#  Interplay between optomechanics and the dynamical Casimir effect

Alessandro Ferreri ID Institute for Quantum Computing Analytics (PGI-12),
Forschungszentrum Jülich, 52425 Jülich, Germany  Hannes Pfeifer ID Institute
of Applied Physics, University of Bonn, 53115 Bonn, Germany  Frank K. Wilhelm
Institute for Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich,
52425 Jülich, Germany  Sebastian Hofferberth ID Institute of Applied Physics,
University of Bonn, 53115 Bonn, Germany  David Edward Bruschi ID Institute for
Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich, 52425 Jülich,
Germany

###### Abstract

We develop a model of a quantum field confined within a cavity with a movable
wall where the position of the wall is quantized. We obtain a full description
of the dynamics of both the quantum field and the confining wall depending on
the initial state of the whole system. Both the reaction and back-reaction of
the field on the wall, and the wall on the field, can be taken into account,
as well as external driving forces on both the cavity and the wall. The model
exactly reproduces the resonant cavity mode stimulation due to the periodic
motion of the mirror (dynamical Casimir effect), as well as the standard
radiation pressure effects on the quantized wall (optomechanics). The model
also accounts for the interplay of the two scenarios. Finally, the time
evolution of the radiation force shows the interplay between static and
dynamical Casimir effect.

##  I Introduction

Optomechanics is the study of the light-matter interaction, which focuses its
effort on the study of the interaction between the modes of the
electromagnetic field and the vibrational modes of macroscopic bodies, such as
reflective mirrors [1, 2, 3, 4]. The employment of optomechanical systems for
technological purposes [5], such as laser cooling [6, 7, 8, 9, 10] and quantum
sensing [11, 12], as well as the possibility to involve macroscopic objects in
the investigation of quantum physics, such as studies of entanglement [13,
14], has lead to a growing interest in this field throughout the last two
decades [15, 16, 17].

The optomechanical coupling arises due to a nonlinear interaction wherein the
excitation (or relaxation) of the vibrational mode of the mirror is determined
by the radiation pressure induced by the presence of photons within the
cavity. The most common approach to the analysis of optomechanical
interactions treats the mirror and the electromagnetic field as two
independent quantum harmonic oscillators [18, 19, 20]. In this approach,
modelling the electromagnetic field as a single harmonic oscillator turns out
to be extremely helpful for the description of the effects of cavity losses
[21]. Moreover, it can be employed in order to investigate phenomena of vacuum
squeezing acting on either the mechanical mode [22, 23] or the cavity field.
The latter case is known as the dynamical Casimir effect (DCE) [24], which can
be understood through the lens of optomechanics as the resonant exchange of
quantum excitations between the mechanical mode and one, or two, cavity modes
of the electromagnetic field [25, 26, 27, 28].

The single mode description of the field trapped in a cavity conveniently
simplifies the study of the system in many scenarios. Nevertheless, the
investigation of the multimode nature of the quantum field in the framework of
optomechanics leads to a more comprehensive interpretation of the interplay
between the quantum degrees of freedom of the wall and the quantum fluctuation
of the field [29]. Concretely, this translates into the quantization of the
position of the wall that is classically determined by Dirichlet boundary
conditions for the field [30, 31, 32], and subsequently the quantization of
the fluctuations of the length which become coupled to the creation and
annihilation operators of the cavity modes [29]. This additional quantum
degree of freedom can be exploited to investigate many interesting features of
this system, such as the correction to both the vacuum state and its energy of
the multimode cavity field caused by the motion of the wall [33, 34, 35].

In this work we propose an approach to extend the standard optomechanical
radiation-pressure coupling to include the mode-mixing and squeezing field
terms in the Hamiltonian of the system. We specialize our computations to a
real scalar field fulfilling static Dirichlet boundary conditions without loss
of generality. Our goal is achieved by adding a small fluctuation to the
length of the cavity, corresponding to the oscillation amplitude of the wall,
and then performing the spatial integration of the Hamiltonian density to
lowest order in the fluctuation length. This additional degree of freedom is
then quantized, and the resulting quantum Hamiltonian reproduces the standard
optomechanical one with the addition of all cavity modes [29].

We compute the time evolution of the whole system including possible time-
dependent external drives acting on two different modes of the cavity field,
as well as on the mechanical degree of freedom. Moreover, we consider an input
state that accounts for the effects of displacement, squeezing and thermal
fluctuation of the mechanical mode [36]. One of our most interesting results
is the excitation of the vacuum state of the cavity mode by means of a time-
dependent displacement acting on the mechanical degree of freedom. We
demonstrate that our model allows to formally recover the DCE as a result of a
time-dependent external drive, without necessarily introducing the dynamics
into the boundary conditions of the field. Furthermore, our approach allows us
not only to recover well-known results, such as the DCE caused by both the
resonant oscillating wall and thermal fluctuations of the mechanical state
[28], but also to unveil novel features, such as the DCE in the presence of
squeezed thermal input states for the mechanical mode, the impact of external
drives on the dynamics, as well as to estimate the Casimir force between the
two cavity walls throughout the dynamics.

The paper is structured as follows. In Section II we introduce the system of
interest. In particular, starting from the Lagrangian of the confined field,
we present the quantization process, as well as the time evolution formalism.
In Section III, we describe the dynamics of relevant quantities, such as the
position of the moving wall, the number of photons within the cavity and the
force between the two mirrors composing the cavity, showing the interplay
between the Casimir force and the time-dependent radiation pressure due to the
arising number of photons. In Section IV we discuss a list of possible
experimental realisations of our system as well as an extension of our model
to a scalar massive field. In Section V we include some considerations as well
as an outlook. Finally, we summarize in Section VI.

##  II Theoretical model

Here we introduce the model that will be used in this work. Standard
approaches to quantum fields confined in cavities can be found in the
literature [29].

###  II.1 Classical Lagrangian and Hamiltonian density

Let us consider a massless real scalar field $\phi(t,x)$ in $1+1$ dimensional
flat spacetime with coordinates $(t,x)$.111We use Einstein’s summation
convention. The classical Lagrangian for the field density is

$\displaystyle\mathcal{L}(t,x)=\frac{\hbar}{2}\partial_{\mu}\phi\partial^{\mu}\phi.$
(1)

The Lagrangian provides the classical field equation, i.e., the Klein Gordon
equation, which reads $\partial_{t}^{2}\phi-\partial_{x}^{2}\phi$=0. We then
impose the static Dirichlet boundary conditions $\phi(t,0)=\phi(t,L)=0$ to
implement the confinement of the field in a cavity of length $L$.

Since we are interested in the time evolution of the system, we introduce a
basis for the field modes
$\phi_{n}(t,x)=\sqrt{\frac{c}{\omega_{n}L}}e^{-i\omega_{n}t}\sin(\frac{n\pi}{L}x)$,
where $\omega_{n}:=\frac{n\pi c}{L}$ is the frequency of mode
$n\in\mathbb{N}$. The field can therefore be expanded as

$\displaystyle\phi(t,x)=$
$\displaystyle\sum_{n=1}\left[\alpha_{n}\,\phi_{n}(t,x)+\alpha_{n}^{*}\,\phi^{*}_{n}(t,x)\right],$
(2)

where the constants $\alpha_{n}$ are Fourier coefficients.

We can construct the Hamiltonian density
$\mathcal{H}:=\Pi\partial_{t}\phi-\mathcal{L}$ by firstly finding the
conjugate momentum $\Pi(t,x):=-\partial_{t}\phi(t,x)$, then by substituting
the expression of $\partial_{t}\phi$ in terms of the momentum into
$\mathcal{H}$, thereby obtaining
$\mathcal{H}(t,x)=\frac{1}{2}\left[\Pi^{2}(t,x)+(\partial_{x}\phi(t,x))^{2}\right]$.
Finally, in the Schrödinger picture the explicit expression in term of the
field expansion coefficients therefore reads

$\displaystyle\mathcal{H}=$
$\displaystyle-\frac{\hbar}{2L}\sum_{nm}\sqrt{\omega_{n}\omega_{m}}\left[\alpha_{n}-\alpha_{n}^{*}\right]\left[\alpha_{m}-\alpha_{m}^{*}\right]s_{n}(x)s_{m}(x)$
$\displaystyle+\frac{\hbar}{2L}\sum_{nm}\sqrt{\omega_{n}\omega_{m}}\left[\alpha_{n}+\alpha_{n}^{*}\right]\left[\alpha_{m}+\alpha_{m}^{*}\right]c_{n}(x)c_{m}(x)$

where we have introduced $s_{n}(x):=\sin(\frac{n\pi}{L}x)$ and
$c_{n}(x):=\cos(\frac{n\pi}{L}x)$ here only for the sake of presentation.

###  II.2 Quantization

So far we have dealt with a classical system. We now wish to quantize the
system, and in particular the Hamiltonian, with classical Hamiltonian density
(II.1). To obtain our goal we adopt the following four-step strategy:

  * i)

In Equation (II.1) we substitute $L\rightarrow L+\delta L$, where $\delta
L/L\ll 1$, and Taylor-expand up to the first order in $\delta L/L$;

  * ii)

We integrate the Hamiltonian density from 0 to $L$ and obtain the classical
Hamiltonian $H:=\int_{0}^{L}\mathcal{H}\,dx$;

  * iii)

We quantize both the field Fourier coefficients $\alpha_{n}$ and the position
of the wall $\delta L$:

$\displaystyle\alpha_{n}\rightarrow$ $\displaystyle\hat{a}_{n},$
$\displaystyle\alpha_{n}^{*}\rightarrow$ $\displaystyle\hat{a}^{\dagger}_{n},$
$\displaystyle\delta L\rightarrow$ $\displaystyle\delta
L_{0}(\hat{b}^{\dagger}+\hat{b}).$ (4)

Here $\delta L_{0}$ is a constant and represents the zero-point fluctuations
of the harmonic oscillator $\hat{b}$ (details on this topic are left to the
literature [4]), and the operators all satisfy the canonical commutation
relations $[\hat{a}_{n},\hat{a}_{n^{\prime}}^{\dagger}]=\delta_{nn^{\prime}}$
and $[\hat{b},\hat{b}^{\dagger}]=1$, while all other commutators vanish. We
have therefore assumed that the (small) displacement of one of the two cavity
walls is characterized by a harmonic degree of freedom with intrinsic
frequency $\omega$ and annihilation and creation operators $\hat{b}$ and
$\hat{b}^{\dagger}$ respectively.

  * iv)

Finally, we normal order the quantum Hamiltonian $\hat{H}$ that we obtained
this way and we introduce the dimensionless amplitude $\epsilon:=\delta
L_{0}/L\ll 1$ which will act as our perturbative parameter.

The system is pictorially illustrated in Figure 1.

Figure 1: Schematic picture of the system. A 1+1 dimensional scalar field is
confined in a cavity of length $L$ which possesses a movable wall. The
fundamental oscillation amplitude is proportional to the zero point
fluctuation $\delta L_{0}$ of the quantum harmonic oscillator.

The procedure described here allows us to obtain the total Hamiltonian
$\hat{H}(t)$ of the form

$\displaystyle\hat{H}(t)=$
$\displaystyle\hat{H}_{0}+\hat{H}_{\textrm{dr}}(t)+\epsilon\hat{H}_{\textrm{I}},$
(5)

where each term reads

$\displaystyle\hat{H}_{0}:=$
$\displaystyle\sum_{n}\hbar\omega_{n}\,\hat{a}_{n}^{\dagger}\hat{a}_{n}+\hbar\omega\hat{b}^{\dagger}\hat{b},$
$\displaystyle\hat{H}_{\textrm{dr}}(t):=$
$\displaystyle\hat{H}_{\textrm{dk}}(t)+\hat{H}_{\textrm{dk'}}(t)+\hat{H}_{\textrm{db}}(t),$
$\displaystyle\hat{H}_{\textrm{I}}:=$
$\displaystyle-2\sum_{n}\hbar\omega_{n}\,\hat{a}_{n}^{\dagger}\hat{a}_{n}\hat{X}_{b}-\sum_{n}\hbar\omega_{n}\,\left(\hat{a}_{n}^{{\dagger}2}+\hat{a}_{n}^{2}\right)\hat{X}_{b}$
$\displaystyle-4\sum_{\begin{subarray}{c}n,m\\\ n\neq
m\end{subarray}}(-1)^{n+m}\hbar\sqrt{\omega_{n}\omega_{m}}\,\hat{X}_{n}\hat{X}_{m}\hat{X}_{b}.$
(6)

Here, the added term $\hat{H}_{\textrm{dr}}(t)$ includes potential external
driving forces that can be applied to both the cavity field, for example to
the modes $k$ and $k^{\prime}$, and the mechanical mode $b$,
$\hat{H}_{\textrm{dj}}(t)=2\lambda_{xj}(t)\hat{X}_{j}+2\lambda_{pj}(t)\hat{P}_{j}$
with $j=k,k^{\prime}$ or $b$. We have also compacted our notation by
introducing the quadrature position and momentum operators
$\hat{X}_{b}=\frac{1}{2}(\hat{b}^{\dagger}+\hat{b})$,
$\hat{P}_{b}=\frac{i}{2}(\hat{b}^{\dagger}-\hat{b})$,
$\hat{X}_{n}=\frac{1}{2}(\hat{a}_{n}^{\dagger}+\hat{a}_{n})$, and
$\hat{P}_{n}=\frac{i}{2}(\hat{a}_{n}^{\dagger}-\hat{a}_{n})$.

We want to emphasize that we perform both the substitution and the Taylor
expansion with respect to $\delta L/L$ _before_ the spatial integration of the
Hamiltonian density. Interestingly, this approach spontaneously provides both
the standard optomechanical interaction Hamiltonian
$\hat{H}_{\textrm{OM}}=-2\epsilon\sum_{n}\omega_{n}\,\hat{a}_{n}^{\dagger}\hat{a}_{n}\hat{X}_{b}$
commonly found in the literature [4], and the cavity multimode mono- and two-
colour squeezing interaction Hamiltonian
$H_{\textrm{sq}}=-\sum_{n}\hbar\omega_{n}\,\left(\hat{a}_{n}^{{\dagger}2}+\hat{a}_{n}^{2}\right)\hat{X}_{b}-4\sum_{n\neq
m}(-1)^{n+m}\hbar\sqrt{\omega_{n}\omega_{m}}\,\hat{X}_{n}\hat{X}_{m}\hat{X}_{b}$,
which induces the dynamical Casimir effect [29, 33].

###  II.3 Time evolution operator

In the last section we derived the Hamiltonian of the system. In general, such
Hamiltonian can depend on time either due to the modulation of the coupling
constants $g_{n}:=\epsilon\omega_{n}$ (which has been considered in the
literature [4, 37]), or the presence of time-dependent external drives [38].
Here we do not consider coupling modulation and instead investigate the
dynamics of a system described by the Hamiltonian (5). The time evolution
operator $\hat{U}(t)$ induced by a time-dependent Hamiltonian $\hat{H}(t)$
reads

$\displaystyle\hat{U}(t)=\overset{\leftarrow}{\mathcal{T}}\exp\left[-\frac{i}{\hbar}\int_{0}^{t}\,dt^{\prime}\,\hat{H}(t^{\prime})\right],$
(7)

where $\overset{\leftarrow}{\mathcal{T}}$ stands for the time-ordering
operator.

It is not difficult to show that

$\displaystyle\hat{U}(t)=\hat{U}_{0}(t)\hat{U}_{\textrm{dr}}^{(+)}(t)\hat{U}_{\textrm{dr}}^{(-)}(t)\hat{U}_{\textrm{I}},$
(8)

modulo an overall complex phase that has no physical significance. This
expression is specified by the following unitary operators

$\displaystyle\hat{U}_{0}(t):=$
$\displaystyle\exp\left[-i/\hbar\,\hat{H}_{0}t\right],$ (9)
$\displaystyle\hat{U}_{\textrm{dr}}^{(+)}(t):=$
$\displaystyle\exp\left\\{-2i\left[\Lambda_{xb}(t)\hat{X}_{b}+\Lambda_{xk}(t)\hat{X}_{k}\right.\right.$
$\displaystyle\left.\left.+\Lambda_{xk^{\prime}}(t)\hat{X}_{k^{\prime}}\right]\right\\},$
(10) $\displaystyle\hat{U}_{\textrm{dr}}^{(-)}(t):=$
$\displaystyle\exp\left\\{-2i\left[\Lambda_{pb}(t)\hat{P}_{b}+\Lambda_{pk}(t)\hat{P}_{k}\right.\right.$
$\displaystyle\left.\left.+\Lambda_{pk^{\prime}}(t)\hat{P}_{k^{\prime}}\right]\right\\},$
(11) $\displaystyle\hat{U}_{\textrm{I}}(t):=$
$\displaystyle\overset{\leftarrow}{\mathcal{T}}e^{-\frac{i}{\hbar}\epsilon\int_{0}^{t}dt^{\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})}.$
(12)

We also need the definition of the auxiliary functions

$\displaystyle\Lambda_{pj}(t):=$
$\displaystyle\int_{0}^{t}dt^{\prime}\left(\lambda_{xj}(t^{\prime})\sin(\omega
t^{\prime})-\lambda_{pj}(t^{\prime})\cos(\omega t^{\prime})\right),$
$\displaystyle\Lambda_{xj}(t):=$
$\displaystyle\int_{0}^{t}dt^{\prime}\left(\lambda_{xj}(t^{\prime})\cos(\omega
t^{\prime})+\lambda_{pj}(t^{\prime})\sin(\omega t^{\prime})\right),$ (13)

and $j\equiv k,k^{\prime}$ or $b$, and we also have

$\hat{\tilde{H}}_{\textrm{I}}(t):=\hat{U}^{\dagger}_{\textrm{d}}(t)\hat{U}^{\dagger}_{0}(t)\hat{H}_{\textrm{I}}(t)\hat{U}_{0}(t)\hat{U}_{\textrm{d}}(t),$
(14)

with
$\hat{U}_{\textrm{d}}(t):=\hat{U}_{\textrm{dr}}^{(+)}(t)\hat{U}_{\textrm{dr}}^{(-)}(t)$.
The explicit expression of $\hat{\tilde{H}}_{\textrm{I}}(t)$ is cumbersome and
is provided in Appendix A.

###  II.4 Time evolution and the initial state

We are now able to determine the time evolution of the expectation value of
any observable $\hat{A}$. In the Heisenberg picture, this is given by

$\displaystyle
A(t)=\textrm{Tr}(\hat{U}^{\dagger}(t)\hat{A}(0)\hat{U}(t)\hat{\rho}(0)),$ (15)

where $\hat{\rho}(0)$ is the initial state of the total cavity field and wall
system. We choose to work with the following general initial state:

$\displaystyle\hat{\rho}(0)=\lvert\mu_{k}\rangle\langle\mu_{k}\rvert\otimes\lvert\mu_{k^{\prime}}\rangle\langle\mu_{k^{\prime}}\rvert\otimes\hat{\rho}^{\textrm{rest}}\otimes\hat{\rho}_{\textrm{m}}^{\textrm{(DST)}},$
(16)

where $\hat{\rho}^{\textrm{rest}}:=\prod_{n\neq k,k^{\prime}}\lvert
0_{n}\rangle\langle 0_{n}\rvert$ is the vacuum state of all modes except two
modes $k,k^{\prime}$. Concretely, we have assumed that two specific cavity
modes $k,k^{\prime}$ are initially found in a coherent state defined by
$\hat{a}_{k}|\mu_{k}\rangle=\mu_{k}|\mu_{k}\rangle$ and with coherent
parameter $\mu_{k}\in\mathbb{R}$ (analogously for $k^{\prime}$), whereas the
mechanical quantum mode of the wall is prepared in a displaced squeezed
thermal (DST) state
$\hat{\rho}_{\textrm{m}}^{\textrm{(DST)}}=D(\beta)S(\zeta)\rho_{m}(T)S^{\dagger}(\zeta)D^{\dagger}(\beta)$,
where T is the temperature, and $\beta=\lvert\beta\rvert e^{i\theta}$ and
$\zeta=re^{i\phi}$ are the coherent and the squeezing parameter of the
mechanical degree of freedom respectively [36].

The total number of excitations at $t=0$ is therefore
$N_{tot}(0)=N_{k}(0)+N_{k^{\prime}}(0)+N_{b}(0)$, where
$N_{k}(0)=\mu_{k}^{2}$, $N_{k^{\prime}}(0)=\mu_{k^{\prime}}^{2}$, and

$\displaystyle N_{b}(0)=\lvert\beta\rvert^{2}+\sinh^{2}r+N_{T}\cosh(2r),$ (17)

with $N_{T}=\sinh^{2}(r_{\textrm{T}})$ and the parameter $r_{\textrm{T}}$
defined by $\tanh
r_{\textrm{T}}:=\exp[-\frac{\hbar\omega}{2k_{\textrm{B}}T}]$.

Concretely, we can compute the time evolution of $\hat{A}$ by Taylor expanding
the unitary operator $U_{I}(t)$ up to the second order in $\epsilon$,

$\displaystyle U_{I}(t)\simeq$ $\displaystyle
1-\frac{i\epsilon}{\hbar}\int_{0}^{t}dt^{\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})-\frac{\epsilon^{2}}{\hbar^{2}}\int_{0}^{t}dt^{\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})\int_{0}^{t^{\prime}}dt^{\prime\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime\prime}).$
(18)

From now on, we assume that all quantities are truncated at second order.

Following this procedure we obtain

$\displaystyle A(t)\simeq$ $\displaystyle
A^{(0)}(t)+\frac{i}{\hbar}\epsilon\Delta
A^{(1)}(t)+\frac{1}{\hbar^{2}}\epsilon^{2}\Delta A^{(2)}(t),$ (19)

where each term has the expression

$\displaystyle A^{(0)}(t)=$
$\displaystyle\textrm{Tr}(\hat{\tilde{A}}(t)\hat{\rho}(0)),$ $\displaystyle
A^{(1)}(t)=$
$\displaystyle\textrm{Tr}\left(\left[\int_{0}^{t}dt^{\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime}),\hat{\tilde{A}}(t)\right]\hat{\rho}(0)\right),$
$\displaystyle A^{(2)}(t)=$
$\displaystyle-\textrm{Tr}\left(\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime\prime})\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})\hat{\tilde{A}}(t)\hat{\rho}(0)\right)$
$\displaystyle-\textrm{Tr}\left(\int_{0}^{t}dt^{\prime}\int_{0}^{t^{\prime}}dt^{\prime\prime}\hat{\tilde{A}}(t)\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})\hat{\tilde{H}}_{\textrm{I}}(t^{\prime\prime})\hat{\rho}(0)\right)$
$\displaystyle+\textrm{Tr}\left(\int_{0}^{t}dt^{\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime})\hat{\tilde{A}}(t)\int_{0}^{t}dt^{\prime\prime}\hat{\tilde{H}}_{\textrm{I}}(t^{\prime\prime})\hat{\rho}(0)\right),$
(20)

and where
$\hat{\tilde{A}}(t):=\hat{U}_{\textrm{d}}^{\dagger}(t)\hat{U}_{0}^{\dagger}(t)\hat{A}(t)\hat{U}_{0}(t)\hat{U}_{\textrm{d}}(t)$.

###  II.5 Time average of measurable quantities

We anticipate that the observables computed in our work will depend on time
through complicated oscillatory functions. While this functional dependence on
time can apparently obfuscate the physical significance, we can extract
meaningful significance by taking the time average of such quantities. This is
particularly useful because the time average in general eliminates
oscillations around mean values.

Given the expectation vale $A(t)$ of an observable $\hat{A}(t)$, we define the
time average $\langle A\rangle_{\tau}$ of the expectation value $A(t)$ as

$\displaystyle\langle A\rangle_{\tau}:=\frac{1}{\tau}\int_{0}^{\tau}dt\,A(t).$
(21)

These time averaged quantities will be computed case by case below.

##  III Dynamics of the system

In the following analysis, we focus our investigation on three quantities: the
position of the oscillating wall, the number of photon for the single mode k,
and the radiation pressure within the cavity.

###  III.1 Dynamics of the wall

We start by studying the dynamics of the wall. We are therefore interested in
calculating the average position $x(t)$ of the wall at time t. This is given
by

$\displaystyle x(t)=L+\delta
L_{0}\textrm{Tr}(\hat{U}^{\dagger}(t)\hat{(}\hat{b}+\hat{b}^{\dagger})\hat{U}(t)\hat{\rho}(0)),$
(22)

which has the general expression
$x(t)=x^{(0)}(1+\epsilon\tilde{x}^{(1)}+\epsilon^{2}\tilde{x}^{(2)})$. Notice
here that, since the length has a dimension, it must be rescaled in order to
have dimensionless perturbative contributions. In the following we consider
the terms $\tilde{x}^{(n)}$ as the rescaled corrections to the zero order. In
our frame of reference, the second wall, which remains static, is assumed to
be located at $x=0$.

####  III.1.1 Zero order: $x^{(0)}(t)$

We are now able to investigate the dynamics of the wall starting from the zero
order contribution. It is immediate to show that the trajectory of the movable
mirror at lowest order is

$\displaystyle x^{(0)}(t)=$ $\displaystyle L.$ (23)

This is expected and, for our purposes, not of particular significance.

####  III.1.2 First order: $\tilde{x}^{(1)}(t)$

We continue by looking at the first order, i.e., the terms that contribute as
a consequence of the back reaction of the field on the cavity wall. It is easy
to find

$\displaystyle\tilde{x}^{(1)}(t)=$ $\displaystyle
2\lvert\beta\rvert\cos(\omega t-\theta)+2\xi(t),$ (24)

where the function $\xi(t)$ is defined in Appendix A.

The dynamics is triggered by two terms. In particular, the first one accounts
for the harmonic oscillation of the mirror, and directly stems from the
quantization of its mechanical motion; indeed, the oscillation amplitude is
strictly connected to the number of phonons due to the factor
$\lvert\beta\rvert$, whereas both the initial position and velocity are
established by the coherent phase $\theta$, which also determines the
orientation of the phonon momentum. There are a few interesting values that
the phase can assume. For instance, if $\theta=0$ (or $\pi$), the mirror is
initially pushed (or pulled) at maximal amplitude $\delta
L_{0}\lvert\beta\rvert$ and the canonical momentum of phonons is zero. If
$\theta=\pm\pi/2$ instead, the mirror initially moves at maximal velocity, and
the phonons have positive ($\theta=+\pi/2$) or negative ($\theta=-\pi/2$)
momentum. The second term is the contribution of the mechanical drive to the
motion of the wall, and its explicit expression depends on our choice of
$\lambda_{xb}(t)$ and $\lambda_{pb}(t)$.

Let us define the external drive as follows:

$\displaystyle\lambda_{xb}(t)=$ $\displaystyle-\frac{g\Omega}{2}e^{-\Omega
t}\cos(\omega t),$ $\displaystyle\lambda_{pb}(t)=$
$\displaystyle-\frac{g\Omega}{2}e^{-\Omega t}\sin(\omega t),$ (25)

where $g$ is the coupling constant of the external drive and $\Omega$ has
units of frequency. Accordingly, the equation of motion (24) becomes

$\displaystyle\tilde{x}^{(1)}(t)=$ $\displaystyle
2\lvert\beta\rvert\cos(\omega t-\theta)+g\sin\left(\omega
t\right)\left(1-e^{-\Omega t}\right).$ (26)

The specific choice of the external drive generates a sinusoidal modulation of
the wall when $\Omega t\gg 1$; moreover, it ensures both
$\tilde{x}^{(1)}(0)=0$ and $\dot{\tilde{x}}^{(1)}(0)=0$ when $\beta=0$, namely
a smooth transition from the static to the dynamical regime. The constant $g$
determines both the intensity and the initial direction of the modulation.
Throughout this paper we will always assume two conditions:
$\Omega\neq\omega_{n}$ and $\Omega\gg\omega$. The first condition prevents any
resonances between the frequency $\Omega$ and any cavity modes of the field,
whereas the second one allows us to neglect the exponential decay $e^{-\Omega
t}$ in our formulas.

####  III.1.3 Second order: $\tilde{x}^{(2)}(t)$

We proceed to the second order corrections. Unfortunately, the presence of
cavity drives makes the second order corrections quite difficult to compute,
therefore we opt for discussing the scenario without cavity drives for this
case. We report the full formula (67) in Appendix B. Thus, henceforth we
impose
$\lambda_{xk}(t)=\lambda_{xk^{\prime}}(t)=\lambda_{pk}(t)=\lambda_{pk^{\prime}}(t)\equiv
0$.

We will focus our analysis on the specific case when one of the possible
resonances between the cavity modes $\omega_{n}$ and the mechanical frequency
$\omega$ occurs. From Equation (67) we identify three possible resonance
conditions:

  * i)

Degenerate resonance (also known as single mode squeezing):
$\omega=2\omega_{k}$;

  * ii)

Nondegenerate resonance (also known as two mode squeezing):
$\omega=\omega_{k}+\omega_{k^{\prime}}$;

  * iii)

Intermode coupling (also known as mode mixing):
$\omega=\omega_{k}-\omega_{k^{\prime}}$.

We stress that only a multimode description of the cavity field as the one
presented here can take the second and the third resonances into account [24,
32].

We assume to operate in the degenerate case i), where $\omega=2\omega_{k}$ for
a chose mode $k$. This choice allows us to employ Equation (67) to obtain

$\displaystyle\tilde{x}^{(2)}(t)\simeq$
$\displaystyle\mu_{k}^{2}\left(1-\cos\left(2\omega_{k}t\right)+\omega_{k}t\sin(2\omega_{k}t)\right).$
(27)

In this formula all off-resonant terms were neglected, since they are by a
factor $\omega_{k}$ or even $\omega_{k}^{2}$ smaller than the resonant ones.

We note that also the second order contributions (27) show a linear growth of
the oscillation amplitude. The occurrence of this linear growth stems
specifically from both the choice of an initial coherent state for the cavity
mode and the resonance between the excited cavity mode and mechanical
oscillation.

The time average of the correction $\tilde{x}^{(2)}(t)$ for large enough times
reads

$\displaystyle\langle\tilde{x}^{(2)}\rangle_{\tau}=$
$\displaystyle\mu_{k}^{2}\left(1-\frac{1}{2}\cos(2\omega_{k}\tau)\right).$
(28)

Interestingly, we note that since the $|\cos\alpha|\leq 1$, we have that
$\langle\tilde{x}^{(2)}\rangle_{\tau}\geq 0$ for all times. This implies that,
on average to this order, the wall of the cavity is always pushed outwards,
and we trace this effect to the back reaction of resonance between the cavity
mode and the harmonic oscillator. This effect is subdominant to the one
obtained at first order, which is expected since back reaction on the wall in
the form of radiation pressure occurs as a higher order than the direct effect
of driving the wall with an external force, or with an initial displacement.

####  III.1.4 Role of the phase in the initial wall displacement

We are now able to discuss the role played by the phase $\theta$ throughout
the dynamics of the system. As already pointed out in Section III.1.1, the
coherent phase $\theta$ determines both the expected position and momentum of
coherent phonons. We now focus on two specific values.

Purely real mechanical coherent parameter. This situation occurs if
$\theta=n\pi$ with $n\in\mathbb{Z}$. Manipulating the overall expression for
$x(t)$ we find

$\displaystyle x(t)\simeq$ $\displaystyle
L\left[1+\epsilon\left(2(-1)^{n}\lvert\beta\rvert-\epsilon\mu_{k}^{2}\right)\cos(2\omega_{k}t)+\epsilon^{2}\mu_{k}^{2}\right.$
$\displaystyle\left.+\epsilon\sin(2\omega_{k}t)\left(\epsilon\mu_{k}^{2}\omega_{k}t+g\right)\right].$
(29)

Although this expression clearly highlights the impossibility to annul the
linear growth of the oscillation amplitude, the mechanical drive could
partially inhibit it as long as $g$ is negative. Moreover, by properly tuning
the coherent parameter $\beta$, we could suppress the additional beat caused
by the term proportional to $\cos(2\omega_{k}t)$.

Purely imaginary mechanical coherent parameter. We now prepare the harmonic
oscillator in a coherent state with purely imaginary parameter: this occurs if
$\theta=(n+1/2)\pi$, with $n\in\mathbb{Z}$. This choice allows us to rewrite
the equation of motion of the wall as following:

$\displaystyle x(t)\simeq$ $\displaystyle
L\left[1+2\epsilon^{2}\mu_{k}^{2}\sin^{2}\left(\omega_{k}t\right)\right.$
$\displaystyle+\left.\epsilon\sin(2\omega_{k}t)\left(\epsilon\mu_{k}^{2}\omega_{k}t+2(-1)^{n}\lvert\beta\rvert+g\right)\right].$
(30)

The action of the external drive does not differ from the case reported above,
and therefore we can momentarily switch it off.

In order to appreciate the role played by phonons throughout the dynamics in
more detail, we distinguish two possible scenarios: either $\theta=+\pi/2$,
achieved by assuming $n=0$, merely leading to an increment of the oscillation
amplitude; or $\theta=-\pi/2$, namely $n=1$, which leads to a reduction of the
oscillation. We interpret the latter case as follows; phonons, having maximal
negative momentum, initially boost the oscillation of the wall leftwards; on
the other hand, the resonance between cavity and mechanical mode induces a
positive (rightward) push of the wall by means of the radiation pressure
within the cavity due to the coherent photons; the damping of the oscillation
of the wall in fact ensues from the combination of these two effects, and the
damping time is found to be
$\bar{t}^{\,(1)}_{f}=N_{b}(0)/\lvert\beta\rvert\mu_{k}^{2}\omega_{k}$ with
$\bar{t}=\epsilon t$. In section (III.2.2) we will see that the attenuation of
the motion is connected with both a reduction in time of the number of
phonons, and the corresponding enhance of the photon number.

####  III.1.5 Third order: $x^{(3)}(t)$

In the last section we studied the motion of the wall when one cavity mode is
initially found in a coherent state. When the cavity field is initially in its
vacuum state, i.e., $\mu_{k}=0$, the first and second order corrections due to
the cavity field initial state (namely proportional to the photon number
$\mu_{k}^{2}$) vanish, and third order effects become dominant. We note that
these effects do not arise due to genuine third order contributions, but due
to a combination of first and second order dynamical ones. For this reason, it
is relevant to compute and analyze them. While it is true that they are
greatly suppressed in the perturbative series, they will give us additional
information regarding the dynamics.

The initial state for this case is $\hat{\rho}(0)=\prod_{n}\lvert
0_{n}\rangle\langle
0_{n}\rvert\otimes\hat{\rho}_{\textrm{m}}^{\textrm{(DST)}}$, where now all the
field modes are initially in the vacuum state. In this scenario, we present
only the case of degenerate resonance where $\omega=2\omega_{k}$ and no
external drive act on the cavity field,
$\lambda_{xk}(t)=\lambda_{xk^{\prime}}(t)=\lambda_{pk}(t)=\lambda_{pk^{\prime}}(t)\equiv
0$. The correction to the third order therefore reads

$\displaystyle x^{(3)}(t)\simeq$
$\displaystyle-\frac{\omega_{k}^{2}t^{2}}{4}\left(2\lvert\beta\rvert\cos(2\omega_{k}t-\theta)+g\sin(2\omega_{k}t)\right).$
(31)

Employing this formula within the full expression of the average position
gives us the final expression

$\displaystyle\frac{x(t)}{L}\simeq$ $\displaystyle
1+2\epsilon\lvert\beta\rvert\cos\theta\cos(2\omega_{k}t)\Gamma_{k}(t)$
$\displaystyle+$
$\displaystyle\epsilon(g+2\lvert\beta\rvert\sin\theta)\sin\left(2\omega_{k}t\right)\Gamma_{k}(t),$
(32)

where we have introduce the important quantity

$\displaystyle\Gamma_{k}(t):=$ $\displaystyle
1-\frac{\epsilon^{2}\omega_{k}^{2}t^{2}}{4}\approx
e^{-\frac{\epsilon^{2}\omega_{k}^{2}t^{2}}{4}}.$ (33)

Recall that, since we are working in perturbation theory, our expressions are
correct and can be employed for all times where $1-\Gamma_{k}(t)\ll 1$.

Interestingly, we can immediately observe the gradual reduction of the
modulation amplitude since $\Gamma_{k}(t)$ decreases with time. We will see in
section III.2.3 that this effect stems from the resonant conversion of
mechanical excitations (phonons) into photons. As done before, we want to
distinguish two cases of interest, namely when $\beta$ is real or imaginary.

Real mechanical coherent parameter. This occurs when $\theta=n\pi$ with
$n\in\mathbb{Z}$. We can rewrite Equation (32) as

$\displaystyle\frac{x(t)}{L}\simeq$ $\displaystyle
1+\epsilon\left[2(-1)^{n}\lvert\beta\rvert\cos(2\omega_{k}t)+g\sin\left(2\omega_{k}t\right)\right]\Gamma_{k}(t).$
(34)

We note that the phononic contribution and the action of the drive give rise
to two oscillating terms which are dephased by $\pi/2$, and whose amplitude
decrease in time.

Purely imaginary mechanical coherent parameter. If we assume that $\beta$ is
purely imaginary in Equation (32), $\theta=\left(n+\frac{1}{2}\right)$ with
$n\in\mathbb{Z}$, we can immediately observe that all contributions oscillate
with the same phase:

$\displaystyle\frac{x(t)}{L}\simeq$ $\displaystyle
1+\epsilon\left(g+2(-1)^{n}\lvert\beta\rvert\right)\Gamma_{k}(t)\sin\left(2\omega_{k}t\right).$
(35)

This means that we can fully control the oscillation amplitude of the cavity
wall. In particular, by properly tuning both the drive coefficient and the
phononic coherent parameter, we can amplify or deamplify the oscillation, or
even suppress it by imposing $g=-2(-1)^{n}\lvert\beta\rvert$.

###  III.2 Number of photons

The dynamics of the mirror is strictly connected with the amount of
excitations present in the system. Therefore, we proceed to study the average
photon number
$N_{k}(t):=\textrm{Tr}\bigl{[}\hat{U}^{\dagger}(t)\hat{a}_{k}^{\dagger}\hat{a}_{k}\hat{U}(t)\hat{\rho}(0)\bigr{]}$
of mode k at time t. We will make use of the initial state (16).

####  III.2.1 Zero order: $N_{k}^{(0)}(t)$

We start from the lowest order contribution, which provides an accurate
estimation of the photon number as long as the total oscillation amplitude is
extremely small. We recall that this is a consequence of the fact that we have
obtained a dynamical mirror implemented by a harmonic oscillator by assuming
that the total deviation from the length $L$ of the cavity is small.

The cavity mode k is initially prepared in the coherent state
$\rho_{k}=\lvert\mu_{k}\rangle\langle\mu_{k}\rvert$ and we consider a nonzero
external drive. The total amount of photons to lowest order is

$\displaystyle N_{k}^{(0)}(t)=$
$\displaystyle(\mu_{k}+\Lambda_{pk}(t))^{2}+\left(\Lambda_{xk}(t)\right)^{2},$
(36)

where we recognize the contribution from both the initial amount of photons
$\mu_{k}^{2}$ in the coherent state, the direct excitation of the cavity mode
due to the external drive $\lambda_{xk}(t)$ and $\lambda_{pk}(t)$, and their
interplay.

####  III.2.2 First order: $N_{k}^{(1)}(t)$

The motion of the wall stimulates the reaction of the field, which leads to
the modification of the photon number at the first order in $\epsilon$. Here
we estimate the correction of the photon number determined by the dynamics of
the mirror. In order to simplify the expressions and facilitate the comparison
with our results in Section III.1.3, we switch off all cavity drives as well
as the coherent parameter $\mu_{k^{\prime}}$ in Equation (16), and present the
photon number in case of degenerate resonance. We stress that off-resonant
terms are smaller than the resonant ones by factors $\omega_{k}$ or
$\omega_{k}^{2}$, and can hence be neglected for cavities with large
frequencies. The full expression of the first order correction with drives and
the coherent excitation of two cavity modes can be found in Appendix C. We
have

$\displaystyle
N^{(1)}_{k}(t)\simeq-\mu_{k}^{2}\omega_{k}t\left(2\lvert\beta\rvert\sin\theta+g\right),$
(37)

where the mechanical drive is again represented by Equation (25). This formula
expresses the resonant exchange of excitations between the mechanical degree
of freedom and the mode of the cavity field $k$ when such resonant cavity mode
is initially found in a coherent state. Hence, this correction would vanish if
we switched the coherent parameter $\mu_{k}$ off.

We can immediately recognise the contribution of the initial phononic state
and the action of the drive. The former strictly depends on the coherent
parameter suggesting that neither the thermal fluctuation nor the squeezing
play a concrete role at the first order. We notice that it vanishes whenever
$\beta$ is a real number, namely when $\theta=n\pi$ with $n\in\mathbb{Z}$,
whereas it grows linearly whenever $\beta$ is purely imaginary
($\theta=\pm\pi/2$). The second term depends on the amplitude of the external
drive and can take positive or negative values.

A direct comparison of this result with both Equation (29) and Equation (30)
suggests that the gradual enhancement of the oscillation amplitude due to the
degenerate resonance condition takes place without exchange of excitations
between the cavity mode $k$ and the mechanical degree of freedom as soon as
$g=-2\sin\theta\lvert\beta\rvert$; whereas it occurs with exchange of
excitations if $\beta$ is a pure complex number, with
$g\neq-2\sin\theta\lvert\beta\rvert$. Moreover, the efficiency of the exchange
of excitations is maximized if $\Im\\{\beta\\}=g$. This behaviour is confirmed
by the investigation of the phonon number, which in case of degenerate
resonance reads $N_{b}(t)=N_{b}^{(0)}(t)+\epsilon\tilde{N_{b}}^{(1)}(t)$, with

$\displaystyle N_{b}^{(0)}(t)\simeq$ $\displaystyle
N_{b}(0)+g\lvert\beta\rvert\sin\theta+\frac{g^{2}}{4}$ (38)
$\displaystyle\tilde{N}_{b}^{(1)}(t)\simeq$
$\displaystyle\frac{\mu_{k}^{2}\omega_{k}t}{2}(2\lvert\beta\rvert\sin\theta+g),$
(39)

where the initial number of phonons is given by Equation (17).

We stress that the direction of conversion (phonons “down-converted” into
photon pairs or photon pairs “up-converted” into phonons) strongly depends on
the interplay between the initial phononic coherent state and the action of
mechanical drive. This is in contrast with what we will see soon in the next
section, by discussing the second order correction by switching off the
coherent parameter $\mu_{k}$. Interestingly, by switching off the mechanical
drive it is immediate to obtain the relation

$\displaystyle\langle N_{k}\rangle_{\tau}+2\langle N_{b}\rangle_{\tau}\simeq$
$\displaystyle N_{k}(0)+2N_{b}(0),$ (40)

which expresses the excitations conservation throughout the dynamics, expected
in a lossless system without time-dependent external inputs.

We finally take advantage of our multimode model and compute the number of
photons for the mode $k$ in case of non-degenerate resonance between the
mechanical mode and two different modes $k$ and $k^{\prime}$ of the cavity
field both initially found in a coherent state with parameter $\mu_{k}$ and
$\mu_{k^{\prime}}$ respectively. We have

$\displaystyle N^{(1)}_{k}(t)\simeq$
$\displaystyle(-1)^{1+k+k^{\prime}}\mu_{k}\mu_{k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}t\left(2\lvert\beta\rvert\sin\theta+g\right).$
(41)

It is evident that this expression is the two-mode equivalent of Equation
(37).

####  III.2.3 Second order: $N_{k}^{(2)}(t)$

Finally, we conclude the analysis of the photon number by computing the
relevant contribution to the number of photons at second order in $\epsilon$
when the field is initially found in its vacuum state. In this case,
$N^{(2)}_{k}(t)=0$ and we note that the first non-zero contribution occurs at
second order. Therefore, the existence of such photons is a pure quantum
effect.

The second order contribution
$N^{(2)}_{k}(t):=\frac{\omega_{k}^{2}t^{2}}{2}\Delta N^{(2)}_{k}(t)$ to the
number of photons reads

$\displaystyle N^{(2)}_{k}(t)=$
$\displaystyle\frac{\omega_{k}^{2}t^{2}}{2}\left(\lvert\beta\rvert^{2}\Delta
N^{(2)}_{\beta,k}(t)+\sinh r\,\Delta N^{(2)}_{\textrm{sq},k}(t)\right.$
$\displaystyle+N_{T}\Delta N^{(2)}_{T,k}(t)+N_{T}\sinh r\Delta
N^{(2)}_{\textrm{sq},T,k}(t)$ $\displaystyle\left.+\Delta
N^{(2)}_{\textrm{vac},k}(t)\right)+N_{\textrm{md},k}^{(2)}(t),$ (42)

where we distinguished the single contributions according to the physical
origin. Here we want to give a physical interpretation of the various elements
of this formula, and we leave the explicit form of these terms to Appendix D.
It is crucial to note that (42) is exact and does not assume any particular
resonant regime. Therefore, all time dependent quantities on the right hand
side have nontrivial expressions that, in general, oscillate in time.

The first contribution is $\Delta N^{(2)}_{\beta,k}(t)$, where the index
$\beta$ emphasizes that this term stems from the coherent state of the
mechanical drive. We notice that this term reproduces the dynamical Casimir
effect induced by the harmonic oscillation of the wall. In order to recover
the photon number expected by the DCE, we impose the degenerate resonant
condition and neglect any off-resonant terms, thereby obtaining and expression
for the contribution
$N^{(2)}_{\beta,k}(t):=\frac{\omega_{k}^{2}t^{2}}{2}\Delta
N^{(2)}_{\beta,k}(t)$, which reads

$\displaystyle N^{(2)}_{\beta,k}(t)\simeq$
$\displaystyle\lvert\beta\rvert^{2}\omega_{k}^{2}t^{2}.$ (43)

This directly shows the connection between the increasing photon number and
the amount of phonons in the coherent state, a connection previously
highlighted in the literature [28, 32]. Since the motion of the wall only
depends on the displacement of the mechanical degree of freedom apart from the
external drive, see Equation (32), we claim that the pair production strictly
stems from the “classical” oscillation of the wall as long as
$\lvert\beta\rvert\gg 1$. This is exactly the regime that reproduces the
conditions necessary for the DCE to occur.

The next three terms in bracket, namely $\Delta N^{(2)}_{\textrm{sq},k}(t)$,
$\Delta N^{(2)}_{T,k}(t)$ and $\Delta N^{(2)}_{\textrm{sq},T,k}(t)$, are
determined by the initial phononic state, in particular by its thermal and
squeezing features. We impose the degenerate resonant condition and compute
the total contribution
$N^{(2)}_{\textrm{sq},T,k}(t):=\frac{\omega_{k}^{2}t^{2}}{2}(\Delta
N^{(2)}_{\textrm{sq},k}(t)+\Delta N^{(2)}_{T,k}(t)+\Delta
N^{(2)}_{\textrm{sq},T,k}(t))$, which reads

$\displaystyle N^{(2)}_{\textrm{sq},T,k}(t)\simeq$
$\displaystyle\omega_{k}^{2}t^{2}\left(\sinh^{2}r+N_{T}+2N_{T}\sinh^{2}r\right).$
(44)

Interestingly, since the average motion of the mirror neither depends on
squeezing of the mechanical state, nor on its thermal fluctuations, the
presence of such contributions in (44) and therefore in (42) means that both
the thermal fluctuations of the phonons and the squeezing of the mechanical
mode induce an excitation of the resonant cavity mode, aside from the motion
of the mirror ascribable to a classical harmonic oscillation.

We want to stress that all contributions analysed so far arise as a direct
consequence of the specific initial phononic state. The term of the
interaction Hamiltonian causing the resonant pair creation, proportional to
$\hat{a}^{{\dagger}2}\hat{b}$, highlights the excitation transfer between the
resonant cavity mode and the harmonic oscillator. This motivates us the
compute the second order contribution to the number $N_{b}(t)$ of phonons at
time $t$, which reads

$\displaystyle\Delta N_{b}^{(2)}(t)=-N_{b}(0)\frac{\omega_{k}^{2}t^{2}}{2},$
(45)

and we find once more that, as long as the mechanical drive is switched off,
the excitations conservation expressed by Equation (40) holds. This means that
the increasing number of photons in Equation (43) and (44) corresponds to a
gradual reduction of the number of phonons. This is exactly what we would
expect if the DCE was produced not by an external drive of the wall, but by an
initial state of the harmonic oscillator different from the vacuum state.

The term $\Delta N^{(2)}_{\textrm{vac},k}(t)$ in (42) stems from the
quantization of the mechanical motion of the wall. Although its contribution
is negligible at resonance, its existence is consistent with our quantum
description of the mechanical motion of the wall, and it is a direct
consequence of vacuum fluctuations.

The last term $N_{\textrm{md},k}^{(2)}(t)$ of Equation (42) originates from
the mechanical drive. By assuming it in Equation (25) and imposing the
degenerate resonance, we can compute the number of photons
$N_{\textrm{md},k}^{(2)}(t)$ that arises from the action of the mechanical
drive:

$\displaystyle N_{\textrm{md},k}^{(2)}(t)\simeq$
$\displaystyle\frac{g\omega_{k}^{2}t^{2}}{4}(4\lvert\beta\rvert\sin\theta+g).$
(46)

The first contribution explicitly depends on the real part of the coherent
parameter, whereas the second one expresses the possibility to generate
photons by forcing the motion of the mirror with the only use of an external
drive at resonance with the cavity mode. Note that, if $\beta=0$, photons are
created in the cavity, but they do not come at the expense of the depletion of
the initial phonons, rather they derive from phonons generated at time $t$ by
the mechanical drive itself. This amount of phonons equals the last term in
Equation (38). Along with $N^{(2)}_{\beta,k}(t)$, this term gives rise to the
standard DCE following a periodical oscillation of the mirror.

The total number of photons, in the case of degenerate resonance, is achieved
by collecting and time-averaging all contributions computed above. We find

$\displaystyle\langle N_{k}^{(2)}\rangle_{\tau}\simeq$
$\displaystyle\frac{\omega_{k}^{2}\tau^{2}}{3}\left(\lvert\beta\rvert^{2}\cos^{2}\theta+\sinh^{2}r+N_{T}+2N_{T}\sinh^{2}r\right)$
$\displaystyle+\frac{\omega_{k}^{2}\tau^{2}}{12}\left(g+2\lvert\beta\rvert\sin\theta\right)^{2},$
(47)

where we recovered the explicit interplay between the oscillation induced by
the presence of coherent phonons and the motion forced by the external drive
that already emerged in Equation (32). As expected, the gradual reduction of
the oscillation amplitude, predicted by taking into account the correction on
the position of the wall at the third order in $\epsilon$, is brought about by
the resonant conversion of phonons into photons. Such phonons can already be
present in the initial phonon state, or generated by the external drive, as
seen in Equation (38).

Since the contribution in Equation (46) depends on the coherent phase
$\theta$, we report two specific scenarios.

Purely real mechanical coherent parameter. Whenever $\beta$ is a real number,
namely for $\theta=n\pi$ and $n\in\mathbb{N}$, the output number of photons in
Equation (47) reads

$\displaystyle\langle N_{k}^{(2)}\rangle_{\tau}\simeq$
$\displaystyle\frac{\omega_{k}^{2}\tau^{2}}{3}\left(N_{b}(0)+\frac{g^{2}}{4}\right).$
(48)

This formula describes the dynamical Casimir effect when the quadratic
contribution in $\omega_{k}$ of the initial phonon number and the action of
the external drive interfere constructively with each other.

Purely imaginary mechanical coherent parameter. If $\beta$ is a imaginary
quantity, $\theta=(n+1/2)\pi$ with $n\in\mathbb{Z}$, the photon number reduces
to

$\displaystyle\langle N_{k}^{(2)}\rangle_{\tau}\simeq$
$\displaystyle\frac{\omega_{k}^{2}\tau^{2}}{3}\left(\sinh^{2}r+N_{T}+2N_{T}\sinh^{2}r\right)$
$\displaystyle+\frac{\omega_{k}^{2}\tau^{2}}{12}\left(g+2(-1)^{n}\lvert\beta\rvert\right)^{2},$
(49)

In contrast to the previous case, our results in Equation (35) and Equation
(49) suggest that we can inhibit any contribution of the phononic displacement
to the dynamics of the wall, as well as to the resonant generation of photons,
by properly tailoring $g$, thereby letting photons arise from both the
phononic thermal fluctuation and the intensity of the mechanical squeezing.

We conclude this section by printing the number of photons that are found when
the mechanical frequency is non-degenerate resonant with two different modes
$k$ and $k^{\prime}$ of the cavity field:

$\displaystyle\langle N_{k}^{(2)}\rangle_{\tau}\simeq$
$\displaystyle\bar{N}_{b}(\theta)\frac{\omega_{k}\omega_{k^{\prime}}\tau^{2}}{3},$
(50)

where we have defined
$\bar{N}_{b}(\theta):=N_{b}(0)+g^{2}/4+g\lvert\beta\rvert\sin\theta$.

###  III.3 Force between the mirrors

The resonance between the cavity mode $k$ and the mechanical oscillation of
the wall induces an excitation in the mode, leading to generation of photons,
as we have shown above. We expect that such increment of the photon number
directly affects the radiation pressure within the cavity acting on the wall,
which coincides with the force between the two mirrors in a (1+1)-dimensional
scenario. This force can immediately be defined from first principles [4], and
it reads $\hat{F}\equiv-\frac{d\hat{H}(t)}{dL}$. In our case, it has the
expression

$\displaystyle\hat{F}=$
$\displaystyle\frac{2\hbar}{L}\sum_{n,m}(-1)^{n+m}\sqrt{\omega_{n}\omega_{m}}\,\hat{X}_{n}\hat{X}_{m}-\frac{\lambda_{xb}(t)}{\delta
L_{0}},$ (51)

where we used the fact that $\epsilon=\delta L_{0}/L$. For completeness, we
restored the vacuum energy of the Hamiltonian to obtain this expression.

We want to estimate the average value of the force when the mode of the
oscillating mirror is resonant with the mode $k$ of the field. In this case,
we find

$\displaystyle\langle F\rangle_{\tau}\simeq$ $\displaystyle
F_{\textrm{vac}}+\bar{N}_{b}(\theta)\frac{\epsilon^{2}\hbar\omega_{k}^{3}\tau^{2}}{6L},$
(52)

where $F_{\textrm{vac}}:=\sum_{n}\frac{\hbar\omega_{n}}{2L}$ is the (formally
divergent) vacuum contribution and, as usual, we neglect all off-resonant
terms. Clearly, the normal ordering would have removed the presence of
$F_{\textrm{vac}}$. However, since we are investigating the time evolution of
the radiation pressure in a cavity with no photons at $t=0$, we want to take
advantage of our multimode description in order to include the zero-point
fluctuations.

We will adopt the reasoning used in the literature [39] in order to
reinterpret the divergent term in Equation (52). As a first step, we suppose
to have a third (static) mirror located on the right hand side of the movable
mirror, say at $L_{2}$ with $L<L_{2}$. We can therefore employ the same model
used so far in order to describe the field in the region $L<x<L_{2}$. In
particular, the force acting on the movable wall due to the presence of this
second cavity is

$\displaystyle F_{2}(t)=$ $\displaystyle
F_{\textrm{vac},2}-\sum_{n}\frac{\hbar\omega_{n}^{\prime}}{2(L_{2}-L)}-F_{\textrm{osc},2}(t),$
(53)

where $\omega_{n}^{\prime}=n\pi c/(L_{2}-L)$, we have introduced the vacuum
fluctuation contribution
$F_{\textrm{vac},2}:=-\sum_{n}\frac{\hbar\omega_{n}^{\prime}}{2(L_{2}-L)}$ and
we have introduce the function
$F_{\textrm{osc},2}(t):=\sum_{n}\frac{\hbar\delta
L_{0}^{2}\omega_{n}^{\prime}\varphi_{1}(t)}{(L_{2}-L)^{2}}+\sum_{\begin{subarray}{c}n,m\\\
n\neq m\end{subarray}}\frac{\hbar\delta
L_{0}^{2}\sqrt{\omega_{n}^{\prime}\omega_{m}^{\prime}}\varphi_{2}(t)}{(L_{2}-L)^{2}}$.
The term $F_{2\textrm{osc}}(t)$ includes all time dependent oscillating terms
via the functions $\varphi_{1}(t)$ and $\varphi_{2}(t)$. These oscillating
terms are proportional to $\delta L_{0}^{2}$, and therefore they are
negligible off-resonance. The function $F_{\textrm{osc},2}(t)$ would given us
a term identical (with opposite sign) to the second one of Equation (52) if
$L_{2}=2L$. However, we want to take the limit $L_{2}\rightarrow\infty$, and
therefore we need to keep $L_{2}$ as a free variable. Therefore we can safely
neglect $F_{\textrm{osc},2}(t)$.

As a second step, we impose a cut-off of the form $e^{-\gamma\omega_{n}}$ in
the divergent terms of (52) and (53). This is a standard procedure and it
allows us to neglect any contribution from high frequency modes that cause
divergences. Since $\sum_{n}ne^{-\alpha n}=e^{\alpha}/(e^{\alpha}-1)^{2}$, we
assume that $\gamma\pi c/L\ll 1$ in order to include as many modes as we can.
We therefore can expand the divergent terms with cutoff in (52) and (53) and
obtain

$\displaystyle F_{\textrm{vac}}\simeq$ $\displaystyle\frac{\hbar
c}{2\pi\gamma^{2}}-\frac{\hbar\pi c}{24L^{2}},$ $\displaystyle
F_{\textrm{vac},2}\simeq$ $\displaystyle-\frac{\hbar
c}{2\pi\gamma^{2}}+\frac{\hbar\pi c}{24(L-L_{2})^{2}}.$ (54)

The total force acting on the movable wall is clearly the sum of the two
contributions $F_{\textrm{vac}}$ and $F_{\textrm{vac},2}$. We can now safely
remove the cutoff $\gamma$ by setting $\gamma=0$, and we can remove the wall
inserted at $L_{2}$ by taking the limit $L_{2}\rightarrow\infty$. We finally
have

$\displaystyle\langle F\rangle_{\tau}\simeq$ $\displaystyle-\frac{\hbar\pi
c}{24L^{2}}+\bar{N}_{b}(\theta)\frac{\epsilon^{2}\hbar\omega_{k}^{3}\tau^{2}}{6L},$
(55)

The first term of this expression is the well-known Casimir force between two
perfectly conducting plates [39]. This force can be interpreted as arising
from the zero-point fluctuations of the field inside the cavity, and therefore
it holds also at $t=0$. On the contrary, we attribute the remaining positive
term to the repulsive force that arises as a consequence of the increasing
radiation pressure that forms inside the cavity. Since initially the cavity
modes do not contain excitations, this contribution is a direct consequence of
the resonant amplification of the field mode $k$. In other words, Equation
(55) describes the interplay between static and dynamical Casimir effect, and
it highlights the dependence of the force on both the initial phononic state
and the intensity of the external drive.

##  IV Applications and extensions

We now briefly discuss potential applications and extensions to our work.

###  IV.1 Prospective experimental platforms

We expect that observing the predicted effects in an experimental system will
be challenging as the different resonance scenarios i) to iii) in Section
III.1.3 require a mechanical mode with high frequency exceeding the optical
mode frequencies – i) and ii) – or matching their difference – iii). Whilst i)
and ii) are only realistic in microwave photonic systems, the latter poses a
more modest constraint on the involved modes. For a Fabry-Pérot type optical
resonator this condition can be met, if the mechanical resonator interacts
with modes of a long optical cavity of length $L$ that are spaced by the free
spectral range $\nu_{\text{FSR}}=c/2L$.

A drawback of this approach is the reduced optomechanical interaction strength
that scales with $1/L$ for systems with mechanical resonators that only
locally interact with the involved optical modes. Therefore, platforms that
make use of extended mechanical resonances as reported in photonic crystal
fibers (PCFs) [40, 41, 42] or optomechanical crystals (OMCs) [43] are more
promising. As still evenly spaced optical modes are required, the Fabry-Pérot
type approach should nevertheless be maintained. For PCFs this can be realized
by employing a reflective coating at the end facets of a fiber piece [44] of
length $L$ as shown in Figure 2 (a). For a GHz range mechanical mode [40, 41]
the required length will be on the order of $0.1\text{\,}\mathrm{m}$ to
realize evenly spaced optical modes within a low dispersion region of the
guided mode spectrum. This length will be reduced for OMCs as silicon exhibits
a higher refractive index and as higher frequency acoustic modes are realized
[45]. Still, up to millimeter long waveguide paths that support both optical
and acoustic modes with photonic and phononic reflectors at the ends will be
required [46, 47] for example using two-dimensional OMC realizations [48, 45]
as sketched in Figure 2 (b). For a close to ideal mode overlap of a low-
dispersion optical and a $\Gamma$-point mechanical mode, the resulting vacuum
optomechanical coupling strength $g_{0}$ scales with the number of required
unit cells $n_{\text{uc}}$ as $1/\sqrt{n_{\text{uc}}}$ as the effective mass
of the mechanical mode linearly increases with $n_{\text{uc}}$. For
$n_{\text{uc}}\sim 10^{3}$ and a waveguide optomechanical coupling of
$\approx$1.5\text{\,}\mathrm{MHz}$$ [45], this would result in
$g_{0}\approx$50\text{\,}\mathrm{kHz}$$. Such approaches, especially if slight
optimizations are conducted, will therefore result in experimentally
accessible systems for resonance scenario iii).

Figure 2: Potential experimental platforms that allow for intermode coupling,
where a mechanical resonance frequency matches the optical mode spacing. A
photonic crystal fiber (PCF) piece with reflectively coated fiber ends as
depicted in (a) can support optical modes with a spacing $c/2L$ that matches
the frequency of acoustic breathing modes of the core
($\sim$\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$) of the core for
$L\sim$0.1\text{\,}\mathrm{m}$$. (b) shows a similar scenario for a
optomechanical crystal platform realization, where an extended waveguide of
length $L$ supports both optical and acoustic waveguide modes. In (c) an
electromechanical platform is depicted featuring a single mechanical resonance
coupled to multiple microwave (MW) circuits. The wide range of available MW
resonance frequencies would also allow the realizations the degenerate or
nondegenerate resonance, however, with an experimentally more limited number
of MW resonators.

Electromechanical systems using superconducting microwave resonators [49, 50]
with a displacement dependent capacitance provide experimental platforms with
smaller photonic mode frequencies. With using higher frequency mechanical
modes this would allow realizations of the other resonance scenarios. The
limitation however will be the number of microwave modes coupled to the same
mechanical element meeting the resonance condition. Several tuned and
simultaneously coupled microwave resonators will be required as indicated in
Figure 2 (c). This strongly restricts the number of involved modes in contrast
to Fabry-Pérot type photonic resonators, where a large number of evenly spaced
optical modes naturally occurs.

Whilst not giving the direct interpretation as a classical dynamic Casimir
effect as in a system of two opposing mirrors, analogous electro- or
optomechanical systems support experimentally accessible regimes that can
realize all three resonance scenarios. The dynamics of involved photonic modes
would be reproduced by such systems providing means for single mode squeezing,
two mode squeezing or mode mixing applications.

###  IV.2 Scalar massive field

The procedure presented in Section II can straightforwardly be extended to the
case of confined massive field. Let us suppose that a massive real scalar
field $\phi(t,x)$ is confined in a static cavity. The classical Lagrangian
density $\mathcal{L}(t,x)$ in $1+1$ dimensions reads

$\displaystyle\mathcal{L}=\frac{\hbar}{2}\left(\frac{\partial^{2}\phi}{c^{2}\partial
t^{2}}-\frac{\partial^{2}\phi}{\partial
x^{2}}-\frac{M^{2}c^{2}}{\hbar^{2}}\phi^{2}\right),$ (56)

where $M$ is the mass. The classical field is solution to the
(1+1)-dimensional Klein Gordon equation for a massive field that reads
$(\square-M^{2}c^{2}/\hbar^{2})\phi(t,x)=0$. We again make the ansatz
$\phi_{n}(t,x)=e^{-i\omega_{n}t}\sin(p_{n}x)$, where the dispersion relation
now reads $\omega_{n}=(\hbar^{2}c^{2}p_{n}^{2}+M^{2}c^{4})^{\frac{1}{2}}$,
with wave vector $p_{n}:=\frac{n\pi}{L}$. The explicit expression of both the
field $\phi(t,x)$ and its conjugate momentum $\Pi(t,x)$ do not differ from
those found before, except for the massive term in the frequencies
$\omega_{n}$.

We can therefore apply the same procedure and find the classical density
Hamiltonian of the massive field, which reads
$\mathcal{H}=\frac{1}{2}\left(\Pi^{2}(t,x)+(\partial_{x}\phi(t,x))^{2}+\frac{M^{2}c^{2}}{\hbar^{2}}\phi^{2}(t,x)\right)$.
This, in turn, allows us to compute the Hamiltonian operator
$\hat{H}=\hat{H}_{0}+\epsilon\hat{H}_{\textrm{I}}$, where

$\displaystyle\hat{H}_{0}:=$
$\displaystyle\sum_{n}\hbar\omega_{n}\,\hat{a}_{n}^{\dagger}\hat{a}_{n}+\hbar\omega\hat{b}^{\dagger}\hat{b},$
$\displaystyle\hat{H}_{\textrm{I}}:=$ $\displaystyle-2\sum_{n}\frac{\hbar
c^{2}p_{n}^{2}}{\omega_{n}}\,\hat{a}_{n}^{\dagger}\hat{a}_{n}\hat{X}_{b}-\sum_{n}\frac{\hbar
c^{2}p_{n}^{2}}{\omega_{n}}\,\left(\hat{a}_{n}^{{\dagger}2}+\hat{a}_{n}^{2}\right)\hat{X}_{b}$
$\displaystyle-4\sum_{n\neq m}(-1)^{n+m}\frac{\hbar
c^{2}p_{n}p_{m}}{\sqrt{\omega_{n}\omega_{m}}}\,\hat{X}_{n}\hat{X}_{m}\hat{X}_{b}.$
(57)

We notice that $\hat{H}_{0}$ is formally identical to the free Hamiltonian of
the massless case, whereas the interaction Hamiltonian $\hat{H}_{\textrm{I}}$
differs from Equation (II.2) only in the coupling constant, which now includes
the dispersion relation for the massive field. For this reason, the extension
of our previous results to the massive scenario can be performed
straightforwardly. As a single example, we report the number of massive
excitations generated at time $t$ in case of degenerate resonance with a
cavity mode $k$, when the input state is given by
$\hat{\rho}(0)=\prod_{n}\lvert 0_{n}\rangle\langle
0_{n}\rvert\otimes\hat{\rho}_{\textrm{m}}^{\textrm{(DST)}}$ in the vacuum
state of the massive field. To second order in $\epsilon$ we obtain

$\displaystyle N_{k}^{(2)}(t)\simeq$
$\displaystyle\frac{\epsilon^{2}c^{4}p_{k}^{4}t^{2}}{\omega_{k}^{2}}N_{b}(0).$
(58)

##  V Considerations and outlook

###  V.1 Considerations

We now proceed with some considerations regarding our work.

First, we notice that the combination $\epsilon\omega_{k}$ in our Hamiltonian
exactly corresponds to the vacuum optomechanical coupling strength [4]. This
allows us to interpret $\delta L_{0}$ as the zero-point fluctuation of the
quantum harmonic oscillator, namely $\delta L_{0}\equiv
x_{ZPF}=\sqrt{\hbar/(2M\omega)}$ (with $M$ mass of the mirror), whose order of
magnitude is typically much less than the length of the whole cavity [2]. This
argument reinforces the perturbative approach employed in this work.

Figure 3: Second order correction $\Delta N_{\beta,k}^{(2)}(t)$ of the average
number of excitations in mode $k$ as a function of the dimensionless
oscillation frequency $\tilde{\omega}:=\frac{L}{\pi c}\omega$ of the wall at
different dimensionless times $\tilde{t}:=\frac{\pi c}{L}t$. We have plotted:
(a) at time $\tilde{t}=30$, (b) at time $\tilde{t}=50$, and (c) at time
$\tilde{t}=100$. In all figures, the four peaks correspond to the resonant
frequency at $\tilde{\omega}=2\tilde{\omega}_{1}=2$,
$\tilde{\omega}=\tilde{\omega}_{1}+\tilde{\omega}_{2}=3$,
$\tilde{\omega}=\tilde{\omega}_{1}+\tilde{\omega}_{3}=4$ and
$\tilde{\omega}=\tilde{\omega}_{1}+\tilde{\omega}_{4}=5$. The mass of the
mirror is $M=10^{-16}$kg.

We also want to point out that the inverse of the coupling strength determines
the regime of validity of our perturbative approach, as observed in Equation
(33). This formula expresses the reduction of the oscillation amplitude at
resonance, and it holds as long as $t\ll t_{c}\equiv 2/\epsilon\omega_{k}$. In
typical optomechanical experiments this quantity ranges between
$10^{-6}$s-$10^{-2}$s [51, 9, 8, 52], but it can dramatically reduce in
superconducting quantum-interference devices [53, 54]. On the other hand, the
inverse of the oscillation frequency, $1/\omega$, gives an estimation of the
scale time for resonant peak in the photon number to emerge. Such peak
increases its intensity rapidly in time, as shown in Figure 3, where the
correction $\Delta N_{\beta,k}^{(2)}(t)$ in Equation (69) is plotted by
scanning the oscillation frequency at different time, where both frequency and
time are normalized as following: $\tilde{\omega}=\frac{L}{\pi c}\omega$ and
$\tilde{t}=\frac{\pi c}{L}t$. The graph shows how the sinc functions of
Equation (69) tend to a Dirac delta function at resonance when $\tilde{t}\gg
1$, or $t\gg 2/\omega_{k}$.

Now we want to give an estimation of the order of magnitude of the second
order correction on the Casimir force. The expected dominant repulsive
behaviour of the force emerges at $L<L_{c}$, where the critical length $L_{c}$
is defined by $\langle F(L_{c})\rangle_{\tau}=0$, namely

$\displaystyle
L_{c}=\left(\frac{4\pi\bar{N}_{b}(\theta)c\,\hbar\,\tau^{2}}{M}\right)^{\frac{1}{3}}.$
(59)

Moreover, the new trend of the Casimir force suggests the presence of a
minimum $\langle F(L_{min})\rangle_{\tau}$ that is reached at
$L_{min}=(5/2)^{1/3}L_{c}$ and has the expression

$\displaystyle\langle F(L_{min})\rangle_{\tau}\simeq$
$\displaystyle-\frac{\hbar\pi c}{40L_{min}^{2}}.$ (60)

In order to provide a concrete evaluation of these quantities, we switch off
the external drive, fix $T=0$ and $r=0$. Moreover, we fix a reference length
of the cavity when the wall is at rest $L_{0}=10\mu$m [33], and assume that
the oscillating mirror has mass $M\approx 10^{-16}$kg. We want to resonantly
excite the lowest mode of the cavity, namely $\omega=2\omega_{1}$. As can be
seen in Figure (4)(a), where we plotted the trend of the Casimir force by
varying the distance between the two walls at different time, no inversion of
the sign of the force is observed. This suggests that correction obtained
above to the standard Casimir force does not play an appreciable role when
$t\ll t_{c}$, namely where the radiation pressure is weak due to the low
amount of generated photons. On the contrary, the repulsive trend becomes
appreciable with the increase of the phonon number, as shown in Figure 4(b)
and (c), where the inversion point is visible for short distances. In
particular, this amounts to $L_{c}/L_{0}\simeq 0.11$ in Figure 4(b) and to
$L_{c}/L_{0}\simeq 0.14$ in Figure 4(c). Furthermore, the total number of
photons at $t=10^{-6}$ ranges from $N_{k}\approx 10^{-6}$ in Figure 4(a) up to
$N_{k}\approx 10^{-4}$ in Figure 4(c).

Figure 4: The Casimir force between the two wall of the cavity at two
different times, namely $t=0\,s$ (solid red line), and $t=10^{-6}\,s$ (blue
dashed line). Each panel has been obtained using different coherent phonon
number: (a) $\lvert\beta\rvert^{2}=1$, (b) $\lvert\beta\rvert^{2}=50$, and (c)
$\lvert\beta\rvert^{2}=100$. We are working here in the degenerate resonant
regime, where the oscillation frequency $\omega$ reads $\omega=2\omega_{1}$.
Other parameters are the following: cavity length $L=10\mu$m, mirror mass
$M=10^{-16}$kg.

One important aspect of the the model proposed above is that it allows us to
compute the excitation transfer between the wall and the field. As already
known [28], this is carried out by the squeezing term in the interaction
Hamiltonian proportional to $(\hat{a}_{k}^{{\dagger}})^{2}\hat{b}$, and can
occur when the phonon state is initially found in an excited state. However,
we want to emphasize that our model predicts the resonant excitation of the
cavity mode by means of an external drive acting on the phononic degree of
freedom. In particular, the term $(g\omega_{k}t/2)^{2}$ in Equation (46) stems
from a time-dependent displacement term in the Hamiltonian, acting on the
quantized mechanical mode.

The specific choice of the external drive in Equation (25) gives rise to a
smooth transition from a static to a dynamical regime, which terminates once
the sinusoidal motion of the wall is stabilized. A similar equation of motion
has been previously employed [32], where the dynamics is not included in the
Hamiltonian but enters the mathematical description of the system in terms of
time-dependent Dirichlet boundary conditions of the equation of motion. In our
model, the use of a time-dependent displacement acting on the mechanical
degree of freedom avoids the requirement of solving differential equations
with time-dependent boundary conditions. As expected, we can recover both the
equation of motion and the number of photons in [32] by imposing either $g=1$
and $\beta=0$ or $g=0$ and $\beta=\frac{1}{2}e^{i\pi/2}$.

In addition, by assuming
$\omega_{k}^{2}t^{2}\lvert\beta\rvert^{2}=\omega_{k}^{2}t^{2}g^{2}/4=N_{0}/2$,
$r=0$ and $T=0$ in Equation (47), our model predicts an interference of the
photon number of the form $N_{k}=N_{0}(1+\sin\theta)$, caused by the interplay
between coherent phonons and mechanical drive, whose pattern depends on the
coherent phase $\theta$. In particular, an effect of amplification occurs at
$\theta=\pi/2$, whereas the inhibition of the output radiation is expected
whenever $\theta=-\pi/2$. Coherently, such values correspond to the maximal
value of the oscillation amplitude and the arrest of the oscillating wall
respectively, see Equation (26). Since the DCE as a squeezing phenomenon
stimulates the creation of photon pairs, and the above-described modulation of
$N_{0}$ involves both photons at the same time, by scanning the phase $\theta$
we can reproduce the interference pattern expected by a SU(1,1) interferometer
characterized by two optical parametric amplifiers in the low gain regime [55,
56, 57, 58]. We leave the study of this aspect to future work.

###  V.2 Outlook

In this work we showed that the quantization of the harmonic oscillation of a
cavity wall translates into the presence of a further “mechanical” quantum
degree of freedom, which in turn gives rise to both the optomechanical
coupling and specific terms determining the exchange of quantum excitations
between photons and phonons. More specifically, we have seen that, in a
multimode description of the cavity field, the latter degrees of freedom are
responsible for effects of multimode squeezing, which can emerge by properly
tuning the oscillation frequency of the harmonic oscillator, or its external
driving force.

The description of the whole system that we have here achieved is general
enough to be exported to many systems, as well as to be used for modelling of
concrete physical systems. Apart from the theoretical and foundational
knowledge gained by following the procedure proposed here, we also highlight
the potential implications for future research. For example, interesting
applications can be found in continuous variables quantum information and
computing, where single- and two-mode quantum gates on bosonic degrees of
freedom are necessary as ingredients for the achievement of desired tasks
[59]. Universal quantum computing also requires nonlinear bosonic gates, i.e.,
unitary operations that are induced by Hermitian operators that are at least
cubic in the creation and annihilation operators (or, equivalently, in the
quadrature operators). While single- and two-mode gates have already been
discussed in the context of resonances for field modes of cavity fields [60,
61, 62], our current work moves beyond the ideal scenario where abstract
boundary conditions are varied as functions of time and allows to concretely
study the gate-induced operations in relation to the additional interplay
between the different degrees of freedom. While our work does not include
decoherence or dissipation, which are are significant and unavoidable in
cavity systems [21, 25], we believe that the current study provides the first
steps towards more concrete realizations. However, we expect that the dynamic
effects, as show in Fig. 3, will persist in a full treatment as their time-
scales are faster than the dissipation or decoherence of optical or mechanical
modes reached in current experimental systems. For this reason, we leave it to
future work to include the effects of cavity loss and more realistic cavity
designs.

##  VI Conclusion

In this paper we have studied the dynamics of a quantum field trapped within a
cavity with a moving quantized wall. Our approach tackles the problem at the
fundamental Hamiltonian level and allows us to obtain a Hamiltonian that
includes the interaction between the field and the degree of freedom of the
wall avoiding the need for time-dependent (Dirichlet) boundary conditions. The
Hamiltonian contains both the standard optomechanical interaction term as well
as the terms responsible for field dynamics, thereby allowing us to explore
each regime separately, together with their interplay. We achieved this result
by adding external driving terms for both the field and mirror-displacement,
and by considering specifically tailored initial states of the whole system.

We employed our model to estimate the time evolution of the position of the
wall, the average photon number in the cavity, and the pressure that acts on
the wall. We focused our efforts on the resonant regime, where either the
intrinsic frequency of the wall or the frequency of the external drive are
twice the frequency of one of the cavity modes. This regime appears naturally
in many physical systems such as, for example, degenerate parametric down-
conversion [63, 64], and four-wave mixing [65, 66]. We were able to reproduce
the standard optomechanical regime or the dynamical Casimir effect regime by
specifically tailoring the initial state, and the external mechanical drive.
We also computed the excitation transfer between the cavity modes and the
vibrational mode of the wall, which occurs at the interplay between these two
scenarios. When the wall is initially displaced but not driven, we found that
this excitation exchange translates into the photon pair production via DCE by
means of the conversion of phonons into photons, leading to a decrease in
amplitude of the wall oscillations. Similarly, the action of an external
drive, initially pushing the wall at the resonant frequency with one of the
cavity mode, causes the reaction of the field, which again translates into the
vacuum squeezing of the resonant mode (DCE).

Furthermore, we took advantage of the multimode structure of the cavity field
in order to study the time evolution of the radiation pressure when the cavity
field is initially prepared in its vacuum state, therefore obtaining the
combined action of the static Casimir force and the repulsive radiation
pressure caused by the increase of cavity photons by means of the dynamical
Casimir effect. In addition, we extended our model to the case of massive
scalar fields, for which the results are formally equivalent to those achieved
in the massless scenario and therefore do not imply qualitative differences,
and we also proposed a physical system based on a photonic crystal fiber for
potential concrete implementation of the techniques that we have developed.
Finally, applications of this model include single- and two-mode quantum gates
acting on the field modes in the cavity. These operations can be used as core
ingredients for quantum information processing with continuous variables, for
example in quantum computing and the realization of multimode quantum gates.

The proposed approach is derived from first principles that follow a general
procedure, and it can therefore be used to tackle other applications where a
boundary is quantized without the need of solving time-dependent differential
equations. We believe that this work provides an important addition to the
study of the interplay of quantum field theory and optomechanics, as well as
novel theoretical tools to be used in the development of future quantum
technologies.

###### Acknowledgements.

We thank Roberto Passante and Lucia Rizzuto for interesting discussions, as
well as Franco Nori, Vincenzo Macrì, and Jorma Louko for the useful comments.
This work has received funding from the European Union’s Horizon 2020 program
under the ERC consolidator grant RYD-QNLO (Grant No. 771417). F.K.W., A.F.,
and D.E.B. acknowledge the program Geqcos, sponsored by the German Federal
Ministry of Education and Research, under the funding program “quantum
technologies–from basic research to the market”, No. 13N15685. H.P. and S.H.
acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy, Cluster of
Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1,
390534769.

## References

  * Meystre [2013] P. Meystre, Annalen der Physik 525, 215 (2013).
  * Kippenberg and Vahala [2008] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008).
  * Kippenberg and Vahala [2007] T. Kippenberg and K. Vahala, Opt. Express 15, 17172 (2007).
  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Rev. Mod. Phys. 86, 1391 (2014).
  * Barzanjeh  _et al._ [2021] S. Barzanjeh, A. Xuereb, S. Gröblacher, M. Paternostro, C. A. Regal, and E. M. Weig, Nature Physics , 1 (2021).
  * Genes  _et al._ [2008] C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, Phys. Rev. A 77, 033804 (2008).
  * Marquardt  _et al._ [2007] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, 093902 (2007).
  * Chan  _et al._ [2011] J. Chan, T. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, Nature 478, 89 (2011).
  * Teufel  _et al._ [2011] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Nature 475, 359 (2011).
  * Lai  _et al._ [2018] D.-G. Lai, F. Zou, B.-P. Hou, Y.-F. Xiao, and J.-Q. Liao, Phys. Rev. A 98, 023860 (2018).
  * Armata  _et al._ [2017a] F. Armata, L. Latmiral, A. D. K. Plato, and M. S. Kim, Phys. Rev. A 96, 043824 (2017a).
  * Rademacher  _et al._ [2020] M. Rademacher, J. Millen, and Y. L. Li, Advanced Optical Technologies 9, 227 (2020).
  * Lee  _et al._ [2011] K. C. Lee, M. R. Sprague, B. J. Sussman, J. Nunn, N. K. Langford, X.-M. Jin, T. Champion, P. Michelberger, K. F. Reim, D. England, D. Jaksch, and I. A. Walmsley, Science 334, 1253 (2011).
  * Riedinger  _et al._ [2018] R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, Nature 556, 473 (2018).
  * Bruschi [2019] D. E. Bruschi, Journal of Mathematical Physics 60, 062105 (2019).
  * Metcalfe [2014] M. Metcalfe, Applied Physics Reviews 1, 031105 (2014).
  * Vanner  _et al._ [2011] M. R. Vanner, I. Pikovski, G. D. Cole, M. S. Kim, Č. Brukner, K. Hammerer, G. J. Milburn, and M. Aspelmeyer, Proceedings of the National Academy of Sciences 108, 16182 (2011).
  * Qvarfort  _et al._ [2019] S. Qvarfort, A. Serafini, A. Xuereb, D. Rätzel, and D. E. Bruschi, New Journal of Physics 21, 055004 (2019).
  * Qvarfort  _et al._ [2020] S. Qvarfort, A. Serafini, A. Xuereb, D. Braun, D. Rätzel, and D. E. Bruschi, Journal of Physics A: Mathematical and Theoretical 53, 075304 (2020).
  * Qvarfort  _et al._ [2021a] S. Qvarfort, A. D. K. Plato, D. E. Bruschi, F. Schneiter, D. Braun, A. Serafini, and D. Rätzel, Phys. Rev. Research 3, 013159 (2021a).
  * Qvarfort  _et al._ [2021b] S. Qvarfort, M. R. Vanner, P. F. Barker, and D. E. Bruschi, Phys. Rev. A 104, 013501 (2021b).
  * Bruschi and Xuereb [2018] D. E. Bruschi and A. Xuereb, New Journal of Physics 20, 065004 (2018).
  * Wang  _et al._ [2019] X. Wang, W. Qin, A. Miranowicz, S. Savasta, and F. Nori, Phys. Rev. A 100, 063827 (2019).
  * Dodonov [2020] V. Dodonov, Physics 2, 67 (2020).
  * Macrì  _et al._ [2018] V. Macrì, A. Ridolfo, O. Di Stefano, A. F. Kockum, F. Nori, and S. Savasta, Phys. Rev. X 8, 011031 (2018).
  * Qin  _et al._ [2019] W. Qin, V. Macrì, A. Miranowicz, S. Savasta, and F. Nori, Phys. Rev. A 100, 062501 (2019).
  * Settineri  _et al._ [2019] A. Settineri, V. Macrì, L. Garziano, O. Di Stefano, F. Nori, and S. Savasta, Phys. Rev. A 100, 022501 (2019).
  * Del Grosso  _et al._ [2019] N. F. Del Grosso, F. C. Lombardo, and P. I. Villar, Phys. Rev. A 100, 062516 (2019).
  * Law [1995] C. K. Law, Phys. Rev. A 51, 2537 (1995).
  * Alves  _et al._ [2003] D. T. Alves, C. Farina, and P. A. M. Neto, Journal of Physics A: Mathematical and General 36, 11333 (2003).
  * Dalvit  _et al._ [2006] D. A. R. Dalvit, F. D. Mazzitelli, and X. O. Millán, Journal of Physics A: Mathematical and General 39, 6261 (2006).
  * Crocce  _et al._ [2001] M. Crocce, D. A. R. Dalvit, and F. D. Mazzitelli, Phys. Rev. A 64, 013808 (2001).
  * Butera and Passante [2013] S. Butera and R. Passante, Phys. Rev. Lett. 111, 060403 (2013).
  * Armata and Passante [2015] F. Armata and R. Passante, Phys. Rev. D 91, 025012 (2015).
  * Armata  _et al._ [2017b] F. Armata, M. S. Kim, S. Butera, L. Rizzuto, and R. Passante, Phys. Rev. D 96, 045007 (2017b).
  * Marian and Marian [1993] P. Marian and T. A. Marian, Phys. Rev. A 47, 4474 (1993).
  * Cirio  _et al._ [2017] M. Cirio, K. Debnath, N. Lambert, and F. Nori, Phys. Rev. Lett. 119, 053601 (2017).
  * Di Stefano  _et al._ [2019] O. Di Stefano, A. Settineri, V. Macrì, A. Ridolfo, R. Stassi, A. F. Kockum, S. Savasta, and F. Nori, Phys. Rev. Lett. 122, 030402 (2019).
  * Plunien  _et al._ [1986] G. Plunien, B. Müller, and W. Greiner, Physics Reports 134, 87 (1986).
  * Kang  _et al._ [2010] M. S. Kang, A. Brenn, and P. S. J. Russell, Physical review letters 105, 153901 (2010).
  * Kang  _et al._ [2011] M. S. Kang, A. Butsch, and P. S. J. Russell, Nature Photonics 5, 549 (2011).
  * Butsch  _et al._ [2012] A. Butsch, M. S. Kang, T. Euser, J. Koehler, S. Rammler, R. Keding, and P. S. J. Russell, Physical Review Letters 109, 183904 (2012).
  * Eichenfield  _et al._ [2009] M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, Nature 462, 78 (2009).
  * Jia  _et al._ [2020] K. Jia, X. Wang, D. Kwon, J. Wang, E. Tsao, H. Liu, X. Ni, J. Guo, M. Yang, X. Jiang, J. Kim, S.-n. Zhu, Z. Xie, and S.-W. Huang, Physical Review Letters 125, 143902 (2020).
  * Ren  _et al._ [2020] H. Ren, M. H. Matheny, G. S. MacCabe, J. Luo, H. Pfeifer, M. Mirhosseini, and O. Painter, Nature communications 11, 1 (2020).
  * Fang  _et al._ [2016] K. Fang, M. H. Matheny, X. Luan, and O. Painter, Nature Photonics 10, 489 (2016).
  * Fang  _et al._ [2017] K. Fang, J. Luo, A. Metelmann, M. H. Matheny, F. Marquardt, A. A. Clerk, and O. Painter, Nature Physics 13, 465 (2017).
  * Safavi-Naeini  _et al._ [2014] A. H. Safavi-Naeini, J. T. Hill, S. Meenehan, J. Chan, S. Gröblacher, and O. Painter, Physical Review Letters 112, 153603 (2014).
  * Wilson  _et al._ [2011] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, nature 479, 376 (2011).
  * Nation  _et al._ [2012] P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Rev. Mod. Phys. 84, 1 (2012).
  * Verhagen  _et al._ [2012] E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, Nature 482, 63 (2012).
  * Murch  _et al._ [2008] K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Nature Physics 4, 561 (2008).
  * Johansson  _et al._ [2009] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Phys. Rev. Lett. 103, 147003 (2009).
  * Johansson  _et al._ [2010] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Phys. Rev. A 82, 052509 (2010).
  * Yurke  _et al._ [1986] B. Yurke, S. L. McCall, and J. R. Klauder, Physical Review A 33, 4033 (1986).
  * Luo  _et al._ [2021] K.-H. Luo, M. Santandrea, M. Stefszky, J. Sperling, M. Massaro, A. Ferreri, P. R. Sharapova, H. Herrmann, and C. Silberhorn, Phys. Rev. A 104, 043707 (2021).
  * Manceau  _et al._ [2017] M. Manceau, F. Khalili, and M. Chekhova, New Journal of Physics 19, 013014 (2017).
  * Ferreri  _et al._ [2021] A. Ferreri, M. Santandrea, M. Stefszky, K. H. Luo, H. Herrmann, C. Silberhorn, and P. R. Sharapova, Quantum 5, 461 (2021).
  * Weedbrook  _et al._ [2012] C. Weedbrook, S. Pirandola, R. García-Patrón, N. J. Cerf, T. C. Ralph, J. H. Shapiro, and S. Lloyd, Rev. Mod. Phys. 84, 621 (2012).
  * Bruschi  _et al._ [2013a] D. E. Bruschi, J. Louko, D. Faccio, and I. Fuentes, New Journal of Physics 15, 073052 (2013a).
  * Friis  _et al._ [2012] N. Friis, M. Huber, I. Fuentes, and D. E. Bruschi, Phys. Rev. D 86, 105003 (2012).
  * Bruschi  _et al._ [2013b] D. E. Bruschi, A. Dragan, A. R. Lee, I. Fuentes, and J. Louko, Phys. Rev. Lett. 111, 090504 (2013b).
  * Couteau [2018] C. Couteau, Contemporary Physics 59, 291 (2018).
  * Rubin  _et al._ [1994] M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122 (1994).
  * Abrams and Lind [1978] R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94 (1978).
  * Yuen and Shapiro [1979] H. P. Yuen and J. H. Shapiro, Opt. Lett. 4, 334 (1979).

##  Appendix A Interaction Hamiltonian

The effective interaction Hamiltonian calculated via Equation (14) is

$\displaystyle\hat{\tilde{H}}_{\textrm{I}}(t)=$
$\displaystyle-\hbar\epsilon\left[\vartheta(\omega_{k},\omega_{k^{\prime}},t)+\sum_{n}\frac{\omega_{n}}{2}\,\left(2\hat{a}_{n}^{\dagger}\hat{a}_{n}+e^{2i\omega_{n}t}\hat{a}_{n}^{{\dagger}2}+e^{-2i\omega_{n}t}\hat{a}_{n}^{2}\right)\right.$
$\displaystyle+\sum_{\begin{subarray}{c}n,m\\\ n\neq
m\end{subarray}}(-1)^{n+m}\frac{\sqrt{\omega_{n}\omega_{m}}}{2}\left(e^{i\omega_{n}t}\hat{a}_{n}^{\dagger}+e^{-i\omega_{n}t}\hat{a}_{n}\right)\left(e^{i\omega_{m}t}\hat{a}_{m}^{\dagger}+e^{-i\omega_{m}t}\hat{a}_{m}\right)$
$\displaystyle-2\sum_{\begin{subarray}{c}n\\\ n\neq
k\end{subarray}}(-1)^{n+k}\sqrt{\omega_{n}\omega_{k}}\psi_{k}(t)\left(e^{i\omega_{n}t}\hat{a}_{n}^{{\dagger}}+e^{-i\omega_{n}t}\hat{a}_{n}\right)-2\sum_{\begin{subarray}{c}n\\\
n\neq
k^{\prime}\end{subarray}}(-1)^{n+k^{\prime}}\sqrt{\omega_{n}\omega_{k^{\prime}}}\psi_{k^{\prime}}(t)\left(e^{i\omega_{n}t}\hat{a}_{n}^{{\dagger}}+e^{-i\omega_{n}t}\hat{a}_{n}\right)$
$\displaystyle+i\omega_{k^{\prime}}\left(e^{2i\omega_{k^{\prime}}t}\Lambda_{k^{\prime}}^{-}(t)\hat{a}_{k^{\prime}}^{\dagger}-e^{-2i\omega_{k^{\prime}}t}\Lambda_{k^{\prime}}^{-}(t)\hat{a}_{k^{\prime}}-i\omega_{k^{\prime}}\Lambda_{pk^{\prime}}(t)(\hat{a}_{k^{\prime}}^{\dagger}+\hat{a}_{k^{\prime}})-\Lambda_{xk^{\prime}}(t)(\hat{a}_{k^{\prime}}^{\dagger}-\hat{a}_{k^{\prime}})\right)$
$\displaystyle\left.+i\omega_{k}\left(e^{2i\omega_{k}t}\Lambda_{k}^{-}(t)\hat{a}_{k}^{\dagger}-e^{-2i\omega_{k}t}\Lambda_{k}^{-}(t)\hat{a}_{k}-i\Lambda_{pk}(t)(\hat{a}_{k}^{\dagger}+\hat{a}_{k})-\Lambda_{xk}(t)(\hat{a}_{k}^{\dagger}-\hat{a}_{k})\right)\right]\left(e^{i\omega
t}\hat{b}^{\dagger}+e^{-i\omega t}\hat{b}+2\xi(t)\right).$ (61)

where we have defined the following auxiliary functions:

$\displaystyle\Lambda_{j}^{-}(t)=$
$\displaystyle\Lambda_{xj}(t)-\Lambda_{pj}(t),$ (62) $\displaystyle\xi(t)=$
$\displaystyle\cos(\omega t)\Lambda_{pb}(t)-\sin(\omega t)\Lambda_{xb}(t),$
(63) $\displaystyle\psi_{2}(t)=$
$\displaystyle\cos(\omega_{k}t)\Lambda_{pk}(t)+\sin(\omega_{k}t)\Lambda_{xk}(t),$
(64) $\displaystyle\psi_{3}(t)=$
$\displaystyle\cos(\omega_{k^{\prime}}t)\Lambda_{pk^{\prime}}(t)+\sin(\omega_{k^{\prime}}t)\Lambda_{xk^{\prime}}(t),$
(65) $\displaystyle\vartheta(\omega_{k},\omega_{k^{\prime}},t)=$
$\displaystyle-\omega_{k}\left([\Lambda_{pk}(t)]^{2}+[\Lambda_{xk}(t)]^{2}\right)-2\omega_{k}\sin(2\omega_{k}t)\Lambda_{pk}(t)\Lambda_{xk}(t)-\omega_{k}\cos(2\omega_{k}t)\left([\Lambda_{xk}(t)]^{2}+[\Lambda_{pk}(t)]^{2}\right)$
$\displaystyle-\omega_{k^{\prime}}\left([\Lambda_{pk^{\prime}}(t)]^{2}+[\Lambda_{xk^{\prime}}(t)]^{2}\right)-2\omega_{k^{\prime}}\sin(2\omega_{k^{\prime}}t)\Lambda_{pk^{\prime}}(t)\Lambda_{xk^{\prime}}(t)-\omega_{k^{\prime}}\cos(2\omega_{k^{\prime}}t)\left([\Lambda_{xk^{\prime}}(t)]^{2}+[\Lambda_{pk^{\prime}}(t)]^{2}\right)$
$\displaystyle+(-1)^{k+k^{\prime}}4\sqrt{\omega_{k}\omega_{k^{\prime}}}\left(\sin(\omega_{k}t)\sin(\omega_{k^{\prime}}t)[\Lambda_{xk^{\prime}}(t)]^{2}+\cos(\omega_{k}t)\cos(\omega_{k^{\prime}}t)[\Lambda_{pk^{\prime}}(t)]^{2}\right),$

with $j\equiv k,k^{\prime}$ or $b$.

##  Appendix B Position of the wall: second order correction

The correction to the position of the wall estimated to the second order in
$\epsilon$ is performed by calculating the second term of Equation (20) when
$\hat{A}\equiv\hat{X}$. This gives:

$\displaystyle\frac{\Delta x^{(2)}(t)}{\delta L_{0}}=$
$\displaystyle\frac{\omega_{k}\mu_{k}^{2}t}{2}\left[\omega
t\operatorname{sinc}^{2}\left(\frac{\omega
t}{2}\right)+\sin\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+\sin\left[\frac{(2\omega_{k}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\right]$
$\displaystyle+\frac{\omega_{k^{\prime}}\mu_{k^{\prime}}^{2}t}{2}\left[\omega
t\operatorname{sinc}^{2}\left(\frac{\omega
t}{2}\right)+\sin\left[\frac{(2\omega_{k^{\prime}}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k^{\prime}}+\omega)t}{2}\right]+\sin\left[\frac{(2\omega_{k^{\prime}}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k^{\prime}}-\omega)t}{2}\right]\right]$
$\displaystyle-(-1)^{k+k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\mu_{k}\mu_{k^{\prime}}t\left[\sin\left[\frac{(\omega_{k}-\omega_{k^{\prime}}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}-\omega)t}{2}\right]\right.$
$\displaystyle+\sin\left[\frac{(\omega_{k}-\omega_{k^{\prime}}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}+\omega)t}{2}\right]+\sin\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t}{2}\right]$
$\displaystyle\left.+\sin\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t}{2}\right]\right]$
$\displaystyle+i(-1)^{k+k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\left[\mu_{k^{\prime}}\left(e^{i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k}(t^{\prime})e^{i(\omega_{k^{\prime}}-\omega)t^{\prime}}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k}(t^{\prime})e^{-i(\omega_{k^{\prime}}-\omega)t^{\prime}}\right.\right.$
$\displaystyle\left.+e^{i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k}(t^{\prime})e^{i(\omega_{k^{\prime}}+\omega)t^{\prime}}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k}(t^{\prime})e^{-i(\omega_{k^{\prime}}+\omega)t^{\prime}}\right)+\mu_{k}\left(e^{i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k^{\prime}}(t^{\prime})e^{i(\omega_{k}-\omega)t^{\prime}}\right.$
$\displaystyle\left.\left.-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k^{\prime}}(t^{\prime})e^{-i(\omega_{k}-\omega)t^{\prime}}+e^{i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k^{\prime}}(t^{\prime})e^{i(\omega_{k}+\omega)t^{\prime}}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\psi_{k^{\prime}}(t^{\prime})e^{-i(\omega_{k}+\omega)t^{\prime}}\right)\right]$
$\displaystyle+$ $\displaystyle\mu_{k}\omega_{k}\left[i\left(e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{pk}(t^{\prime})e^{i\omega
t^{\prime}}-e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{pk}(t^{\prime})e^{-i\omega
t^{\prime}}\right)+\frac{1}{2}\left(e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k}^{-}(t^{\prime})e^{i(2\omega_{k}+\omega)t^{\prime}}\right.\right.$
$\displaystyle\left.\left.+e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k}^{-}(t^{\prime})e^{i(2\omega_{k}+\omega)t^{\prime}}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k}^{-}(t^{\prime})e^{-i(2\omega_{k}-\omega)t^{\prime}}-e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k}^{-}(t^{\prime})e^{i(2\omega_{k}-\omega)t^{\prime}}\right)\right]$
$\displaystyle+$
$\displaystyle\mu_{k^{\prime}}\omega_{k^{\prime}}\left[i\left(e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{pk^{\prime}}(t^{\prime})e^{i\omega
t^{\prime}}-e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{pk^{\prime}}(t^{\prime})e^{-i\omega
t^{\prime}}\right)+\frac{1}{2}\left(e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k^{\prime}}^{-}(t^{\prime})e^{i(2\omega_{k^{\prime}}+\omega)t^{\prime}}\right.\right.$
$\displaystyle\left.\left.+e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k^{\prime}}^{-}(t^{\prime})e^{i(2\omega_{k^{\prime}}+\omega)t^{\prime}}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k^{\prime}}^{-}(t^{\prime})e^{-i(2\omega_{k^{\prime}}-\omega)t^{\prime}}-e^{i\omega
t}\int_{0}^{t}dt^{\prime}\Lambda_{k^{\prime}}^{-}(t^{\prime})e^{i(2\omega_{k^{\prime}}-\omega)t^{\prime}}\right)\right]$
$\displaystyle+\frac{1}{2i}\left(e^{i\omega
t}\int_{0}^{t}dt^{\prime}\vartheta(\omega_{k},\omega_{k^{\prime}},t^{\prime})e^{-i\omega
t}-e^{-i\omega
t}\int_{0}^{t}dt^{\prime}\vartheta(\omega_{k},\omega_{k^{\prime}},t^{\prime})e^{i\omega
t}\right).$ (67)

##  Appendix C Number of photons: first order correction

The correction to the photon number in the mode $k$ estimated up to the first
order is performed by calculating the second term of Equation (20) when
$\hat{A}\equiv\hat{a}_{k}^{\dagger}\hat{a}_{k}$. This reads:

$\displaystyle\Delta N_{k}^{(1)}(t)=$
$\displaystyle-2\mu_{k}^{2}\omega_{k}\left[\lvert\beta\rvert
t\sin\left[\frac{(2\omega_{k}+\omega)t-2\theta}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+\lvert\beta\rvert
t\sin\left[\frac{(2\omega_{k}-\omega)t+2\theta}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\right.$
$\displaystyle\left.+2\int_{0}^{t}dt^{\prime}\xi(t^{\prime})\sin(2\omega_{k}t^{\prime})\right]+(-1)^{k+k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\mu_{k}\mu_{k^{\prime}}\lvert\beta\rvert
t\left[\frac{4}{\lvert\beta\rvert
t}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})\sin\left(\omega_{k^{\prime}}t^{\prime}\right)\cos\left(\omega_{k}t^{\prime}\right)\right.$
$\displaystyle+\sin\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t-2\theta}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t}{2}\right]+\sin\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t+2\theta}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t}{2}\right]$
$\displaystyle\left.+2\cos\left[\frac{(\omega
t-2\theta)}{2}\right]\sin\left[\frac{(\omega_{k}-\omega_{k^{\prime}})t}{2}\right]\left(\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}+\omega)t}{2}\right]+\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}-\omega)t}{2}\right]\right)\right]$
$\displaystyle+4(-1)^{k+k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\mu_{k}\int_{0}^{t}dt^{\prime}\sin(\omega_{k}t^{\prime})\psi_{k^{\prime}}(t^{\prime})\left(2\cos(\omega
t^{\prime}-\theta)+\xi(t^{\prime})\right)$
$\displaystyle+4\omega_{k}\mu_{k}\int_{0}^{t}dt^{\prime}\left[\lvert\beta\rvert\Lambda_{xk}(t^{\prime})\cos(\omega
t^{\prime}-\theta)+\Lambda_{k}^{-}(t^{\prime})\cos(2\omega_{k}t^{\prime})\cos(\omega
t^{\prime}-\theta)+\Lambda_{k}^{-}\xi(t^{\prime})\cos(2\omega_{k}t^{\prime})-\Lambda_{xk}(t^{\prime})\xi(t^{\prime})\right]$
$\displaystyle-2\Lambda_{xk}(t)\left\\{2\mu_{k}\omega_{k}t\cos\left[\frac{(\omega
t-2\theta)}{2}\right]\operatorname{sinc}\left(\frac{\omega
t}{2}\right)+2\mu_{k}\omega_{k}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})+2\mu_{k}\omega_{k}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})\cos(2\omega_{k}t^{\prime})\right.$
$\displaystyle+\mu_{k}\lvert\beta\rvert\omega_{k}t\cos\left[\frac{(2\omega_{k}+\omega)t+2\theta}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+\mu_{k}\lvert\beta\rvert\omega_{k}t\cos\left[\frac{(2\omega_{k}-\omega)t-2\theta}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]$
$\displaystyle+(-1)^{k+k^{\prime}}\mu_{k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\lvert\beta\rvert
t\left(\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t}{2}\right]\cos\left[\frac{(\omega_{k}+\omega_{k^{\prime}}+\omega)t+2\theta}{2}\right]\right.$
$\displaystyle+\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}+\omega)t}{2}\right]\cos\left[\frac{(\omega_{k}-\omega_{k^{\prime}}+\omega)t+2\theta}{2}\right]+\operatorname{sinc}\left[\frac{(\omega_{k}-\omega_{k^{\prime}}-\omega)t}{2}\right]\cos\left[\frac{(\omega_{k}-\omega_{k^{\prime}}-\omega)t-2\theta}{2}\right]$
$\displaystyle\left.+\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t}{2}\right]\cos\left[\frac{(\omega_{k}+\omega_{k^{\prime}}-\omega)t+2\theta}{2}\right]+\frac{8}{\lvert\beta\rvert}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})\cos(\omega_{k}t^{\prime})\cos(\omega_{k^{\prime}}t^{\prime})\right)$
$\displaystyle-2(-1)^{k+k^{\prime}}\sqrt{\omega_{k}\omega_{k^{\prime}}}\int_{0}^{t}dt^{\prime}\psi_{k^{\prime}}(t^{\prime})\big{(}4\lvert\beta\rvert\cos(\omega_{k}t^{\prime})\cos(\omega
t^{\prime}-\theta)+\xi(t^{\prime})\cos(\omega_{k}t^{\prime})\big{)}$
$\displaystyle\left.+2\omega_{k}\int_{0}^{t}dt^{\prime}\left[\Lambda_{pk}(t^{\prime})(\lvert\beta\rvert\cos(\omega
t^{\prime}-\theta)+\xi(t^{\prime}))+\Lambda_{k}^{-}(t^{\prime})\sin(2\omega_{k}t^{\prime})\left(2\lvert\beta\rvert\cos(\omega
t^{\prime}-\theta)+\xi(t^{\prime})\right)\right]\right\\}.$ (68)

##  Appendix D Number of photons: second order correction

The correction to the number of photons up to the second order, when the
initial state is $\hat{\rho}(0)=\prod_{n}\lvert 0_{n}\rangle\langle
0_{n}\rvert\otimes\hat{\rho}_{\textrm{m}}^{\textrm{(DST)}}$, can be expressed
as a combination of terms, collected in Equation (42). In more details:

$\displaystyle\Delta N^{(2)}_{\beta,k}(t)=$ $\displaystyle
2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]+2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+4\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\cos(\omega
t-2\theta)$ $\displaystyle+\sum_{n\neq
k}\frac{\omega_{n}}{\omega_{k}}\left\\{\left(\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]+\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\right)\right.$
$\displaystyle\left.+2\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\cos(\omega
t-2\theta)\right\\},$ (69)

is the contribution stemming from the coherent state of the quantum harmonic
oscillator;

$\displaystyle\Delta N^{(2)}_{\textrm{vac, k}}(t)=$ $\displaystyle
2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+\sum_{n\neq
k}\frac{\omega_{n}}{\omega_{k}}\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right],$
(70)

is the correction due to the quantum vacuum state of the mechanical
oscillation of the wall, and represents the contribution to the number of
photons due to the fluctuation of the phonon number;

$\displaystyle\Delta N^{(2)}_{\textrm{sq},k}(t)=$ $\displaystyle
2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]\sinh
r+2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\sinh r$
$\displaystyle-4\cosh
r\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\cos(\omega
t-\phi)$ $\displaystyle+\sum_{n\neq
k}\frac{\omega_{n}}{\omega_{k}}\left\\{\left(\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]+\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\right)\sinh
r\right.$
$\displaystyle\left.-2\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\cosh
r\cos(\omega t-\phi)\right\\}$ $\displaystyle\Delta N^{(2)}_{T,k}(t)=$
$\displaystyle
2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+2\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]$
$\displaystyle+\sum_{n\neq
k}\frac{\omega_{n}}{\omega_{k}}\left(\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]+\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\right)$
$\displaystyle\Delta N^{(2)}_{\textrm{sq},T,k}(t)=$ $\displaystyle 4\,\sinh
r\left[\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]+\operatorname{sinc}^{2}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\right]$
$\displaystyle-4\,\cosh
r\,\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]\cos(\omega
t-\phi)$ $\displaystyle+\sum_{n\neq
k}\frac{\omega_{n}}{\omega_{k}}\left\\{2\,\sinh
r\left(\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]+\operatorname{sinc}^{2}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\right)\right.$
$\displaystyle\left.-4\,\cosh
r\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]\cos(\omega
t-\phi)\right\\},$ (71)

are the corrections due to squeezing alone, initial temperature in the
mechanical element alone, and the interaction of the two; and finally

$\displaystyle N_{\textrm{md}}^{(2)}(t)=$ $\displaystyle
2\epsilon^{2}\omega_{k}^{2}t\left\\{\beta\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{-2i\omega_{k}t^{\prime}}\left[\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]e^{i\frac{(2\omega_{k}+\omega)t}{2}}+\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]e^{i\frac{(2\omega_{k}-\omega)t}{2}}\right]\right.$
$\displaystyle+\left.\beta^{*}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{2i\omega_{k}t^{\prime}}\left[\operatorname{sinc}\left[\frac{(2\omega_{k}+\omega)t}{2}\right]e^{-i\frac{(2\omega_{k}+\omega)t}{2}}+\operatorname{sinc}\left[\frac{(2\omega_{k}-\omega)t}{2}\right]e^{-i\frac{(2\omega_{k}-\omega)t}{2}}\right]\right\\}$
$\displaystyle+\sum_{n\neq
k}\frac{\epsilon^{2}\omega_{k}\omega_{n}t}{2}\left\\{\beta\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{-i(\omega_{k}+\omega_{n})t^{\prime}}\left[\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]e^{i\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}}\right.\right.$
$\displaystyle\left.+\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]e^{i\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}}\right]$
$\displaystyle+\left.\beta^{*}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{i(\omega_{k}+\omega_{n})t^{\prime}}\left[\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}\right]e^{-i\frac{(\omega_{k}+\omega_{n}+\omega)t}{2}}+\operatorname{sinc}\left[\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}\right]e^{-i\frac{(\omega_{k}+\omega_{n}-\omega)t}{2}}\right]\right\\}$
$\displaystyle+4\epsilon^{2}\omega_{k}^{2}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{-2i\omega_{k}t^{\prime}}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{2i\omega_{k}t^{\prime}}+\sum_{n\neq
k}\epsilon^{2}\omega_{k}\omega_{n}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{-i(\omega_{k}+\omega_{n})t^{\prime}}\int_{0}^{t}dt^{\prime}\xi(t^{\prime})e^{i(\omega_{k}+\omega_{n})t^{\prime}},$
(72)

is the number of photons due to the presence of the mechanical drive.