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arXiv:2406.09230 · Classical and Quantum Gravity

Correlations and signaling in the Schrödinger–Newton model

Jacek Gruca, Ankit Kumar, Ray Ganardi, P. Arumugam, Karolina Kropielnicka, Tomasz Paterek

Published Nov 6, 2024• 5 citations
arXivHTMLPDF

Abstract

Abstract The Schrödinger–Newton (SN) model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the SN interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the SN model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.

DOI
10.1088/1361-6382/ad8f8a

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Abstract The Schrödinger–Newton (SN) model is a semi-classical theory in which, in addition to mutual attraction, massive quantum particles interact with their own gravitational fields. While there are many studies on the phenomenology of single particles, correlation dynamics in multipartite systems is largely unexplored. Here, we show that the SN interactions preserve the product form of the initial state of a many-body system, yet on average agreeing with classical mechanics of continuous mass distributions. This leads to a simple test of the model, based on verifying bipartite gravitational evolution towards non-product states. We show using standard quantum mechanics that, with currently accessible single-particle parameters, two masses released from harmonic traps get correlated well before any observable entanglement is accumulated. Therefore, the SN model can be tested with setups aimed at observation of gravitational entanglement with significantly relaxed requirements on coherence time. We also present a mixed-state extension of the model that avoids superluminal signaling.
Abstract We probe the dynamics of a modified form of the Schrödinger–Newton (SN) system of gravity coupled to single particle quantum mechanics. At the masses of interest here, the ones associated with the onset of ‘collapse’ (where the gravitational attraction is competitive with the quantum mechanical dissipation), we show that the Schrödinger ground state energies match the Dirac ones with an error of . At the Planck mass scale, we predict the critical mass at which a potential collapse could occur for the self-coupled gravitational case, Planck mass, and show that gravitational attraction opposes Gaussian spreading at around this value, which is a factor of two higher than the one predicted (and verified) for the SN system. Unlike the SN dynamics, we do not find that the self-coupled case tends to decay towards its ground state; there is no collapse in this case.
The Schrödinger--Newton (SN) equation provides a semiclassical framework for the evolution of self-gravitating of massive quantum systems. We propose a two-body Schrödinger--Newton model that separates local nonlinear self-localization from the nonseparable Newtonian pair potential. Analytically, we show that the nonlinear self-field preserves the Schmidt spectrum, whereas direct entanglement generation arises from the nonseparable pair potential. Using numerical simulations in a regularized one-dimensional geometry, we find that entanglement generation depends sensitively on the initial spatial configuration and on the mass ratio. Highly localized, self-bound wavepackets experience minimal entanglement growth during scattering. Spatial delocalization and kinetic dispersion broaden the interaction region, amplifying the entangling power of the pair potential and exciting higher-order spatial modes. For dispersive Gaussian initial states, mass asymmetry shatters the lighter particle, producing Wigner negativity and rapid entanglement growth, whereas stationary SN profiles strongly suppress this effect. Stationary SN profiles isolate the bare pair-potential contribution; dispersive Gaussian initial states inflate it.
Abstract No experiment to date has provided evidence for quantum features of the gravitational interaction. Recently proposed tests suggest looking for the generation of quantum entanglement between massive objects as a possible route towards the observation of such features. Motivated by advances in optical cooling of mirrors, here we provide a systematic study of entanglement between two masses that are coupled gravitationally. We first consider the masses trapped at all times in harmonic potentials (optomechanics) and then the masses released from the traps. This leads to the estimate of the experimental parameters required for the observation of gravitationally induced entanglement. The optomechanical setup demands LIGO-like mirrors and squeezing or long coherence times, but the released masses can be light and accumulate detectable entanglement in a timescale shorter than their coherence times. No macroscopic quantum superposition develops during the evolution. We discuss the implications from such thought experiments regarding the nature of the gravitational coupling.
The Schrödinger-Newton equation aims at describing the dynamics of massive quantum systems subject to the gravitational self-interaction. As a deterministic nonlinear quantum wave equation, it is generally believed to conflict with the relativistic no-signalling principle. Here we challenge this viewpoint and show that it is of key importance to study the quantitative and operational character of the superluminal effects. To this end we employ a rigorous formalism of probability measures on spacetime and quantify the probability of a successful superluminal bit transfer via the single-particle Schrödinger-Newton equation. We demonstrate that such a quantity decreases with the increasing size and mass of the system. Furthermore, we prove that the Einstein-Dirac system, which yields the Schrödinger-Newton equation in the non-relativistic limit, is perfectly compatible with the relativistic causal structure. Our study demonstrates that the Schrödinger-Newton equation, which is by construction non-relativistic, is in fact `more compatible' with the no-signalling principle than the ordinary free Schrödinger equation.
The necessity of quantising the gravitational field is still subject to an open debate. In this paper we compare the approach of quantum gravity, with that of a fundamentally semi-classical theory of gravity, in the weak-field non-relativistic limit. We show that, while in the former case the Schrödinger equation stays linear, in the latter case one ends up with the so-called Schrödinger-Newton equation, which involves a nonlinear, non-local gravitational contribution. We further discuss that the Schrödinger-Newton equation does not describe the collapse of the wave-function, although it was initially proposed for exactly this purpose. Together with the standard collapse postulate, fundamentally semi-classical gravity gives rise to superluminal signalling. A consistent fundamentally semi-classical theory of gravity can therefore only be achieved together with a suitable prescription of the wave-function collapse. We further discuss, how collapse models avoid such superluminal signalling and compare the nonlinearities appearing in these models with those in the Schrödinger-Newton equation.
In order to empirically test the quantum nature of gravity, it is essential to explore the construction of classical gravity theories that are as consistent with experiments as possible. In particular, the classical gravity field must receive input regarding matter distribution. Previously, such input has been constructed by taking expectation values of the matter density operator on the quantum state, or by using the outcomes of all measurements being performed on the quantum system---or by using information obtained by auxiliary observers (like those that lead to the continuous spontaneous localization and Diosi-Penrose collapses) that continuously monitor the quantum dynamics. We propose a framework that unifies these models, and argue that the Causal Conditional Formulation of Schr\"odinger-Newton (CCSN) theory, which takes classical inputs only from experimental and environmental channels---without auxiliary observers---is a minimum model within this framework. Since CCSN can be viewed as a quantum feedback control scheme, it can be made explicitly causal and free from pathologies that previously plagued Schr\"odinger-Newton theories. Since classical information from measurement results are used to generate classical gravity, CCSN can mimic quantum gravity better than one would naively expect for a classical theory---making it more subtle to perform tests of the quantum nature of gravity. We predict experimental signatures of CCSN in two concrete scenarios: (i) a single test mass continuously monitored by light, and (ii) two objects interacting via mutual gravity, each monitored separately. In case (i), we show that the mass-concentration effect of self classical gravity still makes CCSN much easier to test than testing the establishment of mutual entanglement, yet the signatures are more subtle than previously thought for classical gravity theories. Using time-delayed measurements and nonstationary measurements, which delay or suspend the flow of classical information into classical gravity, one can make CCSN more detectable. In case (ii), we show that mutual gravity generated by CCSN can lead to correlations that largely mimic signatures of quantum entanglement in steady-state measurements. Rigorous protocols that rule out local operation and classical communication channels, which are experimentally more challenging than simply testing steady-state entanglement, must be applied in order to completely rule out CCSN.