Can gravity mediate the transmission of quantum information?

Abstract

We propose an experiment to test the non-classicality of the gravitational interaction. We consider two optomechanical systems that are perfectly isolated, except for a weak gravitational coupling. If a suitable resonance condition is satisfied, an optical signal can be transmitted from one system to the other over a narrow frequency band, a phenomenon that we call gravitationally induced transparency. In this framework, the challenging problem of testing the quantum nature of gravity is mapped to the easier task of determining the non-classicality of the gravitationally-induced optical channel: If the optical channel is not entanglement-breaking, then gravity must have a quantum nature. This approach is applicable without making any assumption on the, currently unknown, correct model of gravity in the quantum regime. In the second part of this work, we model gravity as a quadratic Hamiltonian interaction (e.g. a weak Newtonian force), resulting in a Gaussian thermal attenuator channel between the two systems. Depending on the strength of thermal noise, the system presents a sharp transition from an entanglement-breaking to a non-classical channel capable not only of entanglement preservation but also of asymptotically perfect quantum communication.

Figures (7)

• Figure 1: (Top) The apparatus required to observe the gravitationally-induced transparency (GIT) phenomenon is based on two opto-mechanical systems, $S_1$ and $S_2$. Each system is composed of an optical cavity mode $a_j$ that is coupled with a mechanical resonator $b_j$, where $j=1,2$. By tuning the mode frequencies and the optomechanical coupling $g$, gravity can induce an effective quantum channel $\Phi$ from the optical input of $S_1$ to the optical output of $S_2$. Any quantum optics experiment demonstrating that $\Phi$ is not entanglement-breaking would imply that gravity is a non-classical phenomenon. (Bottom) Effective transmissivity and output noise as a function of the input probe frequency $\omega$ (relative to the cavity frequency). When the transmissivity is higher than the output noise, the channel $\Phi$ is non-classical. This plot is based on the quadratic gravitational interaction defined in Eq. \ref{['eq:interaction']} and on the following physical parameters: $\omega_B=2 \pi\times0.03$ Hz, $\gamma=10^{-14} \omega_B$, $\kappa=0.1 \omega_B$, $\lambda=3.58\times 10^{-6}$ Hz, $g=1.84 \times 10^{-4}$ Hz, and $T=1$ mK. For more details, see Fig. \ref{['fig:git']} in Supplemental Material supplemental_material\ref{['app:full_solution']}.
• Figure 2: Non-classicality analysis of the GIT channel in the parameter space $(\omega_B, Q=\omega_B/\gamma)$, according to Eq. \ref{['eq:optimal_ratio']} and Eqs. (\ref{['eq:criterion_A']}-\ref{['eq:criterion_C']}). We assume the mechanical degrees of freedom are two nearby gold spheres at a temperature of $T=1$ mK, such that $\lambda \simeq \pi G \rho_{\rm Au}/(6\omega_B)$, where $\rho_{\rm Au}$ is the gold mass density. See Supplemental Material supplemental_material\ref{['app:analysis_of_technical_requirements']} for more details. The transmissivity $\eta$ of the effective channel is $\approx 1$ in the low-frequency region, but it is extremely small ($\lesssim 10^{-23}$) in the high-frequency region. This fact makes the high-frequency region theoretically valid but experimentally problematic. The red star corresponds to the parameters of Fig. \ref{['fig:scheme']}.
• Figure S.1: Spectral analysis of the gravitationally-induced transparency (GIT) phenomenon. The transmissivity $\eta(\omega)$ (full line) and the output noise $[1 - \eta(\omega)] N(\omega)$ (dashed line) for different frequencies of the input mode $a_{\rm in_1}(\omega)$. The frequency is defined in interaction picture, such that the origin $\omega=0$ is centered at the cavity frequency $\omega_A$. The frequency window between the two dotted lines is the estimated transparency linewidth according to Eq. \ref{['eq:linewidth']}. The light-green region highlights the frequency window in which the effective transmission channel $\Phi_\omega=\mathcal{E}_{\eta, N}$ is a non-classical quantum attenuator according to \ref{['eq:general_ratio']}, i.e., where the transmissivity exceeds the output noise. The physical parameters used in this plot are: $\omega_B=2 \pi\times0.03$ Hz, $\gamma=10^{-14} \omega_B$, $\kappa=0.1 \omega_B$, $\lambda= w_G^2/\omega_B=3.58\times 10^{-6}$ Hz, $N_T=[\exp(\omega_B/w_T) - 1]^{-1}=6.94 \times 10^8$, where $w_G$ is given by Eq. \ref{['eq:grav_w']} assuming two nearby gold spheres and where $w_T$ is given by \ref{['eq:grav_w']} assuming a temperature of $T=1$ mK. The optomechanical coupling rate is set to its optimal value $g_{\rm opt}=(\sqrt{\kappa}/2)(\gamma^2 + 4 \lambda^2)^{1/4}\simeq 1.84 \times 10^{-4}$ Hz, according to Eq. \ref{['eq:optimal_parameters']}.
• Figure S.2: Non-classicality analysis of the gravity-induced channel in the $(\omega_B, Q)$ parameter space according to Eq. \ref{['eq:parameters_criterion']} and to the non-classicality criteria $A$, $B$ and $C$, defined in the main text in Eqs. (\ref{['eq:criterion_A']}-\ref{['eq:criterion_C']}). This figure only assumes a linearized Newtonian force and quantum theory, without further experimental limitations apart from the gravitational and environmental critical frequencies reported in Eqs. \ref{['eq:grav_w']} and \ref{['eq:env_w']}. In principle, there are two non-classical regions: one at very low frequencies (strong gravitational coupling) and one at very high frequencies (negligible thermal noise). However, as shown in Fig. \ref{['fig:transmissivity']}, the transmissivity $\eta$ of the effective channel is $\approx 1$ in the low-frequency region, but it is extremely small ($\lesssim 10^{-23}$) in the high-frequency region. This fact makes the high-frequency region theoretically valid but experimentally problematic. The red star corresponds to the parameters of Fig. \ref{['fig:git']}.
• Figure S.3: Effective optimal transmissivity $\eta_{\rm opt}$ of the gravity-induced optical channel in the $(\omega_B, Q)$ parameter space. This figure assumes a linearized Newtonian force and an experimental apparatus characterized by the gravitational critical frequency reported in Eq. \ref{['eq:grav_w']}. The transmissivity is $\approx 1$ in the low-frequency region, but it becomes extremely small in the high-frequency region, limiting experimental accessibility. The red star corresponds to the parameters of Fig. \ref{['fig:git']}. The black dotted lines are the borders between the quantum and classical regions analyzed in Fig. \ref{['fig:quality_frequency_space']}.