# Dissipative dynamics of a particle coupled to field via internal degrees of
freedom

Kanupriya Sinha<EMAIL_ADDRESS>Department of Electrical Engineering,
Princeton University, Princeton, New Jersey 08544, USA  Adrián Ezequiel Rubio
López<EMAIL_ADDRESS>Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences,
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria  Yiğit Subaşı<EMAIL_ADDRESS>Computer, Computational and
Statistical Sciences Division, Los Alamos National Laboratory, Los Alamos, NM
87545, USA

###### Abstract

We study the non-equilibrium dissipative dynamics of the center of mass of a
particle coupled to a field via its internal degrees of freedom. We model the
internal and external degrees of freedom of the particle as quantum harmonic
oscillators in 1+1 D, with the internal oscillator coupled to a scalar quantum
field at the center of mass position. Such a coupling results in a nonlinear
interaction between the three pertinent degrees of freedom – the center of
mass, internal degree of freedom, and the field. It is typically assumed that
the internal dynamics is decoupled from that of the center of mass owing to
their disparate characteristic time scales. Here we use an influence
functional approach that allows one to account for the self-consistent
backaction of the different degrees of freedom on each other, including the
coupled non-equilibrium dynamics of the internal degree of freedom and the
field, and their influence on the dissipation and noise of the center of mass.
Considering a weak nonlinear interaction term, we employ a perturbative
generating functional approach to derive a second order effective action and a
corresponding quantum Langevin equation describing the non-equilibrium
dynamics of the center of mass. We analyze the resulting dissipation and noise
arising from the field and the internal degree of freedom as a composite
environment. Furthermore, we establish a generalized fluctuation-dissipation
relation for the late-time dissipation and noise kernels. Our results are
pertinent to open quantum systems that possess intermediary degrees of freedom
between system and environment, such as in the case of optomechanical
interactions.

##  I Introduction

Dissipative non-equilibrium dynamics of complex quantum systems comprising
different degrees of freedom is a subject of significant interest from the
perspective of theory of open quantum systems, as well as, emerging quantum
information applications and devices BPBook ; Weiss ; Clerk20 . While the
dissipative dynamics of a reduced quantum system coupled linearly to its
environment has been studied extensively, the case of a nonlinear coupling
between a system and its environment via an intermediate degree of freedom and
its effect on the resulting dynamics is seldom discussed HPZ93 ; Maghrebi16 ;
Lampo16 . A commonly employed assumption is that the time-scales associated
with the different degrees of freedom are sufficiently disparate so that their
dynamics can be treated as being effectively decoupled from each other
Stenholm86 ; Lan08 ; Haake . Such an assumption precludes the possibility of
studying the rich interplay between the different degrees of freedom, and its
effect on the non-equilibrium dynamics of the system of interest. Moreover, an
adiabatic elimination of fast variables does not allow one to see the effects
of the non-equilibrium dynamics and quantum fluctuations of the fast degrees
of freedom on the system dissipation and noise.

A canonical example of such a complex open quantum system is the
optomechanical interaction between a neutral polarizable particle and a field,
for instance, a moving atom, nanoparticle, or molecule interacting with the
electromagnetic (EM) field Wieman ; Yin ; Molecules . As an essential feature
these systems possess internal electronic degrees of freedom that interact
with the EM field and are located at the center of mass position represented
by the mechanical degree of freedom (MDF). While the MDF of the particle does
not couple to the EM field directly, its internal degrees of freedom (IDF)
interact with the EM field at the center of mass position, thus mediating an
effective coupling between the MDF and the field. Considering the MDF as the
system of interest, with the EM field and the IDF acting as an environment,
such an effective interaction leads to dissipation and noise in system
dynamics. There is an interesting interplay between the three degrees of
freedom (the MDF, the IDF, and the EM field) that takes place in such a
scenario as has been previously explored in KSthesis ; MOF1 ; MOF2 ; Wang14 .
Considering the dynamics from an approach that includes the self-consistent
backaction of all the degrees of freedom on each other allows one to examine
the role of IDF in the center of mass dynamics. For example, it has been shown
that the IDF can facilitate a transfer of correlations between the field and
center of mass in the case of optomechanical interactions KSthesis ; MOF2  and
affect the center of mass decoherence Brun16 . On the other hand, it has also
been demonstrated that including the quantized center of mass motion can
affect the radiation reaction and thermalization of the IDF AERL19 .

We study here a minimal model that captures the interplay between different
degrees of freedom in a composite system and the nonequilibrium dissipative
center of mass dynamics. Such a model forms the basis for optomechanical
interactions between neutral particles and fields, and more generally applies
to open quantum systems that couple to an environment via internal degrees of
freedom. The remainder of this paper is organized as follows. In Section II we
describe the model in consideration, in Section III we describe the derivation
of a second order effective action for the center of mass by integrating out
the IDF and the field using a perturbative influence functional approach HPZ93
. In Section IV we write an effective Langevin equation for the center of
mass, discuss the resulting dynamics and present a generalized fluctuation-
dissipation relation. We summarize our findings and present an outlook of the
work in Section V.

Figure 1: Schematic representation of a mechanical oscillator of mass $M$ and
frequency $\Omega$ coupled to a massless scalar field $\Phi$ via an IDF. The
internal oscillator with position coordinate $Q$ is described as a harmonic
oscillator of mass $m$ and frequency $\omega_{0}$ and couples to the scalar
field $\Phi\left({x},t\right)$ at the center of mass position ${X}$ of the
mechanical oscillator.

##  II Model

Let us consider a particle with its center of mass described by a mechanical
oscillator of mass $M$ and frequency $\Omega$, and its polarization described
by an IDF as a harmonic oscillator of mass $m$ and resonance frequency
$\omega_{0}$. The composite system is assumed to be interacting with a quantum
field, which we consider to be a scalar field $\Phi(x,t)$ in $1+1$ D. The
total action of the system can be written as

$\displaystyle S=S_{M}+S_{I}+S_{F}+S_{\mathrm{int},}$ (1)

where

$\displaystyle S_{M}$
$\displaystyle=\int\mathrm{d}\tau\left(\frac{1}{2}M\dot{X}^{2}-\frac{1}{2}M\Omega^{2}X^{2}\right)$
(2)

refers to the free action for the mechanical oscillator with
$\left\\{X,\dot{X}\right\\}$ referring to the center of mass position and
velocity, while

$\displaystyle S_{I}$
$\displaystyle=\int\mathrm{d}\tau\left(\frac{1}{2}m\dot{Q}^{2}-\frac{1}{2}m\omega_{0}^{2}Q^{2}\right)$
(3)

corresponds to the free action for the internal degree of freedom with
amplitude $Q$, and

$\displaystyle S_{F}$
$\displaystyle=\int\mathrm{d}\tau\int\mathrm{d}{x}\frac{\epsilon_{0}}{2}\left[\left(\partial_{t}\Phi(x,\tau)\right)^{2}-\left({\mathbf{\partial}_{x}}\Phi(x,\tau)\right)^{2}\right]$
(4)

is the free action for the scalar field $\Phi\left(x,t\right)$, $\epsilon_{0}$
being the vacuum permittivity. We consider $\hbar=c=1$ throughout this paper.
The correspondence with the EM field is established by identifying the scalar
field $\Phi\left({x},t\right)$ as the vector potential.

Considering that the particle interacts with the field via its IDF, at the
position determined by the MDF, the interaction action is given as

$\displaystyle S_{\mathrm{int}}$
$\displaystyle=\int\mathrm{d}\tau\int\mathrm{d}{x}\lambda
Q\Phi({x},\tau)\delta\left({x}-X(\tau)\right)$ (5)

with the strength of the interaction defined by $\lambda$. We note that the
above interaction action is nonlinear, with the two degrees of freedom and the
field interacting together. Such a model, previously referred to as the
mirror-oscillator-field model has been studied in the context of describing
optomechanical interactions from a microscopic perspective MOF1 ; MOF2 ;
KSthesis . It provides a self-consistent treatment of the different degrees of
freedom involved, in that one does not need to impose the boundary conditions
on the field by hand, and the mechanical effects of the field are consistently
described upon tracing over the IDF.

In typical experimental systems of relevance, such as those of atoms and
nanoparticles confined in traps, the center of mass motion is restricted to a
region of the space smaller than the wavelengths of the field that interact
resonantly with the internal degrees. Thus motivated by practical
considerations, we assume that the motion of the MDF is restricted to small
displacements about the average equilibrium position $X_{0}$, such that we can
expand the interaction action to first order in displacements from the
equilibrium as follows

$\displaystyle S_{\mathrm{int}}$
$\displaystyle\approx\int\mathrm{d}\tau\left[\lambda Q\Phi(X_{0},\tau)+\lambda
Q\partial_{x}\Phi(X_{0},\tau)\left(X-X_{0}\right)\right].$ (6)

The first term in the above action represents the linear interaction between
the IDF and the field, similar to the $\sim{\mathbf{p}}\cdot{\mathbf{A}}$
coupling in atom-field interactions. The second term stands for the tripartite
interaction between the IDF, the MDF and the field. Thus, as we show in the
following, the effective coupling of the center of mass to the field via the
IDF leads to its dissipative dynamics. From now on, for simplicity and without
loss of generality, we assume $X_{0}=0$.

###  II.1 Scalar field as bath of harmonic oscillators

The above separation of the total interaction action (Eq.(6)) suggests
splitting the field into two separate oscillator baths such that one of the
baths interacts with the IDF, while the other is coupled to both the internal
and mechanical degrees. We thus make an eigenmode decomposition of the field
to rewrite it in terms of a bath of oscillators as follows QBM3

$\displaystyle\Phi({x},\tau)=\sum_{n}\sqrt{\frac{1}{\epsilon_{0}{L}}}\left[q_{n}^{(-)}\cos({\omega}_{n}{x})+q_{n}^{(+)}\sin({\omega}_{n}{x})\right],$
(7)

where $L$ is the mode volume of the field, $\omega_{n}$ refers to the mode
frequency, and $q_{n}^{(+)}$ and $q_{n}^{(-)}$ refer to two independent set of
eigenmodes of the field, assuming periodic boundary conditions. One can thus
rewrite the free field action as as a sum $S_{F}=S_{F}^{(+)}+S_{F}^{(-)}$,
where

$\displaystyle
S_{F}^{(\pm)}=\int\mathrm{d}\tau\sum_{n}\frac{1}{2}\left[\left.\dot{q}_{n}^{(\pm)}\right.^{2}-\omega_{n}^{2}\left.{q}_{n}^{(\pm)}\right.^{2}\right]$
(8)

with $S_{F}^{(\pm)}$ corresponding to two separate baths of $(+)$ and $(-)$
oscillators. Thus, for $X_{0}=0$ the interaction action in Eq.(6) can be
written as a sum of two separate interaction terms
$S_{\mathrm{int}}=S_{\mathrm{int}}^{(-)}+S_{\mathrm{int}}^{(+)}$, with

$\displaystyle S_{\mathrm{int}}^{(-)}$
$\displaystyle=\int\mathrm{d}\tau\lambda Q\sum_{n}q_{n}^{(-)}$ (9)
$\displaystyle S_{\mathrm{int}}^{(+)}$
$\displaystyle=\int\mathrm{d}\tau\lambda Q\sum_{n}\omega_{n}Xq_{n}^{(+)}.$
(10)

Thus the interaction action $S_{\mathrm{int}}^{(-)}$ linearly couples the
$(-)$ bath of oscillators of the scalar field to the IDF, leading to the
dissipative dynamics of the IDF. The interaction action
$S_{\mathrm{int}}^{(+)}$ couples the center of mass to the $(+)$ bath of
oscillators via the IDF. We note that $S_{\mathrm{int}}^{(+)}$ is essentially
responsible for all the mechanical effects of the field, such as radiation
pressure due to optical fields and vacuum-induced forces PierreBook ;
PeterQOBook . For instance, one can see that eliminating the IDF from the
interaction action $S_{\mathrm{int}}^{(+)}$ yields an effective nonlinear
interaction between the MDF and the field, similar to the intensity-position
coupling in optomechanical interactions KSthesis . We also remark here that
the above separation of the field into two uncorrelated baths is akin to the
Einstein-Hopf theorem that establishes the statistical independence of the
blackbody radiation field and its derivative EinHopf1 ; EinHopf3 .

##  III Effective action derivation from perturbative influence functional
approach

The nonlinearity of the interaction action $S_{\mathrm{int}}^{(+)}$ limits the
possibility of obtaining an exact analytical solution for the center of mass
dynamics. We therefore follow a perturbative generating functional approach as
prescribed in HPZ93 , assuming that the characteristic nonlinear coupling
strength $\left(\sim\lambda\omega_{n}\right)$ can be treated perturbatively to
write the resulting dynamics of the center of mass.

Let us consider the evolution of the system density matrix $\hat{\rho}_{M}$ as
follows

$\displaystyle\hat{\rho}_{M}\left(t\right)=\mathrm{Tr}_{I}\mathrm{Tr}_{F^{+}}\mathrm{Tr}_{F^{-}}\left[\hat{\mathcal{U}}(t,0)\left(\hat{\rho}_{M}(0)\otimes\hat{\rho}_{I}(0)\otimes\hat{\rho}_{F}^{(+)}(0)\otimes\hat{\rho}_{F}^{(-)}(0)\right)\hat{\mathcal{U}}^{\dagger}(t,0)\right],$
(11)

where we have assumed that the density operators for the different degrees of
freedom are initially uncorrelated with each other, with $\hat{\rho}_{I}$ and
$\hat{\rho}_{F}^{(\pm)}$ referring to the density matrices for the IDF and the
$(\pm)$ bath, and $\hat{\mathcal{U}}(t,0)$ corresponding to the time evolution
operator. We write the density matrices in a coordinate representation such
that

$\displaystyle\hat{\rho}_{M}(\tau)=$
$\displaystyle\int\mathrm{d}X\mathrm{d}X^{\prime}\rho_{M}\left(X,X^{\prime};\tau\right)\left|X\right\rangle\left\langle
X^{\prime}\right|,$ (12) $\displaystyle\hat{\rho}_{I}(\tau)=$
$\displaystyle\int\mathrm{d}Q\mathrm{d}Q^{\prime}\rho_{I}\left(Q,Q^{\prime};\tau\right)\left|Q\right\rangle\left\langle
Q^{\prime}\right|,$ (13) $\displaystyle\hat{\rho}_{F}^{(\pm)}(\tau)=$
$\displaystyle\prod_{n}\int\mathrm{d}q_{n}^{(\pm)}\mathrm{d}{q^{(\pm)}_{n}}^{\prime}\rho_{F}^{(\pm)}\left(\left\\{q_{n}^{(\pm)},{q^{(\pm)}_{n}}^{\prime}\right\\};\tau\right)\left|q_{n}^{(\pm)}\right\rangle\left\langle{q^{(\pm)}_{n}}^{\prime}\right|,$
(14)

such that the system evolution in Eq. (11) can be rewritten as

$\displaystyle\rho_{M}\left({X}_{f},{X}_{f}^{\prime};t\right)=$
$\displaystyle\prod_{m,n}\int\mathrm{d}Q_{f}\mathrm{d}q^{(+)}_{nf}\mathrm{d}q^{(-)}_{mf}\int\mathrm{d}{X}_{i}\mathrm{d}Q_{i}\mathrm{d}q^{(+)}_{ni}\mathrm{d}q^{(-)}_{mi}\int\mathrm{d}{X}_{i}^{\prime}\mathrm{d}Q_{i}^{\prime}\mathrm{d}{q^{(+)}_{ni}}^{\prime}\mathrm{d}{q^{(-)}_{mi}}^{\prime}$
$\displaystyle\left[\rho_{M}\left({X}_{i},{X}_{i}^{\prime};0\right)\otimes\rho_{I}\left(Q_{i},Q_{i}^{\prime};0\right)\otimes\rho_{F}\left(\left\\{q^{(+)}_{ni},{q^{(+)}_{ni}}^{\prime}\right\\};0\right)\otimes\rho_{F}\left(\left\\{q^{(-)}_{mi},{q^{(-)}_{mi}}^{\prime}\right\\};0\right)\right.$
$\displaystyle\left.\mathcal{J}\left(X_{f},Q_{f},\left\\{{q_{nf}^{(+)}},{q_{mf}^{(-)}}\right\\};t\big{|}X_{i},Q_{i},\left\\{{q_{ni}^{(+)}},{q_{mi}^{(-)}}\right\\};0\right)\mathcal{J}^{\dagger}\left(X^{\prime}_{f},Q_{f},\left\\{{q_{nf}^{(+)}},{q_{mf}^{(-)}}\right\\};t\big{|}X^{\prime}_{i},Q^{\prime}_{i},\left\\{{q_{ni}^{(+)}}^{\prime},{q_{mi}^{(-)}}^{\prime}\right\\};0\right)\right],$
(15)

where $\left\\{{X}_{i},{X}_{i}^{\prime}\right\\}$ and
$\left\\{{X}_{f},{X}_{f}^{\prime}\right\\}$ refer to the initial and final
coordinates corresponding to the center of mass variables respectively,
$\left\\{Q_{i},Q_{i}^{\prime};Q_{f},Q_{f}^{\prime}\right\\}$ are those for the
internal oscillator, and
$\left\\{q_{ni}^{(\pm)},{q_{ni}^{(\pm)}}^{\prime};q_{nf}^{(\pm)},{q_{nf}^{(\pm)}}^{\prime}\right\\}$
are the initial and final coordinates for the $n^{\mathrm{th}}$ oscillator of
the $(\pm)$ bath. We assume that the initial states
$\rho_{F}^{(\pm)}\left(\left\\{q^{(\pm)}_{ni},{q^{(\pm)}_{ni}}^{\prime}\right\\};0\right)$
of the field and $\rho_{I}\left(Q_{i},Q_{i}^{\prime};0\right)$ of the IDF are
thermal with a temperature $T_{F}$ and $T_{I}$ respectively. The forward
propagator is defined as
$\mathcal{J}\left(X_{f},Q_{f},\left\\{q_{nf}^{(+)},q_{mf}^{(-)}\right\\};t\big{|}X_{i},Q_{i},\left\\{q_{ni}^{(+)},q_{mi}^{(-)}\right\\};0\right)\equiv\prod_{m,n}\left\langle{X}_{f}\right|\left\langle
Q_{f}\right|\left\langle q^{(+)}_{nf}\right|\left\langle
q^{(-)}_{mf}\right|\hat{\mathcal{U}}\left(t,0\right)\left|q^{(-)}_{mi}\right\rangle\left|q^{(+)}_{ni}\right\rangle\left|Q_{i}\right\rangle\left|{X}_{i}\right\rangle$.
Thus the first and last terms in the integral refer to the forward and
backward propagators that can be expressed in path integral representation to
write the evolution as

$\displaystyle\rho_{M}\left({X}_{f},{X}_{f}^{\prime};t\right)=\int\mathrm{d}{X}_{i}\int\mathrm{d}{X}_{i}^{\prime}\,\rho_{M}\left({X}_{i},{X}_{i}^{\prime};0\right)\int_{{X}(0)={X}_{i}}^{{X}(t)={X}_{f}}\mathcal{D}{X}\int_{{X}^{\prime}(0)={X}^{\prime}_{i}}^{{X}^{\prime}(t)=X^{\prime}_{f}}\mathcal{D}{X}^{\prime}e^{i\left(S_{M}[{X}]-S_{M}[{X}^{\prime}]\right)}\mathcal{F}\left[{X},{X}^{\prime}\right],$
(16)

where $\mathcal{F}\left[{X},{X}^{\prime}\right]$ refers to the influence
functional of the field and the internal oscillator acting on the MDF
FeynmanTrick , which can be written explicitly as

$\displaystyle\mathcal{F}\left[{X},{X}^{\prime}\right]\equiv$
$\displaystyle\prod_{m,n}\int\mathrm{d}Q_{f}\mathrm{d}q^{(+)}_{nf}\mathrm{d}q^{(-)}_{mf}\int\mathrm{d}Q_{i}\mathrm{d}q^{(+)}_{ni}\mathrm{d}q^{(-)}_{mi}\int\mathrm{d}Q_{i}^{\prime}\mathrm{d}{q^{(+)}_{ni}}^{\prime}\mathrm{d}{q^{(-)}_{mi}}^{\prime}\rho_{I}\left(Q_{i},Q_{i}^{\prime};0\right)\otimes\rho_{F}^{(+)}\left(\left\\{q^{(+)}_{ni},{q^{(+)}_{ni}}^{\prime}\right\\};0\right)\otimes\rho_{F}^{(-)}\left(\left\\{q^{(-)}_{mi},{q^{(-)}_{mi}}^{\prime}\right\\};0\right)$
$\displaystyle\int_{Q(0)=Q_{i}}^{Q(t)=Q_{f}}\mathcal{D}Q\int_{Q^{\prime}(0)=Q^{\prime}_{i}}^{Q^{\prime}(t)=Q_{f}}\mathcal{D}Q^{\prime}e^{i\left(S_{I}[Q]-S_{I}[Q^{\prime}]\right)}\int_{q^{(+)}_{n}(0)=q^{(+)}_{ni}}^{q^{(+)}_{n}(t)=q^{(+)}_{nf}}\mathcal{D}q_{n}^{(+)}\int_{{q^{(+)}_{n}}^{\prime}(0)={q^{(+)}_{ni}}^{\prime}}^{{q^{(+)}_{n}}^{\prime}(t)=q^{(+)}_{nf}}\mathcal{D}{q^{(+)}_{n}}^{\prime}e^{i\left(S_{F}^{(+)}\left[\left\\{q_{n}^{(+)}\right\\}\right]-S_{F}^{(+)}\left[\left\\{{q_{n}^{(+)}}^{\prime}\right\\}\right]\right)}$
$\displaystyle\int_{q^{(-)}_{m}(0)=q^{(-)}_{mi}}^{q^{(-)}_{m}(t)=q^{(-)}_{mf}}\mathcal{D}q_{m}^{(-)}\int_{{q^{(-)}_{m}}^{\prime}(0)={q^{(-)}_{mi}}^{\prime}}^{{q^{(-)}_{m}}^{\prime}(t)=q^{(-)}_{mf}}\mathcal{D}{q^{(-)}_{m}}^{\prime}e^{i\left(S_{F}^{(-)}\left[\left\\{q_{m}^{(-)}\right\\}\right]-S_{F}^{(-)}\left[\left\\{{q_{m}^{(-)}}^{\prime}\right\\}\right]\right)}e^{i\left(S_{\mathrm{int}}^{(-)}\left[Q,\left\\{q_{m}^{(-)}\right\\}\right]-S_{\mathrm{int}}^{(-)}\left[Q^{\prime},\left\\{{q_{m}^{(-)}}^{\prime}\right\\}\right]\right)}$
$\displaystyle
e^{i\left(S_{\mathrm{int}}^{(+)}\left[{X},Q,\left\\{q_{n}^{(+)}\right\\}\right]-S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},Q^{\prime},\left\\{{q_{n}^{(+)}}^{\prime}\right\\}\right]\right)}.$
(17)

Thus we have treated here the MDF as the system and the IDF and the field as
the bath, with the influence functional capturing the influence of the bath on
the evolution of the MDF. We will now evaluate the influence functional in a
perturbative manner, by first tracing out the $(-)$ bath modes that couple
only to the IDF, and then the $(+)$ bath modes and the IDF that couple to the
center of mass.

###  III.1 Tracing over the $(-)$ bath

Let us define ${\mathcal{F}^{(-)}\left[Q,Q^{\prime}\right]}$ as the influence
functional that accounts for the influence of the $(-)$ bath on the IDF as
follows

$\displaystyle\mathcal{F}^{(-)}\left[Q,Q^{\prime}\right]\equiv\prod_{m}\int\mathrm{d}q^{(-)}_{mf}\int\mathrm{d}q^{(-)}_{mi}\mathrm{d}{q^{(-)}_{mi}}^{\prime}\rho_{F}^{(-)}\left(\left\\{q^{(-)}_{mi},{q^{(-)}_{mi}}^{\prime}\right\\};0\right)\int_{q^{(-)}_{m}(0)=q^{(-)}_{mi}}^{q^{(-)}_{m}(t)=q^{(-)}_{mf}}\mathcal{D}q_{m}^{(-)}\int_{{q^{(-)}_{m}}^{\prime}(0)={q^{(-)}_{mi}}^{\prime}}^{{q^{(-)}_{m}}^{\prime}(t)={q^{(-)}_{mf}}^{\prime}}\mathcal{D}{q^{(-)}_{m}}^{\prime}$
$\displaystyle
e^{i\left(S_{F}^{(-)}\left[\left\\{q_{m}^{(-)}\right\\}\right]-S_{F}^{(-)}\left[\left\\{{q_{m}^{(-)}}^{\prime}\right\\}\right]\right)}e^{i\left(S_{\mathrm{int}}^{(-)}\left[Q,\left\\{q_{m}^{(-)}\right\\}\right]-S_{\mathrm{int}}^{(-)}\left[Q^{\prime},\left\\{{q_{m}^{(-)}}^{\prime}\right\\}\right]\right)}.$
(18)

Performing the path integrals by considering an initial thermal state of the
$(-)$ bath oscillators with temperature $T_{F}$ corresponding to the
temperature of the field, one obtains CalzettaHu

$\displaystyle\mathcal{F}^{(-)}\left[Q,Q^{\prime}\right]=e^{{i}S^{(-)}_{\mathrm{I,IF}}\left[Q,Q^{\prime}\right]},$
(19)

where the influence action due to the $(-)$ bath is given as

$\displaystyle
S^{(-)}_{\mathrm{I,IF}}\left[Q,Q^{\prime}\right]\equiv-\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[Q\left(\tau_{1}\right)-Q^{\prime}\left(\tau_{1}\right)\right]\eta^{(-)}\left(\tau_{1}-\tau_{2}\right)\left[Q\left(\tau_{2}\right)+Q^{\prime}\left(\tau_{2}\right)\right]$
$\displaystyle+i\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[Q\left(\tau_{1}\right)-Q^{\prime}\left(\tau_{1}\right)\right]\nu^{(-)}\left(\tau_{1}-\tau_{2}\right)\left[Q\left(\tau_{2}\right)-Q^{\prime}\left(\tau_{2}\right)\right].$
(20)

The above influence action corresponds to the case of quantum Brownian motion
(QBM) model with a linear system-bath coupling HPZ92 . The dissipation and
noise kernels, $\eta^{(-)}(\tau)$ and $\nu^{(-)}(\tau)$ respectively, are

$\displaystyle\eta^{(-)}(\tau)=$
$\displaystyle-\sum_{n}\frac{\lambda^{2}}{2\omega_{n}}\sin\left(\omega_{n}\tau\right){\Theta(\tau)}$
(21) $\displaystyle\nu^{(-)}(\tau)=$
$\displaystyle\sum_{n}\frac{\lambda^{2}}{2\omega_{n}}\coth\left(\frac{\omega_{n}}{2k_{B}T_{F}}\right)\cos\left(\omega_{n}\tau\right).$
(22)

The spectral density associated with the $(-)$ bath is thus

$\displaystyle
J^{(-)}\left(\omega\right)=\sum_{n}\frac{\lambda^{2}}{2\omega_{n}}\delta\left(\omega-\omega_{n}\right).$
(23)

We note from the above that the influence of the $(-)$ bath on the IDF leads
to its QBM dynamics, with a sub-Ohmic spectral density, which necessitates a
non-Markovian treatmeant of the resulting dynamics Weiss .

We now consider the effect of the IDF in turn on the dynamics of the center of
mass. Using Eq. (III.1) and (19), we can rewrite the total influence
functional in Eq. (III) as

$\displaystyle\mathcal{F}\left[{X},{X}^{\prime}\right]=\prod_{n}$
$\displaystyle\int\mathrm{d}Q_{f}\mathrm{d}q^{(+)}_{nf}\int\mathrm{d}Q_{i}\mathrm{d}q^{(+)}_{ni}\int\mathrm{d}Q_{i}^{\prime}\mathrm{d}{q^{(+)}_{ni}}^{\prime}\,\rho_{I}\left(Q_{i},Q_{i}^{\prime};0\right)\otimes\rho_{F}^{(+)}\left(\left\\{q^{(+)}_{ni},{q^{(+)}_{ni}}^{\prime}\right\\};0\right)$
$\displaystyle\int_{Q(0)=Q_{i}}^{Q(t)=Q_{f}}\mathcal{D}Q\int_{Q^{\prime}(0)=Q^{\prime}_{i}}^{Q^{\prime}(t)=Q^{\prime}_{f}}\mathcal{D}Q^{\prime}e^{i\left(S_{I}[Q]-S_{I}[Q^{\prime}]\right)}\int_{q_{n}^{(+)}(0)=q_{ni}^{(+)}}^{q_{n}^{(+)}(t)=q_{nf}^{(+)}}\mathcal{D}q_{n}\int_{{q^{(+)}_{n}}^{\prime}(0)={q^{(+)}_{ni}}^{\prime}}^{{q^{(+)}_{n}}^{\prime}(t)=q^{(+)}_{nf}}\mathcal{D}{q^{(+)}_{n}}^{\prime}$
$\displaystyle
e^{i\left(S_{F}\left[\left\\{q_{n}^{(+)}\right\\}\right]-S_{F}\left[\left\\{{q_{n}^{(+)}}^{\prime}\right\\}\right]\right)}e^{i\left(S_{\mathrm{int}}^{(+)}\left[{X},Q,\left\\{q_{n}^{(+)}\right\\}\right]-S^{(+)}_{\mathrm{int}}\left[{X}^{\prime},Q^{\prime},\left\\{{q_{n}^{(+)}}^{\prime}\right\\}\right]\right)}e^{iS_{\mathrm{I,IF}}^{(-)}\left[Q,Q^{\prime}\right]},$
(24)

where the integrations over the IDF and the $(+)$ bath of oscillators remain.
Thus far the above influence functional is exact in that we have not made any
additional approximations with regard to the strength of coupling when tracing
over the $(-)$ bath.

###  III.2 Tracing over the $(+)$ bath and the internal degree of freedom

Next we would like to integrate out the (+) bath and the IDF, which are
nonlinearly coupled to the system, using a perturbative generating functional
approach. The generating functional is simply the influence functional of the
environment where the bath oscillators are linearly coupled to the system. We
define the influence action $S_{\mathrm{M,IF}}\left[X,X^{\prime}\right]$
corresponding to the influence functional
$\mathcal{F}\left[{X},{X}^{\prime}\right]$ in Eq.(III.1) such that
$\mathcal{F}\left[{X},{X}^{\prime}\right]\equiv e^{iS_{\mathrm{M,IF}}}$. The
influence action up to second order in the coupling strength $\lambda$ can be
obtained as (see appendix B for a detailed derivation) HPZ93 :

$\displaystyle S_{\mathrm{M,IF}}^{(2)}[{X},{X}^{\prime}]\approx$
$\displaystyle\left\langle
S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\rangle_{0}-\left\langle
S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\rangle_{0}$
$\displaystyle+\frac{i}{2}\left(\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\\}^{2}\right\rangle_{0}-\left\\{\left\langle
S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\rangle_{0}\right\\}^{2}\right)$
$\displaystyle+\frac{i}{2}\left(\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\\}^{2}\right\rangle_{0}-\left\\{\left\langle
S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\rangle_{0}\right\\}^{2}\right)$
$\displaystyle-i\left(\left\langle
S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\rangle_{0}-\left\langle
S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\rangle_{0}\left\langle
S_{\mathrm{int}}^{(+)}\left[{X}^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\rangle_{0}\right),$ (25)

where we have defined ${\bf J}\equiv\left(J,\left\\{J_{n}\right\\}\right)$,
${\bf
J^{\prime}}\equiv\left(J^{\prime},\left\\{J^{\prime}_{n}\right\\}\right)$, and
the expectation values
$\left\langle\mathcal{O}[{J},{J^{\prime}}]\right\rangle_{0}\equiv\left.\mathcal{O}[{J},{J^{\prime}}]\mathcal{G}^{(1)}[{J},{J^{\prime}}]\right|_{{J}={J^{\prime}}=0}$.
The generating functional $\mathcal{G}^{(1)}[{J},{J^{\prime}}]$ for the $(+)$
bath and the IDF is defined as

$\displaystyle\mathcal{G}^{(1)}[J,J^{\prime},\left\\{J_{n},J^{\prime}_{n}\right\\}]\equiv\prod_{n}\mathcal{F}^{(1)}_{n}\left[J_{n},J^{\prime}_{n}\right]\mathcal{F}^{(1)}_{I}\left[J,J^{\prime}\right]$
(26)

with $\mathcal{F}^{(1)}_{I}\left[J,J^{\prime}\right]$ and
$\mathcal{F}^{(1)}_{n}\left[J_{n},J^{\prime}_{n}\right]$ as the influence
functionals for the internal oscillator and the $n^{\mathrm{th}}$ oscillator
of the $(+)$ bath respectively, with a linear coupling to the corresponding
source current terms $\left\\{J,J^{\prime}\right\\}$ and
$\left\\{J_{n},J^{\prime}_{n}\right\\}$. An explicit derivation of the
generating functional is given in Appendix A.

We can calculate the second order influence action in Eq. (III.2) as shown in
Appendix C to obtain a form similar to that of linear QBM dynamics HPZ92 :

$\displaystyle S_{\mathrm{M,IF}}^{(2)}\left[X,X^{\prime}\right]=$
$\displaystyle-\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t_{1}}\mathrm{d}t_{2}\,\left[X(t_{1})-X^{\prime}(t_{1})\right]\eta^{(2)}(t_{1},t_{2})\left[X(t_{2})+X^{\prime}(t_{2})\right]$
$\displaystyle+i\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t_{1}}\mathrm{d}t_{2}\,\left[X(t_{1})-X^{\prime}(t_{1})\right]\nu^{(2)}(t_{1},t_{2})\left[X(t_{2})-X^{\prime}(t_{2})\right],$
(27)

where dissipation and noise kernels are defined as

$\displaystyle\eta^{(2)}(t_{1},t_{2})=$
$\displaystyle\frac{1}{2}\eta^{(+)}\left(t_{1}-t_{2}\right)\left\\{\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle\right\\}+\frac{1}{4}\nu^{(+)}\left(t_{1}-t_{2}\right){g}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right),$
(28) $\displaystyle\nu^{(2)}(t_{1},t_{2})=$
$\displaystyle\frac{1}{2}\nu^{(+)}\left(t_{1}-t_{2}\right)\left\\{\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle\right\\}-\frac{1}{4}\eta^{(+)}\left(t_{1}-t_{2}\right){g}\left(t_{1}-t_{2}\right),$
(29)

with ${G}(t)=g(t)\Theta(t)$ the propagator for the IDF defined as in Eq.(67).
The noise correlation of the IDF is given by

$\displaystyle\nu_{GG}(t_{1},t_{2})\equiv\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{t}\mathrm{d}\tau_{2}{G}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(t_{2}-\tau_{2}\right).$
(30)

The noise arising from the dispersion in the initial conditions of the IDF is
captured in the term $\sim\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$, where
$Q_{h}\left(t\right)$ is the classical solution to the homogeneous Langevin
equation (Eq. (66)) corresponding to the dissipative dyanmics of IDF in the
presence of the $(-)$ bath. The average $\left\llangle\dots\right\rrangle$ is
taken over initial position and momentum distribution of the IDF, as defined
in Eq. (65).

The dissipation and noise kernels associated with the $(+)$ bath, defined
based on a bilinear system-bath coupling are

$\displaystyle\eta^{(+)}\left(\tau\right)\equiv$
$\displaystyle-\frac{1}{2}\sum_{n}\lambda^{2}\omega_{n}\sin\left(\omega_{n}\tau\right){\Theta(\tau)}$
(31) $\displaystyle\nu^{(+)}\left(\tau\right)\equiv$
$\displaystyle\frac{1}{2}\sum_{n}\lambda^{2}\omega_{n}\coth\left(\frac{\omega_{n}}{2k_{B}T_{F}}\right)\cos\left(\omega_{n}\tau\right),$
(32)

similar to those of the $(-)$ bath (Eq. (21) and Eq. (22)) with a coupling
$\lambda\rightarrow\lambda\omega_{n}$. The spectral density associated with
the $(+)$ bath is thus

$\displaystyle
J^{(+)}\left(\omega\right)=\sum_{n}\frac{\lambda^{2}\omega_{n}}{2}\delta\left(\omega-\omega_{n}\right),$
(33)

corresponding to an Ohmic case CalzettaHu . We note that while the two baths
$(+)$ and $(-)$ correspond to the same field, due to a gradient coupling of
the center of mass to the field the $(+)$ bath has an Ohmic spectral density
while the $(-)$ bath is sub-Ohmic (see Eq. (23)).

One can note a few important features from the influence action (Eq. (27)) and
the dissipation and noise kernels therein (Eq. (28) and Eq. (29)):

  * •

The overall structural similarity of the second order influence action to that
of the linear QBM extends to structure of the dissipation and noise kernels as
well. Meaning that writing the interaction Hamiltonian as
$H\sim\hat{X}\hat{\mathcal{B}}$, where $\hat{\mathcal{B}}$ is the total bath
operator (possibly nonlinear), one can write an effective action of the same
form as linear QBM with dissipation and noise kernels given in terms of the
expectation values of the commutator and anticommutator of the two-time bath
correlation functions
$\eta\left(\tau\right)=\left\langle\left[\hat{\mathcal{B}}\left(\tau\right),\hat{\mathcal{B}}\left(0\right)\right]\right\rangle$
and
$\nu\left(\tau\right)=\left\langle\left\\{\hat{\mathcal{B}}\left(\tau\right),\hat{\mathcal{B}}\left(0\right)\right\\}\right\rangle$.
This is a more general property of bilinear system-bath couplings. The
influence functional of Eq.(III.1) for $X=0=X^{\prime}$ results in a Gaussian
path integral, so it is reasonable to obtain that the second order results
with a linear QBM form. One can further verify as a check that the unitarity
of the evolution implies $S_{\mathrm{M,IF}}^{(2)}\left[X,X\right]=0$.

  * •

The dissipation kernel contains two parts that can be interpreted as: (1)
coming from the dissipation kernel of the $(+)$ bath
$(\eta^{(+)}(t_{1}-t_{2}))$ combined with the noise from the IDF
$(\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$, and (2) linear
propagator of the IDF $(G\left(t_{1}-t_{2}\right))$ combined with the noise
from the $(+)$ bath $(\nu^{(+)}\left(t_{1}-t_{2}\right))$. Notice that the
quantity $\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ stands for the
full quantum correlations of the IDF under the influence of the $(-)$ bath
alone. The term $\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ accounts for the
relaxation of the initial conditions’ contribution, thus carrying the
information about the initial state of the IDF and determined only by the
dissipation caused by the $(-)$ bath. On the other hand, the term
$\nu_{GG}\left(t_{1},t_{2}\right)$ stands for the fluctuations on the IDF
generated by the fluctuations of the $(-)$ bath.

  * •

The noise kernel has the combined noise from both the IDF
$(\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ and that from
the $(+)$ bath $(\nu^{(+)}\left(t_{1}-t_{2}\right))$. In addition to that the
noise term also contains the two dissipation kernels from the IDF and the
$(+)$ bath.

##  IV Langevin equation and fluctuation-dissipation relation

Having determined the influence action for the center of mass, we can now turn
to studying its dynamics. This section is devoted to the deduction of an
effective Langevin equation of motion for the center of mass and a
corresponding generalized fluctuation-dissipation relation in the long-time
limit.

###  IV.1 Effective equation of motion for the center of mass

Considering the influence action obtained as a result of tracing out the IDF
and the field (Eq.(27)), the total effective action for the center of mass
variables up to second order reads:

$\displaystyle
S_{\mathrm{M,eff}}\left[X,X^{\prime}\right]=S_{M}\left[X\right]-S_{M}\left[X^{\prime}\right]+S_{\mathrm{M,IF}}^{(2)}\left[X,X^{\prime}\right].$
(34)

The above action is quadratic and has the same form as in the case of linear
QBM CalzettaHu . One can thus deduce an equation of motion for the center of
mass by extremizing the above action to obtain

$\displaystyle\ddot{X}\left(t\right)+\Omega^{2}X\left(t\right)+\frac{2}{M}\int_{0}^{t}d\tau~{}\eta^{(2)}(t,\tau)~{}X(\tau)=0.$
(35)

A Langevin equation can be worked out by implementing the Feynman and Vernon
procedure for Gaussian path integrals FeynmanTrick ; Behunin1 . First, we
notice that $iS_{\mathrm{M,IF}}^{(2)}=i{\rm Re}[S_{\mathrm{M,IF}}^{(2)}]-{\rm
Im}[S_{\mathrm{M,IF}}^{(2)}]$, and that both, the real and imaginary parts of
the effective action are quadratic functionals of $\\{X,X^{\prime}\\}$. Thus,
${\rm exp}(-{\rm Im}[S_{\mathrm{M,IF}}^{(2)}])$ is a Gaussian functional since
the kernel owing to the quadratic nature of the action. Considering that a
Gaussian functional can be written in terms of its functional Fourier
transform (which is also a Gaussian functional), we can write ${\rm exp}(-{\rm
Im}[S_{\mathrm{M,IF}}^{(2)}])$ as an influence functional over a new variable
$\xi(t)$

$\displaystyle
e^{iS_{\mathrm{M,IF}}^{(2)}\left[X,X^{\prime}\right]}=\int\mathcal{D}\xi~{}\mathcal{P}\left[\xi\right]e^{i\int_{0}^{t}dt_{1}\left[X(t_{1})-X^{\prime}(t_{1})\right]\left[\xi(t_{1})-\int_{0}^{t_{1}}dt_{2}\eta^{(2)}(t_{1},t_{2})\left[X(t_{2})+X^{\prime}(t_{2})\right]\right]},$
(36)

where the distribution of the new functional variable is given by

$\displaystyle\mathcal{P}\left[\xi\right]=e^{-\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\xi(t_{1})\left[4\nu^{(2)}(t_{1},t_{2})\right]^{-1}\xi(t_{2})}.$
(37)

It is worth noting that the new variable $\xi\left(t\right)$ comes to replace
the kernel describing the quantum and thermal fluctuations of the composite
environment and drive the dynamics as an external stochastic force. To recover
the influence action as in Eq. (36), one needs to integrate over $\xi$ given
the functional distribution $\mathcal{P}[\xi]$, which is positive definite
since the noise kernel $\nu^{(2)}\left(t_{1},t_{2}\right)$ is symmetric and
positive. Thus, the stochastic variable $\xi\left(t\right)$ acts as a
classical fluctuating force, which can be interpreted as a noise with a
probability distribution $\mathcal{P}[\xi]$. Furthermore, due to the
Gaussianity of $\mathcal{P}\left[\xi\left(t\right)\right]$ the noise is
completely characterized by its first and second moments:

$\displaystyle\left\langle\xi\left(t\right)\right\rangle_{\xi}=0,$
$\displaystyle\left\langle\xi(t_{1})\xi(t_{2})\right\rangle_{\xi}=4\nu^{(2)}(t_{1},t_{2}),$
(38)

where
$\left\langle...\right\rangle_{\xi}=\int\mathcal{D}\xi\mathcal{P}\left[\xi\right](...)$.
Now, we can re-define an effective action for the center of mass including
this new variable as:

$\displaystyle\tilde{S}_{\mathrm{M,eff}}\left[X,X^{\prime},\xi\right]=S_{M}\left[X\right]-S_{M}\left[X^{\prime}\right]+{\mathrm{Re}}\left[S_{\mathrm{M,IF}}^{(2)}\left[X,X^{\prime}\right]\right]+\int_{0}^{t}dt_{1}\left[X(t_{1})-X^{\prime}(t_{1})\right]\xi(t_{1}).$
(39)

Finally, associated with this new action we can derive an effective equation
of motion for the center of mass $(\delta\tilde{S}_{\mathrm{M,eff}}/\delta
X_{\Delta})|_{X_{\Delta}=0}=0$, which now gives the Langevin equation:

$\displaystyle\ddot{X}\left(t\right)+\Omega^{2}X\left(t\right)+\frac{2}{M}\int_{0}^{t}d\tau~{}\eta^{(2)}(t,\tau)~{}X(\tau)=\xi\left(t\right),$
(40)

subjected to the statistical properties of $\xi\left(t\right)$ as in Eq.
(IV.1). We can see that averaging the above equation over the stochastic force
($\left\langle...\right\rangle_{\xi}$) reduces to Eq.(35), which we can
interpret as the Langevin equation of motion after averaging over the possible
noise realizations. As has been shown previously in CRV , the correlations of
the system observables can be obtained from the solutions of such a Langevin
equation.

###  IV.2 Generalized fluctuation-dissipation relation

We now analyze the relations between the dissipation and noise kernels of the
composite environment. Considering the definitions of the dissipation and
noise kernels given in Eqs.(28) and (29), respectively, we first take the
long-time limit. In this limit, the dissipation and noise kernels simplify to

$\displaystyle\eta^{(2)}(t_{1},t_{2})\longrightarrow$
$\displaystyle\left[\frac{1}{2}\eta^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}\left(t_{1},t_{2}\right)+\frac{1}{4}g\left(t_{1}-t_{2}\right)\nu^{(+)}\left(t_{1}-t_{2}\right)\right]\Theta\left(t_{1}-t_{2}\right),$
(41) $\displaystyle\nu^{(2)}(t_{1},t_{2})\longrightarrow$
$\displaystyle\frac{1}{2}\nu^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}\left(t_{1},t_{2}\right)-\frac{1}{4}g\left(t_{1}-t_{2}\right)\eta^{(+)}\left(t_{1}-t_{2}\right),$
(42)

where the term $\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ associated with
the relaxation of the IDF vanishes in the late time limit, and the kernel
$\nu_{GG}(t_{1},t_{2})$ reads:

$\displaystyle\nu_{GG}(t_{1},t_{2})\equiv\int_{0}^{\infty}\mathrm{d}\tau_{1}\int_{0}^{\infty}\mathrm{d}\tau_{2}{G}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(t_{2}-\tau_{2}\right).$
(43)

Notice that this limit holds regardless of the initial state of the IDF. This
is in agreement with the notion that the dynamics at late-times is determined
by the field fluctuations only.

It can be shown that the Fourier and Laplace transforms of the dissipation and
noise kernels corresponding to the individual baths can be related in terms of
the following fluctuation-dissipation relations (see Appendix D for details)

$\displaystyle\bar{\nu}^{(\pm)}(\omega)=$
$\displaystyle-2\coth\left(\frac{\omega}{2k_{B}T_{F}}\right){\rm
Im}\left[\bar{\eta}^{(\pm)}(\omega)\right]$ (44)
$\displaystyle\bar{\nu}_{GG}(\omega)=$
$\displaystyle\coth\left(\frac{\omega}{2k_{B}T_{F}}\right){\rm
Im}[\bar{G}(\omega)],$ (45)

where $\bar{F}(\omega)$ represent the Fourier transforms of $F(t)$, defined as
$\bar{F}(\omega)=\int_{-\infty}^{+\infty}{dt}e^{i\omega t}F(t)$. Note that
both $\eta^{(\pm)}$ and $G$ are causal functions, as is also true for the
dissipation kernel $\eta^{(2)}$.

One can thus rewrite Eqs.(41) and (42) in terms of the Fourier transformed
kernels as follows

$\displaystyle\bar{\eta}^{(2)}(\omega)=$
$\displaystyle\frac{1}{2}\left[\bar{\nu}_{GG}\ast\bar{\eta}^{(+)}\right](\omega)+\frac{1}{4}\left[\bar{G}\ast\bar{\nu}^{(+)}\right](\omega),$
(46) $\displaystyle\bar{\nu}^{(2)}(\omega)=$
$\displaystyle\frac{1}{2}\left[\bar{\nu}_{GG}\ast\bar{\nu}^{(+)}\right](\omega)+\left[{\rm
Im}\left(\bar{G}\right)\ast{\rm
Im}\left(\bar{\eta}^{(+)}\right)\right](\omega).$ (47)

The convolution product is defined for two functions $[A\ast
B](\omega)\equiv\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}A(\omega-\omega^{\prime})B(\omega^{\prime})$,
which satisfies $[A\ast B](\omega)=[B\ast A](\omega)$.

Using the fluctuation-dissipation relations for the kernels of the two baths
and the IDF, we can write the dissipation kernel for the MDF in terms of those
of the IDF and the baths in frequency domain as follows

$\displaystyle{\rm Im}\left[\bar{\eta}^{(2)}(\omega)\right]=$
$\displaystyle\int_{0}^{+\infty}\frac{d\omega^{\prime}}{2\pi}\frac{1}{2}\left[\frac{1}{2}\left\\{\bar{\nu}^{(+)}(\omega-\omega^{\prime})-\bar{\nu}^{(+)}(\omega+\omega^{\prime})\right\\}{\rm
Im}[\bar{G}(\omega^{\prime})]+\left\\{\bar{\nu}_{GG}(\omega-\omega^{\prime})-\bar{\nu}_{GG}(\omega+\omega^{\prime})\right\\}{\rm
Im}\left[\bar{\eta}^{(+)}(\omega^{\prime})\right]\right].$ (48)

We note from the above equation that the contributions to the total
dissipation come from the three body processes between the MDF, IDF and the
$(+)$ bath, with the propagator of the IDF combining with the noise of the
$(+)$ bath and vice versa. Each term in the above equation accounts for the
four possible processes involved in the dynamics at second order, wherein one
of the thermal baths contributes the initial excitation, while the other bath
acts as an extra channel for partial energy exchange.

Furthermore, the above kernel can be alternatively expressed in a compact way
as follows

$\displaystyle{\rm Im}\left[\bar{\eta}^{(2)}(\omega)\right]=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}\frac{1}{2}\left[\coth\left(\frac{\omega-\omega^{\prime}}{2k_{B}T_{F}}\right)-\coth\left(\frac{\omega^{\prime}}{2k_{B}T_{F}}\right)\right]{\rm
Im}[\bar{G}(\omega-\omega^{\prime})]~{}{\rm
Im}\left[\bar{\eta}^{(+)}(\omega^{\prime})\right].$ (49)

We can similarly express the noise kernel in terms of the dissipation kernels
of the IDF and the two baths as follows

$\displaystyle\bar{\nu}^{(2)}(\omega)=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}\frac{1}{2}\left[1-\coth\left(\frac{\omega-\omega^{\prime}}{2k_{B}T_{F}}\right)\coth\left(\frac{\omega^{\prime}}{2k_{B}T_{F}}\right)\right]{\rm
Im}[\bar{G}(\omega-\omega^{\prime})]~{}{\rm
Im}\left[\bar{\eta}^{(+)}(\omega^{\prime})\right].$ (50)

It is not possible to find simple relation between the above tranforms of the
dissipation and noise kernels $\\{\bar{\eta}^{(2)},\bar{\nu}^{(2)}\\}$ because
of the presence of the integrals on $\omega^{\prime}$, which are related to
the convolution products. We therefore define new kernels
$D^{(2)}(\omega,\omega^{\prime})$ and $N^{(2)}(\omega,\omega^{\prime})$ that
depend on two frequency variables, accounting for processes where a given
frequency-component contribution is modified by other frequencies such that
$\bar{\eta}^{(2)}(\omega)=\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}D^{(2)}(\omega,\omega^{\prime})$
and
$\bar{\nu}^{(2)}(\omega)=\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}N^{(2)}(\omega,\omega^{\prime})$.
Thus we can find a relation between these new kernels that reads

$\displaystyle
N^{(2)}\left(\omega,\omega^{\prime}\right)=2\coth\left[\frac{\omega-2\omega^{\prime}}{2k_{B}T}\right]{\rm
Im}\left[D^{(2)}(\omega,\omega^{\prime})\right],$ (51)

or inversely

$\displaystyle{\rm
Im}\left[D^{(2)}(\omega,\omega^{\prime})\right]=\frac{1}{2}\tanh\left[\frac{\omega-2\omega^{\prime}}{2k_{B}T}\right]N^{(2)}\left(\omega,\omega^{\prime}\right).$
(52)

The introduction of an extra variable $\omega^{\prime}$ stands for the
complexity of the environment in terms of its energy exchange with the system.
The nonlinear coupling allows the two parts of the environment (the field and
the IDF) to simultaneously exchange energy with the system. From these
relations, it becomes simple to prove that

$\displaystyle{\rm Im}\left[\bar{\eta}^{(2)}(\omega)\right]=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}\frac{1}{2}\tanh\left[\frac{\omega^{\prime}}{2k_{B}T}\right]N^{(2)}\left(\omega,\frac{\omega-\omega^{\prime}}{2}\right),$
(53)

and

$\displaystyle\bar{\nu}^{(2)}(\omega)=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{d\omega^{\prime}}{2\pi}2\coth\left[\frac{\omega^{\prime}}{2k_{B}T}\right]D^{(2)}\left(\omega,\frac{\omega-\omega^{\prime}}{2}\right).$
(54)

The above equation corresponds to a generalized fluctuation-dissipation
relation (FDR) for the late-time dissipation and noise kernels.

We can physically interpret the results by comparing the present situation to
the case of linear QBM. The FDR that holds between the dissipation and noise
kernels in QBM can be stated in frequency-domain for a single frequency
component $\omega$ in the general form that reads
$[{\textit{N}oise}](\omega)=[{\textit{T}hermalFactor}](\omega)\times[{\textit{D}issipation}](\omega)$,
wherein a single frequency connects both sides of the relation. In our case,
the nonlinear nature of the coupling precludes such a relation at a single
frequency. Physically such a nonlinear coupling leads to processes wherein an
excitation from one of the baths interacts with the system while perturbed by
the other bath. We can thus formulate a generalized FDR Eq. (51) by
introducing two frequency variables to account for the nonlinearity.

##  V Discussion

We have derived the non-equilbrium dissipative center of mass dynamics of a
particle interacting with a field via its IDF. The model considered in this
paper underlies microscopic optomechanical interactions between neutral
particles and fields, and is generally applicable to open quantum systems that
possess intermediary quantum degrees of freedom between system and bath. We
show that the field can be separated into two baths, referred to as $(+)$ and
$(-)$ baths, with the $(-)$ bath coupled linearly to the IDF, and the $(+)$
bath coupled nonlinearly to both IDF and MDF (see Eq. (9) and Eq. (10)). Such
a decomposition allows one to systematically trace over $(-)$ bath and express
its resulting influence on the IDF in terms of a exact second order effective
action as given by Eq. (20). As the nonlinear coupling between the MDF, IDF
and the $(+)$ bath poses a challenge in terms of writing the exact dynamics of
the MDF, we use a perturbative influence functional approach to trace over the
IDF and $(+)$ bath to obtain a second order effective action for the MDF. We
find that the effective action, the dissipation and noise kernels resulting
from the composite environment (Eq. (27), (28) and (29) respectively) are
structurally similar to linear QBM dynamics. This can be attributed to the
quadratic nature of coupling between the system variable and the composite
bath that goes as $\sim X\mathcal{B}$, where the bath operator
$\mathcal{B}\left(\sim Q\sum_{n}q_{n}^{(+)}\right)$ can be nonlinear. We find
a Langevin equation of motion for the MDF, describing its dissipative non-
equilibrium dynamics (Eq. (40)). The dissipation and noise kernels at late
time can be related via a generalized FDR as given by Eq. (51).

This work highlights three aspects:

  1. 1.

Dissipation and noise of the open quantum system in the presence of an IDF –
It can be seen from the dissipation and noise kernels (Eq. (28) and (29))
arising from the composite bath of the IDF+field that the total dissipation of
the MDF is a combination of the dissipation kernel corresponding to the IDF
and noise kernel corresponding to the $(+)$ bath, and vice versa. Similarly
the noise kernel involves a combination of the noise of the IDF and that of
the $(+)$ bath. Additionally, it also includes a contribution resulting from
the combination of the dissipation of the IDF and that of the $(+)$ bath,
which is needed to obtain a generalized FDR for the kernels of our composite
environment. As discussed in Sec. III.2, the dissipation and noise kernels are
similar in structure to those of linear QBM up to the second order
perturbative in action. The present approach can be extended to higher orders
in a systematic way, thus extending the results of Ref.HPZ93  to complex
environments.

  2. 2.

Coupled non-equilibrium dynamics of different degrees of freedom with self-
consistent backaction – The equation of motion for the MDF given in Eq. (40)
describes the quantum center of mass motion including the self-consistent
backaction of the various degrees of freedom on each other. The approach based
on the separation of the field into two uncorrelated baths is akin to the
Einstein-Hopf theorem that establishes the statistical independence of the
blackbody radiation field and its derivative EinHopf1 ; EinHopf3 . The
influence of the $(-)$ bath on the IDF is included via the effective action
Eq. (20), and the perturbative second order effective action in Eq. (27)
includes the influence of the $(+)$ bath and the IDF (affected by the $(-)$
bath). In this way, the quantum field influences the MDF’s dynamics by two
means, by its direct interaction as the $(+)$ bath and through the
fluctuations of the IDF as the $(-)$ bath. Furthermore, considering the zero-
temperature limit of the dynamics, we find that a mechanical oscillator
interacting with the vacuum field exhibits dissipative dynamics as a result of
the quantum fluctuations of its composite nonlinear environment. Such a
vacuum-induced noise poses a fundamental constraint on preparing mechanical
objects in quantum states KSYS1 ; Skatepark .

  3. 3.

Fluctuation-dissipation relations for a system coupled to a composite
environment  – We find a generalized FDR between the late-time dissipation and
noise kernels, that has a similar structure to linear QBM (see Eq. (51))
albeit involving two frequency variables due to the nonlinearity of the
interaction. This allows one to interpret the single-frequency response of the
effective kernels for the MDF in terms of contributions from various frequency
components of the IDF and field. Our result provides an example of FDR for an
open quantum system with nonlinear dynamics when linear response theory is no
longer valid HPZ93 ; JT20 .

As a future prospect our results can be extended to studying the non-
equilibrium dynamics in a wide range of physical systems, particularly when
the time scales of the different degrees of freedom are no longer disparate.
In the context of atom-field interactions, it has been shown for example that
the internal and external degrees of freedom of atoms in a standing laser wave
can exhibit synchronization Argonov05 . Furthermore, it is possible for the
IDF dynamics to be sufficiently slow for a dark internal state which can lead
to highly correlated internal-external dynamics as in the case of velocity
selective coherent population trapping scheme Aspect88 ; Aspect89 , and long-
lived internal states in atomic clock transitions Boyd . It has also been
experimentally observed that the internal and external degrees of freedom can
be efficiently coupled in the presence of colored noise Machluf10 . On the
other hand, when the center of mass dynamics is fast enough to be comparable
to the internal dynamics, one requires a careful consideration of the coupled
internal-external dynamics. Such a situation can arise in the case of
dynamical Casimir effect (DCE) Lin18 , when considering the self-consistent
backaction of the field and the IDF on the mirror Butera ; Belen19 . Finally,
our results can be extended to analyze the thermalization properties and non-
equilibrium dynamics of the center mass of nanoparticles Yin ; AERL18 .

##  VI Acknowledgments

We are grateful to Bei-Lok Hu, Esteban A. Calzetta, and Peter W. Milonni for
insightful discussions. Y.S. acknowledges support from the Los Alamos National
Laboratory ASC Beyond Moore’s Law project and the LDRD program. A.E.R.L. wants
to thank the AMS for the support.

##  Appendix A Generating functional derivation

The generating functionals corresponding to the IDF and the $(+)$ bath can be
explicitly written as

$\displaystyle\mathcal{F}^{(1)}_{I}\left[J,J^{\prime}\right]=\int\mathrm{d}Q_{f}\int\mathrm{d}Q_{i}\int\mathrm{d}Q^{\prime}_{i}\,\rho_{I}\left(Q_{i},Q^{\prime}_{i};0\right)\int_{Q(0)=Q_{i}}^{Q(t)=Q_{f}}\mathcal{D}Q\int_{Q^{\prime}(0)=Q^{\prime}_{i}}^{Q^{\prime}(t)=Q_{f}}\mathcal{D}Q^{\prime}$
$\displaystyle
e^{i\left(S_{I}[Q]-S_{I}[Q^{\prime}]+S_{\mathrm{I,IF}}^{(-)}[Q,Q^{\prime}]\right)}$
$\displaystyle
e^{i\int_{0}^{t}\mathrm{d}\tau\left[J(\tau)Q(\tau)-J^{\prime}(\tau)Q^{\prime}(\tau)\right]}$
(55) $\displaystyle\mathcal{F}^{(1)}_{n}\left[J_{n},J^{\prime}_{n}\right]=$
$\displaystyle\int\mathrm{d}q_{nf}^{(+)}\int\mathrm{d}q_{ni}^{(+)}\int\mathrm{d}{q_{ni}^{(+)}}^{\prime}\,\rho_{F,n}^{(+)}\left(\left\\{q_{ni}^{(+)},{q_{ni}^{(+)}}^{\prime}\right\\};0\right)$
$\displaystyle\int_{q_{n}^{(+)}(0)=q^{(+)}_{ni}}^{q^{(+)}_{n}(t)=q^{(+)}_{nf}}\mathcal{D}q^{(+)}_{n}\int_{{q_{n}^{(+)}}^{\prime}(0)={q_{ni}^{(+)}}^{\prime}}^{{q_{n}^{(+)}}^{\prime}(t)=q^{(+)}_{nf}}\mathcal{D}{{q_{n}}^{(+)}}^{\prime}e^{i\left(S_{F,n}^{(+)}[\left\\{q_{n}^{(+)}\right\\}]-S_{F,n}^{(+)}[\left\\{{q_{n}^{(+)}}^{\prime}\right\\}]\right)}e^{i\int_{0}^{t}\mathrm{d}\tau\left[J_{n}(\tau)q_{n}^{(+)}(\tau)-J^{\prime}_{n}(\tau){{q_{n}^{(+)}}^{\prime}}(\tau)\right]},$
(56)

where $\rho_{F,n}^{(+)}\left(q_{ni}^{(+)},{q_{ni}^{(+)}}^{\prime};0\right)$
represents the initial state of the $n^{\mathrm{th}}$ oscillator of the $(+)$
bath, and $S_{F,n}^{(+)}$ is the corresponding free action. We can then see
from Eq. (III.1) and Eq. (19) that the generating functional for the $(+)$
bath is simply the influence functional for linear QBM given by

$\displaystyle\mathcal{F}^{(1)}_{n}\left[J_{n},J^{\prime}_{n}\right]=e^{iS_{\mathrm{IF},n}^{(+)}[J_{n},J_{n}^{\prime}]},$
(57)

where influence action $S_{\mathrm{IF},n}^{(+)}[J_{n},J_{n}^{\prime}]$ is
defined as

$\displaystyle
S^{(+)}_{\mathrm{IF},n}\left[J_{n},J_{n}^{\prime}\right]\equiv-\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[J_{n}\left(\tau_{1}\right)-J_{n}^{\prime}\left(\tau_{1}\right)\right]\eta_{n}^{(+)}\left(\tau_{1}-\tau_{2}\right)\left[J_{n}\left(\tau_{2}\right)+J_{n}^{\prime}\left(\tau_{2}\right)\right]$
$\displaystyle+i\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[J_{n}\left(\tau_{1}\right)-J_{n}^{\prime}\left(\tau_{1}\right)\right]\nu_{n}^{(+)}\left(\tau_{1}-\tau_{2}\right)\left[J_{n}\left(\tau_{2}\right)-J_{n}^{\prime}\left(\tau_{2}\right)\right],$
(58)

with the dissipation and noise kernels

$\displaystyle\eta_{n}^{(+)}(\tau)=$
$\displaystyle-\frac{1}{2\omega_{n}}\sin\left(\omega_{n}\tau\right)$ (59)
$\displaystyle\nu_{n}^{(+)}(\tau)=$
$\displaystyle\frac{1}{2\omega_{n}}\coth\left(\frac{\omega_{n}}{2k_{B}T_{F}}\right)\cos\left(\omega_{n}\tau\right).$
(60)

Now to evaluate the generating functional for the IDF in terms of
$\left\\{J,J^{\prime}\right\\}$, we follow the approach in CRV . Consider the
effective action pertaining to the IDF in Eq. (A)

$\displaystyle S_{I,\mathrm{eff}}\left[Q,Q^{\prime},J,J^{\prime}\right]=$
$\displaystyle
S_{I}\left[Q\right]-S_{I}\left[Q^{\prime}\right]+S_{I,\mathrm{IF}}^{(-)}\left[Q,Q^{\prime}\right]+\int_{0}^{t}\mathrm{d}\tau\left[J(\tau)Q(\tau)-J^{\prime}(\tau)Q^{\prime}(\tau)\right]$
(61) $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}\tau\left[\frac{1}{2}M\dot{Q}^{2}-\frac{1}{2}M\omega_{0}^{2}Q^{2}+JQ\right]-\left[\frac{1}{2}M\dot{Q}^{\prime
2}-\frac{1}{2}M\omega_{0}^{2}Q^{\prime 2}+J^{\prime}Q^{\prime}\right]$
$\displaystyle-\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[Q(\tau_{1})-Q^{\prime}(\tau_{1})\right]\eta^{(-)}\left(\tau_{1}-\tau_{2}\right)\left[Q(\tau_{2})+Q^{\prime}(\tau_{2})\right]$
$\displaystyle+i\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}\left[Q(\tau_{1})-Q^{\prime}(\tau_{1})\right]\nu^{(-)}\left(\tau_{1}-\tau_{2}\right)\left[Q(\tau_{2})-Q^{\prime}(\tau_{2})\right]$
(62) $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}\tau\left[\frac{1}{2}M\dot{Q}_{\Sigma}(\tau)\dot{Q}_{\Delta}(\tau)-\frac{1}{2}M\omega_{0}^{2}Q_{\Sigma}(\tau)Q_{\Delta}(\tau)+Q_{\Sigma}(\tau)J_{\Delta}(\tau)+Q_{\Delta}(\tau)J_{\Sigma}(\tau)\right]$
$\displaystyle-\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}Q_{\Delta}(\tau_{1})\eta^{(-)}\left(\tau_{1}-\tau_{2}\right)Q_{\Sigma}(\tau_{2})+i\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{\tau_{1}}\mathrm{d}\tau_{2}Q_{\Delta}(\tau_{1})\nu^{(-)}\left(\tau_{1}-\tau_{2}\right)Q_{\Delta}(\tau_{2}),$
(63)

where we have defined the new coordinates as $Q_{\Sigma}\equiv Q+Q^{\prime}$,
$Q_{\Delta}\equiv Q-Q^{\prime}$,
$J_{\Sigma}\equiv\left(J+J^{\prime}\right)/2$, and
$J_{\Delta}\equiv\left(J-J^{\prime}\right)/2$. The dissipation and noise
kernels $\eta^{(-)}\left(t\right)$ and $\nu^{(-)}\left(t\right)$ are as
defined in Eq. (21) and Eq. (22).

Rewriting the generating functional in terms of the relative and center of
mass coordinates, using the approach outlined in CRV , Eq. (A) can be
simplified to obtain

$\displaystyle\mathcal{F}^{(1)}_{I}\left[J,J^{\prime}\right]=$
$\displaystyle\left\llangle e^{2iJ_{\Delta}\cdot\,Q_{h}}\right\rrangle
e^{-2\left(J_{\Delta}\cdot\,{G}\right)\cdot{\nu}^{(-)}\cdot\left(J_{\Delta}\cdot\,{G}\right)^{T}}e^{2iJ_{\Delta}\cdot{G}\cdot
J_{\Sigma}},$ (64)

where

$\displaystyle\left\llangle\dots\right\rrangle=\int\mathrm{d}Q^{i}\mathrm{d}\Pi^{i}W\left(Q^{i},\Pi^{i}\right)(\dots),$
(65)

denotes the average over the initial Wigner distribution $W(Q^{i},\Pi^{i})$.
$Q_{h}$ is the solution to the homogeneous Langevin equation

$\displaystyle{L}\cdot Q_{h}=0,$ (66)

with initial conditions given by $\left\\{Q^{i},\Pi^{i}\right\\}$ and the
differential operator $L(t,t^{\prime})=m(\frac{d^{2}}{dt^{\prime
2}}+\omega_{0}^{2})\delta(t-t^{\prime})+2\eta^{(-)}(t-t^{\prime})$. The
Green’s function corresponding to the Langevin operator ${{L}}$ is defined as

$\displaystyle{L}\cdot{G}={\delta},$ (67)

where $A\cdot B\equiv\int_{0}^{t}\mathrm{d}\tau A(\tau)B(\tau)$.

##  Appendix B Perturbative effective action derivation

Now having obtained the total generating functional as defined in Eq. (26)
which contains the influence of the IDF (Eq. (64)) and the (+) bath (Eq. (A))
on the system, we substitute and back in the total generating functional to
obtain the effective action perturbatively up to second order in the system-
bath coupling parameter in a term by term manner as follows.

$\displaystyle F[X,X^{\prime}]=$
$\displaystyle\left.\exp{\left\\{{i}\left(S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta
J},\left\\{\frac{1}{i}\frac{\delta}{\delta
J_{n}}\right\\}\right]-S_{\mathrm{int}}^{(+)}\left[X^{\prime},\frac{-1}{i}\frac{\delta}{\delta
J^{\prime}},\left\\{\frac{-1}{i}\frac{\delta}{\delta
J^{\prime}_{n}}\right\\}\right]\right)\right\\}}\mathcal{G}^{(1)}[J,J^{\prime},\left\\{J_{n},J^{\prime}_{n}\right\\}]\right|_{{\bf
J}={\bf J^{\prime}}=0}$
$\displaystyle\equiv\exp{\left\\{{i}S_{\mathrm{M,IF}}[X,X^{\prime}]\right\\}}.$
(68)

So far the above expression would yield the exact effective action for the
given system-bath interaction since we have not invoked any weak-coupling
approximations yet. We will now make a perturbative expansion of the
interaction action up to second order in the system-bath coupling strength in
the exponent. Let us consider
$S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta
J},\left\\{\frac{1}{i}\frac{\delta}{\delta
J_{n}}\right\\}\right]\equiv\varepsilon\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta
J},\left\\{\frac{1}{i}\frac{\delta}{\delta J_{n}}\right\\}\right]$ and
$S_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta
J^{\prime}},\left\\{-\frac{1}{i}\frac{\delta}{\delta
J^{\prime}_{n}}\right\\}\right]\equiv\varepsilon^{\prime}\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta
J^{\prime}},\left\\{-\frac{1}{i}\frac{\delta}{\delta
J^{\prime}_{n}}\right\\}\right]$, where the dimensionless parameters
$\varepsilon$ and $\varepsilon^{\prime}$ characterize the system-bath coupling
strength that we expand the influence action about

$\displaystyle
S_{\mathrm{M,IF}}^{(2)}[X,X^{\prime}]\approx\left.S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}+\varepsilon\left.\frac{\delta}{\delta\varepsilon}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}+\varepsilon^{\prime}\left.\frac{\delta}{\delta\varepsilon^{\prime}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}$
$\displaystyle+\frac{\varepsilon^{2}}{2}\left.\frac{\delta^{2}}{\delta\varepsilon^{2}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}+\frac{\varepsilon^{\prime
2}}{2}\left.\frac{\delta^{2}}{\delta\varepsilon^{\prime
2}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}+{\varepsilon\varepsilon^{\prime}}\left.\frac{\delta^{2}}{\delta\varepsilon\delta\varepsilon^{\prime}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}$
(69)

Let us consider the above expression term by term

  1. 1.

$\left.S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}=0$,
this can be understood as the influence action corresponding to a non-
interacting bath, which trivially vanishes.

  2. 2.

$\begin{aligned}
\left.\frac{\delta}{\delta\varepsilon}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}&=\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf J}={\bf
J^{\prime}}=0}\end{aligned}$

  3. 3.

$\begin{aligned}
\left.\frac{\delta}{\delta\varepsilon^{\prime}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}&=-\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf
J}={\bf J^{\prime}}=0}\end{aligned}$

  4. 4.

$\begin{aligned}
\left.\frac{\delta^{2}}{\delta\varepsilon^{2}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}={i}\left(\left.\left[\left(\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right)^{2}\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right]\right|_{{\bf
J}={\bf
J^{\prime}}=0}-\left[\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf J}={\bf
J^{\prime}}=0}\right]^{2}\right)\end{aligned}$

  5. 5.

$\begin{aligned} \left.\frac{\delta^{2}}{\delta\varepsilon^{\prime
2}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}=i&\left(\left.\left[\left(\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right)^{2}\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right]\right|_{{\bf
J}={\bf J^{\prime}}=0}\right.\\\
&\left.-\left[\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf
J}={\bf J^{\prime}}=0}\right]^{2}\right)\end{aligned}$

  6. 6.

$\begin{aligned}
\left.\frac{\delta^{2}}{\delta\varepsilon\delta\varepsilon^{\prime}}S_{\mathrm{M,IF}}[X,X^{\prime}]\right|_{\varepsilon=\varepsilon^{\prime}=0}=&-i\left(\left.\left[\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right]\right|_{{\bf
J}={\bf J^{\prime}}=0}\right.\\\
&\left.-\left[\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf J}={\bf
J^{\prime}}=0}\right]\left[\left.\tilde{S}_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\mathcal{G}^{(1)}[\bf{J},\bf{J^{\prime}}]\right|_{{\bf
J}={\bf J^{\prime}}=0}\right]\right)\end{aligned}$

Putting all the terms together we can rewrite the influence action up to
second order in the system-bath coupling as in Eq (III.2).

##  Appendix C Calculation of averages

###  C.1 First order average

The first order average in the influence action can be calculated as follows

$\displaystyle\left\langle
S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{J}},\left\\{\frac{1}{i}\frac{\delta}{\delta{J_{n}}}\right\\}\right]\right\rangle_{0}$
$\displaystyle=\left.S_{\mathrm{int}}^{(+)}\left[{X},\frac{1}{i}\frac{\delta}{\delta{J}},\left\\{\frac{1}{i}\frac{\delta}{\delta{J_{n}}}\right\\}\right]\mathcal{G}^{(1)}[{J},{J^{\prime}},\left\\{J_{n},J_{n}^{\prime}\right\\}]\right|_{{\bf
J}={\bf J^{\prime}}=0}$
$\displaystyle=\left.\int_{0}^{t}\mathrm{d}\tau\sum_{n}\lambda\omega_{n}\left[\frac{1}{i}\frac{\delta}{\delta{J_{n}(\tau)}}\right]\left[\frac{1}{i}\frac{\delta}{\delta{J(\tau)}}\right]{X}(\tau)\prod_{p}F^{(1)}_{p}[J_{p},J_{p}^{\prime}]F^{(1)}_{I}[J,J^{\prime}]\right|_{{\bf
J}={\bf J^{\prime}}=0}$ (70)

where we have used the generating functional as defined in Eq. (26), and the
interaction action from Eq. (10). Now one can note that the first order
derivative of the influence action which is quadratic in
$\left\\{J_{n},J_{n}^{\prime}\right\\}$ brings in a factor that is linear in
$\left\\{J_{n},J_{n}^{\prime}\right\\}$, and thus vanishes when we set
$J_{n}=J_{n}^{\prime}=0$. Thus, we obtain that

$\displaystyle\left\langle
S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{J}},\left\\{\frac{1}{i}\frac{\delta}{\delta{J_{n}}}\right\\}\right]\right\rangle_{0}=\left\langle
S_{\mathrm{int}}^{(+)}\left[X^{\prime},\frac{1}{i}\frac{\delta}{\delta{J^{\prime}}},\left\\{\frac{1}{i}\frac{\delta}{\delta{J^{\prime}_{n}}}\right\\}\right]\right\rangle_{0}=0.$
(71)

###  C.2 Second order averages

We calculate the second order terms in the influence action as follows:

  1. 1.

$\displaystyle\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\\}^{2}\right\rangle_{0}=$
$\displaystyle\left(\int_{0}^{t}\mathrm{d}t_{1}\sum_{n}\lambda\omega_{n}\left[\frac{1}{i}\frac{\delta}{\delta{J(t_{1})}}\right]\left[\frac{1}{i}\frac{\delta}{\delta{J_{n}(t_{1})}}\right]X(t_{1})\right)$
$\displaystyle\left(\int_{0}^{t}\mathrm{d}t_{2}\sum_{m}\lambda
k_{m}\left[\frac{1}{i}\frac{\delta}{\delta{J{}(t_{2})}}\right]\left[\frac{1}{i}\frac{\delta}{\delta{J_{m}(t_{2})}}\right]X(t_{2})\right)\left.F^{(1)}_{I}[J,J^{\prime}]\,\prod_{p}F^{(1)}_{p}[J_{p},J_{p}^{\prime}]\right|_{{\bf
J}={\bf J^{\prime}}=0}$ (72)
$\displaystyle=\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{m,n}\lambda^{2}\omega_{m}\omega_{n}X(t_{1})X(t_{2})\,\zeta_{I}\left(t_{1},t_{2}\right)\,\sum_{p}\zeta^{(p)}_{m,n}\left(t_{1},t_{2}\right)$
(73)

where we have defined the influence of the (+) bath and the IDF as

$\displaystyle\zeta^{(p)}_{m,n}(t_{1},t_{2})\equiv$
$\displaystyle\left.\frac{\delta}{\delta
J_{n}(t_{1})}\left\\{\frac{\delta}{\delta
J_{m}(t_{2})}\left[F^{(1)}_{p}[J_{p},J^{\prime}_{p}]\right]\right\\}\right|_{J_{p}=J^{\prime}_{p}=0}$
$\displaystyle=$
$\displaystyle-{i}\delta_{mp}\delta_{np}\left[\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)+\eta_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
$\displaystyle-\delta_{mp}\delta_{np}\left[\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
(74) $\displaystyle=$
$\displaystyle\delta_{mp}\delta_{np}\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)-\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right]$
(75) $\displaystyle\zeta_{I}(t_{1},t_{2})\equiv$
$\displaystyle\frac{\delta}{\delta J(t_{1})}\left\\{\frac{\delta}{\delta
J(t_{2})}\left[F^{(1)}_{I}[J,J^{\prime}]\right]\right\\}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{t}\mathrm{d}\tau_{2}\left({G}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(t_{2}-\tau_{2}\right)+{G}\left(t_{2}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(t_{1}-\tau_{2}\right)\right)\right]$
$\displaystyle-\frac{i}{2}\left[{G}\left(t_{1}-t_{2}\right)+{G}\left(t_{2}-t_{1}\right)\right]-\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle.$ (76)

Let us further define an odd function ${g}\left(t\right)$ such that
${G}\left(t_{1}-t_{2}\right)\equiv{g}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)$,
and

$\displaystyle\nu_{GG}\left(t_{1},t_{2}\right)\equiv$
$\displaystyle\int_{0}^{t_{1}}\mathrm{d}\tau_{1}\int_{0}^{t_{2}}\mathrm{d}\tau_{2}\left[{g}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){g}\left(t_{2}-\tau_{2}\right)\right],$
(77)

such that we can rewrite $\zeta_{I}\left(t_{1},t_{2}\right)$ as

$\displaystyle\zeta_{I}\left(t_{1},t_{2}\right)=-\nu_{GG}\left(t_{1},t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)-\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle.$ (78)

We have made use of the fact that the function
$g\left(s\right)=-g\left(-s\right)$ is odd, and the noise kernel
$\nu^{(+)}\left(s\right)=\nu^{(+)}\left(-s\right)$ is even.

This allows us to rewrite Eq.(73) as

$\displaystyle\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\\}^{2}\right\rangle_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X(t_{1})X(t_{2})\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)-\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right]$
$\displaystyle\left[-\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right]$
(79) $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X(t_{1})X(t_{2})\left[i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.$
$\displaystyle\left.-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right],$
(80)

where
$\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\equiv\nu_{GG}\left(t_{1},t_{2}\right)+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$.

  2. 2.

$\displaystyle\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\\}^{2}\right\rangle_{0}=\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{m,n}\lambda^{2}\omega_{m}\omega_{n}X^{\prime}(t_{1})X^{\prime}(t_{2})\,\widehat{\zeta}_{I}\left(t_{1},t_{2}\right)\,\sum_{p}\widehat{\zeta}^{(p)}_{m,n}\left(t_{1},t_{2}\right),$
(81)

where

$\displaystyle\widehat{\zeta}^{(p)}_{m,n}(t_{1},t_{2})\equiv$
$\displaystyle\left.\frac{\delta}{\delta
J^{\prime}_{n}(t_{1})}\left\\{\frac{\delta}{\delta
J^{\prime}_{m}(t_{2})}\left[F^{(1)}_{p}[J_{p},J^{\prime}_{p}]\right]\right\\}\right|_{J_{p}=J^{\prime}_{p}=0}$
$\displaystyle=$
$\displaystyle{i}\delta_{mp}\delta_{np}\left[\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)+\eta_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
$\displaystyle-\delta_{mp}\delta_{np}\left[\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
(82) $\displaystyle=$
$\displaystyle\delta_{mp}\delta_{np}\left[i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)-\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right]$
(83) $\displaystyle\widehat{\zeta}_{I}(t_{1},t_{2})\equiv$
$\displaystyle\frac{\delta}{\delta
J^{\prime}(t_{1})}\left\\{\frac{\delta}{\delta
J^{\prime}(t_{2})}\left[F^{(1)}_{I}[J,J^{\prime}]\right]\right\\}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{t}\mathrm{d}\tau_{2}\left({G}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(\tau_{2}-t_{2}\right)+{G}\left(t_{2}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(\tau_{2}-t_{1}\right)\right)\right]$
$\displaystyle+\frac{i}{2}\left[{G}\left(t_{1}-t_{2}\right)+{G}\left(t_{2}-t_{1}\right)\right]-\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ (84)
$\displaystyle=$
$\displaystyle-\nu^{\prime}_{GG}\left(t_{1},t_{2}\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)$
(85)

Substituting back in Eq.(81)

$\displaystyle\left\langle\left\\{S_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]\right\\}^{2}\right\rangle_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X^{\prime}(t_{1})X^{\prime}(t_{2})\left[i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)-\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right]$
$\displaystyle\left[-\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right]$
(86) $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X^{\prime}(t_{1})X^{\prime}(t_{2})\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.$
$\displaystyle\left.-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right].$
(87)

  3. 3.

$\displaystyle\left\langle
S_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\rangle_{0}=\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{m,n}\lambda^{2}\omega_{m}\omega_{n}X(t_{1})X^{\prime}(t_{2})\,\widetilde{\zeta}_{I}\left(t_{1},t_{2}\right)\,\sum_{p}\widetilde{\zeta}^{(p)}_{m,n}\left(t_{1},t_{2}\right)$
(88)

where

$\displaystyle\widetilde{\zeta}^{(p)}_{m,n}(t_{1},t_{2})\equiv$
$\displaystyle\left.\frac{\delta}{\delta
J_{n}(t_{1})}\left\\{\frac{\delta}{\delta
J^{\prime}_{m}(t_{2})}\left[F^{(1)}_{p}[J_{p},J^{\prime}_{p}]\right]\right\\}\right|_{J_{p}=J^{\prime}_{p}=0}$
$\displaystyle=$
$\displaystyle-i\delta_{mp}\delta_{np}\left[\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)-\eta_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
$\displaystyle+\delta_{mp}\delta_{np}\left[\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\Theta\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{2}-t_{1}\right)\Theta\left(t_{2}-t_{1}\right)\right]$
(89) $\displaystyle=$
$\displaystyle\delta_{mp}\delta_{np}\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right],$
(90)

and

$\displaystyle\widetilde{\zeta}_{I}(t_{1},t_{2})\equiv$
$\displaystyle\frac{\delta}{\delta J(t_{1})}\left\\{\frac{\delta}{\delta
J^{\prime}(t_{2})}\left[F^{(1)}_{I}[J,J^{\prime}]\right]\right\\}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\int_{0}^{t}\mathrm{d}\tau_{1}\int_{0}^{t}\mathrm{d}\tau_{2}\left({G}\left(t_{1}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(\tau_{2}-t_{2}\right)+{G}\left(t_{2}-\tau_{1}\right)\nu^{(-)}\left(\tau_{1}-\tau_{2}\right){G}\left(\tau_{2}-t_{1}\right)\right)\right]$
$\displaystyle+\frac{i}{2}\left[{G}\left(t_{1}-t_{2}\right)-{G}\left(t_{2}-t_{1}\right)\right]+\left\llangle
Q_{h}\left(t_{1}\right)Q_{h}\left(t_{2}\right)\right\rrangle$ (91)
$\displaystyle=$
$\displaystyle\nu^{\prime}_{GG}\left(t_{1},t_{2}\right)-\frac{i}{2}g\left(t_{1}-t_{2}\right).$
(92)

Substituting the above in Eq.(88)

$\displaystyle\left\langle
S_{\mathrm{int}}^{(+)}\left[X^{\prime},-\frac{1}{i}\frac{\delta}{\delta{\bf
J^{\prime}}}\right]S_{\mathrm{int}}^{(+)}\left[X,\frac{1}{i}\frac{\delta}{\delta{\bf
J}}\right]\right\rangle_{0}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X(t_{1})X^{\prime}(t_{2})\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right]\left[\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+\frac{i}{2}g\left(t_{1}-t_{2}\right)\right]$
(93) $\displaystyle=$
$\displaystyle\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\sum_{p}\lambda^{2}k_{p}^{2}X(t_{1})X^{\prime}(t_{2})\left[-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.$
$\displaystyle\left.-\frac{1}{2}g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right].$
(94)

Putting together Eqs. (1), (2), and (3) in Eq.(III.2), we obtain the second
order influence action as

$\displaystyle S_{\mathrm{M,IF}}^{(2)}\left[X,X^{\prime}\right]$
$\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{p}\lambda^{2}k_{p}^{2}\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\left[X(t_{1})X(t_{2})\left\\{i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+X^{\prime}(t_{1})X^{\prime}(t_{2})\left\\{-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.-2X(t_{1})X^{\prime}(t_{2})\left\\{-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{2}g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right]$
(95) $\displaystyle=$
$\displaystyle\frac{i}{8}\sum_{p}\lambda^{2}k_{p}^{2}\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\left[\left\\{X_{\Sigma}(t_{1})+X_{\Delta}(t_{1})\right\\}\left\\{X_{\Sigma}(t_{2})+X_{\Delta}(t_{2})\right\\}\left\\{i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right.\right.$
$\displaystyle\left.\left.+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+\left\\{X_{\Sigma}(t_{1})-X_{\Delta}(t_{1})\right\\}\left\\{X_{\Sigma}(t_{2})-X_{\Delta}(t_{2})\right\\}\left\\{-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right.\right.$
$\displaystyle\left.\left.+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{1}{2}{g}\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\mathrm{sign}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.-2\left\\{X_{\Sigma}(t_{1})+X_{\Delta}(t_{1})\right\\}\left\\{X_{\Sigma}(t_{2})-X_{\Delta}(t_{2})\right\\}\left\\{-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{1}{2}g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right]$
(96) $\displaystyle=$
$\displaystyle\frac{i}{4}\sum_{p}\lambda^{2}k_{p}^{2}\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}$
$\displaystyle\left[X_{\Sigma}(t_{1})X_{\Sigma}(t_{2})\left\\{-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+X_{\Delta}(t_{1})X_{\Delta}(t_{2})\left\\{2\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)-i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right.\right.$
$\displaystyle\left.\left.-\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+X_{\Sigma}(t_{1})X_{\Delta}(t_{2})\left\\{i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\left(\mathrm{sign}\left(t_{1}-t_{2}\right)-1\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\left(\mathrm{sign}\left(t_{1}-t_{2}\right)-1\right)\right.\right.$
$\displaystyle\left.\left.+\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{1}{2}g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+X_{\Delta}(t_{1})X_{\Sigma}(t_{2})\left\\{i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\left(\mathrm{sign}\left(t_{1}-t_{2}\right)+1\right)+\frac{i}{2}{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\left(\mathrm{sign}\left(t_{1}-t_{2}\right)+1\right)\right.\right.$
$\displaystyle\left.\left.-\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-\frac{1}{2}g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right]$
(97) $\displaystyle=$
$\displaystyle\frac{i}{4}\sum_{p}\lambda^{2}k_{p}^{2}\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}\left[X_{\Delta}(t_{1})X_{\Delta}(t_{2})\left\\{2\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.+X_{\Delta}(t_{1})X_{\Sigma}(t_{2})\left\\{2i\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)+i{g}\left(t_{1}-t_{2}\right)\nu_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\Theta\left(t_{1}-t_{2}\right)\right],$
(98)

where we have defined $X_{\Sigma}\left(t\right)\equiv
X\left(t\right)+X^{\prime}\left(t\right)$, and $X_{\Delta}\left(t\right)\equiv
X\left(t\right)-X^{\prime}\left(t\right)$, and for the last line we have used
the property
$\int_{0}^{t}dt_{1}\int_{0}^{t}dt_{2}f\left(t_{1}\right)f\left(t_{2}\right)F\left(t_{1}-t_{2}\right)H\left(t_{1},t_{2}\right)=0$
provided that $F\left(t_{2}-t_{1}\right)=-F\left(t_{1}-t_{2}\right)$ and
$H\left(t_{2},t_{1}\right)=H\left(t_{1},t_{2}\right)$. Using the definitions
of the dissipation and noise kernels for the (+) bath as in Eq.(31) and Eq.
(32)

$\displaystyle
S_{\mathrm{M,IF}}^{(2)}=\int_{0}^{t}\mathrm{d}t_{1}\int_{0}^{t}\mathrm{d}t_{2}$
$\displaystyle\left[\frac{i}{4}X_{\Delta}(t_{1})X_{\Delta}(t_{2})\left\\{2\nu^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)-g\left(t_{1}-t_{2}\right)\eta_{p}^{(+)}\left(t_{1}-t_{2}\right)\right\\}\right.$
$\displaystyle\left.-\frac{1}{4}X_{\Delta}(t_{1})X_{\Sigma}(t_{2})\left\\{{g}\left(t_{1}-t_{2}\right)\nu^{(+)}\left(t_{1}-t_{2}\right)+2\eta^{(+)}\left(t_{1}-t_{2}\right)\nu_{GG}^{\prime}\left(t_{1},t_{2}\right)\right\\}\Theta\left(t_{1}-t_{2}\right)\right],$
(99)

which reduces to Eq. (27).

##  Appendix D Fourier transform of the dissipation and noise kernels

The Fourier transform of the dissipation and noise kernels corresponding to
the baths associated with the scalar field $\\{\eta^{(\pm)},\nu^{(\pm)}\\}$
are

$\displaystyle\bar{\eta}^{(\pm)}(\omega)=$ $\displaystyle i{\rm
Im}\left[\bar{\eta}^{(\pm)}(\omega)\right]=-i\frac{\pi}{2}\left[J^{(\pm)}(\omega)-J^{(\pm)}(-\omega)\right],$
(100) $\displaystyle\bar{\nu}^{(\pm)}(\omega)=$
$\displaystyle\pi\coth\left(\frac{\omega}{2k_{B}T_{F}}\right)\left[J^{(\pm)}(\omega)-J^{(\pm)}(-\omega)\right].$
(101)

From these expressions, we immediately obtain a fluctuation-dissipation
relation for the $\left(\pm\right)$ baths as in Eq.(44).

We now consider the kernel $\nu_{GG}(t_{1},t_{2})$ in the late-time limit
(Eq.(43)) as follows:

$\displaystyle\nu_{GG}(t_{1},t_{2})=\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}\bar{\nu}^{(-)}\left(\omega\right)\int_{0}^{t_{1}}\mathrm{d}\tau_{1}~{}G\left(t_{1}-\tau_{1}\right)e^{-i\omega\tau_{1}}\int_{0}^{t_{2}}\mathrm{d}\tau_{2}~{}G\left(t_{2}-\tau_{2}\right)e^{i\omega\tau_{2}},$
(102)

where we have written $\nu^{(-)}(t)$ in terms of its Fourier transform. The
time integrals on a finite interval are convolutions that can be re-cast in
terms of the Laplace transform of $G$ (defined as
$\tilde{F}(z)=\int_{0}^{+\infty}{dt}e^{-zt}F(t)$) via its inverse as:

$\displaystyle\int_{0}^{t}\mathrm{d}\tau~{}G\left(t-\tau\right)e^{\pm
i\omega\tau}=\int_{l-i\infty}^{l+i\infty}\frac{dz}{2\pi
i}~{}e^{zt}\frac{\tilde{G}(z)}{(z\mp i\omega)},$ (103)

where $l$ has to be larger than the real parts of each of the poles of
$\tilde{G}$ to ensure causality. Specifically, as $\tilde{G}$ has poles
$\\{z_{k}\\}$ with negative or vanishing real parts (${\rm Re}(z_{k})<0$),
implying $l>0$. Using Cauchy’s theorem, the causal property of the retarded
propagator allow us to express the convolution in terms of a sum of the
residue of the poles of $\tilde{G}(z)/(z\mp i\omega)$ as:

$\displaystyle\int_{0}^{t}\mathrm{d}\tau~{}G\left(t-\tau\right)e^{\pm
i\omega\tau}=e^{\pm i\omega
t}\bar{G}(\mp\omega)+\sum_{k}e^{z_{k}t}\frac{\mathcal{R}_{k}}{(z_{k}\mp
i\omega)},$ (104)

with $\mathcal{R}_{k}\equiv{\rm Res}[\tilde{G}(z),z_{k}]$ and where we have
used the relation between the Fourier and Laplace transforms of causal
functions $\tilde{G}(\pm i\omega)=\bar{G}(\mp\omega)$.

Considering that ${\rm Re}(z_{k})<0$, it is easy to realize that in the large-
time limit $t\rightarrow+\infty$:

$\displaystyle\int_{0}^{t}\mathrm{d}\tau~{}G\left(t-\tau\right)e^{\pm
i\omega\tau}\xrightarrow{t\rightarrow+\infty}e^{\pm i\omega
t}\bar{G}(\mp\omega),$ (105)

which immediately allow us to obtain for the long-time limit of Eq.(102):

$\displaystyle\nu_{GG}(t_{1},t_{2})\rightarrow\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}e^{-i\omega(t_{1}-t_{2})}\left|\bar{G}(\omega)\right|^{2}\bar{\nu}^{(-)}\left(\omega\right),$
(106)

where we have used that
$\bar{G}\left(-\omega\right)=\bar{G}^{*}\left(\omega\right)$ given that $G$ is
real. Similarly, we note that the last expression is a function of
$t_{1}-t_{2}$ only, which is generally not the case during the course of the
evolution. In the late-time limit, we can define the Fourier transform of the
kernel as
$\bar{\nu}_{GG}(\omega)\equiv|\bar{G}\left(\omega\right)|^{2}\bar{\nu}^{(-)}\left(\omega\right)$.
We note that $\bar{\nu}_{GG}(\omega)$ is real and even, as expected.

Furthermore, considering Eq.(67), we have that the Fourier transform of the
retarded propagator reads:

$\displaystyle\bar{G}(\omega)=\frac{1}{m\omega_{0}^{2}-m\omega^{2}+2\bar{\eta}^{(-)}(\omega)}.$
(107)

Noticing that $\bar{\eta}^{(-)}(\omega)$ has real and imaginary parts, we
leave the real part as a renormalization of the frequency of the center of
mass, while the imaginary part accounts for the dissipation. After
implementing this, we consider the renormalized frequency
$\omega_{R}^{2}\equiv\omega_{0}^{2}+2{\rm Re}[\bar{\eta}^{(-)}(\omega)]/m$, so
we can re-cast the Fourier transform of the retarded propagator as:

$\displaystyle\bar{G}(\omega)=\frac{1}{m\omega_{R}^{2}-m\omega^{2}+2i{\rm
Im}[\bar{\eta}^{(-)}(\omega)]},$ (108)

from which we obtain that ${\rm
Im}[\bar{G}(\omega)]=-2|\bar{G}(\omega)|^{2}{\rm
Im}[\bar{\eta}^{(-)}(\omega)]$. Combining this with the fluctuation-
dissipation relation for the bath $(-)$ of Eq.(44), we can directly arrive at
the fluctuation-dissipation relation between
$\bar{\nu}_{GG}\left(\omega\right)$ and $\bar{G}$ as in Eq. (45).

## References

  * (1) H.-P. Breuer, and F. Petruccione, Theory of open quantum systems (Oxford University Press, New York, 2002).
  * (2) U. Weiss, Quantum Dissipative Systems (World Scientific, 1992).
  * (3) A. A. Clerk, K. W. Lehnert, P. Bertet, J. R. Petta, and Y. Nakamura, Hybrid quantum systems with circuit quantum electrodynamics,  Nat. Phys. 16, 257 (2020).
  * (4) B. L. Hu, J. P. Paz, and Y. Zhang, Quantum Brownian motion in a general environment. II. Nonlinear coupling and perturbative approach, Phys. Rev. D 47, 1576 (1993).
  * (5) M. F. Maghrebi, M. Krüger, and M. Kardar, Flight of a heavy particle nonlinearly coupled to a quantum bath,  Phys. Rev. B 93, 014309 (2016).
  * (6) A. Lampo, S. H. Lim, J. Wehr, P. Massignan, and M. Lewenstein, Lindblad model of quantum Brownian motion, Phys. Rev. A 94, 042123 (2016).
  * (7) S. Stenholm, The semiclassical theory of laser cooling,  Rev. Mod. Phys. 58, 699 (1986).
  * (8) F. Haake, Systematic adiabatic elimination for stochastic processes, Z. Physik B - Condensed Matter 48, 31 (1982).
  * (9) Y. Lan, T. C. Elston, and G. A. Papoian, Elimination of fast variables in chemical Langevin equations, J. Chem. Phys. 129, 214115 (2008).
  * (10) Carl E. Wieman, D. E. Pritchard, and D. J. Wineland, Atom cooling, trapping, and quantum manipulation, Rev. Mod. Phys. 71, S253 (1999).
  * (11) Z.-Q. Yin, A. A. Geraci, and T. Li, Optomechanics of Levitated dielectric particles,  Int. J. Mod. Phys. B 27, 1330018 (2013).
  * (12) Y. Y. Fein, P. Geyer, P. Zwick, F. Kiałka, S. Pedalino, M. Mayor, S. Gerlich, and M. Arndt, Quantum superposition of molecules beyond 25 kDa, Nat. Phys. 15, 1242 (2019).
  * (13) K. Sinha, A Microscopic Model for Quantum Optomechanics,  Doctoral Thesis (2015).
  * (14) C. R. Galley, R. O. Behunin, and B. L. Hu, Oscillator-field model of moving mirrors in quantum optomechanics,  Phys. Rev. A 87, 043832(2013).
  * (15) K. Sinha, S.-Y. Lin, and B. L. Hu, Mirror-field entanglement in a microscopic model for quantum optomechanics,  Phys. Rev. A 92, 023852 (2015).
  * (16) Q. Wang and W. G. Unruh, Motion of a mirror under infinitely fluctuating quantum vacuum stress,  Phys. Rev. D 89, 085009 (2014).
  * (17) T. A. Brun and L. Mlodinow, Decoherence by coupling to internal vibrational modes,  Phys. Rev. A 94, 052123 (2016).
  * (18) A. E. Rubio López and O. Romero-Isart, Radiation Reaction of a Jiggling Dipole in a Quantum Electromagnetic Field,  Phys. Rev. Lett. 123, 243603 (2019).
  * (19) B. L. Hu and A. Matacz, Quantum Brownian motion in a bath of parametric oscillators: A model for system-field interactions,  Phys. Rev. D 49, 6612 (1994).
  * (20) P. Meystre and M. Sargent, Elements of Quantum Optics  (Springer-Verlag, Berlin, 2007).
  * (21) P. W. Milonni, An introduction to quantum optics and quantum fluctuations (Oxford University Press, 2019).
  * (22) A. Einstein and L. Hopf, Statistische Untersuchung der Bewegung eines Resonators in einem Strahlungsfeld,  Ann. Phys. 33, 1105 (1910).
  * (23) P. W. Milonni, Quantum mechanics of the Einstein-Hopf model,  Am. J. Phys. 49, 177 (1981).
  * (24) R. P. Feynman and F. L. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Ann. Phys. (N.Y.), 24, 118 (1963).
  * (25) E.A. Calzetta and B.-L. Hu, Nonequilibrium Quantum Field Theory (Cambridge University Press, Cambridge, England, 2008).
  * (26) B. L. Hu, J. P. Paz, and Y. Zhang, Quantum Brownian motion in a general environment: Exact master equation with nonlocal dissipation and colored noise, Phys. Rev. D 45, 2843 (1992).
  * (27) R. O. Behunin and B.-L. Hu, Nonequilibrium forces between atoms and dielectrics mediated by a quantum field, Phys. Rev. A 84, 012902 (2011).
  * (28) E. Calzetta, A. Roura, and E. Verdaguer,Stochastic description for open quantum systems,  Physica A 319, 188 (2003).
  * (29) K. Sinha and Y. Subaşı, Quantum Brownian motion of a particle from Casimir-Polder interactions, Phys. Rev. A 101, 032507 (2020).
  * (30) H. Pino, J. Prat-Camps, K. Sinha, B. P. Venkatesh, and O. Romero-Isart, On-chip quantum interference of a superconducting microsphere,  Quantum Sci. Technol. 3, 025001 (2018).
  * (31) J.-T. Hsiang and B. L. Hu, Fluctuation-dissipation relation from the nonequilibrium dynamics of a nonlinear open quantum system, Phys. Rev. D 101, 125003 (2020).
  * (32) V. Yu. Argonov and S. V. Prants, Synchronization of internal and external degrees of freedom of atoms in a standing laser wave,  Phys. Rev. A 71, 053408 (2005).
  * (33) A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Laser Cooling below the One-Photon Recoil Energy by Velocity-Selective Coherent Population Trapping,  Phys. Rev. Lett. 61, 826 (1988).
  * (34) A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis,  J. Opt. Soc. Am. B 6, 2112 (1989).
  * (35) M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, J. Ye, Optical Atomic Coherence at the 1-Second Time Scale, Science 314, 1430 (2006).
  * (36) S. Machluf, J. Coslovsky, P. G. Petrov, Y. Japha, and R. Folman, Coupling between Internal Spin Dynamics and External Degrees of Freedom in the Presence of Colored Noise, Phys. Rev. Lett. 105, 203002 (2010).
  * (37) S.-Y. Lin, Unruh-DeWitt detectors as mirrors: Dynamical reflectivity and Casimir effect,  Phys. Rev. D 98, 105010 (2018).
  * (38) S. Butera and I. Carusotto, Mechanical backreaction effect of the dynamical Casimir emission,  Phys. Rev. A 99, 053815 (2019).
  * (39) M. Belén Farías, C. D. Fosco, Fernando C. Lombardo, and Francisco D. Mazzitelli, Motion induced radiation and quantum friction for a moving atom, Phys. Rev. D 100, 036013 (2019).
  * (40) A. E. Rubio López, C. Gonzalez-Ballestero, and O. Romero-Isart, Internal quantum dynamics of a nanoparticle in a thermal electromagnetic field: A minimal model, Phys. Rev. B 98, 155405 (2018).