Detecting Nonclassical Correlations in Levitated Cavity Optomechanics

Nonclassical optomechanical correlations enable optical control of mechanical motion beyond the limitations of classical driving. Here we investigate the feasibility of using pulsed cavity optomechanics to create and verify nonclassical phase-sensitive correlations between light and the motion of a levitated nanoparticle in a realistic scenario. We show that optomechanical two-mode squeezing can persist even at the elevated temperatures of state-of-the-art experimental setups. We introduce a detection scheme based on optical homodyning that allows the revealing of nonclassical correlations without full optomechanical state tomography. We provide an analytical treatment using the rotating-wave approximation (RWA) in the resolved-sideband regime and prove its validity with a full numerical solution of the Lyapunov equation beyond the RWA. We build on parameters of current experiments for our analysis and conclude that the observation of nonclassical correlations, which are essential for quantum sensing, quantum engines, and quantum simulations with levitated nanoparticles, is possible with state-of-the-art capabilities. The general treatment can be applied to other optomechanical platforms.

Figure 1

A scheme of the pulsed protocol to generate and verify nonclassical correlations between the levitated nanoparticle and light. (a) A sketch of the proposed setup. A subwavelength particle ($P$), trapped within a single-mode cavity, is coupled to its mode via radiation pressure at rate $g$. (b)–(d) Two light pulses, blue and red detuned enter the cavity to interact with the particle and are subsequently routed to the homodyne detector (HD) via the circulator ($C$). The cyan eightlike symbol denotes nonclassical correlations between different parts of the system. (e) Depending on the detuning $\mathrm{\Delta }={\omega }_{\mathrm{cav}}-{\omega }_p$ of the pump, different types of interaction take place between the center-of-mass motion of the particle and the intracavity field.

Figure 2

(a) Robustness of the nonclassicality of the correlations between light and mechanical motion of the levitating nanoparticle [see Fig. 1 (c)]. Two-mode squeezing $S_{\mathrm{gen}}$ (dark blue) and logarithmic negativity $E_N$ (light red) as functions of the heating rate $\mathrm{\Gamma }$ (logarithmic scale, in units of $\kappa$) assuming an initial mechanical occupation of $n_0={10}^4$. The dashed line shows the two-mode squeezing computed without the rotating-wave approximation. The error bars on the blue curve are obtained from the numerical simulation (see the text). (b) Two-mode squeezing as a function of the coupling rate. The dark green line shows the adiabatic regime; light yellow, full solution. The two solutions coincide well at values corresponding to the weak coupling where the adiabatic approximation is valid. Numerical parameters are $g=0.6\kappa$, $\tau =8/\kappa$, $\mathrm{\Omega }=2\kappa$.

Figure 3

Detection of nonclassical correlations with homodyne measurement on leaking light. (a) Comparison of the two-mode squeezing of the leaking entangling pulse and mechanical motion [$S{}^{\mathrm{PM}}$; dark blue; Fig. 1] and two-mode squeezing of two pulses [$S{}^{\mathrm{PP}}$, light red; Fig. 1] as a function of the heating rate. Full and dashed lines show cases of detection efficiency $\eta =0.8$ and $\eta =0.4$, respectively. (b) Detectable two-mode squeezing $-10{\log }_{10}\mathrm{Var}X{}_{\mathrm{gen}}[{\varphi }^{\mathrm{opt}},{\theta }_B,{\theta }_R{}^{\mathrm{opt}}]$ as a function of the homodyne angle ${\theta }_B$ for two other angles set to their optimal values. Heating rate $\mathrm{\Gamma }=0.1\kappa$. Different colors show the initial occupation of mechanics (from wider to narrower curve) $n_0=1$ for dark green, $n_0=10$ for blue, and $n_0=100$ for light yellow. Error bars are obtained by numerical simulations.