Directly Measuring Gravitational Self-Interaction

This paper suggests a clever experimental method for investigating gravitational self-interaction and determining if the nature of gravity is quantum or classical. It might be possible with a very precise double slit experiment that could be performed in a fairly ordinary physics laboratory.

The Schrodinger-Newton equation has frequently been studied as a nonlinear modification of the Schrodinger 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 interference of massive particles is studied by numerically solving the Schrodinger-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.

Sourav Kesharee Sahoo, Ashutosh Dash, Radhika Vathsan, Tabish Qureshi, "Testing Gravitational Self-interaction via Matter-Wave Interferometry" arXiv:2203.01787 (March 3, 2022).

The introduction to the body text of the paper explains that:

The emergence of classicality from quantum theory is an issue which has plagued quantum mechanics right from its inception. Schr¨odinger equation is linear, and thus allows superposition of any two distinct solutions. However, in our familiar classical world, the superposition of macroscopically distinct states, e.g., the state corresponding to two well separated distinct positions of a particle, is never observed. Taking into account environment induced decoherence, one may argue that pure superposition states do not survive for long, and the interaction with the environment causes the off-diagonal elements of the system reduced density matrix to vanish. The remaining diagonal terms are then interpreted as classical probabilities. However, decoherence is based on unitary quantum evolution and if one tried to explain how a single outcome results for a particular measurement, one will eventually be forced to resort to some kind of many worlds interpretation. 

Another class of approaches to address this issue invokes some kind of nonlinearity in quantum evolution, which may cause macroscopic superposition states to dynamically evolve into one macroscopically distinct state. Different theories attribute the origin of the nonlinearity to different sources, for instance, an inherent nonlinearity in the evolution equation, or gravitational self-interaction. 

Considerable effort has been put into finding ways to test any non-linearity which may lead to the destruction of superpositions. For example, an experiment in space was proposed, which would involve preparing a macroscopic mirror in a superposition state. The problem with such experiments, even if they are successfully realized, is that it is difficult to rule out the role of decoherence in destroying the superposition. What is sorely needed, is an effect which can distinguish between the effect of nonlinearity of the Schr¨odinger equation and the effect of environment-induced decoherence. This is the issue we wish to address in this work. 

In 1984 L. Diosi introduced a gravitational self-interaction term in the Schr¨odinger equation in order to constrain the spreading of the wave packet with time. The resulting integro-differential equation, the Schr¨odinger-Newton (S-N) equation compromised the linearity of quantum mechanics but provided localized stationary solutions. It was R. Penrose who used the S-N equation to unravel quantum state reduction phenomenon. He proposed that macroscopic gravity could be the reason for the collapse of the wave function as the wave packet responds to its self gravity. The effect of gravity and self-gravity on quantum systems have been studied by several authors. 

The coupling of classical gravity to a quantum system also addresses the question of whether gravity is fundamentally quantum or classical. While theories of semiclassical gravity have faced several theoretical objections, the ultimate test would be experimental. In such a context, providing an experimental route to test the effect of S-N non-linearity in a simple quantum mechanical context is valuable. 

In the present work, we focus on evolution of a superposition state through the non-linear Schr¨odinger-Newton equation. Any signature of non-linearities due to the gravitational self-interaction in the variation of fringe width with mass should give us an experimental handle on separating the effect of decoherence from gravitational state reduction.