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Nature Communications

Nonlinear optical simulation of the post-Newton Schrödinger equation

Omer Paz, Yonatan Ben-Haim, Shay Rakia, Rivka Bekenstein

Published May 2, 2025• 11 citations

Abstract

One of the grand open problems of modern physics is the unification of Einstein’s general relativity and quantum mechanics. This challenge has been approached by the greatest physicists but there is still no complete theory and experimental evidence remains out of reach. The Newton–Schrödinger equation (NSE) offers insight into this puzzle, as it describes a quantum wavefunction under self-gravity dynamics. Having been studied theoretically for decades, optical experiment of this nonlinear model was demonstrated only in 2015. Although the NSE can be generalized to post-Newtonian gravity approximating general relativity, all experiments simulating nonlinear gravity so far have been limited to Newtonian gravity. Here we present experimental emulation of post-Newtonian dynamics by probing a new physical regime of nonlinearity that mimics larger masses in gravity. We find soliton solutions of the post-Newtonian–Schrödinger equations, distinct from their Newtonian counterparts and demonstrate them experimentally. We observe rich beam evolution requiring previously unconsidered nonlinear terms, thereby opening up new experimental capabilities for simulating wavefunction dynamics in the settings of general relativity. An optical emulation experiment of a quantum wave function dynamics under self-induced post Newtonian gravitational potential is presented, including observation of solitons of this system and rich beam evolution due to high-order nonlinear effects.

DOI: 10.1038/s41467-025-59384-z

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One of the grand open problems of modern physics is the unification of Einstein’s general relativity and quantum mechanics. This challenge has been approached by the greatest physicists but there is still no complete theory and experimental evidence remains out of reach. The Newton–Schrödinger equation (NSE) offers insight into this puzzle, as it describes a quantum wavefunction under self-gravity dynamics. Having been studied theoretically for decades, optical experiment of this nonlinear model was demonstrated only in 2015. Although the NSE can be generalized to post-Newtonian gravity approximating general relativity, all experiments simulating nonlinear gravity so far have been limited to Newtonian gravity. Here we present experimental emulation of post-Newtonian dynamics by probing a new physical regime of nonlinearity that mimics larger masses in gravity. We find soliton solutions of the post-Newtonian–Schrödinger equations, distinct from their Newtonian counterparts and demonstrate them experimentally. We observe rich beam evolution requiring previously unconsidered nonlinear terms, thereby opening up new experimental capabilities for simulating wavefunction dynamics in the settings of general relativity. An optical emulation experiment of a quantum wave function dynamics under self-induced post Newtonian gravitational potential is presented, including observation of solitons of this system and rich beam evolution due to high-order nonlinear effects.

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.

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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.

The Schr\"odinger-Newton model describes self-gravitating quantum particles, and it is often cited to explain the gravitational collapse of the wave function and the localization of macroscopic objects. However, this model is completely nonrelativistic. Thus, in order to study whether the relativistic effects may spoil the properties of this system, we derive a modification of the Schr\"odinger-Newton equation by considering certain relativistic corrections up to the first post-Newtonian order. The construction of the model begins by considering the Hamiltonian of a relativistic particle propagating on a curved background. For simplicity, the background metric is assumed to be spherically symmetric, and it is then expanded up to the first post-Newtonian order. After performing the canonical quantization of the system, and following the usual interpretation, the square of the module of the wave function defines a mass distribution, which in turn is the source of the Poisson equation for the gravitational potential. As in the nonrelativistic case, this construction couples the Poisson and the Schr\"odinger equations and leads to a complicated nonlinear system. Hence, the dynamics of an initial Gaussian wave packet is then numerically analyzed. We observe that the natural dispersion of the wave function is slower than in the nonrelativistic case. Furthermore, for those cases that reach a final localized stationary state, the peak of the wave function happens to be located at a smaller radius. Therefore, the relativistic corrections effectively contribute to increase the self-gravitation of the particle and strengthen the validity of this model as an explanation for the gravitational localization of the wave function.

Current models of quantum interference experiments in external gravitational fields lack a common framework: while matter-wave interferometers are commonly described using the Schrödinger equation with a Newtonian potential, gravitational effects in quantum optics are modeled using either post-Newtonian metrics or highly symmetric exact solutions to Einstein's field equations such as those of Schwarzschild and Kerr. To coherently describe both kinds of experiments, this paper develops a unified framework for modeling quantum interferometers in general stationary space-times. This model provides a rigorous description and coherent interpretation of the effects of classical gravity on quantum probes.

While it is widely believed that gravity should ultimately be treated as a quantum theory, there remains a possibility that general relativity should not be quantized. If this is the case, the coupling of classical gravity to the expectation value of the quantum stress-energy tensor will naturally lead to nonlinearities in the Schrodinger equation. By numerically investigating time evolution in the nonrelativistic "Schrodinger-Newton" approximation, we show that such nonlinearities may be observable in the next generation of molecular interferometry experiments.