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arXiv:2310.18072 · Physical Review A

Probing the Schrödinger-Newton equation in a Stern-Gerlach-like experiment

Gabriel H. S. Aguiar, George E. A. Matsas

Published Mar 25, 2024• 3 citations
arXivHTMLPDF

Abstract

Explaining the behavior of macroscopic objects from the point of view of the quantum paradigm has challenged the scientific community for the past century. A mechanism of gravitational self-interaction, governed by the so-called Schr\"odinger-Newton equation, is among the proposals that aim to shed some light on it. Despite all efforts, this mechanism has been proven difficult to probe. Here, we consider a Stern-Gerlach-like experiment to try it out. The Schr\"odinger-Newton equation can be analytically solved under certain proper conditions, and a change-of-phase effect induced by the gravitational self-interacting potential can be calculated.

DOI
10.1103/physreva.109.032223

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Explaining the behavior of macroscopic objects from the point of view of the quantum paradigm has challenged the scientific community for the past century. A mechanism of gravitational self-interaction, governed by the so-called Schr"odinger-Newton equation, is among the proposals that aim to shed some light on it. Despite all efforts, this mechanism has been proven difficult to probe. Here, we consider a Stern-Gerlach-like experiment to try it out. The Schr"odinger-Newton equation can be analytically solved under certain proper conditions, and a change-of-phase effect induced by the gravitational self-interacting potential can be calculated.
There is no consensus among today's physicists about how to describe the gravitational interaction properly in a quantum framework. We propose in this paper an experimental test aimed at revealing the existence of a non-linear self-interaction à la Schrodinger-Newton (S-N). In this test, a mesoscopic spin 1/2 microsphere is freely falling in a Humpty-Dumpty Stern-Gerlach interferometer. After clarifying the role of the scaling of the interaction in function of the amplitudes of the up and down spin components of the microsphere, it is shown that self-gravity induces a measurable phase shift between them, which paves the way to experimental tests. It is also shown that if we consider two distinct microspheres falling in parallel, the entangling power of the S-N interaction is exactly equal to zero.
The Stern-Gerlach effect, found a century ago, has become a paradigm of quantum mechanics. Unexpectedly, until recently, there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Several theoretical studies have explained why a Stern-Gerlach interferometer is a formidable challenge. Here, we provide a detailed account of the realization of a full-loop Stern-Gerlach interferometer for single atoms and use the acquired understanding to show how this setup may be used to realize an interferometer for macroscopic objects doped with a single spin. Such a realization would open the door to a new era of fundamental probes, including the realization of previously inaccessible tests at the interface of quantum mechanics and gravity.
The Stern-Gerlach effect, found a century ago, has become a paradigm of quantum mechanics. Unexpectedly, until recently, there has been little evidence that the original scheme with freely propagating atoms exposed to gradients from macroscopic magnets is a fully coherent quantum process. Several theoretical studies have explained why a Stern-Gerlach interferometer is a formidable challenge. Here, we provide a detailed account of the realization of a full-loop Stern-Gerlach interferometer for single atoms and use the acquired understanding to show how this setup may be used to realize an interferometer for macroscopic objects doped with a single spin. Such a realization would open the door to a new era of fundamental probes, including the realization of previously inaccessible tests at the interface of quantum mechanics and gravity.
The Schr\"odinger-Newton equation has frequently been studied as a nonlinear modification of the Schr\"odinger equation incorporating gravitational self-interaction. However, there is no evidence yet as to whether nature actually behaves this way. This work investigates a possible way to experimentally test gravitational self-interaction. The effect of self-gravity on the interference of massive particles is studied by numerically solving the Schr\"odinger-Newton equation for a particle passing through a double-slit. The results show that the presence of gravitational self-interaction has an effect on the fringe width of the interference that can be tested in matter-wave interferometry experiments. Notably, this approach can distinguish between gravitational self-interaction and environment-induced decoherence, as the latter does not affect the fringe width. This result will also provide a way to test if gravity requires to be quantized on the scale of ordinary quantum mechanics.
The nonlinear Schr\"odinger-Newton equation, a prospective semiclassical alternative to a quantized theory of gravity, predicts a gravitational self-force between the two trajectories corresponding to the two zzz-spin eigenvalues for a particle in a Stern-Gerlach interferometer. To leading order, this force results in a relative phase between the trajectories. For the experimentally relevant case of a spherical particle with localized wave function, we present a rederivation of that phase which is both rigorous in its approximations and concise, allowing for simple but accurate experimental predictions.
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.