G. L. Klimchitskaya, V. M. Mostepanenko, R. I. P. Sedmik, and H.
AbeleSymmetry11(3)4072019Prospects for Searching Thermal Effects, Non-
Newtonian Gravity and Axion-Like Particles: Cannex Test of the Quantum Vacuum

# Prospects for Searching Thermal Effects, Non-Newtonian Gravity and Axion-
Like Particles: Cannex Test of the Quantum Vacuum

Galina L. Klimchitskaya1,2,111Correspondence<EMAIL_ADDRESS>1Central Astronomical Observatory at Pulkovo of the Russian Academy of
Sciences, Saint Petersburg 196140, Russia;  Vladimir M.
<EMAIL_ADDRESS>1Central Astronomical Observatory at
Pulkovo of the Russian Academy of Sciences, Saint Petersburg 196140, Russia;
René I. P<EMAIL_ADDRESS>1Central Astronomical
Observatory at Pulkovo of the Russian Academy of Sciences, Saint Petersburg
196140, Russia;  Hartmut<EMAIL_ADDRESS>1Central
Astronomical Observatory at Pulkovo of the Russian Academy of Sciences, Saint
Petersburg 196140, Russia;

###### Abstract

We consider the Cannex (Casimir And Non-Newtonian force EXperiment) test of
the quantum vacuum intended for measuring the gradient of the Casimir pressure
between two flat parallel plates at large separations and constraining
parameters of the chameleon model of dark energy in cosmology. A modification
of the measurement scheme is proposed that allows simultaneous measurements of
both the Casimir pressure and its gradient in one experiment. It is shown that
with several improvements the Cannex test will be capable to strengthen the
constraints on the parameters of the Yukawa-type interaction by up to an order
of magnitude over a wide interaction range. The constraints on the coupling
constants between nucleons and axion-like particles, which are considered as
the most probable constituents of dark matter, could also be strengthened over
a region of axion masses from 1 to 100 meV.

###### doi:

10.3390/sym11030407

###### pacs:

quantum vacuum; Casimir pressure; axion; non-Newtonian gravity

##  1 Introduction

Since the development of quantum field theory, it has been appreciated that
the quantum vacuum is a fundamental type of physical reality which potentially
contains all varieties of elementary particles and their interactions.
Although an infinitely large energy density of zero-point oscillations of
quantum fields (the so-called virtual particles) could be considered to be
catastrophic [1], convenient self-consistent procedures have been elaborated
on how to make it equal to zero in the empty Minkowski space and take it into
account when calculating the probabilities of arbitrary processes in the
framework of the Standard Model. In doing so, the quantum vacuum is usually
responsible for some part of the measured quantity (for instance, the Lamb
shift), whereas the rest of it is determined by the ordinary (real) particles.

There is, however, one physical phenomenon, where the measured quantity is
determined entirely by the quantum vacuum. This is the Casimir effect arising
in the quantization volumes restricted by some material boundaries or in
cosmological models with nontrivial topology [2, 3, 4, 5]. An essence of this
effect is that although the vacuum energy in restricted or topologically
nontrivial volumes remains infinitely large, it becomes finite when
subtracting the vacuum energy of empty topologically trivial Minkowski space.
The negative derivative of the obtained finite vacuum energy with respect to
the length parameter (either a separation between the boundary surfaces or a
scale of the topologically nontrivial manifold) results in the Casimir force,
which generalizes the familiar van der Waals force in the case of larger
separations when one should take into consideration the effects of
relativistic retardation [5].

Recently it was understood that the quantum vacuum may be responsible for an
impressive phenomenon in nature, i.e., for acceleration of expansion of the
Universe [6]. This can be explained by an impact of the energy density of
quantum vacuum (which is often referred to as dark energy) corresponding to a
nonzero renormalized value of the cosmological constant in the Einstein
equations of general relativity theory [7].

Another subject is that at sufficiently short separations between the boundary
surfaces, vacuum forces are stronger than Newtonian gravitation. In this case,
they form a background for testing new physics, such as Yukawa-type
corrections to Newton’s law of gravitation arising due to exchange of light
hypothetical scalar particles [8] or due to spontaneous compactification of
extra spatial dimensions at the low-energy compactification scale [9]. Forces
of this kind would alter the energy eigenstates of a neutron in the gravity
potential of the Earth, and are searched for by a technique called Gravity
Resonance Spectroscopy [10, 11, 12] by the qBounce collaboration. On the
background of Casimir forces, one could also search for axions and other
axion-like particles [13, 14, 15, 16, 17, 18, 19, 20] which are considered as
hypothetical constituents of dark matter. It is remarkable that taken
together, dark energy and dark matter contribute for more than 95% of the
energy of the Universe, leaving less than 5% to the forms of energy we are
presently capable to observe directly [6].

Precise measurements of the Casimir force revealed a problem that the
experimental data agree with theoretical predictions of the fundamental
Lifshitz theory only under the condition that in calculations one disregards
the relaxation properties of conduction electrons and the conductivity at a
constant current for metallic and dielectric boundary surfaces, respectively
(see review in [5, 21] and more modern experiments [22, 23, 24, 25, 26, 27]).
Theoretically, it was shown that an inclusion of the relaxation properties of
conduction electrons and the conductivity at a constant current in
computations results in a violation of the Nernst heat theorem for the Casimir
entropy (see review in [5, 21] and further results [28, 29, 30, 31]). Taking
into account that both the relaxation properties of conduction electrons for
metals and the conductivity at a constant current for dielectrics are well
studied really existing phenomena, there must be profound physical reasons for
disregarding them in calculations of the Casimir force caused by the zero-
point and thermal fluctuations of the electromagnetic field.

All precise experiments on measuring the Casimir interaction mentioned above
have been performed in the sphere-plate geometry at surface separations below
a micrometer. In this paper, we consider the Cannex (Casimir And Non-Newtonian
force EXperiment) that was designed to test the quantum vacuum in the
configuration of two parallel plates at separations up to
$15\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [32, 33]. In addition to the
already discussed possibility of testing the nature of dark energy [32], we
consider here the potentialities of this experiment for searching thermal
effects in the Casimir force, stronger constraints on Yukawa-type corrections
to Newton’s gravitational law, and on the coupling constants of axion-like
particles. For this purpose, a modification in the measurement scheme is
proposed, which allows simultaneous measurement of both the Casimir pressure
and its gradient. It is shown that after making several improvements to the
setup, the Cannex test would be capable of performing the first observation of
thermal effects in the Casimir interaction and to strengthen the presently
available limits on Yukawa-type corrections to Newtonian gravity by up to a
factor of 10. Stronger limits could also be obtained on the coupling constants
of axion-like particles to nucleons within a wide range of axion masses.

The paper is organized as follows. In Section 2 we briefly describe the
experimental setup with two parallel plates and possible improvements. Section
3 is devoted to the calculation of thermal effects in the Casimir pressure and
its gradient at separations relevant for Cannex. In Section 4 the prospective
constraints on non-Newtonian gravity and axion-like particles, which could be
obtained from the improved setup, are found. Section 6 contains our
conclusions and discussion.

##  2 Experimental Setup with Improved Precision

We recently demonstrated the feasibility of Casimir pressure gradient
measurements with the parallel plate Cannex setup [33]. In the present
article, we propose several improvements to the experiment that will lead to a
significant increase in sensitivity and reduce the influence of systematic
effects.

In our setup shown in Figure 1, the parallel plate geometry is implemented by
a rigid vertical SiO2 cylinder (lower plate) of radius $R=5.742\,$mm separated
by a vacuum gap of width $a$ from a movable (upper) sensor plate that can be
described as a lumped mass-spring system [32]. This sensor with an interacting
area $A$, elastic constant $k$, and effective mass $m_{\rm eff}$ has a free
resonance frequency $\omega_{0}=\sqrt{k/m_{\rm eff}}$. In [33] an
interferometric detection scheme was used to measure the gradient of the
Casimir pressure. Here, we propose to implement the same scheme to measure
both pressure gradients and the pressure acting between the two flat plates.
Gradients $P^{\prime}(a)\equiv\partial P/\partial a$ are detected using a
phase-locked loop (PLL) that senses shifts

$\displaystyle\Delta\omega=\sqrt{\omega_{0}^{2}-\frac{A}{m_{\rm
eff}}\frac{\partial P}{\partial a}}-\omega_{0}$ (1)

of the sensor’s resonance frequency—a technique widely used in the literature
[34, 35, 36, 37, 38].

Figure 1: Simplified two-dimensional schematic representation of our setup
with included improvements proposed in the main text. The material composition
of the surfaces is shown separately on the left.

In the absence of thermal drifts, the pressure $P$ between our flat surfaces
can be measured by monitoring the extension $\Delta a=PA/k$ of the spring-mass
system using the same interferometer. For the determination and maintenance of
parallelism, however, we recently used a feedback mechanism based on the
capacitance between the two plates. While this scheme has been demonstrated to
work in principle, practice has shown that vibrations in combination with the
high $Q$-factor of our sensor lead to unacceptably long integration times, and
associated susceptibility to thermal drifts [33]. To improve the performance
and reach the full potential of the setup, we propose to replace the
capacitive scheme by an interferometric one. As shown in Figure 1, three Fabry
Pérot interferometers below the sensor plate (created by the end faces of
optical fibers and the reflective surface of the sensor plate) monitor the
distance $a$ at different positions around the lower plate. (Please note that
in the two-dimensional scheme in the figure, only two of these cavities are
depicted.) The three fiber ends can be polished together with the lower plate
to exactly match the surface of the latter. In such a system, the three lower
interferometers could measure the tilt and frequency shift at all times.
Synchronously, the upper (fourth) interferometer, being mechanically connected
to the sensor frame, can be used to measure the extension $\Delta a$ of the
sensor and, hence, the pressure acting on it.

Besides this conceptual change several technical improvements are planned,
which will result in the following: First, we aim to improve the seismic
attenuation by a factor of 10 around $f_{0}=\omega_{0}/(2\pi)$ with respect to
the present performance by means of active control techniques [39]. This would
not only reduce the direct influence on measurements via non-linear effects
and rms noise, but would also improve the stability of various feedback
circuits in a nontrivial way. Second, a newly designed distributed thermal
control concept will guarantee mK stability throughout the setup, thereby
eliminating drift and uncertainty in the sensor characteristics. Third,
electrostatic patch effects that create a systematic pressure background could
be reduced using in situ Ar ion bombardment [38]. In the following, we analyze
the major sources of experimental uncertainty and show the predicted effect of
the mentioned improvements.

###  2.1 Sources of Error

Here, we consider experimental errors in the pressure and its gradient due to
vibrations, the determination of displacement, and frequency shift of our
sensor, variations in the temperature, tilt of the plates, and discuss the
role of electrostatic patch effects.

Previously, we identified vibrations as a major source of error and gave
respective tolerable limits for pressure measurements [33]. Using a five-axis
seismic attenuation system, we achieved a damping factor of $60$ in vertical
direction around the sensor resonance $f_{0}=10.24\,$Hz. The residual
vibrations caused at times a peak displacement noise $\delta a_{p}\approx
300\,$nm of the sensor relative to the lower plate, which proved to be a
severe nuisance during the measurements. For pressure gradients, the main
influence is via non-linear effects [34, 40]. For a function $\phi(a)$
representing either the pressure ($\phi=P$) or its gradient ($\phi=\partial
P/\partial a$), we expand

$\displaystyle\phi(a+\delta a)\approx\phi(a)+\langle\delta
a\rangle\frac{\partial}{\partial a}\phi(a)+\frac{1}{2}\langle\delta
a^{2}\rangle\frac{\partial^{2}}{\partial a^{2}}\phi(a)$ (2)

for small sensor movements $\delta a$. Recognizing that $\langle\delta
a\rangle\equiv 0$, as usual for stochastic fluctuations, the main contribution
to the error comes from the second order term. Thus, for the non-linear shifts
in the pressure and its gradient, we therefore have

$\displaystyle\delta P_{\rm nl}\approx\frac{1}{2}\langle\delta
a^{2}\rangle\frac{\partial^{2}}{\partial
a^{2}}P(a)\,,\quad\text{and}\quad\delta\left(\frac{\partial P}{\partial
a}\right)_{\rm nl}\approx\frac{1}{2}\langle\delta
a^{2}\rangle\frac{\partial^{3}}{\partial a^{3}}P(a)\,,$ (3)

respectively. Similarly, the rms pressure and pressure gradient (entering
measurements mainly as a nuisance within the mechanical bandwidth of the
sensor $2\delta f_{\rm BW}=2.8\,$mHz around $f_{0}$) are computed from the
spectral sensor movement $\delta a(f)$ via

$\displaystyle\delta\phi_{\rm rms}\approx\frac{\partial\phi(a)}{\partial
a}\left[\int_{f_{0}-\delta f_{\rm BW}}^{f_{0}+\delta f_{\rm BW}}\\!{\rm
d}f\,\delta a^{2}(f)\right]^{1/2}\,.$ (4)

We note that vibrational noise also hampers the convergence of various
feedback circuits and thereby influences the achievable sensitivity in a
nontrivial way. Experience has shown that such effects are negligible below a
peak amplitude of $\sim 20\,$nm.

Another fundamental source of error is the uncertainty in various parameters
involved in the evaluation of the frequency shift. We consider the following
calibration procedure. At large separation $a_{cal}\approx
80\,$$\text{\,}\mathrm{\SIUnitSymbolMicro m}$ we apply the AC electrostatic
excitation voltages $V_{\rm ex}$ and $V_{\rm AC}$ driving the sensor resonance
and the surface potential compensation circuit, respectively [33]. At this
separation, the Casimir pressure $P$ is negligibly small with respect to the
electrostatic pressure $P_{\rm ES}$. The free resonance frequency

$\displaystyle\omega_{0}=\left(\omega_{0P}^{2}+\frac{A}{m_{\rm
eff}}\left[\left.\frac{\partial P_{\rm ES}}{\partial
a}\right|_{a_{cal}}+\left.\frac{\partial P}{\partial a}\right|_{a_{\rm
cal}}\right]\right)^{1/2}\quad\text{with}\quad\frac{\partial P_{\rm
ES}}{\partial a}=\frac{\varepsilon_{0}}{2a_{\rm cal}^{3}}\left(V_{\rm
AC}^{2}+V_{\rm ex}^{2}\right)$ (5)

is determined from a measurement of the resonance frequency $\omega_{0P}$
under the influence of the well-known electrostatic and Casimir forces. Here,
$\varepsilon_{0}$ is the dielectric permittivity of vacuum. The effective mass
$m_{\rm eff}$ is determined during a separate sweep recording $\omega_{0P}$ as
a function of an applied DC electrostatic potential. Eventually, we evaluate
the resonance frequency shift measured at different separations to obtain the
Casimir pressure gradient from Equation (1) after subtraction of the
electrostatic pressure gradient $\partial P_{\rm ES}/\partial a$. For pressure
measurements at the same separations, we evaluate the detected extension
$\Delta a$ of the sensor that is related to the total pressure $P_{\rm
tot}=P+P_{\rm ES}=\Delta a\,\omega_{0}^{2}m/A$. The achievable sensitivity for
both types of measurement is limited by the uncertainties in all experimental
quantities entering the respective evaluation. Numerical values for these and
some other uncertainties considered below are given in Table 1.

Variations in the temperature contribute in two ways to the experimental
error. First, different material expansion coefficients influence the
separation between the two interacting surfaces by roughly $63\,$nm/K—an
effect most influential at smaller $a$. Second, the temperature influences the
Youngs modulus of our sensor, which leads to an additional error in $a$, but
also offsets the resonance frequency and, thereby, mimics a pressure gradient.

We also consider errors due to tilt of the plates with respect to each other.
For small angles $\delta\alpha$ between the plates, we may estimate the
influence on the pressure and its gradient by averaging

$\displaystyle\delta\phi(a,\delta\alpha)\approx\int_{A}\\!{\rm
d}A\,\phi[a(x,y,\delta\alpha)]\,,$ (6)

over the sensor area $A$ with the local separation $a(x,y,\delta\alpha)$
deviating from its nominal value $a$ due to the tilt. Numerical calculations
show that the respective relative corrections to the pressure and its gradient
are both of the order $(\delta\alpha R/a)^{2}$, in agreement with the
literature [5]. For the achievable values of $\delta a$ given in Table 1,
these corrections are negligible.

As has been discussed in Ref. [33], electrostatic effects can have a
significant influence. While we recently compared our measurements with the
model [41], we now use the model [42] that has been shown to describe the
observed forces more realistically. The average patch size
$\langle\ell\rangle=0.82\,$$\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and the
value for $\langle\ell^{2}\rangle$ mentioned in Table 1 were derived from the
auto-correlation of actual Kelvin probe data for our surfaces. As
$\langle\ell\rangle$ is much smaller than the plate separation, we can use the
approximation

$\displaystyle P_{patch}(a)\approx\frac{3\zeta(3)\varepsilon_{0}V_{\rm
rms}^{2}\langle\ell^{2}\rangle}{4a^{4}}\,.$ (7)

Here, $V_{\rm rms}=0.634\,$mV, and $\zeta(z)$ is the Riemann zeta function.
Please note that $P_{\rm patch}$ represents a systematic effect that can be
characterized and removed from experimental data.

Table 1: Values for the experimental parameters and uncertainties for the setup in its present configuration and after the proposed improvements. Parameter | Present Value | Improved Value |
---|---|---|---
Separation uncertainty $\delta a_{\rm cal}$ | 2.0 | 0.5 | nm
Mass calibration error $\delta m_{\rm eff}$ | 1.0 | 0.01 | %
Area uncertainty $\delta A$ | $5\times 10^{-4}$ | $5\times 10^{-4}$ | cm2
Applied AC voltages uncertainty | 10 | 1 | $\text{\,}\mathrm{\SIUnitSymbolMicro V}$
Frequency measurement error | $10^{-5}$ | $4\times 10^{-7}$ | Hz
Vibration amplitude at $f_{0}$ | 300 | 20 | nm
Patches $V_{\rm rms}$ | 1.28 | 0.64 | V
Patches $\langle\ell^{2}\rangle$ | 7.3 | 2.4 |  $\text{\,}\mathrm{\SIUnitSymbolMicro m}$2
Tilt angle $\delta\alpha$ | 3.0 | 0.1 | $\text{\,}\mathrm{\SIUnitSymbolMicro rad}$
Temperature stability | 3.0 | 0.5 | mK

###  2.2 Sensitivity Estimation

Based on the models described in Section 2.1, we have calculated the expected
level of experimental uncertainty in measurements of the pressure and its
gradient for both the present situation and presuming a successful
implementation of the proposed improvements. For these calculations, we have
assumed the parameters listed in Table 1. As can be seen in Figure 2a, the
largest uncertainty in force measurements comes from the determination of the
sensor extension, which includes contributions from the calibration and
interferometry. Vibrations play, under the assumption that the proposed
measures work as expected, a minor role. For force gradients (Figure 2b) the
determination of the frequency shift is the limiting factor at separations
larger than $\sim$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, while at smaller
separations sensitivity is limited by vibrations. Temperature variations are
influential on force gradient measurements for all separations, as they modify
the Youngs modulus of our sensor. For comparison, we also plot the achievable
uncertainty for the present version of the setup (dashed upper red lines),
which are determined by the same factors as for the improved version. Please
note that the patch pressure can be characterized separately and removed from
the data.

Figure 2: Estimated experimental uncertainties for the present and improved
versions of the setup as a function of separation between the two parallel
plates. (a) Pressure measurements using the upper interferometer. (b) Pressure
gradient measurements using the lower interferometer.

The results for the improved uncertainties in Figure 2 are based on very
conservative estimates. In the experiment, especially at larger separations,
the uncertainty in the pressure could be reduced statistically by longer
measurements. For further calculations in Section 4, we therefore assume a
pressure sensitivity of
$1\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}^{2}$ (corresponding to
$0.1\text{\,}\mathrm{pN}$) at $a\geq$4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$, which is slightly higher than the value corresponding to the bottom red
line in Figure 2a. The uncertainty in the pressure gradient is mainly
determined by the resolution of the PLL’s frequency measurement. Here, the
possibility for a statistical reduction of the uncertainty may be more
problematic and therefore we use the pressure gradient sensitivity given by
the bottom red line in Figure 2b for calculations in Section 4.

##  3 Possibilities to Measure Thermal Effects in the Casimir Force

As mentioned in Section 1, most of the already performed experiments on
measuring the Casimir interaction exploited the sphere-plate geometry. In
doing so, the measured quantity was either the Casimir force acting between a
sphere and a plate (in the static measurement scheme) or its gradient (in the
dynamic measurement scheme). Due to the proximity force approximation, the
latter quantity can be recalculated into the effective Casimir pressure
between two parallel plates [5, 21]. There is only one modern experiment on
the direct measurement of the Casimir pressure between two parallel plates
[43], but it is not of sufficient precision to observe the thermal effects (an
attempt to measure the Casimir effect between two parallel Al-coated plates at
separations larger than a few micrometers was unsuccessful due to the presence
of large background forces [44]).

The distinctive feature of the Cannex test of the quantum vacuum is that it
can be adapted for simultaneous measurements of the Casimir pressure between
two parallel plates and its gradient (see Section 2).

The Lifshitz formula for the Casimir pressure between two material plates
spaced at a separation $a$ at temperature $T$ is given by [5, 21]

${P}(a)=-\frac{k_{B}T}{\pi}\sum_{l=0}^{\infty}\vphantom{\sum}{}^{{}^{\prime}}\int_{0}^{\infty}k_{\bot}dk_{\bot}q_{l}\sum_{\alpha}\frac{r^{2}_{\alpha}(i{\xi}_{l},k_{\bot})}{e^{2aq_{l}}-r^{2}_{\alpha}(i{\xi}_{l},k_{\bot})}.$
(8)

Here, it is assumed that the parallel plates made of a nonmagnetic material
described by the dielectric permittivity $\varepsilon(\omega)$ are in thermal
equilibrium with the environment at temperature $T$ and the following
notations are introduced. The Boltzmann constant is $k_{B}$, the prime on the
summation sign divides the term with $l=0$ by $2$, $k_{\bot}$ is the magnitude
of the projection of the wave vector on the plane of the plates,
$\xi_{l}=2{\pi}k_{B}Tl/\hbar$ with $l=0,\,1,\,2,\,\ldots\,$ are the Matsubara
frequencies , and $q_{l}=(k_{\bot}^{2}+\xi_{l}^{2}/c^{2})^{1/2}$.

The reflection coefficients $r_{\alpha}$ are defined for two independent
polarizations of the electromagnetic field, transverse magnetic ($\alpha={\rm
TM}$) and transverse electric ($\alpha={\rm TE}$). Explicitly they are given
by

$r_{\rm
TM}(i\xi_{l},k_{\bot})=\frac{\varepsilon_{l}q_{l}-k_{l}}{\varepsilon_{l}q_{l}+k_{l}},\qquad
r_{\rm TE}(i\xi_{l},k_{\bot})=\frac{q_{l}-k_{l}}{q_{l}+k_{l}},$ (9)

where

$k_{l}\equiv
k(i\xi_{l},k_{\bot})=\left(k_{\bot}^{2}+\varepsilon_{l}\frac{\xi_{l}^{2}}{c^{2}}\right)^{1/2}{\\!\\!\\!\\!},\qquad\varepsilon_{l}\equiv\varepsilon(i\xi_{l}).$
(10)

By differentiating Equation (8) with respect to separation between the plates
$a$, one obtains the gradient of the Casimir pressure

${P}^{\prime}(a)=\frac{2k_{B}T}{\pi}\sum_{l=0}^{\infty}\vphantom{\sum}{}^{{}^{\prime}}\int_{0}^{\infty}k_{\bot}dk_{\bot}q_{l}^{2}\sum_{\alpha}\frac{r^{2}_{\alpha}(i{\xi}_{l},k_{\bot})e^{2aq_{l}}}{\left[e^{2aq_{l}}-r^{2}_{\alpha}(i{\xi}_{l},k_{\bot})\right]^{2}}.$
(11)

Equations (8) and (11) take an exact account of the effects of finite
conductivity of the plate metal. As to corrections due to surface roughness,
at separations exceeding $3~{}\upmu$m they are much smaller than an error in
the pressure measurements [5].

For numerical computations it is convenient to introduce the dimensionless
variables

$y=2aq_{l},\qquad\zeta_{l}=\frac{2a\xi_{l}}{c}.$ (12)

In terms of these variables Equations (8) and (11) take the form

${P}(a)=-\frac{k_{B}T}{8\pi
a^{3}}\sum_{l=0}^{\infty}\vphantom{\sum}{}^{{}^{\prime}}\int_{\zeta_{l}}^{\infty}y^{2}dy\sum_{\alpha}\frac{r^{2}_{\alpha}(i{\zeta}_{l},y)e^{-y}}{1-r^{2}_{\alpha}(i{\zeta}_{l},y)e^{-y}}\,,$
(13)

and

${P}^{\prime}(a)=\frac{k_{B}T}{8\pi
a^{4}}\sum_{l=0}^{\infty}\vphantom{\sum}{}^{{}^{\prime}}\int_{\zeta_{l}}^{\infty}y^{3}dy\sum_{\alpha}\frac{r^{2}_{\alpha}(i{\zeta}_{l},y)e^{-y}}{\left[1-r^{2}_{\alpha}(i{\zeta}_{l},y)e^{-y}\right]^{2}},$
(14)

respectively, and the reflection coefficients from Equation (9) take the form

$r_{\rm
TM}(i\zeta_{l},y)=\frac{\varepsilon_{l}y-\sqrt{y^{2}+(\varepsilon_{l}-1)\zeta_{l}^{2}}}{\varepsilon_{l}y+\sqrt{y^{2}+(\varepsilon_{l}-1)\zeta_{l}^{2}}},\qquad
r_{\rm
TE}(i\zeta_{l},y)=\frac{y-\sqrt{y^{2}+(\varepsilon_{l}-1)\zeta_{l}^{2}}}{y+\sqrt{y^{2}+(\varepsilon_{l}-1)\zeta_{l}^{2}}}.$
(15)

For application to the experimental setup of Cannex described in Section 2,
computations should be made for Au plates. The dielectric permittivity of Au
along the imaginary frequency axis is obtained by means of the Kramers-Kronig
relations using the available tabulated optical data for the complex index of
refraction extrapolated down to zero frequency [5, 21]. According to Section
1, this extrapolation can be made either by means of the Drude model taking
into account the relaxation properties of conduction electrons or the plasma
model disregarding these relaxation properties. In dimensionless variables,
the dielectric permittivity of the Drude model along the imaginary frequency
axis is given by

$\varepsilon_{l}^{(D)}=1+\frac{{\tilde{\omega}_{p}}^{2}}{\zeta_{l}(\zeta_{l}+\tilde{\gamma})}\,,$
(16)

where the dimensionless plasma frequency $\tilde{\omega}_{p}$ and relaxation
parameter $\tilde{\gamma}$ are connected with the dimensional ones by
$\tilde{\omega}_{p}=2a\omega_{p}/c$ and $\tilde{\gamma}=2a\gamma/c$. For Au
the standard values $\hbar\omega_{p}=9.0~{}$eV and $\hbar\gamma=35~{}$meV are
used here. The dielectric permittivity of the plasma model
$\varepsilon_{l}^{(p)}$ is obtained from Equation (16) by putting
$\tilde{\gamma}=0$. Please note that for computations at separations $a\geq
3~{}\upmu$m performed below the optical data contribute negligibly small, so
that the obtained gradients of the Casimir pressure are mostly determined by
the extrapolations to lower frequencies.

Before presenting the computational results, we note that in the high-
temperature limit $T\gg T_{\rm cr}=\hbar c/(2k_{B}a)$ all the terms in
Equations (13) and (14) with $l\geq 1$ are exponentially small and both the
Casimir pressure and its gradient are given predominantly by the terms with
$l=0$. The zero-frequency term of the Lifshitz formula takes different forms
depending on the extrapolation used. If the plasma model
$\varepsilon_{l}^{(p)}$ is used for extrapolation, one obtains

$\displaystyle P_{(p)}(a)$ $\displaystyle=-\frac{k_{B}T}{16\pi
a^{3}}\left[\int_{0}^{\infty}\\!\\!dy\frac{y^{2}e^{-y}}{1-e^{-y}}+\int_{0}^{\infty}\\!\\!dy\frac{y^{2}r_{\rm
TE}^{2}(0,y)e^{-y}}{1-r_{\rm TE}^{2}(0,y)e^{-y}}\right]\,$ (17)

and

$\displaystyle P_{(p)}^{\prime}(a)$ $\displaystyle=\frac{k_{B}T}{16\pi
a^{4}}\left[\int_{0}^{\infty}\\!\\!dy\frac{y^{3}e^{-y}}{(1-e^{-y})^{2}}+\int_{0}^{\infty}\\!\\!dy\frac{y^{3}r_{\rm
TE}^{2}(0,y)e^{-y}}{[1-r_{\rm TE}^{2}(0,y)e^{-y}]^{2}}\right]\,,$ (18)

where in accordance to Equation (15)

$r_{\rm
TE}(0,y)=\frac{y-\sqrt{y^{2}+{\tilde{\omega}_{p}}^{2}}}{y+\sqrt{y^{2}+{\tilde{\omega}_{p}}^{2}}}.$
(19)

Calculating the first integrals on the right-hand side of Equations (17) and
(18) and expanding the second ones in the powers of a small parameter
$1/\tilde{\omega}_{p}$, we find [5]

$P_{(p)}(a)=-\frac{k_{B}T\zeta(3)}{4\pi
a^{3}}\left(1-\frac{6}{\tilde{\omega}_{p}}+\frac{48}{{\tilde{\omega}_{p}}^{2}}\right),\quad
P_{(p)}^{\prime}(a)=\frac{3k_{B}T\zeta(3)}{4\pi
a^{4}}\left(1-\frac{8}{\tilde{\omega}_{p}}+\frac{80}{{\tilde{\omega}_{p}}^{2}}\right)\,.$
(20)

If, however, the Drude model $\varepsilon_{l}^{(D)}$ is used for
extrapolation, one arrives at the result

$P_{(D)}(a)=-\frac{k_{B}T}{16\pi
a^{3}}\int_{0}^{\infty}\\!\\!dy\frac{y^{2}e^{-y}}{1-e^{-y}}=-\frac{k_{B}T\zeta(3)}{8\pi
a^{3}}\,$ (21)

and

$P_{(D)}^{\prime}(a)=\frac{k_{B}T}{16\pi
a^{4}}\int_{0}^{\infty}\\!\\!dy\frac{y^{3}e^{-y}}{(1-e^{-y})^{2}}=\frac{3k_{B}T\zeta(3)}{8\pi
a^{4}}.$ (22)

Please note that at $T=300\,$K Equations (20)–(22) are applicable under a
condition

$a\gg\frac{\hbar c}{2k_{B}T}\approx 4~{}\upmu\mbox{m}.$ (23)

Figure 3: (a) The magnitude of the Casimir pressure between two parallel Au
plates and (b) the pressure gradient are shown as functions of separation by
the red and blue solid lines computed at $T=300~{}$K using the plasma and
Drude extrapolations of the optical data, respectively. The same quantities
computed at $T=0$ are shown by the dashed lines. The region of separations
from 5 to 10 $\upmu$m is shown in the insets using the homogeneous scale.

Taking into account, however, that the powers in all exponentially small terms
with $l\geq 1$ depend on $-2\pi T/T_{\rm cr}$, one can see that at $T=300~{}$K
Equations (20)–(22) lead to rather precise results at all separations
exceeding 6 or $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$.

The computational results for the magnitude of the Casimir pressure and its
gradient as functions of separation between the plates are presented in Figure
3

by the red and blue solid lines computed at $T=300~{}$K using extrapolations
of the optical data of Au by means of the plasma and Drude models,
respectively. For comparison purposes, the computational results at $T=0$ are
presented by the dashed lines. They are obtained by Equations (13) and (14)
where summation over the discrete Matsubara frequencies is replaced with a
continuous integration. It is taken into account that with decreasing
temperature the relaxation parameter $\gamma$ quickly decreases towards a very
small residual value at $T=0$ that is determined by the defects of the crystal
lattice. As a result, the values of $P(a)$ and $P^{\prime}(a)$ computed at
$T=0$ using the plasma and Drude models coincide at high accuracy.

For a better visualization, in the insets to Figure 3a,b the computational
results in the separation region from 5 to
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ are presented using a uniform scale
on the vertical axis. It is clearly seen that the theoretical predictions from
using the plasma and Drude model extrapolations of the optical data can be
discriminated if to take into account the errors in measuring $P(a)$ and
$P^{\prime}(a)$ in the improved version of Cannex discussed in Section 2.
Please note that if the Drude model extrapolation is used the thermal effect
in the Casimir pressure and its gradient vanishes at approximately 6.4 and
$8.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ separations, respectively, where
the blue and dashed lines intersect.

To determine the specific role of thermal effects in the Casimir pressure and
its gradient, we have also computed the relative thermal corrections defined
as

$\delta_{T}P(a)=\frac{P(a,T)-P(a,0)}{P(a,0)},\qquad\delta_{T}P^{\prime}(a)=\frac{P^{\prime}(a,T)-P^{\prime}(a,0)}{P^{\prime}(a,0)}\,,$
(24)

where on the right-hand sides we indicated the dependence on temperature
explicitly. Computational results for $\delta_{T}P$ and $\delta_{T}P^{\prime}$
are presented in Figure 4a,b

Figure 4: The relative thermal corrections to (a) the Casimir pressure between
two parallel Au plates and (b) its gradient is shown as functions of
separation by the red and blue lines computed at $T=300~{}$K using the plasma
and Drude extrapolations of the optical data, respectively.

as functions of separation by the red and blue lines for the cases when
extrapolation of the optical data for Au to lower frequencies is made by means
of the plasma and Drude models, respectively. As is seen in these figures,
within the separation range from 3 to $15~{}\upmu$m the thermal effects make a
considerable contribution to the Casimir pressure and its gradient. If the
plasma model extrapolation is used, it reaches 440% and 240% of the zero-
temperature pressure and its gradient, respectively, at $a=15~{}\upmu$m. When
using the extrapolation by means of the Drude model, the relative thermal
effect in the Casimir pressure varies from approximately $-$34% at
$a=3~{}\upmu$m to 130% at $a=15~{}\upmu$m and in the pressure gradient from
approximately $-$40% at $a=4~{}\upmu$m to 60% at $a=15~{}\upmu$m. In this
case, the thermal effect in the Casimir pressure vanishes at $a\approx
6.4~{}\upmu$m and in its gradient at $a\approx 8.7~{}\upmu$m in agreement with
Figure 3a,b, respectively .

##  4 Prospective Constraints on Non-Newtonian Gravity and Axion-Like
Particles

As mentioned in Section 1, the Casimir forces originating from the quantum
vacuum form a background for testing the Yukawa-type corrections to Newton’s
gravitational law and for searching the axion-like particles. It is convenient
to parametrize the Yukawa-type potential between two point masses $m_{1}$ and
$m_{2}$ situated at the points $\mbox{\boldmath$r$}_{1}$ and
$\mbox{\boldmath$r$}_{2}$ as [8]

$V_{\rm Yu}(r)\equiv V_{\rm
Yu}(|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|)=-\alpha\frac{Gm_{1}m_{2}}{|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|}\exp\\!\left(-\frac{(|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|}{\lambda}\right).$
(25)

Here, $\alpha$ and $\lambda$ are the interaction constant and the range of
Yukawa interaction, and $G=6.674\times
10^{-11}\,\mbox{m}^{3}\,\mbox{kg}^{-1}\,\mbox{s}^{-2}$ is the Newtonian
gravitational constant (we note that in the experimental configuration under
consideration one can neglect by the Newtonian gravitational pressure because
it is less than an error in measurements of the Casimir pressure).

The Yukawa-type pressure between the top and bottom plates in the experimental
setup of Section 2 should be calculated taking into account the layer
structure of both plates shown in Figure 1. The top plate is made of high-
resistivity Si of density $\rho_{\rm Si}=2.33~{}\mbox{g\,cm}^{-3}$ coated with
a layer of Cr of density $\rho_{\rm Cr}=7.15~{}\mbox{g\,cm}^{-3}$ and
thickness $d_{\rm Cr}=5~{}$nm followed by a layer of Au of density $\rho_{\rm
Au}=19.3~{}\mbox{g\,cm}^{-3}$ and thickness $d_{\rm
Au}^{t}=$200\text{\,}\mathrm{n}\mathrm{m}$$. The thickness of Si substrate
($100~{}\upmu$m) is sufficiently large to treat it as a semispace. The bottom
plate is made of SiO2 quartz crystal with the density
$\rho_{q}=2.64\,\mbox{g\,cm}^{-3}$ coated with a layer of Cr of thickness
$d_{\rm Cr}=5\,$nm followed by a layer of Au of thickness $d_{\rm
Au}^{b}=$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. The thickness of SiO2
substrate ($6\,$mm) again allows to consider it as a semispace.

Now we assume that one mass $\rho dV_{1}$ belongs to the top plate and another
one $\rho dV_{2}$ to the bottom one and integrate Equation (25) over the
volumes of both parallel plates separated by a distance $a$ taking into
account their layer structure. Calculating the negative derivative of the
obtained interacting energy with respect to $a$, we find the Yukawa force and
finally the pressure [45]

$P_{\rm Yu}(a)=-2\pi G\alpha\lambda^{2}e^{-a/\lambda}\Phi(\lambda),$ (26)

where the function $\Phi$ is defined as

$\displaystyle\Phi(\lambda)=$ $\displaystyle\left[\rho_{\rm Au}-(\rho_{\rm
Au}-\rho_{\rm Cr})\exp\,\left(-\frac{d_{\rm
Au}^{t}}{\lambda}\right)-(\rho_{\rm Cr}-\rho_{\rm
Si})\exp\,\left(-\frac{d_{\rm Au}^{t}+d_{\rm Cr}}{\lambda}\right)\right]$
$\displaystyle\times\left[\rho_{\rm Au}-(\rho_{\rm Au}-\rho_{\rm
Cr})\exp\,\left(-\frac{d_{\rm Au}^{b}}{\lambda}\right)-(\rho_{\rm
Cr}-\rho_{q})\exp\,\left(-\frac{d_{\rm Au}^{b}+d_{\rm
Cr}}{\lambda}\right)\right].$ (27)

The gradient of the Yukawa pressure is obtained by differentiating Equation
(26) with respect to $a$

$P_{\rm Yu}^{\prime}(a)=2\pi G\alpha\lambda e^{-a/\lambda}\Phi(\lambda).$ (28)

Now the constraints on the parameters $\alpha$ and $\lambda$ of Yukawa-type
interactions can be obtained from the inequalities

$\left|P_{\rm Yu}(a)\right|<\Delta P(a),\qquad P_{\rm Yu}^{\prime}(a)<\Delta
P^{\prime}(a),$ (29)

where $\Delta P$ and $\Delta P^{\prime}$ are the total experimental errors in
the measured Casimir pressure and its gradient estimated in Section 2. The
meaning of Equation (29) is that the experimental data are found in agreement
with theoretical predictions for the Casimir pressure and its gradient and no
extra contribution of unknown origin was observed.

To estimate the strength of prospective constraints, which can be obtained
from the improved version of Cannex, we use the thickness of the top and
bottom Au layers $d_{\rm Au}^{t}=200~{}$nm and $d_{\rm Au}^{b}=1~{}\upmu$m and
the total experimental errors $\Delta
P(a)=$1\text{\,}\mathrm{n}\mathrm{N}\mathrm{/}\mathrm{m}^{2}$$ and $\Delta
P^{\prime}(a)$ corresponding to the bottom red line in Figure 2b. The
computational results for $\alpha$ and $\lambda$ obtained by using the first
and second inequalities in Equation (29) are shown by the red lines in Figure
5a,b, respectively. In so doing, the regions of $(\alpha,\lambda)$-planes
above each line are excluded and below each line are allowed.

Figure 5: Constraints on the parameters of Yukawa-type interaction obtained
from two Casimir-less experiments, from measuring the difference in lateral
forces, from the torsion pendulum experiment and from the Cavendish-type
experiments are shown by the top and bottom blue lines, green, black and gray
lines, respectively. The proposed constraints which could be obtained from the
Cannex test of the quantum vacuum when measuring (a) the Casimir pressure and
(b) its gradient are indicated by the red lines.

For comparison purposes, we present in Figure 5a,b the strongest constraints
obtained in the same interaction range from other experiments. The top and
bottom blue lines demonstrate the constraints found from the results of the
Casimir-less experiment [46] and its improved version [47], respectively. Both
experiments were performed by means of a micromechanical torsional oscillator.
The green line shows the constraints obtained from measuring the difference in
lateral forces [48]. The constraints indicated by the black line are found in
[49] from the torsion pendulum experiment. Finally, the constraints of the
gray line follow from the Cavendish-type experiments performed at short
separations [50, 51, 52].

As is seen in Figure 5a,b, the strongest constraints obtained up to date
follow from the improved Casimir-less experiment [47] (the bottom blue line).
It is seen also that the largest strengthening of these constraints, which
could be reached from the Cannex test of quantum vacuum, follows from
measurements of the Casimir pressure (this is because the pressure gradient is
linear in $\lambda$ whereas the pressure is quadratic in $\lambda$). The
prospective constraints are stronger by up to a factor of 10 over a wide
interaction range.

Next we consider the prospective constraints on the axion-to-nucleon coupling
constants which could be obtained from the Cannex test of the quantum vacuum.
Taking into account that the exchange of one axion between two nucleons
results in the spin-dependent interaction potential [53] and the test bodies
in this experiment are not polarized, any additional force of the axion origin
could arise due to two-axion exchange.

Below we deal with axion-like particles coupled to nucleons by means of
pseudo-scalar interaction Lagrangian [54]. In this case, the effective
potential between two nucleons, spaced at the points $\mbox{\boldmath$r$}_{1}$
and $\mbox{\boldmath$r$}_{2}$ of the top and bottom plates, arising due to
exchange of two axions, takes the form [53, 55]

$V_{an}(r)\equiv
V_{an}(|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|)=-\frac{g_{an}^{4}}{32\pi^{3}}\frac{\hbar^{2}m_{a}}{m^{2}}(\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2})^{-2}{K}_{1}\\!\left(\frac{2m_{a}c|\mbox{\boldmath$r$}_{1}-\mbox{\boldmath$r$}_{2}|}{\hbar}\right).$
(30)

Here, $g_{an}$ is the dimensionless coupling constant of an axion to a nucleon
(we assume that the coupling constants to a neutron and a proton are equal
[53]), the mean of the proton and neutron masses is denoted as $m$, the axion
mass is $m_{a}$, ${K}_{1}(z)$ is the modified Bessel function of the second
kind and it is assumed that $r\gg\hbar/(mc)$.

Similar to the case of a Yukawa-type potential, the additional pressure
between the test bodies due to two-axion exchange, is obtained by integrating
Equation (30) over the volumes of both plates with account of their layer
structure, calculate the negative derivative of the obtained result with
respect to $a$ and finally find the pressure (see [15] for details)

$P_{an}(a)=-\frac{\hbar^{3}}{2m^{2}m_{\rm
H}^{2}c}\int_{1}^{\infty}du\frac{\sqrt{u^{2}-1}}{u^{2}}\Psi\left(\frac{m_{a}cu}{\hbar}\right)\exp\,\left(-\frac{2m_{a}cau}{\hbar}\right).$
(31)

Here, $m_{\rm H}$ is the mass of atomic hydrogen and the function $\Psi(x)$ is
defined as

$\displaystyle\Psi(x)=\left[C_{\rm Au}(1-e^{-2xd_{\rm Au}^{t}})+C_{\rm
Cr}e^{-2xd_{\rm Au}^{t}}(1-e^{-2xd_{\rm Cr}})+C_{\rm Si}e^{-2x(d_{\rm
Au}^{t}+d_{\rm Cr})}\right]$ $\displaystyle~{}~{}\times\left[C_{\rm
Au}(1-e^{-2xd_{\rm Au}^{b}})+C_{\rm Cr}e^{-2xd_{\rm Au}^{b}}(1-e^{-2xd_{\rm
Cr}})+C_{q}e^{-2x(d_{\rm Au}^{b}+d_{\rm Cr})}\right],$ (32)

where the coefficients $C$ for each material are given by

$C=\rho\frac{g_{an}^{2}}{4\pi}\left(\frac{Z}{\upmu}+\frac{N}{\upmu}\right).$
(33)

In Equation (33), $\rho$ is the density of the respective material already
indicated above, $Z$ and $N$ are the number of protons and the mean number of
neutrons in the atoms of materials with mean mass $M$ and $\upmu=M/m_{\rm H}$.
For materials under consideration $Z/\upmu=0.40422$, 0.46518, 0.50238, and
0.503205 for Au, Cr, Si, and SiO2, respectively [8]. The values of $N/\upmu$
are 0.60378, 0.54379, 0.50628, and 0.505179 for the same respective materials
[8].

The gradient of the pressure due to two-axion exchange is obtained from
Equation (31) by the differentiation with respect to $a$

$P_{an}^{\prime}(a)=\frac{m_{a}\hbar^{2}}{m^{2}m_{\rm
H}^{2}}\int_{1}^{\infty}du\frac{\sqrt{u^{2}-1}}{u}\Psi\left(\frac{m_{a}cu}{\hbar}\right)\exp\,\left(-\frac{2m_{a}cau}{\hbar}\right).$
(34)

The constraints on the parameters of axion-nucleon interaction, $g_{an}$,
$m_{a}$, are obtained from the inequalities

$\left|P_{an}(a)\right|<\Delta P(a),\qquad P_{an}^{\prime}(a)<\Delta
P^{\prime}(a),$ (35)

similar to Equation (29).

To find the strongest prospective constraints on the axion-to-nucleon
interaction, we have used the same parameters of the improved setup, as listed
above when considering the Yukawa interaction. The computational results for
$g_{an}^{2}/(2\pi)$ as a function of $m_{a}c^{2}$ obtained by using the first
and second inequalities in Equation (35) are shown by the red lines in Figure
6a,b, respectively. Again, the space above each line is excluded (will be
excluded, in the case of the red line) by the results of the respective
experiment.

The strongest laboratory constraints obtained up to date [17] at the same
region of axion masses are shown by the blue line. They follow from the
improved Casimir-less experiment [47]. For comparison purposes, the
constraints following from the Cavendish-type experiment [56, 57],
measurements of the effective Casimir pressure [15, 58, 59], and the lateral
Casimir force between corrugated surfaces [16, 60, 61] are shown by the gray,
black, and green lines, respectively. As can be seen in Figure 6a,b, the
stronger constraints again follow from measuring the Casimir pressure.
According to Figure 6a, the Cannex test of the quantum vacuum could strengthen
the presently known constraints by up to a factor of 3.

Figure 6: Constraints on the axion-to-nucleon coupling constant obtained from
the improved Casimir-less experiment, the Cavendish-type experiment,
measurements of the effective Casimir pressure, and the lateral Casimir force
are shown by the blue, gray, black, and green lines, respectively. The
proposed constraints which could be obtained from the Cannex test of the
quantum vacuum when measuring (a) the Casimir pressure and (b) its gradient
are indicated by the red lines.

##  5 Discussion

In the foregoing, we have considered the Cannex test of the quantum vacuum
intended to measure the gradient of the Casimir pressure between two parallel
plates at separations exceeding a few micrometers. As already discussed in the
literature [32, 33], this experiment would be capable to place stronger
constraints on the chameleon model of dark energy and, thus, bring new
information concerning the origin of vacuum energy and the value of the
cosmological constant. Here, we proposed a modification of the Cannex setup,
which allows for simultaneous measurements of both the pressure and its
gradient. We also considered several improvements in the already existing
setup which will allow for a more precise determination of several parameters.
According to our results, with these improvements the Cannex test of the
quantum vacuum could be used to directly measure for the first time the
thermal effects in the Casimir pressure and its gradient at separations from 5
to 10 micrometers. We have also shown that this experiment could strengthen
the presently known best constraints on the parameters of non-Newtonian
gravity by up to a factor of 10 over a wide interaction range. The constraints
on the axion-to-nucleon coupling constant could be strengthened by up to a
factor of 3 in the region of axion masses from 1 to $100\,$meV.

##  6 Conclusions

To conclude, Cannex can be considered as a promising new laboratory experiment
for investigating unusual features of the quantum vacuum, thermal effects in
the Casimir forces, non-Newtonian gravity and properties of axion-like
particles being hypothetical constituents of dark matter. Together with other
laboratory experiments, such as the demonstration of the dynamical Casimir
effect in superconducting circuits [62, 63], the large-scale accelerator
projects, and astrophysical data, it may serve as a useful tool in resolving
fundamental problems of the quantum vacuum.

#### Author Contributions:

Investigation, G.L.K., V.M.M., R.I.P.S., and H.A.; writing, G.L.K., V.M.M.,
and R.I.P.S.; project administration, H.A.

#### Funding:

This research was partially funded by the Russian Foundation for Basic
Research grant number 19-02-00453 A.

#### Acknowledgments:

V.M.M. was partially supported by the Russian Government Program of
Competitive Growth of Kazan Federal University. G.L.K. and V.M.M. are grateful
to the Atominstitut of the TU Wien, where this work was performed, for kind
hospitality. R.I.P.S. was supported by the TU Wien.

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