A truly relativistic gravity mediated entanglement protocol using superpositions of rotational energies

TL;DR

The authors propose a truly relativistic quantum gravity–mediated entanglement test by placing two massive rotors in superpositions of rotational energy, leveraging mass–energy equivalence so rotational energy sources gravity. They derive the entangling phase $\phi=\frac{G I^2 \omega^4 T}{4 \hbar c^4 r}$ for rotational-energy superpositions, discuss a concrete protocol to realize such superpositions via electric and magnetic dipoles, and analyze three major operational limits—tiny $1/c^4$-suppressed phase, centrifugal deformation, and radiation- induced decoherence—identifying parameter regimes where entanglement could be observed. The paper further argues that disc-shaped rotors can widen the feasible regime and situates the work as a meaningful, though technically demanding, test of gravity in the quantum regime that lacks an electromagnetic analogue. Overall, while extremely ambitious, the scheme would probe the GR–QM interface by testing whether gravity can be sourced by energy in a quantum superposition, with implications for whether gravity must be quantum or classical in nature.

Abstract

Experimental proposals for testing quantum gravity-induced entanglement of masses (QGEM) typically involve two interacting masses which are each in a spatial superposition state. Here, we propose instead a QGEM experiment with two particles which are each in a superposition of rotational states, this amounts to a superposition of mass through mass-energy equivalence. In sharp contrast to the typical protocols studied, our proposal is genuinely relativistic. It does not consider a quantum positional degree of freedom but relies on the fact that rotational energy gravitates: the effect we consider disappears in the limit where the speed of light c approaches infinity. Furthermore, this approach would test a feature unique to gravity since it amounts to sourcing a spacetime in superposition due to a superposition of 'charge'.

Figures (5)

• Figure 1: QGEM protocols aim to witness whether the gravitational interaction between two masses can be in a superposition state. The typical approach considers (a) preparing each mass in a superposition of locations. Another possibility is to (b) prepare each particle in a superposition of mass-states. Concretely, we show howl this can be achieved by (c) preparing each particle in a superposition of rotational energies and exploiting the equivalence between mass and (rotational) energy.
• Figure 2: A dipole field can be used to spin-up a dipole moment. The dipole moment in the $x-y$ plane has polar angle $\theta$. When $\pi/2<\theta<3\pi/2$ a field along $-\hat{y}$ is applied, otherwise a field along $\hat{y}$ is applied. This ensures the torque is always about $\hat{z}$. The process can be reversed.
• Figure 3: (a) The shaded regions represent parameter regimes which satisfy the different requirements we consider. The shaded regions overlap for particles with radii around $0.1m$ radius and angular velocities around $2\pi\times1Hz$, indicating the scheme may be viable for these parameters. (b) It becomes more difficult to prepare superpositions of orientations (separated by angle $\theta_0$) during step (2) as the particle size and the moment of inertia is increased. (c) The magnetic dipole moment $m$ is limited by the particle volume and remanence magnetisation for small particles, and by the requirement that photons are not emitted for fast-spinning particles. These figures consider using two solid spheres.
• Figure S4: Dynamics during ramp-up (step 3) are slower than described in the main text. (a) Ramp-up (showing $\theta$) for initial angle $\theta_0=10^{-6}$. (b) Ramp-up (showing $\omega=\dot{\theta}$) for different initial angles. The curves are flattish until time $t_0$. Smaller $\theta_0$ values give longer $t_0$ values. (c) The inverse relation between $t_0$ and $\theta_0$ is shown more clearly. The time $t_0'$ at which $\theta$ reaches $\pi/2$ behaves similarly to $t_0$. (d) The angular velocity at time $t_0'$ is insensitive to $\theta_0$.
• Figure S5: Conducting the protocol using disc-shaped particles enlarges the overlap between the regimes satisfying the requirements we studied.