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Review

Fifty Years of the Dynamical Casimir Effect

Institute of Physics and International Center for Physics, University of Brasilia, P.O. Box 04455, Brasília 70919-970, Brazil
Submission received: 31 December 2019 / Revised: 8 February 2020 / Accepted: 10 February 2020 / Published: 14 February 2020
(This article belongs to the Special Issue The Quantum Vacuum)

Abstract

:
This is a digest of the main achievements in the wide area, called the Dynamical Casimir Effect nowadays, for the past 50 years, with the emphasis on results obtained after 2010.

1. Introduction

Fifty years ago, in June 1969, G.T. Moore finished his PhD thesis (prepared under the guidance of H.N. Pendleton III and submitted to the Brandeis University), part of which was published in the next year as [1]. Using a simplified one-dimensional model, Moore showed that motions of ideal boundaries of a one-dimensional cavity could result in a generation of quanta of the electromagnetic field from the initial vacuum quantum state. A few years later, DeWitt [2] demonstrated that moving boundaries could induce particle creation from a vacuum in a single-mirror set-up. A more detailed study was performed by Fulling and Davies [3,4]. Thus, step by step, year by year, more and more authors followed this direction of research. In 1989, two names were suggested for such kinds of phenomena: dynamic (or non-adiabatic) Casimir effect [5] and Nonstationary Casimir Effect [6]. Being supported in part by the authority of Schwinger [7], the first name gradually acquired the overwhelming popularity, so that now we have an established direction in the theoretical and experimental physics, known under the general name Dynamical Casimir Effect (DCE). This area became rather large by now: more than 300 papers containing the words “dynamical Casimir” have been published already, including more than 100 publications during the past decade. Moore’s paper [1] has been cited more than 400 times, and some authors use the name “Moore effect” instead of DCE (other names were “Mirror Induced Radiation” or “Motion Induced Radiation”).
To combine different studies under the same “roof”, it seems reasonable to assume the following definition of the Dynamical Casimir Effect: Macroscopic phenomena caused by changes of vacuum quantum states of fields due to fast time variations of positions (or properties) of boundaries confining the fields (or other parameters). Such phenomena include, in particular, the modification of the Casimir force for moving boundaries. However, the most important manifestation is the creation of the field quanta (photons) due to the motion of neutral boundaries. The most important ingredients of the DCE are quantum vacuum fluctuations and macroscopic manifestations. The reference to vacuum fluctuations explains the appearance of Casimir’s name (by analogy with the famous static Casimir effect, which is also considered frequently as a manifestation of quantum vacuum fluctuations), although Casimir himself did not write anything on this subject. Therefore, the DCE can be considered as the specific subfield of a much bigger physical area, known nowadays under the name Casimir Physics. This whole area is outside the present study, so that we give only a few references to the relevant reviews and books [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].
In turn, the subject of the DCE can be divided in several sub-areas. In the strict (narrow) sense, one can think about the “single mirror DCE” or “cavity DCE”. In the most wide sense, the DCE can be related to the amplification of quantum (vacuum) fluctuations in macroscopic systems, and many authors studied analogs of the DCE in parametrically excited media. The “intermediate” area of quantum circuits with time-dependent parameters is also frequently considered as belonging to the DCE. The aim of our review is to describe the main achievements in all those sub-areas for the past 50 years, as well as still unsolved and challenging problems. This is not the first review on the subject. The early history, including studies on the classical fields with moving boundaries, can be found in [25,26,27,28]. The most recent reviews were published almost a decade ago [29,30,31]. Therefore, here I tried to collect a more or less complete list of references to the publications of the decade after 2009. As for earlier papers, I have included only a small part of them, to maintain a reasonable length of this review.

2. One-Dimensional Models with Moving Boundaries

One-dimensional “toy” models were considered by many authors because they are much simpler than the realistic 3D ones; therefore, they admit exact and relatively simple analytical solutions, which can provide some insight into the real effects.

2.1. Classical Fields

The history of studies on classical fields in time-dependent domains began almost 100 years ago. The first exact solution of the wave equation (with c = 1 )
2 A / t 2 2 A / x 2 = 0 ,
in a time-dependent domain 0 < x < L ( t ) , satisfying the boundary conditions
A ( 0 , t ) = A ( L ( t ) , t ) = 0 ,
was obtained by Nicolai [32] for
L ( t ) = L 0 ( 1 + α t ) .
The solution was interpreted in terms of the transverse vibrations of a string with a variable length. A few years later, these results were published in [33], where the extension to the case of electromagnetic field was also made. A similar treatment was given by Havelock [34] in connection with the problem of radiation pressure. However, this subject hardly attracted the attention of researchers for several decades. The splash of new studies happened in the 1960s, probably due to the invention of lasers. We cite here only a few of relevant papers [35,36,37,38]. Much more references can be found in [26,39,40].

2.2. Quantum Fields

In the quantum case, one of the most interesting problems is the creation of quanta of different fields from the initial vacuum (ground) state due to the time variation of some parameters. The generation of massive particles by time-dependent gravitational fields was considered for the first time, probably, by Schrödinger [41]. Later on, this problem was treated by Imamura [42], Parker [43,44], and Zel’dovich [45]. However, the main subject of the present paper is the creation of massless quanta (photons). In this case, the simplest model is the so-called “scalar electrodynamics”, where the main object is the single component of the vector potential operator, perpendicular to the axis x (and parallel to the infinite plane surfaces confining the field). This operator satisfies, in the Heisenberg picture, the same Equation (1) as the classical field vector. If the field is confined between two ideal surfaces (ideal conducting walls) with coordinates x = 0 and x = L ( t ) > 0 , then the condition of vanishing the electric field at each surface, in the frame moving with the wall, results in the boundary conditions (2). It is assumed usually that both walls were at rest for t 0 , so that initially the field can be written in the form
A ( x , t < 0 ) = n = 0 c n sin ( n π x / L 0 ) exp ( i ω n t ) , ω n = n π / L 0 ,
where coefficients c n are complex numbers in the classical case and operators in the quantum case. Then, one has to find the function (operator) A ( x , t ) for t > 0 . This problem can be solved in the frameworks of two approaches.

2.2.1. Moore’s Approach

The specific feature of the wave equation in the single spatial dimension (1) is the existence of the well known form of the general solution: A ( x , t ) = f ( x t ) + g ( x + t ) , where f and g can be arbitrary functions. How can these functions be chosen to satisfy the additional boundary conditions (2)? Moore [1] has found a complete set of solutions to the problem, satisfying the initial condition (4), in the form
A n ( x , t ) = C n exp i π n R ( t x ) exp i π n R ( t + x ) ,
where function R ( ξ ) must satisfy the functional equation
R ( t + L ( t ) ) R ( t L ( t ) ) = 2 .
As a matter of fact, a quite similar approach was used in [32,33,34], although for a linear function L ( t ) only. Independently, Equation (6) was obtained by Vesnitskii [38].
It is easy to find an exact solution to Equation (6) for the uniform law of motion of the boundary (3). This was done by many authors, starting from [32,33,34]. We give it in the form obtained in [37]:
R α ( ξ ) = 2 ln | 1 + α ξ | ln | ( 1 + v ) / ( 1 v ) | , v = α L 0 .
Evidently, function (7) goes to R 0 ( ξ ) = ξ / L 0 if α 0 . For an arbitrary nonrelativistic law of motion, one can find the solution in the form of the expansion over subsequent time derivatives of the wall displacement. Such an idea goes to the papers [1,38] whose results (with some corrections) can be written in the form [26]
R ( ξ ) = ξ λ ( ξ ) 1 2 ξ 2 λ ˙ ( ξ ) + 1 6 ξ λ ¨ ( ξ ) ξ 2 L 2 ( ξ ) + , λ ( ξ ) L 1 ( ξ ) .
In the special case of L ( t ) = L 0 / ( 1 + α t ) , when λ ¨ ( ξ ) 0 , Equation (8) yields another exact solution, R ( ξ ) = L 0 1 ξ + 1 2 α ξ 2 . Unfortunately, the expansions such as Label (8) cannot be used in the long-time limit ξ , since the terms proportional to the derivatives of λ ( ξ ) (which are supposed to be small corrections) become bigger than the unperturbed term ξ λ ( ξ ) .
Several exact solutions to the Moore equation were found with the aid of the “inverse” method [40,46,47], when one chooses some reasonable function R ( ξ ) and determines the corresponding law of motion of the boundary L ( t ) using the consequence of Equation (6),
L ˙ ( t ) = R [ t L ( t ) ] R [ t + L ( t ) ] R [ t L ( t ) ] + R [ t + L ( t ) ] .
To solve differential Equation (9) with some simple functions, R ( ξ ) is more easier than solving the functional Equation (6) for the given function L ( t ) . However, the dependence L ( t ) does not appear to be admissible from the point of view of physics (the velocity may occur greater than the speed of light, or some discontinuities may arise) for any simple function R ( ξ ) . A large list of simple functions R ( ξ ) (rational, exponential, logarithmic, hyperbolic, trigonometrical, and inverse trigonometrical) and, corresponding to them, functions L ( t ) can be found in [40]. The cases considered in [47] correspond to some monotonous displacements of the mirror from the initial to final positions. Typical functions R ( ξ ) used in [47] were some combinations of ξ / L 0 and some trigonometric functions, such as sin ( m π ξ / L 0 ) . However, none of these functions can be used in the parametric resonance case. Exact solutions for some specific trajectories of the single mirror were found in [48,49,50,51,52,53,54,55,56,57,58,59]. Two mirrors moving with constant accelerations in opposite directions were considered in [60,61]. Massless spin-1/2 fields in a 1D box with moving boundaries were studied in [62], while the single mirror case was considered in [63]. The solutions of the 1D Klein–Gordon equation with one uniformly moving boundary (and another boundary at rest) were found in [64,65].
The asymptotic solution of the Moore equation for L ( t ) = L 0 1 + ϵ sin ( π q t / L 0 ) , with q = 1 , 2 , and | ϵ | 1 , was found in [66,67]. For ϵ t 1 , it has the form (here L 0 = 1 )
R ( t ) = t 2 π q Im ln 1 + ζ + exp ( i π q t ) ( 1 ζ ) , ζ = exp ( 1 ) q + 1 π q ϵ t ,
which clearly demonstrates that the asymptotic mode structure in the resonance case is quite different from the mode structure inside the cavity with unmoving walls. The solution (10) was improved in [68]. The additional term
Δ R ( t ) = 2 ζ ϵ ( 1 ) q z sin ( q π z ) 1 + ζ 2 + ( 1 ζ 2 ) cos ( q π z ) , 1 z = t 2 p 1 , p = 0 , 1 , 2 ,
is negligible in the long-time limit, but it is crucial for the solution to satisfy the correct boundary condition at short times. The case of two moving boundaries, left L ( t ) and right R ( t ) , was studied in [69]. Then, Equations (5) and (6) can be generalized as follows:
A n ( x , t ) = C n exp i π n F ( t x ) exp i π n G ( t + x ) ,
G ( t + R ( t ) ) F ( t R ( t ) ) = 2 , G ( t + L ( t ) ) F ( t L ( t ) ) = 0 .
The solutions generalizing (10) and (11) were found in [69] for the boundaries oscillating with equal resonance frequencies, but different amplitudes and with some phase difference. Numerical solutions of Equations (13) were presented in [70]. The authors of [71,72] pointed out on a possible physical realization of one-dimensional models in the case of TEM modes in cylindrical waveguides. The methods of characteristics and circle maps were applied in [73,74,75]. The one-dimensional cavity with one and two oscillating mirrors was considered within the framework of the “optical” approach in [76]. Generalizations of Moore’s approach to the one-dimensional vibrating cavities were considered in [77]. A one-dimensional uniformly contracting cavity was studied in [78]. The computer program for the numerical solution of the Moore equation was presented in [79]. The Floquet map was applied in [80].

2.2.2. The Modification of the Casimir Force

How can the function R ( ξ ) be used? The first application is connected with the modification of the static Casimir force due to the motion of boundaries. It was shown in [3,81] that the force pressing the moving wall (more precisely, the T 11 component of the energy-momentum tensor of the field) is given by a simple formula (for the initial vacuum state of the field)
F = g [ t L ( t ) ] g [ t + L ( t ) ] ,
where
g ( ξ ) = 1 24 π R ( ξ ) R ( ξ ) 3 2 R ( ξ ) R ( ξ ) 2 + π 48 R ( ξ ) 2 .
For the wall at rest, when R 0 ( ξ ) = ξ / L 0 , one gets the following expression for the stationary Casimir force in one dimension [3,81] (returning to the dimensional variables):
F = π c / ( 24 L 0 2 ) .
If the distance between the boundaries L ( t ) slowly varies with time (in the sense that | d L / d t | c ), then the approximate solution (8) yields [6] the following generalization of formula (16):
F = π c 24 L 2 ( t ) 1 + L ˙ c 2 7 3 1 π 2 L L ¨ c 2 2 3 2 π 2 .
The Casimir force in a cavity moving with a constant acceleration was calculated in [82].
In the parametric resonance case, the force averaged over the period of oscillations was calculated in [67] (in the dimensionless units with L 0 = 1 ):
F = π 24 q 2 + 1 2 1 q 2 ζ + ζ 1 , ζ = exp ( 1 ) q + 1 π q ϵ t .
For q 2 , we do not have attraction, but an exponentially increasing pressure on the oscillating wall due to the creation of real photons in the cavity. An effective action approach to the problem of dynamical Casimir force in 1+1 dimensions was developed in [83].

2.2.3. Generation of Quanta inside the 1D Cavity with Moving Boundary

The knowledge of function R ( ξ ) also permits one to calculate the mean number of created quanta inside a nonstationary cavity. The general scheme is as follows [1,81]. Taking into account only the electromagnetic modes whose vector potential is directed along z-axis (“scalar electrodynamics”), one can write down the field operator in the Heisenberg representation A ^ ( x , t ) at t 0 (when both the plates were at rest at the positions x l e f t = 0 and x r i g h t = L 0 ) as (we assume c = = 1 )
A ^ i n = 2 n = 1 1 n sin n π x L 0 b ^ n exp i ω n t + h . c .
where b ^ n means the usual annihilation photon operator and ω n = π n / L 0 . The choice of coefficients in Equation (19) corresponds to the standard form of the field Hamiltonian (in the Gaussian units)
H ^ 1 8 π 0 L 0 d x A / t 2 + A / x 2 = n = 1 ω n b ^ n b ^ n + 1 / 2 .
For t > 0 , the field operator can be written as
A ^ ( x , t ) = 2 n = 1 1 n b ^ n ψ ( n ) ( x , t ) + h . c . ,
where functions ψ ( n ) ( x , t ) are given by formula (5) with C n = i / 2 (in order to satisfy the initial conditions). If the wall stops after some time T, then the field operator can be expanded again over the complete set of sine functions, like in Equation (19), but with the “physical” operators a ^ m and a ^ m instead of b ^ m and b ^ m . The two sets of operators are related by means of the Bogoliubov transformation
a ^ m = n = 1 b ^ n α n m + b ^ n β n m * , m = 1 , 2 , ,
which implies the unitarity conditions
m = 1 α n m * α k m β n m * β k m = δ n k , n = 1 α n m * α n j β n m * β n j = δ m j , n = 1 β n m * α n k β n k * α n m = 0 .
If the final position of moving boundary coincides with initial position L 0 , then the Bogoliubov coefficients can be found in the form [84] (here c = 1 )
α m n β m n = 1 2 m / n t / L 0 1 t / L 0 + 1 d x exp i π n R L 0 x m x .
The mean number of photons in the mth mode equals the average value of the operator a ^ m a ^ m in the initial state | in , since namely this operator has a physical meaning at t > T :
N m in | a ^ m a ^ m | in = n | β n m | 2 + n , k α n m * α k m + β n m * β k m b ^ n b ^ k + 2 Re β n m α k m b ^ n b ^ k .
The first sum on the right-hand side of (25) describes the effect of the photon creation from vacuum, while the other sums are different from zero only in the case of a non-vacuum initial state of the field.
In the resonance case, solution (10) leads after long calculations to the following simple formula for the rate of photon generation in the mth mode, when the wall vibrates at the twice frequency of the first resonator eigenmode, in the limit ϵ t 1 [67]:
d N m / d t | ϵ | 1 ( 1 ) m / ( m π ) .
The energy density inside the cavity can be calculated as [3,81] T 00 ( x , t ) = g ( t + x ) g ( t x ) , with the same function g ( ξ ) (15). It was shown in [68,85,86,87] that, for q 2 , the energy density grows exponentially in the form of q traveling wave packets, which become narrower and higher as time increases. The total energy also grows exponentially. For two resonantly oscillating boundaries (with equal frequencies), the asymptotic photon production rate is given by the same formula (26), where ϵ should be replaced with ϵ ˜ = ϵ L + ( 1 ) q + 1 ϵ R cos ( ϕ ) , where ϵ L , R are relative oscillation amplitudes of each boundary and ϕ the phase difference [69].
The influence of initial states of the field and different boundary conditions on the energy density and radiation force was studied analytically and numerically in [88,89,90,91]. The case of two relativistic moving mirrors was studied in [92].

2.2.4. Expansions over the Instantaneous Basis

Another approach to the problem consists of expanding the function ψ ( n ) ( x , t ) in Equation (21) in a series with respect to the instantaneous basis [93,94,95,96,97]:
ψ ( n ) ( x , t > 0 ) = k = 1 Q k ( n ) ( t ) L 0 L ( t ) sin π k [ x u ( t ) ] L ( t ) , n = 1 , 2 ,
with the initial conditions Q k ( n ) ( 0 ) = δ k n , Q ˙ k ( n ) ( 0 ) = i ω n δ k n , k , n = 1 , 2 , . This way, we satisfy automatically both the boundary conditions, A ( u ( t ) , t ) = A ( u ( t ) + L ( t ) , t ) = 0 , and the initial condition (4). Putting expression (27) into the wave Equation (1), one can arrive after some algebra at an infinite set of coupled differential equations [97,98,99]
Q ¨ k ( n ) + ω k 2 ( t ) Q k ( n ) = 2 j = 1 g k j ( t ) Q ˙ j ( n ) + j = 1 g ˙ k j ( t ) Q j ( n ) + O g k j 2 ,
ω k ( t ) = k π / L ( t ) , g k j ( t ) = g j k ( t ) = ( 1 ) k j 2 k j L ˙ + u ˙ ϵ k j j 2 k 2 L ( t ) , ϵ k j = 1 ( 1 ) k j , j k .
For u = 0 (the left wall at rest), the equations like (28)–(29) were derived in [95,100].
If the wall comes back to its initial position L 0 after some interval of time T, then the right-hand side of Equation (28) disappears, so, at t > T , one can write
Q k ( n ) ( t ) = ξ k ( n ) e i ω k ( t + δ T ) + η k ( n ) e i ω k ( t + δ T ) , k , n = 1 , 2 ,
where ξ k ( n ) and η k ( n ) are some constant complex coefficients. To find these coefficients, one has to solve an infinite set of coupled Equation (28) ( k = 1 , 2 , ) with time-dependent parameters; moreover, each equation also contains an infinite number of terms. However, the problem can be essentially simplified, if the walls perform small oscillations at the frequency ω w close to some unperturbed field eigenfrequency: L ( t ) = L 0 1 + ε L sin p ω 1 ( 1 + δ ) t , u ( t ) = ε u L 0 sin p ω 1 ( 1 + δ ) t + φ . Assuming | ε L | , | ε u | ε 1 , it is natural to look for the solutions to Equation (28) in the form
Q k ( n ) ( t ) = ξ k ( n ) e i ω k ( 1 + δ ) t + η k ( n ) e i ω k ( 1 + δ ) t ,
allowing now the coefficients ξ k ( n ) and η k ( n ) to be slowly varying functions of time. Then, using the standard method of slowly varying amplitudes [101], one can arrive [97] at the following set of equations, containing only three terms with simple time independent coefficients in the right-hand sides:
d d τ ξ k ( n ) = ( 1 ) p ( k + p ) ξ k + p ( n ) ( k p ) ξ k p ( n ) + 2 i γ k ξ k ( n ) ,
d d τ η k ( n ) = ( 1 ) p ( k + p ) η k + p ( n ) ( k p ) η k p ( n ) 2 i γ k η k ( n ) ,
τ = ε ω 1 t / 2 , γ = δ / ε , ξ k ( n ) ( 0 ) = δ k n , η k ( n ) ( 0 ) = 0 .
Equations (32) and (33) correspond to the case of u = 0 (the left wall at rest). Actually, the term u ˙ in g k j ( t ) does not make any contribution to the simplified equations of motion for even values of integer parameter p, so that the rate of change of the cavity length L ˙ / L 0 is only important in this case. On the contrary, if p is an odd number, then the field evolution depends on the velocity of the center of the cavity v c = u ˙ + L ˙ / 2 , but does not depend on L ˙ alone [102]. Therefore, one should simply replace L ˙ / L 0 by 2 v c / L 0 , if p is an odd number.
The uncoupled Equations (32) and (33) hold only for k p . This means that they describe the evolution of all the Bogoliubov coefficients, only if p = 1 . Then all the functions η k ( n ) ( t ) are identically equal to zero due to the initial conditions (34); consequently, no photon can be created from vacuum (in the lowest order approximation with respect to the small parameter ε ). This special “semi-resonance” case was studied in detail in [97,103], where exact analytical solutions to Equation (32) were obtained in terms of the Jacobi polynomials. It was demonstrated that the total number of photons is an integral of motion in this specific case. (A similar phenomenon in the classical case was discussed in [46], whereas the quantum case was also considered in [96]). The exact formula for the total energy in all cavity modes for p = 1 (normalized by the fundamental cavity mode energy ω 1 ) is as follows:
E ( τ ) = E ( 0 ) + 2 sinh 2 ( a τ ) a 2 E ( 0 ) γ 2 Im ( G 1 ) + Re ( G 1 ) sinh ( 2 a τ ) 2 a ,
a = 1 γ 2 , G 1 = 2 n = 1 n ( n + 1 ) i n | b ^ n b ^ n + 1 | i n .
It can be shown that | E ˙ ( 0 ) | E ( 0 ) . Thus, the total energy grows exponentially, when τ 1 (provided γ < 1 ), although it can decrease at τ 1 , if E ˙ ( 0 ) = Re ( G 1 ) < 0 . Since the total number of photons is constant, such a behavior is explained by the effect of pumping the highest modes at the expense of the lowest ones (in the classical case this effect was noticed in 1960s: see [39]). This results in an effective cooling of the lowest electromagnetic modes. The absence of photon creation in the fundamental mode was related to the S L ( 2 , R ) symmetry of the problem in Ref. [104].
If p 2 , we have, in addition to (32) and (33), p 1 pairs of coupled equations for the coefficients with lower indexes 1 k p 1 ,
d d τ ξ k ( n ) = ( 1 ) p ( k + p ) ξ k + p ( n ) ( p k ) η p k ( n ) + 2 i γ k ξ k ( n ) ,
d d τ η k ( n ) = ( 1 ) p ( k + p ) η k + p ( n ) ( p k ) ξ p k ( n ) 2 i γ k η k ( n ) .
In this case, some functions η k ( n ) ( t ) are not equal to zero at t > 0 , thus we have the effect of photon creation from the vacuum. It is convenient to introduce a new set of coefficients ρ k ( n ) , whose lower indexes run over all integers from to :
ρ k ( n ) = ξ k ( n ) k > 0 0 k = 0 η k ( n ) k < 0
Then, Equations (32), (33), (37) and (38) can be combined into a single set of equations ( k = ± 1 , ± 2 , ) [97]
d d τ ρ k ( n ) = ( 1 ) p ( k + p ) ρ k + p ( n ) ( k p ) ρ k p ( n ) + 2 i γ k ρ k ( n ) , ρ k ( n ) ( 0 ) = δ k n .
A remarkable feature of the set of Equation (40) is that its solutions satisfy exactly the unitarity conditions (23), which can be rewritten as ( m , j , n , k = 1 , 2 , )
m = m ρ m ( n ) * ρ m ( k ) = n δ n k n = 1 m n ρ m ( n ) * ρ j ( n ) ρ m ( n ) * ρ j ( n ) = δ m j n = 1 1 n ρ m ( n ) * ρ j ( n ) ρ j ( n ) * ρ m ( n ) = 0 .
Exact solutions to the set (40) were found in [97] in terms of the Gauss hypergeometric function. They can be expressed also in terms of the complete elliptic integrals, if p = 2 [105]. The most interesting consequences of these solutions are as follows.
There is no photon creation in the modes with numbers p , 2 p , . In the short-time limit ( τ 1 ), N ˙ j + p q ( v a c ) τ 2 q + 1 , but the photon generation rate in each mode tends to the constant value in the long-time limit:
d d τ N j + p q ( v a c ) 2 a p 2 sin 2 ( π j / p ) π 2 ( j + p q ) , a p τ 1 .
The total number of photons created from vacuum in all modes increases in time approximately quadratically, both in the short-time and in the long-time limits (although with different coefficients):
N ¨ ( v a c ) = 1 3 p ( p 2 1 ) , | a p τ | 1 , N ¨ ( v a c ) = 2 a 2 p 3 / π 2 , a p τ 1 , a > 0 .
At the same time, the total number of “nonvacuum” photons increases in time only linearly at a p τ 1 .
The total energy in all modes for the initial vacuum state of field is given by the exact formula [97]
E ( v a c ) ( τ ) = p 2 1 12 a 2 sinh 2 ( p a τ ) .
The total energy increases exponentially at τ , provided γ < 1 . In the special case of γ = 0 , such a behavior of the total energy was obtained also in the frameworks of other approaches in [100,106,107,108]. Note that the total “vacuum” and “nonvacuum” energies increase exponentially with time, if γ < 1 , whereas the total number of photons increases only as τ 2 and τ , respectively, under the same conditions. This happens because the number of effectively excited modes increases in time exponentially [109]. The parametric instability in systems with moving boundaries was studied in [110,111,112,113].
The solutions found in [97] were used in [114,115] to study the effects of squeezing and the formation of narrow packets inside the 1D cavity with oscillating walls. The emission of narrow packets outside the 1D Fabry–Perot cavity was studied in [116,117,118]. The influence of the dispersion of the reflection coefficient of the fixed mirror (with the ideal moving one) and the frequency detuning on the formation of packets inside the 1D cavity was investigated for different initial conditions in [119,120,121]. The comparisons of numerical and analytical calculations in the one-dimensional case were made in [122,123,124]. It was shown that the coincidence of results is excellent for small enough values of parameter ε < 10 5 (remember that the realistic values are expected to be less than 10 7 ). However, strong differences were observed for big values ε > 10 2 . For other approaches to solving equations like (28), see, e.g., [99,125,126]. Recently, these equations were solved numerically in the 1D and 3D cases, for one and two moving boundaries, in the study [127].
The case of mixed boundary conditions—Dirichlet condition ϕ ( t , L ( t ) ) = 0 for the moving wall and Neumann condition x ϕ ( t , 0 ) = 0 for the wall at rest—was solved, following the method of [105], in papers [128,129]. In this case, photons can be produced in all modes, differently from the Dirichlet–Dirichlet boundary conditions used in [105].
The analytical results shown above were obtained in the first order approximation with respect to the small parameter ε . Taking into account the higher order approximations, the resonance frequencies can be made smaller than the fundamental frequency ω 1 . For example, it was shown that, in the n-th order approximation, the resonances at the frequency 2 ω ˜ 1 / n are possible [130] (the frequency ω ˜ 1 is slightly shifted from ω 1 due to nonlinear effects). In particular, the photon creation from vacuum at the frequency close to ω 1 becomes possible in the second order approximation.

2.2.5. Quantum Regime of the Wall Motion

In all papers cited above, the law of motion of the wall (boundary) was prescribed. The inclusion of the moving boundary as a part of the dynamical system was made by Barton and Calogeracos [131,132]. However, they considered the boundary as a classical particle. The back reaction of the field on the motion of a classical wall was taken into account in [133,134]. The resonant energy exchange between a moving boundary and cavity modes was studied in [135]. Another approach was used in Ref. [136], where the mirror was replaced by an ensemble of electrons and ions, bounded by some effective parabolic potential.
The quantized motion of the boundary was considered in [137,138,139,140,141]. The authors of [139,140,141] considered a scalar field in a one-dimensional cavity with one fixed and one mobile wall; the latter was bound to an equilibrium position by a harmonic potential and its mechanical degrees of freedom were treated quantum mechanically. The presence of the moving wall yielded an effective interaction between the field modes, described by means of the interaction Hamiltonian
H ^ i n t = b ^ + b ^ j k C k j N a ^ k + a ^ k a ^ j + a ^ j ,
where b ^ and a ^ j are bosonic operators, describing the mobile mirror and field modes, respectively, and N is the normal ordering operator. Note that a simplified version of (44), H ^ i n t = G b ^ + b ^ a ^ a ^ , was used to describe the ponderomotive effects of a strong laser field in cavities with oscillating walls, but under the condition that the mechanical oscillator frequency is many orders of magnitude smaller than the cavity frequency, so that the DCE is negligible [142,143,144,145]. The recent progress in studies of such fully quantized optomechanical systems in connection with the DCE can be seen in [146,147,148,149,150,151,152].

2.3. Partially Transparent Mirrors and General Boundary Conditions in 1D

The DCE with partially reflecting mirrors in the one-dimensional models was considered in [53,102,117,131,132,153,154,155,156,157,158,159]. The comparison of the Dirichlet and Neumann boundary conditions in terms of the solutions to Moore’s Equation (6) was performed in paper [72]. Another approach to this problem was used in [160]. The so-called Robin boundary conditions of the form
ϕ γ ( t ) ϕ / x = 0 | at   the   boundary
were studied in connection with the DCE in [131,132,161,162,163,164]. The simplest model uses the complex reflection and transmission coefficients in the form [131,132]:
r = i γ ω + i γ , t = ω ω + i γ .
This is equivalent to the model of the boundary as an equivalent delta-potential or a jellium-type plasma sheet of infinitely small thickness. Such kind of models was used e.g., in Refs. [165,166]. Quantum and classical effects produced by thin sheets of electrons, working as relativistic mirrors, were considered in [167,168,169].
The further generalization to the δ δ potential was considered in [170]. In this model, the reflection and transmission coefficients have the form
r = 2 ω λ i γ ω λ 2 + 1 + i γ , t = ω λ 2 + 1 ω λ 2 + 1 + i γ .
It was shown that a partially reflecting single moving mirror can produce a larger number of particles in comparison with a perfect one. The interference between the motion of a single mirror and the time-dependent Robin parameter was studied in [171]. The problem of creation of a time-dependent boundary was addressed in [172]. The catastrophic generation of quanta due to instantaneous appearance and disappearance of a wall in a cavity was shown in [173]. The explosive photon production due to instantaneous changes from the Neumann to the Dirichlet BC (or reversely) was considered in [174]. (A similar unphysical behavior was discovered in [175] in the case of instantaneous displacements of boundaries.) Microscopic models of mirrors as oscillators coupled bilinearly to quantum fields were considered in [176,177,178,179,180].

3. Three-Dimensional Models with Moving Boundaries

3.1. Single Mirror DCE

The effect of photon production from vacuum, caused by single mirrors moving in the three-dimensional space with relativistic velocities or with great accelerations, was studied in the papers [181,182,183,184,185,186]. The nonrelativistic motion of the plane mirror was considered in [187,188,189,190,191]. The radiation from dynamically deforming mirrors with different boundary conditions was calculated in [192,193,194]. Graphene-like mirrors were considered in [195]. Classical analogs of the DCE due to the oscillating motion of the plane surfaces containing dipole layers were considered in [196,197]. For the most recent publications, one can see [198,199,200].
Various quantum effects arising due to the motion of dielectric boundaries in three dimensions, including the modification of the Casimir force and creation of photons, have been studied in [201,202,203,204,205,206,207,208].

Quantum Friction for Moving Surfaces

The first calculations of the forces acting on the single mirrors moving with nonrelativistic velocities due to the vacuum or thermal fluctuations of the field were performed in the framework of the spectral approach (using the fluctuation–dissipation theorem) in [209] (three-dimensional case, the force proportional to the fifth-order derivative of the coordinate) and in [210] (one-dimensional model, the force proportional to the third-order derivative of the coordinate). These studies were continued in [25,153,211,212,213,214,215,216,217], where it was assumed that the velocity of the boundary is perpendicular to the surface.
The frictional force proportional to the constant relative velocity of two parallel plates, when this velocity is also parallel to the surfaces, was calculated by Teodorovich [218] in the van der Waals regime (i.e., neglecting the retardation effects). More general results were obtained by Levitov [219] and Polevoi [220]. In particular, Levitov had shown that the linear dependence on the velocity disappears at zero absolute temperature, but the friction force is still nonzero: it is proportional to v 3 for a very small separation between the plates, while it becomes velocity independent (like the dry friction) for a big enough separation. Later on, the theory of “Casimir friction” was the subject of studies [221,222,223,224,225,226,227,228], with controversial results. Moreover, the existence of such a friction force was questioned in [229] and especially in [230]. The latter paper triggered hot discussions [231,232,233,234,235]. More recent results can be found in [236,237,238,239,240,241,242,243,244,245,246,247,248,249,250]. In particular, it was shown in [246] that the friction force between two graphene sheets is nonzero only if the relative velocity is larger than the Fermi velocity of the charge carriers in graphene.
In 1971, Zel’dovich [251,252] predicted that a rotating object can amplify certain incident waves under the condition ω L Ω < 0 , where ω is the frequency of the field mode, L is its azimuthal quantum number, and Ω is the angular velocity of the rotating body (a cylinder). Furthermore, he conjectured that such an object should spontaneously emit radiation for these selected modes. This idea was further developed in [253]. The effect was called “superradiance” in [254,255] and “spontaneous superradiance” in [256]. Recently, a similar phenomenon—the friction force on the rotating bodies due to the interaction with vacuum (or thermal) fluctuations of the EM field—was studied in detail in papers [257,258,259,260]. Analogs in superfluids were considered in [261]. The wave instability of the electromagnetic field inside a rotating cylinder, resulting in exponential growth with time, was studied in [262].

3.2. Cavity DCE

3.2.1. Effective Hamiltonians

The instantaneous basis approach, described above, corresponds to the Heisenberg description of the quantum systems. It was believed for some time after Moore [1] that it is the only possible approach to the systems with moving boundaries. However, an equivalent Schrödinger description (although approximate) also exists. It is based on using some effective Hamiltonians [94,95,96,98,99,100,156,175,263,264,265,266,267]. The general structure of the effective infinite-dimensional quadratic Lagrangians and Hamiltonians arising in the canonical approach to the dynamical Casimir effect was analyzed and classified in [98,263].
Following [268], let us suppose that the set of Maxwell’s equation in a medium with time-independent parameters and boundaries can be reduced to an equation of the form K ^ ( { L } ) F α ( r ; { L } ) = ω α 2 ( { L } ) F α ( r ; { L } ) , where { L } means a set of parameters (for example, the distance between the walls or the dielectric permittivity inside the cavity), ω α ( { L } ) is the eigenfrequency of the field mode labeled by the number (or a set of numbers) α and F α ( r ; { L } ) is some vector function describing the EM field (e.g., the vector potential). In the simplest cases, K ^ ( { L } ) is reduced to the Laplace operator. Usually, the operator K ^ ( { L } ) is self-adjoint, and the set of functions { F α ( r ; { L } ) } is orthonormal and complete in some sense.
Now, suppose that parameters L 1 , L 2 , , L n become time-dependent. If one can still satisfy automatically the boundary conditions, expanding the field F ( r , t ) over “instantaneous” eigenfunctions F ( r , t ) = α q α ( t ) F α ( r ; { L ( t ) } ) (this is true, e.g., for the Dirichlet boundary conditions, which are equivalent in some cases to the TE polarization of the field modes), then the dynamics of the field is described completely by the generalized coordinates q α ( t ) , whose equations of motion can be derived from the effective time-dependent Hamiltonian [98]
H = 1 2 α p α 2 + ω α 2 { L ( t ) } q α 2 + k = 1 n L ˙ k ( t ) L k ( t ) α β p α m α β ( k ) q β ,
m α β ( k ) = m β α ( k ) = L k d V F α r ; { L } L k F β r ; { L } .
Consequently, the field problem can be reduced to studying the dynamics of the infinite set of harmonic oscillators with time-dependent frequencies and bilinear specific (coordinate–momentum) time-dependent coupling. Methods of diagonalization of some specific Hamiltonians like (48) were considered in [269,270].

3.2.2. Parametric Oscillator Model

From the point of view of applications to the DCE, the most important cases are those where the parameters L k ( t ) vary in time periodically. In the case of small harmonic variations at the frequency close to the double unperturbed eigenfrequency of some mode 2 ω 0 , the equations of motion resulting from Hamiltonian (48) can be solved approximately with the aid of the method of slowly varying amplitudes [101]. If the difference ω α ω β is not close to 2 ω 0 for all those modes which have nonzero (or not very small) coupling coefficients m α β , then only the selected mode with label 0 can be excited in the long-time limit, and one can consider only this single resonance mode [27,105], whose excitation is described by the Hamiltonian
H = p 0 2 + ω 0 2 { L ( t ) } q 0 2 / 2 .
The theory of quantum nonstationary harmonic oscillator has been well developed since its foundation by Husimi in 1953 [271] (see, e.g., [272,273,274] for the reviews and references). It appears that all dynamical properties of the quantum oscillator are determined by the fundamental set of solutions of the classical equation of motion
ε ¨ + ω 2 ( t ) ε = 0 .
For harmonic variations of the frequency in the form ω ( t ) = ω 0 1 + 2 κ cos ( 2 ω 0 t ) , with | κ | 1 , Equation (51) can be solved approximately using, e.g., the method of averaging over fast oscillations or the method of slowly varying amplitudes [101]. The final result is a simple formula for the number of quanta created from the initial vacuum state [105,275]:
N = sinh 2 ω 0 κ t .
Here, one can notice the striking difference between the parabolic dependence on time of the total number of created quanta in the 1D case, according to Equation (42), and the exponential dependence (52) in the 3D case. This is explained by the crucial difference in the eigenmode spectra: it is equidistant in the 1D case, but strongly non-equidistant in the generic 3D case. A smooth transition from one situation to another was demonstrated in [276]. Anharmonic and random periodic displacements of boundaries were studied in [277].
In the literature on the DCE, many authors frequently use an equivalent form of the single-mode effective Hamiltonian (50) in terms of annihilation and creation operators, derived by Law [96],
H ^ = ω ( t ) a ^ a ^ i χ t a ^ 2 a ^ 2 , χ t = ( d ω / d t ) / ( 4 ω ) , = 1 .
Algebraic methods of solving the Schrödinger equation with Hamiltonian (53) were used in [278]. Recently, the harmonic oscillator model was re-discovered and applied to the circuit QED in [279].

3.2.3. The Role of Intermode Interactions

The single-mode approximation of the generic Hamiltonian (48) does not work, if the frequency difference between specific modes equals (almost) exactly twice the resonance frequency of some mode. For example, such a resonance coupling between two modes is possible in cubical cavities [280]. This case was studied in [280,281]. It was shown that the number of photons in both the coupled modes grows exponentially with time in the long time limit ω 0 κ t 1 , but the rate of photon generation (the argument of the exponential function) turns out to be two times smaller than the value of this rate in the absence of the resonance coupling. (Actually, this rate depends on the concrete values of the coupling coefficients m α β , but in any case it cannot exceed the ‘uncoupled’ values [281].) This example indicates that the resonance coupling between the modes should be avoided in order to achieve the maximal photon generation rate, at least in the case of TE modes. A detailed numerical study of this case for different sizes of the rectangular cavities was performed in [282]. The authors of paper [166] used numerical methods, taking into account the interaction between 50 lowest coupled modes in the rectangular cavity, bisected by a “plasma sheet” with a periodically varying number of free carriers. Some results of that paper show that the intermode coupling can increase the number of photons in the modes of EM field with the TM polarization.
The parametric excitation of the resonantly coupled modes of the scalar field in the cubic cavity was studied in [283]. The case of the TE polarization of the vector EM field was also considered there. The “swinging” cubic cavity, performing small amplitude periodical rotations along the z-axis, θ ( t ) = ϵ sin ( Ω t ) , was considered in [284] (in the model of scalar field). The frequency Ω was chosen in such a way that three modes, ( 1 , 1 , 1 ) , ( 1 , 2 , 1 ) and ( 2 , 1 , 1 ) , were resonantly coupled.
The role of intermode interactions was also studied in paper [285]. There, the evolution of the classical electromagnetic field inside the cylindrical (rectangular) cavity with ideal boundaries was studied analytically and numerically in the case, when the conductivity σ and dielectric permittivity ϵ of a thin slab attached to the base of the cylinder can vary with time. It was shown that the single-mode model can be justified, if the perturbation of the field is small due to the smallness of σ and δ ϵ . However, no amplification can be achieved in this case for microwave fields, if σ and δ ϵ are due to the creation of free carriers inside the slab.

3.2.4. Time-Dependent Casimir Force

Remember that the famous Casimir formula [286] for the attraction force per unit area between two ideal infinite plates in three dimensions reads
F / S = π 2 c / ( 240 L 0 4 ) .
It was generalized, using the Green function method, to the case of uniform motion of one boundary (in the direction perpendiculat to its surface) in [287] for the scalar field and in [288] for the vector electromagnetic field. The results of two models turned out to be qualitatively different. Namely, the scalar model predicted the increase of the absolute value of the Casimir force in the nonrelativistic case:
F s c a l = π 2 480 L 4 ( t ) 1 + 8 3 v c 2 + , v / c 1 .
On the contrary, the account of the TM modes of the electromagnetic field results in the decrease of the absolute value of the Casimir force in the nonrelativistic case:
F E M = π 2 240 L 4 ( t ) 1 + v c 2 2 3 10 π 2 + , v / c 1 .
Moreover, as a matter of fact, the force practically does not depend on the velocity (with accuracy about 10%), decreasing monotonously (by the absolute value) to the ultra-relativistic limit
F E M = 3 8 π 2 L 4 ( t ) 1 + 1 16 1 v 2 / c 2 2 + , 1 ( v / c ) 2 1 .
However, the force can be significantly amplified under the resonance conditions, either in the L C -contour [209] or in the Fabry–Perot cavity [210] (see also [84]). The time-dependent force between vibrating plates was considered in [289].

3.2.5. Vector Fields in 3D Cavities

Quantum properties of the electromagnetic field between two infinite plates, moving with constant relative velocity, with account of the field polarization, were considered in [290]. The photon generation in the TE and TM modes of electromagnetic field between two parallel vibrating plates was considered in [265]. It was shown that the photon generation rate in the TM modes is higher by one order of magnitude than that in the TE modes (see also [266]). The case of a three-dimensional rectangular cavity divided into two parts by an ideal mirror, which suddenly disappears, was considered in [291]. TE and TM modes in a resonant cavity bisected by a plasma sheet were considered in [166]. The DCE for the Dirac field inside the 3D cubic box with oscillating walls was studied in [292]. The spherical and cylindrical geometries were considered in [72,293,294,295,296,297,298]. General boundary conditions in 3D cavities were used in [299,300]. The scattering approach was applied to the DCE problem in [301]. Numerical algorithms were elaborated in [302].

3.2.6. Saturated Regimes

An unlimited growth of the number of generated “Casimir photons” happens in the simplified models only. The maximal number of quanta that could be generated under realistic conditions is limited due to many factors. One such factor is related to unavoidable nonlinearities in real systems [303,304,305,306]. Another mechanism, related to the temporal coherence, was studied in [307]. The saturation due to the finite reflectance spectral band of mirrors was considered in [308].

4. General Parametric DCE

4.1. Circuit DCE

In view of great difficulties in observing weak manifestations of the DCE, several authors came to the idea of simulating (modeling) this effect in more simple arrangements. Probably, the simplest possibility is to use some electrical circuits [309]. The idea to use quantum resonant oscillatory contours or Josephson junctions with time-dependent parameters (capacitance, inductance, magnetic flux, critical current, etc.) was put forward by Man’ko many years ago [310]. More elaborated proposals in the same direction were presented in [311,312,313,314,315]. The idea to use a superconducting coplanar waveguide in combination with a Josephson junction was developed in [316,317,318,319], and the experiments were reported in [320,321,322,323]. Their success is related to the possibility of achieving the effective velocity of the boundary up to 25% of the light velocity in vacuum (although the analogy with the motion of real boundaries is not perfect). A detailed review of this approach was given in [31]. The experiments [320,321,322,323] were performed in the frequency interval of a few GHz. Their arrangements can be considered as realizations of the one-dimensional models with time-dependent parameters. The experiment [320] was performed with an open strip-line waveguide, where the photon generation was achieved due to periodical fast changes of the boundary conditions on one side of the line. Although that boundary conditions can be interpreted as equivalent to a high effective velocity of an oscillating boundary, such an interpretation has a limited range of validity. The second experiment [322] used the parametric resonance excitation of quanta due to changes of the effective speed of light. This arrangement can be considered as a one-dimensional system with periodically varying distributed parameters. The results of that experiments stimulated many theoretical papers, suggesting further improvements of the experimental schemes [324,325,326,327,328,329,330,331,332,333,334,335,336,337]. The circuit QED with “artificial atoms” (qubits) was the subject of studies [338,339,340,341,342,343,344,345,346,347,348,349,350]. The most recent review on parametric effects in circuit QED can be found in [351].

4.2. Analogs of DCE in Condensed Matter

Analogs of the DCE in Bose–Einstein condensates and ultracold gases with time-dependent parameters were considered in [352,353,354,355,356,357,358,359,360,361,362,363,364]. An obvious advantage of replacing EM waves with their sound analogs is a possibility of achieving high ratios of effective velocities to the sound speed, including the supersonic regimes [358,360]. The use of plasmon resonances in metallic nanoparticles surrounded by an amplifying medium, excited by femtosecond lasers, was suggested in [365]. The DCE with polaritons was studied in [366,367,368,369,370], and the DCE with magnons in [371,372]. The DCE for phonons was considered in [373,374,375,376].

4.3. DCE and Atomic Excitations

Many papers were devoted to the interaction between the “Casimir photons” and multilevel systems (“atoms” or “qubits”) inside cavities or quantum circuits [306,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408]. In particular, two-level atoms (qubits) were considered in [306,378,380,381,382,384,386,391,392,394,400,401,403,404,408]. Two two-level atoms were studied in [390,396,405,407], and ensembles of many two-level atoms in [383,385,388,395,397,398,401,406]. Three-level systems were considered in [379,387,389,408]. The case of N-level atoms in equally spaced, resonant ladder configuration, was investigated in [388]. The circuit DCE with two-level “atoms” was studied in [409]. The case of a two-level atom near a single accelerated mirror was studied in [410]. A possibility of generation of quantum field states with the hyper-Poissonian statistics, as well as some other “exotic” quantum states, due to the DCE in the presence of “atoms”, was shown in [411,412]. The “anti-dynamic” Casimir effect (coherent annihilation of system excitations from common initial states due to parametric modulation) was predicted and studied in papers [304,340,413,414,415,416,417,418]. The effect of generation of pairs of atomic excitations instead of photons (“inverse DCE”) was considered in [304]. A possibility of exciting the Rydberg atoms passing near the boundary with oscillating parameters (“oscillating effective mirror”) was studied in [419]. The influence of the oscillating boundary on the spontaneous emission rate of an atom placed nearby this boundary was considered in [420,421]. The influence of the DCE on the fidelity of the quantum state transfer between two qubits in the ultrastrong coupling regime was studied in [405]. The generation of photons by accelerated neutral objects in cavities with static walls was considered, e.g., in [422].

5. Experimental Proposals for the Cavity DCE

5.1. Difficulties with Real Moving Boundaries

According to formula (52), a possibility of experimental verification of the DCE depends on the amplitude of the frequency variation Δ ω = 2 κ ω 0 . The main difficulty is due to the very high frequency 2 ω 0 . The most exciting dream is to observe the ”Casimir light” in the visible part of the electromagnetic spectrum. However, it seems to be very improbable using real mirrors oscillating at the frequency about 10 15 Hz, due to the practical impossibility of exciting and maintaining the high frequency oscillations of the suspended plate with a large amplitude and for a long time.
It seems that the only possible way to realize the real motion of material boundaries at high frequencies is not to move the whole mirror, but to cause its surface to perform harmonic vibrations with the aid of some mechanism, e.g., using the piezo-effect. The amplitude of such vibrations Δ L is connected with the maximal relative deformation δ in a standing acoustic wave inside the wall as δ = ω w Δ L / v s , where v s 5 · 10 3 m/s is the sound velocity. Since usual materials cannot bear deformations exceeding the value δ m a x 10 2 , the velocity of the boundary cannot exceed the value v m a x δ m a x v s 50 m/s (independent on the frequency). The maximal possible frequency variation amplitude Δ ω can be evaluated as Δ ω v s δ m a x / ( 2 L ) , where L is the cavity size. For the optical frequencies, taking 2 L > 1 μ m, we obtain Δ ω < 5 × 10 7 s 1 , whereas, for the microwave frequencies (in the GHz band) with L 1 cm, we have Δ ω < 10 3 s 1 . Since the time of excitation t must be bigger than 1 / Δ ω , the quality factor of the cavity Q must be not less than Q m i n ω 0 / Δ ω ( L / λ ) 4 π c / ( v s δ ) 10 8 ( L / λ ) . Consequently, there are two main challenges: how to excite high frequency surface oscillations and how to maintain the high quality factor in the regime of strong surface vibrations. The excitation of high amplitude surface vibrations at the optical frequencies seems very problematic. Therefore, hardly the ”Casimir light” in the visible region can be generated in systems with really moving boundaries. However, this seems to be possible in other schemes, where changes of some parameters can be interpreted as variations of an ”effective length” of the cavity.
The GHz frequency band seems more promising. In such a case, the dimensions of cavities must be of the order of few centimeters. Superconducting cavities with the quality factors exceeding 10 10 in the frequency band from 1 to 50 GHz have been available for a long time. Therefore, the most difficult problem is to excite the surface vibrations. At lower frequencies, it was solved long ago. Only recently, a significant progress was achieved in fabrication of the so called “film bulk acoustic resonators” (FBARs): piezoelectric devices working at the frequencies from 1 to 3 GHz [423]. They consist of an aluminum nitride (AlN) film of thickness corresponding to one half of the acoustic wavelength, sandwiched between two electrodes. It was suggested [424,425,426] to use such kind of devices to excite the surface vibrations of cavities in order to observe the DCE. However, no experimental results in this direction were reported until now. For the most recent proposals, see [427,428].

5.2. Simulations with Semiconductor Slabs

In view of difficulties of the excitation of oscillating motion of real boundaries, the ideas concerning the imitation of this motion attracted more and more attention with the course of time. The first concrete suggestion was made by Yablonovitch [5], who proposed to use a medium with a rapidly decreasing in time refractive index (”plasma window”) to simulate the so-called Unruh effect. In addition, he pointed out that fast changes of dielectric properties can be achieved in semiconductors illuminated by subpicosecond optical pulses and supposed that ”the moving plasma front can act as a moving mirror exceeding the speed of light.” Similar ideas and different possible schemes based on fast changes of the carrier concentration in semiconductors illuminated by laser pulses were discussed in [429,430,431]. Yablonovitch [5,429] put emphasis on the excitation of virtual electron–hole pairs by optical radiation tuned to the transparent region just below the band gap in a semiconductor photodiode. He showed that big changes of the real part of the dielectric permittivity could be achieved in this way.
The key idea of the experiment named “MIR” in the university of Padua [432,433] was to imitate the motion of a boundary, using an effective “plasma mirror” formed by real electron–hole pairs in a thin film near the surface of a semiconductor slab, illuminated by a periodical sequence of short laser pulses. If the interval between pulses exceeds the recombination time of carriers in the semiconductor, a highly conducting layer will periodically appear and disappear on the surface of the slab. This can be interpreted as periodical displacements of the boundary. The basic physical idea was nicely explained in [433]: ”…this effective motion is much more convenient than a mechanical motion, since in a metal mirror only the conduction electrons reflect the electromagnetic waves, whereas a great amount of power would be wasted in the acceleration of the much heavier nuclei.” However, attempts to implement this idea in practice [434] met severe difficulties due to high losses in semiconductor materials. A rough explanation was given in [285,435]. It seems that the necessary condition for the photon generation, in addition to a high value of the slab conductivity, is the big negative change of the real part of the dielectric permeability. Probably, this regime was not reached in the experiments. An idea to use the resonance between the field mode and cyclotron transitions inside a semiconductor heterostructure in a strong and rapidly varying magnetic field was suggested in [436].

5.3. Simulations with Linear and Nonlinear Optical Materials

The main mechanism of the DCE is amplification of vacuum fluctuations due to fast variations of instantaneous eigenfrequences of the field modes. These variations can happen either due to change of real dimensions of the cavity confining the field (DCE in narrow sense) or due to changes of the effective (optical) length, when dielectric properties of the medium inside the cavity depend on time (DCE in wide sense). This second possibility was investigated theoretically by many authors [96,275,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460]. The main problem is how to realize fast variations of the dielectric permeability in real experiments. The idea to use laser beams passing through a material with nonlinear optical properties was considered in [5,429,438,439,440,448], and concrete experimental schemes were proposed in [461,462,463,464]. Although those schemes are rather different (Dezael and Lambrecht [461] and Hizhnyakov [463] considered nonlinear crystals with the second-order nonlinearity, whereas Faccio and Carusotto [462] proposed to exploit the third-order Kerr effect), their common feature is the prediction of generation of infrared [461,462] or even visible [463] photons, whose frequency is of the same order as the frequency of the laser beam. The experiment based on the suggestion [462] was realized recently [465]. A simulation of the DCE in photonic lattices or photonic crystals was suggested in papers [466,467].
The evaluation of a possibility to obtain parametric amplification of the microwave vacuum field, using a reentrant cavity enclosing a nonlinear crystal with a strong third-order nonlinearity, whose refractive index is modulated by near infrared high-intensity laser pulses, was performed in [468,469]. Such a configuration seems to be more adequate for the simulation of the DCE because the effective time-varying length of the cavity is created by infrared quanta, whereas an antenna put inside the cavity can select microwave (RF) quanta only, whose frequency is five orders of magnitude smaller. This could help experimentalists to get rid of various spurious effects due to other possible mechanisms of photon generation.

5.4. Interaction with Detectors

An important part of the experiments on the DCE is how to detect the “Casimir photons”. This problem was considered in [105,411,470,471,472,473,474,475,476,477,478,479,480,481]. Detectors modeled by harmonic oscillators were studied in [105,471,475,476,477,478,479]. Two-level and multilevel detectors were considered in [471,477,481]. Electron beams as detectors were suggested in [472]. Rydberg atoms as detectors were proposed in [474].

6. DCE and Other Quantum Phenomena

Various phenomena connected with the DCE were investigated in [482,483,484,485,486]. In particular, the DCE in curved spacetimes or in the presence of gravitational fields was studied in [487,488,489,490,491]. The role of the dynamical Casimir effect in the cosmological problems was studied in [158,492]. The DCE in an infinite uniformly accelerated medium was considered in [493]. The influence of gravitational fields on the DCE was studied in [494]. Many of these studies are connected with the general problem of the quantum radiation produced by time-varying metrics. It takes the origin in the papers published about 50 years ago: [2,4,43,44,45,495,496,497]. For the recent studies one can see Refs. [454,498] The differences between the DCE and the Unruh effect [499,500,501] were discussed, e.g., in studies [31,80,500,502,503,504,505]. Intersections between the DCE and quantum thermodynamics were discussed in [138,415,417,506]. Interferometers and detectors of weak signals with moving mirrors were considered in [61,507,508]. Possible (although looking fantastic at the moment) applications of the DCE to the space flights were discussed in [509].

6.1. Damping and Decoherence

The necessity of taking into account the effects of damping in the DCE was clear from the very beginning. Various estimations of the influence of these effects on the photon production rate inside the cavity and in quantum circuits were made in [27,102,348,349,510,511,512,513,514,515,516].
The closely connected problem of decoherence due to the DCE was considered for the first time in [517,518]. For other publications in this direction, see, e,g., [380,519,520,521,522].

6.2. Entanglement

The problem of entanglement creation by means of the DCE was put forward in the study [523]. Various aspects of the problem of entanglement in systems with moving mirrors (or their analogs) were considered in Refs. [143,144,333,346,361,363,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539]. The quantum discord in the DCE was the subject of studies [333,540]. Other aspects of the quantum information theory were investigated in connection with the DCE in [541,542,543]. Possible applications of the DCE in quantum simulations were discussed in [544].

6.3. Other Dynamical Effects

The Dynamical Lamb Effect, introduced in [545], consists of the excitation of atoms inside a cavity with moving walls, without creation of real photons in the cavity. For further studies, one can consult papers [348,349,523,533,534,535,546,547,548].
The Dynamical Casimir–Polder Effects are related to the influence of the EM field fluctuations on the interaction between atomic-size objects (or between such objects and walls) in non-stationary situations. The term was coined in papers [549,550]. For further developments, one can consult studies [419,551,552,553,554,555,556,557,558].

7. Conclusions

This short review demonstrates an impressive expansion of the front of research related to the DCE (or its analogs) in many different areas that happened during the past decade. Studies in the area of quantum circuits come first (Section 4.1), stimulated by experiments [320,322,323]. They are followed by many results and suggestions that connect the DCE with the condensed matter (Section 4.2) and atomic physics (Section 4.3). Here, the first experiment was reported in [353]. The first experimental results related to the analog of the DCE in an optical fiber were also obtained recently [465]. Thus, the fascinating dynamical Casimir physics continues to attract many scientists. However, an observation of the “real” DCE in cavities with moving boundaries is still a challenge (or dream).

Funding

This research received no external funding.

Acknowledgments

Partial support of the Brazilian funding agency CNPq is acknowledged. I thank the Guest Editors of this special issue, Giuseppe Ruoso and Roberto Passante, for the invitation to submit this paper and for their great patience waiting for it.

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Moore, G.T. Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity. J. Math. Phys. 1970, 11, 2679–2691. [Google Scholar] [CrossRef]
  2. DeWitt, B.S. Quantum field theory in curved spacetime. Phys. Rep. 1975, 19, 295–357. [Google Scholar] [CrossRef]
  3. Fulling, S.A.; Davies, P.C.W. Radiation from a moving mirror in two-dimensional space-time: Conformal anomaly. Proc. R. Soc. Lond. A 1976, 348, 393–414. [Google Scholar] [CrossRef]
  4. Davies, P.C.W.; Fulling, S.A. Radiation from moving mirrors and from black holes. Proc. R. Soc. Lond. A 1977, 356, 237–257. [Google Scholar] [CrossRef]
  5. Yablonovitch, E. Accelerating reference frame for electromagnetic waves in a rapidly growing plasma: Unruh-Davies-Fulling-De Witt radiation and the nonadiabatic Casimir effect. Phys. Rev. Lett. 1989, 62, 1742–1745. [Google Scholar] [CrossRef]
  6. Dodonov, V.V.; Klimov, A.B.; Man’ko, V.I. Nonstationary Casimir effect and oscillator energy level shift. Phys. Lett. A 1989, 142, 511–513. [Google Scholar] [CrossRef]
  7. Schwinger, J. Casimir energy for dielectrics. Proc. Nat. Acad. Sci. USA 1992, 89, 4091–4093. [Google Scholar] [CrossRef] [Green Version]
  8. Plunien, G.; Müller, B.; Greiner, W. The Casimir effect. Phys. Rep. 1986, 134, 87–193. [Google Scholar] [CrossRef]
  9. Milonni, P.W.; Shih, M.L. Casimir forces. Contemp. Phys. 1992, 33, 313–322. [Google Scholar] [CrossRef]
  10. Milonni, P.W. The Quantum Vacuum: An Introduction to Quantum Electrodynamics; Academic Press: New York, NY, USA, 1993. [Google Scholar]
  11. Lamoreaux, S.K. Resource letter GF-1: Casimir force. Am. J. Phys. 1999, 67, 850–861. [Google Scholar] [CrossRef]
  12. Bordag, M.; Mohideen, U.; Mostepanenko, V.M. New developments in the Casimir effect. Phys. Rep. 2001, 353, 1–205. [Google Scholar] [CrossRef] [Green Version]
  13. Milton, K.A. The Casimir Effect: Physical Manifestations of Zero-Point Energy; World Scientific: Singapore, 2001. [Google Scholar] [CrossRef]
  14. Feinberg, J.; Mann, A.; Revzen, M. Casimir effect: The classical limit. Ann. Phys. 2001, 288, 103–136. [Google Scholar] [CrossRef] [Green Version]
  15. Milton, K.A. The Casimir effect: Recent controversies and progress. J. Phys. A: Math. Gen. 2004, 37, R209–R277. [Google Scholar] [CrossRef] [Green Version]
  16. Lamoreaux, S.K. The Casimir force: Background, experiments, and applications. Rep. Prog. Phys. 2005, 68, 201–236. [Google Scholar] [CrossRef]
  17. Brown-Hayes, M.; Brownell, J.H.; Dalvit, D.A.R.; Kim, W.J.; Lambrecht, A.; Lombardo, F.C.; Mazzitelli, F.D.; Middleman, S.M.; Nesvizhevsky, V.V.; Onofrio, R.; et al. Thermal and dissipative effects in Casimir physics. J. Phys. A: Math. Gen. 2006, 39, 6195–6208. [Google Scholar] [CrossRef] [Green Version]
  18. Farina, C. The Casimir effect: Some aspects. Braz. J. Phys. 2006, 36, 1137–1149. [Google Scholar] [CrossRef] [Green Version]
  19. Buhmann, S.Y.; Welsch, D.-G.; Dispersion forces in macroscopic quantum electrodynamics. Prog. Quantum Electron. 2007, 31, 51–130. [Google Scholar] [CrossRef] [Green Version]
  20. Bordag, M.; Klimchitskaya, G.L.; Mohideen, U.; Mostepanenko, V.M. Advances in the Casimir Effect; Oxford University Press: Oxford, UK, 2009. [Google Scholar] [CrossRef]
  21. Babb, J.F. Casimir effects in atomic, molecular, and optical physics. Adv. At. Mol. Opt. Phys. 2010, 59, 1–20. [Google Scholar] [CrossRef] [Green Version]
  22. Lambrecht, A.; Reynaud, S. Casimir effect: theory and experiments. Int. J. Mod. Phys. A 2012, 27, 1260013. [Google Scholar] [CrossRef]
  23. Palasantzas, G.; Dalvit, D.A.R.; Decca, R.; Svetovoy, V.B.; Lambrecht, A. Special issue on Casimir physics. J. Phys. Condens. Matter 2015, 27. no. 21. [Google Scholar] [CrossRef] [Green Version]
  24. Simpson, W.M.R.; Leonhardt, U. Force of the Quantum Vacuum: An Introduction to Casimir Physics; World Scientific: Singapore, 2015. [Google Scholar] [CrossRef]
  25. Jaekel, M.T.; Reynaud, S. Movement and fluctuations of the vacuum. Rep. Prog. Phys. 1997, 60, 863–887. [Google Scholar] [CrossRef] [Green Version]
  26. Dodonov, V.V. Nonstationary Casimir effect and analytical solutions for quantum fields in cavities with moving boundaries. In Modern Nonlinear Optics, Part 1; Evans, M.W., Ed.; John Wiley and Sons, Inc.: New York, NY, USA, 2001; pp. 309–394. [Google Scholar]
  27. Dodonov, V.V.; Dodonov, A.V. Quantum harmonic oscillator and nonstationary Casimir effect. J. Russ. Laser Res. 2005, 26, 445–483. [Google Scholar] [CrossRef]
  28. Dodonov, V.V. Dynamical Casimir effect: Some theoretical aspects. J. Phys. Conf. Ser. 2009, 161, 012027. [Google Scholar] [CrossRef]
  29. Dodonov, V.V. Current status of the Dynamical Casimir Effect. Phys. Scr. 2010, 82, 038105. [Google Scholar] [CrossRef] [Green Version]
  30. Dalvit, D.A.R.; Maia Neto, P.A.; Mazzitelli, F.D. Fluctuations, dissipation and the dynamical Casimir effect. In Casimir Physics; Dalvit, D., Milonni, P., Roberts, D., da Rosa, F., Eds.; Springer: Berlin, Germany, 2011; pp. 419–457. [Google Scholar] [CrossRef] [Green Version]
  31. Nation, P.D.; Johansson, J.R.; Blencowe, M.P.; Nori, F. Stimulating uncertainty: Amplifying the quantum vacuum with superconducting circuits. Rev. Mod. Phys. 2012, 84, 1–24. [Google Scholar] [CrossRef] [Green Version]
  32. Nicolai, E.L. On transverse vibrations of a portion of a string of uniformly variable length. Ann. Petrograd Polytechn. Inst. 1921, 28, 273–287. (In Russian) [Google Scholar]
  33. Nicolai, E.L. On a dynamical illustration of the pressure of radiation. Philos. Mag. 1925, 49, 171–177. [Google Scholar] [CrossRef]
  34. Havelock, T.H. Some dynamical illustrations of the pressure of radiation and of adiabatic invariance. Philos. Mag. 1924, 47, 754–771. [Google Scholar] [CrossRef]
  35. Balazs, N.L. On the solution of the wave equation with moving boundaries. J. Math. Anal. Appl. 1961, 3, 472–484. [Google Scholar] [CrossRef] [Green Version]
  36. Greenspan, H.P. A string problem. J. Math. Anal. Appl. 1963, 6, 339–348. [Google Scholar] [CrossRef] [Green Version]
  37. Baranov, R.I.; Shirokov, Y.M. Electromagnetic field in an optical resonator with a movable mirror. JETP 1968, 26, 1199–1202. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/26/6/p1199?a=list.
  38. Vesnitskii, A.I. A one-dimensional resonator with movable boundaries. Radiophys. Quant. Electron. 1971, 14, 1124–1129. [Google Scholar] [CrossRef]
  39. Krasilnikov, V.N. Parametric Wave Phenomena in Classical Electrodynamics; St. Petersburg University Publ.: St. Petersburg, Russia, 1996. (In Russian) [Google Scholar]
  40. Vesnitskii, A.I. Waves in Systems with Moving Boundaries and Loads; Fizmatlit: Moscow, Russia, 2001. (In Russian) [Google Scholar]
  41. Schrödinger, E. The proper vibrations of the expanding universe. Physica 1939, 6, 899–912. [Google Scholar] [CrossRef]
  42. Imamura, T. Quantized meson field in a classical gravitational field. Phys. Rev. 1960, 118, 1430–1434. [Google Scholar] [CrossRef]
  43. Parker, L. Quantized fields and particle creation in expanding universes, I. Phys. Rev. 1969, 183, 1057–1068. [Google Scholar] [CrossRef]
  44. Parker, L. Quantized fields and particle creation in expanding universes, II. Phys. Rev. D 1971, 3, 346–356. [Google Scholar] [CrossRef]
  45. Zel’dovich, Y.B. Particle production in cosmology. JETP Lett. 1970, 12, 307–311, Reprinted in Selected Works of Yakov Borisovich Zeldovich. Volume 2: Particles, Nuclei, and the Universe; Ostriker, J.P., Barenblatt, G.I., Sunyaev, R.A., Eds.; Princeton University Press: Princeton, NJ, USA, 1993; pp. 223–227. Available online: http://www.jetpletters.ac.ru/ps/1734/article_26352.shtml.
  46. Vesnitskii, A.I. The inverse problem for a one-dimensional resonator the dimensions of which vary with time. Radiophys. Quant. Electron. 1971, 14, 1209–1215 doiorg/101007/BF01035071. [Google Scholar] [CrossRef]
  47. Castagnino, M.; Ferraro, R. The radiation from moving mirrors: The creation and absorption of particles. Ann. Phys. 1984, 154, 1–23. [Google Scholar] [CrossRef]
  48. Walker, W.R.; Davies, P.C.W. An exactly soluble moving mirror problem. J. Phys. A: Math. Gen. 1982, 15, L477–L480. [Google Scholar] [CrossRef]
  49. Carlitz, R.D.; Willey, R.S. Reflections on moving mirrors. Phys. Rev. D 1987, 36, 2327–2335. [Google Scholar] [CrossRef] [PubMed]
  50. Hotta, M.; Shino, M.; Yoshimura, M. Moving mirror model of Hawking evaporation. Prog. Theor. Phys. 1994, 91, 839–869. [Google Scholar] [CrossRef]
  51. Nikishov, A.I.; Ritus, V.I. Emission of scalar photons by an accelerated mirror in 1+1-space and its relation to the radiation from an electrical charge in classical electrodynamics. JETP 1995, 81, 615–624. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/81/4/p615?a=list.
  52. Ritus, V.I. Functional identity of the spectra of Bose- and Fermi radiation of an accelerated mirror in 1 + 1 space and the spectra of electric and scalar charges in 3 + 1 space, and its relation to radiative reaction. JETP 1996, 83, 282–293. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/83/2/p282?a=list.
  53. Obadia, N.; Parentani, R. Uniformly accelerated mirrors. I. Mean fluxes. Phys. Rev. D 2003, 67, 024021. [Google Scholar] [CrossRef] [Green Version]
  54. Ford, L.H.; Roman, T.A. Energy flux correlations and moving mirrors. Phys. Rev. D 2004, 70, 125008. [Google Scholar] [CrossRef] [Green Version]
  55. Good, M.R.R.; Anderson, P.R.; Evans, C.R. Time dependence of particle creation from accelerating mirrors. Phys. Rev. D 2013, 88, 025023. [Google Scholar] [CrossRef] [Green Version]
  56. Good, M.R.R.; Yelshibekov, K.; Ong, Y.C. On horizonless temperature with an accelerating mirror. JHEP 2017, 3, 013. [Google Scholar] [CrossRef] [Green Version]
  57. Good, M.R.R.; Linder, E.V. Slicing the vacuum: New accelerating mirror solutions of the dynamical Casimir effect. Phys. Rev. D 2017, 96, 125010. [Google Scholar] [CrossRef] [Green Version]
  58. Good, M.R.R.; Linder, E.V. Eternal and evanescent black holes and accelerating mirror analogs. Phys. Rev. D 2018, 97, 065006. [Google Scholar] [CrossRef] [Green Version]
  59. Good, M.R.R.; Linder, E.V. Finite energy but infinite entropy production from moving mirrors. Phys. Rev. D 2019, 99, 025009. [Google Scholar] [CrossRef] [Green Version]
  60. Hosoya, A. Moving mirror effects in hadronic reactions. Prog. Theor. Phys. 1979, 61, 280–293. [Google Scholar] [CrossRef] [Green Version]
  61. Obadia, N.; Parentani, R. Uniformly accelerated mirrors. II. Quantum correlations. Phys. Rev. D 2003, 67, 024022. [Google Scholar] [CrossRef] [Green Version]
  62. Mazzitelli, F.D.; Paz, J.P.; Castagnino, M.A. Fermions between moving boundaries. Phys. Lett. B 1987, 189, 132–136. [Google Scholar] [CrossRef]
  63. Horibe, M. Thermal radiation of fermions by an accelerated wall. Prog. Theor. Phys. 1979, 61, 661–671. [Google Scholar] [CrossRef]
  64. Koehn, M. Solutions of the Klein–Gordon equation in an infinite square-well potential with a moving wall. EPL 2012, 100, 60008. [Google Scholar] [CrossRef]
  65. Bialynicki-Birula, I. Solutions of the d’Alembert and Klein–Gordon equations confined to a region with one fixed and one moving wall. EPL 2013, 101, 60003. [Google Scholar] [CrossRef] [Green Version]
  66. Dodonov, V.V.; Klimov, A.B. Long-time asymptotics of a quantized electromagnetic field in a resonator with oscillating boundary. Phys. Lett. A 1992, 167, 309–313. [Google Scholar] [CrossRef]
  67. Dodonov, V.V.; Klimov, A.B.; Nikonov, D.E. Quantum phenomena in resonators with moving walls. J. Math. Phys. 1993, 34, 2742–2756. [Google Scholar] [CrossRef]
  68. Dalvit, D.A.R.; Mazzitelli, F.D. Renormalization-group approach to the dynamical Casimir effect. Phys. Rev. A 1998, 57, 2113–2119. [Google Scholar] [CrossRef] [Green Version]
  69. Dalvit, D.A.R.; Mazzitelli, F.D. Creation of photons in an oscillating cavity with two moving mirrors. Phys. Rev. A 1999, 59, 3049–3059. [Google Scholar] [CrossRef] [Green Version]
  70. Li, L.; Li, B.Z. Numerical solutions of the generalized Moore’s equations for a one-dimensional cavity with two moving mirrors. Phys. Lett. A 2002, 300, 27–32. [Google Scholar] [CrossRef]
  71. Crocce, M.; Dalvit, D.A.R.; Lombardo, F.C.; Mazzitelli, F.D. 2005 Hertz potentials approach to the dynamical Casimir effect in cylindrical cavities of arbitrary section. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S32–S39. [Google Scholar] [CrossRef] [Green Version]
  72. Dalvit, D.A.R.; Mazzitelli, F.D.; Millán, X.O. The dynamical Casimir effect for different geometries. J. Phys. A: Math. Gen. 2006, 39, 6261–70. [Google Scholar] [CrossRef]
  73. de la Llave, R.; Petrov, N.P. Theory of circle maps and the problem of one-dimensional optical resonator with a periodically moving wall. Phys. Rev. E 1999, 59, 6637–6651. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  74. Petrov, N.P.; de la Llave, R.; Vano, J.A. Torus maps and the problem of a one-dimensional optical resonator with a quasiperiodically moving wall. Phys. D 2003, 180, 140–184. [Google Scholar] [CrossRef]
  75. Petrov, N.P. The dynamical Casimir effect in a periodically changing domain: A dynamical systems approach. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S89–S99. [Google Scholar] [CrossRef]
  76. Wȩgrzyn, P. An optical approach to the dynamical Casimir effect. J. Phys. B: Atom. Mol. Opt. Phys. 2006, 39, 4895–4903. [Google Scholar] [CrossRef] [Green Version]
  77. Wȩgrzyn, P. Exact closed-form analytical solutions for vibrating cavities. J. Phys. B: Atom. Mol. Opt. Phys. 2007, 40, 2621–2640. [Google Scholar] [CrossRef]
  78. Fedotov, A.M.; Lozovik, Y.E.; Narozhny, N.B.; Petrosyan, A.N. 2006 Dynamical Casimir effect in a one-dimensional uniformly contracting cavity. Phys. Rev. A 2006, 74, 013806. [Google Scholar] [CrossRef] [Green Version]
  79. Alves, D.T.; Granhen, E.R. A computer algebra package for calculation of the energy density produced via the dynamical Casimir effect in one-dimensional cavities. Comp. Phys. Commun. 2014, 185, 2101–2114. [Google Scholar] [CrossRef]
  80. Martin, I. Floquet dynamics of classical and quantum cavity fields. Ann. Phys. 2019, 405, 101–129. [Google Scholar] [CrossRef] [Green Version]
  81. Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge Univ. Press: Cambridge, UK, 1982. [Google Scholar]
  82. Beyer, H.; Nitsch, J. A note on a Casimir effect in a uniformly accelerated reference frame. Found. Phys. 1990, 20, 459–469. [Google Scholar] [CrossRef]
  83. Nagatani, Y.; Shigetomi, K. Effective theoretical approach to back reaction of the dynamical Casimir effect in 1+1 dimensions. Phys. Rev. A 2000, 62, 022117. [Google Scholar] [CrossRef] [Green Version]
  84. Dodonov, V.V.; Klimov, A.B.; Man’ko, V.I. Generation of squeezed states in a resonator with a moving wall. Phys. Lett. A 1990, 149, 225–228. [Google Scholar] [CrossRef]
  85. Wu, Y.; Chan, K.W.; Chu, M.-C.; Leung, P.T. Radiation modes of a cavity with a resonantly oscillating boundary. Phys. Rev. A 1999, 59, 1662–1666. [Google Scholar] [CrossRef]
  86. Wȩgrzyn, P.; Róg, T. Vacuum energy of a cavity with a moving boundary. Acta Phys. Pol. B 2001, 32, 129–146. [Google Scholar]
  87. Wȩgrzyn, P. Parametric resonance in a vibrating cavity. Phys. Lett. A 2004, 322, 263–269. [Google Scholar] [CrossRef] [Green Version]
  88. Wȩgrzyn, P.; Róg, T. Photons produced inside a cavity with a moving wall. Acta Phys. Pol. B 2003, 34, 3887–3900. [Google Scholar]
  89. Alves, D.T.; Granhen, E.R.; Lima, M.G.; Silva, H.O.; Rego, A.L.C. Time evolution of the energy density inside a one-dimensional non-static cavity with a vacuum, thermal and a coherent state. J. Phys. Conf. Ser. 2009, 161, 012032. [Google Scholar] [CrossRef]
  90. Alves, D.T.; Granhen, E.R.; Lima, M.G.; Rego, A.L.C. Quantum radiation force on a moving mirror for a thermal and a coherent field. J. Phys. Conf. Ser. 2009, 161, 012033. [Google Scholar] [CrossRef]
  91. Alves, D.T.; Granhen, E.R.; Silva, H.O.; Lima, M.G. Quantum radiation force on the moving mirror of a cavity, with Dirichlet and Neumann boundary conditions for a vacuum, finite temperature, and a coherent state. Phys. Rev. D 2010, 81, 025016. [Google Scholar] [CrossRef]
  92. Alves, D.T.; Granhen, E.R.; Pires, W.P. Quantum radiation reaction force on a one-dimensional cavity with two relativistic moving mirrors. Phys. Rev. D 2010, 82, 045028. [Google Scholar] [CrossRef] [Green Version]
  93. Grinberg, G.A. A method of approach to problems of the theory of heat conduction, diffusion and the wave theory and other similar problems in presence of moving boundaries and its applications to other problems.
  94. Razavy, M.; Terning, J. Quantum radiation in a one-dimensional cavity with moving boundaries. Phys. Rev. D 1985, 31, 307–313. [Google Scholar] [CrossRef]
  95. Calucci, G. Casimir effect for moving bodies. J. Phys. A: Math. Gen. 1992, 25, 3873–3882. [Google Scholar] [CrossRef]
  96. Law, C.K. Effective Hamiltonian for the radiation in a cavity with a moving mirror and a time-varying dielectric medium. Phys. Rev. A 1994, 49, 433–437. [Google Scholar] [CrossRef]
  97. Dodonov, V.V. Resonance photon generation in a vibrating cavity. J. Phys. A: Math. Gen. 1998, 31, 9835–9854. [Google Scholar] [CrossRef]
  98. Schützhold, R.; Plunien, G.; Soff, G. Trembling cavities in the canonical approach. Phys. Rev. A 1998, 57, 2311–2318. [Google Scholar] [CrossRef] [Green Version]
  99. Ji, J.-Y.; Jung, H.-H.; Soh, K.-S. Interference phenomena in the photon production between two oscillating walls. Phys. Rev. A 1998, 57, 4952–4955. [Google Scholar] [CrossRef] [Green Version]
  100. Law, C.K. Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. Phys. Rev. A 1995, 51, 2537–2541. [Google Scholar] [CrossRef]
  101. Landau, L.D.; Lifshitz, E.M. Mechanics; Pergamon: Oxford, UK, 1969. [Google Scholar]
  102. Lambrecht, A.; Jaekel, M.T.; Reynaud, S. Motion induced radiation from a vibrating cavity. Phys. Rev. Lett. 1996, 77, 615–618. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  103. Dodonov, V.V. Resonance excitation and cooling of electromagnetic modes in a cavity with an oscillating wall. Phys. Lett. A 1996, 213, 219–225. [Google Scholar] [CrossRef]
  104. Wȩgrzyn, P. Quantum energy in a vibrating cavity. Mod. Phys. Lett. A 2004, 19, 769–774. [Google Scholar] [CrossRef] [Green Version]
  105. Dodonov, V.V.; Klimov, A.B. Generation and detection of photons in a cavity with a resonantly oscillating boundary. Phys. Rev. A 1996, 53, 2664–2682. [Google Scholar] [CrossRef] [PubMed]
  106. Law, C.K. Resonance response of the quantum vacuum to an oscillating boundary. Phys. Rev. Lett. 1994, 73, 1931–1934. [Google Scholar] [CrossRef]
  107. Cole, C.K.; Schieve, W.C. Radiation modes of a cavity with a moving boundary. Phys. Rev. A 1995, 52, 4405–4415. [Google Scholar] [CrossRef]
  108. Méplan, O.; Gignoux, C. Exponential growth of the energy of a wave in a 1D vibrating cavity: Application to the quantum vacuum. Phys. Rev. Lett. 1996, 76, 408–410. [Google Scholar] [CrossRef]
  109. Klimov, A.B.; Altuzar, V. Spectrum of photons generated in a one-dimensional cavity with oscillating boundary. Phys. Lett. A 1997, 226, 41–45. [Google Scholar] [CrossRef]
  110. Cooper, J. Long time behavior and energy growth for electromagnetic waves reflected by a moving boundary. IEEE Trans. Antennas Prop. 1993, 41, 1365–1370. [Google Scholar] [CrossRef]
  111. Dittrich, J.; Duclos, P.; Šeba, P. nstability in a classical periodically driven string. Phys. Rev. E 1994, 49, 3535–3538. [Google Scholar] [CrossRef]
  112. Dittrich, J.; Duclos, P.; Gonzalez, N. Stability and instability of the wave equation solutions in a pulsating domain. Rev. Math. Phys. 1998, 10, 925–962. [Google Scholar] [CrossRef]
  113. Wu, K.; Zhu, W.D. Parametric instability in a taut string with a periodically moving boundary. ASME J. Appl. Mech. 2014, 81, 061002. [Google Scholar] [CrossRef]
  114. Dodonov, V.V.; Andreata, M.A. Squeezing and photon distribution in a vibrating cavity. J. Phys. A: Math. Gen. 1999, 32, 6711–6726. [Google Scholar] [CrossRef] [Green Version]
  115. Andreata, M.A.; Dodonov, V.V. Energy density and packet formation in a vibrating cavity. J. Phys. A: Math. Gen. 2000, 33, 3209–3223. [Google Scholar] [CrossRef]
  116. Lambrecht, A.; Jaekel, M.-T.; Reynaud, S. Generating photon pulses with an oscillating cavity. Europhys. Lett. 1998, 43, 147–152. [Google Scholar] [CrossRef] [Green Version]
  117. Lambrecht, A.; Jaekel, M.-T.; Reynaud, S. Frequency up-converted radiation from a cavity moving in vacuum. Eur. Phys. J. D 1998, 3, 95–104. [Google Scholar] [CrossRef] [Green Version]
  118. Lambrecht, A. Electromagnetic pulses from an oscillating high-finesse cavity: Possible signatures for dynamic Casimir effect experiments. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S3–S10. [Google Scholar] [CrossRef]
  119. Rosanov, N.N.; Fedorov, E.G.; Matskovsky, A.A. Parametric generation of radiation in a dynamic cavity with frequency dispersion. Quantum Electron. 2016, 46, 13–15. [Google Scholar] [CrossRef]
  120. Rosanov, N.N.; Fedorov, E.G. Influence of frequency detunings and form of the initial field distribution on parametric generation of radiation in a dynamic cavity. Opt. Spectrosc. 2016, 120, 803–807. [Google Scholar] [CrossRef]
  121. Fedorov, E.G.; Rosanov, N.N.; Malomed, B.A. The evolution of field distribution in a dynamic cavity. Opt. Spectrosc. 2017, 123, 454–462. [Google Scholar] [CrossRef]
  122. Ruser, M. Vibrating cavities: A numerical approach. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S100–S115. [Google Scholar] [CrossRef] [Green Version]
  123. Ruser, M. Numerical approach to the dynamical Casimir effect. J. Phys. A: Math. Gen. 2006, 39, 6711–6723. [Google Scholar] [CrossRef] [Green Version]
  124. Alves, D.T.; Granhen, E.R.; Silva, H.O.; Lima, M.G. Exact behavior of the energy density inside a one-dimensional oscillating cavity with a thermal state. Phys. Lett. A 2010, 374, 3899–3907. [Google Scholar] [CrossRef] [Green Version]
  125. Ji, J.-Y.; Jung, H.-H.; Park, J.-W.; Soh, K.-S. Production of photons by the parametric resonance in the dynamical Casimir effect. Phys. Rev. A 1997, 56, 4440–4444. [Google Scholar] [CrossRef] [Green Version]
  126. Ji, J.-Y.; Soh, K.-S.; Cai, R.-G.; Kim, S.P. Electromagnetic fields in a three-dimensional cavity and in a waveguide with oscillating walls. J. Phys. A: Math. Gen. 1998, 31, L457–L462. [Google Scholar] [CrossRef] [Green Version]
  127. Villar, P.I.; Soba, A.; Lombardo, F.C. Numerical approach to simulating interference phenomena in a cavity with two oscillating mirrors. Phys. Rev. A 2017, 95, 032115. [Google Scholar] [CrossRef] [Green Version]
  128. Alves, D.T.; Farina, C.; Granhen, E.R. Dynamical Casimir effect in a resonant cavity with mixed boundary conditions. Phys. Rev. A 2006, 73, 063818. [Google Scholar] [CrossRef]
  129. Alves, D.T.; Granhen, E.R. Energy density and particle creation inside an oscillating cavity with mixed boundary conditions. Phys. Rev. A 2008, 77, 015808. [Google Scholar] [CrossRef]
  130. Ordaz-Mendoza, B.E.; Yelin, S.F. Resonant frequency ratios for the dynamical Casimir effect. Phys. Rev. A 2019, 100, 033815. [Google Scholar] [CrossRef]
  131. Barton, G.; Calogeracos, A. On the quantum electrodynamics of a dispersive mirror. I. Mass shifts, radiation, and radiative reaction. Ann. Phys. 1995, 238, 227–267. [Google Scholar] [CrossRef]
  132. Calogeracos, A.; Barton, G. On the quantum electrodynamics of a dispersive mirror. II. The boundary condition and the applied force via Diracs theory of constraints. Ann. Phys. 1995, 238, 268–285. [Google Scholar] [CrossRef]
  133. Oku, K.; Tsuchida, Y. Back-reaction in the moving mirror effects. Prog. Theor. Phys. 1979, 62, 1756–1767. [Google Scholar] [CrossRef] [Green Version]
  134. Colanero, K.; Chu, M.C. Energy focusing inside a dynamical cavity. Phys. Rev. E 2000, 62, 8663–8667. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  135. Cole, C.K.; Schieve, W.C. Resonant energy exchange between a moving boundary and radiation modes of a cavity. Phys. Rev. A 2001, 64, 023813. [Google Scholar] [CrossRef]
  136. Saito, H.; Hyuga, H. Dynamical Casimir effect without boundary conditions. Phys. Rev. A 2002, 65, 053804. [Google Scholar] [CrossRef] [Green Version]
  137. Sinyukov, Y.M. Radiation processes in quantum systems with boundary. J. Phys. A: Math. Gen. 1982, 15, 2533–2545. [Google Scholar] [CrossRef]
  138. Helfer, A.D. Moving mirrors and thermodynamic paradoxes. Phys. Rev. D 2001, 63, 025016. [Google Scholar] [CrossRef] [Green Version]
  139. Butera, S.; Passante, R. Field fluctuations in a one-dimensional cavity with a mobile wall. Phys. Rev. Lett. 2013, 111, 060403. [Google Scholar] [CrossRef] [Green Version]
  140. Armata, F.; Passante, R. Vacuum energy densities of a field in a cavity with a mobile boundary. Phys. Rev. D 2015, 91, 025012. [Google Scholar] [CrossRef] [Green Version]
  141. Armata, F.; Kim, M.S.; Butera, S.; Rizzuto, L.; Passante, R. Nonequilibrium dressing in a cavity with a movable reflecting mirror. Phys. Rev. D 2017, 96, 045007. [Google Scholar] [CrossRef] [Green Version]
  142. Mancini, S.; Man’ko, V.I.; Tombesi, P. Ponderomotive control of quantum macroscopic coherence. Phys. Rev. A 1997, 55, 3042–3050. [Google Scholar] [CrossRef] [Green Version]
  143. Mancini, S.; Giovannetti, V.; Vitali, D.; Tombesi, P. Entangling macroscopic oscillators exploiting radiation pressure. Phys. Rev. Lett. 2002, 88, 120401. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  144. Vitali, D.; Gigan, S.; Ferreira, A.; Böhm, H.R.; Tombesi, P.; Guerreiro, A.; Vedral, V.; Zeilinger, A.; Aspelmeyer, M. Optomechanical entanglement between a movable mirror and a cavity field. Phys. Rev. Lett. 2007, 98, 030405. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  145. Aspelmeyer, M.; Kippenberg, T.J.; Marquardt, F. Cavity optomechanics. Rev. Mod. Phys. 2014, 86, 1391–1452. [Google Scholar] [CrossRef]
  146. Mahajan, S.; Aggarwal, N.; Kumar, T.; Bhattacherjee, A.B.; Mohan, M. Dynamics of an optomechanical resonator containing a quantum well induced by periodic modulation of cavity field and external laser beam. Can. J. Phys. 2015, 93, 716–724. [Google Scholar] [CrossRef] [Green Version]
  147. Macrì, V.; Ridolfo, A.; Di Stefano, O.; Kockum, A.F.; Nori, F.; Savasta, S. Nonperturbative dynamical Casimir effect in optomechanical systems: Vacuum Casimir–Rabi splittings. Phys. Rev. X 2018, 8, 011031. [Google Scholar] [CrossRef] [Green Version]
  148. Jansen, E.; Machado, J.D.P.; Blanter, Y.M. Realization of a degenerate parametric oscillator in electromechanical systems. Phys. Rev. B 2019, 99, 045401. [Google Scholar] [CrossRef] [Green Version]
  149. Di Stefano, O.; Settineri, A.; Macrì, V.; Ridolfo, A.; Stassi, R.; Kockum, A.F.; Savasta, S.; Nori, F. Interaction of mechanical oscillators mediated by the exchange of virtual photon pairs. Phys. Rev. Lett. 2019, 122, 030402. [Google Scholar] [CrossRef] [Green Version]
  150. Butera, S.; Carusotto, I. Mechanical backreaction effect of the Casimir emission. Phys. Rev. A 2019, 99, 053815. [Google Scholar] [CrossRef] [Green Version]
  151. Settineri, A.; Macrì, V.; Garziano, L.; Di Stefano, O.; Nori, F.; Savasta, S. Conversion of mechanical noise into correlated photon pairs: Dynamical Casimir effect from an incoherent mechanical drive. Phys. Rev. A 2019, 100, 022501. [Google Scholar] [CrossRef] [Green Version]
  152. Del Grosso, N.F.; Lombardo, F.C.; Villar, P.I. Photon generation via the dynamical Casimir effect in an optomechanical cavity as a closed quantum system. Phys. Rev. A 2019, 100, 062516. [Google Scholar] [CrossRef] [Green Version]
  153. Jaekel, M.T.; Reynaud, S. Fluctuations and dissipation for a mirror in vacuum. Quant. Opt. 1992, 4, 39–53. [Google Scholar] [CrossRef]
  154. Nicolaevici, N. Quantum radiation from a partially reflecting moving mirror. Class. Quant. Grav. 2001, 18, 619–628. [Google Scholar] [CrossRef]
  155. Obadia, N.; Parentani, R. Notes on moving mirrors. Phys. Rev. D 2001, 64, 044019. [Google Scholar] [CrossRef] [Green Version]
  156. Haro, J.; Elizalde, E. Physically sound Hamiltonian formulation of the dynamical Casimir effect. Phys. Rev. D 2007, 76, 065001. [Google Scholar] [CrossRef] [Green Version]
  157. Haro, J.; Elizalde, E. Black hole collapse simulated by vacuum fluctuations with a moving semitransparent mirror. Phys. Rev. D 2008, 77, 045011. [Google Scholar] [CrossRef] [Green Version]
  158. Elizalde, E. 2008 Dynamical Casimir effect with semi-transparent mirrors, and cosmology. J. Phys. A: Math. Theor. 2008, 41, 164061. [Google Scholar] [CrossRef] [Green Version]
  159. Nicolaevici, N. Semitransparency effects in the moving mirror model for Hawking radiation. Phys. Rev. D 2009, 80, 125003. [Google Scholar] [CrossRef]
  160. Sarabadani, J.; Miri, M.F. Motion-induced radiation in a cavity with conducting and permeable plates. Phys. Rev. A 2007, 75, 055802. [Google Scholar] [CrossRef]
  161. Mintz, B.; Farina, C.; Maia Neto, P.A.; Rodrigues, R.B. Casimir forces for moving boundaries with Robin conditions. J. Phys. A: Math. Gen. 2006, 39, 6559–6565. [Google Scholar] [CrossRef]
  162. Mintz, B.; Farina, C.; Maia Neto, P.A.; Rodrigues, R.B. Particle creation by a moving boundary with a Robin boundary condition. J. Phys. A: Math. Gen. 2006, 39, 11325–11333. [Google Scholar] [CrossRef]
  163. Silva, H.O.; Farina, C. Simple model for the dynamical Casimir effect for a static mirror with time-dependent properties. Phys. Rev. D 2011, 84, 045003. [Google Scholar] [CrossRef] [Green Version]
  164. Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D. Vacuum fluctuations and generalized boundary conditions. Phys. Rev. D 2013, 87, 105008. [Google Scholar] [CrossRef] [Green Version]
  165. Crocce, M.; Dalvit, D.A.R.; Lombardo, F.C.; Mazzitelli, F.D. Model for resonant photon creation in a cavity with time-dependent conductivity. Phys. Rev. A 2004, 70, 033811. [Google Scholar] [CrossRef] [Green Version]
  166. Naylor, W.; Matsuki, S.; Nishimura, T.; Kido, Y. Dynamical Casimir effect for TE and TM modes in a resonant cavity bisected by a plasma sheet. Phys. Rev. A 2009, 80, 043835. [Google Scholar] [CrossRef] [Green Version]
  167. Kulagin, V.V.; Cherepenin, V.A. Generation of squeezed states on reflection of light from a system of free electrons. JETP Lett. 1996, 63, 170–175. [Google Scholar] [CrossRef]
  168. Kulagin, V.V.; Cherepenin, V.A.; Hur, M.S.; Suk, H. Flying mirror model for interaction of a super-intense nonadiabatic laser pulse with a thin plasma layer: Dynamics of electrons in a linearly polarized external field. Phys. Plasmas 2007, 14, 113101. [Google Scholar] [CrossRef] [Green Version]
  169. Bulanov, S.V.; Esirkepov, T.Zh.; Kando, M.; Pirozhkov, A.S.; Rosanov, N.N. Relativistic mirrors in plasmas. Novel results and perspectives. Phys. Uspekhi 2013, 56, 429–464. [Google Scholar] [CrossRef]
  170. Silva, J.D.L.; Braga, A.N.; Alves, D.T. Dynamical Casimir effect with δδ mirrors. Phys. Rev. D 2016, 94, 105009. [Google Scholar] [CrossRef] [Green Version]
  171. Lima Silva, J.D.; Braga, A.N.; Rego, A.L.C.; Alves, D.T. Interference phenomena in the dynamical Casimir effect for a single mirror with Robin conditions. Phys. Rev. D 2015, 92, 025040. [Google Scholar] [CrossRef]
  172. Brown, E.G.; Louko, J. Smooth and sharp creation of a Dirichlet wall in 1+1 quantum field theory: How singular is the sharp creation limit? JHEP 2015, 8, 061. [Google Scholar] [CrossRef] [Green Version]
  173. Harada, T.; Kinoshita, S.; Miyamoto, U. Vacuum excitation by sudden appearance and disappearance of a Dirichlet wall in a cavity. Phys. Rev. D 2016, 94, 025006. [Google Scholar] [CrossRef] [Green Version]
  174. Miyamoto, U. Explosive particle creation by instantaneous change of boundary conditions. Phys. Rev. D 2019, 99, 025012. [Google Scholar] [CrossRef] [Green Version]
  175. Sassaroli, E.; Srivastava, Y.N.; Widom, A. Photon production by the Dynamical casimir effect. Phys. Rev. A 1994, 50, 1027–1034. [Google Scholar] [CrossRef] [PubMed]
  176. Galley, C.R.; Behunin, R.O.; Hu, B.L. Oscillator-field model of moving mirrors in quantum optomechanics. Phys. Rev. A 2013, 87, 043832. [Google Scholar] [CrossRef] [Green Version]
  177. Wang, Q.; Unruh, W.G. Motion of a mirror under infinitely fluctuating quantum vacuum stress. Phys. Rev. D 2014, 89, 085009. [Google Scholar] [CrossRef] [Green Version]
  178. Volovik, G.E.; Zubkov, M.A. Mirror as polaron with internal degrees of freedom. Phys. Rev. D 2014, 90, 087702. [Google Scholar] [CrossRef] [Green Version]
  179. Wang, Q.; Unruh, W.G. Mirror moving in quantum vacuum of a massive scalar field. Phys. Rev. D 2015, 92, 063520. [Google Scholar] [CrossRef] [Green Version]
  180. Lin, S.-Y. Unruh-DeWitt detectors as mirrors: Dynamical reflectivity and Casimir effect. Phys. Rev. D 2018, 98, 105010. [Google Scholar] [CrossRef] [Green Version]
  181. Candelas, P.; Deutsch, D. On the vacuum stress induced by uniform acceleration or supporting the ether. Proc. R. Soc. Lond. A 1977, 354, 79–99. [Google Scholar] [CrossRef]
  182. Frolov, V.P.; Serebriany, E.M. Quantum effects in systems with accelerated mirrors. J. Phys. A: Math. Gen. 1979, 12, 2415–2428. [Google Scholar] [CrossRef]
  183. Frolov, V.P.; Serebriany, E.M. Quantum effects in systems with accelerated mirrors: II. electromagnetic field. J. Phys. A: Math. Gen. 1980, 13, 3205–3211. [Google Scholar] [CrossRef]
  184. Anderson, W.G.; Israel, W. Quantum flux from a moving spherical mirror. Phys. Rev. D 1999, 60, 084003. [Google Scholar] [CrossRef] [Green Version]
  185. Frolov, V.; Singh, D. Quantum radiation of uniformly accelerated spherical mirrors. Class. Quantum Grav. 2001, 18, 3025–3038. [Google Scholar] [CrossRef] [Green Version]
  186. Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D. 2007 Quantum dissipative effects in moving mirrors: A functional approach. Phys. Rev. D 2007, 76, 085007. [Google Scholar] [CrossRef] [Green Version]
  187. Ford, L.H.; Vilenkin, A. Quantum radiation by moving mirrors. Phys. Rev. D 1982, 25, 2569–2575. [Google Scholar] [CrossRef]
  188. Maia Neto, P.A. Vacuum radiation pressure on moving mirrors. J. Phys. A: Math. Gen. 1994, 27, 2167–2180. [Google Scholar] [CrossRef]
  189. Maia Neto, P.A.; Machado, L.A.S. Radiation reaction force for a mirror in vacuum. Braz. J. Phys. 1995, 25, 324–334. [Google Scholar]
  190. Maia Neto, P.A.; Machado, L.A.S. Quantum radiation generated by a moving mirror in free space. Phys. Rev. A 1996, 54, 3420–3427. [Google Scholar] [CrossRef]
  191. Mendonça, J.P.F.; Maia Neto, P.A.; Takakura, F.I. Quantum photon emission from a moving mirror in the nonperturbative regime. Opt. Commun. 1999, 160, 335–343. [Google Scholar] [CrossRef] [Green Version]
  192. Miri, F.; Golestanian, R. Motion-induced radiation from a dynamically deforming mirror. Phys. Rev. A 1999, 59, 2291–2294. [Google Scholar] [CrossRef] [Green Version]
  193. Montazeri, M.; Miri, M.F. Motion-induced radiation from a dynamically deforming mirror: Neumann boundary condition. Phys. Rev. A 2005, 71, 063814. [Google Scholar] [CrossRef]
  194. Sarabadani, J.; Miri, M.F. Mechanical response of the quantum vacuum to dynamic deformations of a cavity. Phys. Rev. A 2006, 74, 023801. [Google Scholar] [CrossRef]
  195. Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D.; Remaggi, M.L. Quantum dissipative effects in graphenelike mirrors. Phys. Rev. D 2013, 88, 105004. [Google Scholar] [CrossRef] [Green Version]
  196. Fosco, C.D.; Mazzitelli, F.D. Radiation from a moving planar dipole layer: Patch potentials versus dynamical Casimir effect. Phys. Rev. A 2014, 89, 062513. [Google Scholar] [CrossRef] [Green Version]
  197. Fosco, C.D.; Lombardo, F.C. Oscillating dipole layer facing a conducting plane: A classical analogue of the dynamical Casimir effect. Eur. Phys. J. C 2015, 75, 598. [Google Scholar] [CrossRef] [Green Version]
  198. Stargen, D.J.; Kothawala, D.; Sriramkumar, L. Moving mirrors and the fluctuation-dissipation theorem. Phys. Rev. D 2016, 94, 025040. [Google Scholar] [CrossRef] [Green Version]
  199. Arkhipov, M.V.; Babushkin, I.; Pul’kin, N.S.; Arkhipov, R.M.; Rosanov, N.N. On the emission of radiation by an isolated vibrating metallic mirror. Opt. Spectrosc. 2017, 122, 670–674. [Google Scholar] [CrossRef]
  200. Fosco, C.D.; Remaggi, M.L.; Rodríguez, M.C. Vacuum fluctuation effects due to an Abelian gauge field in 2+1 dimensions, in the presence of moving mirrors. Phys. Lett. B 2019, 797, 134838. [Google Scholar] [CrossRef]
  201. Barton, G.; Eberlein, C. On quantum radiation from a moving body with finite refractive index. Ann. Phys. 1993, 227, 222–274. [Google Scholar] [CrossRef]
  202. Salamone, G.M. Virtual-photon-cloud creation and emission of radiation from a dielectric slab in arbitrary motion. Phys. Rev. A 1994, 49, 2280–2289. [Google Scholar] [CrossRef] [PubMed]
  203. Salamone, G.M.; Barton, G. Quantum radiative reaction on a dispersive mirror in one dimension. Phys. Rev. A 1995, 51, 3506–3512. [Google Scholar] [CrossRef] [PubMed]
  204. Barton, G. The quantum radiation from mirrors moving sideways. Ann. Phys. 1996, 245, 361–388. [Google Scholar] [CrossRef] [Green Version]
  205. Barton, G.; North, C.A. Peculiarities of quantum radiation in three dimensions from moving mirrors with high refractive index. Ann. Phys. 1996, 252, 72–114. [Google Scholar] [CrossRef]
  206. Gütig, R.; Eberlein, C. Quantum radiation from moving dielectrics in two, three, and more spatial dimensions. J. Phys. A: Math. Gen. 1998, 31, 6819–6838. [Google Scholar] [CrossRef] [Green Version]
  207. Frolov, V.; Singh, D. Quantum effects in the presence of expanding semi-transparent spherical mirrors. Class. Quant. Grav. 1999, 16, 3693–3716. [Google Scholar] [CrossRef] [Green Version]
  208. Frolov, V.; Singh, D. Quantum radiation of a uniformly accelerated refractive body. Class. Quantum Grav. 2000, 17, 3905–3916. [Google Scholar] [CrossRef] [Green Version]
  209. Braginsky, V.B.; Khalili, F.Y. Friction and fluctuations produced by the quantum ground state. Phys. Lett. A 1991, 161, 197–201. [Google Scholar] [CrossRef]
  210. Jaekel, M.T.; Reynaud, S. Motional Casimir force. J. Phys. I (France) 1992, 2, 149–165. [Google Scholar] [CrossRef]
  211. Jaekel, M.T.; Reynaud, S. Causality, stability and passivity for a mirror in vacuum. Phys. Lett. A 1992, 167, 227–232. [Google Scholar] [CrossRef] [Green Version]
  212. Jaekel, M.T.; Reynaud, S. Friction and inertia for a mirror in a thermal field. Phys. Lett. A 1993, 172, 319–324. [Google Scholar] [CrossRef] [Green Version]
  213. Jaekel, M.T.; Reynaud, S. Inertia of Casimir energy. J. Phys. I (France) 1993, 3, 1093–1104. [Google Scholar] [CrossRef] [Green Version]
  214. Eberlein, C. Radiation-reaction force on a moving mirror. J. Phys. I (France) 1993, 3, 2151–2159. [Google Scholar] [CrossRef]
  215. Maia Neto, P.A.; Reynaud, S. Dissipative force on a sphere moving in vacuum. Phys. Rev. A 1993, 47, 1639–1646. [Google Scholar] [CrossRef]
  216. Alves, D.T.; Farina, C.; Maia Neto, P.A. Dynamical Casimir effect with Dirichlet and Neumann boundary conditions. J. Phys. A: Math. Gen. 2003, 36, 11333–11342. [Google Scholar] [CrossRef]
  217. Alves, D.T.; Granhen, E.R.; Lima, M.G. Quantum radiation force on a moving mirror with Dirichlet and Neumann boundary conditions for a vacuum, finite temperature, and a coherent state. Phys. Rev. D 2008, 77, 125001. [Google Scholar] [CrossRef] [Green Version]
  218. Teodorovich, E.V. On the contribution of macroscopic van Der Waals interactions to frictional force. Proc. R. Soc. Lond. A 1978, 362, 71–77. [Google Scholar] [CrossRef]
  219. Levitov, L.S. Van der Waals friction. Europhys. Lett. 1989, 8, 499–504. [Google Scholar] [CrossRef]
  220. Polevoǐ, V.G. Tangential molecular forces caused between moving bodies by a fluctuating electromagnetic field. JETP 1990, 71, 1119–1124. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/71/6/p1119?a=list.
  221. Mkrtchian, V.E. Interaction between moving macroscopic bodies: viscosity of the electromagnetic vacuum. Phys. Lett. A 1995, 207, 299–302. [Google Scholar] [CrossRef]
  222. Golestanian, R.; Kardar, M. Mechanical response of vacuum. Phys. Rev. Lett. 1997, 78, 3421–3425. [Google Scholar] [CrossRef] [Green Version]
  223. Pendry, J.B. Shearing the vacuum—quantum friction. J. Phys. Condens. Matter 1997, 9, 10301–10320. [Google Scholar] [CrossRef]
  224. Golestanian, R.; Kardar, M. Path integral approach to the dynamic Casimir effect with fluctuating boundaries. Phys. Rev. A 1998, 58, 1713–1722. [Google Scholar] [CrossRef] [Green Version]
  225. Volokitin, A.I.; Persson, B.N.J. Theory of friction: The contribution from a fluctuating electromagnetic field. J. Phys. Condens. Matter 1999, 11, 345–359. [Google Scholar] [CrossRef]
  226. Kardar, M.; Golestanian, R. The “friction” of vacuum and other fluctuation-induced forces. Rev. Mod. Phys. 1999, 71, 1233–1245. [Google Scholar] [CrossRef] [Green Version]
  227. Kenneth, O.; Nussinov, S. Small object limit of the Casimir effect and the sign of the Casimir force. Phys. Rev. D 2002, 65, 085014. [Google Scholar] [CrossRef] [Green Version]
  228. Volokitin, A.I.; Persson, B.N.J. Near-field radiative heat transfer and noncontact friction. Rev. Mod. Phys. 2007, 79, 1291–1329. [Google Scholar] [CrossRef]
  229. Schaich, W.L.; Harris, J. Dynamic corrections to Van der Waals potentials. J. Phys. F Met. Phys. 1981, 11, 65–78. [Google Scholar] [CrossRef]
  230. Philbin, T.G.; Leonhardt, U. No quantum friction between uniformly moving plates. New J. Phys. 2009, 11, 033035. [Google Scholar] [CrossRef]
  231. Pendry, J.B. Quantum friction–fact or fiction? New J. Phys. 2010, 12, 033028. [Google Scholar] [CrossRef]
  232. Leonhardt, U. Comment on ‘Quantum Friction–Fact or Fiction?’. New J. Phys. 2010, 12, 068001. [Google Scholar] [CrossRef]
  233. Pendry, J.B. Reply to comment on ‘Quantum friction-fact or fiction?’. New J. Phys. 2010, 12, 068002. [Google Scholar] [CrossRef]
  234. Volokitin, A.I.; Persson, B.N.J. Comment on ‘No quantum friction between uniformly moving plates’. New J. Phys. 2011, 13, 068001. [Google Scholar] [CrossRef]
  235. Philbin, T.G.; Leonhardt, U. Reply to comment on ‘No quantum friction between uniformly moving plates’. New J. Phys. 2011, 13, 068002. [Google Scholar] [CrossRef]
  236. Dedkov, G.V.; Kyasov, A.A. Conservative-dissipative forces and heating mediated by fluctuation electromagnetic field: Two plates in relative nonrelativistic motion. Surf. Sci. 2010, 604, 562–567. [Google Scholar] [CrossRef] [Green Version]
  237. Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D. Quantum dissipative effects in moving imperfect mirrors: Sidewise and normal motions. Phys. Rev. D 2011, 84, 025011. [Google Scholar] [CrossRef] [Green Version]
  238. Despoja, V.; Echenique, P.M.; Šunjić, M. Nonlocal microscopic theory of quantum friction between parallel metallic slabs. Phys. Rev. B 2011, 83, 205424. [Google Scholar] [CrossRef] [Green Version]
  239. Horsley, S.A.R. Canonical quantization of the electromagnetic field interacting with a moving dielectric medium. Phys. Rev. A 2012, 86, 023830. [Google Scholar] [CrossRef] [Green Version]
  240. Maslovski, S.I.; Silveirinha, M.G. Quantum friction on monoatomic layers and its classical analog. Phys. Rev. B 2013, 88, 035427. [Google Scholar] [CrossRef]
  241. Mkrtchian, V.E.; Henkel, C. On non-equilibrium photon distributions in the Casimir effect. Ann. Phys. 2014, 526, 87–101. [Google Scholar] [CrossRef] [Green Version]
  242. Høye, J.S.; Brevik, I. Casimir friction at zero and finite temperatures. Eur. Phys. J. D 2014, 68, 61. [Google Scholar] [CrossRef] [Green Version]
  243. Silveirinha, M.G. Theory of quantum friction. New J. Phys. 2014, 16, 063011. [Google Scholar] [CrossRef] [Green Version]
  244. Volokitin, A.I.; Persson, B.N.J. Quantum Vavilov-Cherenkov radiation from shearing two transparent dielectric plates. Phys. Rev. B 2016, 93, 035407. [Google Scholar] [CrossRef] [Green Version]
  245. Milton, K.A.; Høye, J.S.; Brevik, I. The reality of Casimir friction. Symmetry 2016, 8, 29. [Google Scholar] [CrossRef] [Green Version]
  246. Farías, M.B.; Fosco, C.D.; Lombardo, F.C.; Mazzitelli, F.D. Quantum friction between graphene sheets. Phys. Rev. D 2017, 95, 065012. [Google Scholar] [CrossRef] [Green Version]
  247. Dedkov, G.V.; Kyasov, A.A. Friction Force and Radiative Heat Exchange in a System of Two Parallel Plates in Relative Motion: Corollaries of the Levine–Polevoi–Rytov Theory. Phys. Solid State 2018, 60, 2349–2357. [Google Scholar] [CrossRef]
  248. Despoja, V.; Echenique, P.M.; Šunjić, M. Quantum friction between oscillating crystal slabs: Graphene monolayers on dielectric substrates. Phys. Rev. B 2018, 98, 125405. [Google Scholar] [CrossRef] [Green Version]
  249. Butera, S.; Carusotto, I. Quantum fluctuations of the friction force induced by the dynamical Casimir emission. EPL 2019, 128, 24002. [Google Scholar] [CrossRef]
  250. Farías, M.B.; Lombardo, F.C.; Soba, A.; Villar, P.I.; Decca, R.S. Towards detecting traces of non-contact quantum friction in the corrections of the accumulated geometric phase. arXiv 2019, arXiv:1912.05513. [Google Scholar] [CrossRef]
  251. Zel’dovich, Ya.B. Generation of waves by a rotating body. JETP Lett. 1971, 14, 180–181, Reprinted in Selected Works of Yakov Borisovich Zeldovich. Volume 2: Particles, Nuclei, and the Universe; Ostriker, J.P., Barenblatt, G.I., Sunyaev, R.A., Eds.; Princeton University Press: Princeton, NJ, USA, 1993; pp. 251–254. Available online: http://www.jetpletters.ac.ru/ps/1604/article_24607.shtml.
  252. Zel’dovich, Ya.B. Amplification of cylindrical electromagnetic waves reflected from a rotating body. JETP 1972, 35, 1085–1087. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/35/6/p1085?a=list.
  253. Starobinskiǐ, A.A. Amplification of waves during reflection from a rotating “black hole”. JETP 1973, 37, 28–32. Available online: http://www.jetp.ac.ru/cgi-bin/e/index/e/37/1/p28?a=list.
  254. Press, W.H.; Teukolsky, S.A. Floating orbits, superradiant scattering and black-hole bomb. Nature 1972, 238, 211–212. [Google Scholar] [CrossRef]
  255. Unruh, W.G. Second quantization in the Kerr metric. Phys. Rev. D 1974, 10, 3194–3205. [Google Scholar] [CrossRef]
  256. Bekenstein, J.D.; Schiffer, M. The many faces of superradiance. Phys. Rev. D 1998, 58, 064014. [Google Scholar] [CrossRef] [Green Version]
  257. Manjavacas, A.; García de Abajo, F.J. Thermal and vacuum friction acting on rotating particles. Phys. Rev. A 2010, 82, 063827. [Google Scholar] [CrossRef] [Green Version]
  258. Maghrebi, M.F.; Jaffe, R.L.; Kardar, M. Spontaneous Emission by Rotating Objects: A Scattering Approach. Phys. Rev. Lett. 2012, 108, 230403. [Google Scholar] [CrossRef] [Green Version]
  259. Zhao, R.; Manjavacas, A.; García de Abajo, F.J.; Pendry, J.B. Rotational quantum friction. Phys. Rev. Lett. 2012, 109, 123604. [Google Scholar] [CrossRef] [Green Version]
  260. Maghrebi, M.F.; Jaffe, R.L.; Kardar, M. Nonequilibrium quantum fluctuations of a dispersive medium: Spontaneous emission, photon statistics, entropy generation, and stochastic motion. Phys. Rev. A 2014, 90, 012515. [Google Scholar] [CrossRef] [Green Version]
  261. Calogeracos, A.; Volovik, G.E. Rotational quantum friction in superfluids: Radiation from object rotating in superfluid vacuum. JETP Lett. 1999, 69, 281–287. [Google Scholar] [CrossRef] [Green Version]
  262. Lannebère, S.; Silveirinha, M.G. Wave instabilities and unidirectional light flow in a cavity with rotating walls. Phys. Rev. A 2016, 94, 033810. [Google Scholar] [CrossRef] [Green Version]
  263. Johnston, H.; Sarkar, S. A re-examination of the quantum theory of optical cavities with moving mirrors. J. Phys. A Math. Gen. 1996, 29, 1741–1746. [Google Scholar] [CrossRef]
  264. Chizhov, A.V.; Schrade, G.; Zubairy, M.S. Quantum statistics of vacuum in a cavity with a moving mirror. Phys. Lett. A 1997, 230, 269–275. [Google Scholar] [CrossRef]
  265. Mundarain, D.F.; Maia Neto, P.A. Quantum radiation in a plain cavity with moving mirrors. Phys. Rev. A 1998, 57, 1379–1390. [Google Scholar] [CrossRef] [Green Version]
  266. Uhlmann, M.; Plunien, G.; Schützhold, R.; Soff, G. Resonant cavity photon creation via the dynamical Casimir effect. Phys. Rev. Lett. 2004, 93, 193601. [Google Scholar] [CrossRef] [Green Version]
  267. Levchenko, A.V.; Trifanov, A.I. Regimes of photon generation in Dynamical Casimir Effect under various resonance conditions. J. Phys. Conf. Ser. 2015, 643, 012093. [Google Scholar] [CrossRef]
  268. Dodonov, A.V.; Dodonov, V.V. Resonance generation of photons from vacuum in cavities due to strong periodical changes of conductivity in a thin semiconductor boundary layer. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S47–S58. [Google Scholar] [CrossRef]
  269. Wu, Y.; Chu, M.-C.; Leung, P.T. Dynamics of the quantized radiation field in a cavity vibrating at the fundamental frequency. Phys. Rev. A 1999, 59, 3032–3037. [Google Scholar] [CrossRef]
  270. Yang, X.-X.; Wu, Y. Dynamics of the quantized radiation field in an oscillating cavity in the harmonic resonance case. J. Phys. A: Math. Gen. 1999, 32, 7375–7392. [Google Scholar] [CrossRef]
  271. Husimi, K. Miscellanea in elementary quantum mechanics. II. Prog. Theor. Phys. 1953, 9, 381–402. [Google Scholar] [CrossRef]
  272. Dodonov, V.V.; Man’ko, V.I. Invariants and the Evolution of Nonstationary Quantum Systems (Proceedings of the Lebedev Physics Institute vol 183); Nova Science: Commack, NY, USA, 1989. [Google Scholar]
  273. Dodonov, V.V.; Man’ko, O.V.; Man’ko, V.I. Quantum nonstationary oscillator: Models and applications. J. Russ. Laser Res. 1995, 16, 1–56. [Google Scholar] [CrossRef]
  274. Dodonov, V.V. Parametric excitation and generation of nonclassical states in linear media. In Theory of Nonclassical States of Light; Dodonov, V.V., Man’ko, V.I., Eds.; Taylor & Francis: London, UK, 2003; pp. 153–218. [Google Scholar]
  275. Dodonov, V.V.; Klimov, A.B.; Nikonov, D.E. Quantum phenomena in nonstationary media. Phys. Rev. A 1993, 47, 4422–4429. [Google Scholar] [CrossRef] [PubMed]
  276. Dodonov, A.V.; Dodonov, V.V. Dynamical Casimir effect in a cavity with a weakly non-equidistant spectrum. Phys. Lett. A 2012, 376, 1903–1906. [Google Scholar] [CrossRef]
  277. Dodonov, A.V.; Dodonov, E.V.; Dodonov, V.V. Photon generation from vacuum in nondegenerate cavities with regular and random periodic displacements of boundaries. Phys. Lett. A 2003, 317, 378–388. [Google Scholar] [CrossRef] [Green Version]
  278. Fujii, K.; Suzuki, T. Rotating wave approximation of the Law’s effective hamiltonian on the dynamical Casimir effect. Int. J. Geom. Meth. Mod. Phys. 2014, 11, 1450003. [Google Scholar] [CrossRef] [Green Version]
  279. Brown, K.; Lowenstein, A.; Mathur, H. Effect of forcing on vacuum radiation. Phys. Rev. A 2019, 99, 022504. [Google Scholar] [CrossRef] [Green Version]
  280. Crocce, M.; Dalvit, D.A.R.; Mazzitelli, F.D. Resonant photon creation in a three-dimensional oscillating cavity. Phys. Rev. A 2001, 64, 013808. [Google Scholar] [CrossRef] [Green Version]
  281. Dodonov, A.V.; Dodonov, V.V. Nonstationary Casimir effect in cavities with two resonantly coupled modes . Phys. Lett. A 2001, 289, 291–300. [Google Scholar] [CrossRef] [Green Version]
  282. Ruser, M. Numerical investigation of photon creation in a three-dimensional resonantly vibrating cavity: Transverse electric modes. Phys. Rev. A 2006, 73, 043811. [Google Scholar] [CrossRef] [Green Version]
  283. Yuce, C.; Ozcakmakli, Z. The dynamical Casimir effect for two oscillating mirrors in 3D. J. Phys. A: Math. Theor. 2008, 41, 265401. [Google Scholar] [CrossRef]
  284. Yuce, C.; Ozcakmakli, Z. Dynamical Casimir effect for a swinging cavity. J. Phys. A: Math. Theor. 2009, 42, 035403. [Google Scholar] [CrossRef] [Green Version]
  285. Dodonov, V.V.; Dodonov, A.V. Excitation of the classical electromagnetic field in a cavity containing a thin slab with a time-dependent conductivity. J. Russ. Laser Res. 2016, 37, 107–122. [Google Scholar] [CrossRef] [Green Version]
  286. Casimir, H.B.G. On the attraction between two perfectly conducting plates. Proc. K. Ned. Akad. Wet. 1948, 51, 793–795. [Google Scholar]
  287. Bordag, M.; Petrov, G.; Robaschik, D. Calculation of the Casimir effect for scalar fields with the simplest nonstationary boundary conditions. Sov. J. Nucl. Phys. 1984, 39, 828–831. [Google Scholar]
  288. Bordag, M.; Dittes, F.-M.; Robaschik, D. The Casimir effect with uniformly moving mirrors. Sov. J. Nucl. Phys. 1986, 43, 1034–1038. [Google Scholar]
  289. Hanke, A. non-equilibrium casimir force between vibrating plates. PLoS ONE 2013, 8, e53228. [Google Scholar] [CrossRef] [Green Version]
  290. Villarreal, C.; Hacyan, S.; Jáuregui, R. Generation of particles and squeezed states between moving conductors. Phys. Rev. A 1995, 52, 594–601. [Google Scholar] [CrossRef]
  291. Cirone, M.A.; Rza̧źewski, K. Electromagnetic radiation in a cavity with a time-dependent mirror. Phys. Rev. A 1999, 60, 886–892. [Google Scholar] [CrossRef]
  292. Setare, M.R.; Seyedzahedi, A. Fermion particle production in dynamical Casimir effect in a three-dimensional box. Int. J. Mod. Phys. A 2012, 27, 1250176. [Google Scholar] [CrossRef] [Green Version]
  293. Mkrtchian, V.E.; von Baltz, R. Dynamical electromagnetic modes for an expanding sphere. J. Math. Phys. 2000, 41, 1956–1960. [Google Scholar] [CrossRef]
  294. Mazzitelli, F.D.; Millán, X.O. Photon creation in a spherical oscillating cavity. Phys. Rev. A 2006, 73, 063829. [Google Scholar] [CrossRef] [Green Version]
  295. Pascoal, F.; Céleri, L.C.; Mizrahi, S.S.; Moussa, M.H.Y. Dynamical Casimir effect for a massless scalar field between two concentric spherical shells. Phys. Rev. A 2008, 78, 032521. [Google Scholar] [CrossRef] [Green Version]
  296. Pascoal, F.; Céleri, L.C.; Mizrahi, S.S.; Moussa, M.H.Y.; Farina, C. Dynamical Casimir effect for a massless scalar field between two concentric spherical shells with mixed boundary conditions. Phys. Rev. A 2009, 80, 012503. [Google Scholar] [CrossRef]
  297. Naylor, W. Towards particle creation in a microwave cylindrical cavity. Phys. Rev. A 2012, 86, 023842. [Google Scholar] [CrossRef] [Green Version]
  298. Setare, M. R; Seyedzahedi, A. Fermion particle production as a dynamical Casimir effect inside a three- dimensional sphere. J. Phys. Conf. Ser. 2013, 410, 012150. [Google Scholar] [CrossRef]
  299. Rego, A.L.C.; Mintz, B.W.; Farina, C.; Alves, D.T. Inhibition of the dynamical Casimir effect with Robin boundary conditions. Phys. Rev. D 2013, 87, 045024. [Google Scholar] [CrossRef] [Green Version]
  300. Fosco, C.D.; Giraldo, A.; Mazzitelli, F.D. Dynamical Casimir effect for semitransparent mirrors. Phys. Rev. D 2017, 96, 045004. [Google Scholar] [CrossRef] [Green Version]
  301. Maghrebi, M.F.; Golestanian, R.; Kardar, M. Scattering approach to the dynamical Casimir effect. Phys. Rev. D 2013, 87, 025016. [Google Scholar] [CrossRef] [Green Version]
  302. Villar, P.I.; Soba, A. Adaptive numerical algorithms to simulate the dynamical Casimir effect in a closed cavity with different boundary conditions. Phys. Rev. E 2017, 96, 013307. [Google Scholar] [CrossRef] [Green Version]
  303. Srivastava, Y.N.; Widom, A.; Sivasubramanian, S.; Pradeep Ganesh, M. Dynamical Casimir effect instabilities. Phys. Rev. A 2006, 74, 032101. [Google Scholar] [CrossRef] [Green Version]
  304. de Sousa, I.M.; Dodonov, A.V. Microscopic toy model for the cavity dynamical Casimir effect. J. Phys. A: Math. Theor. 2015, 48, 245302. [Google Scholar] [CrossRef]
  305. Román-Ancheyta, R.; González-Gutiérrez, C.; Récamier, J. Influence of the Kerr nonlinearity in a single nonstationary cavity mode. J. Opt. Soc. Am. B 2017, 34, 1170–1176. [Google Scholar] [CrossRef] [Green Version]
  306. Paredes, A.; Récamier, J. Study of the combined effects of a Kerr nonlinearity and a two-level atom upon a single nonstationary cavity mode. J. Opt. Soc. Am. B 2019, 36, 1538–1543. [Google Scholar] [CrossRef]
  307. Mendonça, J.T.; Brodin, G.; Marklund, M. The influence of temporal coherence on the dynamical Casimir effect. Phys. Lett. A 2011, 375, 2665–2669. [Google Scholar] [CrossRef] [Green Version]
  308. Rosanov, N.N.; Matskovskii, A.A.; Malevich, V.L.; Sinitsyn, G.V. Parametric field excitation in a cavity with oscillating mirrors. 2015, Opt. Spectrosc. 2015, 119, 89–91. [Google Scholar] [CrossRef]
  309. Srivastava, Y.; Widom, A. Quantum electrodynamic processes in electrical engineering circuits. Phys. Rep. 1987, 148, 1–65. [Google Scholar] [CrossRef]
  310. Man’ko, V.I. The Casimir effect and quantum vacuum generator. J. Russ. Laser Res. 1991, 12, 383–385. [Google Scholar] [CrossRef]
  311. Man’ko, O.V. Correlated squeezed states of a Josephson junction. J. Kor. Phys. Soc. 1994, 27, 1–4. [Google Scholar]
  312. Segev, E.; Abdo, B.; Shtempluck, O.; Buks, E.; Yurke, B. Prospects of employing superconducting stripline resonators for studying the dynamical Casimir effect experimentally. Phys. Lett. A 2007, 370, 202–206. [Google Scholar] [CrossRef] [Green Version]
  313. Takashima, K.; Hatakenaka, N.; Kurihara, S.; Zeilinger, A. Squeezing of a quantum flux in a double rf-SQUID system. J. Phys. A: Math. Theor. 2008, 41, 164036. [Google Scholar] [CrossRef]
  314. Fujii, T.; Matsuo, S.; Hatakenaka, N.; Kurihara, S.; Zeilinger, A. Quantum circuit analog of the dynamical Casimir effect. Phys. Rev. B 2011, 84, 174521. [Google Scholar] [CrossRef] [Green Version]
  315. Berdiyorov, G.R.; Milošević, M.V.; Savel’ev, S.; Kusmartsev, F.; Peeters, F.M. Parametric amplification of vortex-antivortex pair generation in a Josephson junction. Phys. Rev. B 2014, 90, 134505. [Google Scholar] [CrossRef] [Green Version]
  316. Dodonov, A.V. Photon creation from vacuum and interactions engineering in nonstationary circuit QED. J. Phys. Conf. Ser. 2009, 161, 012029. [Google Scholar] [CrossRef]
  317. Johansson, J.R.; Johansson, G.; Wilson, C.M.; Nori, F. Dynamical Casimir effect in a superconducting coplanar waveguide. Phys. Rev. Lett. 2009, 103, 147003. [Google Scholar] [CrossRef] [Green Version]
  318. Johansson, J.R.; Johansson, G.; Wilson, C.M.; Nori, F. Dynamical Casimir effect in superconducting microwave circuits. Phys. Rev. A 2010, 82, 052509. [Google Scholar] [CrossRef] [Green Version]
  319. Wilson, C.M.; Duty, T.; Sandberg, M.; Persson, F.; Shumeiko, V.; Delsing, P. Photon generation in an electromagnetic cavity with a time-dependent boundary. Phys. Rev. Lett. 2010, 105, 233907. [Google Scholar] [CrossRef] [Green Version]
  320. Wilson, C.M.; Johansson, G.; Pourkabirian, A.; Simoen, M.; Johansson, J.R.; Duty, T.; Nori, F.; Delsing, P. Observation of the dynamical Casimir effect in a superconducting circuit. Nature 2011, 479, 376–379. [Google Scholar] [CrossRef]
  321. Johansson, J.R.; Johansson, G.; Wilson, C.M.; Delsing, P.; Nori, F. Nonclassical microwave radiation from the dynamical Casimir effect. Phys. Rev. A 2013, 87, 043804. [Google Scholar] [CrossRef] [Green Version]
  322. Lähteenmäki, P.; Paraoanu, G.S.; Hassel, J.; Hakonen, P.J. Dynamical Casimir effect in a Josephson metamaterial. Proc. Nat. Acad. Sci. USA 2013, 110, 4234–8. [Google Scholar] [CrossRef] [Green Version]
  323. Svensson, I.-M.; Pierre, M.; Simoen, M.; Wustmann, W.; Krantz, P.; Bengtsson, A.; Johansson, G.; Bylander, J.; Shumeiko, V.; Delsing, P. Microwave photon generation in a doubly tunable superconducting resonator. J. Phys. Conf. Ser. 2018, 969, 012146. [Google Scholar] [CrossRef]
  324. Wustmann, W.; Shumeiko, V. Parametric resonance in tunable superconducting cavities. Phys. Rev. B 2013, 87, 184501. [Google Scholar] [CrossRef] [Green Version]
  325. Rego, A.L.C.; Alves, J.P.daS.; Alves, D.T.; Farina, C. Relativistic bands in the spectrum of created particles via the dynamical Casimir effect. Phys. Rev. A 2013, 88, 032515. [Google Scholar] [CrossRef] [Green Version]
  326. Rego, A.L.C.; Silva, H.O.; Alves, D.T.; Farina, C. New signatures of the dynamical Casimir effect in a superconducting circuit. Phys. Rev. D 2014, 90, 025003. [Google Scholar] [CrossRef] [Green Version]
  327. Lindkvist, J.; Sabin, C.; Fuentes, I.; Dragan, A.; Svensson, I.-M.; Delsing, P.; Johansson, G. Twin paradox with macroscopic clocks in superconducting circuits. Phys. Rev. A 2014, 90, 052113. [Google Scholar] [CrossRef] [Green Version]
  328. Zhang, Y.N.; Luo, X.W.; Zhou, Z.W. Dynamical Casimir effect in dissipative superconducting circuit system. Sci. China - Phys. Mech. Astron. 2014, 57, 2251–2258. [Google Scholar] [CrossRef]
  329. Andersen, C.K.; Mølmer, K. Multifrequency modes in superconducting resonators: Bridging frequency gaps in off-resonant couplings. Phys. Rev. A 2015, 91, 023828. [Google Scholar] [CrossRef] [Green Version]
  330. Doukas, J.; Louko, J. Superconducting circuit boundary conditions beyond the dynamical Casimir effect. Phys. Rev. D 2015, 91, 044010. [Google Scholar] [CrossRef] [Green Version]
  331. Corona-Ugalde, P.; Martín-Martínez, E.; Wilson, C.M.; Mann, R.B. Dynamical Casimir effect in circuit QED for nonuniform trajectories. Phys. Rev. A 2016, 93, 012519. [Google Scholar] [CrossRef] [Green Version]
  332. Lombardo, F.C.; Mazzitelli, F.D.; Soba, A.; Villar, P.I. Dynamical Casimir effect in superconducting circuits: A numerical approach. Phys. Rev. A 2016, 93, 032501. [Google Scholar] [CrossRef] [Green Version]
  333. Samos-Saenz de Buruaga, D.N.; Sabín, C. Quantum coherence in the dynamical Casimir effect. Phys. Rev. A 2017, 95, 022307. [Google Scholar] [CrossRef] [Green Version]
  334. Sabín, C.; Peropadre, B.; Lamata, L.; Solano, E. Simulating superluminal physics with superconducting circuit technology. Phys. Rev. A 2017, 96, 032121. [Google Scholar] [CrossRef]
  335. Lombardo, F.C.; Mazzitelli, F.D.; Soba, A.; Villar, P.I. Dynamical Casimir effect in a double tunable superconducting circuit. Phys. Rev. A 2018, 98, 022512. [Google Scholar] [CrossRef] [Green Version]
  336. Bosco, S.; Lindkvist, J.; Johansson, G. Simulating moving cavities in superconducting circuits. Phys. Rev. A 2019, 100, 023817. [Google Scholar] [CrossRef] [Green Version]
  337. Ma, S.; Miao, H.; Xiang, Y.; Zhang, S. Enhanced dynamic Casimir effect in temporally and spatially modulated Josephson transmission line. Laser Photonics Rev. 2019, 1900164. [Google Scholar] [CrossRef]
  338. Dodonov, A.V. Asymptotic mean excitation numbers due to anti-rotating term (AMENDART) in Markovian circuit QED. J. Phys. Conf. Ser. 2011, 274, 012137. [Google Scholar] [CrossRef]
  339. Dodonov, A.V. Analytical description of nonstationary circuit QED in the dressed-states basis. J. Phys. A: Math. Theor. 2014, 47, 285303. [Google Scholar] [CrossRef]
  340. Veloso, D.S.; Dodonov, A.V. Prospects for observing dynamical and anti-dynamical Casimir effects in circuit QED due to fast modulation of qubit parameters. J. Phys. B: Atom. Mol. Opt. Phys. 2015, 48, 165503. [Google Scholar] [CrossRef] [Green Version]
  341. Felicetti, S.; Sabin, C.; Fuentes, I.; Lamata, L.; Romero, G.; Solano, E. Relativistic motion with superconducting qubits. Phys. Rev. B 2015, 92, 064501. [Google Scholar] [CrossRef] [Green Version]
  342. Hoi, I.-C.; Kockum, A.F.; Tornberg, L.; Pourkabirian, A.; Johansson, G.; Delsing, P.; Wilson, C.M. Probing the quantum vacuum with an artificial atom in front of a mirror. Nature. Phys. 2015, 11, 1045–1049. [Google Scholar] [CrossRef]
  343. Rossatto, D.Z.; Felicetti, S.; Eneriz, H.; Rico, E.; Sanz, M.; Solano, E. Entangling polaritons via dynamical Casimir effect in circuit quantum electrodynamics. Phys. Rev. B 2016, 93, 094514. [Google Scholar] [CrossRef] [Green Version]
  344. Dodonov, A.V.; Militello, B.; Napoli, A.; Messina, A. Effective Landau-Zener transitions in the circuit dynamical Casimir effect with time-varying modulation frequency. Phys. Rev. A 2016, 93, 052505. [Google Scholar] [CrossRef] [Green Version]
  345. Silva, E.L.S.; Dodonov, A.V. Analytical comparison of the first- and second-order resonances for implementation of the dynamical Casimir effect in nonstationary circuit QED. J. Phys. A: Math. Theor. 2016, 49, 495304. [Google Scholar] [CrossRef] [Green Version]
  346. García-Álvarez, L.; Felicetti, S.; Rico, E.; Solano, E.; Sabin, C. Entanglement of superconducting qubits via acceleration radiation. Sci. Rep. 2017, 7, 657. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  347. Gu, X.; Kockum, A.F.; Miranowicz, A.; Liu, Y.-X.; Nori, F. Microwave photonics with superconducting quantum circuits. Phys. Rep. 2017, 718–719, 1–102. [Google Scholar] [CrossRef]
  348. Zhukov, A.A.; Shapiro, D.S.; Remizov, S.V.; Pogosov, W.V.; Lozovik, Y.E. Superconducting qubit in a nonstationary transmission line cavity: Parametric excitation, periodic pumping, and energy dissipation. Phys. Lett. A 2017, 381, 592–596. [Google Scholar] [CrossRef] [Green Version]
  349. Zhukov, A.A.; Remizov, S.V.; Pogosov, W.V.; Shapiro, D.S.; Lozovik, Y.E. Superconducting qubit systems as a platform for studying effects of nonstationary electrodynamics in a cavity. JETP Lett. 2018, 108, 63–70. [Google Scholar] [CrossRef]
  350. Dessano, H.; Dodonov, A.V. One- and three-photon dynamical Casimir effects using a nonstationary cyclic qutrit. Phys. Rev. A 2018, 98, 022520. [Google Scholar] [CrossRef] [Green Version]
  351. Wustmann, W.; Shumeiko, V. Parametric effects in circuit quantum electrodynamics. Low Temp. Phys. 2019, 45, 848–869. [Google Scholar] [CrossRef] [Green Version]
  352. Carusotto, I.; Balbinot, R.; Fabbri, A.; Recati, A. Density correlations and analog dynamical Casimir emission of Bogoliubov phonons in modulated atomic Bose-Einstein condensates. Eur. Phys. J. D 2010, 56, 391–404. [Google Scholar] [CrossRef]
  353. Jaskula, J.-C.; Partridge, G.B.; Bonneau, M.; Lopes, R.; Ruaudel, J.; Boiron, D.; Westbrook, C.I. Acoustic analog to the dynamical Casimir effect in a Bose-Einstein condensate. Phys. Rev. Lett. 2012, 109, 220401. [Google Scholar] [CrossRef] [Green Version]
  354. Balbinot, R.; Fabbri, A. Amplifying the Hawking signal in BECs. Advances High Energy Phys. 2014, 713574. [Google Scholar] [CrossRef]
  355. Mendonça, J.T.; Dodonov, V.V. Time crystals in ultracold matter. J. Russ. Laser Res. 2014, 35, 93–100. [Google Scholar] [CrossRef]
  356. Dodonov, V.V.; Mendonca, J.T. Dynamical Casimir effect in ultra-cold matter with a time-dependent effective charge. Phys. Scr. 2014, T160, 014008. [Google Scholar] [CrossRef]
  357. Mahajan, S.; Aggarwal, N.; Bhattacherjee, A.B.; ManMohan. Dynamical Casimir effect in superradiant light scattering by Bose-Einstein condensate in an optomechanical cavity. Chin. Phys. B 2014, 23, 020315. [Google Scholar] [CrossRef]
  358. Marino, J.; Recati, A.; Carusotto, I. Casimir forces and quantum friction from Ginzburg radiation in atomic Bose-Einstein condensates. Phys. Rev. Lett. 2017, 118, 045301. [Google Scholar] [CrossRef] [Green Version]
  359. Rosanov, N.N.; Vysotina, N.V. Dynamics of hysteresis for a Bose–Einstein condensate soliton in a dynamic trap. Opt. Spectrosc. 2017, 123, 918–927. [Google Scholar] [CrossRef]
  360. Eckel, S.; Kumar, A.; Jacobson, T.; Spielman, I.B.; Campbell, G.K. A rapidly expanding Bose-Einstein condensate: An expanding universe in the lab. Phys. Rev. X 2018, 8, 021021. [Google Scholar] [CrossRef] [Green Version]
  361. Tian, Z.; Cha, S.-Y.; Fischer, U.R. Roton entanglement in quenched dipolar Bose-Einstein condensates. Phys. Rev. A 2018, 97, 063611. [Google Scholar] [CrossRef] [Green Version]
  362. Motazedifard, A.; Dalafi, A.; Naderi, M.H.; Roknizadeh, R. Controllable generation of photons and phonons in a coupled Bose-Einstein condensate-optomechanical cavity via the parametric dynamical Casimir effect. Ann. Phys. 2018, 396, 202–219. [Google Scholar] [CrossRef] [Green Version]
  363. Lange, K.; Peise, J.; Luecke, B.; Gruber, T.; Sala, A.; Polls, A.; Ertmer, W.; Juliá-Díaz, B.; Santos, L.; Klempt, C. Creation of entangled atomic states by an analogue of the Dynamical Casimir effect. New J. Phys. 2018, 20, 103017. [Google Scholar] [CrossRef]
  364. Michael, M.H.; Schmiedmayer, J.; Demler, E. From the moving piston to the dynamical Casimir effect: Explorations with shaken condensates. Phys. Rev. A 2019, 99, 053615. [Google Scholar] [CrossRef] [Green Version]
  365. Lawandy, N.M. 2006 Scattering of vacuum states by dynamic plasmon singularities: Generating photons from vacuum. Opt. Lett. 2006, 31, 3650–2. [Google Scholar] [CrossRef] [PubMed]
  366. Ciuti, C.; Bastard, G.; Carusotto, I. Quantum vacuum properties of the intersubband cavity polariton field. Phys. Rev. B 2005, 72, 115303. [Google Scholar] [CrossRef] [Green Version]
  367. Koghee, S.; Wouters, M. Dynamical Casimir emission from polariton condensates. Phys. Rev. Lett. 2014, 112, 036406. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  368. Koghee, S.; Wouters, M. Dynamical quantum depletion in polariton condensates. Phys. Rev. B 2015, 92, 195309. [Google Scholar] [CrossRef] [Green Version]
  369. Hizhnyakov, V.; Loot, A.; Azizabadi, S.C. Dynamical Casimir effect for surface plasmon polaritons. Phys. Lett. A 2015, 379, 501–505. [Google Scholar] [CrossRef] [Green Version]
  370. Naylor, W. Vacuum-excited surface plasmon polaritons. Phys. Rev. A 2015, 91, 053804. [Google Scholar] [CrossRef] [Green Version]
  371. Saito, H.; Hyuga, H. 2008 Dynamical Casimir effect for magnons in a spinor Bose-Einstein condensate. Phys. Rev. A 2008, 78, 033605. [Google Scholar] [CrossRef] [Green Version]
  372. Zhao, X.-D.; Zhao, X.; Jing, H.; Zhou, L.; Zhang, W. Squeezed magnons in an optical lattice: Application to simulation of the dynamical Casimir effect at finite temperature. Phys. Rev. A 2013, 87, 053627. [Google Scholar] [CrossRef]
  373. Ford, L.H.; Svaiter, N.F. The phononic Casimir effect: An analog model. J, Phys. Conf. Ser. 2009, 161, 012034. [Google Scholar] [CrossRef]
  374. Motazedifard, A.; Naderi, M.H.; Roknizadeh, R. Dynamical Casimir effect of phonon excitation in the dispersive regime of cavity optomechanics. J. Opt. Soc. Am. B 2017, 34, 642–652. [Google Scholar] [CrossRef] [Green Version]
  375. Wang, X.; Qin, W.; Miranowicz, A.; Savasta, S.; Nori, F. Unconventional cavity optomechanics: Nonlinear control of phonons in the acoustic quantum vacuum. Phys. Rev. A 2019, 100, 063827. [Google Scholar] [CrossRef] [Green Version]
  376. Wittemer, M.; Hakelberg, F.; Kiefer, P.; Schröder, J.-P.; Fey, C.; Schẗzhold, R.; Warring, U.; Schaetz, T. Phonon pair creation by inflating quantum fluctuations in an ion trap. Phys. Rev. Lett. 2019, 123, 180502. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  377. Jáuregui, R.; Villarreal, C. Transition probabilities of atomic systems between moving walls. Phys. Rev. A 1996, 54, 3480–3488. [Google Scholar] [CrossRef]
  378. Janowicz, M. Evolution of wave fields and atom-field interactions in a cavity with one oscillating mirror. Phys. Rev. A 1998, 57, 4784–4790. [Google Scholar] [CrossRef]
  379. Carusotto, I.; Antezza, M.; Bariani, F.; De Liberato, S.; Ciuti, C. Optical properties of atomic Mott insulators: From slow light to dynamical Casimir effects. Phys. Rev. A 2008, 77, 063621. [Google Scholar] [CrossRef] [Green Version]
  380. Werlang, T.; Dodonov, A.V.; Duzzioni, E.I.; Villas-Bôas, C.J. 2008, Rabi model beyond the rotating-wave approximation: Generation of photons from vacuum through decoherence. Phys. Rev. A 2008, 78, 053805. [Google Scholar] [CrossRef] [Green Version]
  381. De Liberato, S.; Gerace, D.; Carusotto, I.; Ciuti, C. Extracavity quantum vacuum radiation from a single qubit. Phys. Rev. A 2009, 80, 053810. [Google Scholar] [CrossRef] [Green Version]
  382. Dodonov, A.V. How ’cold’ can a Markovian dissipative cavity QED system be? Phys. Scr. 2010, 82, 038102. [Google Scholar] [CrossRef]
  383. Zhang, X.; Zheng, T.Y.; Tian, T.; Pan, S.M. The dynamical Casimir effect versus collective excitations in atom ensemble. Chin. Phys. Lett. 2011, 28, 064202. [Google Scholar] [CrossRef]
  384. Dodonov, A.V.; Lo Nardo, R.; Migliore, R.; Messina, A.; Dodonov, V.V. Analytical and numerical analysis of the atom-field dynamics in non-stationary cavity QED. J. Phys. B: Atom. Mol. Opt. Phys. 2011, 44, 225502. [Google Scholar] [CrossRef]
  385. Vacanti, G.; Pugnetti, S.; Didier, N.; Paternostro, M.; Palma, G.M.; Fazio, R.; Vedral, V. Photon production from the vacuum close to the superradiant transition: Linking the dynamical Casimir effect to the Kibble-Zurek mechanism. Phys. Rev. Lett. 2012, 108, 093603. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  386. Dodonov, A.V.; Dodonov, V.V. Approximate analytical results on the cavity dynamical Casimir effect in the presence of a two-level atom. Phys. Rev. A 2012, 85, 015805. [Google Scholar] [CrossRef] [Green Version]
  387. Carusotto, I.; De Liberato, S.; Gerace, D.; Ciuti, C. Back-reaction effects of quantum vacuum in cavity quantum electrodynamics. Phys. Rev. A 2012, 85, 023805. [Google Scholar] [CrossRef] [Green Version]
  388. Dodonov, A.V.; Dodonov, V.V. Dynamical Casimir effect in a cavity with an N-level detector or N-1 two-level atoms. Phys. Rev. A 2012, 86, 015801. [Google Scholar] [CrossRef] [Green Version]
  389. Dodonov, A.V.; Dodonov, V.V. Dynamical Casimir effect in a cavity in the presence of a three-level atom. Phys. Rev. A 2012, 85, 063804. [Google Scholar] [CrossRef] [Green Version]
  390. Dodonov, A.V.; Dodonov, V.V. Dynamical Casimir effect in two-atom cavity QED. Phys. Rev. A 2012, 85, 055805. [Google Scholar] [CrossRef] [Green Version]
  391. Fujii, K.; Suzuki, T. An approximate solution of the dynamical Casimir effect in a cavity with a two-level atom. Int. J. Geom. Meth. Mod. Phys. 2013, 10, 1350035. [Google Scholar] [CrossRef] [Green Version]
  392. Garziano, L.; Ridolfo, A.; Stassi, R.; Di Stefano, O.; Savasta, S. Switching on and off of ultrastrong light-matter interaction: Photon statistics of quantum vacuum radiation. Phys. Rev. A 2013, 88, 063829. [Google Scholar] [CrossRef]
  393. Impens, F.; Ttira, C.C.; Maia Neto, P.A. Non-additive dynamical Casimir atomic phases. J. Phys. B: Atom. Mol. Opt. Phys. 2013, 46, 245503. [Google Scholar] [CrossRef] [Green Version]
  394. Sheremetyev, V.O.; Trifanova, E.S.; Trifanov, A.I. Conditional evolution of vacuum state in dynamical Casimir effect. J. Phys. Conf. Ser. 2014, 541, 012105. [Google Scholar] [CrossRef] [Green Version]
  395. Zhang, X.; Yang, H.; Zheng, T.Y.; Pan, S.M. Linking the dynamical Casimir effect to the collective excitation effect at finite temperature. Int. J. Theor. Phys. 2014, 53, 510–518. [Google Scholar] [CrossRef]
  396. Jin, Z.; Han, T.T.; Zheng, T.Y.; Zhang, X.; Pan, S.M.; Asilibieke, B. Influence of interaction of two Atoms on the dynamical Casimir effect. Int. J. Theor. Phys. 2015, 54, 1627–1632. [Google Scholar] [CrossRef]
  397. Aggarwal, N.; Bhattacherjee, A.B.; Banerjee, A.; Mohan, M. Influence of periodically modulated cavity field on the generation of atomic-squeezed states. J. Phys. B: Atom. Mol. Opt. Phys. 2015, 48, 115501. [Google Scholar] [CrossRef]
  398. Asilibieke, B.; Xue, Z.; Jie, F.; Liu, H.; Pan, S.M.; Zheng, T.Y.; Yang, H. Dynamical Casimir effect and collective excitation effect at finite temperature without the rotating-wave approximation. Int. J. Theor. Phys. 2015, 54, 2762–2770. [Google Scholar] [CrossRef]
  399. Trautmann, N.; Hauke, P. Quantum simulation of the dynamical Casimir effect with trapped ions. New J. Phys. 2016, 18, 043029. [Google Scholar] [CrossRef]
  400. Hoeb, F.; Angaroni, F.; Zoller, J.; Calarco, T.; Strini, G.; Montangero, S.; Benenti, G. Amplification of the parametric dynamical Casimir effect via optimal control. Phys. Rev. A 2017, 96, 033851. [Google Scholar] [CrossRef] [Green Version]
  401. Remizov, S.V.; Zhukov, A.A.; Shapiro, D.S.; Pogosov, W.V.; Lozovik, Y.E. Parametrically driven hybrid qubit-photon systems: Dissipation-induced quantum entanglement and photon production from vacuum. Phys. Rev. A 2017, 96, 043870. [Google Scholar] [CrossRef] [Green Version]
  402. Silveri, M.P.; Tuorila, J.A.; Thuneberg, E.V.; Paraoanu, G.S. Quantum systems under frequency modulation. Rep. Prog. Phys. 2017, 80, 056002. [Google Scholar] [CrossRef]
  403. Lo, L.; Law, C.K. Quantum radiation from a shaken two-level atom in vacuum. Phys. Rev. A 2018, 98, 063807. [Google Scholar] [CrossRef] [Green Version]
  404. González-Gutiérrez, C.; de los Santos-Sánchez, O.; Román-Ancheyta, R.; Berrondo, M.; Récamier, J. Lie algebraic approach to a nonstationary atom-cavity system. J. Opt. Soc. Am. B 2018, 35, 1979–1984. [Google Scholar] [CrossRef] [Green Version]
  405. Benenti, G.; Stramacchia, M.; Strini, G. Dynamical Casimir effect and state transfer in the ultrastrong coupling regime. MDPI Proceedings 2019, 12, 12. [Google Scholar] [CrossRef] [Green Version]
  406. Pan, S.M.; Asilibieke, B.; Zheng, L.; Tian, T.; Zhang, X.; Zheng, T.Y. The dynamical Casimir effect in squeezed vacuum state. Int. J. Theor. Phys. 2019, 58, 22–30. [Google Scholar] [CrossRef]
  407. Liu, H.; Wang, Q.; Zhang, X.; Long, Y.M.; Pan, S.M.; Zheng, T.Y.; Sun, C.; Xue, K. The dynamical behaviors of the two-atom and the dynamical Casimir effect in a non-stationary cavity. Int. J. Theor. Phys. 2019, 58, 786–798. [Google Scholar] [CrossRef]
  408. Dodonov, A.V. Dynamical Casimir effect via four- and five-photon transitions using a strongly detuned atom. Phys. Rev. A 2019, 100, 032510. [Google Scholar] [CrossRef] [Green Version]
  409. Abdel-Khalek, S.; Berrada, K. Entanglement, nonclassical properties, and geometric phase in circuit quantum electrodynamics with relativistic motion. Solid State Commun. 2019, 290, 31–36. [Google Scholar] [CrossRef]
  410. Svidzinsky, A.A.; Ben-Benjamin, J.S.; Fulling, S.A.; Page, D.N. Excitation of an atom by a uniformly accelerated mirror through virtual transitions. Phys. Rev. Lett. 2018, 121, 071301. [Google Scholar] [CrossRef] [Green Version]
  411. Dodonov, A.V.; Dodonov, V.V. Strong modifications of the field statistics in the cavity dynamical Casimir effect due to the interaction with two-level atoms and detectors. Phys. Lett. A 2011, 375, 4261–4267. [Google Scholar] [CrossRef]
  412. Benenti, G.; Siccardi, S.; Strini, G. Exotic states in the dynamical Casimir effect. Eur. Phys. J. D 2014, 68, 139. [Google Scholar] [CrossRef] [Green Version]
  413. Motazedifard, A.; Naderi, M.H.; Roknizadeh, R. Analogue model for controllable Casimir radiation in a nonlinear cavity with amplitude-modulated pumping: Generation and quantum statistical properties. J. Opt. Soc. Am. B 2015, 32, 1555–1563. [Google Scholar] [CrossRef] [Green Version]
  414. Monteiro, L.C.; Dodonov, A.V. Anti-dynamical Casimir effect with an ensemble of qubits. Phys. Lett. A 2016, 380, 1542–1546. [Google Scholar] [CrossRef]
  415. Dodonov, A.V.; Valente, D.; Werlang, T. Antidynamical Casimir effect as a resource for work extraction. Phys. Rev. A 2017, 96, 012501. [Google Scholar] [CrossRef] [Green Version]
  416. Dodonov, A.V.; Díaz-Guevara, J.J.; Napoli, A.; Militello, B. Speeding up the antidynamical Casimir effect with nonstationary qutrits. Phys. Rev. A 2017, 96, 032509. [Google Scholar] [CrossRef] [Green Version]
  417. Dodonov, A.V.; Valente, D.; Werlang, T. Quantum power boost in a nonstationary cavity-QED quantum heat engine. J. Phys. A: Math. Theor. 2018, 51, 365302. [Google Scholar] [CrossRef] [Green Version]
  418. Angaroni, F.; Benenti, G.; Strini, G. Applications of Picard and Magnus expansions to the Rabi model. Eur. Phys. J. D 2018, 72, 188. [Google Scholar] [CrossRef]
  419. Antezza, M.; Braggio, C.; Carugno, G.; Noto, A.; Passante, R.; Rizzuto, L.; Ruoso, G.; Spagnolo, S. Optomechanical Rydberg-atom excitation via dynamic Casimir-Polder coupling. Phys. Rev. Lett. 2014, 113, 023601. [Google Scholar] [CrossRef] [Green Version]
  420. Glaetze, A.W.; Hammerer, K.; Daley, A.J.; Platt, R.; Zoller, P. A single trapped atom in front of an oscillating mirror. Opt. Commun. 2010, 283, 758–765. [Google Scholar] [CrossRef] [Green Version]
  421. Ferreri, A.; Domina, M.; Rizzuto, L.; Passante, R. Spontaneous emission of an atom near an oscillating mirror. Symmetry 2019, 11, 1384. [Google Scholar] [CrossRef] [Green Version]
  422. Scully, M.O.; Kocharovsky, V.V.; Belyanin, A.; Fry, E.; Capasso, F. Enhancing acceleration radiation from ground-state atoms via cavity quantum electrodynamics. Phys. Rev. Lett. 2003, 91, 243004. [Google Scholar] [CrossRef] [Green Version]
  423. Tadigadapa, S.; Mateti, K. Piezoelectric MEMS sensors: State-of-the-art and perspectives. Meas. Sci. Technol. 2009, 20, 092001. [Google Scholar] [CrossRef]
  424. Kim, W.-J.; Brownell, J.H.; Onofrio, R. Detectability of dissipative motion in quantum vacuum via superradiance. Phys. Rev. Lett. 2006, 96, 200402. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  425. Brownell, J.H.; Kim, W.J.; Onofrio, R. Modelling superradiant amplification of Casimir photons in very low dissipation cavities. J. Phys. A: Math. Theor. 2008, 41, 164026. [Google Scholar] [CrossRef]
  426. Sanz, M.; Wieczorek, W.; Gröblacher, S.; Solano, E. Electro-mechanical Casimir effect. Quantum 2018, 2, 91. [Google Scholar] [CrossRef]
  427. Wang, H.; Blencowe, M.P.; Wilson, C.M.; Rimberg, A.J. Mechanically generating entangled photons from the vacuum: A microwave circuit-acoustic resonator analog of the oscillatory Unruh effect. Phys. Rev. A 2019, 99, 053833. [Google Scholar] [CrossRef] [Green Version]
  428. Qin, W.; Macrì, V.; Miranowicz, A.; Savasta, S.; Nori, F. Emission of photon pairs by mechanical stimulation of the squeezed vacuum. Phys. Rev. A 2019, 100, 062501. [Google Scholar] [CrossRef] [Green Version]
  429. Yablonovitch, E.; Heritage, J.P.; Aspnes, D.E.; Yafet, Y. Virtual photoconductivity. Phys. Rev. Lett. 1989, 63, 976–979. [Google Scholar] [CrossRef]
  430. Okushima, T.; Shimizu, A. Photon emission from a false vacuum of semiconductors. Japan. J. Appl. Phys. 1995, 34, 4508–4510. [Google Scholar] [CrossRef]
  431. Lozovik, Y.E.; Tsvetus, V.G.; Vinogradov, E.A. Parametric excitation of vacuum by use of femtosecond laser pulses. Phys. Scr. 1995, 52, 184–190. [Google Scholar] [CrossRef]
  432. Braggio, C.; Bressi, G.; Carugno, G.; Lombardi, A.; Palmieri, A.; Ruoso, G.; Zanello, D. Semiconductor microwave mirror for a measurement of the dynamical Casimir effect. Rev. Sci. Instrum. 2004, 75, 4967–4970. [Google Scholar] [CrossRef]
  433. Braggio, C.; Bressi, G.; Carugno, G.; Del Noce, C.; Galeazzi, G.; Lombardi, A.; Palmieri, A.; Ruoso, G.; Zanello, D. A novel experimental approach for the detection of the dynamic Casimir effect. Europhys. Lett. 2005, 70, 754–760. [Google Scholar] [CrossRef] [Green Version]
  434. Agnesi, A.; Braggio, C.; Bressi, G.; Carugno, G.; Galeazzi, G.; Pirzio, F.; Reali, G.; Ruoso, G.; Zanello, D. MIR status report: An experiment for the measurement of the dynamical Casimir effect. J. Phys. A: Math. Theor. 2008, 41, 164024. [Google Scholar] [CrossRef]
  435. Dodonov, V.V. Dynamical Casimir effect meets material science. IOP Conf. Ser. Mater. Sci. Eng. 2019, 474, 012009. [Google Scholar] [CrossRef]
  436. Hagenmuller, D. All-optical dynamical Casimir effect in a three-dimensional terahertz photonic band gap. Phys. Rev. B 2016, 93, 235309. [Google Scholar] [CrossRef]
  437. Bialynicka-Birula, Z.; Bialynicki-Birula, I. Space-time description of squeezing. J. Opt. Soc. Am. B 1987, 4, 1621–1626. [Google Scholar] [CrossRef] [Green Version]
  438. Lobashov, A.A.; Mostepanenko, V.M. Quantum Effects in Nonlinear Insulating Materials in the Presence of a Nonstationary Electromagnetic Field. Theor. Math. Phys. 1991, 86, 303–309. [Google Scholar] [CrossRef]
  439. Lobashov, A.A.; Mostepanenko, V.M. Quantum Effects Associated With Parametric Generation of Light and the Theory of Squeezed States. Theor. Math. Phys. 1991, 88, 913–925. [Google Scholar] [CrossRef]
  440. Hizhnyakov, V.V. Quantum emission of a medium with a time-dependent refractive index. Quant. Opt. 1992, 4, 277–280. [Google Scholar] [CrossRef]
  441. Johnston, H.; Sarkar, S. Moving mirrors and time-varying dielectrics. Phys. Rev. A 1995, 51, 4109–4115. [Google Scholar] [CrossRef]
  442. Artoni, M.; Bulatov, A.; Birman, J. Zero-point noise in a nonstationary dielectric cavity. Phys. Rev. A 1996, 53, 1031–1035. [Google Scholar] [CrossRef]
  443. Saito, H.; Hyuga, H. The dynamical Casimir effect for an oscillating dielectric model. J. Phys. Soc. Jpn. 1996, 65, 3513–3523. [Google Scholar] [CrossRef]
  444. Cirone, M.; Rza̧źewski, K.; Mostowski, J. Photon generation by time-dependent dielectric: A soluble model. Phys. Rev. A 1997, 55, 62–66. [Google Scholar] [CrossRef] [Green Version]
  445. Mendonça, J.T.; Guerreiro, A.; Martins, A.M. Quantum theory of time refraction. Phys. Rev. A 2000, 62. [Google Scholar] [CrossRef]
  446. Braunstein, S.L. A quantum optical shutter. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S28–S31. [Google Scholar] [CrossRef] [Green Version]
  447. Mendonça, J.T.; Guerreiro, A. Time refraction and the quantum properties of vacuum. Phys. Rev. A 2005, 72, 063805. [Google Scholar] [CrossRef]
  448. Hizhnyakov, V.; Kaasik, H. Emission by dielectric with oscillating refractive index. J. Phys. Conf. Ser. 2005, 21, 155–60. [Google Scholar] [CrossRef]
  449. De Liberato, S.; Ciuti, C.; Carusotto, I. Quantum vacuum radiation spectra from a semiconductor microcavity with a time-modulated vacuum Rabi frequency. Phys. Rev. Lett. 2007, 98, 103602. [Google Scholar] [CrossRef] [Green Version]
  450. Bialynicki-Birula, I.; Bialynicka-Birula, Z. 2008 Dynamical Casimir effect in oscillating media. Phys. Rev. A 2007, 78, 042109. [Google Scholar] [CrossRef] [Green Version]
  451. Choi, J.R.; Kim, D.; Chaabi, N.; Maamache, M.; Menouar, S. Zero-point fluctuations of quantized electromagnetic fields in time-varying linear media. J. Korean Phys. Soc. 2010, 56, 775–781. [Google Scholar]
  452. Bei, X.-M.; Liu, Z.-Z. Quantum radiation in time-dependent dielectric media. J. Phys. B: Atom. Mol. Opt. Phys. 2011, 44, 205501. [Google Scholar] [CrossRef]
  453. Ueta, T. The dynamic Casimir effect within a vibrating metal photonic crystal. Appl. Phys. A 2014, 116, 863–871. [Google Scholar] [CrossRef]
  454. Westerberg, N.; Cacciatori, S.; Belgiorno, F.; Dalla Piazza, F.; Faccio, D. Experimental quantum cosmology in time-dependent optical media. New J. Phys. 2014, 16, 075003. [Google Scholar] [CrossRef] [Green Version]
  455. Larré, P.-E.; Carusotto, I. Propagation of a quantum fluid of light in a cavityless nonlinear optical medium: General theory and response to quantum quenches. Phys. Rev. A 2015, 92, 043802. [Google Scholar] [CrossRef] [Green Version]
  456. Lotfipour, H.; Allameh, Z.; Roknizadeh, R.; Heydari, H. Two schemes for characterization and detection of the squeezed light: Dynamical Casimir effect and nonlinear materials. J. Phys. B: Atom. Mol. Opt. Phys. 2016, 49, 065503. [Google Scholar] [CrossRef] [Green Version]
  457. Hasan, F.; O’Dell, D.H.J. Parametric amplification of light in a cavity with a moving dielectric membrane: Landau-Zener problem for the Maxwell field. Phys. Rev. A 2016, 94, 043823. [Google Scholar] [CrossRef] [Green Version]
  458. Ameri, V.; Eghbali-Arani, M.; Soltani, M. Perturbative approach to dynamical Casimir effect in an interface of dielectric mediums. Eur. Phys. J. D 2016, 70, 254. [Google Scholar] [CrossRef] [Green Version]
  459. Westerberg, N.; Prain, A.; Faccio, D.; Öhberg, P. Vacuum radiation and frequency-mixing in linear light-matter systems. J. Phys. Commun. 2019, 3, 065012. [Google Scholar] [CrossRef]
  460. Hizhnyakov, V. Emission of photon pairs in optical fiber - effect of zero-point fluctuations. arXiv 2019, arXiv:1907.04321. [Google Scholar]
  461. Dezael, F.X.; Lambrecht, A. Analogue Casimir radiation using an optical parametric oscillator. EPL 2010, 89, 14001. [Google Scholar] [CrossRef]
  462. Faccio, D.; Carusotto, I. Dynamical Casimir Effect in optically modulated cavities. EPL 2011, 96, 24006. [Google Scholar] [CrossRef]
  463. Hizhnyakov, V.; Kaasik, H.; Tehver, I. Spontaneous nonparametric down-conversion of light. Appl. Phys. A 2014, 115, 563–568. [Google Scholar] [CrossRef]
  464. Hizhnyakov, V.; Loot, A.; Azizabadi, S.C. Enhanced dynamical Casimir effect for surface and guided waves. Appl. Phys. A 2016, 122, 333. [Google Scholar] [CrossRef]
  465. Vezzoli, S.; Mussot, A.; Westerberg, N.; Kudlinski, A.; Saleh, H.D.; Prain, A.; Biancalana, F.; Lantz, E.; Faccio, D. Optical analogue of the dynamical Casimir effect in a dispersion-oscillating fibre. Commun. Phys. 2019, 2, 84. [Google Scholar] [CrossRef]
  466. Román-Ancheyta, R.; Ramos-Prieto, I.; Perez-Leija, A.; Kurt, B.; León-Montiel, R.d.J. Dynamical Casimir effect in stochastic systems: Photon harvesting through noise. Phys. Rev. A 2017, 96, 032501. [Google Scholar]
  467. Tanaka, S.; Kanki, K. The dynamical Casimir effect in a dissipative optomechanical cavity interacting with photonic crystal. Physics 2020, 2, 34–48. [Google Scholar] [CrossRef] [Green Version]
  468. Dodonov, V.V. Dynamical Casimir effect in microwave cavities containing nonlinear crystals. J. Phys. Condens. Matter 2015, 27, 214009. [Google Scholar] [CrossRef] [PubMed]
  469. Braggio, C.; Carugno, G.; Borghesani, A.F.; Dodonov, V.V.; Pirzio, F.; Ruoso, G. Generation of microwave fields in cavities with laser-excited nonlinear media: Competition between the second- and third-order optical nonlinearities. J. Opt. 2018, 20, 095502. [Google Scholar] [CrossRef] [Green Version]
  470. Grove, P.G. On the detection of particle and energy fluxes in two dimensions. Class. Quantum Grav. 1986, 3, 793–800. [Google Scholar] [CrossRef]
  471. Dodonov, V.V. Photon creation and excitation of a detector in a cavity with a resonantly vibrating wall. Phys. Lett. A 1995, 207, 126–132. [Google Scholar] [CrossRef]
  472. Sarkisyan, E.M.; Petrosyan, K.G.; Oganesyan, K.B.; Hakobyan, A.A.; Saakyan, V.A.; Gevorkian, S.G.; Izmailyan, N.S.; Hu, C.K. Detection of Casimir photons with electrons. Laser Phys. 2008, 18, 621–624. [Google Scholar] [CrossRef]
  473. Braggio, C.; Bressi, G.; Carugno, G.; Della Valle, F.; Galeazzi, G.; Ruoso, G. Characterization of a low noise microwave receiver for the detection of vacuum photons. Nucl. Instr. Meth. Phys. Res. A 2009, 603, 451–455. [Google Scholar] [CrossRef]
  474. Kawakubo, T.; Yamamoto, K. Photon creation in a resonant cavity with a nonstationary plasma mirror and its detection with Rydberg atoms. Phys. Rev. A 2011, 83, 013819. [Google Scholar] [CrossRef] [Green Version]
  475. de Castro, A.S.M.; Cacheffo, A.; Dodonov, V.V. Influence of the field-detector coupling strength on the dynamical Casimir effect. Phys. Rev. A 2013, 87, 033809. [Google Scholar] [CrossRef] [Green Version]
  476. Dodonov, A.V.; Dodonov, V.V. Photon statistics in the dynamical Casimir effect modified by a harmonic oscillator detector. Phys. Scr. 2013, T153, 014017. [Google Scholar] [CrossRef] [Green Version]
  477. Dodonov, A.V. Continuous intracavity monitoring of the dynamical Casimir effect. Phys. Scr. 2013, 87, 038103. [Google Scholar] [CrossRef] [Green Version]
  478. de Castro, A.S.M.; Dodonov, V.V. Parametric excitation of a cavity field mode coupled to a harmonic oscillator detector. J. Phys. A: Math. Theor. 2013, 46, 395304. [Google Scholar] [CrossRef]
  479. de Castro, A.S.M.; Dodonov, V.V. Continuous monitoring of the dynamical Casimir effect with a damped detector. Phys. Rev. A 2014, 89, 063816. [Google Scholar] [CrossRef]
  480. Miroshnichenko, G.P.; Trifanova, E.S.; Trifanov, A.I. An indirect measurement protocol of intracavity mode quadratures dispersion in dynamical Casimir effect. Eur. Phys. J. D 2015, 69, 137. [Google Scholar] [CrossRef] [Green Version]
  481. Angaroni, F.; Benenti, G.; Strini, G. Reconstruction of electromagnetic field states by a probe qubit. Eur. Phys. J. D 2016, 70, 225. [Google Scholar] [CrossRef] [Green Version]
  482. Mendonça, J.T. Time refraction and the perturbed quantum vacuum. J. Rus. Laser Res. 2011, 32, 445–453. [Google Scholar] [CrossRef]
  483. Friis, N.; Lee, A.R.; Louko, J. Scalar, spinor, and photon fields under relativistic cavity motion. Phys. Rev. D 2013, 88, 064028. [Google Scholar] [CrossRef] [Green Version]
  484. Benenti, G.; Strini, G. Dynamical Casimir effect and minimal temperature in quantum thermodynamics. Phys. Rev. A 2015, 91, 020502. [Google Scholar] [CrossRef] [Green Version]
  485. Merlin, R. Orthogonality catastrophes in quantum electrodynamics. Phys. Rev. A 2017, 95, 023802. [Google Scholar] [CrossRef] [Green Version]
  486. de Melo e Souza, R.; Impens, F.; Maia Neto, P.A. Microscopic dynamical Casimir effect. Phys. Rev. A 2018, 97, 032514. [Google Scholar] [CrossRef] [Green Version]
  487. Ottewill, A.C.; Takagi, S. Radiation by moving mirrors in curved space-time. Prog. Theor. Phys. 1988, 79, 429–441. [Google Scholar] [CrossRef] [Green Version]
  488. Ruser, M.; Durrer, R. Dynamical Casimir effect for gravitons in bouncing braneworlds. Phys. Rev. D 2000, 76, 104014. [Google Scholar] [CrossRef] [Green Version]
  489. Céleri, L.C.; Pascoal, F.; Moussa, M.H.Y. Action of the gravitational field on the dynamical Casimir effect. Class. Quantum Grav. 2009, 26, 105014. [Google Scholar] [CrossRef]
  490. Zhitnitsky, A.R. Dynamical Casimir effect in a small compact manifold for the Maxwell vacuum. Phys. Rev. D 2015, 91, 105027. [Google Scholar] [CrossRef] [Green Version]
  491. Lock, M.P.E.; Fuentes, I. Dynamical Casimir effect in curved spacetime. New J. Phys. 2017, 19, 073005. [Google Scholar] [CrossRef]
  492. Brevik, I.; Milton, K.A.; Odintsov, S.D.; Osetrin, K.E. Dynamical Casimir effect and quantum cosmology. Phys. Rev. D 2000, 62, 064005. [Google Scholar] [CrossRef] [Green Version]
  493. Rudnicki, L.; Bialynicki-Birula, I. Dynamical Casimir effect in uniformly accelerated media. Opt. Commun. 2010, 283, 644–649. [Google Scholar] [CrossRef] [Green Version]
  494. Sorge, F.; Wilson, J.H. Casimir effect in free fall towards a Schwarzschild black hole. Phys. Rev. D 2019, 100, 105007. [Google Scholar] [CrossRef] [Green Version]
  495. Hawking, S.W. Black hole explosions? Nature 1974, 248, 30–31. [Google Scholar] [CrossRef]
  496. Hawking, S.W. Particle creation by black holes. Commun. Math. Phys. 1975, 43, 199–220. [Google Scholar] [CrossRef]
  497. Frolov, V.P. Black holes and quantum processes in them. Sov. Phys. Uspekhi 1976, 19, 244–262. [Google Scholar] [CrossRef]
  498. Bialynicki-Birula, I.; Bialynicka-Birula, Z. Electromagnetic radiation by gravitating bodies. Phys. Rev. A 2008, 77, 052103. [Google Scholar] [CrossRef] [Green Version]
  499. Unruh, W.G. Notes on black-hole evaporation. Phys. Rev. D 1976, 14, 870–892. [Google Scholar] [CrossRef] [Green Version]
  500. Crispino, L.C.B.; Higuchi, A.; Matsas, G.E.A. The Unruh effect and its applications. Rev. Mod. Phys. 2008, 80, 787–838. [Google Scholar] [CrossRef] [Green Version]
  501. Ben-Benjamin, J.S.; Scully, M.O.; Fulling, S.A.; Lee, D.M.; Page, D.N.; Svidzinsky, A.A.; Zubairy, M.S.; Duff, M.J.; Glauber, R.; Schleich, W.P.; et al. Unruh acceleration radiation revisited. Int. J. Mod. Phys. A 2019, 34, 1941005. [Google Scholar] [CrossRef]
  502. Sorge, F. Dynamical Casimir Effect in a kicked box. Int. J. Mod. Phys. A 2006, 21, 6173–6182. [Google Scholar] [CrossRef]
  503. Mendonça, J.T.; Brodin, G.; Marklund, M. Vacuum effects in a vibrating cavity: Time refraction, dynamical Casimir effect, and effective Unruh acceleration. Phys. Lett. A 2008, 372, 5621–5624. [Google Scholar] [CrossRef] [Green Version]
  504. Guerreiro, A. On the quantum space-time structure of light. J. Plasma Phys. 2010, 76, 833–843. [Google Scholar] [CrossRef] [Green Version]
  505. Good, M.R.R. On spin-statistics and Bogoliubov transformations in flat space-time with acceleration conditions. Int. J. Mod. Phys. A 2013, 28, 1350008. [Google Scholar] [CrossRef] [Green Version]
  506. Fei, Z.; Zhang, J.; Pan, R.; Qiu, T.; Quan, H.T. Quantum work distributions associated with the dynamical Casimir effect. Phys. Rev. A 2019, 99, 052508. [Google Scholar] [CrossRef] [Green Version]
  507. Luis, A.; Sánchez-Soto, L.L. Multimode quantum analysis of an interferometer with moving mirrors. Phys. Rev. A 1992, 45, 8228–8234. [Google Scholar] [CrossRef] [PubMed] [Green Version]
  508. Brif, C.; Mann, A. Quantum statistical properties of the radiation field in a cavity with a movable mirror. J. Opt. B Quantum Semiclass. Opt. 2000, 2, 53–61. [Google Scholar] [CrossRef]
  509. Maclay, G.J.; Forward, R.L. A Gedanken Spacecraft that Operates Using the Quantum Vacuum (Dynamic Casimir Effect). Found. Phys. 2004, 34, 477–500. [Google Scholar] [CrossRef] [Green Version]
  510. Dodonov, V.V. Dynamical Casimir effect in a nondegenerate cavity with losses and detuning. Phys. Rev. A 1998, 58, 4147–4152. [Google Scholar] [CrossRef]
  511. Dodonov, V.V. Time-dependent quantum damped oscillator with ‘minimal noise’: Application to the nonstationary Casimir effect in nonideal cavities. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S445–S451. [Google Scholar] [CrossRef]
  512. Dodonov, V.V.; Dodonov, A.V. The nonstationary Casimir effect in a cavity with periodical time-dependent conductivity of a semiconductor mirror. J. Phys. A: Math. Gen. 2006, 39, 6271–6281. [Google Scholar] [CrossRef]
  513. Dodonov, V.V. Photon distribution in the dynamical Casimir effect with an account of dissipation. Phys. Rev. A 2009, 80, 023814. [Google Scholar] [CrossRef]
  514. Lombardo, F.C.; Mazzitelli, F.D. The quantum open systems approach to the dynamical Casimir effect. Phys. Scr. 2010, 82, 038113. [Google Scholar] [CrossRef] [Green Version]
  515. Dodonov, V.V. Quantum damped nonstationary oscillator and Dynamical Casimir Effect. Rev. Mex. Fis. S 2011, 57, 120–127. [Google Scholar]
  516. Settineri, A.; Macrí, V.; Ridolfo, A.; Di Stefano, O.; Kockum, A.F.; Nori, F.; Savasta, S. Dissipation and thermal noise in hybrid quantum systems in the ultrastrong-coupling regime. Phys. Rev. A 2018, 98, 053834. [Google Scholar] [CrossRef] [Green Version]
  517. Dalvit, D.A.R.; Maia Neto, P.A. Decoherence via the dynamical Casimir effect. Phys. Rev. Lett. 2000, 84, 798–801. [Google Scholar] [CrossRef] [Green Version]
  518. Maia Neto, P.A.; Dalvit, D.A.R. Radiation pressure as a source of decoherence. Phys. Rev. A 2000, 62, 042103. [Google Scholar] [CrossRef] [Green Version]
  519. Schützhold, R.; Tiersch, M. Decoherence versus dynamical Casimir effect. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S120–S125. [Google Scholar] [CrossRef] [Green Version]
  520. Dodonov, V.V.; Andreata, M.A.; Mizrahi, S.S. Decoherence and transfer of quantum states of field modes in a one-dimensional cavity with an oscillating boundary. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S468–479. [Google Scholar] [CrossRef]
  521. Céleri, L.C.; Pascoal, F.; de Ponte, M.A.; Moussa, M.H.Y. Number of particle creation and decoherence in the nonideal dynamical Casimir effect at finite temperature. Ann. Phys. 2009, 324, 2057–2073. [Google Scholar] [CrossRef] [Green Version]
  522. Román-Ancheyta, R.; de los Santos-Sánchez, O.; González-Gutiérrez, C. Damped Casimir radiation and photon correlation measurements. J. Opt. Soc. Am. B 2018, 35, 523–527. [Google Scholar] [CrossRef] [Green Version]
  523. Narozhny, N.B.; Fedotov, A.M.; Lozovik, Y.E. Dynamical Casimir and Lamb effects and entangled photon states. Laser Phys. 2003, 13, 298–304. [Google Scholar]
  524. Dodonov, A.V.; Dodonov, V.V.; Mizrahi, S.S. Separability dynamics of two-mode Gaussian states in parametric conversion and amplification. J. Phys. A: Math. Gen. 2005, 38, 683–696. [Google Scholar] [CrossRef]
  525. Andreata, M.A.; Dodonov, V.V. Dynamics of entanglement between field modes in a one-dimensional cavity with a vibrating boundary. J. Opt. B Quantum Semiclass. Opt. 2005, 7, S11–S20. [Google Scholar] [CrossRef]
  526. Bruschi, D.E.; Louko, J.; Faccio, D. Entanglement generation in relativistic cavity motion. J. Phys. Conf. Ser. 2013, 442, 012024. [Google Scholar] [CrossRef]
  527. Bruschi, D.E.; Louko, J.; Faccio, D.; Fuentes, I. Mode-mixing quantum gates and entanglement without particle creation in periodically accelerated cavities. New J. Phys. 2013, 15, 073052. [Google Scholar] [CrossRef]
  528. Busch, X.; Parentani, R. Dynamical Casimir effect in dissipative media: When is the final state nonseparable? Phys. Rev. D 2013, 88, 045023. [Google Scholar] [CrossRef] [Green Version]
  529. Busch, X.; Parentani, R.; Robertson, S. Quantum entanglement due to a modulated dynamical Casimir effect. Phys. Rev. A 2014, 89, 063606. [Google Scholar] [CrossRef] [Green Version]
  530. Felicetti, S.; Sanz, M.; Lamata, L.; Romero, G.; Johansson, G.; Delsing, P.; Solano, E. Dynamical Casimir effect entangles artificial atoms. Phys. Rev. Lett. 2014, 113, 093602. [Google Scholar] [CrossRef] [Green Version]
  531. Finazzi, S.; Carusotto, I. Entangled phonons in atomic Bose-Einstein condensates. Phys. Rev. A 2014, 90, 033607. [Google Scholar] [CrossRef] [Green Version]
  532. Sinha, K.; Lin, S.-Y.; Hu, B.L. Mirror-field entanglement in a microscopic model for quantum optomechanics. Phys. Rev. A 2015, 92, 023852. [Google Scholar] [CrossRef] [Green Version]
  533. Berman, O.L.; Kezerashvili, R.Y.; Lozovik, Y.E. Quantum entanglement for two qubits in a nonstationary cavity. Phys. Rev. A 2016, 94, 052308. [Google Scholar] [CrossRef] [Green Version]
  534. Amico, M.; Berman, O.L.; Kezerashvili, R.Y. Tunable quantum entanglement of three qubits in a nonstationary cavity. Phys. Rev. A 2017, 96, 032328. [Google Scholar] [CrossRef] [Green Version]
  535. Amico, M.; Berman, O.L.; Kezerashvili, R.Y. Dissipative quantum entanglement dynamics of two and three qubits due to the dynamical Lamb effect. Phys. Rev. A 2018, 98, 042325. [Google Scholar] [CrossRef] [Green Version]
  536. Agustí, A.; Solano, E.; Sabín, C. Entanglement through qubit motion and the dynamical Casimir effect. Phys. Rev. A 2019, 99, 052328. [Google Scholar] [CrossRef] [Green Version]
  537. Zhang, X.; Liu, H.; Wang, Z.; Zheng, T. Asymmetric quantum correlations in the dynamical Casimir effect. Sci. Rep. 2019, 9, 9552. [Google Scholar] [CrossRef]
  538. Romualdo, I.; Hackl, L.; Yokomizo, N. Entanglement production in the dynamical Casimir effect at parametric resonance. Phys. Rev. D 2019, 100, 065022. [Google Scholar] [CrossRef] [Green Version]
  539. Cong, W.; Tjoa, E.; Mann, R.B. Entanglement harvesting with moving mirrors. JHEP 2019, 06, 021. [Google Scholar] [CrossRef] [Green Version]
  540. Sabin, C.; Fuentes, I.; Johansson, G. Quantum discord in the dynamical Casimir effect. Phys. Rev. A 2015, 92, 012314. [Google Scholar] [CrossRef] [Green Version]
  541. Benenti, G.; D’Arrigo, A.; Siccardi, S.; Strini, G. Dynamical Casimir effect in quantum-information processing. Phys. Rev. A 2014, 90, 052313. [Google Scholar] [CrossRef] [Green Version]
  542. Stassi, R.; De Liberato, S.; Garziano, L.; Spagnolo, B.; Savasta, S. Quantum control and long-range quantum correlations in dynamical Casimir arrays. Phys. Rev. A 2015, 92, 013830. [Google Scholar] [CrossRef]
  543. Sabin, C.; Adesso, G. Generation of quantum steering and interferometric power in the dynamical Casimir effect. Phys. Rev. A 2015, 92, 042107. [Google Scholar] [CrossRef] [Green Version]
  544. Peropadre, B.; Huh, J.; Sabin, C. Dynamical Casimir effect for Gaussian boson sampling. Sci. Rep. 2018, 8, 3751. [Google Scholar] [CrossRef] [PubMed]
  545. Narozhny, N.B.; Fedotov, A.M.; Lozovik, Y.E. Dynamical Lamb effect versus dynamical Casimir effect. Phys. Rev. A 2001, 64, 053807. [Google Scholar] [CrossRef]
  546. Shapiro, D.S.; Zhukov, A.A.; Pogosov, W.V.; Lozovik, Y.E. Dynamical Lamb effect in a tunable superconducting qubit-cavity system. Phys. Rev. A 2015, 91, 063814. [Google Scholar] [CrossRef] [Green Version]
  547. Zhukov, A.A.; Shapiro, D.S.; Pogosov, W.V.; Lozovik, Y.E. Dynamical Lamb effect versus dissipation in superconducting quantum circuits. Phys. Rev. A 2016, 93, 063845. [Google Scholar] [CrossRef] [Green Version]
  548. Amico, M.; Berman, O.L.; Kezerashvili, R.Y. Dynamical Lamb effect in a superconducting circuit. Phys. Rev. A 2019, 100, 013841. [Google Scholar] [CrossRef] [Green Version]
  549. Passante, R.; Persico, F. Time-dependent Casimir-Polder forces and partially dressed states. Phys. Lett. A 2003, 312, 319–323. [Google Scholar] [CrossRef] [Green Version]
  550. Rizzuto, L.; Passante, R.; Persico, F. Dynamical Casimir-Polder energy between an excited- and a ground-state atom. Phys. Rev. A 2004, 70, 012107. [Google Scholar] [CrossRef] [Green Version]
  551. Westlund, P.O.; Wennerstrom, H. Photon emission from translational energy in atomic collisions: A dynamic Casimir-Polder effect. Phys. Rev. A 2005, 71, 062106. [Google Scholar] [CrossRef]
  552. Vasile, R.; Passante, R. Dynamical Casimir-Polder force between an atom and a conducting wall. Phys. Rev. A 2008, 78, 032108. [Google Scholar] [CrossRef] [Green Version]
  553. Tian, T.; Zheng, T.Y.; Wang, Z.H.; Zhang, X. Dynamical Casimir-Polder force in a one-dimensional cavity with quasimodes. Phys. Rev. A 2010, 82, 013810. [Google Scholar] [CrossRef]
  554. Messina, R.; Vasile, R.; Passante, R. Dynamical Casimir-Polder force on a partially dressed atom near a conducting wall. Phys. Rev. A 2010, 82, 062501. [Google Scholar] [CrossRef] [Green Version]
  555. Dedkov, G.V.; Kyasov, A.A. Dynamical Casimir-Polder atom-surface interaction. Surf. Sci. 2012, 606, 46–52. [Google Scholar] [CrossRef] [Green Version]
  556. Haakh, H.R.; Henkel, C.; Spagnolo, S.; Rizzuto, L.; Passante, R. Dynamical Casimir-Polder interaction between an atom and surface plasmons. Phys. Rev. A 2014, 89, 022509. [Google Scholar] [CrossRef] [Green Version]
  557. Armata, F.; Vasile, R.; Barcellona, P.; Buhmann, S.Y.; Rizzuto, L.; Passante, R. Dynamical Casimir-Polder force between an excited atom and a conducting wall. Phys. Rev. A 2016, 94, 042511. [Google Scholar] [CrossRef] [Green Version]
  558. Passante, R. Dispersion interactions between neutral atoms and the quantum electrodynamical vacuum. Symmetry 2018, 10, 735. [Google Scholar] [CrossRef] [Green Version]

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Dodonov, V. Fifty Years of the Dynamical Casimir Effect. Physics 2020, 2, 67-104. https://0-doi-org.brum.beds.ac.uk/10.3390/physics2010007

AMA Style

Dodonov V. Fifty Years of the Dynamical Casimir Effect. Physics. 2020; 2(1):67-104. https://0-doi-org.brum.beds.ac.uk/10.3390/physics2010007

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Dodonov, Viktor. 2020. "Fifty Years of the Dynamical Casimir Effect" Physics 2, no. 1: 67-104. https://0-doi-org.brum.beds.ac.uk/10.3390/physics2010007

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