Gravity mediated entanglement between light beams as a table-top test of quantum gravity Stefan Aimet1 Hadrien Chevalier1 and M. S. Kim1

2025-05-06 0 0 456.68KB 15 页 10玖币

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Gravity mediated entanglement between light beams

as a table-top test of quantum gravity

Stefan Aimet1,*, Hadrien Chevalier1, and M. S. Kim1

1QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom

*stefan.aimet@gmail.com

ABSTRACT

Over the past century, a large community within theoretical physics has been seeking a uniﬁed framework for quantum gravity.

Yet, to date, there is still no experimental evidence of any non-classical features of gravity. While traditional experimental

proposals would usually require immensely challenging Planck scale experiments, recent table-top protocols based on low-

energy quantum control have opened a new avenue into the investigation of non-classical gravity. An approach that has

sparked high interest, both in terms of experimental feasibility and of theoretical implications, is the indirect witnessing of

non-classical gravity through the detection of its capacity to act as an entangling channel. Most discussions have been centred

on the entanglement generation between two gravitationally coupled massive systems. In this work, we instead examine the

entangling capacity of the gravitational interaction between two light pulses. We explain the main experimental and theoretical

advantages of having a photonic protocol, and lay out the steps leading to the determination of the entangling phase, using the

path integral formalism and linearised gravity. We establish a closed form formula for the entangling phase and provide an

estimated order of magnitude of the average photon number required for the generation of appreciable phase. Using statistical

analysis, we show how entanglement may be certiﬁed with lower phase signal.

Introduction

The two main pillars of theoretical physics, general relativity and quantum theory, have been remarkably successful. Experiments

have agreed with the predictions of the two theories in their respective domains to very high accuracy, but it is still an open

question how exactly to incorporate the two theories into a common framework, which is vaguely labelled as quantum gravity.

The hope is to give a more comprehensive account of gravity valid at very high energies or small distances

. Often general

relativity is described as the theory of the very big, and quantum theory as the theory of the very small scales. Yet, this is not

necessarily true, and there are situations where understanding both frameworks together is paramount. This would include

better insight into early universe cosmology and black holes as well as the uniﬁcation of all interactions

. The enterprise of

formulating such a theory has been a fertile ground for various models, for example string theory

3,4

, loop quantum gravity

causal set theory6and many others.

However, all these approaches towards a theory of quantum gravity have been unguided by experimental progress, as the

gravitational ﬁelds sourced by systems displaying quantum behaviour is mostly insigniﬁcant. Vice versa, quantum deviations to

the classical Newtonian or Einsteinian descriptions are completely insigniﬁcant in the regimes where gravitational effects have

importance. The amplitudes of the gravitational coupling and of known quantum couplings become comparable at the Planck

scale. Due to its remoteness from any current and future achievable energy levels, to say there is little hope for generating

on-demand Planck scale effects in particle accelerators is an understatement. Instead, the natural arena of Planck scale effects

consists of black holes and other naturally occurring high-energy cosmic events. Most conventional proposed experimental

tests

7,8

of quantum gravity include the evaporation of black holes, quantum gravity corrections to the cosmic microwave

background or perhaps tests on the discreteness of space. In recent years, much attention has been given to a completely new

approach involving table-top tests of gravity at low energies. At these low energies, many fundamental theories of quantum

gravity may share the same phenomenology. This means that table-top tests may not rely on testing a speciﬁc model of quantum

gravity, but they are often model-agnostic9.

The history of table-top test proposals claiming to probe quantum gravitational effects

9–16

along with surrounding discussions

and debates

17–25

is a short but rich one. Most notably, the experimental proposals by Bose et al.

and Marletto and

Vedral

introduced the idea of gravitationally coupling two massive systems that are both in spatially separated superposition

states. The gravity mediated entanglement (GME) is then understood as evidence for the quantum nature of gravity. This

implication originated ﬁrst from the quantum information theory framework by using the fact that local operations and classical

communication described by a mediator

cannot generate entanglement between any two systems

Q1,226,27

, and was later

corroborated by using a more general approach

. While there is dissent about the implications of the original protocol,

arXiv:2210.12713v2 [quant-ph] 19 Jan 2025

speciﬁcally regarding the conclusions one can draw from entanglement generation

17,19–22,29

, this experimental proposal using

gravitational cat states has generated a lot of interest in the community.

As innovative as it is, the original massive GME protocol still suffers from several experimental challenges and limitations,

some of which pertain to engineering and experimental capabilities, while others are intrinsic to the protocol. From the

point of view of experimental implementation, among the most critical obstacles, we may mention the coherent control of a

microdiamond of mass on the order of

10−14 kg

, which is several orders of magnitude beyond current cutting-edge quantum

control

30–32

. Furthermore, the mismatch between NV centres and centres of mass comes with an unwanted torque, the absence

of which anyhow does not eliminate the issue of free-rotation

. Another concern is the establishment of statistics which would

require particle recycling and implies potential overheating issues

34–38

. Finally, using matter to carry out the experiment comes

with the unavoidable presence of the Casimir effect

, which can only be dealt with by relying on even greater repeatability.

From a theoretical point of view, as argued in Ref.

, the original experiment with massive objects provides only a playground

to test the non-relativistic Newtonian regime of gravity but fails to recognise any potential features of gravity as a relativistic

quantum ﬁeld. This aspect is one of several that led to objections to the claim that the original massive GME protocol can

certify the non-classicality of gravity.

To resolve some of the challenges faced by the traditional GME protocol using massive objects, this paper suggests the use

of light beams as an alternative physical platform to realise GME. General relativity predicts that radiation, as much as matter,

sources a gravitational ﬁeld, the properties of which have been extensively studied

41–46

. The optical equivalent to the matter

Stern-Gerlach operation which performs a spin-dependent spatial splitting would be polarisation-dependent beam-splitting,

and analogously to the matter experiment where one measures a spin entanglement witness, we would look at a polarisation

entanglement witness.

An obvious advantage of the photonic GME protocol is the lack of any undesirable interactions other than gravity. The

effect of the direct, short-range photon-photon scattering

can be safely neglected as long as the beams are not overlapping,

and for low-energy photons. Thus, we shift the problem from discriminating between different relevant and competitive sources

to discriminating between background and signal. It is not hard to imagine why such a variant may be prohibitively challenging:

the phase signal due to the gravitational coupling of light is expected to be extremely minute. However, our analysis serves as a

ﬁrst attempt to quantify how challenging this approach may be. As stated previously, even if the entanglement generation may

be weaker than in the massive protocol - which by the way also requires unreasonably large masses for appreciable signal-

there is certainly a case for such an investigation in light of near-future experimental and technological advances. Modern

laser technology48,49 grants tunability and control of high-intensity light beams with an unprecedented capacity for empirical

repeatability. As we shall see, efﬁcient generation of large amounts of low-noise data will help alleviate the power requirements.

While a single photon may only have a negligibly small effect on gravity mediated entanglement due to its small coupling to

gravity, the collective effect and reliability of light as a source of entanglement generation may outscore any massive counterpart

as a platform for witnessing GME.

Using light also provides a natural framework for the investigation of both relativistic and quantum effects when testing

quantum gravity. Due to the easier tunability of frequency - as opposed to mass - ﬁner features of gravity and other deviations

from classicality may be more easily probed. Furthermore, light may be viewed as a more convincing candidate, insofar as

gravity can be conclusively inferred to be non-classical when GME experiments are performed within light-crossing times

between spatial branches

. For longer interaction times, classical ﬁelds can mediate entanglement as well without the need for

quantised gravitational degrees of freedom. Christodoulou et al

further showed that a fully local approach towards calculating

the gravitational phase using the path integral formalism (i.e. without assuming any instantaneous interactions such as in the

Newtonian limit) further strengthens the case for demonstrating the non-classicality of gravity. This manifestly local and gauge

invariant way of calculating the induced phases is, of course, the natural arena for considering relativistic light beams.

Incidentally, this work may also be seen as a pathway to demonstrate the gravitational coupling of light beams experimentally

for the ﬁrst time. The experimental veriﬁcation of gravitational coupling of light beams regardless of entanglement, would

by itself be an achievement. Various aspects of detecting the gravitational ﬁeld of light beams were considered in Ref.

Only recently, the role of light has also begun to become more popular in the research community for relativistic GME. For

example, the theoretical possibility of photon-matter entanglement was studied in Ref.

. Similarly, the interaction of photons

was investigated in Refs.

52,53

. In this work, we remain in the original double interferometer setup, and use the path integral

formalism for our calculations.

Our work is structured as follows. We begin by laying out the setup and introducing basic notions and approximations.

We give a brief overview of the path integral description of the experiment, and how one extracts a phase evolution. We also

present some instructive and well-established calculations on the metric perturbations sourced by a single circularly polarised

light pulse and introduce further helpful notations. Building on this basic situation, we construct the metric perturbation for two

counter-propagating pulses and derive the action for two circularly polarised counter-propagating, spatially separated light

pulses. After showing some numerical estimations of the gravitationally generated phase, we undertake a statistical analysis

2/15

of entanglement veriﬁcation, which drastically lowers the required beam power. We discuss some further directions worth

examining, such as improvements, further challenges, and more sophisticated models to describe the photonic GME protocol.

Results

Setup

Figure 1. The Mach-Zehnder interferometer setup of the photonic GME protocol. Two counter-propagating light pulses of

length Ltraverse a polarising beam splitter upon their entry into the interferometer of length Dwhich, for certain states,

entangles their polarisation degree of freedom with their spatial path. With the potential insertion of quarter waveplates each

beam splits into left- and right-handed circular polarisation. The two light pulses interact gravitationally. At the end of the

protocol, each spatial branch coincides again meeting at the second polarising beam splitter that disentangles, after the use of

waveplates, the polarisation from the spatial mode. The entanglement is then witnessed by local polarisation measurements.

Let us consider two Mach-Zehnder interferometers, as shown in Fig. 1, that receive two counter-propagating light pulses as

inputs. The length of each interferometer is

and the transverse separation of the two centres is

. Each input pulse enters

through a polarising beam splitter. The reason why we chose two counter-propagating light pulses is that co-propagating

non-overlapping pulses in vacuum do not interact43,54.

In general, the GME protocol consists of systems

(where

a=1,2

), that are both described by their spatial motion

⃗xsa

a(t)

which depends on a ﬁxed internal degree of freedom

, which may take on one of two values to allow for a system to be put

into a superposition state. In the massive case, this internal degree of freedom may correspond to the spin of a particle whereas

for light, we shall take it to correspond to its polarisation state. Here

|+/−⟩

denotes left/right-handed circular polarisation. The

internal conﬁguration between two of the four different branches is then described by a state

|σ⟩=Na|sa⟩

. After passing the

light pulses through the beam splitter separating light into horizontal or vertical polarisation, a quarter-waveplate may be used

to correspondingly have right-handed and left-handed circular components in output modes. At the end of the gravitational

phase induction, the light pulses pass again through quarter-waveplates before passing through a beam splitter to detect the

entanglement. Per deﬁnition, each beam splitter is assumed to transmit and reﬂect equally with no losses in our setup.

Eventually, after having successfully put each of the two light pulses in a spatial superposition correlated with their

polarisation degree of freedom, each branch will pick up a gravitationally generated phase. Even though light beams in coherent

states

55–58

may contain a high number of photons which allows us to source an appreciable metric perturbation, let us state

right now that they cannot be used for the GME protocol. Coherent states (with no squeezing) incident on a beam splitter do

not become entangled 59. As such, in the remainder of this work, we focus on states for which the beam splitter is entangling,

3/15

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Gravitymediatedentanglementbetweenlightbeamsasatable-toptestofquantumgravityStefanAimet1,*,HadrienChevalier1,andM.S.Kim11QOLS,BlackettLaboratory,ImperialCollegeLondon,LondonSW72AZ,UnitedKingdom*stefan.aimet@gmail.comABSTRACTOverthepastcentury,alargecommunitywithintheoreticalphysicshasbeenseekingauni...

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Gravity mediated entanglement between light beams as a table-top test of quantum gravity Stefan Aimet1 Hadrien Chevalier1 and M. S. Kim1

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