Strong quantum squeezing of mechanical resonator via parametric amplification and coherent feedback

A scheme to achieve strong quantum squeezing of a mechanical resonator in a membrane-in-the-middle optomechanical system is developed. To this end, simultaneous linear and nonlinear coupling between the mechanical resonator and the cavity modes is applied. A two-tone driving light field, comprising unequal red-detuned and blue-detuned sidebands, helps in generating a coherent feedback force through the linear coupling with the membrane resonator. Another driving light field with its amplitude modulated at twice the mechanical frequency drives the mechanical parametric amplification through a second-order coupling with the resonator. The combined effect produces strong quantum squeezing of the mechanical state. The proposed scheme is quite robust to excess second-order coupling observed in coherent feedback operations and can suppress the fluctuations in the mechanical quadrature to far below the zero point and achieve strong squeezing (greater than 10 dB) for realistic parameters.

Figure 1

(a) Schematic of the optomechanical system with two cavity modes ${\omega }_{c1}$ and ${\omega }_{c2}$, a two-tone driving field ${\omega }_{\pm }$, and a resonant driving field ${\omega }_2$. (b) A frequency domain representation of the cavity modes and driving field.

Figure 2

Squeezing level of the motion quadrature variance $\langle X^2\rangle$ as a function of the blue-tone driving strength $G_{+}$ (normalized by the red-tone driving strength $G_{-}$). The dotted line indicates the 3-dB squeezing limit (parameters: $n_{th}={10}^3$, $\kappa =5\times {10}^5$ Hz, $G_{-}={10}^5$ Hz, and $\gamma =1$ Hz).

Figure 3

Purity $P$ of the squeezed mechanical state with respect to the parametric drive strength $\chi$ for different values of $G_{+}/G_{-}$ (the values of the other parameters are the same as those shown in Fig. 2).

Figure 4

Schematic of a position-detuned membrane resonator.

Figure 5

Excess second-order coupling strength $\sigma$ with respect to the deviation $z$ in the membrane position (parameters: $L=10\mathrm{mm}$, $\lambda =1064\mathrm{nm}$, $r=0.35$, $G_{-}=0.1\mathrm{MHz}$, and $G_{+}/G_{-}=0.85$).

Figure 6

Mechanical squeezing level with respect to the excess second-order coupling strength (parameters: $\chi =11\mathrm{kHz}$ and $G_{+}/G_{-}=0.85$; the values of the other parameters are the same as those shown in Fig. 2).