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# Using an atom interferometer to infer gravitational entanglement generation

Daniel Carney  Joint Center for Quantum Information and Computer Science/Joint
Quantum Institute, University of Maryland/NIST, College Park, MD 20742, USA
Fermi National Accelerator Laboratory, Batavia, IL 60510, USA  Holger Müller
Department of Physics, University of California, Berkeley, CA 94720, USA
Jacob M. Taylor  Joint Center for Quantum Information and Computer
Science/Joint Quantum Institute, University of Maryland/NIST, College Park, MD
20742, USA

###### Abstract

If gravitational perturbations are quantized into gravitons in analogy with
the electromagnetic field and photons, the resulting graviton interactions
should lead to an entangling interaction between massive objects. We suggest a
test of this prediction. To do this, we introduce the concept of interactive
quantum information sensing. This novel sensing protocol is tailored to
provable verification of weak dynamical entanglement generation between a pair
of systems. We show that this protocol is highly robust to typical thermal
noise sources. The sensitivity can moreover be increased both using an initial
thermal state and/or an initial phase of entangling via a non-gravitational
interaction. We outline a concrete implementation testing the ability of the
gravitational field to generate entanglement between an atomic interferometer
and mechanical oscillator. Preliminary numerical estimates suggest that near-
term devices could feasibly be used to perform the experiment.

##  I Introduction

If a particle is in a superposition of two locations, will its gravitational
field also be in a superposition, and can this field generate entanglement
with another system? This foundational question feynman1971lectures ;
page1981indirect  has received considerable attention diosi1984gravitation ;
penrose1996gravity ; kafri2015bounds ; bose2017spin ; Marletto:2017kzi ;
Haine:2018bwu ; Chevalier:2020uvv ; Howl:2020isj ; Anastopoulos:2020cdp ;
Matsumura:2020law ; carney2019tabletop . Proposed experimental tests to detect
entanglement due to gravity based on Bell tests (or more generally,
entanglement witnesses HORODECKI19961 ; terhal2000bell ) require performing
measurements on both subsystems and are challenging in practice. As a result,
there is still no direct experimental evidence as to whether gravitational
interactions generate entanglement. Here, we propose a test that only requires
observing a single subsystem feynman1963theory ; caldeira1983path ;
joos1985emergence ; zurek2003decoherence . We show that, if an interaction
(such as gravity) between two systems can cause both decoherence (collapse)
and recoherence (revival) of a subsystem, then for restricted classes of
systems the interaction is _necessarily_ capable of generating entanglement.
We propose a concrete implementation based on atom interferometry
kasevich1992measurement ; santarelli1999quantum ; gross2010nonlinear ;
xu2019probing , in which an atom in a superpositon of being in one of two
interferometer arms interacts with a low-frequency mechanical resonator
lee2020new ; catano2020high ; the signal for entanglement-generation is a
collapse and revival of the atomic interference fringes due to the periodic
motion of the resonator. The experiment does not require preparing a non-
classical state of the oscillator and can in fact be enhanced by placing the
oscillator in a thermal state, which appears to make this experiment feasible
with near-term devices.

The relation of such an experiment to the quantization gravity is a subject of
intense current study belenchia2018quantum ; Christodoulou:2018cmk ;
Marshman:2019sne ; Galley:2020qsf . These experiments operate in a regime
where the energy density (or equivalently, spacetime curvature), is far below
the Planck scale $\rho\ll m_{\rm Pl}/\ell_{\rm Pl}^{3}\sim 10^{123}~{}{\rm
eV}/{\rm cm^{3}}$. Thus the non-linearity of the gravitational interaction is
very weak, and one can treat the metric $g_{\mu\nu}$ as a linear perturbation
around flat spacetime. In this limit, one can quantize the gravitational
perturbations (“gravitons”) in exact analogy with quantum electrodynamics;
graviton exchange generates a two-body Newton potential operator

$V_{N}=-\frac{G_{N}m_{1}m_{2}}{|\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2}|}$
(1)

between a pair of masses, just as photons generate the Coulomb potential
Feynman:1963ax ; t1974one ; deser1974one ; Veltman:1975vx ;
donoghue1994general ; burgess2004quantum . We review some standard
demonstrations of this in appendix A. In equation (1),
$\mbox{\boldmath$x$}_{1,2}$ are the position _operators_ on a pair of masses,
and thus this interaction can generate entanglement. However, there are
dissenting opinions penrose1996gravity ; howl2019exploring ; Tilloy:2019hxe ;
Bruschi:2020xbm  about whether gravity should be quantized in this way, and
indeed one can produce models where classical gravitational interactions can
arise but without generating entanglement Kibble:1979jn ; kafri2014classical ;
Oppenheim:2018igd ; Kent:2020gov , providing substantial motivation to perform
tests of (1).

Figure 1: Implementation of the basic protocol using an atom interferometer
and a suspended pendulum (see Section VI). A trapped atom (labeled A) is
prepared some distance $L$ away from a mechanical resonator (B, here pictured
as a pendulum). The atom is then put into a superposition of two different
locations separated by $\ell$, effecting a Hadamard gate $H$. This generates a
state-dependent force between the atoms and resonator, leading to motion in
opposite directions for some time $\Delta t$. Finally, the atom state is
recombined using the inverse Hadamard gate and measured to check for
decoherence caused by the atom-mechanical interaction. When the resonator
undergoes a complete period of motion, its state no longer depends upon the
atoms and coherence is recovered for the interferometer.

The ability to test such a weak entanglement signal relies entirely on our
central technical result, a novel sensing protocol which we refer to as
_interactive quantum information sensing_. This is a detection scheme tailored
specifically to the verification of weak dynamical entanglement generation.
The traditional methods to detect entanglement in bipartite systems
$H_{A}\otimes H_{B}$ use non-local measurements HORODECKI19961 ;
terhal2000bell , and can be very difficult in practice with noisy systems and
weak entanglement. However, in the past two decades, more sophisticated
methods have been developed to address these types of problems
guhne2009entanglement ; pezze2018quantum . We suggest here a new protocol
which relies on time-dependent measurements on a single subsystem. Within
standard quantum mechanics, system $A$ will decohere—evolve from a pure to
mixed state—if it becomes entangled with another system $B$ which is not
measured feynman1963theory ; caldeira1983path ; joos1985emergence ;
zurek2003decoherence . This loss of coherence can be observed via an
interference measurement on $A$ alone. Simple decoherence could be explained
by entanglement but also by, for example, random classical noise
stern1990phase . However, if the same interaction can cause both decoherence
and recoherence of $A$, in a manner controlled by $B$, then for certain
classes of systems we prove that the interaction is necessarily capable of
generating entanglement between subsystems $A$ and $B$. This protocol provides
an indirect test of the quantum communication capabilities of the two systems,
and is a limited probe of the family of quantum channels associated with the
interaction between the two systems. The interplay between the information
theoretic channel properties and the physical interaction provides our
suggested nomenclature.

We outline the interactive sensing protocol in sections II and III. We find
the remarkable result that using an initial state at high temperature can
_increase_ the sensitivity of the protocol, because it can increase the rate
of entanglement generation and lead to a thermally-enhanced collapse and
revival signal. In section IV we demonstrate that this conclusion is robust to
typical sources of noise, essentially because the test does not involve
producing large superpositions of the non-observed subsystem. In section V we
show how to further enhance the protocol using pre-entangled initial
conditions. Finally, we outline an experimental realization with gravitational
entanglement generation between an atom interferometer and a mechanical
oscillator in section VI, before concluding with a discussion of implications
and loopholes in section VII.

##  II Collapse and revival dynamics

To begin, we illustrate the basic idea of the collapse-and-revival dynamics
with an example. The setup is similar to electron spin echo envelope
modulation rowan1965electron ; dikanov1992electron  and the cavity QED
experiments of Haroche _et al._ raimond2001manipulating . Consider an harmonic
oscillator $B$ coupled to a two-state system $A$ through the Hamiltonian

$H=\omega a^{\dagger}a+g(a+a^{\dagger})\sigma_{z}.$ (2)

In section VI we give an implementation of this Hamiltonian where the
oscillator $B$ is a mechanical resonator, the two-state system $A$ corresponds
to an atom located in one of two spatial locations, and $g\ll\omega$ is set by
the atom-oscillator gravitational interaction (1), so $g$ is proportional to
Newton’s constant $G_{N}$. The essential idea is to do an interferometry
measurement on the two-state system $A$ (the “control”) in the presence of
system $B$ (the “target”). The key is the dynamical response of the target
system $B$ to a superposition of $A$.

(a)  (b)  (c)

Figure 2: Equivalent circuit (left) and phase space (center) descriptions of
the experiment, and schematic interferometric data output (right). In the
circuit, the top line represents the atom and bottom line the resonator. The
large box represents joint evolution of the trapped atom and the resonator,
which can be decomposed into conditional displacement of the resonator,
followed by free evolution and an inverse displacement operator. This sequence
can be visualized in the phase space of the oscillator, where the solid and
dashed lines represent the two oscillator evolutions conditioned on the two
possible atomic locations. Interferometric measurement of the atom population
will show rapid fringes with frequency $\omega_{\rm DC}$ due to any stray DC
accelerations (e.g., due to electric fields, Earth’s gravity, or off-center
location of the resonator or atom), modulated by an overall reduction and then
increase due to the atom-resonator entanglement. Resonator motion over a full
period leads to nominal full recovery of the fringes.

To understand the entanglement dynamics generated by (2), it is useful to note
that the time evolution operator can be re-written

$U(t)=e^{-iHt}=D^{\dagger}\left(\sigma_{z}\lambda\right)e^{-i\omega
a^{\dagger}at}D\left(\sigma_{z}\lambda\right)$ (3)

up to an overall phase, where

$D(\alpha)\equiv\exp\left\\{\alpha a^{\dagger}-\alpha^{*}a\right\\}$ (4)

is the usual displacement operator.111To see this, note that
$D^{\dagger}(\alpha)a^{\dagger}aD(\alpha)=|a+\alpha|^{2}$ and expand the free
evolution operator $e^{-i\omega a^{\dagger}at}$ in the middle of (3). Here and
throughout, we will use the dimensionless quantity

$\lambda\equiv\frac{g}{\omega}.$ (5)

This is the length, measured in units of the zero-point length $x_{0}$, that
the oscillator equilibrium is displaced under the force from the atom. This
ratio will set the scale of all observables considered in this paper.

Observing the collapse-and-revival can be done with a typical interferometric
measurement. Consider starting the full system in its decoupled ground state
$\ket{0}_{A}\otimes\ket{0}_{B}$. The interferometry experiment then proceeds
by performing a Hadamard gate (or any other beamsplitter operation) on the
two-state system $A$, $\ket{0}\to(\ket{0}+\ket{1})/\sqrt{2}$, evolving the
joint system for some time $t$, performing the inverse Hadamard gate to
recombine the two-level system, and then measuring its population.
Mathematically, this proceeds as follows:

$\displaystyle\begin{split}\ket{\psi}&=\ket{0}_{A}\otimes\ket{0}_{B}\\\
&\xrightarrow{H}\frac{\ket{0}_{A}+\ket{1}_{A}}{\sqrt{2}}\otimes\ket{0}_{B}\\\
&\xrightarrow{U_{\rm
int}}\frac{\ket{0}_{A}\ket{\delta}_{B}+\ket{1}_{A}\ket{-\delta}_{B}}{\sqrt{2}}\\\
&\xrightarrow{H^{\dagger}}\ket{0}_{A}\frac{\ket{\delta}_{B}+\ket{-\delta}_{B}}{2}+\ket{1}_{A}\frac{\ket{\delta}_{B}-\ket{-\delta}_{B}}{2}.\end{split}$
(6)

Here, the conditionally-evolved states of the oscillator are simply coherent
states

$\ket{\pm\delta}_{B}=D\left(\pm\lambda(e^{-i\omega t}-1)\right)\ket{0}.$ (7)

If we now measure the two-state system $A$, we find for example that the
probability of being in the $\ket{0}$ state is

$P_{A}(0)=\frac{1}{2}+\frac{1}{2}{\rm
Re}\braket{\delta}{-\delta}_{B}=\frac{1}{2}\left(1+e^{-8\lambda^{2}\sin^{2}(\omega
t/2)}\right).$ (8)

We see that the interference term is reduced, with a period set by the
oscillator frequency $\omega$. In particular, at half-period we have a maximum
reduction of the phase contrast, and after a full period the contrast is
completely restored, as in Fig. 2.

Before moving on, we mention for later use an alternative calculation of the
same effect. Consider the Pauli lowering operator
$\sigma_{-}=(\sigma_{x}-i\sigma_{y})/2$ on the two-level system. The
expectation value $\braket{\sigma_{-}(t)}$ tracks the loss of phase contrast;
we will refer to the absolute value as the interferometric visibility
$V=|\braket{\sigma_{-}}|$. Using the time-evolution operator (3), we have

$\displaystyle\begin{split}\sigma_{-}(t)&=U^{\dagger}(t)\sigma_{-}U(t)\\\
&=D^{\dagger}(-\lambda)e^{i\omega
a^{\dagger}at}D(-\lambda)\sigma_{-}D^{\dagger}(\lambda)e^{-i\omega
a^{\dagger}at}D(\lambda)\\\ &=\sigma_{-}D(2\lambda(1-e^{i\omega
t})).\end{split}$ (9)

This is easy to show by working with explicit components in the $\sigma_{z}$
basis, where $\sigma_{-}=\ket{1}\bra{0}$. With an oscillator initially in the
ground state, this gives

$\braket{\sigma_{-}(\pi/\omega)}=\braket{\sigma_{-}(0)}e^{-8\lambda^{2}},\ \ \
\braket{\sigma_{-}(2\pi/\omega)}=\braket{\sigma_{-}(0)}.$ (10)

Here we see again the loss of phase contrast at half period followed by the
revival at a full period.

Up to this point, we have assumed that the oscillator was initialized in its
ground state $\ket{0}$. In a realistic implementation–particularly one where
the oscillator is a massive mechanical object–the oscillator will instead
start in a mixed state, such as a thermal state, due to its coupling to an
environment. Although one may be concerned that this would destroy the revival
of coherence in the atom, it turns out that not only does the revival persist,
but in fact the relative contrast between decoherence and revival is
_enhanced_ so long as the thermalization time scale remains very long. That
the revival persists is a consequence of the harmonic potential: after a full-
period, the state of the oscillator must return to its initial condition.

To see this, consider first the oscillator initialized to an arbitrary
coherent state $\ket{\alpha}$. Using (9), we have

$\displaystyle\begin{split}\braket{\alpha}{\sigma_{-}(t)}{\alpha}&=e^{-2\lambda[\alpha^{*}(1-e^{i\omega
t})-\alpha(1-e^{-i\omega t})]}\\\ &\times e^{-8\lambda^{2}\sin^{2}(\omega
t/2)}\braket{\sigma_{-}(0)}.\end{split}$ (11)

We see the complete revival after a full period, while at half period we now
pick up a phase involving the initial oscillator momentum
$p_{\alpha}=\alpha+\alpha^{*}$. To obtain the thermal-state result, one can
now average over the coherent states (i.e. use the oscillator density matrix
$\rho_{\rm th}=\int d^{2}\alpha
e^{-|\alpha|^{2}/\bar{n}}/(\pi\bar{n})\ket{\alpha}\bra{\alpha}$, with
$\bar{n}$ the thermal phonon occupancy). The result for the qubit visibility
is

$V_{\rm th}(t)=\exp\left[-8\lambda^{2}(2\bar{n}+1)\sin^{2}(\omega
t/2)\right].$ (12)

In particular, we have $V_{\rm th}(2\pi/\omega)=1$, showing a full revival of
the qubit coherence after a full oscillator period. On the other hand, at
half-period, we have $V_{\rm
th}(\pi/\omega)=\exp\left[-8\lambda^{2}\left(2\bar{n}+1\right)\right]$, an
enhancement to the loss of visibility by a factor of $\bar{n}$. Thus, starting
with a thermal state increases the contrast between the ‘dip’ of coherence
halfway through oscillation and the recovery at full oscillation. The
experiment is _easier_ with a hot oscillator.

##  III Revival verifies entanglement generation

As this example clearly shows, entanglement generation between two systems $A$
and $B$ can cause periodic collapse and revival of $A$’s wavefunction. The
crucial question is then: does observation of this collapse and revival
_necessarily_ require entanglement generation between $A$ and $B$? Our central
result says that the answer is yes, under some particular assumptions. We
characterize this with a theorem:

###### Theorem 1

Let $L$ be a channel on $H_{A}\otimes H_{B}$, where $H_{A}$ is a two-state
system and $H_{B}$ is arbitrary. Assume that

  1. (a)

The channel $L$ generates time evolution, in a manner consistent with time
translation invariance, thus obeying a semigroup composition law $L_{t\to
t^{\prime\prime}}=L_{t\to t^{\prime}}L_{t^{\prime}\to t^{\prime\prime}}$ for
all $t\leq t^{\prime}\leq t^{\prime\prime}$,

  2. (b)

The two-level subsystem $H_{A}$ has its populations preserved under the time
evolution, $\sigma_{z}(t)=\sigma_{z}(0)$, and

  3. (c)

$L$ is a separable channel rains1999rigorous : all of its Krauss operators are
simple products. In particular, this means that any initial separable (non-
entangled) state evolves to a separable state: $\rho(t)=L_{t}[\rho(0)]$ is
separable for all separable initial states $\rho(0)$.

Then the visibility $V(t)=|\braket{\sigma_{-}(t)}|$ is a monotonic function of
time.

Here, we have modeled the time evolution of the $A$-$B$ system as a quantum
channel $L$, a map on density matrices $\rho(t)=L_{t}[\rho(0)]$. For example,
within standard quantum mechanics, the unitary evolution of the universe
($A,B$ and their environment $C$, including the experimentalist) generates
such a channel for the reduced $A-B$ evolution. Suppose that we can
experimentally convince ourselves that time-translation invariance in the form
(a) and population condition (b) hold. Then the theorem says that if $L$
cannot generate entanglement (c), then the only possible evolution for the
qubit $A$ is to have its interferometric visibility decay monotonically. Thus
if we observe non-monotonic visibility like the oscillatory signal described
above, we can conclude that the channel must be capable of generating
entanglement.

We note that non-entangling channels still allow for non-trivial interactions.
For example, semiclassical gravity $G_{\mu\nu}=8\pi\braket{T_{\mu\nu}}$
(appropriately completed by a modified version of the Schrödinger equation) is
of this form kafri2015bounds . On the other hand, the graviton model will
produce an entangling channel.

We now give a proof of this theorem. By assumption (a), there exists a
generator $\mathcal{L}$ of $L_{t}$ of Lindblad form lindblad1976generators ;
gorini1976completely :

$\dot{\rho}=\mathcal{L}\rho=-i[H,\rho]-\sum_{j}\gamma_{j}\left[E_{j}^{\dagger}E_{j}\rho+\rho
E_{j}^{\dagger}E_{j}-2E_{j}\rho E_{j}^{\dagger}\right].$ (13)

These Lindblad operators $E_{j}$ are highly constrained by the separability
assumption, because they cannot be used to generate $A-B$ entanglement. To
make this precise, we write the channel in its Krauss representation
$L[\rho]=\sum_{j\geq 0}L_{j}\rho L_{j}^{\dagger}$. Expanding for small times
and comparing to (13), one finds that the Krauss operators $L_{j}$ take the
form, to lowest order in $dt$,

$L_{0}=1-iHdt+Kdt,\ \ L_{j}=E_{j}\sqrt{dt},\ \
K=-\frac{1}{2}\sum_{j>0}E_{j}^{\dagger}E_{j}.$ (14)

See, for example, chapter 3 of preskill1998lecture . Now we invoke the
separability criterion (c), which says that the Krauss operators for $j>0$
take the form of simple product operators, i.e. $E_{j}=A_{j}\otimes B_{j}$
rains1999rigorous . Furthermore, the separability of $L_{0}$ to order $dt$
means that $L_{0}=(1+A_{0}dt)\otimes(1+B_{0}dt)$ for some $A_{0},B_{0}$, and
this can only be satisfied if both $H$ and $E_{j}^{\dagger}E_{j}$ can be
written as sums of operators acting either on $\mathcal{H}_{A}$ _or_
$\mathcal{H}_{B}$. This in turn requires that for each $j>0$, either
$A_{j}^{\dagger}A_{j}=1_{A}$ or $B_{j}^{\dagger}B_{j}=1_{B}$. Finally, we
impose the requirement (b) that the atom populations are invariant. This means
that $\dot{\sigma}_{z}=0$. The only possible non-trivial interaction term
which satisfies these requirements is $E_{z}=\sigma_{z}\otimes B$, with $B$
any operator on $H_{B}$.

We are then left with the the very simple form of the Lindblad generator:

$\mathcal{L}\rho=-\gamma\left[B^{\dagger}B\rho+\rho
B^{\dagger}B-2B\sigma_{z}\rho\sigma_{z}B^{\dagger}\right]+\mathcal{L}_{A}+\mathcal{L}_{B}.$
(15)

Here $\mathcal{L}_{A(B)}$ are Lindblad operators (including Hamiltonians)
acting only on $\mathcal{H}_{A(B)}$, and $\mathcal{L}_{A}(\sigma_{z})=0$. With
this result for the channel’s structure, we can compute the time derivative of
the interferometric visibility $V(t)=|\braket{\sigma_{-}(t)}|$. Since
$[H,\sigma_{z}]=0$, the most general qubit Hamiltonian is a sum of
$\sigma_{z}$ and the identity. We thus have, in the Heisenberg picture,

$\displaystyle\begin{split}\Braket{\frac{d\sigma_{-}}{dt}}&=-i\braket{[H,\sigma_{-}]}+\gamma\left[\braket{E_{z}^{\dagger}\sigma_{-}E_{z}}-\frac{1}{2}\braket{\left\\{E_{z}^{\dagger}E_{z},\sigma_{-}\right\\}}\right]\\\
&=2(-i\omega_{0}-\gamma)\braket{\sigma_{-}},\end{split}$ (16)

where the oscillatory term is generated by the qubit Hamiltonian. Taking the
absolute value to compute the visibility $V=|\braket{\sigma_{-}}|$ removes the
oscillating phase and we have

$\frac{dV}{dt}=-2\gamma V,$ (17)

so it is monotonically decreasing, as we set out to prove.

##  IV Effects of noise during evolution

The sensing protocol is subject to errors caused by random noise during the
time evolution. In a typical realization, the dominant sources of this
continuous noise will consist of thermal load on the oscillator and dephasing
in the atomic system (from, e.g., background fields and gas weakly measuring
the atomic position joos1985emergence ; gallis1990environmental ). These
sources of noise can be modeled by a Lindblad evolution of the form

$\dot{\rho}=-i[H,\rho]-\sum_{i}\frac{1}{2}\\{L_{i}^{\dagger}L_{i},\rho\\}-L_{i}\rho
L_{i}^{\dagger},$ (18)

where the error operators are
$L_{i}\in\\{\sqrt{\bar{n}\gamma_{m}}a^{\dagger},\sqrt{(\bar{n}+1)\gamma_{m}}a,\sqrt{\gamma_{a}}\sigma_{z}\\}$.
The decay rates of the oscillator and atom are $\gamma_{m},\gamma_{a}$,
respectively, and $\bar{n}$ is the thermal phonon occupancy. This description
should be accurate for times similar to or shorter than the damping time
$1/\gamma_{m}$, and assuming only small changes over time in the mechanical
frequency.

It is possible to analytically solve for the atomic visibility
$\eqref{visibility}$ in the presence of this noise, using an explicit Ohmic
heating model where the bath is taken to be an infinite set of bosonic modes
linearly coupled to the mechanical system. The same displacement-operator
picture used in (3) generalizes to this linear bath (see appendix B). One
finds that the visibility at half and full-period evolution is given by

$\displaystyle\begin{split}V(\pi/\omega)&=\exp[-\pi\gamma_{a}/\omega]\exp[-8\lambda^{2}(2\bar{n}+1)]\\\
V(2\pi/\omega)&=\exp[-2\pi\gamma_{a}/\omega]\exp[-8\lambda^{2}(2\bar{n}+1)/Q].\end{split}$
(19)

Here we have assumed the mechanical damping factor $Q=\omega/\gamma_{m}\gg 1$.

This recovers the previous result for the visibility (12), up to an overall
exponential damping from the atomic dephasing and small correction from
mechanical heating. Neglecting atomic dephasing, the visibility at half period
is exactly the same as (12), while at full period, for $Q\gg
1\gg\bar{n}\lambda^{2}$, we have $V(2\pi/\omega)\approx 1$, i.e. we have full
recovery up to a correction at order $1/Q$. Thus, with a sufficiently high-$Q$
oscillator, and with atomic coherence times longer than the mechanical period
$\gamma_{a}\lesssim\omega$, damping does not pose a substantial barrier to the
experiment.

Before moving on, we consider the effects of decoherence from another
inevitable source: blackbody radiation of the oscillator. Here we are
discussing position superpositions of the oscillator at distances of about
$\lambda x_{0}$. With the sorts of experimental parameters we suggest later,
this will be a length many orders of magnitude smaller than a typical
blackbody photon wavelength (or ambient gas molecule’s de Broglie wavelength).
Thus these interactions will be incapable of efficiently decohering the
oscillator, because they are too long-wavelength to efficiently measure the
oscillator’s position gallis1990environmental .

##  V Protocol linear in the weak coupling

Figure 3: Left: Experimental realization of the atomic system as a lattice
interferometer. The lines marked “x” denote populations that do not interfere.
Right: Some example implementations with one or more mechanical masses
connected rigidly. Small black dots represent the atom. In each case, the
mechanical system is restricted to oscillate along the $z$-axis. More masses
enable a stronger gravitational coupling. A natural limiting case would be to
use a toroidal mass. In the example with a single sphere, we have
$R=\sqrt{L^{2}+(\ell/2)^{2}}$ and $\kappa=1$.

Our basic observable (8) is quadratic in the ratio $\lambda=g/\omega$, which
for a weak coupling is a small dimensionless number. Here we suggest a
“boosted” method in which linear sensitivity can be achieved by first
preparing an _entangled_ state of the atom-oscillator system (as demonstrated,
for example, in karg2020light ; thomas2021entanglement ).

Let $\lambda^{\prime}=g^{\prime}/\omega$, where $g$ is the coupling of
interest (e.g., gravity) and $g^{\prime}$ is some other coupling. Consider
performing a $\pi$ gate with the coupling $V_{\rm
int}=(g+g^{\prime})\sigma_{z}x$. This will produce an initial entanglement set
by displacement operators $D(\pm(\lambda+\lambda^{\prime}))$, as in equation
(9). Turning off the non-gravitational $g^{\prime}$ coupling then leads to
only a partial revival of the atomic signal at later times $t>\pi/\omega$.
This leads to the visibility, for $t>\pi/\omega$,

$V_{\rm b}(t)=\exp\left[-8(2\bar{n}+1)\left(\lambda^{\prime
2}+2\lambda\lambda^{\prime}\sin^{2}\frac{\omega
t}{2}+\lambda^{2}\sin^{2}\frac{\omega t}{2}\right)\right].$ (20)

A detailed calculation is given in appendix (C). For times $0<t<\pi/\omega$,
the visibility is given by the previous result (12) but with
$\lambda\to\lambda+\lambda^{\prime}$.

The observable we are interested in is the difference in visibility at half-
period and full-period:

$\displaystyle\begin{split}\Delta V_{\rm
b}&=V_{b}(2\pi/\omega)-V_{b}(\pi/\omega)\\\
&\approx\exp[-8(2\bar{n}+1)\lambda^{\prime
2}](1-16(2\bar{n}+1)\lambda^{\prime}\lambda+O(\lambda^{2})),\end{split}$ (21)

assuming $\lambda\ll\lambda^{\prime}$. We see again that using an initially
“hot” resonator increases the relative visibility. However, here the
observable is _linear_ in the weak gravitational coupling $\lambda$. We note
that if $\bar{n}$ or $\lambda^{\prime}$ are too large, the signal will be
destroyed by the overall prefactor $e^{-8(2\bar{n}+1)\lambda^{\prime 2}}$. The
optimal solution is to tune the non-gravitational coupling to satisfy
$\lambda^{\prime}_{\rm opt}=1/\sqrt{8(2\bar{n}+1)}$, in which case the
prefactor is order one, and the relative visibility is given roughly by
$\Delta V_{b}\approx\sqrt{8(2\bar{n}+1)}\lambda$. Use of this boosted protocol
substantially improves the viability of an experiment with a weak coupling
$g$. We note that this protocol does not violate our assumptions about time-
translation invariance in Theorem 1: once the extra $g^{\prime}$ coupling is
turned off, the entire system proceeds in a time-independent fashion.

##  VI Experimental implementation with atom interferometry

Figure 4: Logarithm $\log_{10}$ of the visibility change $\Delta V$ as
function of the logarithms of the hold time $\tau$ in seconds and the
temperature $T$ in Kelvin for the unboosted scheme (left) and the boosted
scheme (right). The plots assume $\ell=1\,$mm, $\rho=20$g/cm3, $m=m_{\rm Cs}$.

We now show how to apply our sensing protocol to a test of quantum gravity.
The idea is to realize the qubit in the Hamiltonian (2) as an optical-lattice
atom interferometer xu2019probing  with a hold-time $\tau$ and splitting
$\ell$ between the matter wave packets. The majority of the interferometer
time sees the atoms trapped in one of two different potential wells created by
the lattice. The atom position thus becomes a two-state system with
$\sigma_{z}$-eigenvalues corresponding to the two locations. The mechanical
oscillator has a mass $M$ and fundamental frequency $\omega$. Expanding the
Newtonian atom-oscillator potential (1), we then have the total Hamiltonian

$H=\omega a^{\dagger}a-g\sigma_{z}(a+a^{\dagger}).$ (22)

Here, $a,a^{\dagger}$ are oscillator operators, so the second term represents
the position-position coupling. The coupling strength is

$g=\kappa\frac{G_{N}mM\ell x_{0}}{\hbar R^{3}}$ (23)

where $x_{0}=\sqrt{\hbar/2M\omega}$ is the ground state oscillator
uncertainty, $R$ parametrizes the distance between the oscillator and atom,
and $\kappa$ is a dimensionless number of order one which depends on the
specific oscillator mass geometry (see Fig. 3).

Technical challenge |  Examples |  Possible strategies
---|---|---
Non-gravitational interactions |  Van der Waals, stray fields, scattered laser light |  Superconducting shielding, place atoms in waveguide Xin2018fiber
Mean field shift |  Parasitic atom-atom interactions leading to inhomogeneous dephasing Jannin2015Meanfield  |  Spin-echo techniques Laudat  (see also appendix E), fermionic atoms (e.g., Yb-171 or 173) McAlpine ; Niederriter
Exponential decay of signal |  Atomic dephasing |  Interleaved differential measurement, e.g. by toggling the mass between near and far positions Jaffe2017subgrav
Deviations from harmonicity |  Time-dependent oscillator frequency, anharmonic perturbations |  Keep effective temperature below nonlinear thresholds; change materials, mounting, or frequency
Table 1: Some systematic effects and other perturbations expected in a realistic implementation. Loophole or pathology |  Typical sources |  Problematic behavior allowed |  Possible solutions
---|---|---|---
Non-gravitational interactions between atom and oscillator |  Casimir/van der Waals interactions |  Can generate entanglement (reproduce the full desired signal), can generate extra noise |  Vary parameters (masses and distance) to check proper scaling with $V=G_{N}m_{1}m_{2}/r$ law
Stationarity assumption on bath (and/or experimentalist) violated |  Explicit time-dependence introduced by experimentalist (e.g. spin-echo protocol); low-frequency noise (e.g., gravity gradients, seismic noise) |  Violates assumption of theorem in Sec. III. In principle, could mimic collapse and revival |  Adjust theorem to allow for bath relaxation timescale; experimentally verify Markovian nature of oscillator noise
Non-locality |  Time of interaction for experiment is much longer than light-crossing time $T_{\rm int}\gg T_{com}$ |  Allows for non-local, hidden variable model explaining the entanglement (same as Bell test) |  Long baseline version?
Table 2: Some loopholes and pathologies in our proposed test.

The information sensing protocol requires generation of an initial state
$\ket{0}+\ket{1}$. This can be generated, e.g., by a pair of Raman pulses
separated by a free evolution time xu2019probing , by spin-dependent kicks
Jaffe2018SDK , by optical lattice techniques Pagel2020lattice , or by rapidly
splitting a single-well potential to the double-well. Measuring in the
$\sigma_{z}$ basis at the end of the protocol corresponds to closing the atom
interferometer and counting the atoms in the two output ports. To implement
the “boosted” protocol of section V, we can use a number of non-gravitational
interactions to generate the initial entanglement. For example, a hyperfine or
Rydberg atomic state could be magnetically or optically coupled to the
oscillator. Entanglement of this type has been recently demonstrated
experimentally karg2020light ; thomas2021entanglement .

Let us consider how we can obtain a visibility change that is large enough to
be measured. In order to observe at least one full cycle of decay and revival,
we choose $\omega=2\pi/\tau$ where $\tau$ is the atom hold-time. In this case,
the visibility change is given by

$\displaystyle\begin{split}\Delta V&=\frac{\pi
G_{N}^{2}m^{2}\rho}{3\sqrt{2}\ell\omega^{3}\hbar}\left(8+\bar{n}\right)\xrightarrow[k_{\rm
B}T/\omega\to\infty]{}\frac{\pi}{3\sqrt{2}}K^{2},\\\ \Delta V_{\rm
b}&=2^{1/4}G_{N}m\sqrt{\frac{\pi\rho}{3\ell\omega^{3}\hbar}(8+\bar{n})}\xrightarrow[k_{\rm
B}T/\omega\to\infty]{}\frac{2^{1/4}}{\sqrt{3}}K\end{split}$ (24)

in the unboosted and boosted scheme respectively, where

$\displaystyle\begin{split}K^{2}&=\frac{G_{N}^{2}m^{2}\rho
k_{B}T}{\ell\omega^{4}\hbar^{2}}\\\ &\approx 1.04\times
10^{-14}\left(\frac{T}{300\,{\rm K}}\right)\left(\frac{\ell}{1\,{\rm
mm}}\right)^{-1}\left(\frac{\tau}{10\,{\rm s}}\right)^{4}.\end{split}$ (25)

Here we took a solid density $\rho=20~{}{\rm g/cm}^{3},$ cesium atoms
$m=m_{\rm Cs}=133~{}{\rm amu}$, used the four-sphere configuration (Fig. 3)
for definiteness, and maximized the coupling $g$ for a given splitting $\ell$
by choosing a sphere radius of $R_{s}=\ell/(\sqrt{8})$. Longer atomic
interrogation times $\tau$ are preferable. This would require a
correspondingly low-frequency oscillator, e.g. a mHz-scale torsional pendulum.
While 20 s have been experimentally realized xu2019probing , 100 s may be a
reasonable expectation for the future. Using a small matter-wave splitting
$\ell$ is desirable, but subject to mechanical constraints. Choosing, e.g.,
$\ell=1\,$mm, $L=1/\sqrt{2}\,$ mm and $R_{s}=0.35\,$mm would leave about
$0.15\,$ mm free space between the spheres. For $\tau=100\,$s and $T=300$ K we
obtain $\Delta V\sim 10^{-10}$; but for the boosted scheme, it will be as
large as $\Delta V=7\times 10^{-6}$ (see Fig. 4). At the standard quantum
limit, this can be detected with $5-\sigma$ significance by running the
experiment with $\sim 5\times 10^{11}\,$ atoms (see appendix D for details on
noise scaling with many atoms). Assuming that the experiment has $10^{7}$
atoms per run, and each run takes 2 minutes, this will be possible in two
months total run time.

Remarkably, this suggests that the experiment may be feasible in the near
future. A number of systematic effects and technical issues will need to be
understood. We postpone detailed discussion to future work, but flag some
likely issues and ways to handle them in Table 1.

##  VII Implications, loopholes, and conclusions

Our interactive information sensing protocol is a novel strategy for
verification of dynamical entanglement generation. While a standard Bell-type
test requires measurements on both parts of a bipartite system, our protocol
can verify entanglement generation with only single-body measurements.
Crucially, the test verifies the ability of an interaction channel to generate
entanglement, without needing to directly verify the entanglement of the final
state. However, it is important to note that this test is subject to
loopholes. Some are analogous to those in standard Bell tests and others are
particular to our proposal. We suggest a few of these in Table 2.

In our view, the most important loophole stems from our time-translation
invariance assumption, which we used to write the atom-oscillator dynamics in
Lindblad form (13). Non-Markovian time dependence introduced by an
experimentalist or Maxwell’s demon could, in principle, reproduce the observed
collapse and revival dynamics. One way to improve the situation would be to
reformulate the theorem to include some level of non-Markovianity, for example
a bath relaxation time scale. A more robust option would be to prove
experimentally that it is simply the Markovian thermalizing channel acting on
the mechanical system. Methods for this include precision quantum thermometry
purdy2017quantum , which can support the hypothesis of detailed balance. In
any case, extending the results here beyond the strictly Markovian assumption
will be a crucial next step.

The central technical advances suggested here are the interactive sensing
protocol and the use of atoms as a sensor. The key advantage of the periodic
collapse-and-revival protocol is that it enables a huge enhancement with a
thermal state of the mechanical system; understanding if this can be extended
beyond the specific context here would be very interesting. While using
trapped atoms is perhaps counter-intuitive since it decreases the strength of
the signal (the Newton potential), we emphasize that the extremely long
coherence lifetime and ability to generate spatially well-separated
superpositions of the atoms lead to similar parametric scaling of the overall
signal strength.

We have shown how the interactive sensing protocol can be used to test the
ability of the gravitational field to communicate quantum information. If the
answer is yes, this would constitute the first direct evidence that the
gravitational field itself is a quantum mechanical degree of freedom
carney2019tabletop ; belenchia2018quantum ; Christodoulou:2018cmk ;
Marshman:2019sne ; Galley:2020qsf . On the contrary, if the answer is
negative, the existence of the graviton is ruled out carney2019tabletop . The
simple estimates of section VI suggest that this experiment is feasible with
realistic devices, even in the presence of noise. We will present a more
detailed proposal and analysis of systematic effects in a future paper.

## Acknowledgements

We thank Thomas Guff, Jack Harris, John Kitching, and Jess Riedel for
discussions. H.M. has been supported in part by Jet Propulsion Laboratory
under grant number 1652036, the Office of Naval Research under grant numbers
GRANT12980618 and N00014-20-1-2656, as well as the National Science Foundation
under grant number 1708160.

##  Appendix A Newtonian entanglement from graviton exchange

For completeness, we review here some standard arguments about the
perturbative quantization of gravity and its relation to entanglement
generation via the Newton potential (1). Our goal is to explain the standard
logic by which one treats small fluctuations of the metric as a quantum field
and uses this to make predictions in non-relativistic systems. We do not mean
to say that this derivation somehow proves that this is the correct model of
low-energy quantum gravity—on the contrary, determining if this is the correct
set of predictions is a central objective of the experiment proposed in this
paper.

By far the most common and efficient method to compare a field theoretical
description to the non-relativistic setting relevant to these experiments is
to do a “matching” calculation. For example, one can compute scattering
amplitudes in the field theory, compare these to the same amplitude computed
in a potential scattering model, and thus obtain the effective non-
relativistic potential. Since the scattering states form a complete basis for
the Hilbert space (other than bound states), if these two calculations agree
for all scattering states, we can conclude that the two descriptions are
equivalent quantum-mechanically in the regime in which the calculations match.

Figure 5: Feynman diagrams for single photon and graviton exchange,
respectively.

For example, consider electrodynamics: photons $A_{\mu}$ coupled to a Dirac
fermion $\psi$ (e.g., the electron, with charge $e$). See section 4.8 of
peskin2018introduction  for a textbook treatment of what follows. The
Lagrangian is

$\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\gamma^{\mu}(\partial_{\mu}-ieA_{\mu})\psi$
(26)

with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the
electromagnetic field tensor and $\gamma^{\mu}$ the usual Dirac matrices. With
this Lagrangian we can perturbatively compute the elastic $e^{-}e^{-}\to
e^{-}e^{-}$ scattering amplitude to lowest order in the charge using standard
methods. See the Feynman diagram of figure 5. One finds222There is also a
$u$-channel diagram coming from the indistinguishability of the two electrons,
which will also appear in the potential scattering computation, but this is
not important for our argument here so we drop it for simplicity.

$\braket{\mbox{\boldmath$p$}_{1}^{\prime}\mbox{\boldmath$p$}_{2}^{\prime}}{T}{\mbox{\boldmath$p$}_{1}\mbox{\boldmath$p$}_{2}}=4\pi
e^{2}\frac{N_{\rm EM}}{-t-i\epsilon}$ (27)

where $t=-(p_{1}^{\prime}-p_{1})^{2}$ is the Lorentz-invariant four-momentum
transfer, and the numerator is

$N_{\rm
EM}=\bar{u}(\mbox{\boldmath$p$}_{1}^{\prime})\gamma^{\mu}u(\mbox{\boldmath$p$}_{1})\bar{u}(\mbox{\boldmath$p$}_{2}^{\prime})\gamma_{\mu}u(\mbox{\boldmath$p$}_{2}).$
(28)

Here $u,\bar{u}$ are Dirac spinors. In the non-relativistic limit where the
rest masses dominate over the spatial momenta, the numerator reduces to
$N_{\rm EM}\to 1$ and
$t\to(\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1})^{2}$. In the
center-of-mass frame, this means that the amplitude for the relative momentum
$\mbox{\boldmath$p$}=(\mbox{\boldmath$p$}_{1}-\mbox{\boldmath$p$}_{2})/2$ to
transition to $\mbox{\boldmath$p$}^{\prime}$ is given by

$\braket{\mbox{\boldmath$p$}^{\prime}}{T}{\mbox{\boldmath$p$}}=\frac{4\pi
e^{2}}{(\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$})^{2}},$ (29)

where we used conservation of the total momentum to write
$\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}=\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$}$.

We can then compare this to the amplitude computed non-relativistically with a
Hamiltonian description

$H=H_{1}+H_{2}+V(r)$ (30)

including a central potential
$V(r)=V(|\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2}|)$. In the center of
mass frame, the first Born approximation gives

$\braket{\mbox{\boldmath$p$}^{\prime}}{T}{\mbox{\boldmath$p$}}=\tilde{V}(\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$}),$
(31)

where $\tilde{V}$ is the Fourier transform of the potential. Comparing these
expressions, we see that the effective potential is

$\tilde{V}(\mbox{\boldmath$q$})=\frac{4\pi e^{2}}{\mbox{\boldmath$q$}^{2}},$
(32)

or in position space,

$V(\mbox{\boldmath$x$})=\int\frac{d^{3}\mbox{\boldmath$q$}}{(2\pi)^{3}}e^{i\mbox{\boldmath$q$}\cdot\mbox{\boldmath$x$}}\tilde{V}(\mbox{\boldmath$q$})=\frac{e^{2}}{|\mbox{\boldmath$x$}|},$
(33)

recovering the usual Coulomb force.

We emphasize that this line of argument is at the level of transition
_amplitudes_ $\braket{\psi_{f}}{U(t_{f},t_{i})}{\psi_{i}}$, where $U$ means
time evolution and we take $t_{i}\to-\infty,t_{f}\to+\infty$. Thus, the
potential (33) is the potential operator in the full meaning of the term—it is
a two-body operator and can entangle particles, and so forth. Indeed,
entanglement generated by this Coulomb potential is exactly what underlies,
for example, quantum information processing with chains of trapped ions in
Coulomb crystals.

The same type of matching calculation can be performed in gravity. The
additional complication is that we need to work perturbatively around a
background spacetime, and only quantize the perturbations. For an experiment
well-localized in spacetime like ours, we can expand around a locally flat
spacetime metric $\eta_{\mu\nu}$, by the equivalence principle.333A more
accurate statement would be that we should expand around a Schwarzschild
solution, with Schwarzschild mass given by the mass of Earth. This term would
generate a constant background external potential and is inessential to the
core quantum mechanics of the following argument. We write

$g_{\mu\nu}=\eta_{\mu\nu}+\frac{h_{\mu\nu}}{m_{\rm Pl}}$ (34)

where we scale out a factor of the Planck mass to give $h_{\mu\nu}$ dimensions
of mass (i.e. the same dimensions as a canonical bosonic field in four
dimensions). In this expansion, the Einstein-Hilbert Lagrangian becomes

$\mathcal{L}=\frac{1}{2}\partial_{\alpha}h_{\mu\nu}\partial^{\alpha}h^{\mu\nu}+\cdots+\frac{1}{m_{\rm
Pl}}h_{\mu\nu}T^{\mu\nu},$ (35)

where the dots represent other similar second-derivative kinetic terms for the
perturbations $h_{\mu\nu}$ and $T_{\mu\nu}$ is the matter stress tensor. These
metric perturbations $h_{\mu\nu}$ can be quantized through the exact same
procedure as the electromechanical potential $A_{\mu}$ Feynman:1963ax ;
t1974one ; deser1974one ; Veltman:1975vx . There is a key difference—namely
that the interaction is non-renormalizable—which means that we have an
effective quantum field theory kadanoff1966scaling ; wilson1971renormalization
; weinberg1979phenomenological ; donoghue1994general ; burgess2004quantum
which can only make reliable predictions at energy densities well below the
Planck scale.444To be more precise: the non-renormalizable nature of the
interaction means that we have to include all possible generally-covariant
terms, in particular curvature-curvature couplings, in the action $S=m_{\rm
Pl}^{2}\int d^{4}x\sqrt{-g}[R+m_{\rm
Pl}^{-2}(c_{1}R^{2}+c_{2}R_{\mu\nu}R^{\mu\nu})+m_{\rm
Pl}^{-4}c_{3}R^{3}+\cdots]$. Since this is an infinite series of terms each
with an unknown constant coefficient $c_{i}$, the model becomes non-predictive
once these $R^{n\geq 2}$ terms become important. In the regime of these
experiments $R\ll m_{\rm Pl}^{2}$, so all of these terms are extraordinarily
small and the dynamics is determined entirely by the first term $R$, i.e., the
usual Einstein-Hilbert action. However, we are well within this limit in the
kind of experiment envisaged here, as discussed in the introduction.

Proceeding accordingly, the scattering of a pair of masses (here modelled as
single-particle excitations of a massive spinless field) via gravitons is
given by donoghue1994general ; burgess2004quantum

$\braket{\mbox{\boldmath$p$}_{1}^{\prime}\mbox{\boldmath$p$}_{2}^{\prime}}{T}{\mbox{\boldmath$p$}_{1}\mbox{\boldmath$p$}_{2}}=\frac{4\pi}{m_{\rm
Pl}^{2}}\frac{N_{\rm grav}}{-t-i\epsilon}.$ (36)

The numerator is more complicated due to the tensorial nature of the
interaction,

$\displaystyle\begin{split}N_{\rm grav}&=2(p_{1}\cdot
p_{1}^{\prime})(p_{2}\cdot p_{2}^{\prime})+2(p_{1}\cdot
p_{2}^{\prime})(p_{1}^{\prime}\cdot p_{2})\\\ &\ \ \ +8(p_{1}\cdot
p_{1}^{\prime}+m^{2})(p_{2}\cdot p_{2}^{\prime}+m^{2})\end{split}$ (37)

but reduces in the non-relativistic limit to the simple value $N_{\rm grav}\to
m^{2}$. Recognizing that $m_{\rm Pl}^{2}=1/G_{N}$ in terms of the Newton
constant, we can compare this again to the Born approximation (31), and
determine the effective potential

$\tilde{V}(\mbox{\boldmath$q$})=\frac{4\pi
G_{N}m^{2}}{\mbox{\boldmath$q$}^{2}}$ (38)

which again is just $V(r)=G_{N}m^{2}/r$ in real space. In this way, we see
that “graviton exchange” leads to the Newton potential operator (1) in the
non-relativistic limit.

Finally, we note that can one directly obtain a Hamiltonian operator for the
field theory and read off the non-relativistic potential directly, without
resorting to scattering or other matching calculations. In contrast to the
gauge-invariant scattering amplitude approach, this is complicated by the
gauge symmetries of the model (in both the electrodynamics and gravity cases).
To see how this works, consider the electrodynamical Lagrangian (26). To
perform the transformation from the Lagrangian to Hamiltonian we have to fix a
gauge, say Coulomb gauge $\partial_{i}A^{i}=0$. This gauge leads to a second-
class Dirac constraint $\partial_{i}F^{i0}=-J^{0}$, so that
$\nabla^{2}A^{0}=-J^{0}$, i.e., the $A^{0}$ part of the potential is non-
dynamical and simply fixed by the current

$A^{0}(\mbox{\boldmath$x$},t)=-\int
d^{3}\mbox{\boldmath$y$}\frac{J^{0}(\mbox{\boldmath$y$},t)}{|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}|}.$
(39)

Performing the Legendre transformation to obtain the Hamiltonian, one then
finds a coupling

$\displaystyle\begin{split}H_{\rm Coul}&=-\int
d^{3}\mbox{\boldmath$x$}A^{0}(\mbox{\boldmath$x$})J^{0}(\mbox{\boldmath$x$})\\\
&=\int
d^{3}\mbox{\boldmath$x$}d^{3}\mbox{\boldmath$y$}\frac{J^{0}(\mbox{\boldmath$x$})J^{0}(\mbox{\boldmath$y$})}{|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}|},\end{split}$
(40)

which is just the usual non-relativistic Coulomb interaction, since $J^{0}$ is
the charge density. Again, everything here is at the level of operators. In
the case of gravity, the exactly analogous calculation can be performed and
one finds an instantaneous Newton interaction $H=m_{\rm Pl}^{-2}\int
d^{3}\mbox{\boldmath$x$}d^{3}\mbox{\boldmath$y$}T^{00}(\mbox{\boldmath$x$})T^{00}(\mbox{\boldmath$y$})/{|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}|}$.
See chapter 8 of weinberg1995quantum  for a thorough treatment of this kind of
Hamiltonian approach in the case of electrodynamics, and e.g.
anastopoulos2013master  for the calculation in perturbative gravity.

The fact that the field component responsible for the Coulomb/Newton
interaction is non-dynamical (e.g., $A^{0}$ in the above example) has led some
authors to argue that observing Newtonian entanglement would tell us nothing
about the quantization of the “physical” (i.e., dynamical) degrees of freedom
of the gravitational field; for a prototypical expression of this view, see
Anastopoulos:2018drh . There are, however, strong arguments against this
belenchia2018quantum ; Christodoulou:2018cmk ; Marshman:2019sne ;
Galley:2020qsf , which essentially say that there is no consistent way to have
both an entangling Newton interaction and non-quantized metric fluctuations.
We anticipate substantial further debate on this topic, and will present a
detailed discussion in a separate paper carneyprep .

##  Appendix B Detailed calculation of oscillator noise

To get a quantitative estimate of the effects of thermal loading on the
oscillator, let us assume that we can safely neglect atomic dephasing
$\gamma_{a}t\ll 1$ for the experimental timescale of interest. To develop an
exact result, we will use an input-operator method, in which we include an
explicit heat bath for the oscillator. The Lindblad system (18) can be derived
through this method by tracing out the oscillator bath. See
clerk2010introduction  for a lucid review of this technique.

Let $H_{0}$ denote the Hamiltonian for the oscillator _and its bath_. The
total time evolution operator is then

$U(t)=e^{-i(H_{0}+g\sigma_{z}x)t}=e^{-iH_{0}t}\hat{T}e^{-ig\sigma_{z}\int_{0}^{t}x_{I}(t^{\prime})dt^{\prime}}.$
(41)

Here $x_{I}(t)$ is the oscillator position operator in the interaction picture
and $\hat{T}$ is the time-ordering operator. For the case of a linear bath,
such as that assumed in quantum optics or in quantum Brownian motion, we can
explicitly find $x_{I}$. Writing $x=(a+a^{\dagger})/\sqrt{2}$, we have

$\displaystyle\begin{split}a_{I}(t)&=\exp[-i(\omega+\gamma_{m}/2)t]a(0)\\\
&+\sqrt{\gamma_{m}}\int_{0}^{t}\exp[-i(\omega+\gamma_{m}/2)(t-t^{\prime})]a_{\rm
in}(t^{\prime})dt^{\prime}\end{split}$ (42)

where $a_{\rm in}(t)$ is the vacuum noise fluctuation operator, satisfying
$[a_{\rm in}(t),a_{\rm in}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$. Using
the linearity of this expression and the Baker-Campbell-Hausdorf relation, we
then have that

$\displaystyle\begin{split}&\hat{T}\exp\left(-ig\sigma_{z}\int_{0}^{\tau}x_{I}(t)dt\right)\\\
&=\exp\left(-ig\sigma_{z}\int_{0}^{\tau}x_{I}(t)dt\right)\exp(-ig^{2}C(t))\end{split}$
(43)

where $C(t)$ is a real, time-dependent number, arising from the non-commuting
elements of $x_{I}(t)$.

Having dispensed with the time-ordering, we can now explicitly perform the
time integral (including a change of integration order in the $a_{\rm in}$
term). Dropping the $e^{-ig^{2}C(t)}$ phase, which will cancel out of our
observable, we find the time evolution reduces to a simple product of
displacement operators, one for the oscillator and one for each mode $a_{\rm
in}(t^{\prime})$ for $0\leq t^{\prime}\leq t$, that is

$U(t)=e^{-iH_{0}t}D_{a}[\sigma_{z}\alpha(t)]\prod_{0\leq t^{\prime}\leq
t}D_{a_{in}(t^{\prime})}[\sigma_{z}\alpha_{\rm in}(t^{\prime})]$ (44)

where

$\displaystyle\begin{split}\alpha(t)&=\frac{ig}{i\omega-\gamma_{m}/2}(1-e^{(i\omega-\gamma_{m}/2)t}),\\\
\alpha_{\rm
in}(t^{\prime})&=\frac{ig}{i\omega-\gamma_{m}/2}(1-e^{(i\omega-\gamma_{m}/2)(t-t^{\prime})}).\end{split}$
(45)

Finally, we can evaluate our visibility
$\sigma_{-}(t)=U^{\dagger}(t)\sigma_{-}U(t)$, assuming an initial thermal
state for the oscillator and each bath mode and the $\ket{+}$ state for the
atom. Using the same results for coherent states as above, one finds

$\displaystyle\begin{split}\braket{\sigma_{-}}&=\braket{D_{a}[2\alpha(t)]}\prod_{t^{\prime}}\braket{D_{a_{\rm
in}(t^{\prime})}[\alpha_{\rm in}(t^{\prime})]}\\\
&=\exp\left[-8\lambda^{2}(2\bar{n}+1)f(t)\right]\end{split}$ (46)

with

$\displaystyle\begin{split}f(t)&=\frac{\omega^{2}/4}{\omega^{2}+\gamma_{m}^{2}/4}\Big{(}2-2\cos(\omega
t)e^{-\gamma_{m}t/2}+\gamma_{m}t\\\ &-\frac{8\gamma_{m}}{\omega}\sin(\omega
t)e^{-\gamma_{m}t/2}+O(1/Q^{2})\Big{)}\end{split}$ (47)

where $Q=\omega/\gamma_{m}$ is assumed much larger than one. In particular, at
full and half-period this gives

$\displaystyle\begin{split}V(\pi/\omega)&=\exp[-8\lambda^{2}(2\bar{n}+1)],\\\
V(2\pi/\omega)&=\exp[-8\lambda^{2}(2\bar{n}+1)/Q].\end{split}$ (48)

Here we have assumed the mechanical damping factor $Q=\omega/\gamma_{m}\gg 1$.
Re-inserting the exponential damping factor for atomic dephasing then
reproduces the results in (19).

In Fig. 6, we compare this analytical model with a numerical simulation of the
Lindblad equation (13), showing excellent agreement.

Figure 6: Examples of the signal of interest, the phase contrast
$V=|\braket{\sigma_{-}(t)}|$, compared with its initial value $V(0)=1/2$.
Left: Direct simulation of the Lindblad evolution (18) in blue, and we see
good agreement with our analytic solution including noise (46) in orange,
dashed. Normalizing all units to the oscillator frequency $\omega=1$, here we
use values $g=10^{-2}$ for the gravitational coupling, $\gamma_{m}=5\times
10^{-3}$ for the mechanical damping, and $T=2$ for the temperature. (Numerical
simulation with a much higher $T\gg\omega$ as discussed in the paper is
infeasible due to restrictions on the oscillator Hilbert space dimension).
Same parameters as left figure, but with an initial $\pi$-pulse using a non-
gravitational coupling $g^{\prime}=10^{-1}$. The difference between the first
collapse and revival is now much larger than in the un-boosted protocol, as
predicted in (21).

##  Appendix C Detailed calculation of the boosted protocol

Here we give the full computation of the visibility in our entanglement-
enhanced, “boosted” protocol of Section V. The total evolution is a product of
two unitaries, one for the first half-period under the coupling $g+g^{\prime}$
and the second under only $g$. We write these as

$\displaystyle\begin{split}U_{g+g^{\prime}}&=D^{\dagger}((\lambda+\lambda^{\prime})\sigma_{z})e^{-i\omega
n}D((\lambda+\lambda^{\prime})\sigma_{z}),\\\
U_{g}(t)&=D^{\dagger}(\lambda\sigma_{z})e^{-i\omega
n(t-\pi/\omega)}D(\lambda\sigma_{z}).\end{split}$ (49)

With this notation, the visibility of the atom, given some initial coherent
state $\ket{\alpha}$ for the oscillator, is given by (defining
$\tilde{\lambda}=\lambda+\lambda^{\prime}$ for brevity)

$\displaystyle\begin{split}V_{{\rm
b},\alpha}(t)&=\braket{\alpha}{U^{\dagger}_{g+g^{\prime}}U_{g}(t)\sigma_{-}U_{g}(t)U_{g+g^{\prime}}(t)}{\alpha}\\\
&=\bra{\alpha}D^{\dagger}(-\tilde{\lambda})e^{i\omega
n/\pi}D(-\tilde{\lambda})U^{\dagger}_{g}(t)\\\ &\ \ \
\times\sigma_{-}U_{g}(t)D^{\dagger}(\tilde{\lambda})e^{-i\omega
n/\pi}D(\tilde{\lambda})\ket{\alpha}\\\
&=\braket{\alpha}{D(2\tilde{\lambda})e^{i\omega
n/\pi}U^{\dagger}_{g}(t)\sigma_{-}U_{g}(t)e^{-i\omega
n/\pi}D(2\tilde{\lambda})}{\alpha}\\\
&=\braket{\alpha}{D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega
t})\right]D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]}{\alpha}\\\
&=\bra{0}D(-\alpha)D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]\\\
&\ \ \ \times D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega
t})\right]D(\alpha)\ket{0}\\\
&=e^{\phi}\braket{\alpha-2\tilde{\lambda}+\lambda(1+e^{i\omega
t})}{\alpha+2\tilde{\lambda}-\lambda(1+e^{i\omega t})}\\\
&=e^{2\phi}e^{-|4\tilde{\lambda}^{2}-2\lambda(1+e^{i\omega
t})|^{2}/2}.\end{split}$ (50)

To go from the second to third line, we inserted a pair of identity operators
$1=e^{-i\omega n/\pi}e^{i\omega n/\pi}$ and used $e^{i\omega
n/\pi}D(\tilde{\lambda})e^{-i\omega n/\pi}=D(-\tilde{\lambda})$. From the
third to fourth we used the same trick and the more general time evolution
$e^{i\omega nt}D(\tilde{\lambda})e^{-i\omega nt}=D(\tilde{\lambda}e^{i\omega
t})$. In the last few lines the “phase” is

$\phi=\alpha^{*}(2\tilde{\lambda}-\lambda(1+e^{i\omega t}))/2+{\rm c.c.}.$
(51)

Note that we got two factors of this: one in the fifth line, from the braiding
relation $D(\alpha)D(\beta)=e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}$, and then
another in the subsequent line from the inner product
$\braket{\beta}{\alpha}=e^{-|\beta-\alpha|^{2}/2}e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}$.
Notice also that the second exponential does not depend on the coherent state
parameter $\alpha$. Thus we only need to average this phase term over the
Glauber representation, which gives

$\displaystyle\begin{split}&\int\frac{d^{2}\alpha}{\pi\bar{n}}e^{-|\alpha|^{2}/\bar{n}}e^{2\phi}\\\
&\ \ \ =\exp\left(8\lambda
n(\lambda+2\lambda^{\prime})\cos(t\omega)-8n\left(\lambda^{2}+2\lambda\lambda^{\prime}+2\lambda^{\prime
2}\right)\right),\end{split}$ (52)

where we used the explicit coefficient
$\tilde{\lambda}=\lambda+\lambda^{\prime}$. Doing the same with the second
term in (50), and simplifying the terms, we finally obtain

$\displaystyle\begin{split}&V_{\rm
b}(t)=\int\frac{d^{2}\alpha}{\pi\bar{n}}V_{{\rm b},\alpha}(t)\\\
&=\exp\left[-8(2\bar{n}+1)\left(\lambda^{\prime
2}+2\lambda\lambda^{\prime}\sin^{2}\frac{\omega
t}{2}+\lambda^{2}\sin^{2}\frac{\omega t}{2}\right)\right],\end{split}$ (53)

as quoted in (20). Note the limit $\lambda^{\prime}\to 0$ reproduces the
basic, un-boosted protocol. We show the form of this visibility evolution in
Fig. 6.

##  Appendix D Using many atoms

Figure 7: Spin-echo variant of the basic protocol. After a half-period of
evolution, the two pathways through oscillator phase space are maximally
distant. The atomic positions are then flipped ($X$ gate), followed by another
half-period of evolution. This procedure can be repeated arbitrarily, leading
to a net amplification of the basic protocol. The bottom-right figure shows
the resulting conditioned paths of the oscillator through phase space, through
one iteration.

The sensitivity of the protocol can be substantially improved by moving from
using a single atom to use a collection of atoms, as is typical in an atom
interferometer kasevich1992measurement ; santarelli1999quantum ;
gross2010nonlinear . For simplicity, we will take $g$ the same for all the
atoms, though that is not necessary in practice. In that limit, we can define
$J_{z}=\sum_{j}\sigma_{z}^{j}$ and $J_{-}=\sum_{j}\sigma_{-}^{j}$ as the
collective variables that will enter.

Consider the extension of (9) to the case of $N$ atoms prepared in the initial
state $\ket{+++\cdots}$. The “observable” of interest is $\braket{J_{-}(t)}$.
This is easiest to calculate term by term for each atom. The total time
evolution operator, following the same logic as in (3), is

$U(t)=\exp\left(-i\beta J_{z}^{2}\right)D^{\dagger}(\lambda J_{z})e^{i\omega
a^{\dagger}at}D(\lambda J_{z}),$ (54)

with

$\beta=\frac{g^{2}t}{\omega}.$ (55)

This total $J_{z}^{2}$ term is peculiar to the case of $N>1$ atoms; for $N=1$
it is just an overall phase which we dropped in (3). Here, however, it is a
non-trivial operator, physically representing the ponderomotive squeezing of
the spins due to the gravitational coupling with the oscillator. At time $t>0$
we have, for each $i=1,\ldots,N$, the operator evolution

$\sigma_{-}^{i}(t)=U^{\dagger}(t)\sigma_{-}^{i}U(t).$ (56)

If we define $\tilde{J}_{z}^{i}=J_{z}-\sigma_{z}^{i}$, we can see that we can
write the $J_{z}^{2}$ contribution to (56) as

$\exp\left(i\beta J_{z}^{2}\right)\sigma_{-}^{i}\exp\left(-i\beta
J_{z}^{2}\right)=\exp\left(-2i\beta\tilde{J}_{z}\right)\sigma_{-}^{i}.$ (57)

This prefactor then commutes with the rest of the operators in (56). Using
this and the same basic logic as in (9), we find that all the $i\neq j$ spins
will just give a phase proportional to $\tilde{J}_{z}$:

$\sigma_{-}^{i}(t)=\sigma_{-}^{i}e^{-2i\beta\tilde{J}_{z}}D(-\lambda)D(2\lambda
e^{i\omega t})D(-\lambda).$ (58)

Acting on the initial state $\ket{0,+++\cdots}$ with the oscillator prepared
in $\ket{0}$ and each atom in the $\ket{+}$ state, that is to say with $N$
unentangled atoms, we obtain

$\braket{\sigma_{-}^{i}(t)}=\cos^{N-1}(2\beta)\braket{\sigma_{-}(t)}_{1}$ (59)

where the term $\braket{\sigma_{-}(t)}_{1}$ means the answer with a single
spin, as in equations (9) and (10). For $N\gg 1$ and $\beta\ll 1$ (recall
$\beta=g^{2}t/\omega$, so this condition is certainly satisfied for us), we
can Taylor expand the cosine and match it to an exponential for convenience,
i.e., $\cos^{N-1}(2\beta)\approx e^{-2N\beta^{2}}$. Thus since
$\braket{J_{-}}$ has $N$ terms of the form (59), we finally get

$\braket{J_{-}(t)}=Ne^{-2Ng^{4}t^{2}/\omega^{2}}\braket{\sigma_{-}(t)}_{1}.$
(60)

For example, at a half period and full period we then have

$\displaystyle\begin{split}\braket{J_{-}(\pi/\omega)}&=Ne^{-2\pi^{2}N\lambda^{4}}e^{-8\lambda^{2}},\\\
\braket{J_{-}(2\pi/\omega)}&=Ne^{-4\pi^{2}N\lambda^{4}}.\end{split}$ (61)

Note that this noisy phase is independent of the oscillator’s initial state,
so for example we get the same answer if the oscillator begins in a thermal
state. We see the basic $N$ enhancement to the signal here as the prefactor.
The phase noise scales like $N\lambda^{4}$. For our particular implementation
with parameters like those quoted in (25), we have $\lambda\sim 10^{-13}$, so
for $N\sim 10^{10}$ atoms these phase noise exponentials are completely
negligible. The overall $N$ factor here represents the usual $\sqrt{N}$
statistical enhancement in the signal-to-noise, assuming uncorrelated atom
errors.

The same calculation extends directly to the entanglement-enhanced, $g$-linear
protocol (20). This is clear by the algebraic structure of the argument given
above. Explicitly, we now have two time evolution operators of the form (54),
one with a coupling $g+g^{\prime}$ from $t=-\pi/\omega$ to $t=0$, followed by
another with only the gravitational $g$ coupling from $t=0$ onward. In an
obvious notation we can write

$\sigma^{i}_{-}(t)=U^{\dagger}_{g}(t)U^{\dagger}_{g+g^{\prime}}\sigma_{-}^{i}U_{g+g^{\prime}}U_{g}(t).$
(62)

In these $U$ operators, we have the same phase-noise terms, namely $e^{i\beta
J_{z}^{2}}$ in the $U_{g}$ and another factor $e^{i\beta^{\prime}J_{z}^{2}}$,
with $\beta^{\prime}=-(g+g^{\prime})^{2}\pi/\omega$, from the
$U_{g+g^{\prime}}$ factor. These depend only on the $J_{z}$ operator and thus
commute with the other terms (displacement operators and free oscillator
evolution) in $U_{g}$ and $U_{g+g^{\prime}}$. Thus we get an expression

$\displaystyle\begin{split}\sigma^{i}_{-}(t)&\sim\exp(i(\beta+\beta^{\prime})J_{z}^{2})\sigma_{-}^{i}\exp(-i(\beta+\beta^{\prime})J_{z}^{2})\\\
&=\exp(-2i(\beta+\beta^{\prime})\tilde{J}_{z})\sigma_{-}^{i},\end{split}$ (63)

just as in (57) except now with $\beta$ replaced by

$\beta+\beta^{\prime}=\frac{g^{2}t}{\omega}-\frac{(g+g^{\prime})^{2}\pi}{\omega}.$
(64)

In particular, all the terms other than these phase noise exponentials
contribute as given by (20). The overall signal is still increased linearly in
$N$ as in (61), times a negligible contribution from this ponderomotive
squeezing noise.

##  Appendix E Spin-echo version for faster physical oscillators

The physical oscillator frequency is crucially important to the size of the
observable effect. The interferometric contrast scales as a power of
$\lambda=g/\omega$, so a low-frequency oscillator is ideal. However, in
practice, using a very low-frequency (sub-Hz) oscillator would present
substantial technical problems. This can be alleviated by using a high-
frequency oscillator and a spin-echo like sequence to mimic the effect of a
low-frequency oscillator. Specifically, after every $\pi/\omega$ half-period,
we swap the two atom locations, i.e. perform a $\sigma_{x}$ operation (see
Fig. 7). This produces the evolution

$\displaystyle\begin{split}U&=\sigma_{x}e^{-iH\tau/2}\sigma_{x}e^{-iH\tau/2}\\\
&=e^{-i(H_{p}-V)\tau/2}e^{-i(H_{p}+V)\tau/2}\\\
&=D(\lambda\sigma_{z})e^{-iH_{p}\tau/2}D(-\lambda\sigma_{z})D(-\lambda\sigma_{z})e^{-iH_{p}\tau/2}D(\lambda\sigma_{z})\\\
&=D(\lambda\sigma_{z})D(2\lambda\sigma_{z})D(\lambda\sigma_{z})=D(4\lambda\sigma_{z})\end{split}$
(65)

where we used $\tau=2\pi/\omega$ for the final line. With $N_{\pi}$ iterations
of this, we produce the total evolution
$U(N_{\pi})=D(4N_{\pi}\lambda\sigma_{z})$. Performing $N_{\pi}$ iterations,
followed by a $\sigma_{x}$ operation, followed by $N_{\pi}$ further
iterations, we recover the revival:

$\displaystyle\begin{split}U_{\rm spin-
echo}&=\sigma_{x}D(4N_{\pi}\lambda\sigma_{z})\sigma_{x}D(4N_{\pi}\lambda\sigma_{z})\\\
&=D(-4N_{\pi}\lambda\sigma_{z})D(4N_{\pi}\lambda\sigma_{z})\\\
&=\mathbb{I}.\end{split}$ (66)

In this spin-echo style variant, the wavefunction overlap after a total time
$t=N_{\pi}\tau$, i.e. to the halfway point, is given by

$O=\exp\left[-32\frac{N_{\pi}^{2}g^{2}}{\omega^{2}}\right].$ (67)

Thus we have an effect scaling like $g^{2}/\omega_{\rm eff}^{2}$, with the
effective frequency $\omega_{\rm eff}=2\pi/t=\omega/N_{\pi}$. This means we
can have an effectively slow oscillator (which is beneficial for the signal
strength) while using a faster physical oscillator (beneficial for noise
reasons), at the cost of having to perform some $\sigma_{x}$ operations on the
two-state system.

We note that application of this spin-echo protocol would violate the Markov
assumption used to prove the theorem in section III. Adjusting that proof to
accommodate this specific type of non-Markovian control would be necessary to
draw the same conclusion, namely, that the gravitational entanglement in this
protocol is necessarily due to the mass-atom entangling interaction.

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