\Delta }$ via purely dispersive coupling, for ${\kappa }_c=0.1{\omega }_m, G_{\omega }=0.02{\omega }_m, {\mathrm{\Omega }}_f=-{\omega }_m, {\tau }^{-1}=5{\omega }_m, {\gamma }_m={10}^{-5}{\omega }_m$, and ${\overline{n}}_{\mathrm{th}}=0$. (d) Plots the Wigner function of the mechanical states for the detuning $\mathrm{\Delta }={\omega }_m$.

Figure 10

The time evolution of the mechanical Wigner functions at the phase-space origin $W_m(0,0)$ and the fidelity $F_1$ of the mechanical states with respect to the single-phonon state $|1\rangle$. Here, the solid, dashed, and dotted lines correspond to the plots (d) in Figs. 7, 8, and 9, respectively.

Figure 11

The dependence of the maximal negativity of the mechanical Wigner functions at the phase-space origin $W_m(0,0)$ on the mean thermal phonon number ${\overline{n}}_{\mathrm{th}}$. Here, the plots (a), (b), and (c) correspond to the plots (d) in Figs. 7, 8, and 9, respectively.