^{1}

^{*}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

Edited by: Caslav Brukner, University of Vienna, Austria

Reviewed by: Igor Pikovski, Stockholm University, Sweden; Bhupal Dev, Washington University in St. Louis, United States

This article was submitted to High-Energy and Astroparticle Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The Bose-Marletto-Vedral (BMV) experiment tests a quantum gravitational effect predicted by low energy perturbative quantum gravity. It has received attention because it may soon be within observational reach in the lab. We point out that: (i) in relativistic language, the experiment tests an interference effect between proper-time intervals; (ii) the feasibility study by Bose et al. suggests that current technology could allow to probe differences of such proper-time intervals of the order of 10^{−38} seconds, about twenty orders of magnitude beyond the current resolution of the best atomic clocks; (iii) the difference of proper times approaches Planck time (10^{−44} s) if the masses of the particles in the experiment approach the Planck mass (~micrograms). This implies that the experiment might open a window on the structure of time at the Planck scale. We show that if time differences are discrete at the Planck scale—as research in quantum gravity may suggest—the Planckian discreteness of time would appear as quantum levels of an in principle measurable entanglement entropy.

Bose et al. [

The Bose-Marletto-Vedral (BMV) effect is predicted by low energy perturbative quantum gravity, and hence by any approach to quantum gravity consistent with this low energy expansion, including string theory and loop quantum gravity. It is therefore plausibly real. If detected, it would provide indirect empirical evidence that spacetime geometry does obey quantum mechanics.

On the other hand, the BMV effect is insensitive to the limit

Here, we point out that a refinement of the BMV effect

where the experiment is performed with particles of mass _{Pl} is the Planck mass and _{Pl} the Planck time. This expression shows that the time scales probed are extremely small. With current technology, the BMV effect might be detected in the lab by probing relevant entanglement (generated when δϕ ~ 1) using mesoscopic particles, with masses of the order of a millionth of a Planck mass (

Now, it is often pointed out in quantum gravity research that the Planck time _{Pl} could be a minimal observable time; this follows from relativity plus the fact that many approaches to quantum gravity predict a minimal length [_{Pl}, with integer

That is, a discontinuity in δτ could be detected as a discontinuity in δϕ. As discussed below, such a discretization of the phase could be detected by the Bell–like correlations among the particles' spins, which would acquire a characteristic quantum band structure.

The extremely small time intervals probed by the current proposal to implement the BMV experiment are still too large to see time discreteness. But if the experiment can be pushed to work with more massive particles, further approaching the Planck mass, δτ will approach the Planck time [see (1)]. While the Planck time _{P} is at the—so far—deeply inaccessible scale

The analysis that follows is rough and the effect might be questioned by a more detailed investigation. It may turn out that the BMV apparatus does not measure eigenvalues but rather expectation values, or, that the scale of discreteness for differences in duration is actually smaller that Planckian. Nevertheless, a prospect of experimental access to the scale of the Planck time is so interesting to deserve full attention.

Let us start by describing the version of the BMV experiment of Bose et al. [

After a time

For simplicity, consider the case in which the two particles are kept at a small distance

where ^{iϕ} = ^{imc2τ/ℏ}. Therefore, after a (laboratory frame) time

with respect to the other branches. This equation is equivalent to Equation (1).

After the time

This is an entangled state. The amount of entanglement is measured by the entanglement entropy

where

the trace being on the spin states of one of the two particles. A quick calculation gives

This is correctly a hermitian matrix of unit trace. To compute the entropy we need to diagonalize ρ. A straightforward calculation gives the eigenvalues

When δϕ = 0, the eigenvalues are ρ_{+} = 1 and ρ_{−} = 0, thus giving vanishing entanglement entropy, i.e., there is no interference in the output. When δϕ = π, we have ρ_{+} = 1/2 and ρ_{−} = 1/2; the state is maximally entangled and

See

that follows from (4) and (5). The entanglement entropy can be measured by repeated spin measurements on the recombined particles. A specific method would be the following: for a given

The entanglement entropy for δϕ ∈ {0, 2π}.

Consider now the hypothesis that time is discrete at the Planck scale. We consider here the simplest possible ansatz: that

with a non negative integer _{P} with α a dimensionless positive real parameter, we have that the only values of ϕ that are actually realized are

that is, the phase ends up taking only discrete quantized values, when

The entanglement entropy for δϕ ∈ {0, 2π} under the assumption that

For particles with masses larger that the Planck mass interference is likely to disappear altogether, as is common in interference experiments when the wave frequency is much higher than the relevant scale of the apparatus. In this case wave theory goes to the Eikonal approximation. Wave mechanics goes to classical mechanics. The Compton frequency

of objects with mass larger than the Planck mass is formally larger than the Planck frequency

Notice that in this case an apparatus capable of detecting δϕ ~ 1 is going to be affected by genuine dynamical effects since we can also write

and if the left hand side and the first fraction are of order unit, so must be the second, with the consequence that the duration

The current hope is to realize the BMV experiment in the lab with masses ^{−19}

A relativistic language is not needed to derive the correlations that the BMV experiment is expected to detect. In the non-relativistic language no small time intervals are in play: instead of δϕ = ^{2}δτ/ℏ, the phase reads δϕ = ^{2} makes all the difference, where the relevant time

But, if time discreteness is detected, the non-relativistic language becomes insufficient to describe the relevant physics. Time discreteness, according to current tentative theories, is a genuine relativistic effect arising from quantum gravity. On the other hand, this study does not go beyond the observation that the scales probed are possibly close to the relevant scale for such (naïve) models. Here we do not justify or discuss in depth a possible discretization of time and its implications. The effect discussed here however is a relativistic effect, and we do not think it could be relevant for systems in Newtonian gravity. In contrast for instance to Muller et al. [

As mentioned in the introduction, the analysis given here assumes the discreteness of δτ in Planck time multiples. It is possible, but it is not certain that this is implied by quantum gravity. Two reasons that could question this assumption are the following. First, the spectrum of τ could be less trivial and, as a consequence,

Even with these caveats, the possibility that quantum interference effects could depend on time differences of the order of Planck time, a scale so far considered totally out of reach, definitely deserves attention.

All datasets generated for this study are included in the article/supplementary material.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Pierre Martin-Dussaud and Andrea di Biagio for valuable discussions. The authors acknowledge support from the kind donors to the SM Center for Space, Time and the Quantum. This publication was made possible through the support of the ID 61466 grant from the John Templeton Foundation, as part of the The Quantum Information Structure of Spacetime (QISS) Project (