Response of a mechanical oscillator in an optomechanical cavity driven by a finite-bandwidth squeezed vacuum excitation

In this paper, we theoretically investigate the displacement and momentum fluctuations spectra of the movable mirror in a standard optomechanical system driven by a finite-bandwidth squeezed vacuum light accompanying a coherent laser field. Two cases in which the squeezed vacuum is generated by degenerate and nondegenerate parametric oscillators (DPO and NDPO) are considered. We find that for the case of finite-bandwidth squeezed vacuum injection, the two spectra exhibit unique features, which strongly differ from those of broadband squeezing excitation. In particular, the spectra exhibit a three-peaked and a four-peaked structure, respectively, for the squeezing injection from DPO and NDPO. Besides, some anomalous characteristics of the spectra such as squeezing-induced pimple, hole burning, and dispersive profile are found to be highly sensitive to the squeezing parameters and the temperature of the mirror. We also evaluate the mean-square fluctuations in position and momentum quadratures of the movable mirror and analyze the influence of the squeezing parameters of the input field on the mechanical squeezing. It will be shown that the parameters of driven squeezed vacuum affects the squeezing. We find the optimal mechanical squeezing is achievable via finite-bandwidth squeezed vacuum injection which is affected by the intensity of squeezed vacuum. We also show that the phase of incident squeezed vacuum determines whether position or momentum squeezing occurs. Our proposed scheme not only provides a feasible experimental method to detect and characterize squeezed light by optomechanical systems, but also suggests a way for controllable transfer of squeezing from an optical field to a mechanical oscillator.

Figure 1

Schematic description of the system under consideration. An oscillating mirror with frequency ${\omega }_m$ is coupled via radiation pressure to the cavity field of frequency ${\omega }_0$. The cavity is driven by a laser of frequency ${\omega }_c$, accompanying a much weaker squeezed vacuum field with central frequency ${\omega }_s$.

Figure 2

(a) The displacement and (b) the momentum spectrum (W/Hz) of the moving mirror for the case of DPO (blue line) and (c) the displacement and (d) the momentum spectrum (W/Hz) of the moving mirror for the case of NDPO with $\alpha =2{\kappa }_p$ (black line) versus the normalized frequency $\omega /{\omega }_m$ when $T=100\mathrm{mK}$ and $P=5\mathrm{mW}$. The spectrum (W/Hz) for the case of normal vacuum input is plotted for comparison (red dashed line). The parameters of the squeezed vacuum input are ${\kappa }_p=0.1\kappa ,\epsilon =0.4{\kappa }_p$, and ${\varphi }_0=0$.

Figure 3

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for DPO (blue line) with various values of the pumping rate $\epsilon$: (a) $0.1{\kappa }_p$, (b) $0.2{\kappa }_p$, (c) $0.3{\kappa }_p$, (d) $0.4{\kappa }_p$. Here, we set ${\varphi }_0=\pi$. The spectrum (W/Hz) for the case of normal vacuum input is plotted for comparison (red dashed line). The other parameters are the same as those in Fig. 2.

Figure 4

$M\left(\omega \right)$ (dashed black line) and $N\left(\omega \right)$ (blue line) versus the normalized frequency $\omega /{\omega }_m$ for DPO with the parameters of (a) Fig. 3 and (b) Fig. 3.

Figure 5

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for the NDPO squeezed vacuum (black line) and the normal vacuum state (dashed red line) as input probe fields: (a) $T=100$ mK, $\alpha =5{\kappa }_p$, (b) $T=1\mathrm{mK}, \alpha =5{\kappa }_p$, (c) $T=100\mathrm{mK}, \alpha =0.5{\kappa }_p$. Other parameters are ${\kappa }_p=0.1\kappa ,\epsilon =0.1{\kappa }_p,{\varphi }_0=\pi$, and $P=5$ mW.

Figure 6

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for the case of finite-bandwidth squeezed vacuum with ${\kappa }_p=0.1\kappa$ (blue line), infinite-bandwidth squeezed vacuum with ${\kappa }_p=\kappa$ (black dashed-dotted line), and normal vacuum (dashed red line) for (a) DPO and (b) NDPO. Here, we have set $\epsilon =0.01\kappa , {\varphi }_0=\pi , \alpha =0.3\kappa , T=1\mathrm{mK}$, and $P=5\mathrm{mW}$. Other parameters are the same as those in Fig. 2.

Figure 7

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for the case of DPO squeezed vacuum input with various values of the phase ${\varphi }_0$: (a) $0.75\pi$, (b) $0.85\pi$, and (c) $0.95\pi$ (blue line). The spectrum (W/Hz) for the case of normal vacuum input is plotted for comparison (red dashed line). Here, we have set ${\kappa }_p=0.1\kappa , \epsilon =0.3{\kappa }_p, T=1\mathrm{mK}$, and $P=1\mathrm{mW}$. Other parameters are the same as those in Fig. 2.

Figure 8

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for the case of DPO squeezed vacuum with ${\varphi }_0=0.85\pi ,{\kappa }_p=0.1\kappa$, and different values of $\left|\epsilon \right|$: (a) $0.1{\kappa }_p$, (b) $0.4{\kappa }_p$ (blue line). The red dashed line shows the spectrum (W/Hz) for the case of normal squeezed vacuum input. Other parameters are the same as those in Fig. 2.

Figure 9

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for DPO squeezed vacuum (blue line) as input probe field, with different values of ${\kappa }_p$: (a) $0.1\kappa$, (b) $0.5\kappa$, and (c) $10\kappa$ (broadband squeezed vacuum). Other parameters are $\epsilon =0.1{\kappa }_p,{\varphi }_0=0,P=20\mathrm{mW},$ and $T=100\mathrm{mK}$. The red dashed line shows the spectrum (W/Hz) for the case of normal squeezed vacuum input.

Figure 10

The momentum spectrum (W/Hz) of the moving mirror versus the normalized frequency $\omega /{\omega }_m$ for NDPO squeezed vacuum (black line) as input probe field for different values of ${\kappa }_p$: (a) $0.1\kappa$, (b) $0.5\kappa$, (c) $10\kappa$ (broadband squeezed vacuum). Other parameters are $\epsilon =0.1{\kappa }_p,\alpha =0.2\kappa ,{\varphi }_0=0,P=20\mathrm{mW},$ and $T=100\mathrm{mK}$. The red dashed line shows the spectrum (W/Hz) for the case of normal squeezed vacuum input.

Figure 11

The mean-square fluctuations in momentum of the movable mirror versus the normalized detuning ${\mathrm{\Delta }}_0/{\omega }_m$ for (a) DPO, and (b) NDPO squeezed vacuum as input probe fields with different values of ${\kappa }_p$ (with $\alpha =0.5\kappa ,\epsilon =0.3{\kappa }_p$), and versus $\frac{{\kappa }_p}{\kappa }$ for (c) DPO (d) NDPO squeezed vacuum inputs with different value of $\epsilon$. Here, we have set $\alpha =0.5\kappa ,{\varphi }_0=\pi ,P=5\text{mW},T=1\text{mK}$.

Figure 12

The mean-square fluctuations in (a) displacement and momentum of the movable mirror for the case of DPO squeezed vacuum input with $\epsilon =0.3\kappa$, (b) momentum, and (c) displacement of the movable mirror for the case of NDPO squeezed vacuum input with $\epsilon =0.47\kappa ,\alpha =0.5\kappa$, and different values of $\alpha$ versus the squeezing phase ${\varphi }_0$. Other parameters are ${\kappa }_p=\kappa ,P=5\mathrm{mW}, T=1\mathrm{mK}$.

Figure 13

The mean-square fluctuations in momentum of the movable mirror versus pump laser power ($P$) for (a) DPO and (b) NDPO squeezed vacuum input. Here, we have set ${\kappa }_p=\kappa , \epsilon =0.3\kappa ,\alpha =0.3\kappa , {\varphi }_0=\pi$, and $T=1\mathrm{mK}$.