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arXiv:2502.03489 · Classical and Quantum Gravity

Probing the nonclassical dynamics of a quantum particle in a gravitational field

Martin Plávala, Stefan Nimmrichter, Matthias Kleinmann

Published Nov 4, 2025• 1 citation

Abstract

Abstract In quantum mechanics, the time evolution of particles is given by the Schrödinger equation. It is valid in a nonrelativistic regime where the interactions with the particle can be modelled by a potential and quantised fields are not required. This has been verified in countless experiments when the interaction is of electromagnetic origin, but also corrections due to the quantised field are readily observed. When the interaction is due to gravity, then one cannot expect to see effects of the quantised field in current-technology Earth-bound experiments. However, this does not yet guarantee that in the accessible regime, the time evolution is accurately given by the Schrödinger equation. Here we propose to measure the effects of an asymmetric mass configuration on a quantum particle in an interferometer. For this setup we show that with parameters within experimental reach, one can be sensitive to possible deviations from the Schrödinger equation, beyond the already verified lowest-order regime. Performing this experiment will hence directly test the nonclassical behaviour of a quantum particle in the gravitational field.

DOI: 10.1088/1361-6382/ae1b5c

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Abstract In quantum mechanics, the time evolution of particles is given by the Schrödinger equation. It is valid in a nonrelativistic regime where the interactions with the particle can be modelled by a potential and quantised fields are not required. This has been verified in countless experiments when the interaction is of electromagnetic origin, but also corrections due to the quantised field are readily observed. When the interaction is due to gravity, then one cannot expect to see effects of the quantised field in current-technology Earth-bound experiments. However, this does not yet guarantee that in the accessible regime, the time evolution is accurately given by the Schrödinger equation. Here we propose to measure the effects of an asymmetric mass configuration on a quantum particle in an interferometer. For this setup we show that with parameters within experimental reach, one can be sensitive to possible deviations from the Schrödinger equation, beyond the already verified lowest-order regime. Performing this experiment will hence directly test the nonclassical behaviour of a quantum particle in the gravitational field.

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No experimental evidence exists, to date, whether or not the gravitational field must be quantised. Theoretical arguments in favour of quantisation are inconclusive. The most straightforward alternative to quantum gravity, a coupling between classical gravity and quantum matter according to the semi-classical Einstein equations, yields a nonlinear modification of the Schrödinger equation. Here, effects of this so-called Schrödinger-Newton equation are discussed, which allow for technologically feasible experimental tests.

The interplay between gravity and quantum mechanics remains a central open problem in modern physics. In this talk, I will discuss how a fundamentally classical, local gravitational interaction necessarily induces a diffusive modification of quantum dynamics. This diffusion leads to measurable noise even in the classical regime and provides a novel, experimentally accessible signature that does not require macroscopic quantum superpositions. I will derive a lower bound on the gravitational noise required to prevent entanglement generation and outline an experimental proposal based on torsion pendulums to test this bound. Next, I will consider the Schrödinger-Newton equation—a paradigmatic model of semiclassical gravity—which predicts nonlinear, self-induced gravitational effects. I will show how a periodic modulation of the trap frequency in levitated mechanical oscillators can amplify these otherwise negligible deviations by up to six orders of magnitude. The protocol remains feasible within current technologies and opens new pathways to experimentally test the SN dynamics through precise monitoring of the system’s position variance.

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Semiclassical gravity, in which a classical spacetime is sourced by the quantum expectation value of the stress-energy tensor, is a standard framework for describing the gravitational interaction of quantum matter. In the nonrelativistic limit this approach leads to the Schrödinger-Newton equation, which is often assumed to be consistent at least in the weak-field regime. In this work, we reexamine this assumption for spatial quantum superpositions of massive particles. We show that, when the quantum state is properly normalized, no modification of the Newtonian gravitational potential arises at leading order. However, at first post-Newtonian order the semiclassical coupling generically produces state-dependent contributions involving the mass density and the mass current of the superposition. These terms have a parametric scaling which is different from that of the corresponding relativistic corrections and which does not have Planck mass suppression. Our results therefore impose a strong post-Newtonian consistency constraint on deterministic semiclassical gravity, indicating that sourcing the metric solely by expectation values is insufficient to recover a consistent relativistic weak-field expansion.