Momentum Squeezed State Realized via Optimal Filtering in Optomechanics: Implications for Gravity-Induced Entanglement

Abstract

We analyze the conditional quantum state of a mechanical mirror in an optomechanical system subject to continuous measurement, feedback control, and quantum filtering. We identify a parameter regime in which the mirror exhibits momentum squeezing beyond the standard quantum limit, achieved through an appropriate choice of the homodyne detection angle. In this regime, we show that optimal filtering effectively realizes a free-particle-like conditional state. When this mechanism is applied to a configuration consisting of two optomechanical systems, the resulting momentum squeezing significantly enhances the signal of gravity-induced entanglement (GIE). This enhancement arises because the momentum squeezing not only amplifies the distinction between the common and differential modes, but also, in the high-purity regime, increases the position uncertainty in accordance with the uncertainty principle, thereby enlarging the spatial extent of the quantum superposition. Our results provide new insights into experimental strategies for probing the quantum nature of gravity using optomechanical platforms.

Figures (6)

• Figure 1: Spectral densities of the mechanical position $q$ and momentum $p$ without and with causal Wiener filtering. The parameters are chosen as $m = 100~\mathrm{g}$, $\ell = 10~\mathrm{cm}$, $\Omega/2\pi = 10^{-3}~\mathrm{Hz}$, $\kappa/2\pi = 10^{8}~\mathrm{Hz}$, $\omega_c/2\pi = 2.818\times10^{14}~\mathrm{Hz}$, $\Gamma/2\pi = 10^{-18}~\mathrm{Hz}$, $P_\mathrm{in} = 10^{-5}~\mathrm{W}$, $T = 1~\mathrm{K}$, $\Delta = 0$, and $\theta = -\pi/60$, which yield $\omega_\theta/\omega_m \simeq 0.31$ and $\gamma_\theta/2\pi \simeq 3.0\times10^{-4}~\mathrm{Hz}$. Without filtering, the spectral densities of $(q,p)$ exhibit a peak at the mechanical resonance frequency $\omega_m$ (dashed curves). In contrast, when causal Wiener filtering is applied, the spectral densities of the filtered variables $(\tilde{q},\tilde{p})$ show a shifted peak at $\omega_\theta$ (solid curves).
• Figure 2: Schematic illustration of conditional states in phase space. The blue curve represents the trajectory of the conditional first moments revealed by a continuous measurement process. Information acquired through the measurement reduces the uncertainty, resulting in a blue ellipse that characterizes the conditional uncertainty. The gray circle denotes the unconditional uncertainty obtained by averaging over all possible measurement trajectories.
• Figure 3: Contour plot of $V_{pp}$ in the $(\Delta/\kappa,\,\theta)$ plane. The red solid curve indicates the condition given by Eq. \ref{['deftheta']}, which coincides with the parameter region where momentum squeezing occurs. All parameters are the same as those used in Fig. \ref{['SqqSpp']}, except for $\theta$ and $\Delta$. The red dashed curve denotes the zero-measurement-rate condition, $\lambda_\theta = 0$.
• Figure 4: Contour plot of $V_{pp}$ obtained under measurement at the optimized homodyne angle given in Eq. \ref{['deftheta']}. The colored region indicates the parameter space in which momentum squeezing occurs, while the blue curve denotes the boundary defined by $V_{pp}=1$. The orange dashed line corresponds to $\eta \xi = 1$, which defines the boundary of the approximation adopted in Eqs. \ref{['approx:etaxi<1']} and \ref{['approx:etaxi>1']}. We also note that the approximate formula Eq. \ref{['etaxiap']} is valid for $\eta\ll\xi$ and $2\ll\xi$.
• Figure 5: Configuration of the experimental setup. We consider two cylindrical mirrors A and B with radius $R$ and thickness $h$. The gravitational interaction affects only the differential mechanical mode associated with the relative displacement of the two mirrors.