Cavity-assisted squeezing of a mechanical oscillator

We investigate the creation of squeezed states of a vibrating membrane or a movable mirror in an optomechanical system. An optical cavity is driven by the squeezed light and couples via the radiation pressure to the membrane or mirror, effectively providing a squeezed heat bath for the mechanical oscillator. Under the conditions of laser cooling to the ground state, we find an efficient transfer of squeezing with roughly 60% of light squeezing conveyed to the membrane or mirror (on a dB scale). We determine the requirements on the carrier frequency and the bandwidth of squeezed light. Beyond the conditions of ground-state cooling, we predict mechanical squashing to be observable in current systems.

Figure 1

(Color online) (a) A mechanical mode of a dielectric membrane, oscillating at frequency ${\omega }_{\text{m}}$, is coupled via radiation pressure to the cavity field of frequency ${\omega }_{\text{c}}$. The cavity is driven by a laser of frequency ${\omega }_{\text{l}}$ and, in addition, a much weaker squeezed vacuum field with central frequency ${\omega }_{\text{s}}$. (b) Optimal choice for the frequencies ${\omega }_{\text{l}}$ and ${\omega }_{\text{s}}$, in order to transfer squeezing from the light field to the membrane motion: the laser frequency is chosen such that the cooling sideband is addressed, i.e., ${\omega }_{\text{l}}={\omega }_{\text{c}}-{\omega }_{\text{m}}$, whereas the center frequency of the squeezing has to equal the cavity resonance frequency ${\omega }_{\text{c}}$.

Figure 2

(Color online) Schematic phase-space picture of mechanical squeezing via reservoir engineering with squeezed light: a mechanical resonator in thermal equilibrium with equal variance in conjugate quadratures $\delta X_{\phi =0}$ and $\delta X_{\phi =\pi /2}$ (left) is laser sideband cooled by a coherent driving field at frequency ${\omega }_{\text{l}}$ reduces the mechanical variances $⟨\delta X_{\phi }\delta X_{\phi }⟩$ close to the ground-state variances (center). A second squeezed vacuum field at frequency ${\omega }_{\text{s}}$ drives the system into a state with (anti)squeezed variances in conjugate variables (right).

Figure 3

(Color online) (a) Efficiency of squeezing transfer from light to the membrane (in dB) for the following set of parameters $Q={10}^7$, $m=1\text{ng}$, ${\omega }_{\text{m}0}=2\pi \times 1\text{MHz}$, $\kappa =2\pi \times 380\text{kHz}$, $G=2\pi \times 110\text{kHz}$, and $T=100\text{mK}$. The analytical result of Eq. (14) (dashed line) agrees well with the exact numerical result (solid line). For (b)–(d), the squeezing of light was fixed to 8 dB. (b) Squeezing of the membrane vs deviations from the resonance $\delta ={\Delta }_{\text{s}}+{\omega }_{\text{m}}$. Mechanical squeezing requires the resonance condition $\delta =0$ to be fulfilled within the effective mechanical linewidth ${\gamma }_{\text{eff}}$. (c) Squeezing transfer vs bandwidth of squeezing $b_x$ for various values of the sideband parameter $\eta =\kappa /{\omega }_{\text{m}}$ (full lines). Dashed lines give the corresponding value for white squeezed input noise. Outside the resolved-sideband regime, the bandwidth plays a role in optimizing the transfer of squeezing. (d) Dependence of the generated motional squeezing on the membrane environment temperature, analytical (dashed, blue), and exact numerical (solid, blue) result. The dash-dotted red line shows the mean membrane occupation versus temperature without the squeezed light input for the same parameters.