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Can quantum systems succumb to their own (gravitational) attraction?

Thomas Durt, Ralph Willox, Samuel, Colin

Published Nov 24, 2014• 13 citations
OpenAlex

Abstract

The gravitational interaction is generally considered to be too weak to be easily submitted to systematic experimental investigation in the quantum, microscopic, domain.In this paper we attempt to remedy this situation by considering the gravitational influence exerted by a crystalline nanosphere of mesoscopic size on itself, in the semi-classical, mean field, regime.We study in depth the self-localisation process induced by the corresponding non-linear potential of (gravitational) self-interaction.In particular, we characterize the stability of the associated self-collapsed ground state and estimate the magnitude of the corrections that are due to the internal structure of the object (this includes size-effects and corrections due to the discrete, atomic, structure of the sphere).Finally, we derive an approximated, gaussian, dynamics which mimics several essential features of the selfgravitating dynamics and, based on numerical results derived from this model, we propose a concrete experimental setting which we believe might, in the foreseeable future, reveal the existence of gravitational self-interaction effects.

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Can quantum systems succumb to their own (gravitational) attraction?

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The evolution of the center-of-mass wave function for a mesoscopic particle according to the Schr\"odinger-Newton equation can be approximated by a harmonic potential if the wave function is narrow compared to the size of the mesoscopic particle. It was noticed by Colin et al. [Phys. Rev. A 93, 062102 (2016).] that, in the regime where self-gravitational effects are weak, intermediate and wider wave functions may be approximated by a harmonic potential as well but with a width-dependent coupling, leading to a time evolution that is determined only by a differential equation for the width of a Gaussian wave function as a single parameter. Such an approximation results in considerably less computational effort in order to predict the self-gravitational effects on the wave-function dynamics. Here, we provide an alternative approach to this kind of approximation, including a rigorous derivation of the equations of motion for an initially Gaussian wave packet under the assumption that its shape is conserved. Our result deviates to some degree from the result by Colin et al. [Phys. Rev. A 93, 062102 (2016).], specifically in the limit of wide wave functions.

arXiv:1503.02622•Physical Review A•Aug 1, 2016•12 citations
The gravitational interaction is generally considered to be too weak to be easily submitted to systematic experimental investigation in the quantum, microscopic, domain.In this paper we attempt to remedy this situation by considering the gravitational influence exerted by a crystalline nanosphere of mesoscopic size on itself, in the semi-classical, mean field, regime.We study in depth the self-localisation process induced by the corresponding non-linear potential of (gravitational) self-interaction.In particular, we characterize the stability of the associated self-collapsed ground state and estimate the magnitude of the corrections that are due to the internal structure of the object (this includes size-effects and corrections due to the discrete, atomic, structure of the sphere).Finally, we derive an approximated, gaussian, dynamics which mimics several essential features of the selfgravitating dynamics and, based on numerical results derived from this model, we propose a concrete experimental setting which we believe might, in the foreseeable future, reveal the existence of gravitational self-interaction effects.
The gravitational interaction is generally considered to be too weak to be easily submitted to systematic experimental investigation in the quantum, microscopic, domain. In this paper we attempt to remedy this situation by considering the gravitational influence exerted by a crystalline nanosphere of mesoscopic size on itself, in the semi-classical, mean field, regime. We study in depth the self-localisation process induced by the corresponding non-linear potential of (gravitational) self-interaction. In particular, we characterize the stability of the associated self-collapsed ground state and estimate the magnitude of the corrections that are due to the internal structure of the object (this includes size-effects and corrections due to the discrete, atomic, structure of the sphere). Finally, we derive an approximated, gaussian, dynamics which mimics several essential features of the self-gravitating dynamics and, based on numerical results derived from this model, we propose a concrete experimental setting which we believe might, in the foreseeable future, reveal the existence of gravitational self-interaction effects.
The Schrödinger-Newton equation has gained attention in the recent past as a nonlinear modification of the Schrödinger equation due to a gravitational self-interaction. Such a modification is expected from a fundamentally semi-classical theory of gravity and can, therefore, be considered a test case for the necessity of the quantisation of the gravitational field. Here we provide a thorough study of the effects of the Schrödinger-Newton equation for a micron-sized sphere trapped in a harmonic oscillator potential. We discuss both the effect on the energy eigenstates and the dynamical behaviour of squeezed states, covering the experimentally relevant parameter regimes.
We propose an effective non-relativistic framework in which wave-function collapse emerges as a deterministic dynamical instability induced by gravitational self-interaction and regulated by short-distance repulsion. The dynamics is described by a nonlinear Schrödinger equation supplemented by a phenomenological repulsive sector ensuring regularity at high densities. Using a variational Gaussian ansatz, we derive an explicit effective energy functional and show that extended quantum states lose stability beyond a critical mass scale. This loss of stability is associated with a bifurcation in the reduced dynamical system governing the wave-function width, leading to the emergence of stable localized configurations. Within this picture, collapse corresponds to the dynamical selection of one of these localized attractors, driven by infinitesimal asymmetries in the initial state and occurring without stochastic noise or environmental coupling. The mechanism provides a controlled and quantitative realization of gravity-induced localization, extending Schrödinger--Newton-type models while avoiding their pathological short-distance behavior. Possible implications for mesoscopic systems probing the quantum-to-classical transition are briefly discussed.
This work presents numerical and theoretical evidence for a testable regime in which semiclassical Schrödinger–Newton dynamics predict a state-dependent gravitational self-interaction in mesoscopic quantum systems. Using large-scale Split-Step Fourier Method (SSFM) simulations, we identify a mass window (≈10⁻¹¹–10⁻¹⁰ kg) where quantum coherence and gravitational self-interaction coexist without immediate decoherence. In this regime, the gravitational self-energy depends smoothly on the spatial structure of the quantum state, leading to a differential phase evolution between superposition states with identical external potentials but different mass-density distributions. The predicted phase signal scales as m² and reaches experimentally relevant magnitudes (Δφ ≈ 10⁻³ rad) for millisecond evolution times, making the effect accessible to next-generation matter-wave and optomechanical interferometry. The work systematically validates the numerical approach, demonstrates energy conservation and convergence of the nonlinear dynamics, identifies a stable “breathing-mode” regime near 10⁻¹⁰ kg, and isolates the gravitational origin of the effect via null tests with the Schrödinger–Newton term removed. This preprint makes no claim regarding spacetime engineering, geometry manipulation, or violations of the equivalence principle. It is explicitly framed as a falsifiable test of semiclassical gravity in a previously inaccessible experimental regime. Confirmation or null detection would provide strong constraints on Schrödinger–Newton–type models and inform the ongoing search for the correct quantum–gravity interface.
We investigate the effects of the gravitational field on the quantum dynamics of nonrelativistic particles. We consider NNN nonrelativistic particles, interacting with the linearized gravitational field. Using the Feynman-Vernon influence functional technique, we trace out the graviton field to obtain a master equation for the system of particles to first order in GGG. The effective interaction between the particles as well as the self-interaction is in general non-Markovian. We show that the gravitational self-interaction cannot be held responsible for decoherence of microscopic particles due to the fast vanishing of the diffusion function. For macroscopic particles though, it leads to diagonalization to the energy eigenstate basis, a desirable feature in gravity-induced collapse models. We finally comment on possible applications.
The evolution of the center-of-mass wave function for a mesoscopic particle according to the Schr\"odinger-Newton equation can be approximated by a harmonic potential if the wave function is narrow compared to the size of the mesoscopic particle. It was noticed by Colin et al. [Phys. Rev. A 93, 062102 (2016).] that, in the regime where self-gravitational effects are weak, intermediate and wider wave functions may be approximated by a harmonic potential as well but with a width-dependent coupling, leading to a time evolution that is determined only by a differential equation for the width of a Gaussian wave function as a single parameter. Such an approximation results in considerably less computational effort in order to predict the self-gravitational effects on the wave-function dynamics. Here, we provide an alternative approach to this kind of approximation, including a rigorous derivation of the equations of motion for an initially Gaussian wave packet under the assumption that its shape is conserved. Our result deviates to some degree from the result by Colin et al. [Phys. Rev. A 93, 062102 (2016).], specifically in the limit of wide wave functions.