Theoretical Study of the Squeezed-Light-Enhanced Sensitivity to Gravity-Induced Entanglement via Finite-Time Analysis

Abstract

We investigate the advantage of using squeezed input light for generating gravity-induced entanglement (GIE) through Fourier-domain analysis. Based on the findings of Ref.~\cite{Miki2024}, which demonstrated the feasibility of detecting GIE in optomechanical systems under quantum control, we further demonstrate that squeezed input light can reduce the optical noise in the mechanical conditional state and enhance GIE. Furthermore, we estimate the systematic and statistical errors in the measurement of GIE using the Fourier transformation over a finite measurement time. Based on the error estimations using the signal-to-noise ratio (SNR) in GIE detection, we find that a total measurement time of $10^6\,\mathrm{s}$ is required to achieve ${\rm SNR} = 1$ when using squeezed input light, whereas $10^{6.8}\,\mathrm{s}$ is needed without squeezed input light. This result highlights the effectiveness of optomechanical systems and the critical role of squeezed input light in enhancing the detectability of GIE.

Figures (3)

• Figure 1: Configuration of the optomechanical system of the symmetrical setup. See also the footnote in the next page for "squeezed input light".
• Figure 2: The left panel shows the contour plot of $\log_{10}\langle E_{\rm Fil}(\Omega_+)\rangle$ on the plane of the squeezing parameters $(r, \phi)$, while the right panel shows the same quantity plotted on the plane of $(r, \log_{10}P_{\mathrm{in}}~[\mathrm{W}])$. In the left panel, the input laser power is fixed at $P_{\mathrm{in}} = 10^{-10}$ [W], while in the right panel, the squeezing angle is set to $\phi = \pi/2$. The white regions indicate that the two mechanical mirrors are not entangled. The red line in the right panel represents the condition given in Eq. \ref{['condition1']}. Darker blue regions correspond to a stronger degree of entanglement. The left panel shows that increasing $r$ enhances entanglement when $\phi \simeq \pi/2$. The right panel demonstrates that increasing $r$ suppresses radiation-pressure noise, thereby enabling entanglement generation even at higher input laser powers.
• Figure 3: The left panel compares the logarithm of the degree of entanglement and the associated errors as a function of the measurement time $T$. The blue curve represents $\log_{10}$$\langle E_{d}(\Omega_+, T) \rangle$, which asymptotically converges to the red line indicating the value of $\log_{10} \langle E_{\rm Fil}(\Omega_+) \rangle$ in the limit of infinite measurement time. The green dotted line shows the boundary of the entanglement criterion, $\langle E(\Omega_+) \rangle = 1$, below which the two mirrors are entangled. The orange curve represents the degree of entanglement including statistical errors, given by $\langle E_{d}(\Omega_+, T) \rangle + \Delta E_{\rm stat}^{N_0=4}(\Omega_+, T)$, where $\Delta E_{\rm stat}^{N_0=4}(\Omega_+, T)$ denotes the statistical error for four repeated measurements. The black vertical lines indicate the difference between the blue and orange curves, representing the systematic error for finite measurement time $T$. The right panel shows the contour plot of $\text{SNR}(\Omega_+)$ as a function of the total measurement time $T_{\text{total}}$ and the number of measurements $N_0$. The upper-right region above the red dashed curve satisfies $\text{SNR}(\Omega_+) > 1$. Due to this error, the signal-to-noise ratio (SNR) exceeds unity only when $T_{\text{total}} > 10^6$ seconds. In this figure, we fix $r = 1$, $\phi = \pi/2$, $\gamma_m=6.6\times2\pi\times10^{-6}~\mathrm{Hz}$ and $P_{\mathrm{in}} = 10^{-10}~\mathrm{[W]}$.