Strong mechanical squeezing and its detection

We report an efficient mechanism to generate a squeezed state of a mechanical mirror in an optomechanical system. We use an especially tuned parametric amplifier (PA) inside the cavity and the parametric photon phonon processes to transfer quantum squeezing from photons to phonons with almost 100% efficiency. We get 50% squeezing of the mechanical mirror which is limited by the PA. We present analytical results for the mechanical squeezing thus enabling one to understand the dependence of squeezing on system parameters like gain of PA, cooperativity, and temperature. As in cooling experiments the detrimental effects of mirror's Brownian and zero point noises are strongly suppressed by the pumping power. By judicious choice of the phases, the cavity output is squeezed only if the mirror is squeezed thus providing us a direct measure of the mirror's squeezing. Further considerable larger squeezing of the mirror can be obtained by adding the known feedback techniques.

Figure 1

Sketch of the optomechanical system to prepare the movable mirror in a squeezed state. A PA is placed inside the cavity, and the pump of the PA is not shown. Here the movable mirror is coupled to an optical cavity. This scheme also applies to a mechanical resonator capacitively coupled to a microwave cavity.

Figure 2

Sketch of the physical processes leading to the squeezed phonons. The pump ${\omega }_l$ is red detuned with $\mathrm{\Delta }={\omega }_m$. The system is working in the resolved sideband limit. (a) Shows the tuning of the different frequencies. (b) and (c) The usual optomechanical processes leading to the generation of phonons and then conversion back to photons. (d) Shows how the PA leads to squeezed phonons.

Figure 3

The mean square fluctuation $\langle \delta P{\left(t\right)}^2\rangle$ versus the parametric gain $G$ for different parametric phases $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$ when $C=400$ and $T=0$ K. The flat dotted line represents the momentum variance of the vacuum state ${\langle \delta P{\left(t\right)}^2\rangle }_{\mathrm{vac}}=0.5$. The green dotted, red solid, blue dot-dashed, purple short dashed, cyan middle dashed, and orange long-dashed curves correspond to $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$, respectively.

Figure 4

The mean square fluctuation $\langle \delta P{\left(t\right)}^2\rangle$ versus the cooperativity parameter $C$ for different parametric phases $\theta =0$ (dot-dashed), $\pi /16$ (solid), $\pi /6$ (dashed) when $G=0.46\kappa$ and $T=0$ K. The flat dotted line represents the momentum variance of the vacuum state ${\langle \delta P{\left(t\right)}^2\rangle }_{\mathrm{vac}}=0.5$.

Figure 5

The mean square fluctuation $\langle \delta P{\left(t\right)}^2\rangle$ versus the parametric gain $G$ for different parametric phases $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$ when $C=400$ and $T=0$ K. The flat dotted line represents the momentum variance of the vacuum state ${\langle \delta P{\left(t\right)}^2\rangle }_{\mathrm{vac}}=0.5$. The green dotted, red solid, blue dot-dashed, purple short dashed, cyan middle dashed, and orange long-dashed curves correspond to $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$, respectively.

Figure 6

The mean square fluctuation $\langle \delta P{\left(t\right)}^2\rangle$ versus the parametric gain $G$ for different temperatures of the environment $T=0$ K (solid), 10 mK (dot-dashed), and 20 mK (dashed) when $C=400$ and $\theta =\pi /16$. The flat dotted line represents the momentum variance of the vacuum state ${\langle \delta P{\left(t\right)}^2\rangle }_{\mathrm{vac}}=0.5$.

Figure 7

The spectrum $S_{y\mathrm{out}}\left(\omega \right)$ of the phase fluctuation of the output field versus the frequency $\omega$ when $G=0.49\kappa , \theta =\pi /16$, and $T=0$ K in the absence of the optomechanical coupling $(g=0)$ (blue solid) and in the presence of the optomechanical coupling $(g\ne 0)$ (red dotted). Here the spectrum $S_{y\mathrm{out}}\left(\omega \right)$ for $g=0$ has been divided by 100.

Figure 8

The contour plot of the spectrum $S_{z\mathrm{out}}\left(\omega \right)$ of the quadrature fluctuation of the output field versus the frequency $\omega$ and the phase $\varphi$ when $G=0.49\kappa , \theta =\pi /16$, and $T=0$ K. The lower figure zooms the upper figure for the region around zero.

Figure 9

The mean square fluctuation $\langle \delta y{\left(t\right)}^2\rangle$ versus the parametric gain $G$ for different parametric phases $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$ when $T=0$ K. The flat dotted line represents the phase variance of the vacuum state ${\langle \delta y{\left(t\right)}^2\rangle }_{\mathrm{vac}}=0.5$. The green dotted, red solid, blue dot-dashed, purple short dashed, cyan middle dashed, and orange long-dashed curves correspond to $\theta =0,\pi /16,\pi /6,\pi /4,\pi /3,\pi /2$, respectively.