# Photon generation and entanglement in a double superconducting cavity

Cruz I. Velasco1 ID Nicolás F. Del Grosso1,2 ID Fernando C. Lombardo1,2 ID
Alejandro Soba3 Paula I. Villar1,2 1 Departamento de Física Juan José
Giambiagi, FCEyN UBA, Facultad de Ciencias Exactas y Naturales, Ciudad
Universitaria, Pabellón I, 1428 Buenos Aires, Argentina  2 IFIBA CONICET-UBA,
Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria, Pabellón I,
1428 Buenos Aires, Argentina  3 Centro Atómico Constituyentes, Comisión
Nacional de Energía Atómica, Avenida General Paz 1499, San Martín, Argentina

(today)

###### Abstract

We study the dynamical Casimir effect in a double superconducting cavity in a
circuit quantum electrodynamics architecture. Parameters in the quantum
circuit are chosen in such a way the superconducting cavity can mimic a double
cavity, formed by two perfectly conducting outer walls and a dielectric one,
with arbitrary permittivity separating both halves. We undertake a spectral
analysis of the cavity, showing that the spectrum varies significantly
depending on the values of the susceptibility of the dielectric mirror and the
relative lengths of both cavities. We study the creation of photons when the
walls oscillate harmonically with a small amplitude. Furthermore, we explore
the possibility of entangling two uncoupled cavities, starting from a
symmetric double cavity and having both of its halves become uncoupled at a
later given instant. We consider both cases: (i) when the field is initially
in a vacuum state and (ii) the situation in which photon creation via the
dynamical Casimir effect has already taken place. We show that the cavities
become entangled in both cases but, in the latter, the quantum correlation
between individual modes can be greatly increased at the cost of diminishing
the entanglement between most pairs of modes.

##  I Introduction

The dynamical Casimir effect (DCE), in which the mechanical oscillation of the
position of a mirror inside a cavity produces particles from an initial vacuum
state reviews ; moore ; DeWitt ; FullingDavies ; Dodonov ; dalvit shaker ;
Plunien ; crocceScalar , remains one of the most surprising predictions of
quantum field theory. However, the magnitude of the frequencies at which
mechanical oscillations must occur to produce an appreciable number of
photons, can not be reached with current technology for massive mirrors
fiftyYears . Consequently no observations of this phenomena has taken place to
this date.

To bypass this complication, a lot of systems have been studied as to explore
the possibility of varying the boundary conditions of a field inside the
cavity, without the mechanical movement of the walls. One of the more
promising systems where this can be achieved are quantum superconducting
circuits, which are already being used in quantum information and quantum
computing widely qubit_circuit ; computacionChina ; computacionCanada . In
such a case, an external and time dependent magnetic field is used to vary the
magnetic flux through a superconducting quantum interference device (SQUID),
and through it, change the effective length of a superconducting waveguide,
simulating a moving mirror. This type of systems has led to the observation of
this effect squidDemo . Even so, experiments have yet to show the conversion
of mechanical energy (associated with the movement of mirrors) into photons.

In this work we present a study of a circuit conformed by two waveguides that
are coupled between them by a SQUID and each waveguide has also another SQUID
at its other end. This system, under a certain choice of parameters, can
easily be linked to one composed by a one dimensional double cavity that has
perfect conductors as its external walls and a semitransparent dielectric wall
between them. The two systems share similar generalized boundary conditions
contornosGenerales , which admits an equivalent mathematical description of
both. However, the cavity allows for a physically more intuitive picture.
Another interesting property of these systems is that, through the adjustable
parameters they have, one could, in principle obtain different eigenfrequency
spectrum structures. This is of great importance when studying the DCE, as
different regimes of said structure can result in different results for the
photon production rate crocceScalar .

Finally, we have mentioned that superconducting circuits are relevant for
quantum information. The same can be said for double cavity systems. It is a
well known fact that the quantum vacuum presents entanglement between
spatially separated regions entvacio . In reference HalfEmpty , authors
studied the possibility of harvesting entanglement from a vacuum state in a
simple cavity into a two cavity system by quickly introducing a mirror. This
might result in the generation of two entangled cavities that could be
spatially separated and used in quantum communication applications. The double
cavity system treated in the present manuscript presents a natural framework
to continue the study of this problem. It is also well known that the DCE can
allow for the creation of entangled pair of photons dodonovEntanglement , and
even the redistribution of entanglement between field modes that already
present some degree of quantum correlation entNico . This means that, in
principle, the DCE could be used to improve the results of entanglement
harvesting from an initial vacuum state.

The article is organized as follows. In Sec. II we present the model that will
be used for the rest of the work. We show how the mathematical description of
the two systems is equivalent, under certain choice of parameters set. In Sec.
III, we present an analysis of the eigenfrequency spectrum of the double
cavity for different choices of the electric susceptibility of the dielectric
wall and the positions of the three walls. In Sec. IV, we show the results for
photon creation as a consequence of the harmonic oscillation of the perfectly
conducting walls of the double cavity. We carry out the study both through
analytical and numerical means. For the analytical results, we employ a method
known as multiple scale analysis. The numerical method consists of an
integration of the exact differential equations that govern the state of the
field with time dependent boundary conditions. Further, in Sec. V, we study
the possibility of using the DCE to increase the entanglement between cavities
that have been decoupled with respect to the previously studied case of an
initial vacuum state. Finally, in Sec. VI we present the conclusions of the
work.

##  II The Model

The system is a superconducting circuit of length $L$ schematized in the upper
half of Fig. 1. It is composed of two one-dimensional superconducting
waveguides of length $L_{1}=(L+L_{0})/2$ (left) and $L_{2}=(L-L_{0})/2$
(right) which are coupled by a SQUID in $x=0$. Both waveguides also have
SQUIDs in their other ends, located at $x=-L_{1}$ (in the left’s case) and
$x=L_{2}$ (in the right’s). We consider both of the waveguides to be
characterized by the same capacitance and inductance per unit length, $c_{a}$
and $l$ respectively. In such a situation, it is convenient to describe the
state of the electromagnetic field inside the circuit in terms of the phase
field: $\varphi(x,t)=\int^{t}dt^{\prime}E(x,t^{\prime})$. The Lagrangian
density for the system can be written as squidPauyFer

$\mathcal{L}_{\rm
circuit}=\frac{1}{2}\Big{(}\frac{1}{2e}\Big{)}^{2}(C(\partial_{t}\varphi)^{2}-v_{w}^{2}c_{a}(\partial_{x}\varphi)^{2})-E(t)\varphi^{2}$
(1)

where we have defined $v_{w}=(lc_{a})^{-1/2}$,

$C=(c+2\delta(x)C^{2}_{J}+2\delta(x+L_{1})C_{J}^{1}+2\delta(x-L_{2})C_{J}^{2})$
(2)

and

$\displaystyle
E(t)=\delta(x)E_{J}^{0}\cos(f_{0}(t))+\delta(x+L_{1})E_{J}^{1}\cos(f_{1}(t))$
$\displaystyle+\delta(x-L_{2})E_{J}^{2}\cos(f_{2}(t)).$ (3)

Figure 1: Schematics of the superconducting quantum circuit and its double
cavity analogue.

Here $C^{0}_{J}$ and $E^{0}_{J}$ are, respectively, the Josephson capacitance
and energy of the central SQUID. The SQUIDS at the outer ends of the circuit
are also characterized by their capacitance ($C_{J}^{1,2}$) and energy
($E_{J}^{1,2}$). The Josephson energy of all three SQUIDs can be manipulated
by modifying the magnetic flux that goes through it. The variation of said
magnetic flux is characterized by $f_{i}(t)$ ($i=0,1,2$). In the previous
equations and for the rest of this work we consider $\hbar=c=1$.

The field equation for $\varphi$ can be readily obtained from Eq. (1) and
reads

$\square\varphi(x,t)=0.$ (4)

The field must be continuous and satisfy boundary conditions imposed by the
presence of the three SQUIDs. These conditions can be obtained from Eq. (1)
and consist of discontinuities of the spatial derivative of $\varphi$, in
$x=-L_{1}$, $x=L_{2}$ and $x=0$. Particularly, for $x=0$ we get

$\displaystyle\partial_{x}\varphi(t,0^{+})-\partial_{x}\varphi(t,0^{-})=C_{J}^{0\prime}\partial_{t}^{2}\varphi(t,0)$
$\displaystyle+E_{J}^{0\prime}(t)\varphi(t,0).$ (5)

Herein, we have defined $C_{J}^{0\prime}=C_{J}^{0}/v_{w}^{2}c_{a}$ and
$E_{J}^{0\prime}(t)=(2e)^{2}(2E^{0}_{J}/v_{w}^{2}c_{a})\cos(f_{0}(t))$.

The other two conditions are analogous to Eq. (5) and can be used to decouple
the field inside the circuit from any external system by choosing
$E_{J}^{1}=E_{J}^{2}=E_{J}$ with $e^{2}E_{J}/v_{w}^{2}c_{a}\gg 1$ squidPauyFer
. This will allow us to consider that $\varphi(x,t)=0$ for $x<-L_{1}$ and
$x>L_{2}$. For simplicity, we also consider $C_{J}^{1,2}=0$ and that both
SQUIDs are subjected to a time dependent magnetic flux. Thus, the boundary
condition simplifies, allowing us to assume

$\displaystyle 0=\varphi(\mp
L_{1,2})+\frac{1}{E_{J}^{\prime}(t)}\partial_{x}\varphi(\mp L_{1,2})\approx$
$\displaystyle\approx\varphi\Big{(}\frac{\mp
L-L_{0}}{2}+\frac{1}{E_{J}^{\prime}(t)}\Big{)}.$ (6)

We have defined
$E_{J}^{\prime}(t)=(2e)^{2}\frac{2E_{J}}{v_{w}^{2}c_{a}}\cos(f(t))$ and
$f(t)=f_{1}(t)=f_{2}(t)$. It can be noted that the boundary conditions
introduced in (6) can be interpreted as in-phase changes in the effective
lengths of the two waveguides which can be characterized by

$\Delta L(t)=L_{0}-\frac{2}{E_{J}^{\prime}(t)}.$ (7)

This fact can be seen as a consequence of the mathematical analogy between
superconducting circuits and cavities. For the particular choice of parameters
made for our circuit, an analogous system can be found in a one-dimensional
double cavity of length $L=L_{1}+L_{2}$ (as schematized in Fig. 1), with
perfectly conducting outer walls and a semi-transparent dielectric wall
separating both halves in $x=0$. This wall is considered to be infinitesimally
thin, which is a reasonable approximation if we consider this type of systems
membrana_fina . This configuration results in a left cavity of length $L_{1}$
and a right cavity of length $L_{2}$, both sharing the central dielectric
wall. The dielectric membrane is characterized by its electric permittivity
$\epsilon$. A time dependent electric potential $V(t)$ is also applied on this
wall.

Further, we shall consider a scalar field $\hat{\phi}$ inside of the cavity
(as a simplified model of the electromagnetic field). It can be described by
the following Lagrangian density contornosGenerales

$\mathcal{L}=\frac{1}{2}\Big{(}\varepsilon(\partial_{t}\hat{\phi})^{2}-(\partial_{x}\hat{\phi})^{2}-V(t)\hat{\phi}^{2}\Big{)}$
(8)

where

$\varepsilon=1+\delta(x)\alpha$ (9)

and

$V(t)=\delta(x)v(t).$ (10)

From now on, we will use $\alpha$ and $v(t)$ to characterize the dielectric
membrane. For simplicity, we will refer to $\alpha$ as the wall’s electric
susceptibility, and $v(t)$ as the electric potential applied to it, despite
the difference in units of both parameters with said quantities.

Herein, we have taken a coordinate system with its origin in the position of
the dielectric wall, which results in the position of the perfectly conducting
walls being set at $x=-L_{1}$ for the left one, and $x=L_{2}$ for the one in
the right.

We shall consider the case where the position of the perfectly conducting
walls may vary in time ($L_{1}(t)$, $L_{2}(t)$) while the dielectric wall
stays static in $x=0$. Particularly, we assume that the movement of the
external walls is such that the total length of the double cavity does not
vary, i.e. $L_{1}(t)+L_{2}(t)=L$ remains a constant. Then, it is convenient to
characterize the movement of the walls through a displacement parameter
$\Delta L(t)$ such that,

$\Delta L(t)=L_{1}(t)-L_{2}(t).$ (11)

This definition also allows us to write $L_{1,2}=(L\pm\Delta L)/2$. $\Delta
L(t)$ defined here is completely analogous to the one defined in Eq. (7). It
is important to note that this equivalence means the moving walls can be
simulated by a variation of the magnetic flux through the SQUIDs at the
extremes of the circuit.

From Eq. (8) we can derive the field equation, which is equivalent to that
obtained for the circuit (Eq. (4)). Similarly, the field $\hat{\phi}$ must
also be continuous for all values of $x$ along the double cavity. The boundary
conditions appear as a consequence of the presence of the external perfectly
conducting mirrors and the central dielectric wall. For the former, we assume
Dirichlet conditions in $x=-L_{1}(t)$ and $x=L_{2}(t)$. For the latter,
located in $x=0$ we ask for the continuity of the field and, a discontinuity
of the spatial derivative of the field, obtained from Eq. (8), which reads

$\displaystyle\partial_{x}\hat{\phi}(t,x=0^{+})-\partial_{x}\hat{\phi}(t,x=0^{-})=\alpha\partial_{t}^{2}\hat{\phi}(t,0)$
$\displaystyle+v(t)\hat{\phi}(t,0)).$ (12)

By comparing Eq. (5) with (12), we obtain a direct analogy between the
susceptibility $\alpha$ of the dielectric wall with the capacitance of the
central SQUID $C_{J}^{0\prime}$, and the potential applied to the membrane
$v(t)$ with the Josephson energy of that same SQUID $E_{J}^{0\prime}(t)$.

As we have shown the equivalence between both systems, in the following we
shall refer to the mechanical cavity as it provides a more intuitive picture.

###  II.1 Eigenfunctions

Let us now consider the case where the double-cavity is in a static regime,
meaning neither $\Delta L$ nor $v$ vary in time. In this case, all of the
boundary conditions for the field inside the cavity become independent of
time. In this case, we can find a basis of eigenfunctions that are solutions
to the field equation (Eq. (4)) and satisfy the boundary conditions. These
functions are given by hasan

$\displaystyle
u_{m}=\frac{e^{-i\omega_{m}t}}{\sqrt{2\omega_{m}}N_{m}}\Big{[}\theta(-x)\sin(k_{m}L_{2})\sin(k_{m}(x+L_{1}))$
$\displaystyle-\theta(x)\sin(k_{m}L_{1})\sin(k_{m}(x-L_{2}))\Big{]},$ (13)

where $k_{m}$ corresponds to the eigenfunction’s wave-number, $\omega_{m}$ to
its temporal frequency (which in the one dimensional case is equal to $k_{m}$)
and $N_{m}$ is a normalization factor. The functions form an orthogonal basis
if the following generalized Klein-Gordon inner product is satisfied hasan

$\displaystyle(f,g)_{KG}=i\int_{-L_{1}}^{L_{2}}(\dot{f}(t,x)g^{*}(t,x)$
$\displaystyle-f(t,x)\dot{g}^{*}(t,x))(1+\delta(x)\alpha)dx.$ (14)

The normalization factor $N_{m}$ is defined in such a way that the basis is
orthonormal ($(u_{n},u_{l})_{KG}=\delta_{nl}$) and reads

$\displaystyle
N_{m}^{2}=\frac{1}{2}\Big{(}L_{1}\sin^{2}(k_{m}L_{2})+L_{2}\sin^{2}(k_{m}L_{1})$
$\displaystyle+\frac{1}{k_{m}}\sin(k_{m}L)\sin(k_{m}L_{1})\sin(k_{m}L_{2})\Big{)}.$
(15)

The values of $k_{m}$ are such that the eigenfunctions are consistent with Eq.
(12). This condition implies that

$\cos(k_{m}\Delta
L)-\cos(k_{m}L)=\frac{2\sin(k_{m}L)}{k_{m}\Big{(}\alpha+\frac{v}{k_{m}^{2}}\Big{)}},$
(16)

which defines the admissible values of $k_{m}$. This transcendental equation
will be studied further in Section III.

By using these eigenfunctions we can expand the field operator as Grenier

$\hat{\phi}(t,x)=\sum_{m}(\hat{a}_{m}u_{m}(t,x)+\hat{a}^{\dagger}_{m}u_{m}^{*}(t,x)),$
(17)

where $\hat{a}_{m}$ are the bosonic operators corresponding to the different
photon modes.

If we then consider moving boundaries or a time varying potential applied to
the central wall, the functions $u_{m}(t,x)$ can be expressed in terms of an
instantaneous basis Dodonov ; baseinst

$u_{m}(t,x)=\sum_{n}Q_{n}^{m}(t)\Phi_{n}(x,t)$ (18)

where $Q_{n}^{m}(t)$ are time dependent coefficients and

$\Phi_{n}(x,t)=\frac{1}{N_{n}}\Big{[}\theta(-x)\sin(k_{n}(t)L_{2}(t))\sin(k_{n}(t)(x+L_{1}(t)))-\theta(x)\sin(k_{n}(t)L_{1}(t))\sin(k_{n}(t)(x-L_{2}(t)))\Big{]}.$
(19)

It is important to note that the wave-numbers dependence on time comes from
the time dependence of $\Delta L$ and $v$ in Eq. (16). The functions given in
(19) form an orthonormal basis corresponding to the generalized inner product

$\displaystyle(f,g)=\int_{-L_{1}}^{L_{2}}f(x)g(x)(1+\delta(x)\alpha)dx,$ (20)

that we use throughout the work.

##  III Double cavity spectrum

In order to make a complete description of the double cavity, we need to study
the solutions to Eq. (16). For simplicity, we consider the case where
$v(t)=0$, i.e. there is no potential applied at the dielectric wall, and thus,
the solutions will only depend on $L$, $\Delta L$ and $\alpha$. We assume the
value of $L$ to be fixed, and use it to define the dimensionless quantities
$\Delta L/L$ and $\alpha/L$. Additionally, we consider that the walls of the
cavity are not moving, which implies that $\Delta L$ is constant. We must note
that as Eq. (16) is a transcendental equation, it cannot be solved
analytically for general values of $\Delta L/L$ and $\alpha/L$. There are
however, two sets of values of $\alpha/L$ where approximate analytical
solutions can be found, namely $k_{m}\alpha\ll 1$ and $k_{m}\alpha\gg 1$.
Furthermore, in the limit cases $\alpha/L=0$ and $\alpha/L\rightarrow\infty$,
the analytical solutions become exact.

We note now that in general, we must consider the value of $k_{m}\alpha$
instead of just $\alpha/L$ when approximating because of the way $\alpha$
appears in Eq. (16), multiplied by $k_{m}$. This caveat is particularly
important when dealing with lower values of $\alpha$, as we will see later.

We shall start with the general case of the dielectric wall, and show later
the cases where analytical solutions can be applied.

###  III.1 Dielectric wall

For cases in which the susceptibility is such that $k_{m}\alpha\sim 1$, we
cannot use neither of both approximations and so we must solve the Eq. (16)
numerically. To do this we use a similar method to the one employed in
squidPauyFer . In Fig. 2 we show the first five wave numbers, corresponding to
$k_{0}$, $k_{1}$, $k_{2}$, $k_{3}$, and $k_{4}$ for different lengths of the
left cavity ($L_{1}/L$). All of the wave numbers are also plotted for
different values of the electric susceptibility ($\alpha/L$) according to the
colors detailed in the legend.

Figure 2: Wave-number spectrum as a function of the length of the left cavity
($L_{1}/L$) for different values of $\alpha/L$ between 0 and 120. The figure
shows said values for the first 5 modes
($k_{0}$,$k_{1}$,$k_{2}$,$k_{3}$,$k_{4}$).

In Fig. 2, we can see that for $\alpha/L=0$ the wave-numbers are given by
straight horizontal lines which are integer multiples of $\pi/L$ and thus, for
this case $k_{m}$ results independent of the relative size of the left cavity,
and in extension of $\Delta L/L$. This is consistent with the fact that for
$\alpha/L=0$ the dielectric wall becomes completely transparent, and thus, the
wavenumbers are those corresponding to the case of a single cavity of length
$L$. On the other hand, for high values of the susceptibility
($\alpha/L=120$), we can note that the wave-numbers (except $k_{0}$) are
proportional to either $L_{1}^{-1}$ or $(L-L_{1})^{-1}=L_{2}^{-1}$. In this
case, the value of $\alpha/L$ turns the dielectric wall into an almost
perfectly conducting wall. This results in the system becoming one of two
cavities with perfectly conducting walls at their ends one of which they share
among each other. As a consequence of this, the wavenumbers (with $m>1$)
become either integer multiples of $\pi/L_{1}$ or $\pi/L_{2}$. In the
particular case of the first mode ($k_{0}$), we see that as the value of
$\alpha/L$ increases, its value decreases. As we will see in Section III.3,
this mode becomes irrelevant as the susceptibility becomes larger.

For intermediate values of $\alpha/L$ we get a continuous connection between
the two limit cases where, as the value of $\alpha/L$ increases (starting from
$\alpha/L=0$), the wave-numbers gain a dependence on $L_{1}$ and,
consequently, on $\Delta L$. As the value of the susceptibility rises, the
dependence becomes more apparent, until we get $k_{m}\propto L_{1,2}^{-1}$.
Another feature is that the more energetic modes coincide with the latter
extreme value ($k_{m}\propto L_{1,2}^{-1}$) for lower values of $\alpha/L$
($\alpha/L=0.9$ for $k_{4}$) than the ones that are less energetic
($\alpha/L=3.6$ for $k_{1}$). Finally, the “zeroth”-mode’s wave-number, tends
to vanish at a slower rate than for which the rest obtain their extreme value.
This can be seen, for example, in the fact that while for $\alpha/L=3.6$ all
of the higher modes have already taken their $\propto L_{1,2}^{-1}$ shape,
$k_{0}$ has only decreased to approximately a third of its value at
$\alpha/L=0$.

An important factor to consider, mostly for the study of particle creation via
the DCE, is that of the spacing between modes, as an equidistant spectrum
(i.e. $k_{m+1}-k_{m}$ is constant for a given set of $m$) leads to very
different results than for one that is non-equidistant crocceScalar . From the
previous results, we can see that for both limit cases, we get equidistant
spectra. In the case of $\alpha=0$ we get $k_{m+1}-k_{m}=k_{0}$ for all $m$,
while for $\alpha/L\rightarrow\infty$ two different equidistant spectra
$k_{m+1}^{\pm}-k_{m}^{\pm}=k_{1}^{\pm}$. For more general values of $\alpha$
we again use the numerical solutions to Eq. (16). Fig. 3 shows the difference
between pairs of wave-numbers that, for large enough values of $\alpha/L$,
correspond to localized modes in either the right or left cavity. The values
shown in Fig. 3 correspond to the case where $L_{1}/L=0.6$, the spacing
between the modes will vary with the value of $L_{1}/L$.

Figure 3: Difference between wave-numbers as a function of $\alpha/L$ for
$L_{1}/L=0.6$. The differences considered are those between wave-numbers
corresponding to eigenfunctions that, in the case of
$\alpha/L\rightarrow\infty$, localize either in the left cavity (up) and in
the right cavity (down).

We can see that the behavior both for low and high values of $\alpha/L$ are
what we would expect. In the first case, we tend to have
$k_{m}-k_{n}=(m-n)\pi/L$, while in the latter, we have the same value for the
difference of all subsequent modes corresponding to the same cavity. For
values of $\alpha/L\sim 10$ we get equidistance for the spectra corresponding
to each of the cavities, however, for values of this order, the mode
corresponding to $m=0$ is still far from fulfilling this condition. This is
another consequence of $k_{0}$ reaching its limit value of zero for values of
$\alpha/L$ much higher than the rest of the modes. Anyway, for the rest of the
modes, we can consider that the spectrum inside of each of the perfectly
conducting cavities is equidistant for values of $\alpha/L$ of the order of
10.

In the following, we shall discuss the approximate solutions that can be
obtained for $k_{m}\alpha\ll 1$ and $k_{m}\alpha\gg 1$. These solutions lead
to a more accurate explanation of the behavior of $k_{m}$ as a function of
$\alpha$ and $\Delta L$ related to what we have been discussing in this
section.

###  III.2 Transparent wall

We shall start with $k_{m}\alpha\ll 1$. We consider the dielectric wall is
almost transparent, with $\alpha/L=0$ corresponding to a fully transparent
wall, that corresponds to a system consisting of a single cavity. The
particular case of the transparent wall, can be represented by the equations
$\sin(k_{m}L)=0$, with $k_{m}=(m+1)\pi/L$ as the solutions. As we have already
mentioned, in this case the wave-numbers become independent of $\Delta L$ and
thus, independent of the difference between the individual cavity sizes
($L_{1}$ and $L_{2}$). This is consistent with the fact that as the wall
becomes fully transparent, we are left with a single cavity instead of two.
Furthermore, by introducing $k_{m}=(m+1)\pi/L$ in Eq. (13) we obtain

$u_{m}^{0}(t,x)=\frac{e^{-i\omega_{m}^{0}t}}{\sqrt{2\omega_{m}^{0}}}\sqrt{\frac{2}{L}}\sin\bigg(\frac{(m+1)\pi}{L}\,x\bigg{missing}),$
(21)

where $\omega_{m}^{0}=(m+1)\pi/L$. These eigenfunctions are similar to the
ones corresponding to a one-dimensional single cavity with perfectly
conducting walls. Here, the first spatial derivative of $u_{m}^{0}$ becomes
continuous in $x=0$, as that point looses its physical importance when the
wall is transparent.

For the more general case, we can consider $k_{m}=(m\pi/L)(1+\kappa^{0}_{m})$
with $\kappa^{0}_{m}\ll 1$ and $(m\pi/L)\alpha\ll 1$ and solve Eq. (16) for
$\kappa^{0}_{m}$ to the first order. Hence, we obtain

$\displaystyle
k_{m}=\frac{(m+1)\pi}{L}\bigg{(}1-\frac{\alpha}{2L}\Big{(}1+(-1)^{m}$
$\displaystyle\times\cos\Big(\frac{(m+1)\pi}{L}\Delta
L\Big{missing})\Big{)}\bigg{)}.$ (22)

As we can see in the expression above, the first order term is proportional to
$(m+1)\alpha/L$ which grows as we consider higher wave numbers (higher values
of $m$). This means that, for non-zero values of $\alpha$, this approximation
will be worse for modes that are more energetic. We can see that the behavior
described by Eq. 22 is also similar to the case $\alpha/L=0.03$ for all modes,
where we essentially get oscillations with respect to the horizontal line of
amplitude $\propto\alpha/L$ and a frequency that depends on the mode we are
considering ($\propto(m+1)$).

###  III.3 Perfectly conducting wall

In the opposite case, $k_{m}\alpha\gg 1$, we assume a very reflective
dielectric wall, which may be considered to be an ’imperfect’ conductor (a
mirror). Particularly, the limit case of $\alpha/L\rightarrow\infty$
corresponds to a perfectly conducting wall separating both cavities. In this
case, we obtain a system consisting of two perfectly conducting cavities that
share one of their walls. The transcendental equation here simplifies to
$\cos(k_{m}\Delta L)-\cos(k_{m}L)=0$. This equation has “three” sets of
solutions: the first two solutions of the form $k_{m}^{\pm}=2m\pi/(L\pm\Delta
L)$ and the other one, a trivial solution $k_{0}=0$. By introducing
$k_{m}^{\pm}$ in the eigenmodes of Eq. (13), we get

$\displaystyle u_{m}^{\pm}(t,x)$ $\displaystyle=$
$\displaystyle\frac{e^{-i\omega^{\pm}_{m}t}}{\sqrt{2\omega_{m}^{\pm}}}\sqrt{\frac{4}{L\pm\Delta
L}}$ (23) $\displaystyle\times$ $\displaystyle\theta(\mp
x)\sin\Big(\frac{2m\pi}{L\pm\Delta L}\Big{(}x\pm\frac{L\pm\Delta
L}{2}\Big{)}\Big{missing})$

where $\omega^{\pm}=k_{m}^{\pm}$. Here we can see that depending on which
subset of solutions we choose, the eigenfunctions become localized in either
the left cavity ($x<0$) or the right one ($x>0$). This result will be
important when studying entanglement in Section V.

When considering $(2m\pi/(L\pm\Delta L))\alpha\gg 1$, we can solve the
transcendental equation with $k_{m}^{\pm}=(2m\pi/(L\pm\Delta
L))(1+\kappa_{m}^{\pm})$ for $\kappa^{\pm}\ll 1$ and we find the following
solution up to first order

$k_{m}^{\pm}=\frac{2m\pi}{(L\pm\Delta L)}\Big{(}1\mp\frac{(L\pm\Delta
L)}{2\alpha(m\pi)^{2}}\Big{)}.$ (24)

In this case, the approximations for the more energetic modes will be better
than for those with lower $m$. This can be seen in Eq. (24) where the first
order term is proportional to $L/(m\alpha)$.

As for the solution $k_{0}=0$, we assume $k_{0}=(\pi/L)\kappa_{0}$ with
$\kappa_{0}\ll 1$ (instead of $\alpha/L\gg 1$), and hence obtain a non-trivial
solution

$k_{0}=\sqrt{\frac{4L}{\alpha(L^{2}-\Delta L^{2})}}.$ (25)

We note that this “zeroth” mode wave-number depends differently on $L/\alpha$
in comparison to the other modes. Herein, we do not have a linear dependence
like for $k_{m}^{\pm}$, but $\sqrt{L/\alpha}$ instead. Not only does this mean
that the approximation will be valid for higher values of $\alpha/L$ than for
the rest of the modes, but also that it takes on its limiting value
($k_{0}=0$) for higher values of said parameter. It is important to note that
the eigenfunction corresponding to this wave-number does not localize in
either cavity when the susceptibility increases. Particularly, it decreases in
amplitude, vanishing as $\alpha/L\rightarrow\infty$. The fact that this mode
will not become localized, while also reaching its limit value at a slower
pace, indicates that, although all eigenfunctions corresponding to $m\geq 1$
are localized, there still is a non-localized eigenfunction for $m=0$, that
may allow us to “couple” the two cavities. As we shall see in a later section,
this feature leads to interesting results both for particle creation and
entanglement.

##  IV Particle creation

Previously, we have made a thorough study of the spectrum of the field inside
the double cavity for different values of the significant parameters. We have
further studied the behavior of the field’s eigenfunctions. Hence, in this
section, we can analyze the particle creation process. The results of the
previous section will greatly impact on the creation rate. We shall show the
benefit of having such a system in order to get, for example, the exponential
growth of the number of photons in a one dimensional cavity (parametric
resonance), a result that is not usually obtained in this type of systems
crocceScalar ; crocceVect . We shall also analyze the particular behavior of
the zeroth mode, obtaining interesting solutions that can not be obtained in a
double cavity with three perfectly conducing walls.

We consider the situation where the double cavity is initially at rest, with
the difference between cavity sizes given by $\Delta L(t<0)=L_{0}$. At a given
instant $t=0$ the walls of the cavity oscillate, following a trajectory such
that the distance among the walls is kept fixed. The walls oscillate for a
period of time $t_{f}$, at a constant frequency $\Omega$ and an amplitude
$L\epsilon$, modeled by

$\Delta L(t)=L_{0}+L\epsilon\sin(\Omega t).$ (26)

For times larger than $t=t_{f}$, the walls stop moving and the cavity becomes
static again. For simplicity, we consider that there is no potential
difference being applied over the dielectric wall ($v(t)=0$ for all $t$).

For $t<0$, the cavity can be described in terms of the static basis
$u_{m}(t,x)$ defined in Eq. (13) and bosonic operators that correspond to
photon modes in that region of the space-time, $a_{m}^{in}$. This reads

$\hat{\phi}=\sum_{m}(u_{m}(t,x)\hat{a}^{in\,}_{m}+u_{m}^{*}(t,x)\hat{a}_{m}^{in\,\dagger}).$
(27)

We consider the field to be in a state $\ket{in}$, which has a well-defined
number of photons of well-defined wave-number $k_{m}$ in $t<0$
($\langle\hat{N}_{m}\rangle=\expectationvalue{\hat{a}_{m}^{in\,\dagger}\hat{a}_{m}^{in}}=N_{m}^{0}$).

As the walls move, the original basis gets continually deformed into a new one
satisfying the boundary conditions $u_{m}(t,x)\rightarrow v_{l}(t,x)$. The
field can be expanded in this new basis

$\hat{\phi}=\sum_{l}(v_{l}(t,x)\hat{a}^{out\,}_{l}+v_{l}^{*}(t,x)\hat{a}_{l}^{out\,\dagger}),$
(28)

where we have defined new bosonic operators $\hat{a}_{n}^{out}$ corresponding
to a new notion of particles. The connection between the two basis, is given
by $v_{n}=\sqrt{2\omega_{n}}\sum_{m}(B_{n}^{m}u_{m}+A_{n}^{m}u_{m}^{*})$. This
can be introduced in Eq. (28) to find

$\hat{a}_{l}^{out}=\sqrt{2\omega_{l}}\sum_{m}(B_{l}^{m}\hat{a}_{m}^{in}+A_{l}^{m\,*}a_{m}^{in\,\dagger}),$
(29)

which is known as a Bogoliubov transformation Dodonov ; estadoDeVacio . By
using this relation among operators, we can find the number of particles of
well-defined momentum $k_{n}$ in the region corresponding to $t>t_{f}$

$\langle\hat{N}_{l}\rangle=\sqrt{2\omega_{l}}\sum_{m}((1+N_{m}^{0})|A_{l}^{m}|^{2}+N_{m}^{0}|B_{l}^{m}|^{2}).$
(30)

The result implies that the number of particles inside the cavity can vary
with the movement of the walls, even yet for the case of an initial vacuum
($N_{m}^{0}=0$ for all $m$) particles can be created by the movement of the
external walls. This is what is commonly known as Dynamical Casimir Effect
Dodonov ; crocceScalar .

As for the computation of the number of particles after having moved the
walls, we need to know $A_{l}^{m}$ and $B_{l}^{m}$. To do this, we need to
give a continuous description of the field during the time interval when the
walls move. Hence, we introduce the expansion of the $u_{m}(t,x)$ functions in
the instantaneous basis Eq. (19) into Eq. (27). Further, we introduce that
particular expression of $\hat{\phi}$ into the wave equation. By considering
the fact that the instantaneous basis is orthonormal, we find the following
differential equation for the time dependent coefficients

$\displaystyle\ddot{Q}_{l}^{(m)}+k_{l}^{2}Q_{l}^{(m)}=$
$\displaystyle=-\sum_{n}\Big{(}g_{nl}\Big{(}2\lambda\dot{Q}_{n}^{(m)}+\dot{\lambda}Q_{n}^{(m)}+h_{nl}\lambda^{2}Q_{n}^{(m)}\Big{)}\Big{)}$
(31)

where

$\displaystyle\lambda=\frac{\Delta\dot{L}(t)}{L},$ (32) $\displaystyle
g_{nl}=L(\partial_{\Delta L}\Phi_{n},\Phi_{l})$ (33)

and

$h_{nl}=L^{2}(\partial_{\Delta L}^{2}\Phi_{n},\Phi_{l}).$ (34)

This set of infinite coupled differential equations does not admit exact
analytical solutions and so it must be solved by either employing analytical
approximations or numerical methods perturbaciones ; MSA ; russer . We have
used both of these approaches which we shall briefly discuss in the following.

###  IV.1 Multiple scale analysis

We firstly consider some analytical approximations. In order to employ these
methods, we assume that the walls perform small harmonic oscillations
($\epsilon\ll\min\\{L_{1},L_{2}\\}/L$ in Eq. (26)) and search for solutions of
the form

$Q_{n}^{(m)}=A_{l}^{m}(\tau)e^{i\omega_{l}t}+B_{l}^{m}e^{-i\omega_{l}t},$ (35)

where we have introduced a second slower time scale $\tau=\epsilon t$
following the multiple scale analysis (MSA) procedure MSA . After introducing
these expressions in Eq. (31) and solving for the first order in $\epsilon$,
we obtain a system of infinite coupled differential equations for
$A_{l}^{m}(\tau)$ and $B_{l}^{m}(\tau)$ which reads

$\displaystyle\frac{dA_{l}^{m}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\eta_{l}}{2}\delta(\Omega-2\omega_{l})B_{l}^{m}$ (36)
$\displaystyle-$ $\displaystyle\frac{\Omega}{2\omega_{l}}\sum_{n\neq
l}g_{nl}\big{(}\omega_{n}-\frac{\Omega}{2}\big{)}\delta(\Omega-(\omega_{n}+\omega_{l}))B_{n}^{m}$
$\displaystyle-$ $\displaystyle\frac{\Omega}{2\omega_{l}}\sum_{n\neq
l}g_{nl}\Big{(}\big{(}\omega_{n}+\frac{\Omega}{2}\big{)}\delta(\Omega-(\omega_{l}-\omega_{n}))$
$\displaystyle+$
$\displaystyle\big{(}\omega_{n}-\frac{\Omega}{2}\big{)}\delta(\Omega-(\omega_{n}-\omega_{l})\big{)}\Big{)}A_{n}^{m}$

and

$\displaystyle\frac{dB_{l}^{m}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\eta_{l}}{2}\delta(\Omega-2\omega_{l})A_{l}^{m}$ (37)
$\displaystyle-$ $\displaystyle\frac{\Omega}{2\omega_{l}}\sum_{n\neq
l}g_{nl}(\omega_{n}-\frac{\Omega}{2})\delta(\Omega-(\omega_{n}+\omega_{l}))A_{n}^{m}$
$\displaystyle-$ $\displaystyle\frac{\Omega}{2\omega_{l}}\sum_{n\neq
l}g_{nl}\Big{(}\big{(}\omega_{n}+\frac{\Omega}{2})\delta(\Omega-(\omega_{l}-\omega_{n}))$
$\displaystyle+$
$\displaystyle\big{(}\omega_{n}-\frac{\Omega}{2}\big{)}\delta(\Omega-(\omega_{n}-\omega_{l})\big{)}\Big{)}B_{n}^{m}.$

Additionally, we consider initial conditions $A_{l}^{m}(0)=0$ and
$B_{l}^{m}(0)=\delta_{l}^{m}/\sqrt{2\omega_{l}}$, so that the description of
the field is continuous with the one given by Eq. (27) crocceScalar .

We can note that there are three conditions for the frequency of oscillations
of the walls ($\Omega$ in Eq. (26)) in Eqs. (36) and (37) that provide us with
non-trivial solutions for $A_{l}^{(m)}$ and $B_{l}^{(m)}$. The first of these
conditions is often known as single-mode resonance and reads

$\Omega=2\omega_{n}.$ (38)

Futher, there are two other conditions which are responsible of coupling
different modes of the field, namely

$\displaystyle\Omega$ $\displaystyle=\omega_{m}+\omega_{n}$ (39)
$\displaystyle\Omega$ $\displaystyle=|\omega_{m}-\omega_{n}|.$ (40)

The first two of said conditions, Eqs. (38) and (39), are responsible for the
creation of particles. The third condition Eq. (40), leads to the
redistribution of photons between coupled modes but, by itself, it cannot lead
to particle creation. Once $\Omega$ is set, more than just one of these
conditions can be met. This is determined by the structure of the wave-
number/eigenfrequency spectrum, and greatly affects both the rate at which the
particles are created and the energy of said particles.

It is also important to note that the weight of terms associated with the
single-mode resonance of mode $n$ is given by $\partial_{\Delta L}k_{n}$,
which is to say, the slope of the wave number regarding the displacement
$\Delta L$. This quantity can be obtained from the transcendental equation
(Eq. (16))

$\displaystyle\partial_{\Delta L}k_{n}=k_{n}\sin(k_{n}\Delta
L)\Big{(}\big{(}L+\frac{2}{\alpha k_{n}^{2}}\big{)}\sin(k_{n}L)$
$\displaystyle-\Delta L\sin(k_{n}\Delta L)-\frac{2L}{\alpha
k_{n}}\cos(k_{n}L)\Big{)}^{-1}.$ (41)

On the other hand, the strength of the coupling between modes $n$ and $l$ is
given, to a first order approximation, by $g_{nl}$ (Eq. (33)).

###  IV.2 Numerical method

We can generally solve Eq. (31) by means of numerical methods. Herein, we
consider a method similar to the one used in numerical , where some of us
integrate the differential equations by employing a Runge-Kutta fourth order
method between $t=0$ and $t_{max}>t_{f}$. Hence, for $t>t_{f}$ the analytical
solution to Eq. (31) reads

$Q_{n}^{(m)}(t)=\frac{1}{\sqrt{2\omega_{n}}}(A_{n}^{m}(t_{f})e^{-i\omega_{n}t}+B_{n}^{m}(t_{f})e^{i\omega_{n}t}).$
(42)

Further we multiply by $e^{\pm i\omega_{n}t}$ and take the time-average in
$t_{f}<t<t_{max}$, so as to obtain the values for $A_{n}^{m}(t_{f})$ and
$B_{n}^{m}(t_{f})$. Introducing these values in Eq. (30), we can then
calculate the number of particles in the cavity at $t=t_{f}$, when the cavity
is again at rest. Finally, by considering $t_{f}$ a continuous variable we can
repeat this process for several time intervals and obtain $\langle
N_{n}\rangle$ as a function of $t_{f}$.

It is important to recall that Eq. (31) constitutes an infinite set of coupled
differential equations. In order to perform the numerical integration of said
equations we must consider a finite number of modes of the field, and neglect
the rest. This in itself conforms an approximation, which, in order to be
valid, must fulfill certain conditions. In particular, we consider the
simulations to be valid until we see a considerable number of particles in a
mode that fulfills any of the coupling conditions given by Eqs. (39) and/or
(40) with a mode that is being neglected. The results presented in the
following section have been obtained using the first 15 modes of the double
cavity ($k_{0}$ to $k_{14}$), for both different initial configurations of the
double cavity (meaning different values of $\alpha/L$ and $L_{0}/L$) and for
different values of $\Omega$. We have set $\epsilon=3\times 10^{-3}$ as we
have seen this value is consistent with the small oscillations condition
$(\epsilon\ll\min\\{L_{1},L_{2}\\}/L)$ for all values of $L_{0}$ considered.

###  IV.3 Results and discussion

Figure 4: Number of particles as a function of the oscillation time
($t_{f}/L$) obtained through numerical simulations of the system considering
an external frequency of $\Omega=2\omega_{1}$ both for a non-equidistant
spectrum (a) and an equidistant one (b). In (a), we can observe the results of
the simulation for parameters: $\alpha/L=0.5$ and $L_{0}/L=0.44$. It is easy
to note that we only get single mode resonance for $\omega_{1}$ and no other
first order conditions. On the other hand, in (b) the double cavity considered
has parameters $\alpha/L=125$ and $L_{0}/L=0.44$. In this case, we can
consider all of the modes (with $n>0$) to be localized in a single cavity, the
solid lines representing modes that are localized in the left cavity, while
the dotted ones stand for those that are localized in the right one
(exemplified here by mode 3, the behavior being analogous for the rest of
them). Because of the equidistant spectrum in this second case, in addition to
the single mode resonance for mode 1, we also get frequency subtraction
coupling relations of the form $\Omega=\omega_{n+2}^{+}-\omega_{n}^{+}$ (as
$\omega_{1}=\omega_{1}^{+}$) which allow for the creation of particles in
modes 4, 6, 9 and 12. The rest of the modes corresponding to the left cavity
present no particle creation and their behavior is exemplified here by mode 2.
In both cases the analytical results for $\langle N_{1}\rangle$ are shown by
the black dot-dashed line. In (a) this was calculated by applying Eq. (43),
while in (b) these results are valid only for short oscillation time ($t\ll
300$).

We start with the case of a dielectric wall by setting $\alpha/L=0.5$. For the
lower modes this value of the susceptibility fulfills the condition
$k_{m}\alpha\sim 1$. This results in a non-equidistant frequency spectrum for
the less energetic modes. Firstly, we consider the external frequency to be
satisfying the single-mode resonance condition for mode $1$
($\Omega=2\omega_{1}$). We choose mode 1 instead of the less energetic mode 0
because, as we have discussed, the latter vanishes for large enough values of
the electric susceptibility. The field is considered to be in an initial
vacuum state. The results of this numerical simulation can be seen in Fig. 4
(a). Therein, we can observe the number of particles as a function of time for
this value of $\alpha/L$ and $L_{0}/L=0.44$. It is easy to note that there is
an initially exponential growth in the number of particles only for the
resonant mode. This feature agrees with the result predicted by solving Eqs.
(36) and (37) for this case. Results are shown together with the numerical
results as a black dot-dashed line, where the number of particles is predicted
to grow as

$\langle N_{1}\rangle=\sinh^{2}\Big{(}\frac{L\partial_{\Delta
L}k_{1}(L_{0})}{2}\epsilon t_{f}\Big{)}.$ (43)

We note that this result is consistent with previous results for similar
systems crocceScalar ; russer ; pau shaker . The exponential growth of the
resonant mode $\omega_{1}$ agrees with the result predicted by MSA, all other
modes, do not exhibit exponential growth. However, we can also see some
particle creation process for other modes at later times ($t_{f}>300$). We can
note that the number of photons in more energetic modes start growing,
although they do so at a slower rate than for mode 1. The solutions obtained
via the MSA do not predict photon number growth in modes other than the
resonant one, and the result seen in Fig. 4 corresponds to behaviors that can
be described by higher order methods. In any case, for this choice of
$\alpha/L$ and $L_{0}/L$ we manage to get exponential creation of photons from
an initial vacuum state, which is the expected result for this choice of
$\Omega$ and a non-equidistant spectrum.

Following, we can assume identical values for $L_{0}/L$ and $\Omega$, while
considering a much higher value of the electric susceptibility,
$\alpha/L=125$. The results obtained for this new choice of susceptibility can
be observed in Fig. 4 (b). Apart from the zeroth mode, which is expected not
to be localized, the solid line represents modes localized in the left cavity,
while the dotted lines indicate those localized in the right side.
Additionally, we show the MSA prediction for $\langle N_{1}\rangle$ which is
valid for short time scales ($t_{f}\ll 300$). We can immediately notice a
difference in the behavior of the system from the one observed for
$\alpha/L=0.5$ (Fig. 4 (a)). It is easily seen that for a short time period
($\tau\ll(\epsilon\Omega)^{-1}$), the number of particles in mode $1$ grow
following the MSA prediction, at a rate proportional to $t^{2}$. For longer
time periods, $\langle N_{1}\rangle$ grows at a slower rate ($\propto t$), as
well as those higher energy modes that fulfill the condition given by Eq.
(40).

For this value, all frequencies (other than $k_{0}$) can be considered to take
their limit values given by Eq. 24 and hence, the spectrum in this case
becomes equidistant. As this happens, we find that our choice of $\Omega$ no
longer exclusively fulfills the single-mode resonance condition (Eq. (38)),
but also frequency subtraction coupling conditions (Eq. (40)) for infinite
pairs of left cavity modes. This is a direct consequence of each of the left
cavity’s spectrum equidistance, as
$\omega^{+}_{k+2}-\omega^{+}_{k}=2\omega_{1}=\Omega$. Although the right
cavity is also equidistant, we do not have an equivalent relationship with
$\Omega$, because $L_{2}$ is not a multiple of $L_{1}$.

As we have noted before, the effects associated with the type of coupling
relevant to this case, is the redistribution of energy between coupled modes,
while the resonance condition is the one responsible of particle creation. The
first modes are the ones that are firstly populated, starting with the
resonant mode, while as time goes on, particles are created in more energetic
modes. This is consistent with similar cases that have already been studied in
the bibliography crocceScalar ; crocceVect . It is worthy noting that, for
$\alpha=0$, we cannot get particle creation via single-mode resonance because
$\partial_{\Delta L}k_{m}=0$ for any mode $m$ (see Eq. (41)), as the wave-
numbers of the field become independent from the relative size of the
individual cavities.

We now consider the case where $\Omega=\omega_{2}-\omega_{1}$. As we have
previously mentioned, this type of coupling by itself will not lead to the
creation of particles. Because of this, we consider that initially, mode 1 is
populated by a certain number of photons ($N_{1}^{0}=\langle
N_{1}\rangle(t<0)=50$). Again, we start by considering the case with the
dielectric wall ($\alpha/L\sim 0.5$) and $L_{0}/L=0.04$. This parameters will
yield a non-equidistant spectrum. The results for the simulation of this case
can be seen in Fig. 5 (a). As we can see, in the absence of other coupling
conditions, the photons oscillate harmonically between modes 1 and 2. This
result agrees with the analytical results of Eqs. (36) and (37), which read

$\displaystyle\expectationvalue{N_{1}}=N_{1}^{0}\cos^{2}\Big{(}\frac{g_{21}(\omega_{2}^{2}-\omega_{1}^{2})}{4\sqrt{\omega_{1}\omega_{2}}}\epsilon
t_{f}\Big{)},$ (44)
$\displaystyle\expectationvalue{N_{2}}=N_{1}^{0}\sin^{2}\Big{(}\frac{g_{21}(\omega_{2}^{2}-\omega_{1}^{2})}{4\sqrt{\omega_{1}\omega_{2}}}\epsilon
t_{f}\Big{)}.$ (45)

As we can see, the frequency at which the photons oscillate is proportional to
the coupling coefficient between both modes, as can be expected. The numerical
results also show no growth in $\langle N_{k}\rangle$ for any other mode which
is consistent with the analytical results.

Figure 5: Number of particles as a function of oscillation time ($t_{f}/L$)
obtained through numerical simulations of the system considering an external
frequency of $\Omega=\omega_{2}-\omega_{1}$ both for a non-equidistant
spectrum (a) and an equidistant one (b). In this case the system starts with
an initial number of particles in mode 1 $\langle N_{1}\rangle(t_{f}=0)=50$.
In (a) we see the results for $\alpha/L=0.5$ and $L_{0}/L=0.04$. In this case,
only modes 1 and 2 couple which results in a harmonic migration of the photon
number exclusively between said modes. The dashed-and-dotted black line stands
for the MSA prediction for $\langle N_{1}\rangle$ as seen in Eq. (44). In (b)
we again have the case where modes are localized in either the left (solid
lines) or right (dashed lines) cavity ($\alpha/L=125$ and $L_{0}/L=0.44$). As
a consequence of the spectrum being equidistant in each cavity and as
$\omega_{1,2}=\omega_{1,2}^{+}$ we get frequency subtraction coupling for
every pair of successive modes in the left cavity
($\Omega=\omega_{n+1}^{+}-\omega_{n}^{+}$). This results in the migration of
the particles that are initially in mode 1 to left-cavity higher energy modes
as the wall oscillation time increases. The solid black line represents the
total number of photons ($\langle N\rangle=\sum_{k}\langle N_{k}\rangle$)
inside the double cavity which, as we can see, remains constant for all values
of $t_{f}/L$ considered here.

On the other hand, we can see in Fig. 5 (b) the results for the numerical
simulation with the same choice of $\Omega$ but with $\alpha/L=125$ and
$L_{0}/L=0.72$. In this case, all modes except for mode $0$ are localized in
either the left (solid lines) or right (dashed lines) cavity. Now, for this
particular choice of parameters, the spectrum of both individual cavities is
equidistant. As modes 1 and 2 are localized both in the left cavity, for every
pair of successive modes in the left cavity we get a coupling of the form
$\omega_{n+1}^{+}-\omega_{n}^{+}=\Omega$. This results in the transference of
photons from mode 1 to other modes with higher energy, starting with mode 2
($\omega_{2}=\omega_{2}^{+}$), then going to mode 4
($\omega_{4}=\omega_{3}^{+}$), etc. It must be noted that these results are
also consistent with what we would expect the analytical solution to yield as
the terms that contribute to it in Eqs. (36) and (37) will transfer the
photons between the coupled modes while not creating any particles. This last
fact is shown by the solid black line that stands for the total number of
photons inside the double cavity (i.e. $\langle N\rangle=\sum_{m}\langle
N_{m}\rangle$), which stays constant for all $t_{f}/L$.

Following, we can consider the case where $\Omega=\omega_{1}+\omega_{2}$,
fulfilling the coupling condition given by Eq. (39). As we have previously
mentioned, this results in the creation of photons at a rate that depends on
the value coupling coefficient between these modes $g_{12}$ (Eq. (33)). In the
case of a non-equidistant spectrum, and in the absence of any other condition
from Eqs. (38) or (40), the number of photons is known to grow at the same
exponential rate in both of the coupled modes as

$\langle N_{1}\rangle=\langle
N_{2}\rangle=\sinh^{2}\Big{(}\frac{g_{21}(\omega_{2}^{2}-\omega_{1}^{2})}{4\sqrt{\omega_{1}\omega_{2}}}\epsilon
t\Big{)}.$ (46)

This will be the case for values of $\alpha/L$ and $L_{0}/L$ for which $\alpha
k_{1,2}\sim 1$.

In the particular case of $\alpha=0$, when the wall is completely transparent,
because of the way the external walls move in phase, the system is commonly
known as a shaker pau shaker . In this case, we get resonant coupling between
modes $1$ and $2$, and subtraction coupling for $\omega_{k+5}-\omega_{k}$
which results in particle creation in an infinite number of modes for high
enough times.

For the opposite case of $\alpha/L\gg 1$, we might get different results
depending on the value of $L_{0}$. In this case, the spectra of each of the
individual cavities, becomes equidistant. Even more significant for particle
creation, as the central wall resembles a perfect conductor, the two cavities
are decoupled. This means that if modes 1 and 2 correspond to modes localized
in different cavities, the coefficient $g_{12}=0$, which results in no
particles being created. In Fig. 6, we can see $\langle N_{1}\rangle$ for
$\alpha/L=125$ and two different values of $L_{1}/L$. It must be noted that
although the photon number is shown only for mode 1, in the case of mode 2 the
results are analogous.

Figure 6: Number of particles of well defined frequency $\omega_{1}$ as a
function of time for an electric susceptibility of $\alpha/L=125$. The
external excitation frequency is set at $\Omega=\omega_{1}+\omega_{2}$. The
two values of $L_{1}/L$ correspond to, a case where modes 1 and 2 are
localized in the same cavity ($L_{1}/L=0.72$) and one where they are localized
each in a different cavity ($L_{1}/L=0.53$).

One of these values, $L_{1}/L=0.72$ corresponds to a particular configuration
of the double cavity where modes $1$ and $2$ are localized in the same cavity.
Here we can see $\langle N_{1}\rangle$ grows with time. In this case, we also
have a coupling with more energetic modes which fulfills
$\omega_{k+3}^{+}-\omega_{k}^{+}=\Omega$, resulting in a similar situation to
the one seen in Fig. 4 (b). In the second case, $L_{1}/L=0.53$, we do not see
particle growth in the considered time interval. This is due to the fact that
modes 1 and 2 correspond to different cavities. We must note that the crucial
difference among the two cases is which mode we are labeling as 2. As the
length of the left cavity increases (with the increase of $L_{0}$), the second
mode of that cavity becomes less energetic than the first mode of the right
cavity ($\omega_{2}^{+}<\omega_{1}^{-}$) and then, we label
$\omega_{2}^{+}=\omega_{2}$ when, for lower $L_{0}$, we had
$\omega_{1}^{-}=\omega_{2}$.

As we have already mentioned, in the case where modes 1 and 2 correspond to
different cavities we get $g_{12}=0$ and thus, no particle creation. However,
it might be interesting, from an information point of view, to create
particles in two cavities that are separated by a wall with $\alpha/L\gg 1$.
This is because particles created via a frequency sum coupling condition are
known to be created in entangled pairs. Hence, in principle, achieving this
would allow us to entangle two ”decoupled” cavities via the DCE. In the
following subsection, we propose a way to do this by employing the residual
zeroth mode in a cavity with a high value of $\alpha$.

Finally, we must note that this system allows for further studies in the field
of particle creation. Some interesting examples would be to consider the
movement of the dielectric wall while the perfectly conducting walls stay
still, the consideration of a time-varying electric potential ($v(t)$) applied
to the dielectric wall, or even the combination of these two (or other) time
varying boundary conditions.

###  IV.4 The zeroth mode as a coupling tool

As we have seen in Section III, in cases where $\alpha/L\gg 1$ we get a
different behavior for modes with $m\geq 1$, tending to localize in either the
left or right cavity. The behavior of the mode $m=0$ is different since
instead of localizing, it slowly vanishes as $(\alpha/L)^{-1/2}$. This
exceptional feature allows us to consider a cavity whit a finite value
susceptibility but large enough for us to take $\omega_{m}^{\pm}=2\pi/(L\pm
L_{0})$. This permits a description of the double cavity as a system composed
of two cavities which only ’share’ mode 0, with the rest of the modes being
localized in one or the other.

This scheme would allow to create particles in both cavities fundamental modes
by indirectly coupling them via the zeroth mode. We may achieve said coupling
by choosing a cavity configuration where

$\omega_{1}^{+}+\omega_{0}=\omega_{1}^{-}-\omega_{0}.$ (47)

The strength of the coupling between modes 0 and 1$\pm$ is mediated by
$g_{01}^{\pm}=1/(2\pi(\alpha(L\pm L_{0}))^{1/2})$ which is non-zero for finite
values of $\alpha$. By using Eqs. (24) and (25), we find that this can be
obtained by choosing

$L_{0}/L=1/\sqrt{(1+\pi^{2}\alpha/L)}.$ (48)

The results for the number of particles can be obtained by employing the MSA
method, and the value of $\langle N_{m}\rangle$ obtained doing this can be
seen in Fig. 7. Here we can see that it is possible indeed to create particles
in the different “decoupled” cavities by employing the method above proposed.
The number of photons would grow exponentially. However, the growth occurs at
a very slow rate, and thus it requires to maintain the oscillations for long
time intervals in order to get an appreciable number of particles.

Figure 7: Number of particles as a function of oscillation time $t_{f}/L$ in
the case where the external frequency is $\Omega=\omega_{1}^{+}+\omega_{0}$,
the susceptibility is $\alpha/L=30$ and $L_{0}/L$ corresponding to Eq. (48).
The three modes plotted are those with non-trivial analytic solutions.

Finally, we must note that the zeroth mode could, in principle be used to
overcome one of the practical limitations for the experimental detection of
the mechanical DCE in a cavity. It is known that the values of $\Omega$ needed
to create particles are usually higher than what current technology allows
(the lowest being usually $\Omega=2(\pi/L)$). In this case however, we can
profit by the existence of the zeroth mode. For $\alpha/L=30$ and
$L_{1}/L=0.74$, we get $\omega_{0}=\omega_{1}^{+}/10$. In this case, we can
choose $\Omega=2\omega_{0}$ (which is an order of magnitude smaller than
$2\omega_{1}^{+}$). Since we have $\partial_{\Delta L}k_{0}=\Delta
Lk_{0}/(4L_{1}L_{2})\neq 0$ we can obtain particle creation in mode 0 from
single-mode resonance. There is, however, a persisting problem: the time for
which the walls must be moving at this frequency. Through the MSA we can
estimate that, even for times of $t/L\sim 2000$, we can obtain $\langle
N_{0}\rangle\sim 1$ for this choice of parameters. Going back to Fig. 4 (a),
we can observe that we would get $\langle N_{1}\rangle\sim 1$ for $t/L\sim
300$. This difference is due to the small value of $\partial_{\Delta L}k_{0}$
when $k_{0}$ takes small values and it implies that although oscillations can
be maintained at a considerably smaller frequency, they would have to be
maintained for larger time scales to get a measurable number of photons.

##  V Entanglement

As we have mentioned, superconducting circuits have lately become extremely
relevant in problems relating to quantum information and quantum
communication, mostly because of their role in recent advances in quantum
computing computacionChina ; computacionCanada . Particularly, techniques as
boson sampling bosonSampling  present a promising alternative to achieve
quantum supremacy. Recently, DCE has been studied in systems that employ boson
sampling bosonSamplingDCE . Considering all of this, we might note that the
study of the relationship between the DCE and quantum information in this type
of systems might yield practical applications in the manipulation of
entanglement.

We will start this section by providing the analytical tools to study the
entanglement between pairs of modes of the field inside the double cavity.
With this purpose we will employ gaussian state formalism. Gaussian states are
those for which their characteristic functions and quasiprobability
distributions are given by gaussian functions in phase space gaussianosteoria
. This subset of quantum states includes thermal, coherent, squeezed and
vacuum states.

From now on we will employ a basis of quadrature operators
$R=(q_{0},p_{0},q_{1}.p_{1},...)$, where

$\displaystyle q_{m}=\frac{1}{\sqrt{2}}(a_{m}+a_{m}^{\dagger})$ (49)
$\displaystyle p_{m}=\frac{-i}{\sqrt{2}}(a_{m}-a_{m}^{\dagger}).$ (50)

A given gaussian state $\rho$ can be completely characterized by its
displacement vector gaussianosteoria

$D_{j}=\expectationvalue{R_{j}}_{\rho},$ (51)

and its covariance matrix

$V_{ij}=\frac{1}{2}\expectationvalue{R_{i}R_{j}+R_{j}R_{i}}_{\rho}-D_{i}D_{j}.$
(52)

The particular subset of gaussian states composed of those obtained through
Bogoliubov transformations from an initial vacuum, we find that $D_{j}=0$ for
all $j$, so we shall assume this in the following. If we consider the field
inside the double cavity as our global system, we can express the covariance
matrix of the whole system in terms of $2\times 2$ matrices. The diagonal
blocks correspond to the covariance matrix of a subsystem conformed only by a
single mode $k$ of the field, this being

$V_{k}=\frac{1}{2}\begin{pmatrix}2\expectationvalue{\hat{q}_{k}^{2}}&\expectationvalue{\hat{q}_{k}\hat{p}_{k}+\hat{p}_{k}\hat{q}_{k}}\\\
\expectationvalue{\hat{q}_{k}\hat{p}_{k}+\hat{p}_{k}\hat{q}_{k}}&2\expectationvalue{\hat{p}_{k}^{2}}\end{pmatrix}.$
(53)

The off-diagonal blocks can be though as the correlation between a pair of
modes $j$ and $k$, which reads

$C_{jk}=\frac{1}{2}\begin{pmatrix}2\expectationvalue{\hat{q}_{k}\hat{q}_{j}}&\expectationvalue{\hat{q}_{j}\hat{p}_{k}+\hat{p}_{k}\hat{q}_{j}}\\\
\expectationvalue{\hat{q}_{k}\hat{p}_{j}+\hat{p}_{j}\hat{q}_{k}}&2\expectationvalue{\hat{p}_{k}\hat{p}_{j}}\end{pmatrix},$
(54)

where we have $C_{kj}=C_{jk}^{T}$.

If a system is in a global gaussian state, the states of subsystems are also
gaussian and their covariance matrix is given by the restriction of the
covariance matrix of the whole system to the subsystem that is being
considered entNico .

In this section we start by considering the entanglement between two modes of
the field $j$ and $k$. We thus consider a subsystem consisting of said pair of
modes which is in a gaussian state characterized by

$V_{j|k}=\begin{pmatrix}V_{j}&C_{jk}\\\ C_{jk}^{T}&V_{k}\end{pmatrix}.$ (55)

We call this type of subsystems a bipartite system. In order to quantify the
entanglement between pairs of modes of the field, we employ the logarithmic
negativity, which for bipartite systems can be defined as entNico

$\mathcal{N}_{j,k}=\max\\{0,-\ln(2\nu_{-})\\}$ (56)

for this type of systems. In Eq. (56) we have defined

$\nu_{-}=\sqrt{\frac{\Sigma}{2}-\frac{1}{2}\sqrt{\Sigma^{2}-4\det V_{i|j}}}$
(57)

and

$\Sigma=\det V_{j}\det V_{k}-2\det C_{jk}.$ (58)

###  V.1 Decoupling the systems

In this section, we study the problem of entanglement generation between the
two halves of the double cavity by employing a protocol inspired by that
presented in HalfEmpty . We thus consider the double cavity we have been
discussing in the previous sections of total length $L$, electric
susceptibility $\alpha/L$, and the relative size of the single cavities given
by $\Delta L(t<0)=L_{0}$. Initially ($t<0$, $in$) the cavity is stationary and
the field is in a vacuum state. Starting at $t=0$ and for an interval of time
equal to $t_{f}$, the external perfectly conducing walls will move following a
trajectory given by $\Delta L(0<t<t_{f})=L_{0}+L\epsilon\sin(\Omega t)$. At
$t=t_{f}$ ($out$) the walls stop at their initial position. At this same
instant, we quickly turn on the potential difference applied on the central
wall in such a way that $v\rightarrow\infty$. By looking at Eq. (16) we can
see that suddenly applying an very high potential difference at the wall has
the same effect that taking $\alpha\rightarrow\infty$ if $v$ is high enough
($v/k_{m}\gg 1$). So, this last transformation is equivalent to turning the
central dielectric wall into a perfect conductor, decoupling both cavities.

The description of the modes of the field after the movement of the external
walls and before turning on $v$ is given in terms of the global eigenfunctions
(Eq. (13)). Once $v$ is turned on both halves of the cavity become decoupled,
and so the eigenfunctions now become the localized ones, given by Eq. (23).
The functions of both bases are related to each other by teoriaGlobalLocal

$\displaystyle
u_{l}^{\pm}=\sum_{m}(\xi^{\pm}_{ml}u_{m}(t,x)+\chi^{\pm}_{ml}u_{m}^{*}(t,x)).$
(59)

The expansion coefficients have been defined by using the Klein-Gordon product

$\displaystyle\xi_{ml}^{\pm}=(u_{l}^{\pm},u_{m})_{KG}$ (60)
$\displaystyle\chi_{ml}^{\pm}=-(u_{l}^{\pm},u_{m}^{*})_{KG},$ (61)

evaluating we get

$\displaystyle\xi_{ml}=-e^{-i(\frac{\pi l}{L_{1/2}}-k_{m})\Delta
t}\Big{(}\frac{\pi l}{L_{1/2}}-k_{m}\Big{)}\mathcal{V}^{\pm}_{ml}$ (62)
$\displaystyle\chi_{ml}=e^{-i(\frac{\pi l}{L_{1/2}}+k_{m})\Delta
t}\Big{(}\frac{\pi l}{L_{1/2}}+k_{m}\Big{)}\mathcal{V}^{\pm}_{ml}$ (63)

where $\Delta t=t-t_{f}$ and

$\mathcal{V}_{ml}^{\pm}=(-1)^{l}\frac{\sin(k_{m}L_{1})\sin(k_{m}L_{2})}{\sqrt{2L_{1,2}N_{m}((\frac{\pi
l}{L_{1,2}})^{2}-k_{m}^{2})}}\sqrt{\frac{\pi l}{L_{1,2}k_{m}}}$ (64)

if $k_{m}\neq\pi l/L_{1}$ and $k_{m}\neq\pi l/L_{2}$, while if $k_{m}=\pi
l/L_{1}$

$\mathcal{V}_{ml}^{+}=\frac{L_{1}}{2\pi
N_{m}}\sqrt{\frac{L_{1}}{2}}\sin\Big(\frac{L_{2}}{L_{1}}l\pi\Big{missing}),$
(65)

and if $k_{m}=\pi l/L_{2}$

$\mathcal{V}_{ml}^{-}=\frac{L_{2}}{2\pi
N_{m}}\sqrt{\frac{L_{2}}{2}}\sin\Big(\frac{L_{1}}{L_{2}}l\pi\Big{missing}).$
(66)

Both basis of eigenfunctions are associated to their own bosonic operators,
which correspond to the notion of particle before and after we turn $v$ on.
Said operators can be associated via a Bogoliubov transformation which employ
coefficients $\xi_{ml}^{\pm}$ and $\chi_{ml}^{\pm}$

$\hat{a}_{l\pm}=\sum_{m}(\xi_{ml}^{\pm*}\hat{a}_{m}^{out}-\chi_{ml}^{\pm*}\hat{a}_{m}^{out\,\dagger}).$
(67)

The bosonic opperators $a^{out}$ correspond to the global eigenfunctions for
$t=t_{f}$ and can be readily linked to those corresponding to the initial
state of the system ($a_{m}^{in}$) by Eq. (29). We note that the bogoliubov
trasnformation of Eq. (67), hints at the creation of particles because of
taking $v\rightarrow\infty$ at $t=t_{f}$. Indeed, by substituting
$\sqrt{\omega_{l}}A_{l}^{m}$ and $\sqrt{\omega_{l}}B_{l}^{m}$ by
$\xi^{\pm}_{ml}$ and $\chi^{\pm}_{ml}$ respectively in Eq. (30) we get

$\displaystyle\langle N_{l\pm}\rangle=\frac{4\pi}{(L\pm
L_{0})}\sum_{m}\Big{[}N_{m}^{0}\Big{(}\frac{2\pi l}{(L\pm
L_{0})}-k_{m}\Big{)}^{2}$ (68) $\displaystyle+(1+N_{m}^{0})\Big{(}\frac{2\pi
l}{(L\pm L_{0})}+k_{m}\Big{)}^{2}\Big{]}\mathcal{V}_{ml}^{\pm\,2}.$ (69)

The result above implies that even if no photons are created during the
oscillation of the walls ($0<t<t_{f}$), we will nevertheless get particle
creation as a result of suddenly turning on $v$. In reference HalfEmpty  it
was shown that, in the case of an initial single cavity $\alpha=0$
entanglement can be harvested from the vacuum state.

By replacing $a_{m}^{out}$ with Eq. (29) in Eq. (67), we can relate the local
bosonic operators with the ones corresponding to the initial state of the
system, linking the two transformations that result from moving the walls and
turning $v$ on. This results in

$a_{j\pm}=\sum_{m}[\tilde{\alpha}^{\pm}_{jm}\hat{a}_{m}^{in}+\tilde{\beta}^{\pm*}_{jm}\hat{a}_{m}^{in\,\dagger}]$
(70)

where we have defined

$\displaystyle\tilde{\alpha}_{jm}^{\pm}=\sum_{l}\sqrt{2\omega_{l}}(B_{l}^{m}\xi_{lj}^{\pm*}-A_{l}^{m}\chi_{lj}^{\pm*})$
(71)
$\displaystyle\tilde{\beta}_{jm}^{\pm*}=\sum_{l}\sqrt{2\omega_{l}}(B_{l}^{m}\xi_{lj}^{\pm*}-A_{l}^{m}\chi_{lj}^{\pm*}).$
(72)

Since we aim to study the entanglement between modes localized in different
cavities, in the following we shall consider bipartite systems formed by one
mode $j+$ in the left cavity and another one $k-$ in the right cavity. Taking
this into account, and as a way to simplify notation when calculating the
logarithmic negativity $\mathcal{N}_{j,k}$ we will consider the first index to
always correspond to a mode of the left cavity, while the other one will
correspond to the one in the right.

###  V.2 Results and discussion

The results showcased in this section where obtained by numerically evaluating
the expressions presented before. To do this we must consider a finite number
of modes, in this case taken as $N=200$. We must note that, in the same way as
with the numerical simulations of Section IV taking a finite $N$ represents an
approximation. This approximation can be justified by considering that the
potential difference $v$ cannot be turned on in an infinitely short time. In
practice, this transformation is done in a certain finite time interval
$\delta t$. However, we can consider that for mode $j$ the transformation
happens in a negligible amount of time if $\delta t\ll\omega_{j}^{-1}$
HalfEmpty . This means that while for less energetic modes the transformation
is instantaneous, for those with higher energies the transformation will be
slower. Indeed, for the more energetic ones, the transformation (of turning on
the potential difference) can be considered to be adiabatic and as such it
will neither create particles nor generate entanglement.

Before going on, we also note that the transformation of instantaneously
introducing a potential difference in the position of the wall can be easily
translated into circuits. In this last case, said transformation would consist
in turning $E^{0\prime}(t)$ from $0$ to a value much higher than $1$. This
would indeed be possible as it is similar to what is used to simulate the
perfectly conducting walls at the extremes of the circuit squid1 ; squid2 .

We start by considering the case where the perfectly conducting walls of the
double cavity remain stationary at all times. In this case, the only
transformation that the system will undergo is the turning on of $v$ at $t=0$.
We start by considering the simplest case, where the double cavity is in a
symmetrical configuration ($L_{0}=0$) and the dielectric wall is fully
transparent ($\alpha/L=0$). In Fig. 8 we present the values of the logarithmic
negativity for the first 20 modes localized in the two cavities. This case was
presented in HalfEmpty  and works as a control case in relation for the
variations we will consider later.

Figure 8: Logarithmic negativity for bipartite systems formed by pairs of
modes localized in different cavities for an initial double cavity with
$L_{0}/L=0$ and $\alpha/L=0$. Only the first 20 modes corresponding to each
cavity are shown.

As we can see, the entanglement structure in this case is symmetrical, with
the larger values of $\mathcal{N}_{n,m}$ concentrated among pairs of less
energetic modes. Moreover, we see that the maximum of the entanglement
corresponds to the one between the fundamental modes of each cavity ($m=n=1$).
As the difference between $n$ and $m$ increases the entanglement between modes
tends to vanish at a slow rate.

In Fig. 9 we show $\mathcal{N}_{n,m}$ for two different initial double cavity
configurations with different values for $L_{0}/L$. As we can see, by
increasing the value of $L_{0}$ and as a result, making the two cavities
differ in size, the entanglement looses the symmetrical structure it had in
Fig. 8. This happens because, as $L_{0}/L$ increases so does the length of the
left cavity, $L_{1}$, which results on the frequencies corresponding to the
modes of this cavity being more similar to each other as
$\omega_{n+1}^{+}-\omega_{n}^{+}\propto L_{1}^{-1}$. The opposite happens with
the right cavity, which gets smaller and so the frequencies of its
corresponding modes become more different among each other
$\omega_{m+1}^{-}-\omega_{m}^{-}\propto L_{2}^{-1}$. This results in
$\mathcal{N}_{n,m}$ a slower variation with $n$, and a faster one with $m$. In
the case for $L_{0}=0.6$, we can see that the negativity vanishes for certain
pairs of modes, for example $n=1$ and $m=12$. Apart from this, we can see in
both cases shown that the maximum is still located in $n=m=1$, although its
value is lower than for the symmetric case from $0.1$ to $0.06$.

Figure 9: Logarithmic negativity for bipartite systems formed by pairs of
modes localized in different cavities for an initial double cavity with
$\alpha/L=0$ and different values of $L_{0}/L$. Only the first 20 modes
corresponding to each cavity are shown.

We further consider the case where initially, $L_{0}/L=0$ but, $\alpha/L\neq
0$. The results are shown in Fig. 10. We can see the inclusion of a non-
transparent wall in the initial system results in a decrease in the value of
$\mathcal{N}_{n,m}$ for all bipartite systems, lowering the maximum in $n=m=1$
from $0.1$ to $0.08$. In fact, for many pair of modes the entanglement
vanishes as we increase $\alpha/L$. This can be understood if we consider
that, as we increase the susceptibility of the central wall, the initial
configuration becomes more similar to the final system where the central wall
is essentially a perfect conductor. In fact, as we have previously discussed,
for higher modes, the values of $\alpha/L$ required to get their
eigenfunctions localized are lower. As such, for these modes the
transformation of turning on $v$ is essentially an adiabatic transformation as
neither their eigenfunctions nor eigenfrequencies are affected. This results
in no particles being created in those modes and thus no entanglement with the
modes located in the other cavity. Finally, we note that in these case the
symmetry of the entanglement structure is maintained.

Figure 10: Logarithmic negativity for bipartite systems formed by pairs of
modes localized in different cavities for an initial double cavity with
$L_{0}/L=0$ and different values of $\alpha/L$. Only the first 20 modes
corresponding to each cavity are shown.

Having studied how the initial state of the cavity modifies the entanglement
structure we obtain after separating the two halves, we might now consider the
case where the external walls of the cavity move for a certain amount of time.
We assume that the initial configuration of the cavity is given by the
simplest case of $L_{0}/L=0$ and $\alpha/L=0$. We consider the walls oscillate
at a frequency given by $\Omega=\omega_{1}+\omega_{2}=5\pi/L$. As we mentioned
before, after the oscillation we turn on $v$ decoupling the cavities. So as to
keep this case as simple as possible, we assume a toy model of the field
inside the cavity, for which we dismiss the frequency subtraction couplings
that this choice of $\Omega$ would satisfy. This means the energy delivered to
the field by the movement of the walls only results in particle creation in
modes 1 and 2. It is important to note that although this is not possible in a
one-dimensional cavity it might be in a three-dimensional one.

The non-trivial Bogoliubov coefficients associated to the particle creation
read

$\displaystyle\sqrt{2\omega_{1}}A_{1}^{2}=\sqrt{2\omega_{2}}A_{2}^{1}=\sinh\Big{(}\frac{|\gamma_{12}|}{2}\tau\Big{)}$
(73)
$\displaystyle\sqrt{2\omega_{1}}B_{1}^{1}=\sqrt{2\omega_{2}}B_{2}^{2}=\cosh\Big(\frac{|\gamma_{12}|}{2}\tau\Big{missing}).$
(74)

By introducing the above in equation 72 we can calculate $\mathcal{N}_{n,m}$
through the covariance matrix. In Fig. 11 we observe the results of
$\mathcal{N}_{n,m}$ for this case. We can see that for large enough $t_{f}/L$,
all the bipartite systems reach an asymptotic value of $\mathcal{N}_{n,m}$.
For most subsystems, we get a vanishing value of $\mathcal{N}_{n,m}$.
Although, in Fig. 11 only the cases with $n=m=1,2$ vanish, all of the pairs of
modes that have been omitted show the same behavior for large enough values of
$t_{f}/L$. For the other bipartite systems shown, we see that the negativity
takes on non-zero asymptotic values. The maximum of said values is achieved
for $n=2$ and $m=1$ ($\mathcal{N}_{2,1}\approx 0.49$), followed closely by
$n=1$ and $m=2$ ($\mathcal{N}_{1,2}\approx 0.44$), for the rest of the
subsystems we get comparatively negligible values of negativity. Said values
decrease with the difference between $n$ and $m$.

It should also be noted that while $\mathcal{N}_{1,2}$ grows monotonically,
$\mathcal{N}_{2,1}$ decreases at first, vanishing for a finite time interval,
increasing again for larger values of $t_{f}/L$. This last fact, together with
the fact that $\mathcal{N}_{1,2}\neq\mathcal{N}_{2,1}$ for $t_{f}/L\gg 1$,
show an asymmetry in the entanglement structure, even if the final state of
the system is conformed of two identical cavities. Interestingly, the
asymmetry is dependent upon the initial velocity of the walls. It can be seen
that if we take $\epsilon\rightarrow-\epsilon$ in Eq. (26), meaning that the
walls move initially towards the right instead of towards the left
($L_{1,2}=(L\pm\Delta L)/2$), the roles of $m$ and $n$ in $\mathcal{N}_{n,m}$
are inverted.

Figure 11: Logarithmic negativity for bipartite systems formed by pairs of
modes localized in different cavities for an initial double cavity with
$L_{0}/L=0$ and $\alpha/L=0$, where at $t=0$ the external perfectly conducting
walls oscillate at a frequency $\Omega=\omega_{1}+\omega_{2}$ until $t=t_{f}$.

We can note that the maximum values taken by $\mathcal{N}_{1,2}$ and
$\mathcal{N}_{2,1}$ are many times larger than their initial values. This
shows that the DCE can be used to increase the entanglement between pairs of
modes when compared to the case where there is no particle creation. We also
find a simplification of the entanglement structure, as only two bipartite
systems ($n=1$, $m=2$ and $n=2$, $m=1$) present a non vanishing logarithmic
negativity.

Another approach to entangle modes in different cavities is to use the method
discussed in Section IV.4. We have shown that by coupling the 1st localized
modes in each cavity with the zeroth mode, it was possible to get particle
creation in modes that are localized in both cavities. The fact that the modes
are initially localized requires for $\alpha/L$ to be high in the initial
configuration of the double cavity. This means that turning on $v$ will not
affect any mode besides mode zero, which must vanish when the walls are
perfect conductors. In this case, we might consider that we take
$v\rightarrow\infty$ slowly so as to neglect the entanglement that this non-
localized mode might contribute. Because of this, it is easy to see
$\mathcal{N}_{1,1}$ will be the only quantity that is not zero. In Fig. 12 we
show $\mathcal{N}_{1,1}$ as a function of $t_{f}$.

Figure 12: Logarithmic negativity for bipartite systems formed by pairs of
modes localized in different cavities for an initial double cavity with
$L_{0}/L=0$ and $\alpha/L=30$, where at $t=0$ the external perfectly
conducting walls oscillate at a frequency
$\Omega=\omega_{1}^{+}+\omega_{0}=\omega_{1}^{-}-\omega_{0}$ until $t=t_{f}$.

In this case, we see a generation of entanglement between the two modes which
are initially uncorrelated. As the oscillation time increases we find that the
negativity takes, for sufficiently large $t_{f}/L$, an asymptotic value of
$\mathcal{N}_{1,1}\approx 1.23$. Said value is approximately $2.5$ times
larger than the maximum obtained for the previous case
($\mathcal{N}_{2,1}\approx 0.49$). So, this approach results in stronger
entanglement between localized modes if we consider a single bipartite system.
Here, the structure is further simplified, as the only entangled subsystem is
the one with $n=1$, $m=1$. However, the problem with this case compared with
the previous one is the large values of $t_{f}/L$ needed to get comparatively
higher values of $\mathcal{N}$. To achieve $\mathcal{N}_{1,1}=0.5$, one would
need $t_{f}/L\approx 3000$ which is at least 10 times greater than that needed
to achieve the asymptotic value of $\mathcal{N}_{2,1}\approx 0.49$.

##  VI Conclusions

Throughout this work we have studied the DCE in a double cavity, its link with
quantum superconducting circuits and the possibility of using this effect in
quantum information. We have exhibited the variability of the frequency
spectrum, showing its dependence on the electric susceptibility $\alpha$ and
on the relative size of the individual cavities ($L_{1}/L$ and $L_{2}/L$). We
have shown that for extreme values of $\alpha$ analytical solutions exist for
the eigenfrequencies. We have also obtained the spectrum numerically, showing
the continuous behavior between the two extreme cases of the susceptibility.
Furthermore, we have shown that for high enough values of $\alpha/L$ the modes
localize in either the left or right cavity. This is true except for the least
energetic mode that tends to vanish for high enough values of said parameter,
but at a slower rate than the rest. Further, we have arrived at the conclusion
that depending on the choice of parameters we can obtain both an equidistant
structure ($\alpha/L\gg 1$ and $\alpha/L\ll 1$) as well as a non-equidistant
one ($\alpha/L\sim 1$), which results in a very advantageous characteristic of
the system.

Further, we have studied the particle production via the DCE. This study has
been developed both by analytical approximations (MSA) and exact numerical
integration of the dynamical equations of the system Eq. (31). We have
discussed the three conditions over the external frequency $\Omega$ that can
lead to non-trivial solutions, two of which lead to particle creation
($\Omega=2\omega_{m}$ and $\Omega=\omega_{m}+\omega_{n}$) and one that only
provides photon redistribution ($\Omega=|\omega_{m}-\omega_{n}|$). We have
shown that changes in the structure of the spectrum lead to widely different
results. For example, we have seen that in the non-equidistant case we can get
exponential photon production through single mode resonance in the resonant
mode, at a rate that is proportional to the slope of the wave-number in
relation to changes in wall displacement ($\partial_{\Delta L}k_{l}$). In the
equidistant case, we get photon production in the resonant mode that goes as a
power law in time ($t_{f}^{2}$ for short times and $t_{f}$ for long times),
but we also get particle creation in other, more energetic modes, as a result
of coupling by subtraction condition. Beyond the analytical expressions of the
number of particles as a function of $t_{f}$, the equations that resulted from
this analysis can be used to predict and understand the behavior of the field.
We have also proposed the study of other time-varying boundary conditions,
such as a time-varying potential applied to the dielectric wall. We have
considered the cases in which the zeroth mode can be employed as a coupling
tool between both halves of the cavities, that are essentially decoupled for
large susceptibilities. Furthermore, we have briefly discussed a way to excite
the exponential creation of particles in this mode through single-mode
resonance. It is important to stress that this takes place for values of
$\Omega$ which are one order smaller than what is usually considered for
experimental proposals. However, whenever the zeroth mode is involved, much
longer excitation times are needed to get a non-negligible number of
particles.

Finally, we have analyzed a protocol with which we can induce particle
creation and later decouple the two cavities that conform our system. This
protocol allows to generate a system of two decoupled but entangled cavities,
from an initial double cavity with a semi-transparent wall in the middle. We
have shown how the entanglement gathered from an initial vacuum (in the
absence of particle creation) can be modified by changes in the configuration
of the double cavity, i.e. different values of $\alpha/L$ or $L_{0}/L$.
Additionally, we have shown how the presence of DCE particles produced from
the coupling by resonance condition ($\Omega=\omega_{1}+\omega_{2}$) can be
used to increase the entanglement between certain pairs of modes. For example,
$\mathcal{N}_{1,2}$ increases from $0.06$ to $0.44$. We have also analyzed an
alternative protocol, where we use the zeroth mode as an indirect coupling
tool between modes in different cavities. In such a case, we have seen that
entanglement can be generated between the modes that are localized in the two
cavities and its maximum can be higher than the one obtained for the other
protocol ($\mathcal{N}_{1,1}=1.2$ in this case, and $\mathcal{N}_{1,1}=0.49$
in the other). However, as in the case with particle creation involving the
zeroth mode, the excitation times needed to improve the entanglement between
the different cavity modes are much larger (2 orders of magnitude) than in the
resonant case.

## Acknowledgments

This research was supported by Agencia Nacional de Promoción Científica y
Tecnológica (ANPCyT), Consejo Nacional de Investigaciones Científicas y
Técnicas (CONICET), Universidad de Buenos Aires (UBA). P.I.V. thanks The Abdus
Salam International Center for Theoretical Physics for its support through the
Associate programme. The research of F.C.L. was supported in part by the
National Science Foundation under Grant No. PHY-1748958. We thank D. Blanco,
E.A. Calzetta, and F.D. Mazzitelli for useful comments.

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