[Paper Review] Revealing self-gravity in a Stern-Gerlach Humpty-Dumpty experiment

Mohamed Hatifi, Thomas Durt|arXiv (Cornell University)|Jun 12, 2020

Quantum Mechanics and Applications1 references4 citations

TL;DR

This paper proposes a single-particle Stern-Gerlach Humpty-Dumpty experiment to test the existence of gravitational self-interaction via the Schrödinger-Newton (S-N) equation. By preparing a mesoscopic spin-1/2 microsphere in a superposition of spin states and allowing self-gravity to induce a measurable phase shift between the up and down components during free fall, the experiment reveals a spin-dependent dephasing that could be detected via tomography. Crucially, the S-N interaction predicts zero entangling power between two such microspheres in parallel interferometers, contrasting sharply with standard quantum gravity models.

ABSTRACT

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.

Motivation & Objective

• To propose a feasible experimental test for the existence of non-linear self-gravitational interactions in the quantum regime.
• To address the lack of consensus on how to quantize gravity in the Newtonian limit, particularly within semi-classical approaches like the Schrödinger-Newton framework.
• To distinguish the predictions of the S-N model from standard quantum gravity approaches by examining spin-dependent phase shifts and entanglement behavior.
• To clarify the controversial role of amplitude-dependent scaling in self-gravitational potential and its impact on observable effects.
• To demonstrate that self-gravity induces a measurable phase shift in a single microsphere, while entangling power between two such particles vanishes in the S-N model.

Proposed method

• Model a mesoscopic spin-1/2 microsphere as a rigid, homogeneous sphere with mass m and radius R, using the Schrödinger-Newton equation to describe its self-gravitational interaction.
• Apply the Stern-Gerlach Humpty-Dumpty interferometer setup, where the microsphere's wave packet is split into up and down spin components by a spatially inhomogeneous magnetic field.
• Calculate the phase shift induced by the self-gravitational potential, which depends on the spatial overlap of the spin components and scales with |β+|2 − |β−|2, where β± are the spin amplitudes.
• Use the S-N potential V(r) = −Gm² / r to compute the non-linear self-interaction energy, which contributes to the total Hamiltonian and leads to dephasing.
• Perform numerical estimates of the phase shift using realistic parameters (e.g., m ~ 10⁻¹⁵ kg, R ~ 10⁻⁹ m, T ~ 0.1 s), showing the phase shift is within detectable range.
• Analyze the entangling power between two identical microspheres falling in parallel interferometers, showing it vanishes identically in the S-N model due to the absence of non-local gravitational entanglement.

Experimental results

Research questions

• RQ1 Can the self-gravitational interaction predicted by the Schrödinger-Newton equation produce a measurable phase shift in a single mesoscopic particle's spin state during free fall in a Stern-Gerlach interferometer?
• RQ2 How does the amplitude dependence of the self-gravitational potential affect the observable phase shift between spin components in a superposition state?
• RQ3 Does the semi-classical S-N model predict non-zero entanglement between two massive particles falling in parallel Humpty-Dumpty interferometers, as assumed in standard quantum gravity proposals?
• RQ4 What is the role of nuclear-scale self-interactions in masking or distorting the gravitational phase shift, and can they be neglected in mesoscopic experiments?
• RQ5 How does the S-N model's prediction for entangling power compare with standard quantum gravity models that assume no self-interaction?

Key findings

• The self-gravitational interaction in the S-N model induces a measurable phase shift between the up and down spin components of a single microsphere, proportional to (|β+|2 − |β−|2), which can be detected via spin tomography after recombination.
• The dominant contribution to the phase shift is the self-energy term, estimated as (6/5)(Gm²/ℏR)(T₅ − 2Tₛ), and is not canceled by constant terms as incorrectly claimed in prior work.
• Nuclear-scale self-interactions contribute only a small, negligible correction (ωₛ·t < 10⁻² for t < 10⁻² s), confirming that they do not mask the main gravitational effect.
• The entangling power of the S-N interaction between two identical microspheres in parallel interferometers is exactly zero, in stark contrast to standard quantum gravity models that predict non-zero entanglement.
• The electromagnetic analogy shows that semi-classical self-interaction models (like S-N) are incompatible with established quantum electrodynamics, where self-interaction is ruled out by spectroscopic data.
• The paper concludes that the S-N model leads to distinct, testable predictions—especially the absence of two-body entanglement—making it falsifiable through future experiments.