# Semi-classical gravity phenomenology under the causal-conditional quantum
measurement prescription

Yubao Liu  Center for Gravitational Experiment, Hubei Key Laboratory of
Gravitation and Quantum Physics, School of Physics, Huazhong University of
Science and Technology, Wuhan, 430074, China  Haixing Miao  State Key
Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing, China  Yanbei Chen<EMAIL_ADDRESS>Burke Institute of
Theoretical Physics, California Institute of Technology, Pasadena, CA, 91125,
USA  Yiqiu Ma<EMAIL_ADDRESS>Center for Gravitational Experiment, Hubei Key
Laboratory of Gravitation and Quantum Physics, School of Physics, Huazhong
University of Science and Technology, Wuhan, 430074, China  Department of
Astronomy, School of Physics, Huazhong University of Science and Technology,
Wuhan, 430074, China

###### Abstract

The semi-classical gravity sourced by the quantum expectation value of the
matter’s energy-momentum tensor will change the evolution of the quantum state
of matter. This effect can be described by the Schroedinger-Newton (SN)
equation, where the semi-classical gravity contributes a gravitational
potential term depending on the matter quantum state. This state-dependent
potential introduces the complexity of the quantum state evolution and
measurement in SN theory, which is different for different quantum measurement
prescriptions. Previous theoretical investigations on the SN-theory
phenomenology in the optomechanical experimental platform were carried out
under the so-called post/pre-selection prescription. This work will focus on
the phenomenology of SN theory under the causal-conditional prescription,
which fits the standard intuition on the continuous quantum measurement
process. Under the causal-conditional prescription, the quantum state of the
test mass mirrors is conditionally and continuously prepared by the projection
of the outgoing light field in the optomechanical system. Therefore a
gravitational potential depends on the quantum trajectory is created and
further affects the system evolution. In this work, we will systematically
study various experimentally measurable signatures of SN theory under the
causal-conditional prescription in an optomechanical system, for both the
self-gravity and the mutual gravity scenarios. Comparisons between the SN
phenomenology under three different prescriptions will also be carefully made.
Moreover, we find that quantum measurement can induce a classical correlation
between two different optical fields via classical gravity, which is difficult
to be distinguished from the quantum correlation of light fields mediated by
quantum gravity.

##  I Introduction

Einstein’s General Theory of Relativity reveals the nature of gravity as a
spacetime curvature $G_{\mu\nu}$ that coupled to the matter energy-stress
tensor $T_{\mu\nu}$, which can be summarised as Einstein’s field equation
$G_{\mu\nu}=8\pi T_{\mu\nu}$. In Einstein’s theory, both spacetime geometry
and matter are classical. However, the physical law that governs matter
evolution is quantum mechanics, which means that the energy-stress tensor
should be an operator $\hat{T}_{\mu\nu}$ in the quantum world. Therefore,
quantising the spacetime geometry $G_{\mu\nu}$ is one natural approach to
establishing a consistent description of gravity [1, 2], which is yet to be
successful. On the other hand, there is also an alternative semi-classical
approach in which the spacetime geometry remains classical, while it is
sourced by the quantum expectation of the stress-energy operator, i.e.
$G_{\mu\nu}=8\pi\langle\psi|\hat{T}_{\mu\nu}|\psi\rangle$ ($|\psi\rangle$ is
the quantum state of matter which evolves with the spacetime) as originally
proposed by Möller and Rosenfeld [3, 4].

Although quantum gravity seems the most natural and logical way forward, and
despite many arguments against classical gravity (in particular its
inconsistency with Everett’s relative-state interpretation raised by Page [5,
6, 7]), classical gravity has not been ruled out completely. Unlike the other
three fundamental interactions, there is currently no direct experimental
evidence for the quantumness of the gravitational field due to the demanding
condition for such a test. Therefore, it is meaningful to test the quantumness
of the gravitational field. In history, similar discussions on the quantumness
of electromagnetic (EM) field also motivated the works on experimentally
testing the so-called semi-classical EM theory [8, 9, 10, 11, 12, 13, 14],
which was once thought to be indistinguishable from quantum electrodynamics
[15, 16, 17, 18].

Recent developments in quantum optomechanics provides an experimental platform
for testing physical phenomena at a new interface between quantum and
gravitational physics. [19, 20, 21]. This platform allows the preparation,
manipulation, and characterisation of macroscopic objects near Heisenberg
Uncertainty, and spans a wide range of experimental scenarios, including
levitated nanospheres [22, 23, 24], nanomechanical oscillators [25, 26, 27,
28, 29, 30, 31, 32, 33], membranes [34], and suspended test masses [35, 36,
37].

Experimental opportunities motivate the study of the phenomenology of the
semi-classical gravity of macroscopic, non-relativistic objects. In this
paper, we shall focus on the Schroedinger-Newton theory [38, 39, 40, 41, 42,
43]

$i\hbar\frac{\partial|\Psi\rangle}{\partial
t}=H|\Psi\rangle+V_{N}(|\Psi\rangle)|\Psi\rangle$ (1)

in which the quantum state $|\Psi\rangle$ of a system of particles, as can be
represented by the multi-particle joint wavefunction, is subject to a Newton
gravitational potential $V$ that in turn depends on the state $|\Psi\rangle$.
In Ref. [44, 45], the effect of $V_{N}$ is elaborated for a macroscopic test
mass (a solid), when its Center-Of-Mass (COM) position uncertainty is less
than the zero-point position uncertainty of atoms near their lattice sites. In
phase space, the quantum uncertainty (in terms of the position-momentum
covariance matrix) of the COM evolves at a shifted frequency from that of the
expectation values. Further works discussed possible experimental signatures
of the SN theory [44, 45, 46, 47].

The nonlinearity in $V(|\Psi\rangle)$ brings an ambiguity to SN theory when
describing quantum measurements, and in this process breaking the _time-
reversal symmetry_ of standard quantum mechanics. Let us consider a scattering
amplitude problem: suppose we prepare a system at an initial state
$|i\rangle$, let it evolve for duration $T$, and would like to compute the
probability that it will be found at $|f\rangle$. Even though the SN equation
appears to be time-reversal symmetric, since in general, $|i\rangle$ does not
evolve into $|f\rangle$, when inserting the $|\Psi\rangle$ in $V$, one needs
to specify whether to make $|\Psi\rangle$ agree with $|i\rangle$ at the
initial time, or to make $|\Psi\rangle$ agree with $|f\rangle$ at the final
time. In other words, if we denote with $\hat{U}$ the evolution operator, then
the relation $p_{i\rightarrow f}=|\langle f|\hat{U}|i\rangle|^{2}=|\langle
i|\hat{U}|f\rangle|^{2}$ will not hold since $\hat{U}$ depends on the quantum
state due to the nonlinearity:
$\hat{U}(t,t_{0})=\hat{U}_{|\psi\rangle}(t,t_{0})$ [45]. In standard quantum
mechanics, the wavefunction collapse has three different, but equivalent
prescriptions/interpretations: pre-selection, post-selection, and conditional
collapse, as discussed by Aharonov _et al._ in a milstone paper [48, 49].
However, the equivalence of these prescriptions depends on the linearity of
quantum mechanics, that is, such an equivalence will be broken in SN theory.
For example, for post/pre-selection prescription, we have $p^{\rm
pre/post}_{i\rightarrow f}=|\langle
f|\hat{U}_{|\psi\rangle_{i/f}}|i\rangle|^{2}$ and clearly $p^{\rm
pre}_{i\rightarrow f}\neq p^{\rm post}_{i\rightarrow f}$.

In this paper, we will introduce a version of the so-called causal-conditional
prescription of SN theory [50], which differs from the pre/post-selection
prescriptions, leading to significantly different phenomenology. In the non-
relativistic limit of this prescription, Newton’s potential will be determined
by the instantaneous conditional quantum state of the system. Consider a
system of two macroscopic mirrors (A and B), interacting via their _mutual
gravitational interaction_ , with each of the centre-of-mass position $x_{A}$
and $x_{B}$ monitored by a separate optical field. In previous theoretical
proposals [51, 52], it was assumed that, under SN theory, the motion of mirror
A, driven by quantum radiation-pressure fluctuations, will have a vanishing
quantum expectation, therefore will not drive the motion of mirror B. This
argument predicts zero correlation between the two out-going optical fields
$y_{A}$ and $y_{B}$. In this way, any correlation between the two output
fields can be used to verify the quantum nature of gravity.

However, as we shall see in this paper, under the causal-conditional
prescription, the _continuous quantum measurement_ of mirror motions can
actually induce correlations between the two out-going fields. We can in fact
argue that some degrees of correlation between the two optical fields must
exist in SN theory, in which the quantum state $|\Psi\rangle$ used in
generating Newton’s potential is updated according to measurement results. In
the specific case of the two mirrors under causal-conditional prescription,
the conditional state of mirror A, hence the conditional expectation of
$E(x_{A}|y_{A})$, evolves in a way that depends on measurement results
$y_{A}$, hence A exerts a classical gravitational force on B that is
correlated with $y_{A}$, which in turn establishes a correlation between
$y_{A}$ and $y_{B}$. As it turns out, in comparison with zero correlations,
the _classical correlation_ predicted by the causal conditional prescription
is much more difficult to distinguish from the quantum-gravity-induced
correlations of light fields when the two mirrors are interacting via weak
quantum gravity. This result provides an important caution toward experimental
verification of the quantum nature of gravity.

This paper is structured as follows. Section II will give a general discussion
of the continuous quantum measurement under different prescriptions for SN
theory. Then in Section III and IV, the optomechanical systems with the strong
SN effect (semi-classical self-gravity scenario) and the weak SN effect (semi-
classical mutual-gravity scenario) will be thoroughly analysed, respectively.
Section V will discuss the physical origin of the semi-classical gravity-
induced light field correlations from the aspect of non-linear quantum
mechanics. Finally, in Section VI summary and discussions of this work will be
presented.

##  II Continuous quantum measurement in SN theory

The first step to study the SN phenomenology is to establish a theoretical
description of the continuous quantum measurement in SN theory. As we have
mentioned in the _Introduction_ , the quantum state evolution and measurement
induced state collapse in a non-linear quantum mechanical theory are different
from the standard quantum mechanics. This is because _the symmetry of pre and
post-selection prescriptions in the standard quantum mechanics is no longer
valid_ , which has been extensively discussed and applied to the analysis of
the measurement of a single test mass mirror exerted by its self-gravity in
[44] with the following Hamiltonian:

$\hat{H}=\frac{\hat{p}^{2}}{2M}+\frac{1}{2}M\omega_{\rm{m}}^{2}\hat{x}^{2}+\frac{1}{2}M\omega_{\rm{SN}}^{2}(\hat{x}-\langle\hat{x}\rangle)^{2}-\hbar\alpha\hat{a}_{1}\hat{x},$
(2)

where $\alpha$ is the measurement strength that proportional to the coherent
amplitude of the pumping light and $\hat{a}_{1}$ is the amplitude operator of
the optical flucutations. The $\omega_{\rm SN}=\sqrt{Gm/6\sqrt{\pi}x_{\rm
zp}^{3}}$ is the SN frequency [44], in which the $m,x_{\rm zp}$ are the mass
and zero-point displacement of the crystal lattice oscillation in the test
mass mirror.

In the cases of the pre/post-selection (as discussed in [45]), the evolution
operator $\hat{U}_{|\psi\rangle}={\rm
exp}[-i\int_{0}^{t}\hat{H}(|\psi_{m}(t)\rangle)dt]$ depends on the
initial/final mechanical quantum state. This means that
$\langle\hat{x}\rangle$ in the above Hamiltonian can be treated as
$\langle\psi_{m}(t_{i/f})|\hat{x}(t)|\psi_{m}(t_{i/f})\rangle$ throughout the
entire continuous measurement processes, which is a deterministic c-number.
Therefore this approach linearised the problem.

Figure 1: Upper panel: Different prescription of state evolution and
measurement in the Schroedinger-Newton theory: (a) the post-selection
prescription; (b) the causal-conditional prescription where the measurement-
evolution processes satisfy Eq. (3), where the conditional mechanical quantum
state prepared continuously follows a quantum trajectory; (c) the pre-
selection prescription. Lower panel: Spacetime diagram for quantum
measurement, where the thick blue line represents the worldline of the mirror.

However, for continuous measurement, a more intuitive prescription is the
_casual conditional_ prescription, which can be represented by:

$|\psi_{m}(t_{f})\rangle=\hat{U}_{|\psi_{N-1}\rangle}^{\Delta
t}\hat{P}_{N-1}\hat{U}_{|\psi_{N-1}\rangle}^{\Delta
t}...\hat{P}_{1}\hat{U}_{|\psi_{0}\rangle}^{\Delta
t}|0\rangle_{o}\otimes|\psi_{m}(t_{i})\rangle,$ (3)

where the $\hat{U}_{|\psi_{j}\rangle}^{\Delta t}$ is the SN evolution of
quantum state in a infinitestimal duration $\Delta t$ when the mechanical
state is $|\psi_{j}\rangle$, the $\hat{P}_{j}$ is the projection operator
acting on the light field at time $t_{0}+j\Delta t$. The projective
measurement on the light field will prepare the joint entangled opto-
mechanical state $\hat{U}_{|\psi_{j}\rangle}^{\Delta
t}|\psi_{j}\rangle\otimes|0\rangle_{j}$ onto a conditional mechanical quantum
state $|\psi_{j+1}\rangle$ with measurement record $y_{i}$. This causal-
conditional prescription is equivalent to the pre/post selection prescription
only in standard quantum mechanics. This inequivalency can be seen from the
fact that in the SN theory $\hat{U}_{|\psi_{N}\rangle}^{\Delta
t}...\hat{U}_{|\psi_{0}\rangle}^{\Delta
t}\neq\hat{U}_{|\psi_{0}\rangle}(t_{N},t_{0})$.

A direct result of the causal-conditional prescription is the dependence of
the gravitational field on the stochastic quantum trajectory [53] since the
gravitational field in the SN theory is sourced by the conditional quantum
expectations of the mirror’s physical quantities. In contrast, under the
pre/post-selection prescription, the gravitational field throughout the
continuous quantum measurement process has a deterministic evolution, which
will exhibit a different phenomenology. Interestingly, the gravitational field
evolution under the causal-conditional prescription is somewhat similar to the
quantum gravity, where the gravitational field is sourced by the mirror
exerted by the stochastic quantum radiation pressure noises. These points will
be elaborated in the following sections using the optomechanical systems as an
example.

Optomechanical system is the most promising experimental platform for testing
macroscopic quantum mechanics [21, 54, 55, 25, 26, 56]. In the following, we
will give a complete analysis on the phenomenology of semi-classical gravity
on the optomechanical system, in particular under the causal-conditional
prescription. We will discuss two different scenarios: (1) the optomechanical
system influenced by the mirror’s the semi-classical _self-gravity_ , where
the SN effect is relatively strong since the gravity interaction happens at
length scale $\sim x_{\rm zp}$; (2) the optomechanical system with two mirrors
interacting via _mutual semi-classical gravity_ , where the SN effect is
relatively weak since the gravity interaction happens at the mirror separation
length scale.

##  III Optomechanical system influenced by self-gravity

###  III.1 Results of the Pre/post-selection prescription: an overview

For an optomechanical quantum measurement system, the signatures of SN theory
can be studied through the traditional preparation-evolution-verification
process sketched in [44]. In this scenario, the mirror is first prepared onto
a mechanical squeezed state, then undergoes an SN free evolution and finally
performs quantum tomography on the evolved state. The signature of the SN
effect manifests itself in the free evolution stage, where the (Gaussian)
Wigner function of the squeezed mechanical state rotates in the phase space
around its mean value $(\langle\hat{x}\rangle_{c},\langle\hat{p}\rangle_{c})$
at frequency $\omega_{q}=\sqrt{\omega_{m}^{2}+\omega_{\rm SN}^{2}}$. The
scenario this work focus on is similar to that discussed in [45], where we
directly measure the spectrum of the outgoing optical field that interacts
with the quantum test mass mirror exerted by the semi-classical gravitational
field described by the SN theory.

Figure 2: A single cavity optomechanical system with semi-classical self-
gravity. The mirror is in the classical gravitational potential created by the
quantum expectation value of its stress-energy tensor, thereby having a
Schroedinger-Newton correction to its pendulum frequency. This effect does not
exist when gravity follows quantum mechanical law.

In [45], the signatures of the SN theory are studied under the pre/post-
selection prescription, which can be summarised as a Lorentzian peak in the
spectrum of the outgoing field around $\omega_{q}$ as:

$\Delta
S_{a_{2}a_{2}}(\omega)\approx\beta(\beta+2)\frac{\gamma_{m}^{2}}{\gamma_{m}^{2}+4(\omega-\omega_{q})^{2}},\quad\beta=\frac{\alpha^{2}}{M\hbar\gamma_{m}\omega_{q}},$
(4)

where a high $Q-$oscillator is assumed, i.e $\gamma_{m}\ll\omega_{q}$.

While for the post-selection prescription, the signature of the SN theory is,
on the contrary, a _Lorentzian dip_ in the spectrum of the outgoing field
around $\omega_{q}$:

$\Delta
S_{a_{2}a_{2}}(\omega)\approx-\beta(\beta+2)\frac{\gamma_{m}^{2}}{(1+\beta)^{2}\gamma_{m}^{2}+4(\omega-\omega_{q})^{2}}.$
(5)

Besides, the outgoing field spectrum has another peak at around $\omega_{m}$.
The SN observational feature at around $\omega_{q}$ is because the conditional
mean position of the mechanical quantum state under the continuous quantum
measurement does not coincide with the $\langle\hat{x}\rangle_{\rm pre/post}$
under the pre/post selection prescription. This means that during the quantum
measurement process, the conditional quantum expectation value of mechanical
state feels a restoring force $\propto-m\omega^{2}_{\rm
SN}(\langle\hat{x}\rangle_{\rm pre/post}-\langle\hat{x}\rangle_{c})$ (see Fig.
3).

As we shall see in the next section, the outgoing field spectrum in the case
of the causal-conditional prescription will be different: the peak/dip around
$\omega_{q}$ does not exist. We plot the comparison of the outgoing field
spectrum of these three different quantum measurement prescriptions in Fig. 4.
The numerical analysis is based on the sampling parameters listed in Tab. 1.

Parameters | Symbol | Value
---|---|---
Mirror mass | $M$ | 0.2 kg
Mirror bare frequency | $\omega_{m}/(2\pi)$ | $4\times 10^{-3}\,{\rm Hz}$
SN frequency | $\omega_{\rm SN}/(2\pi)$ | $7.8\times 10^{-2}\,{\rm Hz}$
Quality factor | $Q_{m}$ | $10^{7}$
Mechanical damping | $\gamma_{m}/(2\pi)$ | $4\times 10^{-10}\,{\rm Hz}$
Optical wavelength | $\lambda$ | $1064\,{\rm nm}$
Cavity Finesse | $\mathcal{F}$ | $300$
Intra-cavity power | $P_{\rm cav}$ | $480\,{\rm nW}$
Table 1: The parameters of the optomechanical system with a single mirror
exerted by its semi-classical self-gravity.

###  III.2 Causal-conditional prescription for the semi-classical self-gravity

####  III.2.1 Output optical spectrum

Following the above causal-conditional prescription, we can obtain the
following stochastic master equation (SME) for describing the evolution of
conditional mechanical state under continuous quantum measurement in the SN
theory. The projective measurement result of the optical quadrature fields at
homodyne angle $\theta$ is given as $\hat{y}_{\theta}=1/\sqrt{\Delta
t}\int_{t}^{t+\Delta t}\hat{a}_{\theta}(t^{\prime})dt^{\prime}$, which
satisfies: $y_{\theta}=\alpha\langle\hat{x}\rangle\sin\theta\sqrt{\Delta
t}+\Delta W/\sqrt{2\Delta t}$. For later use, we redefine $\tilde{a}_{\theta}$
as $\hat{y}_{\theta}/\sqrt{\Delta t}$ thereby:

$\tilde{a}_{\theta}=\alpha\langle\hat{x}\rangle_{c}\sin\theta+dW/{\sqrt{2}dt}.$
(6)

The corresponding stochastic master equation is:

$\begin{split}d\hat{\rho}=&-\frac{i}{\hbar}[\hat{H}_{0},\hat{\rho}]dt-\frac{\alpha^{2}}{4}[\hat{x},[\hat{x},\hat{\rho}]]dt-\frac{i\alpha}{\sqrt{2}}\cos\theta[\hat{x},\hat{\rho}]dW\\\
&+\frac{\alpha}{\sqrt{2}}\sin\theta\\{\hat{x}-\langle\hat{x}\rangle,\hat{\rho}\\}dW-\frac{i\gamma_{m}}{2\hbar}\left\\{\hat{x},\\{\hat{p},\hat{\rho}\\}\right\\}dt,\end{split}$
(7)

where $\hat{H}_{0}=\hat{H}+\hat{H}_{\rm SN}$ is the free mechanical
Hamiltonian (including its self-gravitational interaction) in the SN theory,
the second and third term on the right hand side (r.h.s) is the standard
Lindblad term and the Ito-term, respectively. The last term describes the
mechanical thermal dissipation. This mechanical dissipation term is important
since the system can not reach a stationary stochastic process without it.
Physically it is due to the fact that the interaction of light field with the
mechanical motion (when the pumping field is on-resonance with the cavity) can
only re-distribute quantum information without dynamical energy exchange.

Using Eq. (7), the conditional expectation and variance of $\hat{x}$ and
$\hat{p}$ are respectively given as:

$\begin{split}&d\langle\hat{x}\rangle_{c}=\frac{\langle\hat{p}\rangle_{c}}{M}dt+\sqrt{2}\alpha
V^{c}_{xx}\sin\theta dW,\\\
&d\langle\hat{p}\rangle_{c}=-M\omega_{\rm{m}}^{2}\langle
x\rangle_{c}dt-\gamma_{m}\langle\hat{p}\rangle_{c}dt+\sqrt{2}\alpha
V^{c}_{xp}\sin\theta dW\\\
&\qquad\qquad\qquad+\frac{\hbar\alpha}{\sqrt{2}}\cos\theta dW,\end{split}$ (8)

and

$\begin{split}&\dot{V}^{c}_{xx}=\frac{2V^{c}_{xp}}{M}-2\alpha^{2}\sin^{2}\theta
V_{xx}^{c2},\\\
&\dot{V}^{c}_{xp}=\frac{V^{c}_{pp}}{M}+M\omega_{q}^{2}V^{c}_{xx}-2\alpha^{2}\sin^{2}\theta
V^{c}_{xx}V^{c}_{xp}-\alpha^{2}\sin\theta\cos\theta\hbar V_{xx},\\\
&\dot{V}^{c}_{pp}=-2M\omega_{q}^{2}V^{c}_{xp}-2\alpha^{2}\sin^{2}\theta
V_{xp}^{c2}-2\alpha^{2}\sin\theta\cos\theta\hbar V_{xp}\\\
&\qquad\qquad\qquad-\frac{\alpha^{2}\cos^{2}\theta\hbar^{2}}{2}+\frac{1}{2}\alpha^{2}\hbar^{2},\end{split}$
(9)

where $\omega_{q}\equiv\sqrt{\omega_{\rm m}^{2}+\omega_{\rm{SN}}^{2}}$.

It is important to note that the oscillation frequency of the conditional
expectation value of the mechanical displacement $\langle\hat{x}\rangle_{c}$
is $\omega_{m}$ rather than $\omega_{q}$ under the causal-conditional
prescription, which is different from the pre/post-selection prescription.
This can be understood from the fact that the gravitational field at time $t$
in this case is sourced by the conditional mechanical state
$|\psi_{m}(t)\rangle_{c}$, which also follows a random trajectory. The
$\langle x\rangle_{c}$ in this case is always located at the potential minimum
thereby feeling no restoring force. In contrast, the gravitational field under
the pre/post-selection prescriptions is sourced by the deterministic evolving
mechanical state
$\hat{U}_{|\psi_{m}(t_{i/f})\rangle}|\psi_{m}(t_{i/f})\rangle$, therefore
feels a restoring force as discussed in the previous subsection. This is an
important difference, which will change the features of the outgoing light
spectrum.

Figure 3: Physical picture of the phenomenology under different prescriptions.
Upper panel: in the pre/post-selection prescriptions, the gravitational
potential is contributed by a fixed source with $\langle\hat{x}\rangle_{\rm
pre/post}=\langle\psi_{m}(t_{i/f})|\hat{x}(t)|\psi_{m}(t_{i/f})\rangle$, while
the conditional quantum expectation of the mirror position during the
measurement process $\langle\hat{x}\rangle_{c}\neq\langle\hat{x}\rangle_{\rm
pre/post}$, which contributes to a restoring force. Lower panel: under the
causal-conditional prescription, the $\langle\hat{x}\rangle_{c}$ always
locates at the potential minimum, which doesn’t feel a restoring force.

With the above equations, the conditional mean displacement can be formally
solved as (assuming that the phase quadrature is measured $\theta=\pi/2$):

$\begin{split}\langle\hat{x}(t)&\rangle_{c}=e^{-\gamma_{m}t/2}x^{(0)}(t)+\sqrt{2}\alpha\int_{0}^{t}dW(s)e^{-\gamma_{m}(t-s)/2}\\\
&\times\left[V^{c}_{xx}\cos{\omega_{\rm
m}(t-s)}+\frac{V^{c}_{xp}}{M\omega_{\rm m}}\sin{\omega_{\rm m}(t-s)}\right.\\\
&+\left.\left(\frac{V^{c}_{xp}}{4M\omega_{\rm
m}Q_{m}^{2}}+\frac{V^{c}_{xx}}{2Q_{m}}\right)\sin{\omega_{\rm
m}(t-s)}\right],\end{split}$ (10)

where $x^{(0)}(t)=x_{0}\cos(\omega_{\rm m}t)+(p_{0}/M)\sin(\omega_{\rm m}t)$
is the free mechanical motion which will be forgotten for $t\gg 1/\gamma_{m}$,
the terms that $\propto(1/Q_{m},1/Q_{m}^{2})$ relates to the mechanical
dissipation while the second line is purely contributed from the quantum
measurement process.

When the system reaches the steady state, the steady solution of Eq. (9):

$\begin{split}&V^{c}_{xx}=\frac{\hbar}{\sqrt{2}M\omega_{q}}\frac{1}{\sqrt{1+\sqrt{1+\Lambda^{4}}}},\\\
&V^{c}_{xp}=V^{c}_{px}=\frac{\hbar}{2}\frac{\Lambda^{2}}{1+\sqrt{1+\Lambda^{4}}},\\\
&V^{c}_{pp}=\frac{\hbar
M\omega_{q}}{\sqrt{2}}\frac{\sqrt{1+\Lambda^{4}}}{\sqrt{1+\sqrt{1+\Lambda^{4}}}},\end{split}$
(11)

with $\Lambda=\sqrt{\hbar\alpha^{2}/(M\omega_{q}^{2})}$. These formulae give
us a glimpse on the experimental requirements to distinguish quantum gravity
and semi-classical gravity. In case of quantum gravity, all the $\omega_{q}$
in the above formula should be replaced by $\omega_{m}$, therefore taking
$V^{c}_{xx}$ as an example:

$\frac{V^{c\rm SN}_{xx}}{V^{c\rm
QG}_{xx}}=\frac{\omega_{m}}{\omega_{q}}\left[\frac{1+\sqrt{1+\Lambda_{\rm
QG}^{4}}}{1+\sqrt{1+\Lambda_{\rm SN}^{4}}}\right]^{1/2},$ (12)

where we redefine $\Lambda_{\rm
SN/QG}=\sqrt{\hbar\alpha^{2}/(M\omega_{q/m}^{2})}$. It is difficult to test
the quantumness of gravity when the optomechanical interaction is very strong:
$\Lambda_{\rm QG/SN}\gg 1$, since we have $V^{c\rm SN}_{xx}/V^{c\rm
QG}_{xx}\approx\omega_{m}\Lambda_{\rm QG}/\omega_{q}\Lambda_{\rm SN}=1$. This
target could be achieved only when the $\Lambda_{\rm SN/QG}$ takes a moderate
value and $\omega_{m}/\omega_{q}$ is not close to one. The moderate
$\Lambda_{\rm SN/QG}$ indicates that the optomechanical interaction can not be
too strong, thereby low-temperature technology is required to suppress the
thermal environmental effect.

Combining Eqs. (6) (10) and (11), we can compute the auto correlation function
of $\langle\tilde{a}_{2}(t)\tilde{a}_{2}(t+\tau)\rangle$ and moreover the
power density spectrum. After some tedious but straightforward algebra, the
result in the high-$Q$ limit is:

$\begin{split}&S_{a_{2}a_{2}}(\Omega)=\frac{4}{\gamma_{m}^{2}+4(\Omega-\omega_{m})^{2}}\times\\\
&\quad\left[\alpha^{2}\frac{V_{xp}}{M}\left(1-\frac{\Omega^{2}}{\omega_{m}^{2}}\right)+\alpha^{4}\left(\frac{V^{2}_{xp}}{M^{2}\omega_{m}^{2}}+\frac{\Omega^{2}}{\omega_{m}^{2}}V_{xx}^{2}\right)\right]+1,\end{split}$
(13)

where only the quantum noise is given here for illustrative purposes. It is
clear that there is no quantum radiation pressure noise-induced pole around
$\omega_{q}$ in this case, which is different from the pre/post-selection
scenario where the SN signature appears around $\omega_{q}$ due to the reason
discussed before. At the resonance point $\Omega=\omega_{m}$, the difference
between the SN spectrum and QG spectrum has a simple formula:

$\frac{\Delta S^{\rm SN-QG}_{a_{2}a_{2}}(\omega_{m})}{S^{\rm
QG}_{a_{2}a_{2}}(\omega_{m})}=\frac{2}{\sqrt{1+\sqrt{1+\Lambda_{\rm
SN}^{4}}}}\frac{\Lambda_{\rm QG}^{4}Q^{2}}{1+\Lambda_{\rm
QG}^{4}Q^{2}}\left(\frac{\omega_{\rm SN}^{2}}{\omega_{q}^{2}}\right)$ (14)

For a strong optomechanical coupling, we have $\Lambda_{\rm QG/SN}\gg 1$ which
suppressed the difference. For a weak optomechanical coupling with
$\Lambda_{\rm SN/QG}\sim 1$, the difference could be significant for a high Q
oscillator when we ignore the thermal noise as shown in the third panel of
Fig. 4. However, this difference will be almost completely diminish even if
the temperature satisfy $T/Q\sim 10^{-13}$ K.

The outgoing optical phase spectrum (normalised by the spectrum without
gravity effect at $\omega_{m}$) under the different prescriptions is depicted
in Fig.4. The upper panel shows the spectrum around $\omega_{m}$ and the
signal peak appears at $\omega_{m}$ under the causal-conditional prescription,
which corresponds to the resonant oscillation frequency of the
$\langle\hat{x}\rangle$. The post-selection prescription also has a featured
signal near $\omega_{m}$. The spectrum around $\omega_{q}$ is shown in the
middle panel, where there is no featured signal for the causal-conditional
prescription. Moreover, taking into account the thermal noise will make it
very difficult to distinguish the spectrum for quantum gravity and semi-
classical gravity under causal conditional prescription.

Figure 4: The spectrum of the outgoing field $S_{a_{2}a_{2}}$ (quantum noise
only) in the Schroedinger-Newton theory for different prescriptions around
original mechanical frequency $\omega_{m}$ and the SN-modified frequency
$\omega_{q}=\sqrt{\omega_{m}^{2}+\omega_{\rm SN}^{2}}$. All these spectrums
are normalised by the standard quantum mechanics one. It is clear that the
causal-conditional approach has a feature at the $\omega_{m}$ rather than
$\omega_{q}$. The lowest panel shows the ratio between the outgoing spectrum
of SN theory under the causal conditional prescription and the quantum
gravity, where the thermal noise is considered.

####  III.2.2 Pondermotive squeezing

Another phenomenon in this optomechanical system under the causal-condition
prescription is the pondermotive squeezing [57]. Pondermotive squeezing is the
radiation-pressure induced correlation between the phase and amplitude
quadratures of the outgoing optical field, which was experimentally
demonstrated in [36, 37, 30, 28, 58]. When the self-gravity is quantum, to the
leading order, the Hamiltonian has the same form as that when there is no
gravity effect, so does the pondermotive squeezing spectrum.

Figure 5: Pondermotive squeezing spectrum for the optomechanical system with a
single mirror exerted by its self-gravity. Solid curves: Pondermotive
squeezing in the semi-classical SN theory, under the causal-conditional
prescription. Dashed curves: pondermotive squeezing when the gravity is
quantum. The red spot denotes the frequency where there will be a distinctive
feature. Thermal effect is also considered which shows that the semi-classical
gravity and quantum gravity can only be distinguished when the parameters
setting is very extreme.

However, when the self-gravity is semi-classical with Hamiltonian described by
Eq. (2), there will be a different pondermotive squeezing spectrum under the
causal-conditional prescription (see Fig. 5). This difference happens when the
self-gravity effect is strong $\omega_{\rm SN}\geq\omega_{m}$, which will
diminish in case the gravity effect is weak, e.g. in the mutual gravity case
shown later.

We perform the following steps to calculate the pondermotive squeezing effect
under the causal-conditional prescription. The output optical field has a
spectrum given as
$S_{a_{\theta}a_{\theta}}=S_{a_{1}a_{1}}\cos^{2}\theta+S_{a_{2}a_{2}}\sin^{2}\theta+2{\rm
Re}[S_{a_{1}a_{2}}]\sin\theta\cos\theta$ . The $\hat{a}_{\theta}$ is the
$\theta$-quadrature of the output field, in which
$\hat{a}_{\theta}=\hat{a}_{1}\cos\theta+\hat{a}_{2}\sin\theta$ with
$\hat{a}_{1/2}$ represents the amplitude/phase quadrature of the output field,
and the $\theta$ is the homodyne angle. Optimisation to the $\theta$ leads to
the quadrature with minimum uncertainty (i.e. the pondermotive squeezing)
shown in Fig. 5. Taking into account the thermal noise effect, this result
shows that the quantum gravity and semi-classical gravity can be distinguished
using the pondermotive squeezing spectrum for the optomechanical system with
mirror exerted by its self-gravity, only when the environmental temperature
takes an extremely low value.

##  IV Optomechanical system with two mirrors interacting via mutual gravity

###  IV.1 General Discussions

Two mirrors can be coupled via mutual gravitational force. In the discussion
below, we ignore the self-gravity effect and only consider the mutual gravity
for illustrative purpose. In the SN theory, the corresponding Hamiltonian can
be written as [51]:

$\begin{split}\hat{H}_{\rm
SN}=\sum_{A/B}\left[\frac{\hat{p}^{2}_{A/B}}{2m}+\frac{m\omega_{m}^{2}}{2}\hat{x}^{2}_{A/B}-\mathcal{C}(\hat{x}_{A/B}-\langle\hat{x}_{B/A}\rangle_{c})^{2}\right].\end{split}$
(15)

Here, we have $\mathcal{C}=m\tilde{\Lambda}G\rho$ with $\tilde{\Lambda}$
decided by the geometric shape of the mirror, and $\rho$ is the mirror matter
density.

Under the causal-conditional prescription, the quantum expectation value here
is for the conditional quantum state of the mirrors. Since different
measurement schemes of the optical fields will lead to different conditional
mechanical quantum states, the configuration topology of the optomechanical
device is important, as we will show later. In this work, two different
configurations will be studied: (1)  _Folded interferometer configruation_
where the projective measurement is performed on the common and differential
optical fields [59] (see Fig. 6). As we shall see, the folded interferometer
configuration can be mapped onto a single-mirror problem. (2)  _Linear cavity
configuration_ where the projective measurement is directly performed on the
outgoing optical field reflected by each cavity [51] (Fig. 8), which can not
be mapped onto a single-mirror problem and the quantum measurement induced
correlation mentioned in the _Introduction_ will appear. In both two cases,
the interaction Hamiltonian between the optical fields and the mechanical
motion takes the same form

$\hat{H}_{\rm
int}=-\hbar\alpha(\hat{a}_{1}\hat{x}_{A}+\hat{b}_{1}\hat{x}_{B}),$ (16)

where we assumed that the linear optomechanical coupling strengths $\alpha$ of
both mirrors are the same. We also assume that both cavities have no detuning
to the frequency of their pumping lasers, which means there is no dynamical
back-action in the optomechanical system.

Another important point that needs to be discussed is the treatment of
environmental noises such as thermal noise. It is known that all environmental
noises are _fundamentally speaking_ quantum mechanical since these
environmental degrees of freedom are _quantum mechanical_. Since the classical
gravity can not convey quantum mechanical information, the quantum
environmental effect will only apply to the fluctuation of the A/B mirror on
its own thereby does not contribute to the mutual correlation between the
output fields shown in Fig. 8. Mathematically, there will be additional
thermal terms in the evolution equation for the second-order correlation
functions (see the Appendix), while the evolution for the first-order moments
is unchanged. Note that in this case, the mutual correlation of two mirrors
satisfies $V^{c}_{(x/p)_{A}(x/p)_{B}}=0$, which means that the Ricatti
equation that describes the evolution of the second-order moments can be
treated separately.

Parameters | Symbol | Value
---|---|---
Mirror mass | $M$ |  $10^{-3}$ kg
Mirror bare frequency | $\omega_{m}/(2\pi)$ | $0.5\,{\rm Hz}$
SN frequency | $\omega_{g}/(2\pi)$ | $2\times 10^{-4}\,{\rm Hz}$
Quality factor | $Q_{m}$ | $3\times 10^{7}$
Mechanical damping | $\gamma_{m}/(2\pi)$ | $1.67\times 10^{-8}\,{\rm Hz}$
Environmental temperature | $T$ | $300\,{\rm K}$
Optical wavelength | $\lambda$ | $1064\,{\rm nm}$
Cavity Finesse | $\mathcal{F}$ | $4000$
Intra-cavity power | $P_{\rm cav}$ | $2000\,{\rm W}$
Table 2: The parameters of the optomechanical system with two mirrors
interacting via mutual SN gravity.

###  IV.2 Folded interferometer configuration

The conditional mechanical quantum state in the SN theory can also be prepared
by projecting the optical field in the common and differential mode basis. For
example, in a folded interferometer configuration, the optical field being
measured is: $\hat{c}=1/\sqrt{2}(\hat{a}+\hat{b})$ and
$\hat{d}=1/\sqrt{2}(\hat{a}-\hat{b})$. The common and differential motional
degrees of freedom are: $\hat{x}_{\pm}=1/\sqrt{2}(\hat{x}_{A}\pm\hat{x}_{B})$
and $\hat{p}_{\pm}=1/\sqrt{2}(\hat{p}_{A}\pm\hat{p}_{B})$. Written in terms of
these common and differential optical/mechanical modes, the Hamiltonian Eq.
(15) can be transformed into:

$\begin{split}\hat{H}_{\pm}=\frac{\hat{p}^{2}_{\pm}}{2m}+\left(\frac{m\omega_{m}^{2}}{2}-\mathcal{C}\right)\hat{x}^{2}_{\pm}\pm
2\mathcal{C}\hat{x}_{\pm}\langle\hat{x}_{\pm}\rangle-\hbar\alpha\hat{c}_{1}/\hat{d}_{1}\hat{x}_{\pm},\end{split}$
(17)

where $\mathcal{C}$ is the mutual-gravity Schrödinger-Newton coefficient,
$\hat{c}_{1}/\hat{d}_{1}$ means $\hat{c}_{1}$ or $\hat{d}_{1}$. This
Hamiltonian shows that the common and differential modes are completely
decoupled from each other, which means we can treat them independently.

Using the Stochastic master equation, the conditional expectation value can be
obtained as:

$\begin{split}&d\langle\hat{x}_{+}\rangle=\frac{\langle\hat{p}_{+}\rangle}{m}dt+\sqrt{2}\alpha
V^{c}_{x_{+}x_{+}}dW_{c}+\sqrt{2}\alpha V^{c}_{x_{+}x_{-}}dW_{d},\\\
&d\langle\hat{p}_{+}\rangle=-m\omega_{m}^{2}\langle x_{+}\rangle
dt+2\mathcal{C}(\langle\hat{x}_{+}\rangle_{c}-\langle\psi_{m}|\hat{x}_{+}|\psi_{m}\rangle)dt\\\
&\qquad\qquad+\sqrt{2}\alpha V^{c}_{x_{+}p_{+}}dW_{c}+\sqrt{2}\alpha
V^{c}_{p_{+}x_{-}}dW_{d},\\\
&d\langle\hat{x}_{-}\rangle=\frac{\langle\hat{p}_{-}\rangle}{m}dt+\sqrt{2}\alpha
V^{c}_{x_{-}x_{-}}dW_{d}+\sqrt{2}\alpha V^{c}_{x_{-}x_{+}}dW_{c},\\\
&d\langle\hat{p}_{-}\rangle=-m\omega_{m}^{2}\langle\hat{x}_{-}\rangle
dt+2\mathcal{C}(\langle\hat{x}_{-}\rangle_{c}+\langle\psi_{m}|\hat{x}_{-}|\psi_{m}\rangle)dt\\\
&\qquad\qquad+\sqrt{2}\alpha V^{c}_{x_{-}p_{-}}dW_{d}+\sqrt{2}\alpha
V^{c}_{x_{-}p_{+}}dW_{c},\end{split}$ (18)

where $dW_{c/d}$ are the Ito-terms for measuring the optical
common/differential fields. The form of
$\langle\psi_{m}|\hat{x}_{\pm}|\psi_{m}\rangle$ depends on the measurement
prescription. Under pre and post-selection prescriptions, we have
$\langle\psi_{m}(t_{i})|\hat{U}_{|\psi_{m}(t_{i})\rangle}^{\dagger}(t,t_{i})\hat{x}_{\pm}\hat{U}_{|\psi_{m}(t_{i})\rangle}(t,t_{i})|\psi_{m}(t_{i})\rangle$
and
$\langle\psi_{m}(t_{f})|\hat{U}_{|\psi_{m}(t_{f})\rangle}(t,t_{i})\hat{x}_{\pm}\hat{U}^{\dagger}_{|\psi_{m}(t_{f})\rangle}(t,t_{i})|\psi_{m}(t_{f})\rangle$,
respectively. However, under the causal-conditional prescription, we have
$\langle\psi_{m}|\hat{x}_{\pm}|\psi_{m}\rangle=\langle\hat{x}_{\pm}\rangle_{c}$.
Therefore the mutual gravity terms in Eq. (18) can be simplified to be zero
and $4\mathcal{C}\langle\hat{x}_{-}\rangle_{c}$ for
$\langle\hat{p}_{+}\rangle$ and $\langle\hat{p}_{-}\rangle$, respectively.
This means that the conditional mean of the common and differential modes will
evolve with frequency $\omega_{m}$ and
$\omega_{-}=\sqrt{\omega^{2}_{m}-4\mathcal{C}/m}$.

Figure 6: Upper panel: Folded interferometer configuration. The optical field
is projected into the common mode $\hat{c}$ and differential mode $\hat{d}$.
The common and differential motions of these two mirrors are correspondingly
prepared into the quantum states, of which the evolution can be separated.
Lower panel: the gravitational potential felt by the quantum trajectories of
common and differential motions under the pre/post selection prescription.

The above equations for the quantum trajectory are physically transparent, see
the lower panel of Fig. 6. Under the pre/post-selection prescription, the
random quantum trajectory of mirror A/B with the mean displacement
$\langle\hat{x}_{A/B}\rangle_{c}$ will feel a harmonic-like potential
$U_{g}(\langle\psi_{m}|\hat{x}_{A/B}(t)\psi_{m}\rangle)$ generated by the
classical gravitational force of mirror B/A at pre/post-selection state and
their suspension systems, which follows a deterministic evolution. From Fig.
6, it is clear that the potential force felt by common/differential motion
trajectory will drive common/differential motion trajectory themselves, which
contributes to the $\mathcal{C}$-dependence term in Eq. (18). These terms will
change the resonant frequency of the two motional modes. However, under the
causal-conditional prescription, the potential also follows a quantum
trajectory. Therefore there is no additional restoring force contributed by
the mutual gravity for the common motion, which is different from the case of
the differential motion.

This leads to a different result compared to that predicted in [59], which
focuses on the quantum gravity and the semi-classical gravity is only briefly
discussed without specifying the detailed quantum measurement processes. In
[59], the features of quantum gravity manifest in the pondermotive squeezing
of the outgoing field. They show that there is a frequency shift between the
pondermotive squeezing spectrum for the common and differential output fields
in the quantum gravity theory, while there is no such frequency shift for the
semi-classical gravity. In the following, a similar calculation will be
performed for both quantum gravity and the Schroedinger-Newton theory under
the causal-conditional prescription.

Following the method in [59], we can measure the pondermotive squeezing effect
in this mutual-gravity optomechanical system. For the common/differential
output channel, optimisation to the homodyne angle $\theta$ leads to the
quadrature with minimum uncertainty (i.e. the pondermotive squeezing) shown in
Fig. 7 111Note that the work in [59] only present the squeezing level at
$\omega_{m}$ for the quantum gravity case without presenting the full
squeezing spectrum., where we show the minimum quadrature uncertainty (in
terms of the squeezing level) of the output $\hat{c}$ and $\hat{d}$ states
when $T/Q_{m}=10^{-5}{\rm K}$ under the causal-conditional prescription. As
shown clearly in Fig. 7, one can not distinguish semi-classical gravity from
quantum gravity by investigating the pondermotive squeezing spectrum under the
causal-conditional prescription, that is, the frequency shift still exists for
the semi-classical gravity. This result can be understood from the similarity
between Eq. (18) and the Heisenberg equation of motion for the mirrors in the
quantum gravity theory [59]. In addition, there is a very tiny (almost
indistinguishable) difference on the numerical value of the squeezing level
for the semi-classical gravity and quantum gravity, which originates from the
difference of steady solution of second-order conditional moments in these two
cases (for example, the $\omega_{q}$ in Eq. (11) should be replaced by
$(\sqrt{\omega_{m}^{2}-2\mathcal{C}/m},\omega_{-})$ for the differential mode
in the SN theory and quantum gravity, respectively).

For completeness, we present the calculation details and the results for the
semi-classical SN theory under the pre/post selection prescription in the
Supplementary Material of this paper, which would certainly have a different
pondermotive squeezing spectrum compared to that under the causal-conditional
prescription. Moreover, only the spectrum under the pre-selection prescription
fits the results in [59], where there is no frequency shift between the
$\hat{c}$ and $\hat{d}$-spectrum.

In summary, in contrast to the conclusion made in [59], the pondermotive
squeezing is not an ideal figure of merit for testing the semi-classical
gravity theory under the conditional prescription and the quantum gravity.

Figure 7: Upper pannel: The squeezing spectrum of the outgoing field from
common and differential modes, including both the quantum gravity case and the
SN case, calculated using the casual-conditional prescription. Middle pannel:
the dependence of maximum squeezing level on intra-cavity power $P_{\rm cav}$
and the mechanical quality factor $Q_{m}$. Lower panel: The (almost zero)
difference between pondermotive squeezing level $\Delta S^{\rm SN-QG}_{\rm
sq}$ in SN theory and that in quantum gravity, where we focus on the peak
value at $\Omega=\omega_{m}$. We only choose a sub-region of parameter space
in the middle panel with the difference is around $10^{-7}$ dB, while the
value in other region of parameter space is much smaller.

###  IV.3 Linear cavities configuration

Now we switch to the linear cavities configuration shown in Fig. 8, whose main
physical scenario can be summarised as follows. If the gravity is quantum and
thereby can mediate quantum information, there will be a _quantum correlation_
between the two output light fluctuations, which has been studied in [51]. In
contrast, the work in [51] argues that if the gravity is classical and sourced
by the quantum expectation value of the energy-momentum of the mirror A, then
the mirror B will not feel the fluctuating gravity force sourced by A. This is
because the mean value of the first mirror position is zero. Finally, the work
in [51] concludes that there will be no correlation between the two output
light noises.

Figure 8: Correlations induced by quantum measurement. The projection of the
joint optomechanical state onto the optical Hilbert space can produce a
mechanical quantum trajectory, which can source the semi-classical gravity and
affect the motion of the other mirror. The correlation between the mechanical
motion of the two mirrors will manifest itself in the correlation of the
outgoing fields.

However, although there is no quantum correlation between the two output light
fields, there will be _quantum measurement induced classical correlations_
under the causal-conditional prescription. Continuous monitoring of the
mechanical degree of freedom via projective measurement on the optical fields
collapses the joint optomechanical quantum state and thereby preparing the so-
called _conditional quantum state_ of the mirror motion denoted as
$|\psi_{c}\rangle$. The mean value of the mirror displacement
$\bar{x}_{c}=\langle\psi_{c}|\hat{x}|\psi_{c}\rangle$ and momentum
$\bar{p}_{c}=\langle\psi_{c}|\hat{p}|\psi_{c}\rangle$ will follow a stochastic
_quantum trajectory_ [27], which is determined by the measurement records (see
Fig. 8). Therefore even if the gravity is classical (i.e., the mirror motion
is described by SN theory), the “stochastic motion" of the
$(\bar{x}_{c},\bar{p}_{c})$ of mirror A in the phase space can still source a
gravitational force acting on the mirror B (vice versa), which then can be
further recorded by the light field monitoring the position of mirror B. In
this way, the two light fields establish a _classical correlation_ via the
_classical gravitational interaction sourced by the quantum trajectory induced
by the wave function collapse_ , which will be quantitatively studied as
follows.

Suppose we measure the arbitrary quadratures of the outgoing optical fields:

$\begin{split}\hat{a}_{\theta
A}=\hat{a}_{1A}\cos\theta_{A}+\hat{a}_{2A}\sin\theta_{A},\\\ \hat{a}_{\theta
B}=\hat{a}_{1B}\cos\theta_{B}+\hat{a}_{2B}\sin\theta_{B},\end{split}$ (19)

then the quantum trajectories satisfy the following equations:

$d\mathbf{x}_{A/B}=\mathbb{G}_{m}^{q}\mathbf{x}_{A/B}dt+\mathbb{G}_{g}\mathbf{x}_{B/A}dt+\sqrt{2}\alpha\mathbb{V}_{A/B}d\mathbf{W},$
(20)

where
$\mathbf{x}_{A/B}=[\langle\hat{x}_{A/B}\rangle_{c},\langle\hat{p}_{A/B}\rangle_{c}]^{T}$,
$d\mathbf{W}=[dW_{A},dW_{B}]^{T}$. The $\mathbb{G}_{m}^{q}$ is the mechanical
response under the Schroedinger-Newton theory:

$\mathbb{G}_{m}^{q}=\left[\begin{array}[]{cc}0&1/M\\\
-M\omega_{q}^{2}&-\gamma_{m}\end{array}\right],\quad\mathbb{G}_{g}=\left[\begin{array}[]{cc}0&0\\\
M\omega_{g}^{2}&0\end{array}\right],$ (21)

and

$\begin{split}&\mathbb{V}_{A}=\left[\begin{array}[]{cc}V_{x_{A}x_{A}}\sin\theta_{A}&V_{x_{A}x_{B}}\sin\theta_{B}\\\
V_{x_{A}p_{A}}\sin\theta_{A}+(\hbar/2)\cos\theta_{A}&V_{x_{B}p_{A}}\sin\theta_{B}\end{array}\right],\\\
&\mathbb{V}_{B}=\left[\begin{array}[]{cc}V_{x_{A}x_{B}}\sin\theta_{A}&V_{x_{B}x_{B}}\sin\theta_{B}\\\
V_{x_{A}p_{B}}\sin\theta_{A}&V_{x_{B}p_{B}}\sin\theta_{B}+(\hbar/2)\cos\theta_{B}\end{array}\right].\end{split}$
(22)

In the mutual gravity case, we redefine
$\omega_{q}=\sqrt{\omega_{m}^{2}-\omega_{g}^{2}}$, where
$\omega_{g}=\sqrt{2\mathcal{C}/M}$. The $\gamma_{m}$ term here represents the
mechanical loss due to the coupling with the thermal bath, where the
corresponding thermal force term is discussed in detail in the Supplementary
Material.

The second order correlations for the A and B mirrors are 222The lengthy
concrete form will be shown in the appendix.:

$\begin{split}\mathcal{\hat{D}}\mathbf{V}_{A}=-2\alpha^{2}[V_{x_{A}x_{B}}^{2},V_{x_{A}x_{B}}V_{p_{A}x_{B}},V_{x_{B}p_{A}}^{2}]^{T}\\\
\mathcal{\hat{D}}\mathbf{V}_{B}=-2\alpha^{2}[V_{x_{A}x_{B}}^{2},V_{x_{A}x_{B}}V_{p_{A}x_{B}},V_{x_{B}p_{A}}^{2}]^{T}\end{split}$
(23)

where $\mathbf{V}_{A}=[V_{x_{A}x_{A}},V_{x_{A}p_{A}},V_{p_{A}p_{A}}]^{T}$ and
the $\mathcal{\hat{D}}\mathbf{V}_{A}$ represents the same equation as Eq. (11)
(similar for the $\mathbf{V}_{B}$) for arbitrary measurement angle
$\theta_{A},\theta_{B}$. The right-hand side is the contribution of the mutual
correlations between these two mirrors. As we shall show in the Supplementary
Material, the evolution equations for these mutual correlations do not depend
on the elements of $\mathbf{V}_{A/B}$, which is different from the quantum
gravity case. Therefore, in the ideal case, if these two mirrors are _prepared
in the uncorrelated initial state_ , then the mutual correlations are always
zero. In this case, the steady value of the second-order correlation has the
same form as the one given in Eq. (11).

The mutual correlation induced by the classical mutual gravity between the two
mirrors’ quantum trajectories now actually comes from the $d\langle
p\rangle_{A/B}\supset M\omega_{g}^{2}\langle x\rangle_{B/A}dt$ (i.e., the
$\mathbb{G}_{g}$ term in the Eq. (20)), where the
$\langle\hat{x}\rangle_{B/A}$ has randomness induced by the quantum
measurement of light field. For example, if we measure the phase qudrature of
the two outgoing light fields:

$\tilde{a}_{2A/B}=\alpha\langle x_{A/B}\rangle_{c}+dW_{A/B}/\sqrt{2}dt,$ (24)

their mutual correlation is only contributed by the classical gravitational
interaction between the two quantum trajectories. More explicitly, the
conditional mean displacement can be formally solved as:

$\begin{split}&\langle\hat{x}_{A}(t)\rangle_{c}=e^{-\gamma_{m}t/2}x^{(0)}_{A}(t)+\\\
&\sqrt{2}\alpha\int^{t}_{0}dW_{A}(s)\left[V_{x_{A}x_{A}}\cos\omega_{m}(t-s)+V_{x_{A}p_{A}}\frac{\sin\omega_{m}(t-s)}{M\omega_{m}}\right]+\\\
&\sqrt{2}\alpha\int^{t}_{0}dW_{A}(s)\left[V_{x_{A}x_{A}}\cos\omega_{-}(t-s)+V_{x_{A}p_{A}}\frac{\sin\omega_{-}(t-s)}{M\omega_{-}}\right]+\\\
&\sqrt{2}\alpha\int^{t}_{0}dW_{B}(s)\left[V_{x_{B}x_{B}}\cos\omega_{m}(t-s)-V_{x_{B}p_{B}}\frac{\sin\omega_{m}(t-s)}{M\omega_{m}}\right]+\\\
&\sqrt{2}\alpha\int^{t}_{0}dW_{B}(s)\left[V_{x_{B}x_{B}}\cos\omega_{-}(t-s)-V_{x_{B}p_{B}}\frac{\sin\omega_{-}(t-s)}{M\omega_{-}}\right],\\\
\end{split}$ (25)

where $\omega_{-}=\sqrt{\omega^{2}_{m}-2\omega_{g}^{2}}$. The
$\langle\hat{x}_{B}(t)\rangle_{c}$ can be given in a similar way, where the
only difference is to add a minus sign to all the $\cos\omega_{-}(t-s)$ and
$\sin\omega_{-}(t-s)$ terms. For simplicity, we only write down the form in
the $Q_{m}\rightarrow\infty$ approximation. It is clear that the quantum
trajectory of mirror A is driven by the projective measurement on both the two
outgoing fields. Substituting Eq. (25) into Eq.(24), it is easy to see that
there will be a correlation between $\tilde{a}_{2A}$ and $\tilde{a}_{2B}$
since $\tilde{a}_{2A/B}$ is also depend on $dW_{B/A}$. Moreover, there is also
correlation if we measure two different outgoing field quadratures
$\tilde{a}_{1A}$ and $\tilde{a}_{2B}$. This measurement-induced correlation is
different from the quantum optical correlation that was studied in [51] when
the gravity is quantum, since classical gravity can not convey quantum
information.

$\bullet$ Measurement of $(\tilde{a}_{2A},\tilde{a}_{2B})$— The optical
correlation spectrum induced by the quantum measurement of $\tilde{a}_{2A}$,
$\tilde{a}_{2B}$ is (we keep the leading term $\propto\alpha^{4}$):

$\begin{split}S^{\rm
SN}_{a_{A2}a_{B2}}(\Omega)=\frac{2\hbar^{2}\alpha^{4}\omega_{g}^{2}}{M^{2}}&|\chi_{m}(\Omega)\chi_{g}(\Omega)|^{2}\frac{\omega_{q}^{2}-\Omega^{2}}{1+\sqrt{1+\Lambda^{4}}}\\\
&\times\bigg{[}\frac{\Lambda^{4}}{1+\sqrt{1+\Lambda^{4}}}+\frac{2\Omega^{2}}{\omega_{q}^{2}}\bigg{]},\end{split}$
(26)

where $\chi^{-1}_{m}(\Omega)=-\Omega^{2}+\omega_{m}^{2}-i\Omega\gamma_{m}$,
$\chi^{-1}_{g}(\Omega)=-\Omega^{2}+\omega_{-}^{2}-i\Omega\gamma_{m}$ and
$\Lambda=\sqrt{\hbar\alpha^{2}/M\omega_{q}^{2}}$. Later on, we will
characterise the correlation between the two outgoing fields by the
correlation level defined in Eq. (28) (33), which also depends on the outgoing
spectrum. For example, the spectrum of the phase of the outgoing field
$\tilde{a}_{A}$: $S_{a_{A2}a_{A2}}(\Omega)$ when we measure
$(\tilde{a}_{2A},\tilde{a}_{2B})$:

$\begin{split}&S_{a_{A2}a_{A2}}(\Omega)\approx
1+\frac{\hbar^{2}\alpha^{4}}{M^{2}}|\chi_{m}(\Omega)\chi_{g}(\Omega)|^{2}\times\\\
&\frac{\omega_{g}^{4}+(\Omega^{2}-\omega_{q}^{2})^{2}+\gamma_{m}^{2}(\Omega^{2}+\omega_{q}^{2})/2}{1+\sqrt{1+\Lambda^{4}}}\bigg{[}\frac{\Lambda^{4}}{1+\sqrt{1+\Lambda^{4}}}+\frac{2\Omega^{2}}{\omega_{q}^{2}}\bigg{]},\end{split}$
(27)

where we also have $S_{a_{A2}a_{A2}}(\Omega)=S_{a_{B2}a_{B2}}(\Omega)$ because
of symmetry.

To characterise the quantum measurement induced correlation, we define the
correlation level at each frequency $\Omega$ to be (in terms of dB) :

$C^{\rm SN}_{A_{2}B_{2}}(\Omega)=-10\,{\rm Log}_{10}\left[1-\frac{|S^{\rm
SN}_{a_{A2}a_{B2}}(\Omega)|^{2}}{S^{\rm SN}_{a_{A2}a_{A2}}(\Omega)S^{\rm
SN}_{a_{B2}a_{B2}}(\Omega)}\right].$ (28)

With Eq. (26)(27), the terms in the square bracket of the above Eq. (28) is
approximately equal to:

$\begin{split}C^{\rm SN}_{A_{2}B_{2}}&(\Omega)\approx-10\,{\rm Log}_{10}\\\
&\left[1-\left(\frac{2\omega_{g}^{2}(\omega_{q}^{2}-\Omega^{2})}{\omega_{g}^{4}+(\omega_{q}^{2}-\Omega^{2})^{2}+\gamma_{m}^{2}(\Omega^{2}+\omega_{q}^{2})/2}\right)^{2}\right].\end{split}$
(29)

This approximate formula shows that there will be a strong correlation at
$\Omega=(\omega_{m},\omega_{-})$, while zero correlation at
$\Omega=\omega_{q}$, which well fits the peak features shown in Fig. 9
calculated using the exact formula.

The difference between the SN theory under causal-conditional prescription and
the quantum gravity can be approximately represented as (the exact formula is
a bit cumbersome):

$\begin{split}\Delta_{\rm SN-QG}\bigg{[}\frac{S^{\rm
cond}_{A_{2}A_{2}}-S_{A_{2}A_{2}}}{S_{A_{2}A_{2}}}\bigg{]}\approx\left[\frac{\tilde{\omega}^{2}}{\tilde{\omega}^{4}+4(\omega_{m}/\omega_{g})^{2}}\right]^{3}\frac{4\gamma^{2}_{m}}{\omega_{g}^{2}},\end{split}$
(30)

where $\tilde{\omega}^{2}\equiv(\Omega^{2}-\omega_{q}^{2})/\omega_{g}^{2}$ and
$S^{\rm cond}_{A_{2}A_{2}}$ is the conditional variance of the outgoing field
$\tilde{a}_{2}$ when the $\tilde{b}_{2}$ is measured. This difference is
negligibly small.

$\bullet$ Measurement of $(\tilde{a}_{1A},\tilde{a}_{2B})$— The optical
correlation spectrum induced by the quantum measurement of $\tilde{a}_{2A}$,
$\tilde{a}_{2B}$ can be derived as (we keep the leading term
$\propto\alpha^{2}$):

$S^{\rm SN}_{a_{A1}a_{B2}}(\Omega)=S^{\rm
QG}_{a_{A1}a_{B2}}(\Omega)\approx\frac{\hbar\alpha^{2}\omega_{g}^{2}}{M}\chi_{m}(\Omega)\chi_{g}(\Omega).$
(31)

Note that, in this case, the difference between $S^{\rm
SN}_{a_{A1}a_{B2}}(\Omega)$ and $S^{\rm QG}_{a_{A1}a_{B2}}(\Omega)$ is
precisely zero. This can be understood as follows: from Eq. (20)(21)(22), one
can see that the only term that proportional to $dW_{B}$ in the quantum
trajectory of mirror-A has the coefficient equal to $\hbar/2$ when
$\theta_{A}=0$ and $\theta_{B}=\pi/2$, which is independent from the second
order correlation matrix $\mathbb{V}_{A/B}$. However, the difference between
the SN theory under causal conditional prescription and the quantum gravity
theory manifests in the $\mathbb{V}_{A/B}$. Therefore, the correlation between
the $(\hat{a}_{A1},\hat{a}_{B2})$ has no difference under SN theory comparing
to the quantum gravity theory. In this case, the difference of the correlation
level will be dominated by the difference of the outgoing spectrum.

The outgoing spectrum in the SN theory when we measure the
$(\tilde{a}_{1A},\tilde{a}_{2B})$ is:

$\begin{split}&S^{\rm SN}_{a_{B2}a_{B2}}(\Omega)\approx
1+\frac{\hbar^{2}\alpha^{4}}{M^{2}}|\chi_{m}(\Omega)\chi_{g}(\Omega)|^{2}\bigg{[}\omega_{g}^{4}+\\\
&\left.\frac{(\omega_{q}^{2}-\Omega^{2})^{2}+\gamma_{m}^{2}(\Omega^{2}+\omega_{q}^{2})/2}{1+\sqrt{1+\Lambda^{4}}}\left(\frac{\Lambda^{4}}{1+\sqrt{1+\Lambda^{4}}}+\frac{2\Omega^{2}}{\omega_{q}^{2}}\right)\right],\end{split}$
(32)

Similarly, supposing the outgoing field quadratures $\tilde{a}_{1A}$ and
$\tilde{a}_{2B}$ are measured, the correlation level can be defined as:

$C^{\rm SN}_{A_{1}B_{2}}(\Omega)=-10\,{\rm Log}_{10}\left[1-\frac{|S^{\rm
SN}_{a_{A1}a_{B2}}(\Omega)|^{2}}{S_{a_{A1}a_{A1}}(\Omega)S_{a_{B2}a_{B2}}(\Omega)}\right],$
(33)

where $S_{a_{A1}a_{A1}}(\Omega)=1$ since the outgoing amplitude quadrature
does not carry the information of mirror displacement. With Eq. (31) and (32),
the correlation level can be approximately written as:

$\begin{split}C^{\rm SN}_{A_{1}B_{2}}(\Omega)=-10\,{\rm
Log}_{10}\bigg{[}1-\bigg{(}1+\frac{(\omega_{q}^{2}-\Omega^{2})^{2}+\gamma_{m}^{2}(\Omega^{2}+\omega_{q}^{2})/2}{\omega_{g}^{4}(1+\sqrt{1+\Lambda^{4}})}\left[\frac{\Lambda^{4}}{1+\sqrt{1+\Lambda^{4}}}+\frac{2\Omega^{2}}{\omega_{q}^{2}}\right]\bigg{)}^{-1}\bigg{]},\end{split}$
(34)

which demonstrates that the correlation level reaches maximum value when
$\Omega=\omega_{q}$ as exhibits in Fig. 9. We can also derive the difference
between the SN theory and the quantum gravity when we measure the
$(\tilde{a}_{1A},\tilde{a}_{2B})$, for $\Lambda\gg 1$, the result has a simple
form:

$\begin{split}\Delta_{\rm SN-QG}\bigg{[}\frac{S^{\rm
cond}_{A_{2}A_{2}}-S_{A_{2}A_{2}}}{S_{A_{2}A_{2}}}\bigg{]}\approx\frac{1}{2}\frac{\tilde{\omega}^{2}}{\tilde{\omega}^{4}+1}\frac{\gamma_{m}^{2}}{\omega_{g}^{2}}\ll
1,\end{split}$ (35)

while the result for $\Lambda\ll 1$ has a complicated form but also a
negligible magnitude.

Figure 9: The correlation spectrum between the output optical fields from the
left and right cavities induced by the classical gravity, where the effect of
thermal noise has been considered. It is easy to see that there is almost no
difference between the quantum gravity case and the classical gravity case
under the causal-conditional prescription.

The above analysis shows that, the classical gravity induced correlation
spectrum in the SN theory is almost indistinguishable from the quantum gravity
induced correlation spectrum, under the causal-conditional prescription.
Furthermore, considering the effect of thermal noise and finite quality factor
$Q_{m}$, we plot the correlation spectrum $C^{\rm SN}_{A_{2}B_{2}}(\Omega)$
and $C^{\rm SN}_{A_{1}B_{2}}(\Omega)$ in Fig. 9 using the sample parameters
listed in Tab. 2. The thermal noise enhances the indistinguishability of the
SN theory from the quantum gravity. Under the pre-selection prescription,
there will be no correlation of the output light fields when the gravity field
is classical. For completeness, the correlation spectrum under the post-
selection prescription is shown in the Supplementary Material, which is
certainly different from that in quantum gravity theory.

Figure 10: The correlation between the output optical fields from the left and
right cavities induced by the classical gravity, where the effect of thermal
noise has been considered. Upper panel: the dependence of the maximum
correlation value on the mechanical quality factor. Middle panel: the
dependence of the correlation on the mechanical quality factor $Q_{m}$ and the
intracavity power $P_{\rm cav}$. It is almost identical to the result
predicted by quantum gravity. Lower panel: The (almost zero) difference
between the correlation level in SN theory and that in quantum gravity theory.
We only choose a sub-region of parameter space in the middle panel with the
difference is around $10^{-6}-10^{-7}$ dB, while the value in other region of
parameter space is much smaller.

##  V The correlation and entanglement in nonlinear quantum mechanics

The above discussion on the linear cavity systems shows a correlation of the
outgoing light fields, which is similar to the entanglement in the standard
quantum mechanics. Looking into this phenomenon, this section devotes itself
to a discussion of this correlation at the conceptual level.

Suppose we have two quantum systems A and B at an initial product state
$|\Psi\rangle_{i}=|\Psi\rangle_{A}\otimes|\Psi\rangle_{B}$. In the standard
quantum mechanics without interaction between A and B, the state evolution
follows:

$|\Psi\rangle_{i}\rightarrow\hat{U}^{AB}|\Psi\rangle_{A}\otimes|\Psi\rangle_{B}=\hat{U}^{A}|\Psi\rangle_{A}\otimes\hat{U}^{B}|\Psi\rangle_{B},$
(36)

where we have $\hat{U}^{AB}=\hat{U}^{A}\otimes\hat{U}^{B}$. The final state is
still a product state without entanglement. When these systems are coupled in
the standard quantum mechanical way (say we have
$\hat{H}=\hat{H}_{A}+\hat{H}_{B}+\hat{H}_{AB}$), the $\hat{U}^{AB}$ is not
separable and the final state will be an entangled state:

$|\Psi\rangle_{i}\rightarrow\hat{U}^{AB}|\Psi\rangle_{A}\otimes|\Psi\rangle_{B}=\sum_{i}c_{i}|\Psi\rangle_{Ai}\otimes|\Psi\rangle_{Bi}.$
(37)

This formula means that the projective measurement on $|\Psi\rangle_{Ai}$ will
immediately collapse the joint quantum state onto $|\Psi\rangle_{Bi}$, which
exhibits a correlation between the measurement result of systems A and B. A
system consisting of two optomechanical devices coupled via quantum gravity is
in this category.

Moreover, if we have a nonlinear quantum mechanics such as Schroedinger-Newton
theory, the state evolution follows:

$\begin{split}|\Psi\rangle_{i}\rightarrow&\hat{U}^{AB}|\Psi_{A}\rangle\otimes|\Psi_{B}\rangle=\\\
&\hat{U}^{A}(|\Psi_{B}\rangle)|\Psi_{A}\rangle\otimes\hat{U}^{B}(|\Psi_{A}\rangle)|\Psi_{B}\rangle,\end{split}$
(38)

where we have used the example Hamiltonian

$\hat{H}=\hat{H}_{A}+\hat{H}_{B}+\hat{H}^{A}_{\rm
int}(|\Psi_{B}\rangle)+\hat{H}^{B}_{\rm int}(|\Psi_{A}\rangle),$ (39)

with

$\begin{split}\hat{U}^{AB}=&{\rm
exp}\left[-\frac{i}{\hbar}[\hat{H}_{A}+\hat{H}^{A}_{\rm
int}(|\Psi_{B}\rangle)]t\right]\otimes\\\ &{\rm
exp}\left[-\frac{i}{\hbar}[\hat{H}_{B}+\hat{H}^{B}_{\rm
int}(|\Psi_{A}\rangle)]t\right].\end{split}$ (40)

The final state is still a product state, however only in mathematical
appearance. In reality, the quantum states of A and B are correlated subtly as
shown in Eq. (38): measurement on the system state $|\Psi_{A/B}\rangle$ would
induce the change of evolution operator $\hat{U}^{B/A}(|\Psi_{A/B}\rangle)$,
thereby affecting the evolution of $|\Psi_{B/A}\rangle$. Therefore, although
Eq. (38) has the mathematical form of a pure product state, there is still a
correlation between the system A and B, which only exists in the nonlinear
quantum mechanics such as the SN theory.

Under the pre-selection prescription, since we do not measure the initial
state in a real experiment, we have the interaction Hamiltonian as
$\hat{H}^{B/A}_{\rm int}(|\Psi_{A/B}(t=0)\rangle)$. Therefore, this means the
final states $|\Psi_{A/B}(t_{f})\rangle$ do not depend on each other thereby
existing no correlation. However, under the post-selection or causal-
conditional prescription, the interaction Hamiltonian is $\hat{H}^{B/A}_{\rm
int}(|\Psi_{A/B}(t_{f})\rangle)$ or $\hat{H}^{B/A}_{\rm
int}(|\Psi_{A/B}(t_{f})\rangle_{c})$ ($|\Psi_{A/B}(t_{f})\rangle_{c}$ is the
conditional final state generated by continuous quantum measurement),
respectively. Therefore there will be correlations between the two systems due
to the above-discussed reasons. Further investigation of the correlations in
nonlinear quantum mechanics is beyond the scope of this work and will be
written elsewhere.

##  VI Discussion and Summary

Testing the gravitational law in the quantum era now becomes a blooming field,
where many proposals were raised in recently years [62, 63, 64, 65, 66, 67,
68]. These proposals covered many different aspects and the phenomenologies in
the quantum/gravity interface, and triggered many discussions and even debates
on these phenomenologies [69, 70, 71, 72]. This work devotes a deeper
understanding of the phenomenologies of Schroedinger-Newton theory in the
quantum optomechanical system, which is motivated by the theory of semi-
classical gravity. We pointed out that the nonlinear term in the Schroedinger-
Newton equation breaks the time-symmetry of quantum measurement in the
standard quantum mechanics and brings additional complexity. We specifically
analysed the Schroedinger-Newton phenomenology under the causal-conditional
prescription by establishing the stochastic master equation in the
Schroedinger picture. We apply the master equation to study the behaviour of
the optomechanical systems exerted by the single mirror’s self-gravity force
and mutual gravity between the two mirrors, under the continuous quantum
measurement. Our results show that, different from the predictions of the
previous work [51, 59, 45, 44], the semi-classical gravity effect under the
causal-conditional prescription is very difficult to be distinguished from the
quantum gravity effect with ponderomotive squeezing or correlation/spectrum of
outgoing fields, even in the case of optomechanical system exerted by the
single mirror’s (relatively strong) classical self-gravity when we considered
the thermal environment. The previously predicted feature [45] at
$\omega_{q}=\sqrt{\omega_{m}^{2}+\omega_{\rm SN}^{2}}$ diminishes, mainly
because that the continuous quantum measurement induces the collapse of the
joint mirror-light wave function and creates a stochastic quantum trajectory
of the mirror state. This quantum trajectory can also participate in the
classical gravitational interaction process and create correlations, as we
have discussed in Section IV. Since the causal-conditional prescription fits
our intuition about the continuous quantum measurement process, the
phenomenology obtained in this work is an important reference for the
experiment: the phenomena observed using the methods proposed in [44, 45, 51,
59] could not be the sufficient condition to recognize quantum gravity or rule
out SN theory, under the current experimental state-of-arts and the weak
gravitational interactions. It is possible to test the quantum nature of
gravity using the pondermotive squeezing spectrum of the optomechanical system
exerted by the mirror’s self-gravity only if the environmental temperature
takes extremely low values.

Another point that worthy to be discussed is the semi-classical gravity
itself. Semi-classical gravity is usually criticized since it has
contradictions to the many-worlds interpretations [5], and some
inconsistencies when combining a quantum world with a classical space-time
theater [6, 41]. However, as Steven Carlip pointed out [41], _“theoretical
arguments aganist such mixed classical-quantum models are strong, but not
conclusive, and the question is ultimately one for experiment."_. In
particular, the strong argument by Page and Geilker on the contradiction
between semi-classical gravity and the many-world interpretation may diminish
if the wavefunction collapse can be explained within the quantum mechanics,
which is still an open question. As a side-remark, Stamp et.al recently
proposed an alternative approach for reconciling quantum mechanics and
gravity, which is called correlated-worldline (CWL) theory [73, 74, 75]. The
CWL theory is fundamentally a quantum gravity theory, of which the feature is
that the different paths in the path-integral are correlated via gravity. In
the infra-limit, the CWL theory will reduce to the Schroedinger-Newton theory
333Private communications with Philip C. E. Stamp., which will be discussed
elsewhere. Therefore, pursuing the experimental/theoretical research in
testing the quantumness of gravity is still very important, despite those
criticism of semi-classical gravity.

###### Acknowledgements.

We thank Animesh Data, Mikhail Korobko, Sergey Vyachatnin, Enping Zhou, Daiqin
Su, Philip C. E. Stamp and Markus Aspelmeyer for helpful discussions. Y. M. is
supported by the start-up fund provided by Huazhong University of Science and
Technology. H. M. is supported by State Key Laboratory of Low Dimensional
Quantum Physics and the start-up fund from Tsinghua University. Y. C. is
supported by Simons Foundation.

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