# Decoherence of a composite particle induced by a weak quantized
gravitational field

Thiago H. Moreira<EMAIL_ADDRESS>QPequi Group,
Institute of Physics, Federal University of Goiás, Goiânia, Goiás, 74.690-900,
Brazil  Lucas C. Céleri<EMAIL_ADDRESS>QPequi Group, Institute of Physics,
Federal University of Goiás, Goiânia, Goiás, 74.690-900, Brazil

###### Abstract

Despite the fact that we have some proposals for the quantum theory of gravity
like string theory or loop quantum gravity, we do not have any experimental
evidence supporting any of these theories. Actually, we do not have
experimental evidence pointing in the direction that we really need a quantum
description of the gravitational field. In this scenario, several proposals
for experimentally investigating quantum gravitational effects far from Plank
scale have recently appear in literature, like gravitationally induced
entanglement, for instance. An important issue of theses approaches is the
decoherence introduced by the quantum nature not only of the system under
consideration, but also from the gravitational field itself. Here, by means of
the Feynman-Vernon influence functional we study the decoherence of a quantum
system induced by the quantized gravitational field and by its own quantum
nature. Our results may be important in providing a better understanding of
many phenomena like the decoherence induced by the gravitational time-
dilation, the quantum reference frames and the quantum equivalence principle.

##  I Introduction

Ever since the development of quantum mechanics and general relativity in the
first decades of the last century, the construction of a successful quantum
theory of gravity poses a problem to this day. Moreover, we do not have
experimental results to guide such a construction since quantum gravitational
effects are expected to be relevant at the Planck scale, which is far beyond
our current technology. However, in recent years there has been increasing
interest in attempts to probe the quantum nature of gravity in nonrelativistic
quantum systems. The idea is not to give a full description of quantum
gravitational effects, but to establish the necessity or not to quantize the
field and, at the same time, to provide some experimental guidance for the
theoretical construction of the theory. Among these developments we can
mention investigations regarding the possibility of gravitationally mediated
entanglement [1, 2, 3, 4, 5], the consequences of the supperposition of
spacetimes [6] and the study of indefinite causal order in the gravitational
scenario [7, 8], just to mention a few. We point the reader to Ref. [9] for
more references. Recently another proposal has been put forward in Ref. [10,
11, 12] with the goal to investigate the quantum nature of the gravitational
field by detecting the effect of the quantum noise induced by the gravitons on
classical particles in the same spirit of the quantum Brownian motion.

Gravitons appear as a consequence of the quantization of gravitational waves
in a weak-field, perturbative regime. Although far from quantum gravity at the
Planck scale, the detection of its effects on matter would provide enough
evidence to establishing the true necessity of the quantization of gravity,
even without a complete theory. After the observations of gravitational waves
announced by LIGO [13], questions have been raised on the possibility that
these could also reveal quantum effects of the gravitational field. However,
according to arguments by F. Dyson [14], the detection of individual gravitons
was bound to be nearly impossible.

The proposal in Refs. [10, 11, 12] focus instead on detecting the effect on
falling bodies due to the quantization of the gravitational field, which
enters the equation of motion as a kind of quantum noise. Employing the
Feynman-Vernon influence functional formalism [15, 16], the authors obtained a
Langevin-like stochastic equation describing the geodesic deviation of two
falling masses with the gravitons effect entering as a stochastic force. The
noise kernel was computed for four distinct classes of initial graviton
states, namely the Minkowski vacuum as well as thermal, coherent and squeezed
states, with the latter exhibiting an exponential enhancement of the effect.
The same analysis was performed in Ref. [17] by means of a different approach
and later on it was also done in Ref. [18], where all graviton modes and
polarizations were taken into account.

On the other hand, there has also been an increasing interest on investigating
decoherence induced by gravitational fields, both classical and quantum (for a
review, see Ref. [19] and references therein). By gravitational decoherence we
mean a setup in which gravitational perturbations are viewed as an environment
affecting the evolution of quantum particles in superposition in a given basis
that leads to the loss of quantum coherence [20, 21, 22, 23]. There have also
been investigations on the decoherence of a composite particle [24] induced by
the entanglement of the center-of-mass variables with internal degrees of
freedom caused by gravitational time dilation [25, 26] and, more recently, in
any given spacetime manifold described by a generic Riemann curvature tensor
[27].

In this scenario, we study the decoherence of a particle that contains both
external and internal degrees of freedom that are coupled by the presence of a
quantized weak gravitational field. We focus on the decoherence of the
external degrees of freedom, that will be affected by the internal ones and by
the quantum fluctuations of the gravitational field. We treat the problem by
using the Feynman-Vernon influence functional formalism of open quantum
systems, which is closely related to the closed-time-path integral method [15,
16, 28, 29, 30, 31, 32]. We follow the analysis of decoherence induced by
gravitons discussed in Ref. [17] with the difference that we are considering a
particle whose internal degrees of freedom are described quantum mechanically.
Since the gravitational field couples with all degrees of freedom in the same
way, we are effectively dealing with an open quantum system interacting with
two interacting environments. This becomes important in the quest for a deeper
understanding of several phenomena involving quantum systems and the
gravitational field, among which we can mention the decoherence induced by the
gravitational time-dilation [25, 26] and the quantum equivalence principle
[33].

The article is organized as follows. In Sec. II we discuss the total action of
the system and, by treating the field as a quantum environment we obtain the
reduced density matrix of the relevant variables in terms of the gravitational
influence functional. Since the total action is quadratic, we can solve the
problem exactly by encoding all the gravitational influence into the noise and
dissipation kernels. The internal structure of the system is taken into
account in Sec. III, where we split the particle degrees of freedom into
external and internal variables, and integrate the internal ones by applying
the influence functional method once more. In order to make this calculation
possible, we are considering such degrees of freedom to be described by
independent harmonic oscillators in thermal equilibrium. This leads to an
expression for the reduced density matrix of the external degrees of freedom
of the quantum particle. In Sec. IV we take the external degrees of freedom to
be in a superposition of two distinct paths and compute the decoherence rate,
as well as the decoherence time, by considering different initial states for
the quantum gravitational field. Our conclusions and some final remarks are
left to Sec. V. Details of the calculations are presented in the appendices.

Unless stated otherwise, we work in Planck units by setting
$\hbar=c=G=k_{B}=1$. The metric signature is $(-+++)$. Greek indexes in
tensors and vectors runs from $0$ to $3$, while Latin ones takes the spacial
labels $1$, $2$ and $3$.

##  II Gravitational influence functional

The goal of this section is twofold. First, we briefly review the results
obtained in Refs. [10, 11, 12], presenting the system we are considering as
well as setting the notation that will be employed in the rest of the paper.
This will make the present paper self-contained. Additionally, we add to this
formalism the quantum description of the internal degrees of freedom of our
system of interest. Specifically, we consider two freely falling particles on
a spacetime manifold over which a metric $g_{\mu\nu}$ is defined. Our aim here
is to establish the response of the particles to the quantized gravitational
radiation field in order to set up the basis for the decoherence analysis we
present in the sequence.

###  II.1 The classical action

Let us start by writing the classical action of a weak gravitational field
coupled to a pair of free falling massive particles. We write the total action
as $S=S_{\rm matter}+S_{\rm EH}$, with the first term containing the action of
the particles and their interaction with the gravitational field, while the
second term describes the Einstein-Hilbert action of the field.

If the coordinates of the particles are denoted by $\zeta^{\mu}$ and
$\xi^{\mu}$, the action for both particles is given by

$\displaystyle S_{\rm matter}$ $\displaystyle=$ $\displaystyle-
M_{0}\int\differential
t\,\sqrt{-g_{\mu\nu}\dot{\zeta}^{\mu}\dot{\zeta}^{\nu}}$ (1) $\displaystyle+$
$\displaystyle\int\differential t\,L_{\rm
rest}\sqrt{-g_{\mu\nu}\dot{\xi}^{\mu}\dot{\xi}^{\nu}},$

where a dot means differentiation with respect to the coordinate time $t$.
Since the first particle has no internal degrees of freedom we simply choose
its rest Lagrangian as its mass $M_{0}$. The second particle is considered to
be structured, with external and internal degrees of freedom, in such a way we
can write its rest Lagrangian as

$L_{\rm rest}(\varrho,\dot{\varrho}\,\bar{t})=-m_{0}+\lambda L_{\rm
int}(\varrho,\dot{\varrho}\,\bar{t}).$ (2)

In this equation, $m_{0}$ is the mass of the second particle, while $L_{\rm
int}(\varrho,\dot{\varrho}\,\bar{t})$ describes its internal degrees of
freedom with coordinates $\varrho$ and generalised velocities
$\dot{\varrho}=\derivative*{\varrho}{t}$. Also,
$\bar{t}=\derivative*{t}{\tau}$ with $\tau$ being the particle’s proper time
and $\lambda$ is a dimensionless parameter introduced for mathematical
convenience. Additionally, we consider that the first particle is on shell
with worldline $\zeta_{0}^{\mu}(t)$ and $M_{0}\gg L_{\rm rest}$. Furthermore,
we place the first particle at rest at the origin of our coordinate system,
$\zeta_{0}^{\mu}(t)=t\,\delta_{0}^{\mu}$, such that the coordinate time $t$ is
the first particle’s proper time. Since we will focus on the second particle,
from here on we simply refer to it as the system.

Under such assumptions, the first term in the action (1) essentially has no
dynamics. It is then appropriate to think of $(t,\xi^{i})$ as the Fermi normal
coordinates defined with respect to the worldline of the first particle. In
these coordinates, we can write the metric components as [34]

$\begin{split}g_{00}(t,\xi^{i})&=-1-R_{i0j0}(t,0)\xi^{i}\xi^{j}+O(\xi^{3}),\\\
g_{0i}(t,\xi^{i})&=-\frac{2}{3}R_{0jik}(t,0)\xi^{j}\xi^{k}+O(\xi^{3}),\\\
g_{ij}(t,\xi^{i})&=\delta_{ij}-\frac{1}{3}R_{ikjl}(t,0)\xi^{k}\xi^{l}+O(\xi^{3}),\end{split}$
(3)

where $R_{\mu\nu\rho\sigma}$ is the Riemann tensor. In our parameterization,
$\xi^{0}(t)=t$ and thus

$\displaystyle\dot{\tau}$
$\displaystyle\equiv\sqrt{-g_{\mu\nu}(\xi)\dot{\xi}^{\mu}\dot{\xi}^{\nu}}\simeq\sqrt{1+R_{i0j0}(t,0)\xi^{i}\xi^{j}-\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}}$
$\displaystyle\simeq
1-\frac{1}{2}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}+\frac{1}{2}R_{i0j0}(t,0)\xi^{i}\xi^{j},$
(4)

in the nonrelativistic limit.

As usual, we express the weakness of the gravitational field by decomposing
the metric into the flat Minkowski metric $\eta_{\mu\nu}=\rm{diag}(-1,1,1,1)$
and a small perturbation $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, with
$\absolutevalue{h_{\mu\nu}}\ll 1$ .

Now, by taking advantage of the gauge invariance of linearized gravity, we
choose to work in the transverse-traceless gauge such that the metric
perturbation obeys $\bar{h}_{0\nu}=0$, $\eta^{\mu\nu}\bar{h}_{\mu\nu}=0$ and
$\partial^{\mu}\bar{h}_{\mu\nu}=0$, with
$\bar{h}_{\mu\nu}=h_{\mu\nu}-h\,\eta_{\mu\nu}/2$ and $h=h^{\mu}_{\mu}$ [35].
In this gauge, the non-vanishing components of the Riemann curvature tensor
are given by $R_{0i0j}(t,0)=-\ddot{\bar{h}}_{ij}(t,0)/2$, thus implying that
Eq. (II.1) takes the form

$\dot{\tau}\simeq
1-\frac{1}{2}\quantity[\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}+\frac{1}{2}\ddot{\bar{h}}_{ij}(t,0)\xi^{i}\xi^{j}],$

which, by dropping the non-dynamical terms, results in the matter action

$\displaystyle S_{\rm matter}$ $\displaystyle=$
$\displaystyle\lambda\int\differential t\,L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})$ (5) $\displaystyle+$
$\displaystyle\frac{1}{2}\int\differential t\,\quantity[m_{0}-\lambda L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})]\,\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}$
$\displaystyle+$ $\displaystyle\frac{1}{4}\int\differential
t\,\quantity[m_{0}-\lambda L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})]\,\ddot{\bar{h}}_{ij}(t,0)\xi^{i}\xi^{j}.$

Note that the first two terms of this equation represents the free action
(including the interaction between the internal and external degrees of
freedom) of the system in the Minkowisk background, while the last one
represents the interaction of the system with the gravitational perturbation.

We now move to the gravitational field, which is described by the Einstein-
Hilbert action [35]

$S_{\rm EH}=\frac{1}{16\pi}\int\differential^{4}x\,\sqrt{-g}\,R,$

where $g$ stands for the determinant of the metric and $R$ is the Ricci
scalar. Now, up to second order in the perturbation and in the transverse-
traceless gauge, this action reduces to

$S_{\rm
EH}=-\frac{1}{64\pi}\int\differential^{4}x\,\partial_{\mu}\bar{h}_{ij}\partial^{\mu}\bar{h}^{ij},$
(6)

from which we immediately obtain the wave equation $\Box\bar{h}_{ij}=0$. We
may then express the gravitational waves in terms of its two physical degrees
of freedom corresponding to the two polarization components by means of a
Fourier transform [35, 18],

$\bar{h}_{ij}(t,\mathbf{x})=\int\differential^{3}k\,\sum_{s}\epsilon_{ij}^{s}(\mathbf{k})q_{s}(t,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}},$
(7)

where $\epsilon_{ij}^{s}$ denotes the polarization tensor, which satisfies the
normalization
$\epsilon_{ij}^{s}(\mathbf{k})\epsilon^{ij}_{s^{\prime}}(\mathbf{k})=2\delta^{s}_{s^{\prime}}$,
transversality $k^{i}\epsilon_{ij}^{s}(\mathbf{k})=0$ and traceless
$\delta^{ij}\epsilon_{ij}^{s}(\mathbf{k})=0$ conditions. By plugging Eq. (7)
into Eq. (6) we obtain

$S_{\rm EH}=\frac{\pi^{2}}{8}\int
dt\int\differential^{3}k\,\sum_{s}\quantity[\absolutevalue{\dot{q}_{s}(t,\mathbf{k})}^{2}-\mathbf{k}^{2}\absolutevalue{q_{s}(t,\mathbf{k})}^{2}],$

where we used the fact that $\bar{h}_{ij}$ must be real, thus implying that
$\epsilon_{ij}^{s}(\mathbf{k})q_{s}^{*}(t,\mathbf{k})=\epsilon_{ij}^{s}(-\mathbf{k})q_{s}(t,-\mathbf{k})$.

Before writing down the total action, we note that

$\displaystyle\ddot{\bar{h}}_{ij}(t,0)\xi^{i}\xi^{j}$
$\displaystyle=\int\differential^{3}k\,\sum_{s}\epsilon_{ij}^{s}(\mathbf{k})\ddot{q}_{s}(t,\mathbf{k})\xi^{i}\xi^{j}$
$\displaystyle=\frac{1}{2}\int\differential^{3}k\,\sum_{s}\epsilon_{ij}^{s}(\mathbf{k})\quantity[\ddot{q}_{s}(t,\mathbf{k})+\ddot{q}^{*}_{s}(t,\mathbf{k})]\xi^{i}\xi^{j}$
$\displaystyle=\int\differential^{3}k\,\sum_{s}\epsilon_{ij}^{s}(\mathbf{k})\textrm{Re}[\ddot{q}_{s}(t,\mathbf{k})]\xi^{i}\xi^{j},$

which follows from Eqs. (5) and (7). We then see that only the real part of
the amplitude couples to the matter degrees of freedom. We therefore take
these amplitudes to be real and write the total action as

$\displaystyle S$ $\displaystyle=S_{\rm EH}+S_{\rm p}+S_{\rm int}$
$\displaystyle=\frac{\pi^{2}}{8}\int\differential
t\int\differential^{3}k\,\sum_{s}\quantity[\dot{q}^{2}_{s}(t,\mathbf{k})-\mathbf{k}^{2}q_{s}^{2}(t,\mathbf{k})]$
$\displaystyle\hskip 14.22636pt+\frac{1}{2}\int\differential
t\,\quantity[m_{0}-\lambda L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})]\,\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}$
$\displaystyle\hskip 14.22636pt+\lambda\int\differential t\,L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})$ $\displaystyle\hskip
14.22636pt+\frac{m_{0}}{4}\int\differential
t\,\int\differential^{3}k\,\sum_{s}q_{s}(t,\mathbf{k})\,X^{s}(t,\mathbf{k}),$
(8)

where we have defined

$X^{s}(t,\mathbf{k})\equiv\derivative[2]{t}\quantity{\epsilon_{ij}^{s}(\mathbf{k})\xi^{i}\xi^{j}\quantity[1-\frac{\lambda}{m_{0}}L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})]}.$ (9)

Our goal now is to quantize the gravitational field and to study the effect of
its quantum fluctuations on the system of interest.

###  II.2 Stochastic effective action

The interaction between the system and the gravitational field gives rise to a
coupling between the system’s external and internal variables. Ultimately, we
are not interested in what happens to the gravitons themselves, and the
quantized gravitational field shall be treated as an environment. We treat
this problem by using the Feynman-Vernon influence functional approach to open
quantum systems [15, 16]. In this approach, the influence functional is
defined as

$\mathcal{F}[x,x^{\prime}]=e^{iS_{\rm IF}[x,x^{\prime},t]},$ (10)

where $x$ and $x^{\prime}$ are distinct values of the system variables and
$S_{\rm IF}$ is the influence action.

Taking into account that the gravitational part of our system is quadratic in
the relevant variables and the interaction is linear, we obtain the following
result

$\begin{split}S_{\rm IF}^{g}[x,x^{\prime}]=\int\differential t\differential
t\left\\{\frac{1}{2}\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]D^{ijkl}_{\rm
g}(t,t^{\prime})\quantity[x_{kl}(t^{\prime})+x^{\prime}_{kl}(t^{\prime})]\right.\\\
\left.+\frac{i}{2}\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]N^{ijkl}_{\rm
g}(t,t^{\prime})\quantity[x_{kl}(t^{\prime})-x^{\prime}_{kl}(t^{\prime})]\right\\},\end{split}$
(11)

where we have defined

$x_{ij}(t)\equiv\xi_{i}(t)\xi_{j}(t)\quantity[1-\frac{\lambda}{m_{0}}L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})],$ (12)

while

$D^{ijkl}_{\rm
g}(t,t^{\prime})=i\quantity(\frac{m_{0}}{4})^{2}\derivative[2]{t}\derivative[2]{{t^{\prime}}}\int\differential^{3}k\,\sum_{s}\epsilon^{ij}_{s}(\mathbf{k})\epsilon^{kl}_{s}(\mathbf{k})\expectationvalue{\commutator{q_{s}(t,\mathbf{k})}{q_{s}(t^{\prime},\mathbf{k})}}_{\rm
g}\theta(t-t^{\prime}),$ (13a) and $N^{ijkl}_{\rm
g}(t,t^{\prime})=\frac{1}{2}\quantity(\frac{m_{0}}{4})^{2}\derivative[2]{t}\derivative[2]{{t^{\prime}}}\int\differential^{3}k\,\sum_{s}\epsilon^{ij}_{s}(\mathbf{k})\epsilon^{kl}_{s}(\mathbf{k})\expectationvalue{\anticommutator{q_{s}(t,\mathbf{k})}{q_{s}(t^{\prime},\mathbf{k})}}_{\rm
g},$ (13b)

are the (gravitational) dissipation and noise kernels, respectively [18, 32].
The $q$’s stand for position operators in the Heisenberg representation. The
expectation values with the subscript $g$ are computed with respect to the
initial state of the gravitons. Also, in deriving Eq. (11) we have used the
fact that each graviton mode (and polarization) is independent of all the
others and therefore can be treated separately in such a way that the total
influence action is the sum of the action corresponding to each mode (and
polarization) [15, 16]. The details of the calculations are presented in
Appendix A.

Following Feynman and Vernon, we express the noise term in $S_{\rm IF}^{g}$ in
terms of a stochastic variable $\mathcal{N}_{ij}(t)$ using the Gaussian
functional identity [18]

$\begin{split}&e^{-\frac{1}{2}\int
dtdt^{\prime}\,\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]N^{ijkl}_{\rm
g}(t,t^{\prime})\quantity[x_{kl}(t^{\prime})-x^{\prime}_{kl}(t^{\prime})]}\\\
&\hskip 5.69046pt=\mathcal{C}\int\mathcal{D}\mathcal{N}\,e^{-\frac{1}{2}\int
dtdt^{\prime}\,\mathcal{N}_{ij}(t)(N_{g}^{-1})^{ijkl}(t,t^{\prime})\mathcal{N}_{kl}(t^{\prime})}\\\
&\hskip 42.67912pt\times e^{-i\int
dt\,\mathcal{N}^{ij}(t)\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]},\end{split}$
(14)

where $\mathcal{C}$ is a normalization constant. Stochastic averages are
performed considering the Gaussian probability density

$P[\mathcal{N}]=\mathcal{C}\,e^{-\frac{1}{2}\int
dtdt^{\prime}\,\mathcal{N}_{ij}(t)(N_{g}^{-1})^{ijkl}(t,t^{\prime})\mathcal{N}_{kl}(t^{\prime})}.$
(15)

For example, the one- and two-point correlation functions are given by

$\expectationvalue{\mathcal{N}^{ij}(t)}_{\rm
sto}=\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\mathcal{N}^{ij}(t)=0,$ (16a)
$\displaystyle\expectationvalue{\mathcal{N}^{ij}(t)\mathcal{N}^{kl}(t^{\prime})}_{\rm
sto}$
$\displaystyle=\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\mathcal{N}^{ij}(t)\mathcal{N}^{kl}(t^{\prime})$
$\displaystyle=N_{\rm g}^{ijkl}(t,t^{\prime}).$ (16b)

The (gravitational) influence functional then becomes

$\begin{split}&e^{iS_{\rm IF}^{g}[x,x^{\prime}]}\\\ &\hskip
2.84544pt=\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\,e^{-i\int
dt\,\mathcal{N}^{ij}(t)\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]}\\\ &\hskip
14.22636pt\times e^{\frac{i}{2}\int
dtdt^{\prime}\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]D^{ijkl}_{\rm
g}(t,t^{\prime})\quantity[x_{kl}(t^{\prime})+x^{\prime}_{kl}(t^{\prime})]}.\end{split}$
(17)

We now proceed by considering only the external degrees of freedom of our
system since we are interested in the action of the noise coming from its
internal degrees of freedom and from the quantum fluctuations of the
gravitational field. In order to do this, we have to compute the reduced
density matrix of the relevant degrees of freedom while tracing out all the
others.

##  III The external degrees of freedom density matrix

The particle (external + internal degrees of freedom) density matrix at time
$t$ is given by [see Eq. (59)]

$\begin{split}\rho_{\rm
p}(\xi,\varrho,\xi^{\prime},\varrho,t)=\int\differential\xi(0)\differential\xi^{\prime}(0)\differential\varrho(0)\differential\varrho^{\prime}(0)\\\
\times\mathcal{J}(\xi,\varrho,\xi^{\prime},\varrho,t|\xi(0),\varrho(0),\xi^{\prime}(0),\varrho^{\prime}(0),0)\\\
\times\rho_{\rm
p}(\xi(0),\varrho(0),\xi^{\prime}(0),\varrho^{\prime}(0),0),\end{split}$ (18)

where

$\begin{split}&\mathcal{J}(\xi,\varrho,\xi^{\prime},\varrho,t|\xi(0),\varrho(0),\xi^{\prime}(0),\varrho^{\prime}(0),0))\\\
&=\int\mathcal{D}\xi\mathcal{D}\xi^{\prime}\mathcal{D}\varrho\mathcal{D}\varrho^{\prime}\,e^{i\quantity(S_{\rm
sys}[\xi,\varrho]-S_{\rm sys}[\xi^{\prime},\varrho^{\prime}])}e^{iS^{g}_{\rm
IF}[\xi,\varrho,\xi^{\prime},\varrho^{\prime}]},\end{split}$ (19)

with $e^{iS^{g}_{\rm IF}}$ given in Eq. (17) in terms of the variable
$x_{ij}(t)$ that is defined in Eq. (12).

We observe that the term involving the dissipation kernel in Eq. (17) is of
order $O(\xi^{4})$ since $x_{ij}$ is already of order $O(\xi^{2})$. Thus the
leading order contribution comes from the term involving the noise kernel and
we approximate Eq. (17) as

$e^{iS_{\rm
IF}^{g}[x,x^{\prime}]}\simeq\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\,e^{-i\int
dt\,\mathcal{N}^{ij}(t)\quantity[x_{ij}(t)-x^{\prime}_{ij}(t)]}.$ (20)

This means that we are only considering the influence of the gravitational
field encoded in the noise kernel, which depends on the initial state of the
gravitons, as can be seen from the definition (13b).

Within the above approximation we find

$\begin{split}&\mathcal{J}(\xi,\varrho,\xi^{\prime},\varrho,t|\xi(0),\varrho(0),\xi^{\prime}(0),\varrho^{\prime}(0),0))\\\
&=\int\mathcal{D}\xi\mathcal{D}\xi^{\prime}\mathcal{D}\varrho\mathcal{D}\varrho^{\prime}\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\,e^{i\quantity(S_{\rm
eff}[\xi,\varrho,\mathcal{N}]-S_{\rm
eff}[\xi^{\prime},\varrho^{\prime},\mathcal{N}])},\end{split}$ (21)

where

$\begin{split}&S_{\rm eff}[\xi,\varrho,\mathcal{N}]=\int
dt\,\left[\frac{1}{2}m_{0}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}-\mathcal{N}_{ij}\xi^{i}\xi^{j}\right.\\\
&\hskip
5.69046pt\left.-\lambda\quantity(\frac{1}{2}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}-\frac{1}{m_{0}}\mathcal{N}_{ij}\xi^{i}\xi^{j}-1)L_{\rm
int}(\varrho,\dot{\varrho}\bar{t})\right].\end{split}$ (22)

We now assume that initially the external and internal degrees of freedom of
our system are uncorrelated, thus implying that

$\rho_{\rm
p}\left(\xi,\varrho,\xi^{\prime},\varrho^{\prime},0\right)=\rho_{\rm
ext}\left(\xi,\xi^{\prime},0)\rho_{\rm
int}(\varrho,\varrho^{\prime},0\right),$

where $\rho_{\rm ext}$ ($\rho_{\rm int}$) stands for the external (internal)
degrees of freedom density matrix elements and all the variables are written
at the initial time. The interaction with the gravitons couples the external
and internal variables and we are left with the total density matrix (18). We
are interested here in the dynamics of the external degrees of freedom and,
thus, we proceed by taking the partial trace over the internal ones

$\rho_{\rm ext}(\xi,\xi^{\prime},t)=\int\differential\varrho\,\rho_{\rm
p}(\xi,\varrho,\xi^{\prime},\varrho,t).$

resulting in

$\begin{split}\rho_{\rm
ext}&(\xi,\xi^{\prime},t)=\int\differential\xi(0)d\xi^{\prime}(0)\,\rho_{\rm
ext}(\xi(0),\xi^{\prime}(0),0)\\\
&\times\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\int_{\xi(0)}^{\xi}\mathcal{D}\xi\int_{\xi^{\prime}(0)}^{\xi^{\prime}}\mathcal{D}\xi^{\prime}\\\
&\times e^{i\quantity(S_{\rm eff}^{(1)}[\xi,\mathcal{N}]-S_{\rm
eff}^{(1)}[\xi^{\prime},\mathcal{N}])}e^{S_{\rm
IF}^{(\textrm{int})}[\xi,\xi^{\prime},\mathcal{N}]},\end{split}$ (23)

where

$\begin{split}&e^{iS_{\rm
IF}^{(\textrm{int})}[\xi,\xi^{\prime},\mathcal{N}]}\\\ \hskip 5.69046pt&=\int
d\varrho d\varrho(0)d\varrho^{\prime}(0)\,\rho_{\rm
int}(\varrho(0),\varrho^{\prime}(0),0)\\\ &\hskip
5.69046pt\times\int_{\varrho(0)}^{\varrho}\mathcal{D}\varrho\int_{\varrho^{\prime}(0)}^{\varrho}\mathcal{D}\varrho^{\prime}\,e^{i\quantity(S_{\rm
eff}^{(2)}[\xi,\varrho,\mathcal{N}]-S_{\rm
eff}^{(2)}[\xi^{\prime},\varrho^{\prime},\mathcal{N}])},\end{split}$ (24)

while

$S_{\rm eff}^{(1)}=\int
dt\quantity(\frac{1}{2}m_{0}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}-\mathcal{N}_{ij}\xi^{i}\xi^{j}),$
(25a) and $S_{\rm eff}^{(2)}=\lambda\int
dt\quantity(1-\frac{1}{2}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}+\frac{1}{m_{0}}\mathcal{N}_{ij}\xi^{i}\xi^{j})L_{\rm
int}(\varrho,\dot{\varrho}\bar{t}).$ (25b)

Thus, we essentially have the same problem as the one treated in Sec. II,
namely a system interacting with a quantum environment. Therefore, we shall
compute the Feynman-Vernon influence functional once again. It is worth to
remark that the total influence functional (gravitons plus internal degrees of
freedom) is not simply the product of the individual functionals, since the
gravitational field couples with all variables describing the system. When
considering the system of interest to be the external degrees of freedom, we
effectively end with two environments that interact with the system and with
each other. In such a case the additive property of the influence action for
multiple environments does not hold [15].

In order to proceed with our calculations, let us assume that the Lagrangian
describing the internal degrees of freedom is of the form

$L_{\rm
int}(\varrho,\dot{\varrho}\,\bar{t})=\sum_{\alpha}\quantity[\frac{1}{2}\mu_{\alpha}\bar{\varrho}_{\alpha}^{2}-V(\varrho_{\alpha})],$
(26)

where the $\mu$’s are some reduced masses of the system and $V$ is a function
of the coordinates. Since we want to keep terms only up to second order in the
position and velocity coordinates, we may write

$L_{\rm
int}(\varrho,\dot{\varrho}\,\bar{t})\simeq\sum_{\alpha}\quantity(\frac{1}{2}\mu_{\alpha}\dot{\varrho}_{\alpha}^{2}-\vartheta_{\alpha}\varrho_{\alpha}-\frac{1}{2}\mu_{\alpha}\varpi_{\alpha}^{2}\varrho_{\alpha}^{2}),$
(27)

where the $\vartheta$’s and the $\varpi$’s are constants. Then, Eq. (25b)
becomes

$\begin{split}S_{\rm eff}^{(2)}=\sum_{\alpha}\left[\lambda\int
dt\quantity(\frac{1}{2}\mu_{\alpha}\dot{\varrho}_{\alpha}^{2}-\frac{1}{2}\mu_{\alpha}\varpi_{\alpha}^{2}\varrho_{\alpha}^{2})\right.\\\
\left.+\lambda\vartheta_{\alpha}\int
dt\,Y(t)\varrho_{\alpha}(t)\right],\end{split}$ (28a) with
$Y(t)=\frac{1}{2}\delta_{ij}\dot{\xi}^{i}\dot{\xi}^{j}-\frac{1}{m_{0}}\mathcal{N}_{ij}\xi^{i}\xi^{j}-1.$
(28b)

Therefore, we are considering the internal degrees of freedom to be
approximated by a set of independent harmonic oscillators that couples
linearly with the external degrees of freedom. In that case the internal
degrees of freedom influence functional (24) is Gaussian, implying that we can
write the influence action as

$\begin{split}&S^{(\textrm{int})}_{\rm IF}[Y,Y^{\prime}]=\int dtdt^{\prime}\\\
&\times\left\\{\frac{1}{2}\quantity[Y(t)-Y^{\prime}(t)]D_{\rm
int}(t,t^{\prime})\quantity[Y(t^{\prime})+Y^{\prime}(t^{\prime})]\right.\\\
&\hskip 17.07182pt\left.+\frac{i}{2}\quantity[Y(t)-Y^{\prime}(t)]N_{\rm
int}(t,t^{\prime})\quantity[Y(t^{\prime})-Y^{\prime}(t^{\prime})]\right\\},\end{split}$
(29)

with

$D_{\rm
int}(t,t^{\prime})=i\lambda^{2}\sum_{\alpha}\vartheta_{\alpha}^{2}\expectationvalue{\commutator{\varrho_{\alpha}(t)}{\varrho_{\alpha}(t^{\prime})}}_{\rm
int}\theta(t-t^{\prime}),$ (30a) $N_{\rm
int}(t,t^{\prime})=\frac{1}{2}\lambda^{2}\sum_{\alpha}\vartheta_{\alpha}^{2}\expectationvalue{\anticommutator{\varrho_{\alpha}(t)}{\varrho_{\alpha}(t^{\prime})}}_{\rm
int},$ (30b)

where now the $\varrho(t)$’s are operators in the Heisenberg representation.

Similarly to what we did for the term containing the noise kernel for the
gravitational influence functional, we can express the noise term in $S_{\rm
IF}^{(\textrm{int})}$ in terms of a stochastic variable using the same
Gaussian functional identity. Then, this term in the internal degrees of
freedom influence functional will give place to a Gaussian probability density
and a linear term on the $Y$’s variables. Therefore, just like in the
gravitational case, the leading order contributions come from the noise term
and we may take

$\begin{split}&e^{iS_{\rm IF}^{(\textrm{int})}[Y,Y^{\prime}]}\\\ &\hskip
5.69046pt\simeq e^{-\frac{1}{2}\int
dtdt^{\prime}\,\quantity[Y(t)-Y^{\prime}(t)]N_{\rm
int}(t,t^{\prime})\quantity[Y(t^{\prime})-Y^{\prime}(t^{\prime})]},\end{split}$
(31)

resulting in

$\displaystyle\rho_{\rm ext}(\xi,\xi^{\prime},t)$ $\displaystyle=$
$\displaystyle\int d\xi(0)d\xi^{\prime}(0)\,\rho_{\rm
ext}(\xi(0),\xi^{\prime}(0),0)\int_{\xi(0)}^{\xi}\mathcal{D}\xi\int_{\xi^{\prime}(0)}^{\xi^{\prime}}\mathcal{D}\xi^{\prime}\,e^{\frac{i}{2}m_{0}\delta^{ij}\int
dt\,\quantity(\dot{\xi}_{i}\dot{\xi}_{j}-\dot{\xi}_{i}^{\prime}\dot{\xi}_{j}^{\prime})}$
(32) $\displaystyle\times$
$\displaystyle\int\mathcal{D}\mathcal{N}\,P[\mathcal{N}]\,e^{-i\int
dt\,\mathcal{N}^{ij}\quantity(\xi_{i}\xi_{j}-\xi_{i}^{\prime}\xi_{j}^{\prime})}e^{-\frac{1}{2}\int
dtdt^{\prime}\,\quantity[Y(t)-Y^{\prime}(t)]N_{\rm
int}(t,t^{\prime})\quantity[Y(t^{\prime})-Y^{\prime}(t^{\prime})]}.$ (33)

Fallowing the same reasoning presented in the previous section, we consider
only the influence encoded into the noise kernel.

The stochastic averages shown in Eqs. (16) can be employed provided we work on
a perturbative regime (dropping terms of order $\xi^{3}$ and higher). From
this, we obtain the external degrees of freedom density matrix given as

$\displaystyle\rho_{\rm ext}(\xi,\xi^{\prime},t)$ $\displaystyle\simeq\int
d\xi(0)d\xi^{\prime}(0)\,\rho_{\rm
ext}(\xi(0),\xi^{\prime}(0),0)\int_{\xi(0)}^{\xi}\mathcal{D}\xi\int_{\xi^{\prime}(0)}^{\xi^{\prime}}\mathcal{D}\xi^{\prime}\,e^{\frac{i}{2}m_{0}\delta^{ij}\int
dt\,\quantity(\dot{\xi}_{i}\dot{\xi}_{j}-\dot{\xi}_{i}^{\prime}\dot{\xi}_{j}^{\prime})}$
$\displaystyle\hskip
14.22636pt\times\exp\left\\{-\frac{1}{4}\delta^{ij}\delta^{kl}\int
dtdt^{\prime}\,\quantity[\dot{\xi}_{i}(t)\dot{\xi}_{j}(t)-\dot{\xi}_{i}^{\prime}(t)\dot{\xi}_{j}^{\prime}(t)]N_{\rm
int}(t,t^{\prime})\quantity[\dot{\xi}_{k}(t^{\prime})\dot{\xi}_{l}(t^{\prime})-\dot{\xi}_{k}^{\prime}(t^{\prime})\dot{\xi}_{l}^{\prime}(t^{\prime})]\right.$
(34) $\displaystyle\hskip 19.91684pt\left.-\int
dtdt^{\prime}\,\quantity[\xi_{i}(t)\xi_{j}(t)-\xi^{\prime}_{i}(t)\xi^{\prime}_{j}(t)]\quantity[\frac{1}{2}+\frac{1}{m_{0}^{2}}N_{\rm
int}(t,t^{\prime})]N^{ijkl}_{\rm
g}(t,t^{\prime})\quantity[\xi_{k}(t^{\prime})\xi_{l}(t^{\prime})-\xi^{\prime}_{k}(t^{\prime})\xi^{\prime}_{l}(t^{\prime})]\right\\}.$

As we can see, the off-diagonal elements of the density matrix in the
coordinate basis ($\xi\neq\xi^{\prime}$) evolves with a exponentially decaying
amplitude. Therefore, for an initial superposition state of the external
degrees of freedom, the interaction with the internal degrees of freedom
induced by the gravitons as well as the interaction with the gravitons
themselves inevitably leads to lost of quantum coherence. In order to
determine how significant these effects are, in the next section we compute
the decoherence rate.

##  IV Decoherence rate

We assume that the system is in a superposition state of two spatially
separated locations, $\rho_{\rm ext}(0)=\ket{\Psi(0)}\bra{\Psi(0)}$ with

$\ket{\Psi(0)}=\frac{1}{\sqrt{2}}\quantity(\ket{\xi^{(1)}(0)}+\ket{\xi^{(2)}(0)}).$
(35)

Let us say that the superposition state lasts for a time in the interval
$0<t<t_{f}$ and $\xi^{(1)}(t)=\xi^{(2)}(t)$ holds for $t\notin[0,t_{f}]$. This
means that the decoherence turned both trajectories indistinguishable. We can
safely say that the coherence between the two components of this state will be
effectively lost when $\exp\quantity[-\Gamma(t_{f})]<<1$ —since the decay is
exponential as we saw in the last section—, where $\Gamma(t_{f})$ is the
decoherence rate [17]. Such a rate can be estimated from Eq. (III), resulting
in

$\begin{split}\Gamma(t_{f})=\delta^{ij}\delta^{kl}\int_{0}^{t_{f}}dtdt^{\prime}\,V_{i}(t)\Delta
v_{j}(t)N_{\rm int}(t,t^{\prime})\\\ \times V_{k}(t^{\prime})\Delta
v_{l}(t^{\prime})\\\
+2\int_{0}^{t_{f}}dtdt^{\prime}\,\Xi_{i}(t)\Delta\xi_{j}(t)\quantity[1+\frac{1}{m_{0}^{2}}2N_{\rm
int}(t,t^{\prime})]\\\ \times N^{ijkl}_{\rm
g}(t,t^{\prime})\Xi_{k}(t^{\prime})\Delta\xi_{l}(t^{\prime}),\end{split}$ (36)

where

$\Xi_{i}\equiv\frac{1}{2}\quantity(\xi_{i}^{(1)}+\xi_{i}^{(2)}),\hskip
8.5359pt\Delta\xi_{i}\equiv\xi_{i}^{(1)}-\xi_{i}^{(2)},$

and similarly for $V_{i}$ and $\Delta v_{i}$. Since the quantities $V_{i}$ and
$\Xi$ are averages of the paths in superposition, we will assume these to be
time independent [17]. In Appendix B we explicitly compute $N_{\rm int}$ by
considering an Ohmic bath in the high temperature limit. The internal degrees
of freedom noise kernel is given in Eq. (77). In this case, the decoherence
rate becomes

$\begin{split}\Gamma(t_{f})&=\frac{\lambda^{2}\gamma\pi}{\beta}\delta^{ij}\delta^{kl}V_{i}V_{k}\int_{0}^{t_{f}}dt\,\Delta
v_{j}(t)\Delta v_{l}(t)\\\ &\hskip
5.69046pt+2\Xi_{i}\Xi_{k}\int_{0}^{t_{f}}dtdt^{\prime}\,\Delta\xi_{j}(t)N_{\rm
g}^{ijkl}(t,t^{\prime})\Delta\xi_{l}(t^{\prime})\\\ &\hskip
5.69046pt+\frac{4\lambda^{2}\gamma\pi}{\beta
m_{0}^{2}}\Xi_{i}\Xi_{k}\int_{0}^{t_{f}}dt\,\Delta\xi_{j}(t)N_{\rm
g}^{ijkl}(t)\Delta\xi_{l}(t),\end{split}$ (37)

where $N_{\rm g}^{ijkl}(t)\equiv\lim_{t^{\prime}\to t}N_{\rm
g}^{ijkl}(t,t^{\prime})$ while $\gamma$ represents the effective coupling
constant and $\beta$ is the inverse temperature associated with the internal
degrees of freedom.

Let us now consider a specific configuration of the superposition state. Take
the separation $\Delta\xi_{i}(t)=\xi_{i}^{(2)}(t)-\xi_{i}^{(1)}(t)$ to be
given by [17, 29]

$\Delta\xi_{i}(t)=\left\\{\begin{array}[]{ll}2v_{i}t&\textrm{for}\hskip
5.69046pt0<t\leq t_{f}/2\\\ 2v_{i}(t_{f}-t)&\textrm{for}\hskip
5.69046ptt_{f}/2<t<t_{f}\end{array}\right.,$ (38a) such that $\Delta
v_{i}(t)=\left\\{\begin{array}[]{ll}2v_{i}&\textrm{for}\hskip 5.69046pt0<t\leq
t_{f}/2\\\ -2v_{i}&\textrm{for}\hskip
5.69046ptt_{f}/2<t<t_{f}\end{array}\right..$ (38b)

For this specific configuration, the decoherence rate (37) turns out to be

$\begin{split}\Gamma(t_{f})&=8\Xi_{i}v_{j}\Xi_{k}v_{l}\left[\int_{0}^{t_{f}/2}dtdt^{\prime}\,tt^{\prime}N_{\rm
g}^{ijkl}(t,t^{\prime})\right.\\\ &\hskip
28.45274pt+\int_{t_{f}/2}^{t_{f}}dtdt^{\prime}\,(t_{f}-t)(t_{f}-t^{\prime})N_{\rm
g}^{ijkl}(t,t^{\prime})\\\ &\hskip
28.45274pt\left.+2\int_{0}^{t_{f}/2}dt\int_{t_{f}/2}^{t_{f}}dt^{\prime}\,t(t_{f}-t^{\prime})N_{\rm
g}^{ijkl}(t,t^{\prime})\right]\\\ &\hskip
5.69046pt+\frac{16\lambda^{2}\gamma\pi}{\beta
m_{0}^{2}}\Xi_{i}v_{j}\Xi_{k}v_{l}\left[\int_{0}^{t_{f}/2}dt\,t^{2}N_{\rm
g}^{ijkl}(t)\right.\\\ &\hskip
28.45274pt\left.+\int_{t_{f}/2}^{t_{f}}dt\,(t_{f}-t)^{2}N_{\rm
g}^{ijkl}(t)\right].\end{split}$ (39)

In order to proceed, we need to specify the state of the gravitational field.
We show here only the results of the calculations considering four distinct
initial states: $i$) vacuum; $ii$) thermal; $iii$) coherent; $iv$) squeezed.
The details of the calculations can be found in Appendix C

####  IV.0.1 Vacuum state

For gravitons initially in the vacuum state, the noise kernel is given in Eq.
(85), from which we can compute the decoherence rate as

$\Gamma_{\rm(vac)}(t_{f})=\frac{8m_{0}^{2}}{5\pi}\Lambda_{\rm
g}^{2}\mathcal{K}\quantity[G(\Lambda_{\rm
g}t_{f})+\lambda^{2}\kappa\quantity(\Lambda_{\rm g}t_{f})^{3}],$ (40)

where

$\begin{split}G(x)&\equiv 1+\frac{2}{3x}\quantity[\sin
x-8\sin\quantity(\frac{x}{2})]\\\ &\hskip
14.22636pt+\frac{1}{x^{2}}\quantity[\frac{2}{3}\cos
x-\frac{32}{3}\cos\quantity(\frac{x}{2})+10].\end{split}$ (41)

$\mathcal{K}\equiv\mathcal{P}^{ijkl}\Xi_{i}v_{j}\Xi_{k}v_{l}$, while

$\kappa\equiv\frac{\gamma\pi}{108m_{0}^{2}}\frac{\Lambda_{\rm g}}{\beta}.$
(42)

$\mathcal{P}^{ijkl}$ is a function of the Kronecker delta explicitly given in
Appendix C and $\Lambda_{\rm g}$ is an appropriate cutoff. We observe that
when $\lambda=0$ (no internal degrees of freedom), we recover the same
qualitative behavior found in Ref. [17], as expected.

The decoherence rate consists of the sum of two terms, one involving the
decoherence induced exclusively by the quantum gravitational field and another
one that comes from the coupling with both the internal degrees of freedom and
the gravitons. The intensity of the latter term is given by a constant
$\kappa$, which depends on the initial state considered. We expect that this
term gives a greater contribution to the decoherence rate since it involves
the coupling with two interacting environments and thus contains a further
level of coarse-graining. In fact, Fig. 1 shows how, even for small values of
$\kappa$, the behavior of the decoherence rate is dominated by the term
involving not only the gravitons but also the internal degrees of freedom of
the particle.

Figure 1: Vacuum state. Decoherence rate coming from the purely gravitational
fluctuations and due to the presence of the internal degrees of freedom of our
system. This last contribution becomes very important for $\kappa\gtrsim
10^{-4}$.

The constant $\kappa$ quantifies the strength of the decoherence due to the
presence of the internal degrees of freedom. Dropping the exclusively
gravitational contribution, setting $\lambda=1$ and restoring the constants
$\hbar$, $c$, $k_{B}$ and $G$, the contribution to the decoherence rate is
given by

$\Gamma_{\rm(vac)}(t_{f})\simeq\frac{8\mathcal{K}}{5\pi}\quantity(\frac{m_{0}}{M_{\rm
pl}})^{2}\kappa\Lambda_{\rm g}^{2}\quantity(\Lambda_{\rm g}t_{f})^{3},$ (43)

where

$\kappa=\frac{1}{108}\varkappa\Lambda_{\rm g}T,\hskip
28.45274pt\varkappa\equiv\frac{k_{B}\gamma\pi}{\hbar^{3}c^{4}m_{0}^{2}}$

while $M_{\rm pl}=\sqrt{\hbar c/G}$ is the Planck mass.

####  IV.0.2 Thermal state

We now consider that the quantum gravitational field is in a thermal state
with inverse temperature $\beta_{\rm g}^{-1}$. Before computing the kernel, it
is important to observe that what we call temperature of gravitons may not be
interpreted as temperature in a formal sense, but rather as a phenomenological
parameter characterizing the power spectrum of the noise that is not related
to any thermodynamic definition. It is important to keep this distinction in
mind since gravitons interact very weakly and their thermalization cannot be
assumed [20].

That being sad, the noise kernel is given in Eq. (87), thus resulting in the
following expression for the decoherence rate

$\begin{split}&\Gamma_{\rm(th)}(t_{f})=\Gamma_{\rm(vac)}(t_{f})\\\ &\hskip
5.69046pt+\frac{16m_{0}^{2}\pi}{15\beta_{\rm
g}^{2}}\mathcal{K}\,\quantity[G_{\rm th}\quantity(\frac{\pi t_{f}}{\beta_{\rm
g}})+\lambda^{2}\kappa_{\rm th}\quantity(\frac{\pi t_{f}}{\beta_{\rm
g}})^{3}],\end{split}$ (44)

where

$G_{\rm
th}(x)\equiv\frac{1+16e^{x}+26e^{2x}+16e^{3x}+e^{4x}}{(e^{2x}-1)^{2}}-\frac{15}{x^{2}},$
(45a) and $\kappa_{\rm
th}\equiv\frac{4\gamma\pi^{2}}{189m_{0}^{2}\beta\beta_{\rm g}}.$ (45b)

Figure 2 shows the comparison between the distinct contributions for the
decoherence rate coming from the gravitational field and from the internal
degrees of freedom.

Figure 2: Thermal state. Decoherence rate coming from the purely gravitational
fluctuations and due to the presence of the internal degrees of freedom of our
system. This last contribution becomes very important for $\kappa_{\rm
th}\gtrsim 10^{-4}$.

Although we observe a distinct quantitative behaviour with respect to the
vacuum case, the qualitative pattern is the same, with the internal variables
contributing significantly to the decoherence phenomenon. The intensity of
such contribution is obtained by dropping the purely gravitational effect,
setting $\lambda=1$ and restoring all the physical quantities, resulting in
the decoherence rate

$\begin{split}\Gamma_{\rm(th)}(t_{f})&\simeq\Gamma_{\rm(vac)}(t_{f})\\\
&+\frac{16\pi^{4}\mathcal{K}}{15}\quantity(\frac{m_{0}}{M_{\rm
pl}})^{2}\kappa_{\rm th}\quantity(k_{B}T_{\rm g})^{2}\quantity(k_{B}T_{\rm
g}t_{f})^{3},\end{split}$ (46)

with

$\kappa_{\rm th}=\frac{4\pi}{189}\varkappa k_{B}T_{\rm g}T$

####  IV.0.3 Coherent state

For gravitons initially in a coherent sate characterized by the real
displacement parameter $\alpha$, the noise kernel can be found in Eq. (93),
implying the decoherence rate

$\begin{split}&\Gamma_{\rm(coh)}(t_{f})=\Gamma_{\rm(vac)}(t_{f})\\\ &\hskip
5.69046pt+\frac{128m_{0}^{2}\alpha^{2}\tilde{\Lambda}_{\rm
g}^{2}}{15\pi}\mathcal{K}\quantity[G_{\rm coh}^{(I)}(\tilde{\Lambda}_{\rm
g}t_{f})+\lambda^{2}\kappa_{\rm coh}G_{\rm coh}^{(II)}(\tilde{\Lambda}_{\rm
g}t_{f})],\end{split}$ (47)

where

$\begin{split}G_{\rm
coh}^{(I)}(x)\equiv\frac{1}{1152x^{2}}\left\\{1495+126x^{2}-1728\cos\quantity(\frac{x}{2})\right.\\\
+288\cos x-64\cos\quantity(\frac{3x}{2})+9\cos(2x)+18x\sin(2x)\\\
\left.-96x\quantity[9\sin\quantity(\frac{x}{2})-3\sin
x+\sin\quantity(\frac{3x}{2})]\right\\},\end{split}$ (48a)
$\begin{split}G_{\rm
coh}^{(II)}(x)\equiv\frac{1}{12x^{3}}\left[441+2x^{6}+216\quantity(x^{2}-2)\cos
x\right.\\\ \left.+9\quantity(2x^{2}-1)\cos(2x)-36x\quantity(12-2x^{2}+\cos
x)\sin x\right],\end{split}$ (48b) and $\kappa_{\rm
coh}\equiv\frac{\gamma\pi\tilde{\Lambda}_{\rm g}}{192m_{0}^{2}\beta}.$ (48c)

We again observe the same pattern that is illustrated in Fig. 3.

Figure 3: Coherent state. Decoherence rate coming from the purely
gravitational fluctuations and due to the presence of the internal degrees of
freedom of our system. This last contribution becomes very important for
$\kappa_{\rm coh}\gtrsim 10^{-4}$.

For this case, the contribution to the decoherence rate coming from the
internal degrees of freedom takes the form

$\begin{split}\Gamma_{\rm(coh)}(t_{f})&\simeq\Gamma_{\rm(vac)}(t_{f})\\\
&+\frac{128\alpha^{2}\mathcal{K}}{15\pi}\quantity(\frac{m_{0}}{M_{\rm
pl}})^{2}\kappa_{\rm coh}\tilde{\Lambda}_{\rm g}^{2}G_{\rm
coh}^{(II)}(\tilde{\Lambda}_{\rm g}t_{f}),\end{split}$ (49)

with

$\kappa_{\rm coh}=\frac{1}{192}\varkappa\tilde{\Lambda}_{\rm g}T.$

We now move to our last example, the squeezed state.

####  IV.0.4 Squeezed state

Considering that the gravitons are initially in a squeezed sate parametrized
by $\zeta=re^{i\varphi}$, we obtain the noise kernel shown in Eq. (99). From
this we directly obtain the decoherence rate

$\begin{split}&\Gamma_{\rm(sq)}(t_{f})=\cosh 2r\,\Gamma_{\rm(vac)}(t_{f})\\\
&\hskip 5.69046pt-\frac{2m_{0}^{2}\bar{\Lambda}_{\rm
g}^{2}}{135\pi}\mathcal{K}\sinh 2r\\\ &\hskip 14.22636pt\times\quantity[G_{\rm
sq}^{(I)}(\bar{\Lambda}_{\rm g}t_{f};\varphi)+\lambda^{2}\kappa_{\rm sq}G_{\rm
sq}^{(II)}(\bar{\Lambda}_{\rm g}t_{f};\varphi)],\end{split}$ (50)

where

$\begin{split}G_{\rm
sq}^{(I)}(x;\varphi)\equiv\frac{1}{x^{2}}\left[-576\cos\quantity(\frac{x}{2}-\varphi)+216\cos(x-\varphi)\right.\\\
-64\cos\quantity(\frac{3x}{2}-\varphi)+9\cos(2x-\varphi)\\\
+(18x^{2}+415)\cos\varphi\\\
-288x\sin\quantity(\frac{x}{2}-\varphi)+216x\sin(x-\varphi)\\\
\left.-96x\sin\quantity(\frac{3x}{2}-\varphi)+18x\sin(2x-\varphi)\right],\end{split}$
(51a) $\begin{split}G_{\rm
sq}^{(II)}(x;\varphi)\equiv\frac{1}{x^{3}}\left[72(x^{2}-2)\cos(x-\varphi)\right.\\\
+(6x^{2}-3)\cos(2x-\varphi)+147\cos\varphi\\\
-144x\sin(x-\varphi)+24x^{3}\sin(x-\varphi)\\\
\left.-6x\sin(2x-\varphi)-4x^{3}\sin\varphi\right],\end{split}$ (51b) and
$\kappa_{\rm sq}\equiv\frac{3\gamma\pi\bar{\Lambda}_{\rm g}}{2\beta
m_{0}^{2}}.$ (51c)

Figure 4 illustrates the dynamical behaviour of the decoherence rate. Although
we obtained the same pattern here, in this case the contribution coming from
purely gravitational fluctuations are dominant, except for very large
interaction strength $\kappa$.

Figure 4: Squeezed state. The decoherence rate is found to be dominated by the
corrections involving both gravitons and internal degrees of freedom for
$\kappa_{\rm sq}\gtrsim 10^{2}$, in contrast with the cases considered before
where this correction dominates already for $\kappa\gtrsim 10^{-4}$. In fact,
even for $\kappa_{\rm sq}\sim 10^{-2}$ the behavior of the decoherence rate is
completely dominated by the exclusive graviton contribution.

Disregarding the contributions coming solely from the quantum fluctuations of
the gravitational field, we end up with the following expression for the
decoherence rate

$\begin{split}\Gamma_{\rm(sq)}(t_{f})&\simeq\cosh
2r\,\Gamma_{\rm(vac)}(t_{f})\\\ &-\frac{2\mathcal{K}}{135\pi}\sinh
2r\quantity(\frac{m_{0}}{M_{\rm pl}})^{2}\kappa_{\rm sq}\bar{\Lambda}_{\rm
g}^{2}G_{\rm sq}^{(II)}(\bar{\Lambda}_{\rm g}t_{f};\varphi),\end{split}$ (52)

with

$\kappa_{\rm sq}=\frac{3}{2}\varkappa\bar{\Lambda}_{\rm g}T.$

Since we have the decoherence rate, it is straight forward to obtain the
decoherence time, which we present in the sequence for all the considered
initial states.

####  IV.0.5 Decoherence time

As mentioned before, the decoherence time is obtained by setting
$\Gamma(\tau_{\rm dec})=1$. The case of the vacuum and the thermal initial
states of the field directly leads to the decoherence times

$\tau_{\rm
dec}^{(\textrm{vac})}=\quantity(\frac{5\pi}{8\mathcal{K}})^{1/3}\quantity(\frac{M_{\rm
pl}}{m_{0}})^{2/3}\quantity(\frac{1}{\kappa\Lambda_{\rm g}^{5}})^{1/3},$ (53)

and

$\begin{split}\tau_{\rm
dec}^{(\textrm{th})}&=\quantity(\frac{15\pi}{8\mathcal{K}})^{1/3}\quantity(\frac{M_{\rm
pl}}{m_{0}})^{2/3}\\\ &\hskip
28.45274pt\times\quantity(\frac{1}{3\kappa\Lambda_{\rm
g}^{5}+2\pi^{5}\kappa_{\rm th}(k_{B}T_{\rm g})^{5}})^{1/3},\end{split}$ (54)

respectively.

However, for the condition $\Gamma(\tau_{\rm dec})=1$ to hold we must have
both $G_{\rm coh}^{(II)}$ and $G_{\rm sq}^{(II)}$ less than unit for the cases
of the coherent and the squeezed states, respectively [17]. Thus we use
$G_{\rm coh}^{(II)}\simeq x^{3}/3$ and $G_{\rm sq}^{(II)}\simeq 2x^{3}/3$,
which hold for $x<<1$. Under such assumptions, we find the decoherence times
for the coherent and the squeezed initial states to be given by

$\begin{split}\tau_{\rm
dec}^{(\textrm{coh})}&=\quantity(\frac{45\pi}{8\mathcal{K}})^{1/3}\quantity(\frac{M_{\rm
pl}}{m_{0}})^{2/3}\\\ &\hskip
28.45274pt\times\quantity(\frac{1}{9\kappa\Lambda_{\rm
g}^{5}+16\alpha^{2}\kappa_{\rm coh}\tilde{\Lambda}_{\rm
g}^{5}})^{1/3},\end{split}$ (55)

and

$\begin{split}\tau_{\rm
dec}^{(\textrm{sq})}&=\quantity(\frac{405\pi}{4\mathcal{K}})^{1/3}\quantity(\frac{M_{\rm
pl}}{m_{0}})^{2/3}\\\ &\hskip
28.45274pt\times\quantity(\frac{1}{162\kappa\Lambda_{\rm g}^{5}\cosh
2r-\kappa_{\rm sq}\bar{\Lambda}_{\rm g}^{5}\sinh 2r})^{1/3},\end{split}$ (56)

respectively.

As already pointed out by the authors of Ref. [17], the decoherence time is
found to be proportional to the ratio $M_{\rm pl}/m_{0}$. For a point
particle, decoherence does not occur for $m_{0}<<M_{\rm pl}$. However, when
there are internal degrees of freedom the decoherence time is found to be
reduced with respect to the one estimated in Ref. [17]. Specifically when
these degrees of freedom are in thermal equilibrium with a high temperature,
the constants $\kappa$ are very large and the decoherence time becomes even
smaller.

##  V Conclusions

We considered the evolution of the external degrees of freedom of a freely
falling particle under the action of two sources of noise: $i$) the coupling
with its internal degrees of freedom and $ii$) the coupling with the quantized
weak gravitational field. Due to the universal nature of the gravitational
coupling, both environments are not independent. Employing the Feynman-Vernon
influence functional technique we computed the decoherence rate considering a
superposition of the relevant degrees of freedom. From this we estimated the
decoherence time.

For all the cases considered here, the decoherence rate was found to be given
by the sum of two terms. One arising purely from the quantum fluctuations of
the gravitational field and another contribution that comes from the coupling
with both the internal degrees of freedom and the gravitons, whose intensity
depends on the initial state of the gravitational field. In order to reach our
conclusions, we assumed that the temperature associated with the internal
degrees of freedom is high enough so they can be treated as a white noise.
Within this range of intensities analyzed here, we found that the presence of
the dynamical internal degrees of freedom, which couples to the external
variables due to the (quantum) gravitational field, causes a reduction in the
decoherence time with respect to the one estimated in the literature by
considering only the coupling with the gravitons.

As described here, the corrections involving both of these interacting
environments (gravitational field and internal dynamics) dominate except for
the squeezed initial state of the gravitons. This indicates that gravitons in
a squeezed state yields a much higher contribution to decoherence.
Nevertheless, we expected that the intensity of the contribution coming from
the internal dynamics to be much larger than the values considered here since
they are proportional to the gravitational cutoff $\Lambda$ (or graviton
temperature $T_{\rm g}=1/\beta_{\rm g}$ in the case of thermal initial state)
as well as to the temperature of the internal degrees of freedom $T=1/\beta$.
This will turn such contribution even more important.

Our results show that it is fundamental to take into account the quantum
nature of matter as well as the quantum fluctuations of the gravitational
field when considering phenomena like entanglement induced by gravity, quantum
reference frames or decoherence induced by the gravitational time-dilation.
Actually some of these effects can be completely blurred by the quantum
fluctuations coming from the sources we discussed here.

An interesting additional question that deserves investigation is the
contribution from these fluctuations to the entropy production due to the
universal coupling with gravity, as recently discussed in Ref. [37].

## Acknowledgements

This work was supported by the the National Institute for the Science and
Technology of Quantum Information (INCT-IQ), Grant No. 465469/2014-0, by the
National Council for Scientific and Technological Development (CNPq), Grants
No. 308065/2022-0 and Coordination of Superior Level Staff Improvement
(CAPES).

##  Appendix A Open quantum systems and the Feynman-Vernon influence
functional

In this Appendix we present the formalism of the Feynman-Vernon influence
functional in order to study the general dynamics of an open quantum system.
We follow the presentation in Ref. [32]. Let us consider that the system of
interest is described by the variables $x=\quantity{x_{n}}$ and interacts with
an environment described by the variables $q=\quantity{q_{n}}$. The classical
action takes the form

$S[x,q]=S_{\rm sys}[x]+S_{\rm env}[q]+S_{\rm int}[x,q],$ (57)

where the action $S_{\rm int}[x,q]$ describes the interaction between the
system and the environment. The quantum state of the total system is described
by the density matrix with matrix elements
$\rho(xq,x^{\prime}q^{\prime},t)=\matrixelement{xq,t}{\rho}{x^{\prime}q,t}$
depending on both system and environment variables that evolves unitarily
under the total Hamiltonian. However, we are only interested in computing the
expectation values of system observables without taking the environment
specific quantum state into account. The expectation value of such observables
may be computed with the reduced density matrix
$\rho_{r}=\textrm{Tr}_{q}\,\rho$.

Assuming that at $t=0$ the system and environment are uncorrelated,

$\rho(x_{i}q_{i},x_{i}^{\prime}q_{i}^{\prime},0)=\rho_{\rm
sys}(x_{i}x_{i}^{\prime},0)\rho_{\rm env}(q_{i}q_{i}^{\prime},0),$ (58)

and using the path integral representation of transition amplitudes, the
reduced density matrix evolves in time according to

$\rho_{r}(xx^{\prime},t)=\int
dx_{i}dx_{i}^{\prime}\,\mathcal{J}_{r}(xx^{\prime},t|x_{i}x_{i}^{\prime},0)\rho_{\rm
sys}(x_{i}x_{i}^{\prime},0),$ (59)

where the evolution operator for the reduced density matrix is defined by

$\begin{split}\mathcal{J}_{r}(xx^{\prime},t|x_{i}x_{i}^{\prime},0)&\equiv\int_{x_{i}}^{x}\mathcal{D}x\int_{x_{i}^{\prime}}^{x^{\prime}}\mathcal{D}x^{\prime}\,\\\
&\hskip 14.22636pt\times e^{i\quantity(S_{\rm sys}[x]-S_{\rm
sys}[x^{\prime}])}\mathcal{F}[x,x^{\prime}].\end{split}$ (60)

The functional $\mathcal{F}[x,x^{\prime}]$ is the so-called Feynman-Vernon
influence functional [32, 15, 16], and it is given by

$\displaystyle\mathcal{F}[x,x^{\prime}]$ $\displaystyle=e^{iS_{\rm
IF}[x,x^{\prime},t]}$
$\displaystyle=\int_{-\infty}^{\infty}dqdq_{i}dq_{i}^{\prime}\,\rho_{\rm
env}(q_{i}q_{i}^{\prime},0)$ $\displaystyle\hskip
14.22636pt\times\int_{q_{i}}^{q}\mathcal{D}q\,e^{i\quantity(S_{\rm
env}[q]+S_{\rm int}[x,q])}$ $\displaystyle\hskip
14.22636pt\times\int_{q_{i}^{\prime}}^{q}\mathcal{D}q^{\prime}\,e^{-i\quantity(S_{\rm
env}[q^{\prime}]+S_{\rm int}[x^{\prime},q^{\prime}])},$ (61)

where $S_{\rm IF}$ is called the influence action and it encodes all the
influence of the environment on the system.

The path integral representation (A) involves two histories, which may be
described as an integral over single histories defined on a _closed time path_
(CTP). For the remainder of this appendix, we denote $x(t)=x^{1}(t)$,
$x^{\prime}(t)=x^{2}(t)$ and we shall think of $x^{a}$, $a=1,2$, as a single
doublet field defined on a single branch time path. We also define a metric
tensor $c_{ab}=\textrm{diag}\quantity(1,-1)$ that, together with its inverse
$c^{ab}=\textrm{diag}\quantity(1,-1)$ (defined such that
$c^{ad}c_{db}=\delta^{a}_{b}$), may be used to raise and/or lower indices.
Using the Einstein summation convention, the kinetic terms in the system
action will be written as

$\frac{1}{2}\int dt\,c_{ab}\dot{x}^{a}\dot{x}^{b}=\frac{1}{2}\int
dt\,\dot{x}_{a}\dot{x}^{a}=\frac{1}{2}\int
dt\,\quantity(\dot{x}^{2}-\dot{x}^{\prime 2}).$

We will refer to the _CTP action_ $S[x^{a}]\equiv S[x]-S[x^{\prime}]$ without
discriminating the contributions from either branch [32].

###  A.1 The linear coupling model

Let us consider the case where the environment action is quadratic in the $q$
variable(s), the initial environment density matrix is Gaussian and the
interaction term is bilinear,

$S_{\rm int}=\int dt\,x^{a}(t)Q_{a}[q(t)],$ (62)

where the $Q$’s are linear combinations of the $q$’s [32]. Note that, in that
case, Eq. (A) is a functional Fourier transform of a Gaussian functional of
histories $Q(t)$ and $Q^{\prime}(t)$. Since the Fourier transform of a
Gaussian is another Gaussian, we conclude that, under all these assumptions,
the influence action must also be quadratic in $x$ and $x^{\prime}$. Therefore
we write

$S_{\textrm{IF}}=\frac{1}{2}\int
dtdt^{\prime}\,x^{a}(t)\mathbb{M}_{ab}(t,t^{\prime})x^{b}(t^{\prime}),$ (63)

where

$\mathbb{M}_{ab}(t,t^{\prime})=-i\frac{\delta^{2}}{\delta x^{a}(t)\delta
x^{b}(t^{\prime})}\evaluated{e^{iS_{\rm IF}[x^{a}]}}_{x_{a}=0}.$ (64)

Eq. (A) in CPT notation reads

$\begin{split}\mathcal{F}[x^{a}]&=\int_{-\infty}^{\infty}dq^{1}(t)dq^{1}(0)dq^{2}(0)\\\
&\times\int\mathcal{D}q^{a}\,e^{i\quantity(S_{\rm env}[q^{a}]+S_{\rm
int}[x^{a},q^{a}])}\rho_{\rm env}(q^{1}(0),q^{2}(0),0),\end{split}$

where the path integral is taken under the CTP boundary condition
$q^{1}(t)=q^{2}(t)$. A direct variation from this equation with $S_{\rm int}$
given by Eq. (62) yields

$\frac{\delta^{2}}{\delta x^{a}(t)\delta
x^{b}(t^{\prime})}\evaluated{e^{iS_{\rm
IF}[x^{a}]}}_{x_{a}=0}=-\int_{-\infty}^{\infty}dq^{1}(t)dq^{1}(0)dq^{2}(0)\int\mathcal{D}q^{a}\,e^{iS_{\rm
env}[q^{a}]}Q_{a}(t)Q_{b}(t^{\prime})\rho_{\rm env}(q^{1}(0),q^{2}(0),0).$
(65)

After a long calculation [32], we find

$\mathbb{M}_{ab}(t,t^{\prime})=i\matrixquantity(\expectationvalue{T\quantity[Q(t)Q(t^{\prime})]}&-\expectationvalue{Q(t^{\prime})Q(t)}\\\
-\expectationvalue{Q(t)Q(t^{\prime})}&\expectationvalue{\tilde{T}\quantity[Q(t)Q(t^{\prime})]}),$
(66)

where $T$ stands for temporal ordering (the latest time to the left) and
$\tilde{T}$ stands for _anti_ -time ordering (the latest time to the right).
Here, expectation values are understood to be taken with respect to the
initial state of the environment. In this last expression, the $Q^{\prime}$s
are now understood as operators in the Heisenberg representation. Then,
writing the influence action in terms of the original variables, and after
another long calculation which involves only algebraic manipulations, we find

$\begin{split}S_{\rm IF}=\int
dtdt^{\prime}\,\left\\{\frac{1}{2}\quantity[x(t)-x^{\prime}(t)]D(t,t^{\prime})\quantity[x(t^{\prime})+x^{\prime}(t^{\prime})]\right.\\\
\left.+\frac{i}{2}\quantity[x(t)-x^{\prime}(t)]N(t,t^{\prime})\quantity[x(t^{\prime})-x^{\prime}(t^{\prime})]\right\\},\end{split}$
(67)

where we have defined the _dissipation_ $D$ and _noise_ $N$ kernels

$D(t,t^{\prime})=i\expectationvalue{\commutator{Q(t)}{Q(t^{\prime})}}\theta(t-t^{\prime}),$
(68a)
$N(t,t^{\prime})=\frac{1}{2}\expectationvalue{\anticommutator{Q(t)}{Q(t^{\prime})}}.$
(68b)

##  Appendix B Internal degrees of freedom noise kernel

The internal degrees of freedom noise kernel is given by Eq. (30b). Since the
free Lagrangian for each $\varrho_{\alpha}$ is the one of a harmonic
oscillator, we may write the position operators in the Heisenberg picture as

$\varrho_{\alpha}(t)=\sqrt{\frac{1}{2\mu_{\alpha}\varpi_{\alpha}}}\quantity(b_{\alpha}e^{-i\varpi_{\alpha}t}+b_{\alpha}^{\dagger}e^{i\varpi_{\alpha}t}),$

where the $b$’s ($b^{\dagger}$’s) are annihilation (creation) operators. A
direct calculation then yields

$\begin{split}\expectationvalue{\anticommutator{\varrho_{\alpha}(t)}{\varrho_{\alpha}(t^{\prime})}}_{\rm
int}=\frac{2}{\mu_{\alpha}\varpi_{\alpha}^{2}}\expectationvalue{H_{\alpha}}_{\rm
int}\cos\varpi_{\alpha}(t-t^{\prime})\\\
+\frac{1}{\mu_{\alpha}\varpi_{\alpha}}\quantity[\expectationvalue{b_{\alpha}^{2}}_{\rm
int}e^{-i\varpi_{\alpha}(t+t^{\prime})}+\expectationvalue{\quantity(b^{\dagger}_{\alpha})^{2}}_{\rm
int}e^{i\varpi_{\alpha}(t+t^{\prime})}],\end{split}$ (69)

where $H_{\alpha}$ is the free Hamiltonian operator of the $\alpha$th
oscillator with frequency $\varpi_{\alpha}$.

Let us take the initial internal state to be a thermal one with temperature
$1/\beta$, so that $\expectationvalue{b_{\alpha}^{2}}_{\rm
int}=\expectationvalue{\quantity(b^{\dagger}_{\alpha})^{2}}_{\rm int}=0$. The
canonical partition function reads

$Z=\sum_{n=0}^{\infty}\exp\quantity[-\varpi_{\alpha}\beta\quantity(n+\frac{1}{2})]=\frac{e^{-\varpi_{\alpha}\beta/2}}{1-e^{-\varpi_{\alpha}\beta}},$

and therefore

$\expectationvalue{H_{\alpha}}_{\rm
int}=\frac{\varpi_{\alpha}}{2}\coth\quantity(\frac{\varpi_{\alpha}\beta}{2}).$
(70)

We are then left with

$\expectationvalue{\anticommutator{\varrho_{\alpha}(t)}{\varrho_{\alpha}(t^{\prime})}}_{\rm
int}=\frac{1}{\mu_{\alpha}\varpi_{\alpha}}\coth\quantity(\frac{\varpi_{\alpha}\beta}{2})\cos\varpi_{\alpha}(t-t^{\prime}),$
(71)

and Eq. (30b) becomes

$N_{\rm
int}(t,t^{\prime})=\frac{\lambda^{2}}{2}\sum_{\alpha}\frac{\vartheta_{\alpha}^{2}}{\mu_{\alpha}\varpi_{\alpha}}\coth\quantity(\frac{\varpi_{\alpha}\beta}{2})\cos\varpi_{\alpha}(t-t^{\prime}).$
(72)

Let us further assume that the internal frequencies span a continuum, so we
may replace

$\sum_{\alpha}\to\int_{0}^{\infty}d\varpi\,\sigma(\varpi),$ (73)

where $\sigma(\varpi)d\varpi$ is the number of oscillators with frequencies
between $\varpi$ and $\varpi+d\varpi$. Then,

$N_{\rm
int}(t,t^{\prime})=\frac{\lambda^{2}}{2}\int_{0}^{\infty}\frac{d\varpi}{\varpi}\Omega(\varpi)\coth\quantity(\frac{\varpi\beta}{2})\cos\varpi(t-t^{\prime}),$
(74)

with

$\Omega(\varpi)\equiv\frac{\sigma(\varpi)\vartheta^{2}(\varpi)}{\mu_{\varpi}}.$

If we consider the internal degrees of freedom to represent an ohmic bath, we
have [32]

$\Omega(\varpi)=\gamma\varpi^{2}$ (75)

for some constant $\gamma$. For that case, we have

$N_{\rm
int}(t,t^{\prime})=\frac{1}{2}\lambda^{2}\gamma\int_{0}^{\infty}d\varpi\,\varpi\coth\quantity(\frac{\varpi\beta}{2})\cos\varpi(t-t^{\prime}).$
(76)

At high temperature, $\beta<<t-t^{\prime}$, the integral is dominated by low
frequencies, and we find

$N_{\rm
int}(t,t^{\prime})=\frac{\pi\lambda^{2}\gamma}{\beta}\delta(t-t^{\prime}),$
(77)

namely a white noise.

##  Appendix C Gravitational noise kernel

In this appendix we compute the noise kernel enconding the influence of the
quantized gravitational field, defined by Eq. (13b), for different initial
states. In what follows, we shall consider the position operators to be
independent of the polarizations and of the direction of $\mathbf{k}$, that
is, $q_{s}(t,\mathbf{k})=q_{\omega}(t)$, $\omega=\absolutevalue{\mathbf{k}}$.
This means that the angular integral we need to compute is

$\int
d\Omega\sum_{s}\epsilon_{s}^{ij}(\mathbf{k})\epsilon_{s}^{kl}(\mathbf{k}).$

The polarization tensors can be written as [35, 18]

$\begin{split}\epsilon_{ij}^{+}&=\hat{\epsilon}_{i}^{(1)}\hat{\epsilon}_{j}^{(1)}-\hat{\epsilon}_{i}^{(2)}\hat{\epsilon}_{j}^{(2)}\\\
\epsilon_{ij}^{\crossproduct}&=\hat{\epsilon}_{i}^{(1)}\hat{\epsilon}_{j}^{(2)}+\hat{\epsilon}_{i}^{(2)}\hat{\epsilon}_{j}^{(1)},\end{split}$
(78)

where the spatial polarization unit vectors $\hat{\epsilon}^{(1)}$ and
$\hat{\epsilon}^{(2)}$ are orthogonal to the direction of propagation
$\hat{k}=\mathbf{k}/\absolutevalue{\mathbf{k}}$. They satisfy

$\hat{\epsilon}_{i}^{(1)}\hat{\epsilon}_{j}^{(1)}+\hat{\epsilon}_{i}^{(2)}\hat{\epsilon}_{j}^{(2)}=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}\equiv
P_{ij}.$ (79)

Then, a straightforward calculation yields

$\sum_{s}\epsilon_{s}^{ij}\epsilon_{s}^{kl}=P^{ik}P^{jl}+P^{il}P^{jk}-P^{ij}P^{kl}.$
(80)

We can use the following integration over solid angles:

$\int d\Omega\,\hat{k}_{i}\hat{k}_{j}=\frac{4\pi}{3}\delta_{ij},$ $\int
d\Omega\,\hat{k}_{i}\hat{k}_{j}\hat{k}_{k}\hat{k}_{l}=\frac{4\pi}{15}\quantity(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}),$

to obtain

$\int d\Omega
P_{ij}P_{kl}=\frac{8\pi}{5}\delta_{ij}\delta_{kl}+\frac{4\pi}{15}\quantity(\delta_{ij}\delta_{jl}+\delta_{il}\delta_{jk}).$

We finally obtain

$\int
d\Omega\sum_{s}\epsilon_{s}^{ij}(\mathbf{k})\epsilon_{s}^{kl}(\mathbf{k})=\frac{8\pi}{15}\mathcal{P}^{ijkl},$
(81a) where $\mathcal{P}^{ijkl}\equiv
3\quantity(\delta^{ik}\delta^{jl}+\delta^{il}\delta^{jk})-2\delta^{ij}\delta^{kl}.$
(81b)

In order to compute the noise kernel (13b), we will need to evaluate the
expectation value of the anti-commutator of position operators at different
times with respect to the gravitons initial state. Since the free Lagrangian
for each mode of the gravitational field is the one of a harmonic oscillator,
we may write the position operators in the Heisenberg picture as

$q_{\omega}(t)=\sqrt{\frac{1}{2m\omega}}\quantity(a_{\omega}e^{-i\omega
t}+a_{\omega}^{\dagger}e^{i\omega t}),$

where the $a$’s ($a^{\dagger}$’s) are annihilation (creation) operators
satisfying the usual commutation relations. A direct calculation then yields

$\begin{split}&\expectationvalue{\anticommutator{q_{\omega}(t)}{q_{\omega}(t^{\prime})}}_{\rm
g}=\frac{2}{m\omega^{2}}\expectationvalue{H_{\omega}}_{\rm
g}\cos\omega(t-t^{\prime})\\\ &\hskip
5.69046pt+\frac{1}{m\omega}\quantity[\expectationvalue{a_{\omega}^{2}}_{\rm
g}e^{-i\omega(t+t^{\prime})}+\expectationvalue{\quantity(a^{\dagger}_{\omega})^{2}}_{\rm
g}e^{i\omega(t+t^{\prime})}]\end{split}$ (82)

where

$H_{\omega}=\omega\quantity(a_{\omega}^{\dagger}a_{\omega}+\frac{1}{2})=\frac{\omega}{2}\anticommutator{a_{\omega}}{a_{\omega}^{\dagger}}$

is the free Hamiltonian operator of the oscillator with frequency $\omega$. We
may now compute the noise kernel for different initial states of the
gravitational field.

###  C.1 Minkowski vacuum

If the initial state is the vacuum state, we have
$\expectationvalue{a_{\omega}^{2}}_{\rm
g}=\expectationvalue{\quantity(a^{\dagger}_{\omega})^{2}}_{\rm g}=0$ and
$\expectationvalue{H_{\omega}}_{\rm g}=\omega/2$, so that

$\expectationvalue{\anticommutator{q_{\omega}(t)}{q_{\omega}(t^{\prime})}}_{\rm
g}=\frac{1}{m\omega}\cos\omega(t-t^{\prime}).$ (83)

Then, from Eq. (13b) we have

$\begin{split}N_{\textrm{g
(vac)}}^{ijkl}(t,t^{\prime})&=\frac{m_{0}^{2}}{15\pi}\mathcal{P}^{ijkl}\int_{0}^{\infty}d\omega\,\omega^{5}\cos\omega(t-t^{\prime}).\end{split}$

Since the integral over $\omega$ is divergent in its upper limit, we need to
introduce the momentum cutoff $\Lambda_{\rm g}$, so that [18, 17]

$\int_{0}^{\Lambda_{\rm
g}}d\omega\,\omega^{5}\cos\omega(t-t^{\prime})=\Lambda_{\rm
g}^{6}\,F\quantity[\Lambda_{\rm g}(t-t^{\prime})],$ (84a) where $\displaystyle
F(x)$ $\displaystyle\equiv\frac{1}{x^{6}}\int_{0}^{x}dy\,y^{5}\cos y$
$\displaystyle=\frac{1}{x^{6}}\left[\quantity(5x^{4}-60x^{2}+120)\cos
x\right.$ $\displaystyle\hskip
28.45274pt\left.+x\quantity(x^{4}-20x^{2}+120)\sin x-120\right].$ (84b)

Then, the noise kernel for the Minkowski vacuum initial state reads

$N_{\textrm{g (vac)}}^{ijkl}(t,t^{\prime})=\frac{m_{0}^{2}\Lambda_{\rm
g}^{6}}{15\pi}\mathcal{P}^{ijkl}F\quantity[\Lambda_{\rm g}(t-t^{\prime})].$
(85)

###  C.2 Thermal state

If the gravitons are initially in a thermal state with temperature
$1/{\beta_{\textrm{g}}}$, we have $\expectationvalue{a_{\omega}^{2}}_{\rm
g}=\expectationvalue{\quantity(a^{\dagger}_{\omega})^{2}}_{\rm g}=0$ once
again. The canonical partition function reads

$Z_{\rm g}=\sum_{n=0}^{\infty}\exp\quantity[-\omega\beta_{\rm
g}\quantity(n+\frac{1}{2})]=\frac{e^{-\omega\beta_{\rm
g}/2}}{1-e^{-\omega\beta_{\rm g}}},$

from which we find

$\expectationvalue{H_{\omega}}_{\rm g}=-\partialderivative{\beta_{\rm g}}\ln
Z_{\rm g}=\omega\quantity(\frac{1}{2}+\frac{1}{e^{\omega\beta_{\rm g}}-1}).$
(86)

Plugging in Eq. (82), we find, for the thermal initial state,

$\begin{split}\expectationvalue{\anticommutator{q_{\omega}(t)}{q_{\omega}(t^{\prime})}}_{\rm
g}&=\frac{1}{m\omega}\cos\omega(t-t^{\prime})\\\ &\hskip
5.69046pt+\frac{2}{m\omega}\frac{1}{e^{\omega\beta_{\rm
g}}-1}\cos\omega(t-t^{\prime}),\end{split}$

and the noise kernel becomes

$\begin{split}N_{\textrm{g (th)}}^{ijkl}(t,t^{\prime})&=N_{\textrm{g
(vac)}}^{ijkl}(t,t^{\prime})\\\ &\hskip
5.69046pt+\frac{8m_{0}^{2}\pi^{5}}{\beta_{\rm g}^{6}}\mathcal{P}^{ijkl}F_{\rm
th}\quantity[\frac{\pi(t-t^{\prime})}{\beta_{\rm g}}],\end{split}$ (87)

where

$\begin{split}F_{\rm th}(x)&\equiv\frac{1}{x^{6}}\\\ &\hskip
5.69046pt-\frac{1}{15\sinh^{6}x}\quantity(2\cosh^{4}x+11\cosh^{2}x+2).\end{split}$
(88)

###  C.3 Coherent state

For a quantum harmonic oscillator, the coherent state is defined by

$\ket{\alpha_{\omega}}=\mathscr{D}(\alpha_{\omega})\ket{0},$ (89)

where $\mathscr{D}(\alpha_{\omega})$ is called the _coherent-state
displacement operator_ , defined as [36]

$\mathscr{D}(\alpha_{\omega})=\exp(\alpha_{\omega}a_{\omega}^{\dagger}-\alpha_{\omega}^{*}a_{\omega}),$
(90)

with $a_{\omega}$ ($a_{\omega}^{\dagger}$) being the annihilation (creation)
operator. Usually we take the displacement parameter to be independent of
$\omega$, $\alpha_{\omega}=\alpha$. The displacement operator satisfies a
number of properties,

$\begin{split}\mathscr{D}^{\dagger}(\alpha)\mathscr{D}(\alpha)&=1,\\\
\mathscr{D}^{\dagger}(\alpha)a_{\omega}^{n}\mathscr{D}(\alpha)&=(a_{\omega}+\alpha)^{n},\\\
\mathscr{D}^{\dagger}(\alpha)(a_{\omega}^{\dagger})^{n}\mathscr{D}(\alpha)&=(a_{\omega}^{\dagger}+\alpha^{*})^{n},\end{split}$
(91)

for $n\in\mathbb{N}$, from which we find

$\begin{split}\expectationvalue{a_{\omega}^{2}}_{\rm g}&=\alpha^{2},\\\
\expectationvalue{(a_{\omega}^{\dagger})^{2}}_{\rm g}&=(\alpha^{*})^{2},\\\
\expectationvalue{H_{\omega}}_{\rm
g}&=\omega\quantity(\absolutevalue{\alpha}^{2}+\frac{1}{2}),\end{split}$ (92)

where we also used the fact that the coherent state is an eigenstate of the
annihilation operator, $a_{\omega}\ket{\alpha}=\alpha\ket{\alpha}$.

For simplicity, let us assume that the displacement parameter is real. In that
case, we find

$\begin{split}\expectationvalue{\anticommutator{q_{\omega}(t)}{q_{\omega}(t^{\prime})}}_{\rm
g}&=\frac{1}{m\omega}\cos\omega(t-t^{\prime})\\\ &\hskip
5.69046pt+\frac{\alpha^{2}}{m\omega}\cos(\omega t)\cos(\omega
t^{\prime}),\end{split}$

and the noise kernel becomes

$\begin{split}N_{\textrm{g (coh)}}^{ijkl}(t,t^{\prime})&=N_{\textrm{g
(vac)}}^{ijkl}(t,t^{\prime})\\\
&+\frac{m_{0}^{2}\alpha^{2}}{15\pi}\mathcal{P}^{ijkl}\int_{0}^{\tilde{\Lambda}_{\rm
g}}d\omega\,\omega^{5}\cos(\omega t)\cos(\omega t^{\prime}),\end{split}$ (93)

where we regularized the divergent integral by introducing the momentum cutoff
$\tilde{\Lambda}_{\rm g}$.

###  C.4 Squeezed state

For a quantum harmonic oscillator, the squeezed state is defined by

$\ket{\zeta_{\omega}}=\mathscr{S}(\zeta_{\omega})\ket{0},$ (94)

where $\mathscr{S}(\zeta_{\omega})$ is called the _squeeze operator_ , defined
as [36]

$\mathscr{S}(\zeta_{\omega})=\exp[\frac{1}{2}\zeta_{\omega}^{*}a_{\omega}^{2}-\frac{1}{2}\zeta_{\omega}(a_{\omega}^{\dagger})^{2}],$
(95)

where $\zeta_{\omega}$ is the _complex squeeze parameter_

$\zeta_{\omega}=r_{\omega}e^{i\varphi_{\omega}}.$ (96)

Here we take the squeeze parameter to be independent of $\omega$,
$\zeta_{\omega}=\zeta$. The squeeze operator satisfies a number of properties,

$\begin{split}\mathscr{S}^{\dagger}(\zeta)\mathscr{S}(\zeta)&=1,\\\
\mathscr{S}^{\dagger}(\zeta)a_{\omega}^{n}\mathscr{S}(\zeta)&=(a\cosh
r-a^{\dagger}e^{i\varphi}\sinh r)^{n},\\\
\mathscr{S}^{\dagger}(\zeta)(a_{\omega}^{\dagger})^{n}\mathscr{S}(\zeta)&=(a^{\dagger}\cosh
r-a\,e^{-i\varphi}\sinh r)^{n},\end{split}$ (97)

for $n\in\mathbb{N}$, from which we find

$\begin{split}\expectationvalue{a_{\omega}^{2}}_{\rm g}&=-e^{i\varphi}\sinh
r\cosh r,\\\ \expectationvalue{(a_{\omega}^{\dagger})^{2}}_{\rm
g}&=-e^{-i\varphi}\sinh r\cosh r,\\\ \expectationvalue{H_{\omega}}_{\rm
g}&=\frac{\omega}{2}\cosh 2r,\end{split}$ (98)

so that

$\begin{split}\expectationvalue{\anticommutator{q_{\omega}(t)}{q_{\omega}(t^{\prime})}}_{\rm
g}&=\frac{\cosh 2r}{m\omega}\cos\omega(t-t^{\prime})\\\ &\hskip
5.69046pt-\frac{\sinh
2r}{m\omega}\cos\quantity[\varphi-\omega(t+t^{\prime})],\end{split}$

and the noise kernel becomes

$\begin{split}N_{\textrm{g (sq)}}^{ijkl}(t,t^{\prime})&=\cosh 2r\,N_{\textrm{g
(vac)}}^{ijkl}(t,t^{\prime})\\\ &\hskip
5.69046pt-\frac{m_{0}^{2}\bar{\Lambda}_{\rm g}^{6}}{15\pi}\sinh
2r\,\mathcal{P}^{ijkl}F\quantity[\bar{\Lambda}_{\rm
g}(t+t^{\prime});\varphi],\end{split}$ (99)

where we introduced the the momentum cutoff $\bar{\Lambda}_{\rm g}$ and
defined

$\begin{split}&F(x;\varphi)\equiv\frac{1}{x^{6}}\left[\quantity(5x^{4}-60x^{2}+120)\cos(x-\varphi)\right.\\\
&\hskip
5.69046pt\left.+x\quantity(x^{4}-20x^{2}+120)\sin\quantity(x-\varphi)-120\cos\varphi\right].\end{split}$
(100)

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