Boosting Gravity-Induced Entanglement through Parametric Resonance

TL;DR

The paper addresses generating gravity-induced entanglement between two masses by exploiting parametric resonance in two Mathieu oscillators coupled through Newtonian gravity. It develops an analytical framework in which the gravitational interaction yields decoupled modes with shifted Mathieu parameters $a\pm\eta$, and computes entanglement from the covariance matrix of an initial Gaussian state, revealing exponential growth of $E_{\mathcal{N}}$ in unstable regions with rate tied to the Floquet exponent $\mu_I$, with an approximate form $E_{\mathcal{N}}(\tau) \approx \tfrac{3}{8}\eta e^{2\mu_I\tau}$. The study also analyzes environmental decoherence by introducing damping $\gamma$ and white-noise $\mu$, finding that noise suppresses entanglement and that $\mu<\eta$ is needed for observability, while damping has only a modest, pre-threshold effect. The results suggest a practical path to amplify gravity-induced entanglement far below the Planck scale, with experimental routes such as optically induced springs or periodic stiffness modulation, albeit requiring robust decoherence suppression.

Abstract

Establishing quantum gravity theory remains one of the major challenges in modern physics, as the lack of experimental evidence makes it difficult to explore. In response to this challenge, proposals to test quantum entanglement induced by Newtonian gravity in table-top experiments have attracted significant attention as a potentially feasible approach far below the Planck energy scale. In this work, we propose a scheme to amplify gravity-induced entanglement between two masses using parametric resonance. Specifically, we consider two parametrically resonant oscillators interacting through Newtonian gravity, each governed by the Mathieu equation. We analyzed the logarithmic negativity between two oscillators and investigate the effects of random force noise and linear damping. As a result, we find an exponential growth of gravity-induced entanglement between the oscillators, which reflects the dynamical instability of parametric resonant systems.

Figures (6)

• Figure 1: Setup with two parametric resonant oscillators coupled via Newtonian gravity. Each oscillator is assumed to follow the Mathieu equation, which describes a periodical modulation of its effective frequency as $\omega(\tau)=\sqrt{a-2q \cos(2\tau)}$.
• Figure 2: Left panel: Stability diagram of the Mathieu equation \ref{['eq:single-mathieu']} in the $(a,q)$-plane. The instability rate, defined as $\mu_I\equiv -{\rm Im}(\mu)$, is represented by a color scale: blue regions $(\mu_I=0)$ correspond to stable solutions , while red regions $(\mu_I>0.25)$ indicate strong exponential growth. The upper-left region with $2q>a$, where $\omega^2$ becomes negative during part of the cycle, is cross-hatched in black; such cases are challenging to realize experimentally and are not considered in this work (see however Fujita2023Michimura2017). Right panel: Periodic modulation of the oscillator potential for representative parameter points shown in the left panel. The blue, orange and green markers correspond to $(a, q) = (4.8,2.4)$, $(4.8,1.8)$, and $(4.8,0.1)$, respectively. As $q$ increases, the potential becomes shallower at $\tau=\pi(n+\frac{1}{2})$ ($n$ is an integer).
• Figure 3: Logarithmic negativity $E_{\mathcal{N}}$ evaluated at $\tau=3\pi$ for a relatively large gravitational coupling $\eta=0.01$ in the $(a,q)$-plane. The color map, ranging from dark purple to bright yellow, shows the magnitude of the generated entanglement $E_{\mathcal{N}}$. Thin green lines represent contours of the imaginary part of the Floquet exponent $\mu_I$ at intervals of 0.1, and the thick green curves trace $\mu_I=0$, marking the boundary between stable and unstable regions. As expected, $E_{\mathcal{N}}$ increases with the strength of the Mathieu instability $\mu_I$.
• Figure 4: Time evolution of the logarithmic negativity $E_{\mathcal{N}}(\tau)$ with coupling $\eta=10^{-12}$. Blue $(a=4.8,\,q=2.4)$, orange $(a=4.8,\,q=1.8)$, and green $(a=4.8,\,q=0.1)$ indicate strongly unstable, weakly unstable, and stable parameters, respectively. Dashed lines depict the approximate analytic formula in Eq. \ref{['eq:EN-approx-eta']} for the unstable cases. No dashed line is shown for the stable trajectory because the Floquet exponent is purely real ($\mu_I=0$). The blue trajectory reaches the detection threshold $E_{\mathcal{N}}=0.01$ (horizontal red dashed line) at $\tau\simeq 44$ (vertical dot-dashed line), which corresponds to $t \approx 0.044~\mathrm{s}$ for a modulation frequency of $\omega = 1~\mathrm{kHz}$ for instance.
• Figure 5: We plot the ratio $\mathcal{R}$ defined in Eq. \ref{['Ratio R']} as a function of $\tau$ to illustrates how environmental random forces suppress the logarithmic negativity $E_{\mathcal{N}}$. We fix the gravitational coupling at $\eta=10^{-5}$ and use the Mathieu parameters $(a,q)=(4.8,\,2.4)$. The top horizontal line corresponds to $\mu=0$. From top to bottom, the random-noise strength increases as $\mu = 0,\,0.1\,\eta,\,0.2\,\eta,\,\ldots,\,0.9\,\eta$, in steps of $0.1\,\eta$. As $\mu/\eta$ increases, the entanglement is progressively suppressed, and the late-time plateau value of $E_{\mathcal{N}}$ asymptotically approaches $\mathcal{R} = 1-\mu/\eta$.