# Classical Casimir pressure in the presence of axion dark matter

Philippe Brax  Institut de Physique Théorique, Université Paris-Saclay, CEA,
CNRS, F-91191 Gif-sur-Yvette Cedex, France  Pierre Brun  Irfu/Département de
Physique des Particules, Université Paris-Saclay, CEA, F-91191 Gif-sur-Yvette
Cedex, France

###### Abstract

We study the effects of an oscillating axion field on the pressure between two
metallic plates. We consider the situation where a magnetic field parallel to
the plates is present and show that the electric field induced by the coupling
of the axion to photons leads to resonances. When the boundary plates are
perfect conductors, the resonances are infinitely thin whilst they are
broadened when the conductivity of the boundary plates is taken into account.
The resonances take place at the tower of distances close to
$d_{n}=\frac{(2n+1)\pi}{m}$ where $m$ is the axion mass and have a finite
width and height depending on the conductivity. The resulting resonant
pressure on the plate depends on the induced polarisation at the surface of
the plates. We investigate the reach of future Casimir experiments in terms of
the axion mass and the conductivity of the boundary plates. We find that for
large enough conductivities, the axion-induced pressure could be larger than
the quantum Casimir effect between the plates.

##  I Introduction

The standard model contains a few naturalness issues, one of them being the
so-called strong-CP problem. This is an archetypical fine tuning problem : QCD
(Quantum Chromodynamics) is observed to respect the CP symmetry, as verified
at the $10^{-10}$ level by the non-observation of the neutron EDM (Electric
Dipole Moment) Pendlebury  _et al._ (2015). In the strict context of the
standard model, QCD contains a CP-violating $\theta G\tilde{G}$ term, where
$G_{\mu\nu}$ is the gluon field strength and no degree of freedom allows for
the $\theta$ term to relax to zero. The Peccei-Quinn mechanism consists in
introducing a new U(1) symmetry that is broken spontaneously at some high
energy $M$, making the $\theta$ term dynamical and solving the above puzzle
Peccei and Quinn (1977a, b). This is done at the expense of introducing new
fields, e.g. new heavy fermions Kim (1979); Shifman  _et al._ (1980), or
extending the Higgs sector Zhitnitsky (1980); Dine  _et al._ (1981).
Eventually, there remains a single degree of freedom corresponding to a
massless pseudoscalar goldstone boson called the axion Weinberg (1978);
Wilczek (1978). Because of the QCD phase transition, $\langle
q\bar{q}\rangle\neq 0$ three things happen : the axion acquires a small mass,
the $\theta$ term can effectively relax to zero, and the axion $\phi$ acquires
an effective interaction with the electromagnetic field, of the form $\phi/M\
F\tilde{F}$, where $F_{\mu\nu}$ is the electromagnetic field strength.

Independently, observations of the universe at all scales from dwarf galaxies
to the whole Hubble radius favour the presence of a new type of cosmological
fluid, which we call cold dark matter Workman and Others (2022). The energy
density of dark matter today represents about one third of the total energy,
and 84% of the total mass Collaboration (2020). The axions could well be the
source of dark matter. They could have been produced in the early universe
during the Peccei-Quinn symmetry breaking in a thermal way but such a thermal
relic is irrelevant for dark matter as its abundance turns out to be highly
suppressed Kolb and Turner (1990). On the other hand, the zero mode of the
axion field could be misaligned with the minimum of the axion potential whose
origin is non-perturbative. Due to this non-thermal production mechanism, the
axions would be cold in the first place. In that case the axion field
oscillates in its potential and one has approximately $\phi=\phi_{0}\cos(mt)$
with $m$ the axion mass close to the minimum of the axion potential. In this
scenario, it is the classical energy density of the axion field that plays the
role of the energy density for the cold dark matter fluid Arias  _et al._
(2012). In this case, the local dark matter density $\rho_{0}$ is related to
the amplitude of the oscillation $\phi_{0}$ and its frequency $m$ through
$\rho_{0}=\frac{1}{2}m^{2}\langle\phi^{2}\rangle+\frac{1}{2}\langle\dot{\phi}^{2}\rangle=\frac{1}{2}m^{2}\phi_{0}^{2}$.

It is well known that the zero-point energy of the electromagnetic field in a
cavity leads to an attractive force between the walls due to the so-called
Casimir pressure Casimir (1948); Bordag  _et al._ (2009). This is related to
the fact that even if the cavity contains only vacuum, the corresponding
electromagnetic modes inside the cavity are countable, as opposed to outside
the cavity where they are continuous. Suppose one sets up such a cavity
embedded inside a constant magnetic field. The dark matter axion field will
couple to some of the modes of the vacuum and could potentially modify the
value of the Casimir pressure. The question addressed here is whether this
could lead to observable effects in actual experiments. Typically, there are
two potentially important physical effects in this setting. The first one is
classical and comes from the force between the plates due to the induced
polarisation in metals comprising the boundary plates. We discuss this effect
here. The second one is a quantum effect and follows from the shift in the
quantum modes of the electromagnetic field between the plaques due to the
axion-photon coupling. This is left for future work, see Kharlanov and
Zhukovsky (2010); Fukushima  _et al._ (2019); Brevik (2021); Ema  _et al._
(2023). We give an estimate of this effect in the appendix F.

In this paper, we study the effect of the dark matter oscillating axion field
on the Casimir pressure at the classical level. We consider a constant
background magnetic field which triggers the mixing between axions and
photons. As a result an electromagnetic field is induced along the magnetic
field lines. In particular between two metallic parallel plates, we estimate
the effects of this mixing on the Casimir pressure. Intuitively we expect the
axion to induce resonances and thus a modification of the Casimir forces. The
resonant effect on the pressure between the plates depends on the geometry of
the cavity and on the frequency of the oscillating axion field. These
resonances are regularized by the dissipation effect in the metal. To account
for dissipation, we use a classical approach, and leave the full quantum field
theoretic treatment to a future study, where the modification to the quantum
pressure due to the axion field will be discussed. At the classical level, we
find that the resonances are not perfect and have a finite width depending on
the conductivity of the metal. The width of the resonances is also broadened
by the velocity dispersion of the axion in the galactic halo. Our analysis
assumes that the broadening is dominated by the metallic effects. We find that
for distances of the order of a few microns, the axion-induced pressure can
compete with the quantum pressure in $1/d^{4}$, where $d$ is the distance
between the plates. We also notice that in a real material, the quantum
pressure is reduced by the imperfect reflexivity of the boundary plates and we
recall the Lifschitz formalism in an appendix. This would make the emergence
of the axion-induced effects easier. We leave the detailed study of this to
future work. The competitive cases where the axion-induced pressure becomes of
the order of the quantum Casimir effect require conductivities which are
larger than the ones of ordinary metals such as copper. We leave
phenomenlogical investigations on the type of metal and experimental
situations, such as temperature, necessary to maximise the axionic effect to
future work.

In part II, we introduce the model and derive the general equations for an
axion subject to a magnetic field in a geometry with boundaries. The treatment
of dissipation in metal is described there, and the classical equations of
motion are obtained. In part III, a one-sided boundary condition is
considered. This is an intermediate step towards the treatment of the
classical Casimir pressure induced by an axion in the presence of a magnetic
field, and allows one to retrieve known results, such as the ones typically
considered for dish-antenna experiments Horns  _et al._ (2013). Then in part
IV we consider two boundary conditions, i.e. a cavity, with an external
magnetic field parallel to the plates. We will see that in the ideal case with
infinite conductivity, we obtain infinite resonances for discrete plate
separations, described by Dirac distributions. We will see that the resonances
are regularized when a finite conductivity is considered. In the last part we
discuss orders of magnitudes for the modifications of the Casimir forces and
some prospects. We discuss Green’s functions in the one and two side plates,
more details on dissipation, the energy budget, the Lifschitz theory I. D.
Dzyaloshinskii and Pitaevskii (1960, 1961) and the quantum effects in the
appendices.

##  II Axions in a constant magnetic field

###  II.1 The model

We start with the usual Lagrangian of electrodynamics in vacuum coupled to an
axion-like field $\phi$ in natural units

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{\phi}{4M}F^{{\mu\nu}}\tilde{F}_{\mu\nu},$
(1)

where
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$ is
the dual field strength. We define $\epsilon^{0123}=1$. The pseudo-scalar
field $\phi$ will be assumed to be time dependent, as representing the
oscillations of the dark matter field in our environment. The energy scale $M$
is of the order of the scale of the Peccei-Quinn symmetry breaking. For dark
matter axions of mass $m\sim 10^{-5}\;\rm eV$, this scale is at least of the
order of $10^{12}\;\rm GeV$ Workman and Others (2022). The full lagrangian
includes also kinetic terms for the axion, which we disregard here as we are
interested in the conventional electromagnetic field modes. The equations of
motion for the photon field become

$\partial_{\mu}F^{\mu\nu}+\frac{1}{2M}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}(\phi
F_{\rho\sigma})=0.$ (2)

In the Lorentz gauge $\partial_{\mu}A^{\mu}=0$, the first term gives $\Box
A^{\nu}$ and instead of the classical d’Alembert equation we obtain the
modified propagation equation

$\Box A^{\nu}=-\frac{1}{2M}\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}(\phi
F_{\rho\sigma}).$ (3)

We decompose the gauge field into a background field and a perturbation
$A^{\nu}=\bar{A}^{\nu}+a^{\nu}$, where the background field is generated by a
homogeneous and constant magnetic field corresponding to
$\bar{A}^{i}=\frac{1}{2}\epsilon^{ijk}x^{j}B^{k},\bar{A}^{0}=0,$ satisfying
the background wave equation $\Box\bar{A}^{\nu}=0$. The Maxwell’s equation for
the fluctuation of the electromagnetic field becomes

$\Box
a^{\nu}=-\frac{1}{2M}\epsilon^{\mu\nu\rho\sigma}\Big{(}(\partial_{\mu}\phi)F_{\rho\sigma}+\phi\partial_{\mu}F_{\rho\sigma}\Big{)}.$
(4)

In the following, we will assume that the axion field is homogeneous and time-
dependent $\partial_{\mu}\phi=\dot{\phi}\delta_{\mu}^{0}$, corresponding to
the cosmological field associated to dark matter in the environment. Then we
obtain

$\Box
a^{\nu}=-\frac{1}{2M}\Big{(}\epsilon^{0\nu\rho\sigma}\dot{\phi}\;(\bar{F}_{\rho\sigma}+f_{\rho\sigma})+\epsilon^{\mu\nu\rho\sigma}\phi\;(\partial_{\mu}\bar{F}_{\rho\sigma}+\partial_{\mu}f_{\rho\sigma})\Big{)},$
(5)

and upon considering the magnetic field is constant in space,i.e.
$\partial_{\mu}\bar{F}_{\rho\sigma}=0$, we get

${\Box
a^{\nu}=-\frac{1}{2M}\dot{\phi}\epsilon^{0\nu\rho\sigma}(\bar{F}_{\rho\sigma}+f_{\rho\sigma})-\frac{1}{2M}\phi\;\epsilon^{\mu\nu\rho\sigma}\partial_{\mu}f_{\rho\sigma},}$
(6)

where last term vanishes as
$f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$. The two equations of
motion now become

${\Box\vec{a}=-\frac{1}{M}\dot{\phi}\Big{(}\vec{B}+\vec{b}\Big{)}\;\;\;\;\;\text{
and }\;\;\;\;\;\Box a^{0}=0.}$ (7)

They are conveniently written in terms of the perturbations of electric
$e^{i}=-\dot{a}^{i}$ and the magnetic field
$b_{i}=\frac{1}{2}\epsilon_{ijk}f_{jk}$ in vector form as

$\Box\vec{b}=-\frac{1}{M}\dot{\phi}\;\vec{\nabla}\wedge(\vec{B}+\vec{b})\;\;\;\;\;\text{
and
}\;\;\;\;\;\Box\vec{e}=\frac{1}{M}\ddot{\phi}\Big{(}\vec{B}+\vec{b}\Big{)}+\frac{1}{M}\dot{\phi}\Big{(}\dot{\vec{B}}+\dot{\vec{b}}\Big{)}.$
(8)

These equations can also be easily derived from a reduced Lagrangian involving
an external current induced by both the background magnetic field and the
oscillating axion, see appendix C.

###  II.2 The classical field and the introduction of dissipation

The classical equations of motions can be solved by iterations in an inverse
power expansion in $M$. At leading order we have

${\Box\vec{b}=-\frac{1}{M}\dot{\phi}\;\vec{\nabla}\wedge\vec{B}\;\;\;\;\;\text{
and
}\;\;\;\;\;\Box\vec{e}=\frac{1}{M}\ddot{\phi}\vec{B}+\frac{1}{M}\dot{\phi}\dot{\vec{B}}.}$
(9)

Let us denote the axion-induced source term of the electric field by

$\vec{J}=\frac{1}{M}\ddot{\phi}\vec{B}+\frac{1}{M}\dot{\phi}\dot{\vec{B}},$

then the perturbation of the electric field becomes

$\vec{e}(x)=\int\mathrm{d}^{4}u\;G(x;u)\vec{J}(u),$ (10)

in terms of a well-defined Green’s function which depends on the geometry of
the experimental situation under consideration. Here $x^{\mu}$ and $u^{\mu}$
are 4-vectors.

In the cases of interest, we will consider experimental situations where the
background magnetic field is present everywhere in space. The boundary
conditions used to exhibit the Casimir effect correspond to a cavity between
two parallel plates. We will see that in this case both $\vec{e}$ and
$\vec{b}$ are parallel to the plates. Inside the plates, the perturbations of
the electric and magnetic field penetrate only over a length depending on
$\omega_{\rm Pl}^{-1}$, i.e. the skin depth $\delta$.

Before moving to the estimates of the classical pressure in specific cases, we
must discuss the treatment of dissipation inside matter. This is relevant for
the case where the metallic plates are made of non-ideal conductors. For this
we follow the method described in Landau and Lifshitz (1984). Inside matter as
in the case of metals, the relevant vector field for electric phenomena is the
displacement field $\vec{d}=\vec{e}+\vec{p}$, sometimes called the electric
induction field. It differs from the electric field in vacuum due to the
existence of the polarisation $\vec{p}$. The displacement vector is related to
the electric field via a retarded effect

$\vec{d}(\vec{x},t)=\int
d\tau\epsilon(t-\tau)\vec{e}(\vec{x},\tau)\equiv\epsilon\star_{t}\vec{e}(\vec{x},t),$
(11)

or equivalently
$\vec{d}(\vec{x},\omega)=\epsilon(\omega)\vec{e}(\vec{x},\omega)$ in Fourier
space with

$\vec{d}(\vec{x},t)=\int\not{d}\omega\epsilon(\omega)\vec{e}(\vec{x},\omega)e^{-i\omega
t},$ (12)

with $\not{d}=d/2\pi$. Dissipation will be taken into account considering that
the permittivity is not equal to one, and is a complex number. This is because
the propagation equation in matter in the $z$ direction is
$\partial^{2}_{z}e=\epsilon\partial^{2}_{t}e$ and if $\epsilon\in\mathbb{C}$,
then from the dispersion relation $k^{2}=\epsilon\omega^{2}$, $k$ is also a
complex number $k=k^{\prime}+ik^{\prime\prime}$, and a plane wave gets a real
factor that can account for dissipation $e^{-k^{\prime\prime}z}e^{-i(\omega
t-k^{\prime}z)}$.

In the following we are interested in the case of a metal, and we follow the
Drude model. The permittivity relates the polarisation field to the electric
field through $\vec{p}=(\epsilon-1)\vec{e}$. On the one hand one has in the
$x$ direction $p=Nqx$, where $N$ is the density of electrons and $q$ their
charge. The movement of electrons are modeled classically with a frictional
force and a restoring force, such that

$\ddot{\vec{x}}=\frac{q}{m}\left(\vec{e}+\dot{\vec{x}}\wedge\vec{B}\right)-\gamma\dot{\vec{x}}-\omega_{0}^{2}\vec{x},$
(13)

where $\gamma$ is a typical damping time and $\omega_{0}^{2}$ is related to
the confinement of electrons around atoms. Typically we expect that
$\gamma=Nv_{e}\sigma_{\rm int}$ where $\sigma_{\rm int}$ is the cross section
of the moving electrons with the material and $v_{e}=\sqrt{\frac{3T}{m_{e}}}$
is the thermal velocity of the electrons. We take the external first-order
magnetic field in the $x$ direction and we consider waves propagating in the
$z$ direction. Because the electric field perturbation is in the direction of
the external magnetic field, the only displacement of the electrons related to
$\vec{e}$ is in the $x$ direction and as far as the polarisation is concerned,
the magnetic force has no effect. Solving the equation of motion in Fourier
space one gets

$x=\frac{q/m}{\omega_{0}^{2}-\omega^{2}-i\gamma\omega}e.$ (14)

In metals, electrons are free and $\omega_{0}\rightarrow 0$ so it is
disregarded. By identification with the expression $\vec{d}=\vec{e}+\vec{p}$,
one gets for the complex permittivity

$\epsilon=1-\frac{\omega_{\rm Pl}^{2}}{\omega^{2}+i\gamma\omega},$ (15)

where $\omega_{\rm Pl}=\sqrt{Nq^{2}/m}$ is the plasma frequency. The damping
time $\gamma$ is related to the conductivity through

$\sigma=\frac{\omega^{2}_{\rm Pl}}{\gamma}.$ (16)

The integral leading to the displacement field $\vec{d}(\vec{x},t)$ can be
evaluated in the complex plane. In the Drude model, when $t<0$ the contour
must be closed in the upper half plane and as long as
$\vec{e}(\vec{x},\omega)$ has no singularity in the upper half plane we find
that $\vec{d}(\vec{x},t)=0$. On the contrary, when $t\geq 0$, the contour must
be closed in the lower half plane where the permittivity has a pole at
$\omega_{p}=-i\gamma$. Assuming that $|\omega||\epsilon(\vec{x},\omega)|$
converges to zero for large $|\omega|$ so that the integral on the large
circle in the lower half plane vanishes, we find that

$\vec{d}(\vec{x},t)=\sigma\vec{e}(\vec{x},-i\gamma)e^{-\gamma t},$ (17)

implying that the displacement field vanishes after a time $1/\gamma$ and has
an amplitude depending on the conductivity. This is what happens due to
dissipation as long as a permanent regime is not considered.

Including the displacement field, the phenomenological equation for the
electric field is

$-\partial_{0}\left(\epsilon\star_{t}\partial_{0}G\right)+\Delta
G=\delta^{(4)}(x^{\mu}-y^{\mu}).$ (18)

This equation can be solved using the Green’s function of the operator in
Fourier space

$G\equiv(\Delta+\epsilon(\omega)\omega^{2})^{-1}.$ (19)

The Green’s function can be explicitly defined as soon as the boundary
conditions are specified. In the next parts of the paper, different boundary
conditions are considered, and the associated Green’s functions are used to
solve the equations of motions in the vacuum and in the metallic plates.

###  II.3 Classical pressure from the electromagnetic field and the axion
field

In Casimir experiments, the observable is an attractive force between the
plates. This can be described by a negative pressure in vacuum. Before moving
on with the expression of the pressure, several remarks are in order. First of
all, an interesting effect follows from the Maxwell equation in matter. Inside
the plaques we have the identity

$\vec{\nabla}\wedge\vec{b}=\partial_{0}\vec{e}+\vec{j}_{\rm ind},$ (20)

where $\vec{j}_{\rm ind}$ is the induced current. As
$\vec{\nabla}\wedge\vec{b}-\partial_{0}\vec{e}=-\Delta\vec{a}+\partial_{0}^{2}a$,
we find that $\vec{j}_{\rm
ind}=-\partial_{0}(\epsilon\star_{t}\partial_{0}\vec{a}-\partial_{0}\vec{a})$
which coincides with

$\vec{j}_{\rm ind}=\dot{\vec{p}}$ (21)

inside matter. As can be seen this current only exists in a finite width
within the plate, i.e. this is a skin effect. This current is responsible for
the dissipation power $\vec{j}_{\rm ind}.\vec{e}$ in the plate, corresponding
to the loss of energy $\frac{d}{dt}(\frac{\vec{p}.\vec{e}}{2})$ due to the
polarisability of the material. As a result, the plaque heats up due to the
Joule effect. We will give a thorough discussion of dissipative effects in
appendix D.

The oscillating axion field will perturb the electromagnetic modes via the
coupling between the axion and two photons. In Casimir experiments, the
observed effect is a pressure in vacuum. Here there is an additional classical
pressure that will be induced by the coupling of the axion field with the
electromagnetic field. In vacuum, the energy momentum tensor of the
electromagnetic field is given by

$T^{\mu\nu}=F^{\mu\rho}F^{\nu}_{\rho}-\frac{\eta^{\mu\nu}}{4}F^{2},$ (22)

which gives for the pressure against the plates

$\sigma^{v}_{zz}=-T^{zz}=\frac{e_{z}^{2}+b_{z}^{2}}{2}-\frac{e^{2}+b^{2}}{2}.$
(23)

Notice that there is no contribution from the axion term as $F\tilde{F}$ is
topological and independent of the metric. Indeed $T^{\mu\nu}$ is obtained by
the variation of the action with respect to the metric; the $F\tilde{F}$ term
gets a $(-g)^{-1/2}$ term for the Levi-Civita tensor to be covariant, that
cancels with the $\sqrt{-g}$ in the integration measure of the action.
Moreover $e_{z}=b_{z}=0$ implying that the electromagnetic pressure on the
plate due to the axion source is given by

$\sigma^{v}_{zz}=-\frac{e^{2}+b^{2}}{2},$ (24)

evaluated on the plates. This is the pressure due to the vacuum on the
interface between the plaques and the vacuum.

In matter, the energy-momentum tensor of electromagnetism gives the pressure

$\sigma^{m}_{zz}=\frac{e_{z}d_{z}+b_{z}^{2}}{2}-\frac{\vec{e}.\vec{d}+b^{2}}{2}=-\frac{\vec{e}.\vec{d}+b^{2}}{2}.$
(25)

As $e_{z}=0$, the pressure on the plaques is given by the difference from the
two classical pressures inside and outside the plaques. This gives

$P=\sigma_{zz}^{v}-\sigma_{zz}^{m}=-\frac{e^{2}}{2}+\frac{\vec{e}\cdot\vec{d}}{2}=-\frac{1}{2}\vec{e}\cdot\left(\vec{e}-\vec{d}\right)=\frac{\vec{p}.\vec{e}}{2}=\Re\left((\epsilon-1)e^{2}\right).$
(26)

This term is always negative. In the next sections, the explicit determination
of the electric field and the related polarisation field, allows for the
computation of the pressure. In the single-plate case, this leads to a thrust
force on the plate, whereas in the two-plate case, this modifies the Casimir
effect.

##  III Radiation from a single plate

As an application of the Green’s function techniques, we will derive the
expression for the electric field when only one plate is present and the
magnetic field penetrates inside a metal with finite conductivity. This
generalizes the ideal case where the conductivity is taken to be infinite. The
geometry for this section is sketched in Fig. 1.

Figure 1: Geometry and boundary conditions for the one-plate case.

Now we consider the generation of an electric field due to the axion coupling
when the magnetic occupies all space. This is deduced from Eq. 10 as a
function of the Fourier transform in time of the Green’s function evaluated at
the frequency $\omega=m$,

$e(z,t)=J_{0}\Re\left[e^{-imt}\int_{-\infty}^{\infty}dz_{0}G(m,z;z_{0})\right],$
(27)

where in the case of an external magnetic field $B_{0}$ and a homogeneous
axion field, the source term depends on the amplitude

$J_{0}=\frac{m^{2}\phi_{0}B_{0}}{M}.$

The Green’s function is such that it vanishes in vacuum at $z=+\infty$ and in
matter at $z=-\infty$. In vacuum, this selects the modes in $e^{i\omega z}$
where the replacement $\omega^{2}\to\omega^{2}+i\tilde{\epsilon}$ with
$\tilde{\epsilon}\to 0$ guarantees the convergence. The Green’s function is
solution of Eq. 18. To determine explicitly $G$, it is convenient to separate
the cases $z_{0}<0$ and $z_{0}>0$, which is done in the next subsections. More
details on the derivation of the Green’s function will be given in the case of
two-plate systems.

####  III.0.1 Green’s function for $z_{0}<0$

We find that when $z<z_{0}$,

$G(\omega,z;z_{0})=\frac{1}{2\xi(\omega)}\left(1+\frac{\xi(\omega)+i\omega}{\xi(\omega)-i\omega}e^{2\xi(\omega)z_{0}}\right)e^{\xi(\omega)(z-z_{0})},$
(28)

whilst when $z\in[z_{0},0]$,

$G(\omega,z;z_{0})=\frac{e^{\xi(\omega)z_{0}}}{2\xi(\omega)}\left(e^{-\xi(\omega)z}+\frac{\xi(\omega)+i\omega}{\xi(\omega)-i\omega}e^{\xi(\omega)z}\right),$
(29)

and finally when $z>0$,

$G(\omega,z;z_{0})=\frac{e^{i\omega
z+\xi(\omega)z_{0}}}{\xi(\omega)-i\omega},$ (30)

where we defined the function $\xi(\omega)$ as

$\xi(w)=\left(-\epsilon(w)\omega^{2}\right)^{1/2},$ (31)

where the square root is such that the real part is always positive. The same
calculation can be performed when $z_{0}>0$.

####  III.0.2 Green’s function for $z_{0}>0$

In this case, it is shown in appendix A that if $z<0$,

$G(\omega,z;z_{0})=\frac{e^{i\omega
z_{0}+\kappa(\omega)z}}{i\omega+\xi(\omega)},$ (32)

if $z\in[0,z_{0}]$,

$G(\omega,z;z_{0})=\left(\frac{1}{i\omega+\xi(\omega)}-\frac{1}{2i\omega}\right)e^{i\omega(z+z_{0})}+\frac{e^{i\omega(z_{0}-z)}}{2i\omega},$
(33)

and finally when $z>z_{0}$,

$G(\omega,z;z_{0})=\left(\frac{1}{i\omega+\xi(\omega)}-\frac{1}{2i\omega}\right)e^{i\omega(z+z_{0})}+\frac{e^{i\omega(z-z_{0})}}{2i\omega}.$
(34)

####  III.0.3 The electric field in vacuum

In dish antenna experiments, the boundary condition implied by the presence of
the conductor leads to the emergence of an outgoing electric field wave. This
propagating electric field is the signal of interest. To compute its
amplitude, we now have to select the Green’s function for $z>0$, i.e. outside
the plate in equations 30, 33 and 34. As a result we have

$\int_{-\infty}^{\infty}dz_{0}G(m,z;z_{0})=\frac{1}{m^{2}}+\left(\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\right)e^{imz}.$
(35)

Hence there are two components. There is an oscillating electric field at the
frequency $m$ of the axion oscillations. This is due to the presence of the
magnetic field in the vacuum and the $\phi\vec{E}\cdot\vec{B}$ form of the
axion coupling : as soon as a magnetic field line is present, a small electric
field is induced. There is also a propagating wave with frequency $m$, due to
the presence of the boundary condition at $z=0$ and the breaking of
translation invariance in that direction. Explicitly written, the electric
field reads

$e(z,t)=\frac{J_{0}\cos
mt}{m^{2}}+J_{0}\Re\left[\left(\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\right)e^{im(z-t)}\right].$
(36)

It is interesting to take the limit of an ideal metal corresponding to
$\xi(m)\to\infty$. In this case we find that the electric field becomes

$e_{\rm ideal}(z,t)=\frac{J_{0}}{m^{2}}\Re\left[e^{-imt}(1-e^{imz})\right].$
(37)

As expected the oscillating solution is simply

$e_{\rm osc}(t)=\frac{J_{0}\cos mt}{m^{2}},$ (38)

directly from the propagation equation in the absence of $z$ dependence.
Mathematically, this is the solution of the propagation equation including the
source term. The propagating wave satisfies the propagation equation in the
absence of the source term and its amplitude is such that the electric field
vanishes at the surface of the perfect conductor. This specifies a unique
solution. Here this has been obtained using the Green’s function and has been
generalized to the case of a non-perfect conductor. The other component is a
propagating electric field in the $z$ direction,

$e(z,t)_{\rm prop}=\frac{J_{0}}{m^{2}}\cos\left(m(z-t)\right).$ (39)

This expression allows us to retrieve the usual expressions for dish antenna
experiments. The output power per unit area is given by the Poynting vector

$\Pi=\langle e^{2}\rangle=\frac{\rho_{0}B_{0}^{2}}{m^{2}M^{2}}$

where the average is taken over the rapid oscillations. With conventional
units and some typical values for the parameters, one finds

$\Pi_{\rm ideal}=2.76\times 10^{-30}{\;\rm
W/m^{2}}\left(\frac{\rho_{0}}{0.3\rm\;GeV/cm^{3}}\right)\left(\frac{B_{0}}{1\;\rm
T}\right)^{2}\left(\frac{m}{100\;\rm\mu
eV}\right)^{-2}\left(\frac{M}{10^{14}\;\rm GeV}\right)^{-2}.$ (40)

The previous expression is used to estimate the signal power in dish antenna
experiments in the case a perfect conductor setting the boundary condition. It
is interesting to see how the Green’s function method developed here allows
for a generalisation to the case of non-ideal metals. To obtain a useful
formula, one can consider for real metals that
$\omega\ll\gamma\ll\omega_{Pl}$, in which case

$\Pi_{\rm
real}=\frac{\rho_{0}B_{0}^{2}}{M^{2}}\left(\frac{1}{m^{2}}+\frac{1}{\sigma^{2}}\right)=\Pi_{\rm
ideal}+\Pi_{\sigma}.$ (41)

This leads to a tiny correction of the emitted power, which is at first order
independent of the axion mass,

$\Pi_{\sigma}=4.55\times 10^{-35}{\;\rm
W/m^{2}}\left(\frac{\rho_{0}}{0.3\rm\;GeV/cm^{3}}\right)\left(\frac{B_{0}}{1\;\rm
T}\right)^{2}\left(\frac{\sigma}{6\times
10^{8}\;\rm/\Omega/m}\right)^{-2}\left(\frac{M}{10^{14}\;\rm
GeV}\right)^{-2},$

where the reference value for the conductivity is that of Copper. Notice that
a finite conductivity, and thus dissipation leads to an increase of the signal
power. This is rather counter-intuitive, but allowing for $\vec{p}\neq\vec{0}$
leads to more small oscillating dipoles, hence more radiation. As can be seen
from the previous formula, the effect is very small.

####  III.0.4 Pressure on a single plate

The emission of an electromagnetic wave from the surface of the plate extracts
momentum from the electromagnetic field, thus leading to a force per unit
area. This is simply the radiation pressure, and is due to the presence of the
polarization field at the surface of the conductor. The pressure is given by a
time average of Eq. 26,

$P=\frac{\langle\vec{e}(0,t).\vec{p}(0,t)\rangle}{2}$ (42)

where $\vec{p}$ is the polarisation vector, which is deduced from Eq. 12,

$\vec{p}(z,t)=\int\not{d}\omega(\epsilon(\omega)-1)\vec{e}(z,\omega).$ (43)

Using the results from the previous section, we have that $\vec{p}$ is along
$\vec{B}$ with a magnitude

$p(0,t)=J_{0}\Re\big{(}(\epsilon(m)-1)\left(\frac{1}{m^{2}}+\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\right)e^{-imt}\big{)}.$
(44)

So in that case, the averaged pressure on the plaque is given by

$P=\frac{J_{0}^{2}}{2}\Re\left[\epsilon(m)-1\right]\left|\frac{1}{m^{2}}+\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\right|^{2}$
(45)

which is always negative, i.e. the vacuum attracts the plaque. This pressure
is rather small for typical values of the parameters. For instance, for a 50 T
magnetic field, an axion mass of $m=100\;\rm\mu eV$ and a coupling scale
$M=10^{14}\;\rm GeV$, the value of the pressure is of the order of
$10^{-39}\;\rm Pa$. One can imagine using this pressure to produce thrust on a
spaceship. Should a spaceship be equipped with such a system and a 1 $\rm
m^{2}$ metal plate, the pressure converts into a $10^{-40}\;\rm kg$ thrust. As
a comparison, the same plate at the level of the Oort cloud would undergo a
thrust of about $10^{-17}\;\rm kg$ from Solar radiation pressure. On the other
hand as dark matter is present everywhere even in the absence of radiation
from nearby stars, we could imagine that such an effect could be used in total
darkness.

##  IV Classical pressure in an empty cavity

###  IV.1 The ideal case

Let us move on to the simple case of a two-plate system, where the plates are
made of an ideal conductor, i.e. with infinite conductivity. In that case the
electric field do not penetrate the metallic plates, as sketched in Fig. 2.

Figure 2: Geometry and boundary conditions for the two-plate case with perfect
conductors.

Like in the previous situation, the electric field $\vec{e}$ points in the
$\vec{B}$ direction. We denote by $e=\frac{\vec{e}.\vec{B}}{B}$ its magnitude
such that $\Box e=J$. Denoting by $G(z,t;z_{0},t_{0})$ the Green’s function
such that

$-\ddot{G}+G^{\prime\prime}=\delta(t-t_{0})\delta(z-z_{0}),$ (46)

where ${}^{\prime}=d/dz$ we have then

$e(z,t)=\int dz_{0}dt_{0}G(z,t;z_{0},t_{0})J(z_{0},t_{0}).$ (47)

We now perform the computation of the Green’s function in that case.

####  IV.1.1 Determination of the Green’s function

Using Fourier modes in time, the Green’s function verifies

$G^{\prime\prime}+\omega^{2}G=\delta(z-z_{0})e^{i\omega t_{0}}.$ (48)

Denoting by $d$ the distance between the plates in the $z$ direction, the
solution becomes

$\displaystyle 0\leq z\leq z_{0},\ \ G=A\sin\omega z$ $\displaystyle z_{0}\leq
z\leq d,\ \ G=C\sin\omega z+D\cos\omega z.$

We impose the Dirichlet boundary conditions, that is
$e(0,\omega)=e(d,\omega)=0$. This corresponds to the non-penetration of the
electric field inside the plates and the continuity of its parallel component
at the interface. This implies that

$C\sin\omega d+D\cos\omega d=0.$ (49)

We must also have a discontinuity of $G^{\prime}$ with a jump

$[G^{\prime}]_{z_{0}}=e^{i\omega t_{0}}.$ (50)

We find that

$C=\frac{\sin\omega z_{0}}{\omega}\frac{e^{i\omega z_{0}}}{\tan\omega d},\ \
D=-\frac{\sin\omega z_{0}}{\omega}{e^{i\omega z_{0}}}$ (51)

and

$A=\frac{\sin\omega z_{0}}{\omega}\frac{e^{i\omega z_{0}}}{\tan\omega
d}\left(1-\frac{\tan\omega d}{\tan\omega z_{0}}\right).$ (52)

Now we have

$J=\frac{m^{2}B\phi_{0}}{M}\cos mt\equiv J_{0}\frac{e^{imt}+e^{-imt}}{2}.$
(53)

Using

$G(z,t;z_{0},t_{0})=\int\not{d}\omega e^{-i\omega t}G(z,\omega;z_{0},t_{0}),$
(54)

and

$\displaystyle 0\leq z\leq z_{0}\ \ G(z,\omega;z_{0},t_{0})=\frac{e^{i\omega
t_{0}}}{\omega}\left(\frac{\sin\omega z_{0}}{\tan\omega d}-\cos\omega
z_{0}\right)\sin\omega z$ $\displaystyle z_{0}\leq z\leq d\ \
G(z,\omega;z_{0},t_{0})=\sin\omega z_{0}\frac{e^{i\omega
t_{0}}}{\omega}\left(\frac{\sin\omega z}{\tan\omega d}-\cos\omega z\right)$

we first evaluate

$I_{1}=\int\not{d}\omega\frac{e^{i\omega(t_{0}-t)}}{\omega}\frac{\sin\omega
z\sin\omega z_{0}}{\tan\omega d}.$ (56)

The integrand is a meromorphic function in the complex plane with no
singularity at the origin and simple poles for

$\omega_{n}=\frac{n\pi}{d}\text{, with }n\in\mathbb{N}^{\star}.$ (57)

If $t>t_{0}$, we close the contour on a large circle in the lower half plane,
avoiding all the poles with infinitesimal circles above the horizontal axis

$I_{1}=-\pi i\sum_{n\neq 0}\frac{\rm Res_{n}}{2\pi},$ (58)

where ${\rm
Res_{n}}=\frac{e^{i\omega_{n}(t_{0}-t)}}{d\omega_{n}}\sin\omega_{n}z\sin\omega_{n}z_{0}$.
The same method holds when $t<t_{0}$ by closing the contour in the upper half
plane. For the same reason, by going to the complex plane we have

$\int\not{d}\omega\frac{e^{i\omega(t_{0}-t)}}{\omega}\sin\omega z\cos\omega
z_{0}=0,$ (59)

leading to

$G(z,t;z_{0},t_{0})=\sum_{n>0}\frac{\sin\omega_{n}z\sin\omega_{n}z_{0}}{d\omega_{n}}\sin\omega_{n}(t_{0}-t).$
(60)

We see that the Green’s function oscillate at the resonance frequencies
$\omega_{n}$.

####  IV.1.2 Resonances

The final step for this evaluation of the electric field consists in
performing the time integral in Eq. 47, first we obtain

$e(z,t)=-\pi
J_{0}\sum_{n>0}\frac{\sin\omega_{n}t\sin\omega_{n}z}{d\omega_{n}^{2}}(1-\cos\omega_{n}d)\delta(\omega_{n}-m).$
(61)

Notice that the sum vanishes for all even n, implying that

$e(z,t)=-2\pi J_{0}\sum_{p\geq
0}\frac{\sin\omega_{2p+1}t\sin\omega_{2p+1}z}{d\omega_{2p+1}^{2}}\delta(\omega_{2p+1}-m).$
(62)

The delta function selects one specific frequency, at which there is a
resonance when $\omega_{2p+1}=m$. This specifies a number of distances
$d_{p}=(2p+1)\pi/m$ where the electric field is resonant. The energy and
pressure associated with these resonances is ill defined as it involves the
square of Dirac distributions. This resonance is regularized when one takes
into account the finite conductivity of the plates. In particular, we will
find that the finite conductivity leads to a shift of the resonances below the
real axis in the complex plane of pulsations $\omega$. Moreover, the number of
resonances will become finite.

###  IV.2 Non-ideal conductors with no magnetic field inside

As a generalization of the previous situation, and in order to regularise the
resonances obtained before, we now perform the same study with the inclusion
of a finite conductivity for the metallic plates. For pedagogical purpose, the
study is performed in two steps, first one considers no magnetic field
penetrates the metallic plates. This would correspond to superconducting
material and leads to simpler expressions. Then, the more realistic case of a
magnetic field present everywhere, including in the plates, is considered.

####  IV.2.1 The Green’s function

We now introduce a finite conductivity for the boundary metallic plates, and
consider the magnetic field only lies outside the plates. It corresponds to
the situation sketched in Fig. 3. In the next section, we will extend the
setting to the more realistic case where both the conductivity is finite and
the magnetic field penetrates inside the metal.

Figure 3: Geometry and boundary conditions for the two-plate case with finite-
conductivity but no magnetic field.

Inside the plates the permittivity is not equal to one and is given by 15.
Notice that there is a pole in the lower half plane as required by causality,
i.e. the support of the Fourier transform of the permittivity $\epsilon(t)$ as
a function of time is the positive real axis implying that the displacement
field at a given time depends only on the values of the electric field in the
past. In this case, the Green’s function satisfies

$G^{\prime\prime}+\epsilon(\omega)\omega^{2}G=e^{i\omega
t_{0}}\delta(z-z_{0})$ (63)

for $z<0$ and $z>d$, together with

$G^{\prime\prime}+\omega^{2}G=e^{i\omega t_{0}}\delta(z-z_{0})$ (64)

for $0\leq z\leq d$. The values of the Green’s function for different ranges
in $z$ are shown in Tab. 1. The values for $G_{+}$ and $G_{-}$ are specified
below.

Range for $z$ | Expression of the Green’s function
---|---
$z<0$ | $G=G_{-}e^{\xi z},\ z<0$
$0\leq z\leq z_{0}$ | $G=A\sin\omega z+B\cos\omega z$
$z_{0}\leq z\leq d$ | $G=C\sin\omega z+D\cos\omega z$
$z>d$ | $G=G_{+}e^{-\xi(z-d)}$
Table 1: Expressions of the Green’s function in different $z$ ranges.

Notice that here $z_{0}$ lies between $0$ and $d$. The general case is treated
in appendix B. We find that

$G_{-}=\Theta(\omega){e^{i\omega
t_{0}}}\left(\cos\omega(z_{0}-d)-\frac{\xi}{\omega}\sin\omega(z_{0}-d)\right)$
(65)

and

$G_{+}=\frac{\Theta(\omega)}{\sin\omega d}{e^{i\omega t_{0}}}\left(\cos\omega
z_{0}+\frac{\xi}{\omega}\sin\omega z_{0}\right),$ (66)

where $\Theta(\omega)$ is given by

$\Theta(\omega)=\frac{1}{\omega\left((1-\frac{\xi^{2}}{\omega^{2}})\sin\omega
d-2\frac{\xi}{\omega}\cos\omega d\right)}.$ (67)

With this, we can identify

$\displaystyle 0\leq z\leq z_{0},\ G=G_{-}\left(\cos\omega
z+\frac{\xi}{\omega}\sin\omega z\right)$ $\displaystyle z_{0}\leq z\leq d,\
G=G_{+}\left(\cos\omega(z-d)-\frac{\xi}{\omega}\sin\omega(z-d)\right).$ (68)

This Green’s function depends on the frequency $\omega$ and has resonances
when $\Theta(\omega)$ has poles. The nature of the resonances is important as
will be seen below.

####  IV.2.2 The resonances

The function $\Theta$ has poles when $\omega$ verifies

$\tan\omega d=\frac{2\frac{\xi}{\omega}}{1-\frac{\xi^{2}}{\omega^{2}}}.$ (69)

We can distinguish several cases.

When $\omega\gg\omega_{\rm Pl}$, the resonances would be for

$\tan\omega d\simeq 2i,$ (70)

which has no solution. In this limit the Green’s function is

$\displaystyle z\leq z_{0}\
G\simeq\frac{e^{i\omega(t_{0}+z_{0}-z)}}{2i\omega}$ $\displaystyle z\geq
z_{0}\ G\simeq\frac{e^{i\omega(t_{0}+z-z_{0})}}{2i\omega},$ (71)

with no effect from the plates at all, i.e. this is the free propagator in
(1+1) dimensions and the plates are transparent at high frequency. This is the
optical regime of a metal as one expects for frequencies larger than the
plasma frequency.

Below the plasma frequency, there is another characteristic scale given by
$\gamma$. The poles are now obtained as solutions of

$\tan\omega d\simeq-\frac{2}{(-\epsilon)^{1/2}},$ (72)

which gives

$\omega_{n}d\simeq n\pi\left(1-\frac{2}{d\omega_{\rm
Pl}}\right)-\frac{i\gamma}{\omega_{\rm Pl}}$ (73)

as long as $\omega_{n}\gg\gamma$ . When $\omega\ll\gamma$, the poles are
located at

$\omega_{n}d\simeq n\pi-2e^{i\pi/4}\frac{\sqrt{n\pi\gamma}}{\omega_{\rm
Pl}\sqrt{d}}.$ (74)

Resonances exist for $\omega_{n}\ll\gamma$ only if $d\gg\gamma^{-1}$.

As a result all the poles are shifted by the effect of the plasma frequency
and are now below the real axis. Notice that there are only a finite number of
poles below the plasma frequency, whereas there were an infinite number in the
ideal case. Poles below the real axis imply that the Green’s function respects
causality. In fact, the poles of $\Theta$ represent the eigenfrequencies of
the free system with no source

$a^{\prime\prime}+\epsilon(\omega)\omega^{2}a=0.$ (75)

These modes have a temporal dependence in $e^{-i\omega_{n}t}$ which goes to
zero at large time, i.e. the modes are evanescent waves which decay due to the
dissipation induced by a finite conductivity.

####  IV.2.3 The electric field

We can now evaluate the electric field with metallic plates on the boundary.
Formally the electric field is given by

$e(z,t)=J_{0}\Re\left[\int_{0}^{d}dz_{0}\int\not{d}\omega e^{-i\omega
t}\not{\delta}(\omega-m)G(\omega,z;z_{0},0)\right],$ (76)

where $G(\omega,z;z_{0},t_{0})$ was calculated in the previous section. Thanks
to time translation invariance we can always set $t_{0}=0$ and use the
expressions derived in the appendix B obtained by setting $p_{\parallel}=0$ as
we consider that the problem is planar. Notice that the integral on $z_{0}$ is
only between the plates as the magnetic field vanishes in the plates. Now the
integral over $\omega$ is trivial if $G$ has no singularities along the real
axis. We have seen that there are only poles when $\Theta$ diverges. This can
happen when 69 is satisfied, i.e. for a finite number values for $\omega$, or
when $\omega=0$. Close to $\omega=0$, we have
$\Theta\simeq-1/(2\omega^{2}_{\rm Pl}d+\omega_{\rm Pl})$ so no singularity. As
a result when the plasma frequency does not vanish, the Green’s function has
no singularity on the real axis and therefore

$e(z,t)=J_{0}\Re\left[e^{-imt}\int_{0}^{d}dz_{0}G(m,z;z_{0},0)\right].$ (77)

As a result, we have

$z\leq 0,\ \ e(z,t)=J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{m}(\sin
md+\frac{\xi(m)}{m}(1-\cos md))e^{\xi(m)z}\right]$ (78)

and

$z\geq d,\ \ e(z,t)=J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{m}(\sin
md+\frac{\xi(m)}{m}(1-\cos md))e^{-\xi(m)(z-d)}\right],$ (79)

which is symmetric in $z\to d-z$. Notice that the electric field only
penetrates inside the plates over a distance of order $1/\xi(m)$ corresponding
to a skin effect. Finally, we have for $0\leq z\leq d$

$\displaystyle
e(z,t)=J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{m}\left\\{\left(\sin
mz+\frac{\xi(m)}{m}(1-\cos mz)\right)\left(\cos m(z-d)-\frac{\xi(m)}{m}\sin
m(z-d)\right)\right.\right.$ $\displaystyle\left.\left.+\left(\cos
mz+\frac{\xi(m)}{m}\sin mz\right)\left(\sin
m(d-z)+\frac{\xi(m)}{m}\left(1-\cos m(d-z)\right)\right)\right\\}\right],$

which is also symmetric in $z\to d-z$.

In the next section, we generalise this result to the case where the magnetic
field is in all space, i.e. a more realistic situation, and calculate the
classical Casimir pressure induced by the axion.

###  IV.3 Non-ideal conductors with magnetic field in all space

####  IV.3.1 Electric field

The configuration under consideration here is probably the closest to a
possible realistic experimental setup. The cavity is made of conductors with
finite conductivity and the magnetic field is present everywhere. The
situation is sketched in Fig. 4.

Figure 4: Geometry and boundary conditions for the two-plate case with real
conductors.

When the magnetic field is present everywhere, the electric field receives new
contributions $\delta e(z)$, which are given

$\displaystyle\delta
e(z,t)=J_{0}\Re\left[e^{-imt+\xi(m)z}\frac{\Theta(m)}{\xi(m)}\left(1+e^{-\xi(m)z}\left(\cos
md+\frac{\xi(m)}{m}\sin md\right)\right.\right.$
$\displaystyle\left.\left.+\frac{m}{\xi(m)}\left(1-e^{-\xi(m)z}\right)\left(\sin
md-\frac{\xi(m)}{m}\cos md\right)\right)\right]$

and the contribution obtained by $z\to d-z$ when $z\geq d$. Between the
plates, for $z\in[0,d]$ the contribution is given by

$\displaystyle\delta
e(z,t)=J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{\xi(m)}\left(\cos mz+\cos
m(d-z)+\frac{\xi(m)}{m}(\sin mz+\sin m(d-z))\right)\right],$

which is symmetric in $z\to d-z$.

One interesting fact is that there is a non-vanishing oscillating electric
field at infinity, which is not propagating and results from the coupling
between the axion and the external magnetic field. More precisely we get far
inside the plate

$e_{\rm plate}(t)=J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{\xi(m)}\left(2\cos
md+\left(\frac{\xi(m)}{m}-\frac{m}{{\xi}(m)}\right)\sin md\right)\right].$
(83)

When the conductivity is larger than the mass, as in realistic situations,
this simplifies to

$e_{\rm plaque}(t)\approx J_{0}\Re\left[e^{-imt}\frac{\Theta(m)}{m}\sin
md\right],$ (84)

and finally

$e_{\rm plaque}(t)\approx-
J_{0}\Re\left[\frac{e^{-imt}}{\xi^{2}(m)}\right]\simeq-
J_{0}\Re\left[i\frac{e^{-imt}}{m\sigma}\right].$ (85)

This implies that there is a conduction current

$j_{\rm cond}(t)=\frac{J_{0}}{m}\sin mt,$ (86)

the amplitude of which does not depend on the conductivity.

####  IV.3.2 Classical pressure

The experimental observable for the effect under scrutiny here is a
modification of the Casimir pressure in the void between the plates. The
pressure is given by Eq. 42, which comes only from the polarisation effects in
matter given by

$p(d,t)=\frac{\vec{B}.\vec{p}}{B}=\int\not{d}\omega e^{-i\omega
t}(\epsilon(\omega)-1)e(d,\omega).$ (87)

As a result, we only need to evaluate the integral

$e(d,t)=J_{0}\Re\left[e^{-imt}\int_{-\infty}^{\infty}dz_{0}G(m,d;z_{0},0)\right].$
(88)

The expressions that we need are given below. When $z_{0}<0$ we have

$G(m,d;z_{0})=\Theta(m)e^{\xi(m)z_{0}},$ (89)

for $z_{0}\in[0,d]$,

$G(m,d;z_{0})=\Theta(m)\left(\cos mz_{0}+\frac{\xi(m)}{m}\sin mz_{0}\right),$
(90)

and finally for $z_{0}>d$

$G(m,d;z_{0})=\Theta(m)\left(\cos md+\frac{\xi(m)}{m}\sin
md\right)e^{-\xi(z_{0}-d)},$ (91)

which is continuous at $z_{0}=0,d$. Now it is easy to see that
$\int_{-\infty}^{\infty}dz_{0}G(m,d;z_{0},t_{0})$ converges and becomes

$\displaystyle{\cal
G}(m,d)=\int_{-\infty}^{\infty}dz_{0}G(m,d;z_{0},t_{0})=\frac{\Theta(m)}{m}\left[\frac{m}{\xi(m)}\left(1+\cos
md+\frac{\xi(m)}{m}\sin md\right)\right.$ $\displaystyle\left.+\sin
md+\frac{\xi(m)}{m}(1-\cos md)\right].$ (92)

Then the electric field is given by

$e(\omega,d)=\frac{J_{0}}{2}\left({\cal
G}(m,d)\not{\delta}(\omega-m)+\bar{\cal G}(m,d)\not{\delta}(\omega+m)\right),$
(93)

and the polarisability is

$p(t,d)=J_{0}\Re\left[e^{-imt}(\epsilon(m)-1){\cal G}(m,d)\right].$ (94)

As a result the pressure becomes

$\langle P_{z}\rangle=\frac{J_{0}^{2}}{2}\left|{\cal
G}(m,d)\right|^{2}\Re\left[\epsilon(m)-1\right].$ (95)

where

$\Re\left[\epsilon(m)-1\right]=-\frac{\omega^{2}_{\rm Pl}}{m^{2}+\gamma^{2}}.$
(96)

Notice that this is always negative so the pressure is attractive. Moreover
there is still a resonance when $\Theta$ diverges.

##  V Resonances in a Copper cavity

For a fixed mass $m$ when the size of the cavity $d$ takes different values,
the pressure on the plates experiences resonances. The resonances are due to
the vanishing of $\Theta(m)$, which happens for complex values of $m$ with an
imaginary part controlled by the parameter $\gamma$. In real situations,
resonances occur for values close to $m_{n}=n\pi/d$ and their height is
regularized by the dissipation parameter $\gamma$. Effectively, the resonances
on the real axis are obtained for

$\tan m_{n}d\simeq-\Re\left(\frac{2}{\sqrt{-\epsilon(m_{n})}}\right),$ (97)

where $|\epsilon(m_{n})|\gg 1$. Close to the resonances, we have

${\cal
G}(m,d)\simeq\frac{2\Theta(m)}{m_{n}}\frac{\xi(m_{n})}{m_{n}}\left(1-(-1)^{n}\right)$
(98)

from which we recover that the resonances are only present when $n=2p+1$ 111
Of course, the amplitude ${\cal G}$ does not vanish when $n=2p$. The
subdominant terms are small compared to the resonant case when $n=2p+1$., this
is a result we already obtained in the ideal case. For a fixed cavity size,
the behavior of the pressure with the mass of the axion, is not trivial. This
is illustrated in Fig. 5, for which the use of Copper plates is considered,
corresponding to a damping time $\gamma$ parameter of $10^{14}\rm\;Hz$, or
$6.6\times 10^{-11}\;\rm GeV$ in natural units. We consider a 50 T magnetic
field, close to the value of the higher stationary magnetic fields obtained
experimentally, a plate separation of $50\;\rm\mu m$ ($2.5\times 10^{11}\;\rm
GeV^{-1}$ in natural units) and a generic axion scale of $10^{10}\;\rm GeV$.

Figure 5: The Casimir pressure (absolute value) from axions as a function of
the axion mass for a distance $d=50\;\rm\mu m$ cm and $B=50$ T.

In Fig. 5, one can see the resonances, alternatively upward and downward. As
expected the peak separation decreases as the mass increases. Resonances
disappear gradually above $m=2\times 10^{-9}\;\rm GeV$, which is precisely the
plasma frequency for Copper.

For a fixed mass, the dependence of the axion classical pressure with the
distance is displayed in Fig. 6. The pressure varies on 10 orders of magnitude
between upward and downward peaks, around a mean value that does not depend on
the distance. In Fig. 6, an axion mass of $1.26\times 10^{-11}\;\rm GeV$ is
considered, corresponding to the first peak at $50\;\rm\mu m$.

Figure 6: The Casimir pressure (absolute value) from vacuum and from axions as
a function of the distance between the metallic plates when a magnetic field
$B=50$ T is present between the plates, and $m=1.26\times 10^{-11}\;\rm GeV$.

In Fig. 6, the conventional vacuum pressure is shown in comparison to the
axion pressure, with its typical $1/d^{4}$ dependance. At the first peak for
$d=50\;\rm\mu m$, the vacuum pressure is 19 orders of magnitude higher than
the axion pressure, making impossible to detect these axions in a Copper
cavity. As the vacuum pressure drops very quickly with the size of the cavity,
and as the axion field injects energy uniformly, the ratio becomes more
favorable as the size increases. Around a few meters, some axion resonances
are larger than the vacuum pressure. However, no measurement of Casimir
pressure is available at this scale.

The search for axion through Casimir effects at short distances in a Copper
cavity is not very promising as shown above. To search for a signal, the
general rule is to try measuring the Casimir pressure on the largest possible
distance. In addition, we will now see that the use of a reflecting material
with a smaller value for the damping parameter $\gamma$ leads to more narrow
resonances and might provide a higher signal.

##  VI The damping parameter and the width of resonances

In this section we investigate the influence of the damping parameter $\gamma$
on the strength of a potential axion signal. From Eq. 67, we can approximate

$\Theta(m)\simeq\frac{1}{2m_{2p+1}}\sqrt{-\epsilon(m_{2p+1})}\frac{1}{\frac{\sqrt{-\epsilon(m)}}{\Re\left(\sqrt{-\epsilon(m_{2p+1})}\right)}-1}$
(99)

and upon using $\xi(m_{n})/m_{n}\simeq\sqrt{-\epsilon(m_{n})}$ we find that
the Green’s function reduces to

${\cal
G}(m,d)\simeq\frac{1}{m_{2p+1}^{2}}\frac{1}{\frac{\sqrt{-\epsilon(m)}}{\Re\left(\sqrt{-\epsilon(m_{2p+1})}\right)}-1}$
(100)

close to the resonance at $m=m_{2p+1}$. This can be approximated in the two
regimes corresponding to $m_{2p+1}\gg\gamma$ and $m_{2p+1}\ll\gamma$.

The width of the resonances will be controlled by the value of the damping
parameters. Two regimes can be observed depending on the relative value of
$\gamma$ and the axion mass. As the width diminishes, the peak signal
naturally increases.

###  VI.1 Broad resonances $m\ll\gamma$

We first consider when the resonances is given by $m_{2p+1}$ and when
$m_{2p+1}\ll\gamma$. In this case we have

$\sqrt{-\epsilon(m)}\simeq\frac{\omega_{\rm Pl}}{\sqrt{\gamma
m}}e^{-i\pi/4}\left(1+\frac{i\omega}{2\gamma}\right).$ (101)

Notice that this is a complex number with a non-vanishing imaginary part even
when $\gamma$ is vanishingly small. This implies that the width of the
resonance has no direct connection to $\gamma$ as can be seen in the
approximation

${\cal
G}(m,d)\simeq\frac{1}{m_{2p+1}}\frac{1}{\frac{m-m_{2p+1}}{2}+im_{2p+1}}.$
(102)

As the width of the resonance is of order $m_{2p+1}$ and $m_{2p+1}\ll\gamma$,
the width of the resonance is constrained to be small compared to $\gamma$. On
the other hand, as the width is of the order of the location $m=m_{2p+1}$ of
the resonance, this corresponds to a broad resonance. When several resonances
of this type are present, the resonance of lowest order $p=0$ dominates as its
amplitudes is the largest. The resulting pressure becomes

$\langle P_{z}\rangle=-\frac{2\omega_{\rm
Pl}^{2}J_{0}^{2}}{\gamma^{2}m^{2}_{2p+1}}\frac{1}{(m-m_{2p+1})^{2}+4m_{2p+1}^{2}}.$
(103)

At the resonance we have

$\langle P_{z}\rangle=-\frac{2\omega_{\rm
Pl}^{2}J_{0}^{2}}{\gamma^{2}m^{4}_{2p+1}},$ (104)

which shows that the strength of the signal is suppressed by $\gamma^{2}$ for
high values of the damping parameter. So for large $\gamma$ values, the
resonances are broader and with a reduced peak height. This is illustrated in
Fig. 7, where $\gamma=10^{-8}\;\rm GeV$ has been used ($1.5\times 10^{16}$
Hz). It displays the first four resonances of the axion-induced pressure with
respect to the distance between the plates.

Figure 7: First resonances from axions as a function of the distance between
the metallic plates, with an increased value of the damping parameter
$\gamma=10^{-8}\;\rm GeV$, $B=50$ T, and $m=1.26\times 10^{-11}\;\rm GeV$.

###  VI.2 Narrow resonances $\gamma\ll m$

It appears from the previous section that to increase the strength of the
signal, it is better to consider a small damping parameter, i.e. a material
that tends to be a better conductor. We focus now on the case where
$m_{2p+1}\gg\gamma$ for which we obtain

$\sqrt{-\epsilon(m)}\simeq\frac{\omega_{\rm
Pl}}{\sqrt{m}}\left(1-\frac{i\gamma}{2m}\right).$ (105)

In this case, the imaginary part is directly related to $\gamma$. This leads
to

${\cal G}(m,d)\simeq\frac{1}{m_{2p+1}}\frac{1}{m-m_{2p+1}+\frac{i\gamma}{2}}.$
(106)

This is a Breit-Wigner distribution with a width $\Gamma=\gamma$, much smaller
than the location of the resonance at $m=m_{2p+1}$. In this case we obtain the
pressure

$\langle P_{z}\rangle=-\frac{J_{0}^{2}}{2}\frac{\omega_{\rm
Pl}^{2}}{m_{2p+1}^{4}}\frac{1}{(m-m_{2p+1})^{2}+\frac{\gamma^{2}}{4}}.$ (107)

At the resonance we have

$\langle P_{z}\rangle=-{2J_{0}^{2}}\frac{\omega_{\rm
Pl}^{2}}{m_{2p+1}^{4}}\frac{1}{\gamma^{2}}.$ (108)

which diverges in the small $\gamma$ limit where the resonances become
infinitely thin. Fig. 8 shows how resonances get narrower and higher for small
values of $\gamma$. For this figure, $\gamma=10^{-17}\;\rm GeV$ (15 MHz) and
all other parameters are unchanged compared to previous figures.

Figure 8: First resonances from axions as a function of the distance between
the metallic plates, with an decreased value of the damping parameter
$\gamma=10^{-17}\;\rm GeV$, $B=50$ T, and $m=1.26\times 10^{-11}\;\rm GeV$.

###  VI.3 Comparison with the halo velocity distribution

The width of the resonance is limited by the dispersion due to the finite
velocity of dark matter in the galactic halo

$\frac{\Delta m}{m}=\frac{\gamma}{m}\gtrsim v^{2}.$ (109)

From an observational point of view, there is no interest in searching for a
material with arbitrarily small $\gamma$. The natural velocity dispersion will
blur the resonance anyway. As the Casimir effect cannot be seen for distances
less than a typical distance $d_{\rm max}$ which is a fraction of a micron, we
find that the observability of the axion effect compared to the usual Casimir
effect will be possible only for

$\gamma\gtrsim\frac{\pi v^{2}}{d_{\rm max}}$ (110)

For lower values of $\gamma$, the resonances would be affected by the velocity
dispersion mostly. For $d_{\rm max}=50\;\rm\mu m$, the limiting value of
$\gamma$ is $\sim 10^{-17}\;\rm GeV$ (15 MHz). Notice that the uncertainty on
$m$ would also result in an uncertainty on the first resonance $\Delta d$ such
that

$\frac{\Delta d}{d}=\frac{\gamma d}{\pi}\gtrsim v^{2}.$ (111)

##  VII Potential sensitivity of Casimir experiments

To estimate the possible reach of Casimir experiments with a magnetic field,
we assume an experiment could be built with a 50 T magnetic field to perform a
measurement of Casimir pressure between $d=5\;\mu$m and $d=50\;\mu$m. A 50 T
magnetic field is of the order of the highest intensity for stable magnetic
fields produced in the laboratory, see for instance Song  _et al._ (2019) (27
T) and SMHFF (45 T). Running a Casimir experiment in such a high intensity
magnetic field is therefore probably challenging but not impossible. We also
assume that the plates are made out of an extremely good conductor and that
the regularization of the resonances is the result of the natural velocity
dispersion of the dark matter halo. This is independent of the material as
long as the conductivity is high enough. This could be achieved with
superconducting material, or any material with a damping factor of $\gamma\sim
10^{-17}\;\rm GeV$. The plasma frequency is taken equal to $2\times
10^{-9}\;\rm GeV$. As the London length for superconductors lie un the range
10 nm to 1000 nm, this seems achievable Tixador and Brunet (2003). We assume
that a 1% deviation of the vacuum Casimir pressure can be observed. To
determine the sensitivity of such an experiment, we solve numerically the
equation

$P_{\rm axion}\left(m,M\right)=\eta\times P_{\rm vacuum},$ (112)

for the variables $m$ and $M$, where $\eta$ is the sensitivity of the
experiment to the observation of a variation of the standard value of the
pressure. In the following we assume $\eta=1\%$, meaning that a percent-level
deviation to the conventional Casimir pressure is detectable. We consider the
ideal pressure relation

$P_{\rm vacuum}=-\frac{\pi^{2}}{240d^{4}},$ (113)

so by taking the value of the axion-induced pressure at a resonance 108 for a
distance $d$ one gets the value of the sensitivity on the axion-photon
coupling

$g_{\rm sens}=\sqrt{\eta}\frac{\pi}{\sqrt{960}}\frac{\gamma}{\omega_{\rm
P}}\frac{m}{Bd^{2}},$ (114)

where the coupling is simply taken as the inverse of the axion scale $g=1/M$.

The projected constraints are displayed in Fig. 9 in a coupling-mass plane.
The mass range is imposed by the range of distances over which the Casimir
effect can reasonably be probed. There exist several constraints in the
considered mass range, at low mass, the laboratory experiments ALPS Ehret  _et
al._ (2010) and PVLAS Ejlli  _et al._ (2020) constraint the highest values of
the coupling. In the whole range, the blue region is the constraint from the
CAST helioscope cas (2017). The blue regions on the right side of the plot are
direct constraints from telescopes (JWST Janish and Pinetti (2023); Roy  _et
al._ (2023), WINERED Yin  _et al._ (2024), MUSE Regis  _et al._ (2021), VIMOS
Grin  _et al._ (2007)), and indirect constraints from the level of infrared
background in the universe as measured by $\gamma$-rays Bernal  _et al._
(2023). The Casimir type of experiments for axion search obviously require
that axions are the dark matter. Above the dashed red line labelled “"decay
limit", in the redish region the lifetime of the axions is shorter than the
age of the universe so they cannot be the dark matter. Finally The oblique
yellow band labelled “QCD” corresponds to the region where parameters nicely
combine to solve the strong-CP problem. The main result of this paper is the
central light-green region labelled “Casimir $5\;\rm\mu m-50\;\mu m$”. It
contains the parameters that comply with the condition 112 for $\eta=1\%$. The
edge at 2 eV is due to the value of the plasma frequency, above which
resonances disappear.

To understand the shape of the sensitivity curve that is shown below, let us
consider the first resonance. At short distance, the ratio between the vacuum
pressure and the axion pressure is not favorable, because $P_{\rm vacuum}$
decreases very steeply with $d$ and $P_{\rm axion}$ is constant on average.
The mass corresponding to this first resonance is as high as it can get
because of the inverse proportionality between $m_{\rm a}$ and $d$ at
resonance. One can imagine an experiment where $d$ can be varied. When $d$
increases, two things happen : the mass of the first resonance decreases, and
the ratio $P_{\rm axion}/P_{\rm vacuum}$ improves, making the sensitivity
better. In a coupling/mass plane, going from short distance to large distance,
the sensitivity follows a line from a larger mass / larger coupling value, to
a lower mass lower coupling value. The same reasoning is true for all the
other resonances, thus leading to a sawtooth-like line for the sensitivity
curve of Casimir experiments. This is what appears in Fig. 9. The figure also
displays various existing constraints in the axion coupling/mass plane, the
excluded regions being the different blue patches.

Figure 9: Potential reach of Casimir experiments in a mass / coupling plane.
The three green regions correspond to different assumptions on the
experimental parameters (see text).

To show the limits of the current sensitivity estimates, some parameters can
be varied. For example it is expected that the plasma frequency in
superconducting materials depend on the magnetic field Inui (2021). In that
case the plasma frequency could be lower. In Fig. 9, the dark green region at
higher values of the coupling corresponds to resonances of the same width as
for the other curves, but with higher values of the plasma frequency, here
$\omega_{\rm p}=2\times 10^{-11}\;\rm GeV$. On a more optimistic side, If it
were possible to measure the Casimir force on distances up to 500 $\mu$m, the
sensitivity region would enlarge towards lower masses and lower coupling
values, as shown in lighter green in Fig. 9. If it were possible to probe the
Casimir pressure on such large distances, some parameters favored by QCD could
be tested.

Finally let comment briefly on the quantum contribution to the Casimir
pressure from the presence of the rapidly oscillating axion field. The
correction to the Casimir pressure can be estimated by consider a typical one
loop vacuum diagram for the photons with two axionic insertions. At each
vertex and working in the Coulomb gauge, one time derivative acts on the
oscillating axion leading to a factor of $m^{2}$. The diagram itself is
proportional to $\phi_{0}^{2}/M^{2}$. Finally a factor of $d^{2}$ must be
inserted for dimensional reason, see the appendix F for a sketch of the
calculation. The fact that the contribution vanishes when $m=0$ follows from
the fact that for a constant axion field, the $F\tilde{F}$ term in the
Lagrangian is non-dynamical and therefore no pressure can be generated. All in
all we have

$\frac{\delta P_{\rm axion}}{P_{\rm casimir}}={\cal
O}\left(\frac{\phi_{0}^{2}}{M^{2}}m^{2}d^{2}\right)$ (115)

which is negligible for the typical values considered in this paper. A more
thorough analysis is left for the future.

##  VIII Conclusion

We showed that the presence of axionic dark matter in the form of an
oscillating scalar field coupled to electromagnetism modifies the Casimir
effect at the classical level, i.e. it induces a classical pressure between
metallic plates. The main new phenomenon resulting from our study is the
appearance of a series of resonances whose positions depend on the spatial
scale probed by the Casimir experiment. The resonances are regularized by the
finite conductivity of the metallic plates. Estimate of the sensitivity show
that the potential reach of this method could be competitive with other probes
such as helioscopes or haloscopes. This will require experiments sensitive to
long distance Casimir forces, and very high magnetic fields.

###### Acknowledgements.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(Grant agreement No. 865306). We would like to thank Andreas Ringwald for
relevant remarks and suggestions.

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##  Appendix A Radiation from a single plate

As an application of the Green’s function techniques, we will derive the
expression for the electric field when only one plate is present and the
magnetic field penetrate inside a metal with finite conductivity. This
generalises the ideal case where the conductivity is taken to be infinite. We
first consider the generation of an electric due to the axion coupling when
the magnetic field occupies all space. This is given by

$e(z,t)=J_{0}\Re\left[e^{-imt}\int_{-\infty}^{\infty}dz_{0}G(m,z;z_{0},t_{0})\right].$
(116)

where the Green’s function is such that it vanishes in vacuum at $z=+\infty$
and in matter at $z=-\infty$. In vacuum, this selects the modes in $e^{i\omega
z}$ where the replacement $\omega^{2}\to\omega^{2}+i\tilde{\epsilon}$ with
$\tilde{\epsilon}\to 0$ guarantees the convergence.

We now have to select the Green’s function for $z>0$, i.e. outside the plaque.
This gives

$z_{0}<0:\ G(\omega,z;z_{0})=\frac{e^{i\omega
z+\xi(\omega)z_{0}}}{\xi(\omega)-i\omega},$ (117)

and

$z_{0}\in[0,z]:G(\omega,z;z_{0})=\left(\frac{1}{i\omega+\xi(\omega)}-\frac{1}{2i\omega}\right)e^{i\omega(z+z_{0})}+\frac{e^{i\omega(z-z_{0})}}{2i\omega},$
(118)

whilst finally

$z_{0}>z:G(\omega,z;z_{0})=\left(\frac{1}{i\omega+\xi(\omega)}-\frac{1}{2i\omega}\right)e^{i\omega(z+z_{0})}+\frac{e^{i\omega(z_{0}-z)}}{2i\omega}.$
(119)

As a result we have

$\int_{-\infty}^{\infty}dz_{0}G(m,z;z_{0},0)=\frac{1}{m^{2}}+\left(\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\right)e^{imz}.$
(120)

Hence there are two components. There is an oscillating electric field at the
frequency $m$ of the axion oscillations. There is also a propagating wave with
frequency $m$

$e(z,t)=\frac{J_{0}\cos
mt}{m^{2}}+J_{0}\Re\left[\frac{1}{\xi(m)(\xi(m)-im)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}e^{im(z-t)}\right].$
(121)

Let us now assume that the magnetic field is only in the plaque for $z<0$,
then the external electric is obtained by evaluating

$\int_{-\infty}^{0}dz_{0}G(m,z;z_{0},0)=\frac{1}{\xi(m)(\xi(m)-im)}e^{imz},$
(122)

corresponding to a propagating wave

$e(z,t)=J_{0}\Re\left[\frac{1}{\xi(m)(\xi(m)-im)}e^{im(z-t)}\right].$ (123)

As typically $\xi(m)\sim\omega_{\rm Pl}$ for $m\ll\omega_{\rm Pl}$, we have
the order of magnitude estimate

$e(z,t)\simeq\frac{J_{0}}{\omega_{\rm Pl}^{2}}\cos(m(z-t),$ (124)

i.e. a propagating wave of small amplitude.

In the case the magnetic field does not penetrate in the plaque we have

$e(z,t)=\frac{J_{0}\cos
mt}{m^{2}}-J_{0}\Re\left[\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}e^{im(z-t)}\right],$
(125)

where we see that the field vanishes at the surface only in the ideal case.

##  Appendix B The Green’s function in the two-plate case

In this appendix we will give explicit formulae for the Green’s function
satisfying

$-\partial_{0}(\epsilon\star_{t}\partial_{0}G)+\Delta
G=\delta^{(4)}(x^{\mu}-y^{\mu}).$ (126)

In the following we will use time-translation invariance and space-translation
invariance in the $(x,y)$ plane along the plaques to choose
$y^{\mu}=(\vec{0},z_{0},0)$. In terms of Fourier decomposition the Green’s
function satisfies

$\left(\partial_{z}^{2}-p_{\parallel}^{2}+\epsilon(\omega)\omega^{2}\right)G=\delta(z-z_{0}).$
(127)

We define

$\xi=\left(p_{\parallel}^{2}-\epsilon(\omega)\omega^{2}\right)^{1/2},$ (128)

where the square root is such that the real part is always positive. We also
have

$\Delta={\rm sign}(\omega)\left(\omega^{2}-p_{\parallel}^{2}\right)^{1/2},$
(129)

with the sign of $\omega$ when $|\omega|>p_{\parallel}$. We will separate the
$z$-axis into three intervals and give the solution in each case using the
continuity of the Green’s function at the boundaries $z=0$ and $z=d$. The
first derivative is continuous there too whilst there is a jump
$[\frac{dG}{dz}|_{z=z_{0}}=1$ of the first derivative when $G$ itself is
continuous at $z=z_{0}$. Moreover we impose that $\lim_{|z|\to\infty}G=0$.
This determines a unique solution.

###  B.1 $z_{0}\in[0,d]$

When $0\leq z_{0}\leq d$ is between the plates the Green’s function is then
defined by

$G=G_{-}e^{\xi z},\ z<0$ (130)

and

$G=G_{+}e^{-\xi(z-d)},\ z>d,$ (131)

where we have now

$G_{-}=\Theta(\omega,p_{\parallel})\left((\cos\Delta(z_{0}-d)-\frac{\xi}{\Delta}\sin\Delta(z_{0}-d)\right)$
(132)

and

$G_{+}={\Theta(\omega,p_{\parallel})}\left(\cos\Delta
z_{0}+\frac{\xi}{\Delta}\sin\Delta z_{0}\right)$ (133)

and the Green’s function between the plates

$\displaystyle 0\leq z\leq z_{0},\ G=G_{-}\left(\cos\Delta
z+\frac{\xi}{\Delta}\sin\Delta z\right)$ $\displaystyle z_{0}\leq z\leq d,\
G=G_{+}\left(\cos\Delta(z-d)-\frac{\xi}{\Delta}\sin\Delta(z-d)\right),$ .
(134)

where

$\Theta(\omega,p_{\parallel})=\frac{1}{\Delta\left(\left(1-\frac{\xi^{2}}{\Delta^{2}}\right)\sin\Delta
d-2\frac{\xi}{\Delta}\cos\Delta d\right)}.$ (135)

###  B.2 $z_{0}<0$

This case is different from the previous case as

$G=G_{-}e^{\xi z},\ z\leq z_{0},$ (136)

and

$G=Ae^{\xi z}+Be^{-\xi z},\ z\in[z_{0},0],$ (137)

whilst

$\displaystyle 0\leq z\leq d,\ G=G_{+}\left(\left(\sin\Delta
d-\frac{\xi}{\Delta}\cos\Delta d\right)\sin\Delta z+\left(\cos\Delta
d+\frac{\xi}{\Delta}\sin\Delta d\right)\cos\Delta z\right)$ $\displaystyle
z\geq d,\ G=G_{+}e^{-\xi(z-d)}$ . (138)

We find that

$\displaystyle A=\frac{G_{+}}{2}\left(\cos\Delta
d+\frac{\xi}{\Delta}\sin\Delta d+\frac{\Delta}{\xi}\left(\sin\Delta
d-\frac{\xi}{\Delta}\cos\Delta d\right)\right)$ $\displaystyle
B=\frac{G_{+}}{2}\left(\cos\Delta d+\frac{\xi}{\Delta}\sin\Delta
d-\frac{\Delta}{\xi}\left(\sin\Delta d-\frac{\xi}{\Delta}\cos\Delta
d\right)\right)$

and we have

$G_{+}=\Theta(\omega,p_{\parallel})e^{\xi z_{0}}.$ (140)

Finally we find that

$G_{-}=\Theta(\omega,p_{\parallel})\left(\cosh\xi z_{0}\left(\cos\Delta
d+\frac{\xi}{\Delta}\sin\Delta d\right)+\frac{\Delta}{\xi}\sinh\xi
z_{0}\left(\sin\Delta d-\frac{\xi}{\Delta}\cos\Delta d\right)\right)$ (141)

###  B.3 $z_{0}>d$

We have now

$z<0,\ G=G_{-}e^{\xi z}$ (142)

and

$z\in[0,d],\ G=G_{-}\left(\cos\Delta z+\frac{\xi}{\Delta}\sin\Delta z\right),$
(143)

whilst we have

$z\in[d,z_{0}],\ G=\frac{G_{-}}{2}\left(\sin\Delta
d\left(\frac{\xi}{\Delta}+\frac{\Delta}{\xi}\right)e^{-\xi(z-d)}+\left[2\cos\Delta
d+\sin\Delta
d\left(\frac{\xi}{\Delta}-\frac{\Delta}{\xi}\right)\right]e^{\xi(z-d)}\right)$
(144)

and finally

$z>z_{0},\ G=G_{+}e^{-\xi(z-d)},$ (145)

where we find

$G_{-}=\Theta(\omega,p_{\parallel})e^{-\xi(z_{0}-d)}$ (146)

and

$G_{+}=\Theta(\omega,p_{\parallel})\left(\cosh\xi(z_{0}-d)\left(\cos\Delta
d+\frac{\xi}{\Delta}\sin\Delta
d\right)+\frac{\Delta}{\xi}\sinh\xi(z_{0}-d)\left(-\sin\Delta
d+\frac{\xi}{\Delta}\cos\Delta d\right)\right).$ (147)

##  Appendix C Dissipation

In the background magnetic field, the complete equations of motion for
$a^{\mu}$ can be deduced from the complete action 222Notice that this action
is only gauge invariant $a_{\mu}\to a_{\mu}+\partial_{\mu}\alpha$ on shell
when we impose current conservation $\partial_{\mu}{\cal J}^{\mu}=0$. This can
be remedied by introducing a Stückelberg field $\theta$ as  $S=\int
d^{4}x\left(-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}-\frac{\phi}{4M}f^{{\mu\nu}}\tilde{f}_{\mu\nu}-{\cal
J^{\mu}}(a_{\mu}-\partial_{\mu}\theta)\right)$ (148) which transforms as
$\theta\to\theta+\alpha$ under a gauge transformation. The equations of motion
of $\theta$ give $\partial_{\mu}{\cal J}^{\mu}=0$ whilst the value of $\theta$
is left undetermined. This does not matter as the Hamiltonian obtained by
Legendre transform with respect to both $a^{\mu}$ and $\theta$ does not depend
on $\theta$.

$S=\int
d^{4}x\left(-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}-\frac{\phi}{4M}f^{{\mu\nu}}\tilde{f}_{\mu\nu}-{\cal
J}^{\mu}a_{\mu}\right).$ (149)

The magnetic field induces the source term for the gauge field. It is
convenient to rewrite the action in terms of the electric and magnetic fields
using
$-\frac{1}{4}f_{\mu\nu}f^{\mu\nu}=\frac{1}{2}\left(\vec{e}^{2}-\vec{b}^{2}\right)$
and $-\frac{1}{4}f^{{\mu\nu}}\tilde{f}_{\mu\nu}=\vec{e}.\vec{b}$ resulting in
the action

$S=\frac{1}{2}\int
d^{4}x\left(\vec{e}^{2}-\vec{b}^{2}+\frac{2\phi}{M}\vec{e}.\vec{b}-2\vec{J}.\vec{a}\right),$
(150)

from which the equations of motion used in the main text can be obtained.

Inside matter as in the case of metals, the physics is modified. The
permittivity $\epsilon$ of the material and its effect on the electric field
captures in an effective way the interactions between the matter particles and
the photons. This leads to the existence of the displacement vector which is
related to the polarisation of the medium by $\vec{d}=\vec{e}+\vec{p}$. The
displacement vector is related to the electric field via a retarded effect

$\vec{d}(\vec{x},t)=\int d\tau\epsilon(t-\tau)\vec{e}(\vec{x},\tau).$ (151)

A naive action due to the presence of a non-trivial permittivity can be
postulated as

$S=\frac{1}{2}\int
d^{4}x\left(\vec{e}.\vec{d}-\vec{b}^{2}+\frac{2\phi}{M}\vec{e}.\vec{b}-2\vec{J}.\vec{a}\right).$
(152)

As the action is not local anymore, the concept of particle is ill-defined.
Moreover dissipation coming from the imaginary part of $\epsilon(\omega)$
breaks unitarity. Let us use this action naively and get the canonical
momentum associated to the vector potential

$\vec{\pi}=-\frac{1}{2}(\epsilon+\hat{\epsilon})\star_{t}\vec{e}-\frac{\phi}{M}\vec{b},$
(153)

where we have introduced the notation $\hat{\epsilon}(t)=\epsilon(-t)$ and
$\star_{t}$ is the convolution operator in time. The Hamiltonian becomes

$H=\frac{1}{2}\int
d^{3}x\left(\vec{e}.\vec{\hat{d}}+\vec{b}^{2}+2\vec{J}.\vec{a}\right).$ (154)

where we have defined

$\vec{\hat{d}}=\hat{\epsilon}\star_{t}\vec{e}=\int
d\tau\epsilon(\tau-t)\vec{e}(\tau)$ (155)

As can be seen this differs from the usual energy postulated for
electromagnetism

$H_{\rm mat}=\frac{1}{2}\int
d^{3}x\left(\vec{e}.\vec{d}+\vec{b}^{2}+2\vec{J}.\vec{a}\right).$ (156)

The two Hamiltonians reduce to the same expression when
$\epsilon=\hat{\epsilon}$. This condition can be understood after Fourier
transforming in time
$\vec{d}(\omega,\vec{x})=\epsilon(\omega)\vec{e}(\omega,\vec{x})$ where
$\epsilon(\omega)$ is a complex function such that
$\bar{\epsilon}(-\omega)=\epsilon(\omega)$. In a metal, the permittivity has a
pole on the negative imaginary axis. There is dissipation unless
$\epsilon(-\omega)=\epsilon(\omega)$ implying that $\Im\epsilon=0$, i.e. the
imaginary part of $\epsilon(\omega)$ vanishes. When this is the case we have
$\epsilon(t)=\epsilon(-t)$ and the Hamiltonian $H$ coincide with $H_{\rm
mat}$. In general this is not the case and the two Hamiltonians differ.

There is another major issue with the naive action: it does not reproduce the
correct equations of motion. Indeed the Euler-Lagrange for $\vec{a}$ coming
from (152) is

$-\partial_{0}\left(\frac{(\epsilon+\hat{\epsilon})}{2}\star_{t}\partial_{0}\vec{a}\right)+\Delta\vec{a}=-\frac{\dot{\phi}}{M}\vec{b}+\vec{J}$
(157)

which does not coincide with the phenomenological equation of motion

$-\partial_{0}(\epsilon\star_{t}\partial_{0}\vec{a})+\Delta\vec{a}=-\frac{\dot{\phi}}{M}\vec{b}+\vec{J}.$
(158)

This equation is not time-reversal invariant as $\epsilon\neq\hat{\epsilon}$,
i.e. dissipation implies a specific arrow of time. In fact it is interesting
to consider the time-reversed process defined as

$\tilde{a}^{i}(\tilde{t})=a^{i}(-t),$ (159)

where $\tilde{t}=-t$. It satisfies the time reversed equation

$-\partial_{0}(\hat{\epsilon}\star_{\tilde{t}}\partial_{0}\vec{\tilde{a}})+\Delta\vec{\tilde{a}}=-\frac{\dot{\phi}}{M}\vec{b}+\vec{J}.$
(160)

where time derivatives are with respect to $\tilde{t}$. The equation of motion
(157) is the symmetrised in time of (158) and (160). In this case when
$\epsilon(t)=\epsilon(-t)$ the normal process and its time-reversed satisfy
the same equation. This only applies to plasmas and not to metals with a
finite conductivity.

##  Appendix D Energy budget

###  D.1 The small mass limit in the one plaque case

Let us consider the case of $m\ll\gamma\ll\omega_{\rm Pl}$ then we have

$\xi\simeq\sqrt{m\sigma}e^{-i\pi/4}\left(1+i\frac{m}{2\gamma}\right).$ (161)

This implies that

$\frac{1}{m^{2}}+\frac{1}{\xi(m)\left(\xi(m)-im\right)}-\frac{\xi(m)}{m^{2}}\frac{1}{\xi(m)+im}\simeq\frac{-i}{m\sqrt{m\sigma}}e^{i\pi/4}.$
(162)

When the magnetic field is present everywhere

$e(z,t)=\frac{J_{0}}{m^{2}}\cos mt-\frac{J_{0}}{m^{2}}\cos
m(t-z)+\frac{J_{0}}{m\sqrt{m\sigma}}\sin\left(m(t-z)-\frac{\pi}{4}\right),$
(163)

corresponding to the electric field generated in the ideal case complemented
by a term which disappears when the conductivity because infinite. The
pressure on the plaques follows from

$\Re\left[\epsilon(m)-1\right]=-\frac{\omega^{2}_{\rm Pl}}{\gamma^{2}+m^{2}}$
(164)

and become

$\langle P\rangle\simeq-\frac{J_{0}^{2}}{2\gamma m^{3}},$ (165)

which is always negative, corresponding to the fact that the vacuum for $z>0$
attracts the plate situated at $z\leq 0$.

In the single plaque case with a magnetic field only in the metal we have

$e(z,t)=\frac{J_{0}}{m\sigma}\sin m(t-z),$ (166)

which vanishes for a perfect conductor. The pressure on the plaque becomes

$\langle P\rangle\simeq\frac{J_{0}^{2}}{2m^{3}\gamma},$ (167)

i.e. the same result as before when the magnetic field is everywhere.

In both cases the vacuum attracts the plate. This effect will reemerge in the
two plate case where the two plates will attract each other classically.

###  D.2 Energy balance

It is interesting to consider the conservation of energy when one plaque is
present and radiation from the plate takes place. Inside the plaques we have
the Maxwell equation

$\vec{\nabla}\wedge\vec{b}=\partial_{0}\vec{e}+\vec{j}_{\rm ind}+\vec{j}_{\rm
axion},$ (168)

where $\vec{j}_{\rm ind}$ is the induced current

$\vec{j}_{\rm ind}=\dot{\vec{p}},$ (169)

where $\vec{p}$ is the polarisability. This current only exists in a finite
width of the plaque, i.e. this is a skin effect. This current is responsible
for the dissipation power $\vec{j}_{\rm ind}.\vec{e}$ in the plaque
corresponding to the loss of energy. As a result, the plaque heats up due to
the Joule effect. The axionic current is given by

$\vec{j}_{\rm axion}=\frac{\dot{\phi}}{M}\vec{B}.$ (170)

This is complemented with the Bianchi identity

$\partial_{0}\vec{b}+\vec{\nabla}\wedge\vec{e}=0,$ (171)

from which we can get the propagation equation in matter for the magnetic
field

$\left(\epsilon(\omega)\omega^{2}+\Delta\right)\vec{b}=0.$ (172)

It coincides with the propagation equation for the electric field.

Let us now study the local conservation of energy

$E=\frac{\vec{e}^{2}+\vec{b}^{2}}{2},$ (173)

which satisfies

$\frac{dE}{dt}+\vec{\nabla}.\vec{P}_{\rm poynting}=-\left(\vec{j}_{\rm
ind}+\vec{j}_{\rm axion}\right).\vec{e},$ (174)

where the Poynting vector corresponds to the radiated power

$\vec{P}_{\rm poynting}=\vec{e}\wedge\vec{b}.$ (175)

In the absence of currents we have

$\frac{dE}{dt}+\vec{\nabla}.\vec{P}_{\rm poynting}=0,$ (176)

expressing the conservation of energy, i.e. the energy lost by the system is
matched by the flux of radiation. When the axion is present and dissipation
too, we find that the power dissipated by Joule’s effect is

$P_{\rm joule}=\langle\vec{j}_{\rm ind}.\vec{e}\rangle,$ (177)

which has to be positive. We will study this local equation more precisely
below.

In the case treated in the main text with a single plaque and a magnetic field
the polarisability at the surface of the metal does not vanish and becomes

$p(0,t)=-\frac{J_{0}\sigma\cos\left(mt-\frac{\pi}{4}\right)}{m^{2}\sqrt{m\sigma}}$
(178)

in the small mass limit. This induces a surface current

$j_{\rm
ind}(0,t)=\frac{J_{0}\sigma\sin\left(mt-\frac{\pi}{4}\right)}{m\sqrt{m\sigma}}.$
(179)

The power dissipated by Joule’s effect at the surface of the metal is on
average

$P_{\rm joule}=\left\langle j_{\rm
ind}(0,t)e(0,t)\right\rangle=\frac{J_{0}^{2}}{2m^{3}}.$ (180)

Let us now assume that dissipation acts adiabatically over times $t_{\rm
dis}\gg 1/m$, then the average energy decreases as

$\frac{d\langle E\rangle}{dt}=\frac{1}{2m^{3}\sigma}\frac{dJ_{0}^{2}}{dt},$
(181)

i.e. the amplitude of the current inside matter varies in time due to
dissipation.

As we consider the balance equation at $z=0$ we have

$\langle\vec{\nabla}\vec{P}_{\rm
Poynting}\rangle=\left\langle\frac{de}{dz}(0,t)b(0,t)+\frac{db}{dz}(0,t)e(0,t)\right\rangle,$
(182)

where the Bianchi identity gives us that

$b_{y}=-\frac{J_{0}}{m^{2}}\cos
m(t-z)+\frac{J_{0}}{m\sqrt{m\sigma}}\sin\left(m(t-z)-\frac{\pi}{4}\right).$
(183)

Notice that the non-propagating term of $e(z,t)$ in $\cos mt$ corresponds to a
term with a Fourier transform proportional to $\delta(k)$ as a result we can
also write $b_{y}=\frac{k}{\omega}e(k,\omega)$ where the Fourier transform of
the non-propagating part is

$e(\omega,k)\supset\frac{J_{0}}{2m^{2}}\left(\not{\delta}(\omega-m)+\not{\delta}(\omega+m)\right)\not{\delta}(k),$
(184)

whose contribution does not appear in $b_{y}(k,\omega)$ as
$k\not{\delta}(k)\equiv 0$. Hence only the propagating part of $e$ contributes
to $b_{y}$ as $k=\pm m$. We then find that $\langle
b(0,t)\partial_{z}e(0,t)\rangle=0$ and $\langle\vec{\nabla}\vec{P}_{\rm
Poynting}\rangle=\langle\partial_{z}b(0,t)e(0,t)\rangle$ which gives

$\langle\vec{\nabla}\vec{P}_{\rm
Poynting}\rangle=-\frac{J_{0}^{2}}{2\sqrt{2}m^{2}\sqrt{m\sigma}}.$ (185)

Similarly we find that the axion injects energy in the system as

$\langle j_{\rm
axion}(0,t)e(0,t)\rangle=\frac{J_{0}^{2}}{2\sqrt{2}m^{2}\sqrt{m\sigma}},$
(186)

implying that the radiated energy balances the axionic injection

$\langle\vec{\nabla}\vec{P}_{\rm Poynting}\rangle+\langle j_{\rm
axion}(0,t)e(0,t)\rangle=0.$ (187)

This implies that the change of amplitude of the current satisfies

$\frac{d\langle E\rangle}{dt}=-\langle j_{\rm ind}(0,t)e(0,t)\rangle,$ (188)

where the variation of the electromagnetic energy at the surface of the plaque
is due to the Joule effect from the induced current. This would give

$J_{0}^{2}(t)=J_{0}^{2}(0)e^{-\sigma t},$ (189)

corresponding to a dissipation time $t_{\rm dis}=1/\sigma$. Typically for
Copper, we have $\sigma=10^{-8}\;{\rm GeV}$ implying that the dissipation time
appears to be much shorter than the oscillation time $1/m$ as long as $m\leq
10$ eV. In this case, the previous calculation does not apply and one must
take into account both the time evolution of the axions and the photons.

###  D.3 Axion electrodynamics in the plaque

Let us now go beyond the simple hypothesis that $\phi_{0}$ is constant in the
plaque. The Joule’s effect implies that this cannot be the case. We must
analyse the coupled Klein-Gordon equation following the line of Ahonen  _et
al._ (1996)

$\ddot{\phi}-\Delta\phi+m^{2}\phi=-\frac{B}{M}\dot{a}$ (190)

where $e=-\dot{a}$. The equation for the electric field becomes

$\Delta a-\ddot{a}=-\frac{B}{M}\dot{\phi}+\sigma\dot{a},$ (191)

where the term $j_{\sigma}=\sigma e$ is the conduction current.

We can analyse the modes of this system in the plaque by going to Fourier
space. Having two equations with no sources, the non-trivial modes are
obtained by requiring that the determinant of the system vanishes. This gives

$(\omega^{2}-k^{2}+i\sigma\omega)(\omega^{2}-k^{2}-m^{2})=\frac{B^{2}\omega^{2}}{M^{2}}.$
(192)

As we are interested in $B/M\ll m$, the mixing is very small implying that
there are two branches of solutions. The first branch corresponds to

$\omega^{2}-k^{2}+i\sigma\omega\approx 0,$ (193)

corresponding to two modes

$\omega_{1}\approx-i\sigma,\ \omega_{2}\approx-i\frac{k^{2}}{\sigma},$ (194)

where we have assumed that $k\ll\sigma$. Both modes are decaying modes in time
with no time oscillations. So they do not correspond to the case of an initial
oscillatory axion being affected by Joule’s effect. The mode $\omega_{1}$
decays very fast whereas the second one has a long lifetime.

The more interesting modes which would correspond to an initially oscillating
axions are on the second branch close to $\omega^{2}=k^{2}+m^{2}$. The modes
are then

$\omega_{\pm}=\pm\left((k^{2}+m^{2})^{1/2}+\frac{m^{2}}{2\sigma^{2}}\frac{B^{2}}{M^{2}}\frac{1}{\sqrt{k^{2}+m^{2}}}-\frac{B^{2}}{2M^{2}}\frac{i}{\sigma}\right)$
(195)

The physical mode is the decreasing one $\omega_{+}$ whose imaginary part is
in $-\frac{B^{2}}{2M^{2}}\frac{i}{\sigma}$. Notice that when $\sigma$ is very
large, this gives a very large lifetime. This is the surprising result of this
analysis. The mode $\omega_{1}$ corresponds to the intuition that the Joule’s
effect withdraws the energy from a finite source and therefore depletes the
axion and the electric field very rapidly. The mode $\omega_{+}$ corresponds
to the fact that the Joule’s effect removes energy from the system but the
axion keeps replenishing. The two effects compete but the second one with a
long lifetime is the dominant effect.

As a result, the analysis in the plaque confirms that our initial hypothesis
of a constant amplitude for the axion is justified as the variation time due
to the Joule effect is very large. So we are entitled to trust our analysis
and keep that $J_{0}$ is constant on the time scale of the experiment.

##  Appendix E Comparison with the Lifschitz theory

For a metal the Casimir effect depends on the conductivity too. In this case,
the pressure separates in the electric pressure

$P_{z}^{+}=-\int\not{d}\omega\not{d}^{2}p_{\parallel}\frac{\Delta}{{\left(\frac{1+\frac{\xi}{\Delta}}{1-\frac{\xi}{\Delta}}\right)^{2}e^{2\Delta
d}-1}},$ (196)

coming from the $TE$ modes with two plates with the permittivity $\epsilon$.
We have introduced $\Delta=(p^{2}_{\parallel}+\omega^{2})^{1/2}$. This
integral can be written as

$P_{z}^{+}=-\int\not{d}\omega\not{d}^{2}p_{\parallel}\frac{\Delta}{{r_{TE}^{-2}e^{2\Delta
d}-1}},$ (197)

where $r_{TE}$ is the reflection coefficient for the $TE$ modes evaluated for
imaginary frequencies
$r_{TE}(i\omega)=\frac{1-\frac{\xi}{\Delta}}{1+\frac{\xi}{\Delta}}$ where
$\xi=(p_{\parallel}^{2}+\omega^{2}\epsilon(i\omega))^{1/2}$ depends on the
permittivity for imaginary frequencies

$\epsilon(i\omega)=1+\frac{\omega^{2}_{\rm Pl}}{\omega(\omega+\gamma)}.$ (198)

The magnetic pressure is given by

$P_{z}^{-}=-\int\not{d}\omega\not{d}^{2}p_{\parallel}\frac{\Delta}{{\left(\frac{1+\frac{\xi}{\epsilon(i\omega)\Delta}}{1-\frac{\xi}{\epsilon(i\omega)\Delta}}\right)^{2}e^{2\Delta
d}-1}},$ (199)

coming from the $TM$ modes with two plates with the same permittivity
$\epsilon$. Notice that the same integral can be written as

$P_{z}^{-}=-\int\not{d}\omega\not{d}^{2}p_{\parallel}\frac{\Delta}{{r_{TM}^{-2}e^{2\Delta
d}-1}},$ (200)

where $r_{TM}$ is the reflection coefficient for the $TM$ modes evaluated for
imaginary frequencies. In the ideal case where $r_{TE}=r_{TM}=1$, the electric
and magnetic contributions to the Casimir pressure are equal and the total
pressure reduces to

$P_{z}=-2\int\not{d}\omega\not{d}^{2}p_{\parallel}\frac{\Delta}{e^{2\Delta
d}-1},$ (201)

which can be evaluated as

$P_{z}=-\frac{2\times
4\pi}{16(2\pi)^{3}d^{4}}\int_{0}^{\infty}dx\frac{x^{3}}{e^{x}-1}=-\frac{2\times
4\pi}{16(2\pi)^{3}d^{4}}\Gamma(4)\zeta^{(}4)=-\frac{\pi^{2}}{240d^{4}}.$ (202)

This is the usual Casimir pressure in the ideal case. At finite temperature
$T$, the Lifschitz formulae must be computed by replacing the integral
$\int\not{d}\omega\to T\sum_{n>0}$ over the Matsubara frequencies where
$\omega_{n}=2\pi nT$. As the reflection coefficients are less than unity, the
effect of the conductivity of the real plates is to reduce the quantum
pressure between the plates. Finally, let us notice that when $\omega$ becomes
larger than $\omega_{\rm Pl}$, the reflection coefficients converge to zero
implying that the pulsation integral giving rise to the Casimir pressure has a
natural cut-off scale of order $\omega_{\rm Pl}$ above which the integral
receive no contribution. Similarly when the transverse momenta $p_{\parallel}$
are much larger than $\omega$, the TE reflection coefficient vanishes too. In
all cases, when the quantised pulsation
$\omega_{n}\sim\sqrt{p^{2}_{\parallel}+\frac{n^{2}\pi^{2}}{d^{2}}}$ exceeds
the plasma frequency, in particular for large $p_{\parallel}$, the plaques
become transparent implying that $\epsilon=1$ and the absence of Casimir
pressure. As a result the plasma frequency serves as a UV cut-off for energies
and momenta for the Casimir effect in the presence of metallic plates.

##  Appendix F Quantum Effects

The contribution of the axion to the quantum pressure can be evaluated by
considering the corrections of the photon propagator due to the axion. For
each photon polarisation, let us consider the vacuum energy per unit surface
which behaves like

$\frac{E_{\rm
vac}}{S}=\int\not{d}\omega\not{d}^{2}p_{\parallel}\omega^{2}\Delta(\omega,p_{\parallel})$
(203)

where the photon propagator in the cavity with resonances at
$\omega=\omega_{n}(p_{\parallel})\sim
p^{2}_{\parallel}+\frac{n^{2}\pi^{2}}{d^{2}}$, where $\vec{p}_{\parallel}$ is
the transverse momentum to the plates, is given by

$\Delta(\omega,p_{\parallel})\sim\sum_{n}\frac{i}{\omega^{2}-\omega^{2}_{n}(p_{\parallel})+i\epsilon}.$
(204)

A contour integral in the lower half plane gives the usual
$\frac{1}{2}\sum_{n}\int\not{d}^{2}p_{\parallel}\omega_{n}(p_{\parallel})$
before the axionic corrections. As the photon energy momentum tensor is not
corrected by the axion coupling due to its topological nature, we just have to
correct the propagator and average over the fast axion oscillations. The first
correction to the vacuum energy appears after two axionic insertions and gives
a term in

$\frac{\delta E_{\rm vac}}{S}\sim
m^{2}\frac{\phi_{0}^{2}}{2M^{2}}\int\not{d}\omega\not{d}^{2}p_{\parallel}p^{2}_{\parallel}\Delta(\omega,p_{\parallel})(\Delta(\omega+m,p_{\parallel})+\Delta(\omega-m,p_{\parallel}))\Delta(\omega,p_{\parallel})$
(205)

where each vertex bring a spatial and a time derivative. The time derivatives
lead to the prefactor in $m^{2}$ coming from the derivative of the axion field
and the spatial derivative is in $p_{\parallel}$. In this expression, we are
only interesting in the order of magnitude. A more precise calculation using
field theoretic methods from first principle is in progress. Notice that the
axionic insertions shift the energy in the propagator $\omega\to\omega\pm m$.
The integral is dominated by the resonances at $\omega_{n}$ and gives a
leading contribution

$\frac{\delta E_{\rm vac}}{S}\sim
im^{2}\frac{\phi_{0}^{2}}{4M^{2}}\sum_{n}\int\not{d}\omega\not{d}^{2}p_{\parallel}\omega^{2}p^{2}_{\parallel}\frac{1}{(\omega^{2}-\omega^{2}_{n}(p_{\parallel})+i\epsilon)^{2}}\frac{1}{\omega_{n}^{2}(p_{\parallel})-\frac{m^{2}}{4}}$
(206)

The same contour integral in the lower half plane gives a contribution in

$\frac{\delta E_{\rm vac}}{S}|\sim
m^{2}\frac{\phi_{0}^{2}}{4M^{2}}\sum_{n}\int\not{d}\omega\not{d}^{2}p_{\parallel}p^{2}_{\parallel}\frac{1}{\omega_{n}(p_{\parallel})}\frac{1}{\omega_{n}^{2}(p_{\parallel})-\frac{m^{2}}{4}}$
(207)

which is an integral which can be calculated in dimensional regularisation

$\frac{\delta E_{\rm vac}}{S}\sim
m^{2}\frac{\phi_{0}^{2}}{4M^{2}}\sum_{n}\int\not{d}^{2}p_{\parallel}p^{2}_{\parallel}\frac{1}{(p^{2}_{\parallel}+\frac{n^{2}\pi^{2}}{d^{2}})^{3/2}}\sim
m^{2}\frac{\phi_{0}^{2}}{16\pi
M^{2}}\frac{\Gamma(-\frac{3}{2})}{\Gamma(\frac{1}{2})}\sum_{n}\frac{n\pi}{d}$
(208)

when $md\ll 1$. This gives

$\frac{\delta E_{\rm vac}}{S}\sim
m^{2}d^{2}\frac{\phi_{0}^{2}}{16M^{2}}\frac{\Gamma(-\frac{3}{2})}{\Gamma(\frac{1}{2})}\zeta(-1)\frac{1}{d^{3}}$
(209)

and dimensionally we have

$\frac{\delta E_{\rm vac}}{E_{\rm vac}}\sim
m^{2}d^{2}\frac{\phi_{0}^{2}}{M^{2}}$ (210)

as $\frac{E_{\rm vac}}{S}\sim 1/d^{3}$.