# Quantumness of gravity in harmonically trapped particles

Youka Kaku<EMAIL_ADDRESS>Shin’ya Maeda
<EMAIL_ADDRESS>Yasusada Nambu
<EMAIL_ADDRESS>Yuki Osawa
<EMAIL_ADDRESS>Department of Physics, Graduate School of
Science, Nagoya University, Chikusa, Nagoya 464-8602, Japan

(December 10, 2022)

###### Abstract

This study investigates the quantumness of gravity under the setup of the
atomic interferometry from the viewpoint of mass-energy equivalence. We
evaluated interference visibility considering a particle with internal energy
levels in a harmonic trapping potential. As per the result, for a spatially
superposed gravitational source mass, interference visibility exhibits
collapse and revival behavior, which implies that an initial separable
internal state evolves to the entangled state with respect to the degrees of
freedom of the center of mass, the internal energy levels, and the external
source state. In particular, it does not exhibit revival behavior when gravity
is treated as a quantum interaction, while it revives with a finite period for
a semiclassical treatment of gravity. We also examined the temporal behavior
of entanglement negativity and found that the nonrevival behavior of
visibility reflects the creation of the entanglement between the internal
energy state and the external source state which is uniquely induced by the
quantum interaction of gravity in accordance with the weak equivalence
principle.

##  I Introduction

The unification of gravity and quantum theory is one of the most challenging
subjects in modern physics because the experiment to test quantum gravity has
not been realized thus far. Nevertheless, there are several experiments that
focus on the quantum aspect of a particle under the classical external
gravitational field to examine the relation between quantum mechanics and
classical gravity [1]. For example, in the Colella-Overhauser-Werner (COW)
experiment [2], the difference in gravitational potential causes the phase
difference of a quantum particle and leads to gravity-induced quantum
interference. This phenomenon was confirmed in the experiment using a neutron
interferometer [3].

As a further development of quantum experiments under classical gravity, the
authors of paper [4] discuss the decoherence mechanism of a quantum particle
with internal (INT) degrees of freedom using mass-energy equivalence. Herein,
the particle with the INT degrees of freedom is characterized by two systems:
center of mass (CM) system and INT system. According to mass-energy
equivalence, the particle acquires a different mass depending on the INT
energy levels. Therefore, the dynamical variables of the CM and the INT
systems entangle via special relativistic and gravitational couplings, which
is known as universal gravitational decoherence. As a result, when atomic
interferometry is considered, the interference visibility of the INT state
exhibits collapse and revival behavior. Although time dilation due to
classical gravity in a quantum clock system is recently measured in a feasible
laboratory experiment [5], universal gravitational decoherence has not been
confirmed yet. In addition, details regarding the entanglement behavior have
been studied in [6, 7], and the gravitational decoherence in the Ramsey
interferometry is also discussed by Haustein et al. [8].

Although the COW experiment [2] and the gravitational decoherence introduced
by Pikovski et al. [4] treat the quantum system under classical external
gravity, they focus on the quantum aspects of the probe particle rather than
the quantumness of gravity. Recently, as the first step to tackle the
quantumness of gravity, ideas based on quantum information have been proposed
to test the quantumness of low-energy Newtonian gravity in tabletop
experiments. These ideas are referred to as the Bose-Marletto-Vedral (BMV)
proposal [9, 10, 11], and are based on the principle of quantum information
theory, which states that local operations and classical communication (LOCC)
cannot create entanglement between two systems [12]. Based on this principle,
gravity can be clarified as a quantum interaction or not by detecting the
creation of gravity-induced entanglement. The BMV proposal received a lot of
attention, and has stimulated many other related proposals [13, 14, 15]. In
interferometry experiments, the creation of entanglement between the probe and
the environment is reflected in the quantum decoherence of the probe system
[16, 17, 18]. Carney et al. [19] explored the quantum gravity-induced
decoherence in a hybrid system that comprised a massive oscillator and a
source mass particle in cat state, while some comments on this paper are
discussed in [20] that LOCC with stochastic noise in the system can also
reproduce the decoherence.

We note here that the purpose of the BMV proposal and other related proposals
is to confirm if gravity produces quantum entanglement in the nonrelativistic
scale. These proposals just assume that the Newtonian gravitational potential
is treated as a two-body operator between gravitational sources, which is the
consequence of the quantum field theory to explain the gravitational force by
exchanging gravitons. Even though gravity-induced entanglement does not lead
to the quantization of the gravitational field immediately, it is worth a try
to explore as the first step to investigating quantum gravity for the
following two reasons; First, it may be possible to detect gravity-induced
entanglement in the near future as another way to approach the quantum feature
of gravity based on the recent progress in the quantum experiments of
macroscopic objects. For instance, since we need to treat the Planck energy
scale under strong gravity, such as nearby the blackhole, to detect graviton,
it is challenging to obtain direct evidence of graviton with terrestrial
colliders. On the other hand, gravity-induced entanglement is expected to be
the new direction to confirm the quantumness of gravity by focusing on the
gravity of the Planck mass object in the tabletop experiment, although it
cannot support the existence of graviton directly. Second, testing the
gravity-induced entanglement in the nonrelativistic scale gives a valuable
clue to the Newtonian limit of quantum gravity. Respecting the above, we will
investigate gravity-induced quantum entanglement, which we refer to as the
quantumness of gravity throughout this paper.

In this study, we propose a new approach to capture the quantumness of gravity
in the Ramsey interferometry from the point of view of mass-energy
equivalence. We assumed that a probe particle with two energy levels as the
INT degrees of freedom is trapped in the harmonic oscillator potential, and
feels external gravity, which mass source is in a cat state. Based on this
setup, we aim to detect the interference visibility of the probe particle in
Ramsey interferometry and calculate it for two cases; when the external
gravity produces the entanglement between the probe particle system and the
gravitational mass source system, or not. To perform a specific calculation,
we adopt a particular form of gravitational interaction. For the former case,
we assume the first quantization of Newtonian gravity in Eq. (17), which
contains operators of both the particle and the source system and produces
quantum entanglement between them. For the latter case, we assume the
semiclassical gravity[21, 22, 23] which will be introduced in Eq. (22).
Herein, semiclassical means that the gravitational mass source is quantized,
but gravity remains fundamentally classical. As a result, we will show that
the quantumness of gravity is reflected as a nonrevival behavior of the
interference visibility, and that it is related to the creation of genuine
tripartite system entanglement. We will also see that under the leading order
approximation with respect to the separation of the source, this nonrevival
behavior does not appear for other quantum interactions such as the
electromagnetic Coulomb interaction respecting the weak equivalence principle;
Our proposal successfully captures quantum nature unique to gravity.

The structure of the remainder of this paper is as follows. In Sec. II, we
will briefly review the Ramsey interference [24]. In Sec. III, we explore a
particle with INT degrees of freedom in an external gravitational field. In
Sec. IV we introduce the experimental setup, calculate the transition
probability in Ramsey interference following Haustein el al. [8], and reveal
the behavior of the interference visibility under quantum superposition of
gravitational source. In Sec. V, we estimate the creation of entanglement by
calculating the entanglement negativity. Section VI, we discuss the results of
the study and explore the comparison of quantized gravity and other quantum
interactions. Section VII presents a summary of this study. The unit of
$\hbar=1$ was adopted throughout the study.

##  II Ramsey interferometry

In this section, we briefly review the concept of Ramsey interferometry [24,
25, 26]. Considering a particle (atom) with two internal energy levels, the
Hamiltonian of the particle can be expressed as

$\hat{H}=\sum_{j=0,1}\hat{H}_{j}\ket{E_{j}}\bra{E_{j}},\quad\bra{E_{j}}\ket{E_{k}}=\delta_{jk},$
(1)

where $\ket{E_{j}}$ denotes the $j$th internal energy eigenstate, and
$\hat{H}_{j}$ denotes the CM Hamiltonian of the particle associated with the
$j$th internal energy level. The evolution operator of the total state is
expressed as

${\hat{U}}(t)=\exp\left[-i\hat{H}t\right]=\sum_{j}\hat{U}_{j}(t)\ket{E_{j}}\bra{E_{j}},$
(2)

where $\hat{U}_{j}=\exp\left(-i\hat{H}_{j}t\right)$ is the evolution operator
of the particle state with the $j$th internal energy level. The Ramsey
interferometry is performed as per the following steps (Fig. 1):

Figure 1: Schematic representation of Ramsey interferometry using a superposed
particle with different internal energy levels. After evolution, information
on the INT state is obtained from the interference visibility.

Step 0

Prepare the initial state of the particle with the internal energy $E_{0}$:

$\ket{\Psi(0)}=\ket{\psi(0)}\otimes\ket{E_{0}},$ (3)

where $\ket{\psi(0)}$ denotes the initial CM state.

Step 1

Apply a $\pi/2$-pulse to take the initial state into a superposition of two
internal energy eigenstates:111Identical to applying the Hadamard gate
$H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\\ 1&-1\end{bmatrix}$ to a qubit
state.

$\ket{\Psi^{\prime}(0)}=\ket{\psi(0)}\otimes\left(\ket{E_{0}}+\ket{E_{1}}\right)/\sqrt{2}.$
(4)

Step 2

Evolve the superposed state $\ket{\Psi^{\prime}(0)}$ with the evolution
operator mentioned in Eq. (2) up to time $t$:

$\displaystyle\ket{\Psi(t)}$
$\displaystyle=\hat{U}(t)\ket{\Psi^{\prime}(0)}=\biggl{(}\hat{U}_{0}(t)\ket{\psi(0)}\otimes\ket{E_{0}}+\hat{U}_{1}(t)\ket{\psi(0)}\otimes\ket{E_{1}}\biggr{)}/\sqrt{2}.$
(5)

Step 3

Apply the $\pi/2$-pulse again to make the INT state as
$\ket{E_{0}}\rightarrow(\ket{E_{0}}+\ket{E_{1}})/\sqrt{2},~{}\ket{E_{1}}\rightarrow(\ket{E_{0}}-\ket{E_{1}})/\sqrt{2}$:

$\displaystyle\ket{\Psi^{\prime}(t)}$
$\displaystyle=\frac{1}{2}\biggl{[}\hat{U}_{0}(t)\ket{\psi(0)}\otimes(\ket{E_{0}}+\ket{E_{1}})+\hat{U}_{1}(t)\ket{\psi(0)}\otimes(\ket{E_{0}}-\ket{E_{1}})\biggr{]}$
$\displaystyle=\frac{1}{2}\biggl{[}\left(\hat{U}_{0}(t)+\hat{U}_{1}(t)\right)\ket{\psi(0)}\otimes\ket{E_{0}}+\left(\hat{U}_{0}(t)-\hat{U}_{1}(t)\right)\ket{\psi(0)}\otimes\ket{E_{1}}\biggr{]}.$
(6)

Step 4

Measure the occupation probability of the lower energy eigenstate
$\ket{E_{0}}$:

$\displaystyle P(t)$
$\displaystyle=\frac{1}{4}\bra{\psi(0)}(\hat{U}_{0}^{\dagger}+\hat{U}_{1}^{\dagger})(\hat{U}_{0}+\hat{U}_{1})\ket{\psi(0)}$
$\displaystyle=\frac{1}{2}\left(1+\mathrm{Re}\left[\bra{\psi(0)}\hat{U}_{0}^{\dagger}(t)\hat{U}_{1}(t)\ket{\psi(0)}\right]\right)$
$\displaystyle=\frac{1}{2}\bigl{(}1+\mathrm{Re}\left[\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}\bigr{]}\right).$
(7)

Herein, $|\psi_{i}(t)\rangle\equiv\hat{U}_{i}|\psi(0)\rangle$. Expressing
$\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}=|\mathcal{V}(t)|\,e^{i\Theta(t)}$ with
real functions $|\mathcal{V}|$ and $\Theta$, the occupation probability is
expressed as

$P(t)=\frac{1}{2}\left(1+|\mathcal{V}(t)|\cos\Theta(t)\right).$ (8)

Herein, $|\mathcal{V}(t)|$ is the interference visibility, which contains
information regarding the internal energy levels of the particle. For example,
if $H_{i}=E_{i}$, then $\hat{U}_{j}(t)=e^{-iE_{j}t}$ and
$P(t)=\frac{1}{2}(1+\cos(\Delta E\,t)),\Delta E=E_{1}-E_{0}$. By measuring
this probability as the function of time, we can determine an energy gap of
this system. In this case, the interference visibility is equal to one.

##  III particle with internal states in the external gravitational field

We consider a particle with INT degrees of freedom in the external
gravitational field. When the particle is moving slowly in a weak
gravitational field, any internal energy contributes to the total rest mass of
the particle respecting the mass-energy equivalence as follows:

$\displaystyle\hat{m}=m_{0}\hat{I}+\hat{H}_{\text{INT}}/c^{2}$ (9)

Here, $m_{0}$ is the rest mass for the CM system, $H_{\text{INT}}$ is the
Hamiltonian for the INT system, and they satisfy $H_{\text{INT}}\ll
m_{0}c^{2}$. Let us assume the particle to have two internal energy levels,
namely $E_{0}=0$ and $E_{1}=E$. Then by substituting

$\displaystyle\hat{H}_{\text{INT}}=\sum_{j=0,1}E_{j}\ket{E_{j}}\bra{E_{j}},$
(10)

the total rest mass of the particle is given as

$\hat{m}=m_{0}\,\hat{I}+\frac{1}{c^{2}}\sum_{j=0,1}E_{j}\ket{E_{j}}\bra{E_{j}},\quad\hat{I}=\sum_{j=0,1}\ket{E_{j}}\bra{E_{j}}.$
(11)

The total state of the particle system is described by the CM system and the
INT system. The CM system is characterized by its position $\boldsymbol{x}$
and conjugate momentum $\boldsymbol{p}$. The INT system is characterized by
its energy eigenstate $\ket{E_{j}}$. Considering a weak external gravitational
field, the metric is given by

$ds^{2}=-(1+2\Phi(\boldsymbol{x})/c^{2})dt^{2}+(1-2\Phi(\boldsymbol{x})/c^{2})d\boldsymbol{x}^{2},$
(12)

where $\Phi(\boldsymbol{x})$ is the gravitational potential with
$|\Phi|/c^{2}\ll 1$. In general, the Hamiltonian of a free-falling particle
with its mass $m$ on the background spacetime is

$\displaystyle H=\sqrt{-g_{00}(m^{2}c^{4}+c^{2}g_{ij}p^{i}p^{j})}.$

This time, since we consider the particle whose mass depends on the internal
energy level as in (11), and assume that the system is on a less relativistic
scale, we obtain

$\displaystyle H\approx
H_{\text{CM}}+\left(1+\Gamma(\boldsymbol{x},\boldsymbol{p})/c^{2}\right)H_{\text{INT}},\quad\Gamma(\boldsymbol{x},\boldsymbol{p}):=-\frac{\boldsymbol{p}^{2}}{2m_{0}^{2}}+\Phi(\boldsymbol{x})$
(13)

as a lowest order approximation in Taylor expansion of $1/c^{2}\ll 1$ and
$H_{\text{INT}}/(m_{0}c^{2})\ll 1$. Here, $H_{\text{CM}}$ is the CM
Hamiltonian whose explicit form is

$\displaystyle H_{\text{CM}}$
$\displaystyle=m_{0}+\frac{\boldsymbol{p}^{2}}{2m_{0}}+m_{0}\Phi(\boldsymbol{x}),$
(14)

$H_{\text{INT}}$ is the INT Hamiltonian given in (10), and $\Gamma$ is the
red-shift factor. We take a unit of $c=1$ in the following.

Using Eq. (13), if the particle is trapped in a harmonic potential with the
stiffness constant $k$, the total Hamiltonian can be written as follows [8]:

$\displaystyle\hat{H}$
$\displaystyle=\hat{m}+\frac{\hat{p}^{2}}{2\hat{m}}+\hat{m}\,\hat{\Phi}(\hat{x})+\frac{k}{2}\,\hat{x}^{2}$
$\displaystyle=\sum_{j=0,1}\left[m_{j}+\frac{\hat{p}^{2}}{2m_{j}}+m_{j}\hat{\Phi}(\hat{x})+\frac{m_{j}\omega_{j}^{2}}{2}\,\hat{x}^{2}\right]\otimes\ket{E_{j}}\bra{E_{j}}=\sum_{j=0,1}\hat{H}_{j}\otimes\ket{E_{j}}\bra{E_{j}},$
(15)

where $\omega_{j}=(k/m_{j})^{1/2}$. Note that the internal energy level
$E_{0},~{}E_{1}$ are eigenvalues of $H_{\text{INT}}$, and they have nothing to
do with an infinite number of energy eigenvalues of $H_{\text{CM}}$ since the
INT and CM systems are orthogonal. To see that the particle system consists of
two independent systems, CM and INT, explicitly, we can rewrite Eq. (15) as

$\displaystyle\hat{H}=\sum_{j=0,1}\left\\{\sum_{n=0}^{\infty}\left(m_{j}+\hbar\omega_{j}(n+1/2)\right)|n_{j}\rangle\langle
n_{j}|\right\\}\otimes|E_{j}\rangle\langle E_{j}|$ (16)

where $n$ labels the energy eigenvalues of the CM system, and $|n_{j}\rangle$
is the $n$th eigenstate of the CM system for $\hat{H}_{j}$.

Figure 2 displays the setup for the external gravitational potential
$\hat{\Phi}$, which form depends on whether gravity creates the quantum
entanglement or not. The gravitational source mass is placed at $d+\hat{X}$.
In Sec. IV we will consider that the gravitational source is spacially
superposed at $X=\pm\beta$.

Figure 2: Setup for the proposed method. The gravitational source is placed at
$X=\pm\beta$ as a quantum mechanically superposed state (cat state), and a
trapped particle with internal energy levels interacts with it.

We first focus on the case when gravity produces the quantum entanglement
between the particle and the gravitational source systems. We will consider a
specific form of the Newtonian potential which contains both the operators of
the particle and the gravitational source systems, which we call quantized
Newtonian gravity (QG) throughout the text. Note that quantized Newtonian
gravity does not immediately imply the Newtonian limit of quantum gravity
theory, but we adopt it as a particular gravitational interaction that
produces entanglement. The QG potential is given as

$\displaystyle\hat{\Phi}$
$\displaystyle=\frac{-GM}{d+\hat{X}-\hat{x}}\approx-\frac{GM}{d}\left\\{1+\frac{\hat{x}-\hat{X}}{d}+\left(\frac{\hat{x}-\hat{X}}{d}\right)^{2}\right\\}$
(17) $\displaystyle=-A(\hat{X})-B(\hat{X})\,\hat{x}-C\,\hat{x}^{2},$ (18)

where

$\displaystyle
A(\hat{X}):=\frac{GM}{d}\left(1-\frac{\hat{X}}{d}+\frac{\hat{X}^{2}}{d^{2}}\right),\quad
B(\hat{X}):=\frac{GM}{d^{2}}\left(1-\frac{2\hat{X}}{d}\right),\quad
C:=\frac{GM}{d^{3}}.$ (19)

In the second equality of Eq. (17), we performed Taylor expansion for
$d\gg|\langle\hat{X}^{n}\rangle|,|\langle\hat{x}^{n}\rangle|~{}(n=0,1,2,\cdots)$.
Therefore, the Hamiltonian is

$\hat{H}_{\text{QG}}=\sum_{j=0,1}\left[\mathcal{E}_{j}(\hat{X})+\frac{\hat{p}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2}}{2}\left(\hat{x}-\Delta_{j}(\hat{X})\right)^{2}\right]\ket{E_{j}}\bra{E_{j}},$
(20)

where

$\displaystyle\mathcal{E}_{j}(\hat{X}):=m_{j}\left(1-A(\hat{X})-\frac{B^{2}(\hat{X})}{2\omega_{j}^{2}}\right),\quad\Delta_{j}(\hat{X}):=B(\hat{X})/\omega_{j}^{2}.$
(21)

Herein, we redefined a shifted angular frequency as
$\omega_{j}^{2}-C\to\omega_{j}^{2}$. $\mathcal{E}_{j}(\hat{X})$ represents the
offset of the total energy of the CM system with the $j$th INT state, and
$\Delta_{j}(\hat{X})$ represents the shift in the harmonic potential due to
the weak external gravitational field.

The information of source position $\hat{X}$ is included in
$\mathcal{E}_{j}(\hat{X})$ and $\Delta_{j}(\hat{X})$, which indicates the
entanglement between the CM system $\hat{x}$ and the gravitational source (S)
system $\hat{X}$. Moreover, since the INT system also couples with the CM
system and S system as well, the total system is in a tripartite entangled
state. In particular, the entanglement between the INT system and S system is
uniquely induced by the gravitational coupling
$\hat{m}\,\hat{\Phi}(\hat{x},\hat{X})$; No other quantum interaction can
create this entanglement since they do not couple to the mass $\hat{m}$ due to
the weak equivalence principle. Further discussion will be given in Sec. VI.
Figure 3 displays a schematic representation of the potential of Hamiltonian
$\hat{H}_{j}$. Shifts of the vertex height and the symmetry axis of the
potential depend on INT states and this behavior is caused by the coupling
between the CM system and INT system via the weak external gravitational
field. Section IV will discuss that these shifts of the potential result in
collapse and revival of interferometric visibility, which can be interpreted
as the universal decoherence of the INT state via external gravitational field
[6, 4].

Figure 3: Schematic representation of the potential for CM degrees of freedom.
Blue curve: the potential of $\hat{H}_{0}$. Red curve: the potential of
$\hat{H}_{1}$. Shifts in the symmetry axis and the vertex height of the
potential are caused by the differences $\Delta_{1}(X)-\Delta_{0}(X)$ and
$\mathcal{E}_{1}(X)-\mathcal{\hat{E}}_{0}(X)$. The difference in the vertex
height depends on the location of the gravitational source $X$.

In the semiclassical treatment of gravity, hereafter referred to as classical
gravity (CG), $\hat{\Phi}$ is replaced by the expectation value concerning the
state of the gravitational source
$\langle{\hat{\Phi}}\rangle_{S}=\langle{\varphi_{S}}|\hat{\Phi}|{\varphi_{S}}\rangle$
which does not create quantum entanglement between the particle and the
gravitational source systems ,unlike the QG case. Therefore, for
$d\gg|\langle\hat{X}^{n}\rangle_{S}|,|\langle\hat{x}^{n}\rangle|~{}(n=0,1,2,\cdots)$
, the gravitational potential is approximately given by

$\displaystyle\hat{\Phi}$
$\displaystyle=\left\langle\frac{-GM}{d+\hat{X}-\hat{x}}\right\rangle_{S}\approx-\frac{GM}{d}\left\\{1+\frac{\hat{x}-\langle{\hat{X}}\rangle_{S}}{d}+\frac{\hat{x}^{2}-2\langle{\hat{X}}\rangle_{S}\,\hat{x}+\langle{\hat{X}^{2}}\rangle_{S}}{d^{2}}\right\\}$
(22) $\displaystyle=-A-B\,\hat{x}-C\,\hat{x}^{2},$ (23)

where

$\displaystyle
A:=\frac{GM}{d}\left(1-\frac{\langle{\hat{X}}\rangle_{S}}{d}+\frac{\langle{\hat{X}^{2}}\rangle_{S}}{d^{2}}\right),\quad
B:=\frac{GM}{d^{2}}\left(1-\frac{2\langle{\hat{X}}\rangle_{S}}{d}\right),\quad
C:=\frac{GM}{d^{3}}.$ (24)

Therefore, the corresponding Hamiltonian becomes

$\hat{H}_{\text{CG}}=\sum_{j=0,1}\left[\mathcal{E}_{j}+\frac{\hat{p}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2}}{2}\left(\hat{x}-\Delta_{j}\right)^{2}\right]\ket{E_{j}}\bra{E_{j}},$
(25)

where

$\displaystyle\mathcal{E}_{j}=m_{j}\left(1-A-\frac{B^{2}}{2\omega_{j}^{2}}\right),\quad\Delta_{j}=B/\omega_{j}^{2}.$
(26)

In the CG case, CM and INT can entangle through relativistic and classical
gravitational couplings, while INT and S cannot entangle.

Next, we introduce the annihilation operator for the CM system for later use:

$\hat{a}_{j,\hat{X}}=\sqrt{\frac{m_{j}\omega_{j}}{2}}\left(\hat{x}+i\frac{\hat{p}}{m_{j}\omega_{j}}-\Delta_{j}(\hat{X})\right).$
(27)

Therefore, the Hamiltonian for the QG case becomes

$\hat{H}_{\text{QG}}(\hat{X})=\sum_{j=0,1}\left[\mathcal{E}_{j}(\hat{X})+\omega_{j}\left(\hat{a}_{j,\hat{X}}{}^{\\!\\!\\!\\!\\!\\!\\!\\!{\dagger}}\,\,\,\,\,\hat{a}_{j,\hat{X}}+\frac{1}{2}\right)\right]\ket{E_{j}}\bra{E_{j}}.$
(28)

The Hamiltonian of the CG case is obtained by replacing the functions of
$\hat{X}$ with the expectation value for the S state, $f(\hat{X})\to\langle
f(\hat{X})\rangle_{S}$, in Eq. (28).

##  IV Evolution of the particle state and Visibility

In this section, we evaluate the interference visibility. We assume that the
gravitational source state is in superposition of two Gaussian states (See
Fig. 2),

$\ket{\varphi_{S}}=\frac{1}{\sqrt{N}}\left(\ket{\varphi_{-\beta}}+\ket{\varphi_{+\beta}}\right),\quad\varphi_{\pm\beta}(X)=\bra{X}\ket{\varphi_{\pm\beta}}=\left(\frac{1}{\pi\sigma^{2}}\right)^{1/4}e^{-(X\mp\beta)^{2}/(2\sigma^{2})},\quad
N=2(1+e^{-\beta^{2}/\sigma^{2}}),$ (29)

where $\beta$ is the separation of the cat state and $\sigma$ is the width of
each Gaussian state. The evolution operator associated with the $j$th INT
state can be expressed as

$e^{-i\mathcal{E}_{j}(\hat{X})t}\,e^{-i\omega_{j}(\hat{a}_{j,\hat{X}}{}^{\\!\\!\\!\\!\\!\\!\\!\\!{\dagger}}\,\,\,\,\hat{a}_{j,\hat{X}}+1/2)t}=e^{-i\mathcal{E}_{j}(\hat{X})t}\,\hat{U}_{j,\hat{X}}(t),$
(30)

where $\hat{U}_{j,\hat{X}}(t)=e^{-i\hat{h}_{j}(\hat{X})t}$ is the evolution
operator with the harmonic oscillator Hamiltonian
$\hat{h}_{j}(\hat{X}):=\omega_{j}(\hat{a}_{j,\hat{X}}{}^{\\!\\!\\!\\!\\!\\!\\!\\!{\dagger}}\,\,\,\,\hat{a}_{j,\hat{X}}+1/2)$.
As the initial state of the CM system, we prepare the ground state of
$\hat{h}_{0}(X=0)$ as

$\displaystyle\psi_{\text{ini}}(x)=\left(\frac{a_{0}}{\pi}\right)^{1/4}\exp\left[-\frac{a_{0}}{2}(x-\Delta_{0}(0))^{2}\right],\quad
a_{0}=m_{0}\,\omega_{0}.$ (31)

Therefore, the time evolution of the total state associated with the $j$th INT
state becomes

$\ket{\Psi_{j}(t)}=e^{-i\mathcal{E}_{j}(\hat{X})t}\,\hat{U}_{j,\hat{X}}(t)\ket{\psi_{\text{ini}}}\otimes\ket{\varphi_{S}}=\int
dX\,\varphi_{S}(X)\,e^{-i\mathcal{E}_{j}(X)t}\ket{\psi_{j,X}(t)}\otimes\ket{X},$
(32)

Here, we defined
$\ket{\psi_{j,X}(t)}:=\hat{U}_{j,X}(t)\ket{\psi_{\text{ini}}}$. Introducing
the propagator of the harmonics oscillator, we obtain

$\displaystyle K_{j,X}(x,t;y,0):=\bra{x}\hat{U}_{j,X}(t)\ket{y}$
$\displaystyle=\sqrt{\frac{a_{j}}{2\pi
i\sin(\omega_{j}t)}}\exp\left[\frac{ia_{j}}{2\sin(\omega_{j}t)}\biggl{\\{}((x-\Delta_{j}(X))^{2}+(y-\Delta_{j}(X))^{2})\cos(\omega_{j}t)-2(x-\Delta_{j}(X))(y-\Delta_{j}(X))\biggr{\\}}\right],$
(33)

and the wave function of the CM state can be explicitly calculated as [27]

$\displaystyle\psi_{j,X}(x,t)=\bra{x}\hat{U}_{j,X}(t)\ket{\psi_{\text{ini}}}=\int
dy\bra{x}\hat{U}_{j,X}(t)\ket{y}\bra{y}\ket{\psi_{\text{ini}}}=\int
dy\,K_{j,X}(x,t;y,0)\psi_{\text{ini}}(y)$
$\displaystyle=\left(\frac{a_{0}}{\pi}\right)^{1/4}\sqrt{\frac{a_{j}}{a_{j}\cos(\omega_{j}t)+ia_{0}\sin(\omega_{j}t)}}\,\exp\left[-\frac{a_{j}^{2}}{2a_{0}}\frac{R_{j,X}(t,x)+iI_{j,X}(t,x)}{1+(a_{j}^{2}/a_{0}^{2}-1)\cos^{2}(\omega_{j}t)}\right],$
(34)

where $a_{j}:=m_{j}\omega_{j}$ and

$\displaystyle
R_{j,X}(t,x):=\biggl{[}(x-\Delta_{j}(X))-(\Delta_{0}(X)-\Delta_{j}(X))\cos(\omega_{j}t)\biggr{]}^{2},$
(35) $\displaystyle
I_{j,X}(t,x):=-\sin(\omega_{j}t)\left[\frac{a_{0}}{a_{j}}\biggl{\\{}\left((x-\Delta_{j}(X))^{2}+(\Delta_{0}(X)-\Delta_{j}(X))^{2}\right)\cos(\omega_{j}t)-2(x-\Delta_{j}(X))(\Delta_{0}(X)-\Delta_{j}(X))\biggr{\\}}\right.$
$\displaystyle\left.\hskip
250.0pt-\frac{a_{j}}{a_{0}}(x-\Delta_{j}(X))^{2}\cos(\omega_{j}t)\right].$
(36)

To obtain the transition probability Eq. (7), we should evaluate

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}$ $\displaystyle=\int
dX\,|\varphi_{S}(X)|^{2}\,e^{-i(\mathcal{E}_{1}(X)-\mathcal{E}_{0}(X))t}\bra{\psi_{0,X}(t)}\ket{\psi_{1,X}(t)}.$
(37)

The source state is assumed to be prepared as Eq. (29) to reveal quantum
superposition on gravity, and has no dynamics for simplicity of treatment.
Furthermore, the S system is regarded as a two-level state with $X=\pm\beta$.
Therefore, Eq. (32) reduces to the following expression,

$\displaystyle\ket{\Psi_{j}(t)}\approx\frac{1}{\sqrt{N}}\sum_{s=\pm\beta}e^{-i\mathcal{E}_{j}(s)t}\ket{\psi_{j,s}(t)}\otimes\ket{\varphi_{s}}.$
(38)

Herein, we made the assumption that $\psi_{j,X}(t,x)$ and
$e^{-i\mathcal{E}_{j}(X)t}$ in Eq. (32) do not change rapidly with respect to
$X$ within the width $\sigma$ of the Gaussian function in $\varphi_{S}(X)$.
Each assumption requires the following condition respectively,

$\displaystyle
a_{j}\left(\frac{GM}{\omega^{2}d^{3}}\right)^{2}\ll\frac{1}{\sigma^{2}},\quad\frac{Gm_{j}M}{d^{3}}t\ll\frac{1}{\sigma^{2}}.$
(39)

The former inequality is satisfied when the Gaussian dispersion of the CM
system and the source system are in the same order, and the displacement of
the CM state is relatively small compared to $d$:
$\frac{GM}{\omega^{2}d^{3}}\sim\Delta_{j}/d\ll 1$. The latter inequality gives
the upper bound of the valid time range $t$ for the form of the state
expressed in Eq. (38). Then Eq. (37) is approximately given as a summation of
four terms,

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}\approx\frac{1}{N}\\!\\!\sum_{s_{1},s_{2}=\pm\beta}\\!\\!\bra{\varphi_{s_{1}}}\ket{\varphi_{s_{2}}}e^{-i(\mathcal{E}_{1}(s_{2})-\mathcal{E}_{0}(s_{1}))t}\bra{\psi_{0,s_{1}}(t)}\ket{\psi_{1,s_{2}}(t)}.$
(40)

Figure 4 depicts a schematic picture of the Ramsey interferometry under
quantum gravity (QG). Since the CM state depends not only on the INT energy
level $E_{i}$ but also the gravitational source position $X=\pm\beta$, its
time evolution is given by a superposition of four different worldline
branches (Additional details regarding the states involved in the interference
visibility are given in Appendix A).

Figure 4: Schematic picture of Ramsey interferometry in the QG case. Four
different worldline branches are involved in interference visibility.

For the CG case, the Ramsey interferometry is derived analogously by replacing
the functions of $\hat{X}$ with the expectation value of $f(\hat{X})\to\langle
f(\hat{X})\rangle_{S}$. Then, Eq. (37) can be expressed as

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}=e^{-i(\mathcal{E}_{1}-\mathcal{E}_{0})t}\bra{\psi_{0}(t)}\ket{\psi_{1}(t)},$
(41)

where $\mathcal{E}_{j}$ is as per in Eq. (26), and
$\ket{\psi_{j}(t)}:=e^{-i\hat{h}_{j}t}\ket{\psi_{\text{ini}}}$ is the time
evolved CM state with the harmonics oscillator Hamiltonian for CG
$\hat{h}_{j}:=\langle\hat{h}_{j}(\hat{X})\rangle_{S}$. The Ramsey
interferometry under CG is depicted in Fig. 1. We see that the time evolution
of the CM system is given by a superposition of two worldline branches
concerning the INT energy levels $E_{0}$ or $E_{1}$.

Although it is possible to derive an explicit but complicated formula of Eq.
(40) using a straightforward calculation of the Gaussian integral, we only
present the plotting results in subsequent figures Figs. 5, 6, 7, and 11.
Instead, in the following, we derive an approximate analytical form of Eq.
(37) by focusing only up to the leading term of the Taylor expansion of the
gravitational potential expressed in Eq. (18), which helps our qualitative
understanding of gravity-induced decoherence. If we neglect the subleading
terms of the Taylor expansion parameters $x/d$ and $X/d$ in (18), Eq. (19) can
be expressed as

$\displaystyle A(\hat{X})\to\frac{GM}{d}\left(1-\frac{\hat{X}}{d}\right),\quad
B(\hat{X})\to\frac{GM}{d^{2}},\quad C\to 1.$ (42)

Herein, the $\hat{X}$ dependence only appears in $\mathcal{E}_{j}(\hat{X})$,
and the coupling of the CM system operator $\hat{x}$ and the S system operator
$\hat{X}$ disappears as $\Delta_{j}(\hat{X})\to\Delta_{j}$. Although the BMV
proposal [9, 10] focuses on the entanglement between the CM system and S
system, this study aims to investigate the entanglement between the INT system
and S system, which is uniquely induced by gravitational coupling due to the
weak equivalence principle. (Further details will be revisited in Sec. VI.)
Therefore, this approximation is enough to capture the crucial effect of the
quantumness of gravity in our setup of the Ramsey interferometry.

Since $\hat{h}_{j}$ does not depend on $\hat{X}$ for now, the $X$ integration
in Eq. (37) can be simplified to

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}$
$\displaystyle=\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}\times\int
dX\,|\varphi_{S}(X)|^{2}\,e^{-i(\mathcal{E}_{1}(X)-\mathcal{E}_{0}(X))t}.$
(43)

Evaluating the inner product of CM states $\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}$
with (42), we obtain

$\displaystyle\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}=\int
dx\,\psi_{0}^{*}(t,x)\psi_{1}(t,x)=:|\mathcal{V}_{C}(t)|\,e^{i\Theta_{C}(t)},$
(44)

where

$\displaystyle|\mathcal{V}_{C}(t)|$
$\displaystyle=\left(\frac{4a_{0}\,a_{1}}{4a_{0}\,a_{1}\cos^{2}(\omega_{1}t)+(a_{0}+a_{1})^{2}\sin^{2}(\omega_{1}t)}\right)^{1/4}\exp\left[-\frac{2a_{0}\,a_{1}^{2}(\Delta_{0}-\Delta_{1})^{2}\sin^{2}(\omega_{1}t/2)}{a_{0}^{2}+a_{1}^{2}+(a_{0}^{2}-a_{1}^{2})\cos(\omega_{1}t)}\right],$
(45) $\displaystyle\Theta_{C}(t)$
$\displaystyle=-\frac{a_{0}\,a_{1}^{2}\left(\Delta_{0}-\Delta_{1}\right)^{2}\sin(\omega_{1}t)}{a_{0}^{2}+a_{1}^{2}+(a_{0}^{2}-a_{1}^{2})\cos(\omega_{1}t)}-\frac{1}{2}\arg\left[e^{-i\omega_{0}t}\left(2a_{0}/a_{1}\cos(\omega_{1}t)+i\left(1+(a_{0}/a_{1})^{2}\right)\sin(\omega_{1}t)\right)\right].$
(46)

It should be noted that $\mathcal{V}_{C}(t)$ is a $2\pi/\omega_{1}$ periodic
function in time $t$ that reflects the oscillation of the CM state in the
harmonic oscillator potential. The $X$ integration in Eq. (43) reads to

$\displaystyle\int
dX\,|\varphi_{S}(X)|^{2}\,e^{-i(\mathcal{E}_{1}(X)-\mathcal{E}_{0}(X))t}=|\mathcal{V}_{Q}(t)|\,e^{i\Theta_{\mathcal{E}}(t)},$
(47)

where

$\displaystyle|\mathcal{V}_{Q}(t)|$
$\displaystyle=\frac{2}{N}\exp\left[-\left(\frac{GME}{2d}\frac{\sigma}{d}\,t\right)^{2}\right]\left|\cos\left(\frac{GME}{d}\frac{\beta}{d}\,t\right)+e^{-\beta^{2}/\sigma^{2}}\right|,$
(48) $\displaystyle\Theta_{\mathcal{E}}(t)$
$\displaystyle=-(\mathcal{E}_{1}(0)-\mathcal{E}_{0}(0))\,t.$ (49)

Finally, when we treat up to the leading term of $x/d,~{}X/d$, the analytic
expression of Eq. (37) is given by

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}=|\mathcal{V}_{C}(t)\mathcal{V}_{Q}(t)|\,e^{i(\Theta_{C}(t)+\Theta_{\mathcal{E}}(t))}\qquad\text{(QG
case)},$ (50)

where the time-dependent functions
$|\mathcal{V}_{C}|,\Theta_{C},|\mathcal{V}_{Q}|,\Theta_{\mathcal{E}}$ are
given in Eqs. (45), (46), (48), and (49), respectively. For the CG case, the
treatment can be applied by replacing $\hat{X}\to\langle X\rangle=0$, which
reduces the $X$ integration in Eq. (47) to 1, and we get the final expression
for the inner product as

$\displaystyle\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}=|\mathcal{V}_{C}(t)|\,e^{i(\Theta_{C}(t)+\Theta_{\mathcal{E}}(t))}\qquad\text{(CG
case)}.$ (51)

Let us discuss some properties of interference for different treatments of
gravity based on Eqs. (50) and (51) in the following.

First, when there is no external gravitational source ($G=0$ and
$\Delta_{0}=\Delta_{1}=0$), the probability oscillates with a period
determined by the energy gap of the INT system
$\mathcal{E}_{1}-\mathcal{E}_{0}=E$. The envelope of the oscillation is
determined by visibility $|\mathcal{V}_{C}(t)|$, whose time period
$\pi/\omega_{1}$ reflects the evolution of the squeezed state, as seen Eq.
(46) (and also in Appendix A). The decoherence arises from the entanglement
between the CM system and INT systems induced by the kinetic term. In other
words, a particle with a different energy level evolves along a different
branch of world line with a different proper times due to the special
relativistic redshift, as seen in Fig. 1, which results in universal
decoherence [4].

For the CG case,the visibility factor $|\mathcal{V}_{C}(t)|$ contains two
typical periods: (i) $\pi/\omega_{1}$, which corresponds to the period of
squeezed state, and (ii) $2\pi/\omega_{1}$, which corresponds to the period of
coherent state originated from the interaction of the CM state with the
external CG [8] (see also Appendix A). Each periodic decoherence behavior in
CG occurs due to the entanglement between the CM and the INT systems induced
by the kinetic term, and the semiclassical gravitational interaction; the
later refers to universal decoherence caused by a gravitational redshift [4].
In the left panel of Fig. 5, we show the time dependence of the probability
$P(t)$ under CG with a green solid line, which is obtained by evaluating Eq.
(41). It exhibits oscillation with a period $\approx
2\pi/(\mathcal{E}_{1}-\mathcal{E}_{0})$, which roughly corresponds to the
energy gap of the INT system with relativistic and gravitational corrections.
The dashed lines denote its envelope $(1+|\mathcal{V}_{C}(t)|)/2$, and
parameters are chosen as
$(m_{0}^{3}/k)^{1/2}=10,~{}(m_{0}/m_{1})^{1/2}=0.5,~{}Gm_{0}M/(kd^{3})=0.015,~{}m_{0}^{1/4}d=10,~{}\beta/d=0.01,~{}\sigma/d=0.001$.
The visibility again exhibits revival behavior with the $2\pi/\omega_{1}$
period, thereby reflecting semi-classical gravity induced entanglement.

Figure 5: Probability $P(t)$ of the CG case (left panel) and QG case (right
panel). Solid line: time dependence of the probability $P(t)$. Dashed line:
the envelope of $P(t)$, which is $\frac{1}{2}(1+|\mathcal{V}(t)|)$.

Next, we conduct an in-depth evaluation of the visibility of the QG case in
detail as the main result of this study and clarify the effect of the quantum
superposition of the gravitational source on the visibility of the Ramsey
interferometry. As an effect of the quantumness of gravity, the visibility
additionally contains $|\mathcal{V}_{Q}(t)|$. Furthermore, a modulation with a
longer period appears in the visibility due to the following reasons: (i) the
exponential factor in $|\mathcal{V}_{Q}(t)|$ of Eq. (48) causes the temporal
decay of coherence. It contains the gravitational coupling between the source
mass and the INT energy gap and vanishes for $\sigma\to 0$. This indicates the
entanglement induced by the quantum gravitational interaction between the INT
system and the Gaussian dispersion of the S system. Since its decoherence time
scale is given by $t\sim\left(\frac{GME}{2d}\frac{\sigma}{d}\right)^{-1}$, the
system decoheres more rapidly for larger $\sigma$. (ii) The product of two
kinds of periodic functions, namely the cosine function in
$\mathcal{V}_{Q}(t)$ and the function $\mathcal{V}_{C}(t)$ with a period
$2\pi/\omega_{1}$ causes a beat in the visibility. The cosine function in
$\mathcal{V}_{Q}(t)$ comes from the gravitational coupling between the source
mass and the INT energy gap, and it vanishes for $\beta\to 0$. This indicates
the entanglement induced by the quantum gravitational interaction between the
INT system and the distant cat state. Since its decoherence time scale is
given by $t\sim\left(\frac{GME}{d}\frac{\beta}{d}\right)^{-1}$, the system
decoheres more rapidly for larger $\beta$.

Decoherence effect (i) is caused by the Gaussian spread of the source mass
state, and causes nonrevival behavior of the visibility. It is induced by the
entanglement between the INT state $\ket{E_{j}}$ and the Gaussian state $\int
dXe^{-X^{2}/2\sigma^{2}}\ket{X}$ with infinite rank. Decoherence effect (ii)
is induced by the entanglement between the INT state and the distant cat state
$\ket{\pm\beta}$ with rank 2. Therefore, the combination of the INT state and
the distant cat state subsystem mentioned in (ii) is much easier to get
entangled compared to the the combination of the INT state $\ket{E_{j}}$ and
the Gaussian state $\int dXe^{-X^{2}/2\sigma^{2}}\ket{X}$ with infinite rank
mentioned in (i). To summarize, the quantumness of gravity is reflected in the
visibility as a nonrevival behavior of visibility (coherence), which is
induced by the entanglement between the INT system and the S system as
mentioned in Sec. III.

In the right panel of Fig. 5, we showed the time dependence of the probability
$P(t)$ for the QG case with a solid line, and its envelope
$(1+|\mathcal{V}_{C}(t)\mathcal{V}_{Q}(t)|)/2$ with a dotted line. The
parameters are set to
$(m_{0}^{3}/k)^{1/2}=10,~{}(m_{0}/m_{1})^{1/2}=0.5,~{}Gm_{0}M/(kd^{3})=0.015,~{}m_{0}^{1/4}d=10,~{}\beta/d=0.01,~{}\sigma/d=0.001$.
The figure displays the beat in the envelope of its oscillation, and unlike
the CG case, the visibility does not exhibit revival behavior. It should be
noted that the exponential decay in $\mathcal{V}_{QG}(t)$ obtained in Eq. (48)
is not obvious here, since the exponential factor in Eq. (37) is not dominant
in the integral unless a time $t$ violates the condition expressed in Eq.
(39).

Finally, the behavior of visibility is explored by comparing the CG and QG
cases displayed in Fig. 6. The blue and red lines denote the CG case and QG
case respectively. The parameters are identical to those mentioned in Fig. 5.
As per the figure, the visibility of QG returns to nearly one after a period
of $(GME\beta/d^{2})^{-1}$, while its value decays by $\exp\left(-(GME\sigma
t/2d^{2})^{2}\right)$, as estimated in Eq. (48). To summarize, the quantumness
of gravity appears as a nonrevival behavior of interference visibility.
Stronger decoherence in QG compared to that in CG indicates the entanglement
sharing between the particle and S systems.

Figure 6: Visibility for the CG case (blue line) and the QG case (red line).
Owing to the decoherence effect induced by the spread of the source state, the
visibility of QG does not come back to unity and does not show revival
behavior.

##  V Negativity of reduced bipartite states

In this section, we will evaluate the entanglement between the CM, INT, and
the gravitational source systems, which provide a better understanding of the
decoherence in the Ramsey interference mentioned in Sec. IV. The state of the
total system is given by

$\displaystyle\ket{\Psi(t)}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=0,1}e^{-i\mathcal{E}_{j}(\hat{X})t}\,|{E_{j}}\rangle\otimes|{\psi_{j,\hat{X}}(t)}\rangle\otimes|{\varphi_{S}}\rangle.$
(52)

To see the entanglement structure during the collapse and revival of coherence
in Fig. 6, we assumed the same approximation adopted in Sec. IV for Eq. (39).
As preparation, the CM states and the source Gaussian state are rewritten as

$\displaystyle\begin{bmatrix}|{C_{0}}\rangle\\\ |{C_{1}}\rangle\\\
|{C_{2}}\rangle\\\
|{C_{3}}\rangle\end{bmatrix}=\begin{bmatrix}|{C_{J[0,0]}}\rangle\\\
|{C_{J[0,1]}}\rangle\\\ |{C_{J[1,0]}}\rangle\\\
|{C_{J[1,1]}}\rangle\end{bmatrix}:=\begin{bmatrix}|{\psi_{0,-\beta}}\rangle\\\
|{\psi_{0,+\beta}}\rangle\\\ |{\psi_{1,-\beta}}\rangle\\\
|{\psi_{1,+\beta}}\rangle\end{bmatrix},\quad\begin{bmatrix}|{S_{0}}\rangle\\\
|{S_{1}}\rangle\end{bmatrix}:=\begin{bmatrix}|{\varphi_{-\beta}}\rangle\\\
|{\varphi_{+\beta}}\rangle\end{bmatrix},$ (53)

where we define a the function $J[j,k]:=2j+k=\\{0,1,2,3\\}$, and the offset of
the total energy $\mathcal{E}_{J[j,k]}:=\mathcal{E}_{j}((2k-1)\beta)$. The
state of the total system is expressed as

$\displaystyle\ket{\Psi(t)}$
$\displaystyle=\frac{1}{\sqrt{2N}}\sum_{j=0,1}\sum_{k=0,1}e^{-i\mathcal{E}_{J[j,k]}t}\,|{E_{j}}\rangle\otimes|{C_{J[j,k]}}\rangle\otimes|{S_{k}}\rangle.$
(54)

Therefore, the density matrix of the total system is obtained as

$\rho=\ket{\Psi}\bra{\Psi}=\frac{1}{2N}\sum_{j_{1},j_{2}}\sum_{k_{1},k_{2}}e^{-i(\mathcal{E}_{J[j_{1},k_{1}]}-\mathcal{E}_{J[j_{2},k_{2}]})t}\ket{E_{j_{1}}}\bra{E_{j_{2}}}\otimes|{C_{J[j_{1},k_{1}]}}\rangle\langle{C_{J[j_{2},k_{2}]}}|\otimes\ket{S_{k_{1}}}\bra{S_{k_{2}}}.$
(55)

We considered three different reduced bipartite states as follows:

$\displaystyle\rho_{\text{INT:CM}}:=\mathrm{Tr}_{\text{S}}\rho=\frac{1}{2N}\\!\\!\sum_{j_{1}j_{2},k_{1}k_{2}}e^{-i(\mathcal{E}_{J[j_{1},k_{1}]}-\mathcal{E}_{J[j_{2},k_{2}]})t}\bra{S_{k_{2}}}\ket{S_{k_{1}}}\ket{E_{j_{1}}}\bra{E_{j_{2}}}\otimes|{C_{J[j_{1},k_{1}]}}\rangle\langle{C_{J[j_{2},k_{2}]}}|,$
(56)
$\displaystyle\rho_{\text{INT:S}}:=\mathrm{Tr}_{\text{CM}}\rho=\frac{1}{2N}\\!\\!\sum_{j_{1}j_{2},k_{1}k_{2}}e^{-i(\mathcal{E}_{J[j_{1},k_{1}]}-\mathcal{E}_{J[j_{2},k_{2}]})t}\langle{C_{J[j_{2},k_{2}]}}|{C_{J[j_{1},k_{1}]}}\rangle\ket{E_{j_{1}}}\bra{E_{j_{2}}}\otimes\ket{S_{k_{1}}}\bra{S_{k_{2}}},$
(57)
$\displaystyle\rho_{\text{CM:S}}:=\mathrm{Tr}_{\text{INT}}\rho=\frac{1}{2N}\sum_{j,k_{1}k_{2}}e^{-i(\mathcal{E}_{J[j,k_{1}]}-\mathcal{E}_{J[j,k_{2}]})t}\ket{S_{k_{1}}}\bra{S_{k_{2}}}\otimes|{C_{J[j,k_{1}]}}\rangle\langle{C_{J[j,k_{2}]}}|.$
(58)

To evaluate the negativity for the reduced states, we introduced an
orthonormal basis for the S state and the CM state. For the source state,
using the orthonormal basis
$\\{\ket{s_{k}}\\},~{}k=\\{0,1\\},~{}\bra{s_{k_{1}}}\ket{s_{k_{2}}}=\delta_{k_{1}k_{2}}$,
we obtain

$\begin{bmatrix}\ket{S_{0}}\\\
\ket{S_{1}}\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}\sqrt{1+v_{S}}&\sqrt{1-v_{S}}\\\
\sqrt{1+v_{S}}&-\sqrt{1-v_{S}}\end{bmatrix}\begin{bmatrix}\ket{s_{0}}\\\
\ket{s_{1}}\end{bmatrix},\quad
v_{S}=\bra{S_{0}}\ket{S_{1}}=e^{-\beta^{2}/\sigma^{2}}.$ (59)

This relation is written using a $2\times 2$ matrix $S$ as
$\ket{S_{k}}=\sum_{\ell=0,1}S_{k\ell}\ket{s_{\ell}}$. For the CM state
$|{C_{J[j,k]}}\rangle$, the orthonormal basis
$\\{\ket{c_{J}}\\},~{}J=\\{0,1,2,3\\},\bra{c_{J_{1}}}\ket{c_{J_{2}}}=\delta_{J_{1}J_{2}}$
is obtained using the Gram-Schmidt orthonormalization as follows:

$\begin{bmatrix}|{C_{0}}\rangle\\\ |{C_{1}}\rangle\\\ |{C_{2}}\rangle\\\
|{C_{3}}\rangle\end{bmatrix}=\begin{bmatrix}\sqrt{N_{0}}&0&0&0\\\
\bra{c_{0}}\psi_{1}\rangle&\sqrt{N_{1}}&0&0\\\
\bra{c_{0}}\psi_{2}\rangle&\bra{c_{1}}\psi_{2}\rangle&\sqrt{N_{2}}&0\\\
\bra{c_{0}}\psi_{3}\rangle&\bra{c_{1}}\psi_{3}\rangle&\bra{c_{2}}\psi_{3}\rangle&\sqrt{N_{3}}\end{bmatrix}\begin{bmatrix}|{c_{0}}\rangle\\\
|{c_{1}}\rangle\\\ |{c_{2}}\rangle\\\ |{c_{3}}\rangle\end{bmatrix},\quad\quad
N_{J}=1-\sum_{0\leq K\leq J-1}|\langle c_{K}|{\psi_{J}}\rangle|^{2}.$ (60)

Equivalently, the CM state can be expressed as
$|{\psi_{J}}\rangle=\sum_{K=0,1,2,3}\mathcal{U}_{JK}|{c_{K}}\rangle$ using a
$4\times 4$ matrix $\mathcal{U}_{JK}$. Therefore, the reduced states are

$\displaystyle\rho_{\text{INT:CM}}=\frac{1}{2N}\\!\\!\sum_{j_{1}j_{2},k_{1}k_{2}}\\!\\!e^{-i(\mathcal{E}_{J[j_{1},k_{1}]}-\mathcal{E}_{J[j_{2},k_{2}]})t}\left(\sum_{\ell}S^{*}_{k_{1}\ell}\,S_{k_{2}\ell}\right)|{E_{j_{1}}}\rangle\langle{E_{j_{2}}}|\otimes\sum_{K_{1}K_{2}}\mathcal{U}_{J[j_{1},k_{1}]K_{1}}\,\mathcal{U}^{*}_{J[j_{2},k_{2}]K_{2}}|{c_{K_{1}}}\rangle\langle{c_{K_{2}}}|,$
(61)
$\displaystyle\rho_{\text{INT:S}}=\frac{1}{2N}\\!\\!\sum_{j_{1}j_{2},k_{1}k_{2}}\\!\\!e^{-i(\mathcal{E}_{J[j_{1},k_{1}]}-\mathcal{E}_{J[j_{2},k_{2}]})t}\left(\sum_{K}\mathcal{U}^{*}_{J[j_{1},k_{1}]K}\,\mathcal{U}_{J[j_{2},k_{2}]K}\right)|{E_{j_{1}}}\rangle\langle{E_{j_{2}}}|\otimes\sum_{\ell_{1}\ell_{2}}S_{k_{1}\ell_{1}}\,S^{*}_{k_{2}\ell_{2}}|{s_{\ell_{1}}}\rangle\langle{s_{\ell_{2}}}|,$
(62)
$\displaystyle\rho_{\text{CM:S}}=\frac{1}{2N}\\!\\!\sum_{j,k_{1}k_{2}}\\!e^{-i(\mathcal{E}_{J[j,k_{1}]}-\mathcal{E}_{J[j,k_{2}]})t}\sum_{\ell_{1}\ell_{2}}S_{k_{1}\ell_{1}}\,S_{k_{2}\ell_{2}}|{s_{\ell_{1}}}\rangle\langle{s_{\ell_{2}}}|\otimes\sum_{K_{1}K_{2}}\mathcal{U}_{J[j,k_{1}]K_{1}}\,\mathcal{U}^{*}_{J[j,k_{2}]K_{2}}|{c_{K_{1}}}\rangle\langle{c_{K_{2}}}|.$
(63)

The entanglement negativity [28] for a bipartite state is obtained as
$\mathcal{N}=\sum_{\lambda_{i}<0}|\lambda_{i}|$ where $\lambda_{i}$ is the
eigenvalue of the partially transposed density matrix. Logarithmic negativity
is defined by

$N_{E}:=\log_{2}(2\mathcal{N}+1),$ (64)

which quantifies the distillable number of Bell pairs.

Figure 7 displays the time dependence of the logarithmic negativity $N_{E}(t)$
of the reduced bipartite states for the CG and QG cases. The blue, red, and
gray line denote entanglement of the CM:INT system, the INT:S system, and CM:S
systems respectively. Each entanglement is obtained by evaluating Eqs.
(61),(62), and (63). The parameters are identical to those used in Fig. 6,
namely
$(m_{0}^{3}/k)^{1/2}=10,~{}(m_{0}/m_{1})^{1/2}=0.5,~{}Gm_{0}M/(kd^{3})=0.015,~{}m_{0}^{1/4}d=10,~{}\beta/d=0.01,~{}\sigma/d=0.001$.
For the CG case, the CM:INT entanglement emerges and disappears for every
$2\pi/\omega_{1}$ period, and the CM:S and S:INT entanglements remain zero.
Comparing the CM:INT entanglement with the visibility in Fig. 6 reveals that
they both oscillate alternatively with the $2\pi/\omega_{1}$ period.
Therefore, the CM:INT entanglement induced by the relativistic and classical
gravitational redshift is reflected in the revival of visibility in the Ramsey
interference. For the QG case, whole the CM:INT, INT:S, and CM:S entanglements
emerge as time evolves. The CM:S entanglement is relatively smaller than
others, because the CM:S entanglement is obtained from the second order of
Taylor expansion in Eq. (18), whereas the others originated from the first
order. An important effect of the quantumness of gravity appears in the INT:S
entanglement which dominates the CM:INT entanglement alternatively during the
time evolution. As the source state $\ket{\varphi_{\pm\beta}}$ reduces to a
two-level state for $\sigma\to 0$, the INT:S entanglement can achieve nearly
the maximal logarithmic negativity value of one as the reduced state nearly
evolves to the Bell state. Moreover, the beat of the visibility in Fig. 6
corresponds exactly to the envelope of the oscillation of the INT:S
entanglement. If the CM:S entanglement is neglected, we can conclude that the
nonrevival property of the Ramsey interference exactly reflects the creation
of the INT:S entanglement revealing the quantumness of gravity.

Figure 7: Logarithmic negativity of reduced bipartite states for CG case (left
panel) and QG case (right panel). Bipartite subsystems are denoted as CM:INT
(blue line), INT:S (red line) and CM:S (gray line).

As per Fig. 7, CM:S entanglement is negligible under the condition that the
Taylor expansion in Eq. (18) is validated. Now, we again consider up to the
first order of the Taylor expansion in Eq. (18) to neglect the CM:S
entanglement and derive an explicit analytic form of negativity for three
different reduced bipartite states to achieve further qualitative
understanding. In this approximation, since the dependence of $X$ only appears
in $\mathcal{E}_{j}$ [See also (42)], Eq. (52) becomes

$\displaystyle\ket{\Psi(t)}$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{j=0,1}|{E_{j}}\rangle\otimes|{\tilde{C}_{j}}\rangle\otimes|{\tilde{S}_{j}}\rangle,$
(65)

where

$\displaystyle|{\tilde{C}_{j}}\rangle:=|{\psi_{j}(t)}\rangle,\quad|{\tilde{S}_{j}}\rangle:=\int
dX\,e^{-i\mathcal{E}_{j}(X)t}\,\varphi_{S}(X)|{X}\rangle.$ (66)

In particular, in the qubit limit of the CM state and the source state with
$\langle{\tilde{C}_{j}}|\tilde{C}_{k}\rangle\rightarrow\delta_{jk},\langle{\tilde{S}_{j}}|\tilde{S}_{k}\rangle\rightarrow\delta_{jk}$,
we obtain the Greenberger-Horne-Zeilinger (GHZ) state for the total system as
$|{\Psi_{\text{GHZ}}}\rangle=\left(|{000}\rangle+|{111}\rangle\right)/\sqrt{2}$.
Therefore, the existence of genuine tripartite entanglement for the CM-INT-S
state is confirmed. To evaluate the negativity for the reduced states,
orthonormal basis $|{c_{j}}\rangle$ for the CM state and $|{s_{j}}\rangle$ for
the S state, which satisfy
$\bra{c_{j}}\ket{c_{k}}=\bra{s_{j}}\ket{s_{k}}=\delta_{jk}$ are introduced as
follows:

$\displaystyle\begin{bmatrix}|{\tilde{C}_{0}}\rangle\\\
|{\tilde{C}_{1}}\rangle\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}e^{i\theta_{C}/2}\sqrt{1+v_{C}}&e^{i\theta_{C}/2}\sqrt{1-v_{C}}\\\
e^{-i\theta_{C}/2}\sqrt{1+v_{C}}&-e^{-i\theta_{C}/2}\sqrt{1-v_{C}}\end{bmatrix}\begin{bmatrix}|{c_{0}}\rangle\\\
|{c_{1}}\rangle\end{bmatrix},\quad
v_{C}=\left|\langle{\tilde{C}_{0}}|\tilde{C}_{1}\rangle\right|,\quad\theta_{C}=\arg\left[\langle{\tilde{C}_{0}}|\tilde{C}_{1}\rangle\right],$
(67) $\displaystyle\begin{bmatrix}|{\tilde{S}_{0}}\rangle\\\
|{\tilde{S}_{1}}\rangle\end{bmatrix}=\frac{1}{\sqrt{2}}\begin{bmatrix}e^{i\theta_{S}/2}\sqrt{1+v_{S}}&e^{i\theta_{S}/2}\sqrt{1-v_{S}}\\\
e^{-i\theta_{S}/2}\sqrt{1+v_{S}}&-e^{-i\theta_{S}/2}\sqrt{1-v_{S}}\end{bmatrix}\begin{bmatrix}|{s_{0}}\rangle\\\
|{s_{1}}\rangle\end{bmatrix},\quad
v_{S}=\left|\langle{\tilde{S}_{0}}|\tilde{S}_{1}\rangle\right|,\quad\theta_{S}=\arg\left[\langle{\tilde{S}_{0}}|\tilde{S}_{1}\rangle\right].$
(68)

Using the orthonormal basis
$\\{\ket{E_{0}},\ket{E_{1}},\ket{c_{0}},\ket{c_{1}},\ket{s_{0}},\ket{s_{1}}\\}$,
the total state is expressed as

$\rho=\ket{\Psi}\bra{\Psi}=\frac{1}{8}\begin{bmatrix}\mathbf{N}_{C}\otimes\mathbf{N}_{S}&e^{i(\theta_{C}+\theta_{S})}\,\mathbf{N}_{C}\,\mathbf{Z}\otimes\mathbf{N}_{S}\,\mathbf{Z}\\\
e^{-i(\theta_{C}+\theta_{S})}\,\mathbf{Z}\,\mathbf{N}_{C}\otimes\mathbf{Z}\,\mathbf{N}_{S}&\mathbf{Z}\,\mathbf{N}_{C}\,\mathbf{Z}\otimes\mathbf{Z}\,\mathbf{N}_{S}\,\mathbf{Z}\end{bmatrix},$
(69)

where the $2\times 2$ sub-matrices are defined by

$\mathbf{N}_{C}=\begin{bmatrix}1+v_{C}&\sqrt{1-v_{C}^{2}}\\\
\sqrt{1-v_{C}^{2}}&1-v_{C}\end{bmatrix},\quad\mathbf{N}_{S}=\begin{bmatrix}1+v_{S}&\sqrt{1-v_{S}^{2}}\\\
\sqrt{1-v_{S}^{2}}&1-v_{S}\end{bmatrix},\quad\mathbf{Z}=\begin{bmatrix}1&0\\\
0&-1\end{bmatrix}.$ (70)

Therefore, the three different reduced bipartite states are

$\displaystyle\rho_{\text{INT:CM}}=\mathrm{Tr}_{\text{S}}\rho$
$\displaystyle=\frac{1}{4}\begin{bmatrix}\mathbf{N}_{C}&v_{S}\,e^{i(\theta_{C}+\theta_{S})}\,\mathbf{N}_{C}\,\mathbf{Z}\\\
v_{S}\,e^{-i(\theta_{C}+\theta_{S})}\,\mathbf{Z}\,\mathbf{N}_{C}&\mathbf{Z}\,\mathbf{N}_{C}\,\mathbf{Z}\end{bmatrix},$
(71) $\displaystyle\rho_{\text{INT:S}}=\mathrm{Tr}_{\text{CM}}\rho$
$\displaystyle=\frac{1}{4}\begin{bmatrix}\mathbf{N}_{S}&v_{c}\,e^{i(\theta_{C}+\theta_{S})}\,\mathbf{N}_{S}\mathbf{Z}\\\
v_{c}\,e^{-i(\theta_{C}+\theta_{S})}\,\mathbf{Z}\,\mathbf{N}_{S}&\mathbf{Z}\,\mathbf{N}_{S}\,\mathbf{Z}\end{bmatrix},$
(72) $\displaystyle\rho_{\text{CM:S}}=\mathrm{Tr}_{\text{INT}}\rho$
$\displaystyle=\frac{1}{8}\left(\mathbf{N}_{C}\otimes\mathbf{N}_{S}+\mathbf{Z}\,\mathbf{N}_{C}\,\mathbf{Z}\otimes\mathbf{Z}\,\mathbf{N}_{S}\,\mathbf{Z}\right).$
(73)

As per Eq. (73), $\rho_{\text{CM:S}}$ is separable from its structure. The
negativity for reduced bipartite states are

$\displaystyle\mathcal{N}(\rho_{\text{INT:CM}})$
$\displaystyle=\frac{1}{4}\left(-1+v_{S}+\sqrt{(1+v_{S})^{2}-4v_{C}^{2}\,v_{S}}\right),$
(74) $\displaystyle\mathcal{N}(\rho_{\text{INT:S}})$
$\displaystyle=\frac{1}{4}\left(-1+v_{C}+\sqrt{(1+v_{C})^{2}-4v_{S}^{2}\,v_{C}}\right),$
(75) $\displaystyle\mathcal{N}(\rho_{\text{CM:S}})$ $\displaystyle=0.$ (76)

Figure 8 displays the dependence of negativities
$\mathcal{N}(\rho_{\text{INT:CM}})$, $\mathcal{N}(\rho_{\text{INT:S}})$ on
$v_{S}$ and $v_{C}$. For the CG case, since the state is not dependent on $X$,
we have $v_{S}=1$. Therefore, $\mathcal{N}(\rho_{\text{INT:S}})$ becomes zero,
whereas $\mathcal{N}(\rho_{\text{INT:CM}})$ varies between 0 and 1 depending
on values of $v_{C}$. For the QG case, since $v_{S}$ oscillates between 0 and
1, $\mathcal{N}(\rho_{\text{INT:CM}})$ oscillates between $0$ and
$\sqrt{1-v_{C}^{2}}/2$, whereas $\mathcal{N}(\rho_{\text{INT:S}})$ oscillates
between $0$ and $v_{C}/2$; $\mathcal{N}(\rho_{\text{INT:CM}})$ and
$\mathcal{N}(\rho_{\text{INT:S}})$ vary alternatively. This implies a
monogamous relation of quantum entanglement between the INT:CM systems and
INT:S systems, and confirms the existence of genuine tripartite entanglement
of the CM-INT-S system.

For the state expressed in Eq. (69), the negativity for the bipartition
INT:CM+S of the total state is

$\mathcal{N}(\rho_{\text{INT:CM+S}})=\frac{1}{2}\sqrt{1-v_{S}^{2}\,v_{C}^{2}},$
(77)

and it obeys monogamy inequality [29] as follows:

$\mathcal{N}^{2}(\rho_{\text{INT:CM}})+\mathcal{N}^{2}(\rho_{\text{INT:S}})\leq\mathcal{N}^{2}(\rho_{\text{INT:CM+S}}).$
(78)

The difference between the sides of this inequality represents the residual
entanglement which quantifies genuine tripartite entanglement (right panel of
Fig. 8). For the CG case, since $v_{S}=1$, there is no residual entanglement.
For the QG case, since $v_{C}\neq 1$ or $v_{S}\neq 1$, the value of residual
entanglement is nonzero. The point $v_{C}=v_{S}=0$ corresponds to the
maximally entangled GHZ state.

Figure 8: Dependence of negativity on $v_{S}$ and $v_{C}$ for reduced
bipartite states. Left panel: INT:CM. Middle panel: INT:S. Right panel:
residual entanglement, which quantifies genuine tripartite entanglement. For
the CG case, since $v_{S}=1$, $\mathcal{N}(\rho_{\text{INT:S}})$ is always
zero, but $\mathcal{N}(\rho_{\text{INT:CM}})\neq 0$ depending on the value of
$v_{C}$.

Finally, we evaluated the entanglement entropy in our system. In Fig. 9, we
depicted the time dependence of the entanglement entropy between unipartite-
bipartite systems. The left panel shows the CG case, and the right panel shows
the QG case. The orange line shows CM:others systems, the blue line shows
INT:others systems and the green line shows the S:others systems. Parameters
are chosen as the same as in Fig. 6, namely
$(m_{0}^{3}/k)^{1/2}=10,~{}(m_{0}/m_{1})^{1/2}=0.5,~{}Gm_{0}M/(kd^{3})=0.015,~{}m_{0}^{1/4}d=10,~{}\beta/d=0.01,~{}\sigma/d=0.001$.
We see any entanglement entropy is bounded by $\log 2$ (gray dashed line),
which value involves the GHZ state. In the CG case, We can see that the
entanglement entropy of S:others systems vanishes. This means that the source
system is isolated from the particle system due to the classical interaction.
The entanglement entropy of CM:others and INT:others systems take the same
value since they both show the entanglement shared between the CM system and
the INT system. In the QG case, we can see the entanglement is shared between
the whole 3 systems. Especially, the entanglement entropy of INT:others and
S:others almost reach the maximum value $\log 2$ when the visibility is at the
minimum in Fig. 6. This is because the INT system is represented by 2 qubits
and the S state is approximately represented by 2 qubits which nearly reads to
the maximally entangled state.

Figure 9:  Entanglement entropy between unipartite-bipartite systems for CG
case (left panel) and QG case (right panel). Each line shows CM:others systems
(orange line), INT:others systems (blue line), and S:others systems (green
line). The gray dashed line shows $\log 2$; the maximally entangled state.

##  VI Discussion

In this section, we discuss the feasibility of detecting the quantumness of
gravity in our proposal. Let us suppose an experiment being performed using an
${}^{27}\mathrm{Al}^{+}$ quantum clock [30] with a probe laser wavelength
$\lambda=2\pi\hbar c/E=267\,\mathrm{nm}$. A coherent state of the mesoscopic
mass source with $M\sim 1\,\mathrm{ng}$ may be realized in the near future. In
addition, the coherent state can be experimentally realized for
$20\,\mathrm{sec}$ [31]. Apart from that, we assume
$d=200\,\mu\mathrm{m},~{}\sigma=1\,\mu\mathrm{m}$ and
$\beta=10\,\mu\mathrm{m}$ to compare setups of the BMV proposal [9, 10, 11].
Therefore, for a duration of coherence time scale $t$, the fractional change
of the decoherence factor in $|\mathcal{V}_{Q}(t)|$ of Eq. (48) can be
estimated as

$\displaystyle\left(\frac{1}{2\hbar}\frac{GME}{d}\frac{\sigma}{d}\,t\right)^{2}=1.7\times
10^{-34}\left(\frac{M}{10\,\mathrm{ng}}\right)^{2}\left(\frac{\lambda}{267\,\mathrm{nm}}\right)^{-2}\left(\frac{d}{200\,\mu\mathrm{m}}\right)^{-4}\left(\frac{\sigma}{1\,\mu\mathrm{m}}\right)^{2}\left(\frac{t}{20\,\mathrm{sec}}\right)^{2},$
(79)
$\displaystyle\left(\frac{1}{\hbar}\frac{GME}{d}\frac{\beta}{d}\,t\right)^{2}=6.8\times
10^{-32}\left(\frac{M}{10\,\mathrm{ng}}\right)^{2}\left(\frac{\lambda}{267\,\mathrm{nm}}\right)^{-2}\left(\frac{d}{200\,\mu\mathrm{m}}\right)^{-4}\left(\frac{\beta}{10\,\mu\mathrm{m}}\right)^{2}\left(\frac{t}{20\,\mathrm{sec}}\right)^{2}.$
(80)

After performing time Fourier transformation on the probability $P(t)$
obtained using the spectroscopy experiments, the least necessary precision to
detect the QG effect in our proposal is approximately $10^{-32}$, which is
extremely small to be distinguished in the present clock spectroscopy, whose
observation uncertainty is about $10^{-19}$ [30]. Let us discuss our result
Eq. (80) in comparison with visibility change obtained by the setup of Carney
et al. [19] . Carney et al. investigated entanglement between a massive
oscillator and a source mass particle with a cat state, and evaluated the
interference visibility of the particle state. Their estimation of the time
change of the visibility due to quantum gravitational interaction is

$\left(\frac{GMm}{d}\frac{\beta}{d}\frac{x_{0}}{d}\,t\right)^{2}\propto
d^{-6},$ (81)

where $x_{0}$ is the spread of the ground state wave function of the
oscillator. For the optimal values of parameters in their setup, the ratio
provides a value of $\sim 10^{-28}$. Although Eq. (80) exhibits a lower
suppression factor $d^{2}$ compared to (81), the ratio $E/m$ makes the
visibility change smaller compared to that in Eq. (81).

As a crucial issue of this study, we also discuss the uniqueness of gravity
compared to other quantum interactions, such as the electromagnetic Coulomb
force. In our setup, CG creates an entanglement only between the CM and INT
systems, whereas all subsystems share entanglement in the QG case. It should
be noted that, in particular, the entanglement between the INT and S systems
is uniquely induced by the quantumness of gravity, and no other quantum
interactions can establish this entanglement. This is because according to the
weak equivalence principle and the mass-energy equivalence, only gravity can
couple to the energy; other quantum interactions do not have this property
[4]. Figure 10 depicts the entanglement structure for different external
forces, namely CG, QG, and Coulomb force. To capture the uniqueness of the
quantumness of gravity, we should detect entanglement held between the INT
system and the S system.

Figure 10: Entanglement structure for the CG, QG, and Coulomb interaction
cases. Dotted lines represent the existence of couplings between degrees of
freedom, and possible pairs of subsystems sharing entanglement. For the QG
case, the total system shares genuine tripartite entanglement which cannot be
reduced to bipartite entanglement between the reduced bipartite systems. For
the quantum Coulomb case, tripartite entanglement is reduced to bipartite
entanglement between the reduced bipartite systems.

For comparison, let us consider the case when the Coulomb interaction is the
external force instead of the gravitational interaction. To focus on the
connection of mass-energy equivalence with entanglement here, we do not
consider a dipole-photon interaction which introduces a direct coupling
between the INT and S system.222 Note that we have made an assumption in our
setup that the internal degrees of freedom of the particle are labeled only by
the energy levels, which results in the unique correspondence of the INT:S
entanglement and QG. Generally, the internal degrees of freedom of an atom are
labeled not only by energy level but also by spin. In such cases, the dipole-
photon interaction can also create INT:S entanglement besides gravity. A
general form of the dipole-photon interaction is given by
$\hat{\mathbf{d}}\cdot\hat{\mathbf{E}}$, where
$\hat{\mathbf{d}}=d_{+}\ket{E_{1}}\bra{E_{0}}+d_{-}\ket{E_{0}}\bra{E_{1}}$ is
the electric dipole moment operator of the INT system, and $\hat{\mathbf{E}}$
is the external electric field. When the external electric field has the
source (S) dependence as $\hat{\mathbf{E}}(\hat{X})$, this interaction can
provide the INT-S entanglement in the leading order of expansion for the
source separation. The Coulomb interaction is given by

$\displaystyle
V_{\text{Coulomb}}=\frac{qQ}{4\pi\epsilon_{0}|d+\hat{X}-\hat{x}|},$ (82)

where $q$ and $Q$ are the electric charges of the oscillator and the source,
respectively. In comparison with the QG interaction

$\displaystyle V_{\text{QG}}=-\frac{G\hat{m}M}{|d+\hat{X}-\hat{x}|},$ (83)

the Coulomb interaction does not contain the INT operator $\hat{m}$ as a
result of the weak equivalence principle. Then, it is obvious that the Coulomb
interaction does not produce the entanglement between the INT and S systems as
depicted in Fig. 10. The Hamiltonian for the external Coulomb interaction can
be expressed as

$\displaystyle\hat{H}_{\text{Coulomb}}(\hat{X})=\sum_{j=0,1}\left[\mathcal{E}_{j}(\hat{X})+\frac{\hat{p}^{2}}{2m_{j}}+\frac{m_{j}\omega_{j}^{2}}{2}\left(\hat{x}-\frac{qB(\hat{X})}{k}\right)^{2}\right]\ket{E_{j}}\bra{E_{j}},\quad\mathcal{E}_{j}(\hat{X})=m_{j}-qA(\hat{X})-\frac{q^{2}B^{2}(\hat{X})}{2k},$
(84) $\displaystyle
A(\hat{X})=\frac{Q}{4\pi\epsilon_{0}d}\left(1-\frac{\hat{X}}{d}+\frac{\hat{X}^{2}}{d^{2}}\right),\quad
B(\hat{X})=\frac{Q}{4\pi\epsilon_{0}d^{2}}\left(1-\frac{2\hat{X}}{d}\right),\quad
C=\frac{Q}{4\pi\epsilon_{0}d^{3}}.$ (85)

Unlike the gravity case, the symmetry axis of the trapping potential is
independent of the INT state, and the quantity
$\mathcal{E}_{1}(\hat{X})-\mathcal{E}_{0}(\hat{X})$ is independent of
$\hat{X}$. As a result, the visibility of the Coulomb force case exhibits a
different behavior compared to the CG and QG cases. Following the same
procedure used to derive Eq. (50), we obtain the $\pi/\omega_{1}$ periodic
visibility as

$\displaystyle|\bra{\Psi_{0}(t)}\ket{\Psi_{1}(t)}|=\left(\frac{4a_{0}\,a_{1}}{4a_{0}\,a_{1}\cos^{2}(\omega_{1}t)+(a_{0}+a_{1})^{2}\sin^{2}(\omega_{1}t)}\right)^{1/4}.$
(86)

Here, the period $2\pi/\omega_{1}$ of coherent state does not appear since the
displacement factor $\Delta(\hat{X}):=qB(\hat{X})/k\approx Q/d^{2}$ is
independent of the internal energy level. Therefore, up to the leading order
in Eq. (18), we conclude that the nonrevival behavior in visibility uniquely
discriminates the quantumness of gravity. The visibility is more likely to
exhibits revival behavior for the Coulomb interaction case than the QG case
since the former shares less entanglement than the latter, as in Fig. 10. The
left panel of Fig. 11 presents the behavior of the probability for the Coulomb
interaction by evaluating up to the second-order of Taylor expansion in Eq.
(18). The parameters are set to
$(m_{0}^{3}/k)^{1/2}=10,~{}(m_{0}/m_{1})^{1/2}=0.85,~{}qQ/(4\pi\epsilon_{0}kd^{3})=0.15,~{}m_{0}^{1/4}d=10,~{}\beta/d=0.01,~{}\sigma/d=0.001$.
As the figure shows, the visibility exhibits a revival behavior with some
longer period than $\pi/\omega_{1}$ obtained by the first-order approximation
in Eq. (86). This is because the visibility beats due to the addition of two
functions with periods $2\pi/\omega_{0}$ and $2\pi/\omega_{1}$, reflecting the
respective period of the coherent states $\ket{\Psi_{0}(t)}$ and
$\ket{\Psi_{1}(t)}$ as an effect of the external quantum Coulomb force. In the
first-order approximation, the displacement effects of these coherent states
are neglected. Regarding the behavior of negativity, within the first order
approximation, $v_{S}=1$ results in $\mathcal{N}(\rho_{\text{INT:S}})=0$ and
provides a nonzero value of $\mathcal{N}(\rho_{\text{CM:INT}})$ with period
$\pi/\omega_{1}$. The quantumness of Coulomb force appears as a small nonzero
value of $\mathcal{N}_{\text{CM:S}}$, which can be revealed beyond the first-
order approximation, as per the right panel of Fig. 11.

Figure 11: Left panel: time dependence of the visibility $\mathcal{V}(t)$ for
the Coulomb case. Right panel: logarithmic negativity for the Coulomb case;
Red line: bipartite INT:S subsystems. Blue line: bipartite INT:CM subsystem.
Grayline: bipartite CM:S subsystem.

Also, we briefly discuss the case where the harmonic oscillator potential
trapping the particle is proportional to the probe mass [32]:
$\frac{1}{2}\hat{m}\,\omega^{2}\hat{x}^{2}$, where $\omega$ is a constant
parameter. Here, a shift of the potential due to the external gravity becomes
independent of the internal energy level, and we obtain
$\Delta_{1}=\Delta_{0}\neq 0$. Since the displacement effect does not become
obvious in the Ramsey interference, $|\mathcal{V}_{C}(t)|$ reduces to a
$\pi/\omega_{1}$ periodic function rather than $2\pi/\omega_{1}$ periodic
function. Therefore, the visibility revival period becomes $\pi/\omega_{1}$
for the CG case, while it does not exhibit revival for the QG case due to the
quantumness of gravity.

Finally, we comment on the relevance of the alternative theories of gravity.
Since our proposal are focusing only on the nonrelativistic scale, it is
difficult to distinguish alternative theories of gravity when they reduce to
the Newtonian gravity in the Newtonian limit. To distinguish them, we need
experiment on the higher orders of $1/c^{2}$ expansion where the difference
between alternative theories appears. It may be interesting to discuss how the
Ramsey interference differs depending on alternative theories of gravity by
analyzing the higher-order expansions, although the detection is technically
difficult at present.

##  VII Summary

In this study, we investigated the quantumness of gravity in the setup of the
Ramsey interferometry. For the classical external gravitational field, the
interference visibility exhibits oscillatory behavior, and the coherence
between the two energy states exhibits collapse and revival behavior in time.
The decoherence behavior originated from the coupling between the CM state and
INT state due to the mass-energy equivalence principle. On the other hand,
when a gravitational source is in quantum superposition, the visibility
exhibited additional oscillation with a longer period and decay at a long time
scale for the QG case, while it remains $2\pi/\omega_{1}$ periodic in the CG
case. For the zero spread of the source state, the time scale of decoherence
was determined as $t\sim\left(\frac{GME}{d}\frac{\beta}{d}\right)^{-1}$, which
constrains the feasibility of the experimental detection of the quantumness of
gravity. In addition, we found that the spread of the source wave function
provides another decoherence effect that is not periodic in time depending on
the variance of the source Gaussian state $\sigma$.

Regarding the entanglement behavior, the initial separable CM-INT-S system
acquired genuine tripartite entanglement due to the quantumness of gravity. On
the contrary, the quantumness of the Coulomb force cannot acquire the
entanglement between the INT and S systems. This is because the Coulomb
interaction does not couple to mass-energy of the INT state, unlike gravity,
which obeys the weak equivalence principle. Since the nonrevival feature of
the interference visibility reproduce the entanglement between the INT and S
systems, it is possible to distinguish the quantumness of gravity from other
quantum interactions by observing the visibility of Ramsey interference. We
believe that our study is beneficial for further understanding of the quantum
nature of gravity.

###### Acknowledgements.

We would like to thank A. Matsumura and K. Yamamoto for providing their
valuable insight on the subject. This research was supported in part by the
Japan Science and Technology Agency (JST), the Nagoya University
Interdisciplinary Frontier Fellowship (Y.K.), and JSPS KAKENHI Grant No.
19K03866 (Y.N.) and No. 22H05257 (Y.N.).

##  Appendix A WIGNER FUNCTION OF TIME EVOLVED CM STATE

In this section we will investigate the time evolution of the CM state
$\ket{\psi_{j,X}(t)}$ by depicting its Wigner function [33], which gives us an
intuitive understanding of the visibility behavior.

In the QG case, the time-evolved CM state is given by

$\displaystyle\ket{\psi_{j,X}(t)}=\hat{U}_{j,X}(t)\ket{\psi_{\text{ini}}},$
(87)

where
$\hat{U}_{j,X}(t):=e^{-i\omega_{j}(\hat{a}_{j,X}^{\dagger}\,\hat{a}_{j,X}+1/2)t}$,
$\ket{\psi_{\text{ini}}}$ is the ground state of the Hamiltonian
$\omega_{j}(\hat{a}_{0,0}{}^{\\!\\!\\!\\!\\!\\!\dagger}\,\,\,\hat{a}_{0,0}+1/2)$,
and $\hat{a}_{j,X}$ is as per Eq. (27). The relation between the two
annihilation operators $\hat{a}_{0,0}$ and $\hat{a}_{j,X}$ is as follows, by

$\displaystyle\hat{a}_{j,X}=\cosh r_{j}\,\hat{a}_{0,0}+\sinh
r_{j}\,\hat{a}_{0,0}^{\dagger}+\alpha_{j,X},$ (88) $\displaystyle
e^{r_{j}}=\sqrt{\frac{m_{j}\omega_{j}}{m_{0}\omega_{0}}},\quad\alpha_{j,X}=\sqrt{\frac{m_{0}\omega_{0}}{2}}e^{r_{j}}(\Delta_{0}(0)-\Delta_{j}(X)).$
(89)

Using the squeezing operator
$\hat{S}(r_{j})=e^{r_{j}(\hat{a}_{0,0}^{2}-\hat{a}_{0,0}^{\dagger 2})/2}$ and
the displacement operator
$\hat{D}(\alpha)=e^{\alpha(\hat{a}_{0,0}^{\dagger}-\hat{a}_{0,0})}$, the
relation of the two annihilation operators can be rewritten as

$\hat{a}_{j,X}=\hat{S}^{\dagger}(r_{j})\hat{D}^{\dagger}(\alpha_{j,X})\hat{a}_{0,0}\hat{D}(\alpha_{j,X})\hat{S}(r_{j}).$
(90)

Therefore, the initial ground state $\ket{\psi_{\text{ini}}(t)}$ associated
with $\hat{a}_{0,0}$ evolves to become the squeezed coherent state
characterized by parameters $r_{j}$ and $\alpha_{j,X}$ as given below,

$\displaystyle\ket{\psi_{j,X}(t)}=S^{\dagger}(r_{j})D^{\dagger}(\alpha_{j,X})\hat{U}_{0,0}D(\alpha_{j,X})S(r_{j})\ket{\psi_{\text{ini}}}.$
(91)

For the no gravity case, the time-evolved CM state is obtained by replacing
$\alpha_{j,X}\to 0$. For the CG case, the time-evolved CM state is obtained by
replacing $\alpha_{j,X}\to\alpha_{j}$.

Next, we explore the temporal behavior of the CM Wigner function [33]. The
Wigner function is a quasiprobability distribution in the phase space $(x,p)$
and is defined as

$\displaystyle
W_{\psi}(x,p)=\frac{1}{2\pi}\int^{\infty}_{-\infty}d\xi\,\psi\\!\left(x+\frac{\xi}{2}\right)\psi^{*}\\!\left(x-\frac{\xi}{2}\right)e^{i\,\xi\,p},$
(92)

where $\psi(x)$ is the wave function of the CM system. For simplicity, we
focus on the state with $X=\pm\beta$, which is the same condition as we
depicted in Fig. 4. Length unit of $2m_{0}\omega_{0}=1$.

Figure 12 displays the time evolution of the Wigner function of the CM state
for the no gravity case. The blue and red regions respectively denote the
Wigner function of the $\ket{\psi_{0}(t)}$ and $\ket{\psi_{1}(t)}$ states. The
parameters are set to $e^{r_{1}}=1.2,~{}\omega_{1}/\omega_{0}=0.8$. Since
there is no displacement effect due to gravity, the time-evolved state is
simply squeezed due to a special relativistic effect. The two states overlap
for every $\pi/\omega_{1}$ period, which reflects the period of squeezing.
Since the visibility of the Ramsey interference in Eq. (40) contains
$|\bra{\psi_{0}(t)}\ket{\psi_{1}(t)}|$, its revival period $\pi/\omega_{1}$
stems from the squeezing period.

Figure 12: Time evolution of the Wigner function of the CM state for the no
gravity case. A blue region: Wigner function of the $\ket{\psi_{0}(t)}$ state.
A red region: Wigner function of the $\ket{\psi_{1}(t)}$ state. Contour lines
are drawn with $(0.02,0.05,0.08,0.11,0.14)$.

Figure 13 displays the time evolution of the Wigner function of the CM state
for the CG case. The blue and red regions respectively denote the Wigner
function of $\ket{\psi_{0}(t)}$ and $\ket{\psi_{1}(t)}$ states. The parameters
are set to $e^{r_{1}}=1.2,~{}\alpha_{1}=3,~{}\omega_{1}/\omega_{0}=0.8$. The
time evolved state is squeezed by a special relativistic effect and displaced
by a gravitational effect. The two states overlap for every $2\pi/\omega_{1}$
period reflecting the period of coherent state, which results in the
$2\pi/\omega_{1}$ revival period of the visibility.

Figure 13: Time evolution of the Wigner function of the CM state for the CG
case. A blue region: Wigner function of the $\ket{\psi_{0}(t)}$ state. A red
region: Wigner function of the $\ket{\psi_{1}(t)}$ state. Contour lines are
drawn with $(0.02,0.05,0.08,0.11,0.14)$.

Figure 14 displays the time evolution of the Wigner function of the CM state
for the CG case. The blue region with solid contours denotes The Wigner
function of $\ket{\psi_{0,-\beta}(t)}$, the blue region with dashed contours
denotes the Wigner function of $\ket{\psi_{0,+\beta}(t)}$, the red region with
solid contours denotes the Wigner function of $\ket{\psi_{1,-\beta}(t)}$, and
the red region with dashed contours denotes the Wigner function of
$\ket{\psi_{1,+\beta}(t)}$. The parameters are set to
$e^{r_{1}}=1.2,~{}\alpha_{0,0}=1.5,~{}\alpha_{1,0}=3.0,~{}\alpha_{j,+\beta}/\alpha_{j,-\beta}=0.6,~{}\omega_{1}/\omega_{0}=0.8$.
There are four kinds of squeezed-coherent states with $j=0,~{}1$ and
$X=\pm\beta$. The Wigner functions of $\ket{\psi_{0,X}(t)}$ and
$\ket{\psi_{1,X}}$ moves with different the time periods $2\pi/\omega_{0}$ and
$2\pi/\omega_{1}$, respectively, and typically do not coincide. The nonrevival
behavior of the visibility for the QG case is originated from the fact that
these four states do not coincide, as well as the phase difference
$e^{-i\mathcal{E}_{j,X}t}$ in Eq. (40).

Figure 14: Time evolution of the Wigner function of the CM state for the QG
case. Blue region with solid contours: Wigner function of
$\ket{\psi_{0,-\beta}(t)}$. Blue region with dashed contours: Wigner function
of $\ket{\psi_{0,+\beta}(t)}$. Red region with solid contours; Wigner function
of $\ket{\psi_{1,-\beta}(t)}$. Red region with dashed contours: Wigner
function of $\ket{\psi_{1,+\beta}(t)}$. Contour lines are drawn with
$(0.02,0.05,0.08,0.11,0.14)$.

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