PHYSICAL REVIEW D 103, 065005 (2021)

Dynamical Casimir effect in nonlinear vibrating cavities

† Lianna A. Akopyan 1,2,3,* and Dmitrii A. Trunin 1,2, 1Moscow Institute of Physics and Technology, 141700, Institutskii pereulok, 9, Dolgoprudny, Russia 2Institute for Theoretical and Experimental Physics, 117218, B. Cheremushkinskaya, 25, Moscow, Russia 3Russian Quantum Center, 121205, Bolshoy Boulevard, 30, building 1, Moscow, Russia

(Received 7 December 2020; accepted 12 February 2021; published 11 March 2021)

Nonlinear terms in the equations of motion can induce secularly growing loop corrections to correlation functions. Recently such corrections were shown to affect the particle production by a nonuniformly moving ideal mirror. We extend this conclusion to the cases of ideal vibrating cavity and single semitransparent mirror. These models provide natural IR and UV scales and allow a more accurate study of the loop behavior. In both cases we confirm that two-loop correction to the Keldysh propagator quadratically grows with time. This growth indicates a breakdown of the semiclassical approximation and emphasizes that bulk nonlinearities in the dynamical Casimir effect cannot be neglected for large evolution times.

DOI: 10.1103/PhysRevD.103.065005

I. INTRODUCTION of scalar particles by such boundary conditions has been extensively investigated by different approaches to the A nonuniformly accelerated mirror is known to create mode decomposition of the quantized field [10–18] and particles due to fluctuations of the ubiquitous quantum calculation of the effective action [19–21]. Generalizations fields. This effect, which is often referred to as the dynamical of the scalar DCE with imperfect mirrors were considered Casimir effect (DCE), was theoretically predicted 50 years in [22–25]. Moreover, the simplified model (1.1) can be ago by G. T. Moore who considered a simplified model of implemented using superconducting circuits [26–28]; this one-dimensional ideal cavity [1]. Subsequently it was also idea led to the first experimental observations of the extended to a single-mirror case [2–5]. Similarly to the DCE [29,30]. A comprehensive review of the current Hawking [6] and Unruh [7–9] effects, DCE has no classical theoretical and experimental status of the DCE can be counterparts and illustrates the most fundamental features of found in [31,32]. quantum field theory. We emphasize that the existing theoretical and exper- These features are most clearly manifested in a simpli- imental studies of the DCE are mainly limited to semi- fied two-dimensional model which admits relatively classical (tree-level) effects, i.e., they focus on the simple analytical solutions. This is the model of a free linearized Eq. (1.1) or its analogs. At the same time, in massless scalar field with time-dependent Dirichlet nonstationary quantum field theory semiclassical approxi- boundary conditions: mation often fails to provide the complete answer, because quantum loop corrections substantially affect the state of ð∂2 − ∂2Þϕðt; xÞ¼0; ϕ½t; LðtÞ ¼ ϕ½t; RðtÞ ¼ 0; t x the system. Furthermore, loop corrections to the tree-level ð1:1Þ correlation functions, quantum averages and stress-energy flux indefinitely grow with time. Therefore, at large where functions LðtÞ and RðtÞ determine the positions of evolution times such corrections are significant even if two perfectly reflective mirrors at the moment t. Most of the nonlinear terms in the classical equations of motion seem papers on the DCE (including seminal ones [1–4]) are negligible. For example, loop corrections considerably devoted to this simplified model. In particular, the creation affect the particle creation processes in an expanding universe [33–40], strong electric [41–43], scalar [44–46] *[email protected] and gravitational [47] fields. † [email protected] Recently loop corrections were also shown to play an important role in the DCE [48]. Namely, it was shown that Published by the American Physical Society under the terms of in a simplified two-dimensional model of the DCE, which the Creative Commons Attribution 4.0 International license. describes a massless real scalar field on a single ideal mirror Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, background, quantum loop corrections to the Keldysh and DOI. Funded by SCOAP3. propagator indefinitely grow with the evolution time.

2470-0010=2021=103(6)=065005(22) 065005-1 Published by the American Physical Society LIANNA A. AKOPYAN and DMITRII A. TRUNIN PHYS. REV. D 103, 065005 (2021)

This means that at large evolution times the semiclassical two ideal mirrors which corresponds to an infinitely high approximation is not applicable. Furthermore, the Keldysh potential well. In Sec. III we model a nonideal mirror with a propagator is closely related to the stress-energy flux and delta-functional potential. the state of the quantum field, so loop corrections evidently We would like to consider a nonhomogeneous motion affect the particle creation in the DCE. of the mirrors, i.e., such potential Vðt; xÞ that changes with In this paper we extend the results of [48] to two more time in an arbitrary inertial reference frame. Since the realistic models. First, we examine the case of a vibrating Hamiltonian of such a theory explicitly depends on time, cavity bounded by two perfectly reflecting mirrors. On the quantum loop corrections to the tree-level correlation one hand, this model provides a natural IR scale and allows functions should be calculated with the nonstationary a more accurate study of the secular growth. On the other Schwinger–Keldysh diagrammatic technique [49,50].In hand, it has more experimental significance than the single- this subsection we briefly introduce this technique. mirror model. Second, we discuss the case of a single A comprehensive review can be found in [37,51–55]. nonideal (semitransparent) mirror which is well defined in First of all, suppose we know the state of the system (1.2) the UV region. at a moment t0 and would like to calculate the expectation In both cases we calculate two-loop corrections to the value of the operator O at a moment t. In the interaction Keldysh propagator of a two-dimensional massless real picture we have the following expectation value1: scalar field with quartic (λϕ4) self-interaction. This calcu- † lation essentially reduces to the calculation of the energy hOiðtÞ¼hU ð∞;t0ÞT ½OðtÞUð∞;t0Þi: ð1:3Þ level density and anomalous quantum average which are parts of the Keldysh propagator. We find that in both cases Here T denotes the time-ordering and h i ≡ hinjjini, loop corrections quadratically grow with time. Also we where jini is the state of the system at the moment t0. The show that this growth is associated with the violation of the evolution operator in our case has the following form: conformal invariance by the λϕ4 interaction term. This paper is organized as follows. In Secs. IAand IB Z Z λ t1 we briefly review the nonequilibrium Schwinger–Keldysh T − ϕ4 Uðt1;t2Þ¼ exp i 4 dt dx ðt; xÞ : ð1:4Þ diagrammatic technique and discuss the physical picture t2 underlying the scalar DCE. In the following sections we use this technique to calculate loop corrections to the quantum Now it is straightforward to see that loop corrections to averages of a massless scalar field on various nonstationary an arbitrary tree-level correlation function [i.e., correlation backgrounds. In Sec. II we consider the case of two function (1.3) with O ¼ ϕðt1;x1Þϕðt2;x2Þϕðtn;xnÞ] nonuniformly moving ideal mirrors. We employ the geo- reduces to the sum over all possible products of the metrical method of constructing modes to simplify the following four bare propagators: calculation of loop integrals. Also we illustrate this calcu- −− lation by examples of resonant cavity, synchronized and iG12 ≡ hT ϕ1ϕ2i¼θðt1 − t2Þhϕ1ϕ2iþθðt2 − t1Þhϕ2ϕ1i; unsynchronized “broken” mirror trajectories. In Sec. III þþ ≡ T˜ ϕ ϕ θ − ϕ ϕ θ − ϕ ϕ ; we discuss the case of a single nonideal mirror. For iG12 h 1 2i¼ ðt2 t1Þh 1 2iþ ðt1 t2Þh 2 1i þ− −þ simplicity we assume that the proper acceleration of the iG12 ≡ hϕ1ϕ2i; iG12 ≡ hϕ2ϕ1i: ð1:5Þ mirror is much smaller than the energy scale of semi- transparency. Finally, we discuss the results and conclude Here T˜ is the reverse time-ordering and we denote in Sec. IV. In addition, we consider the stationary case in G12 ¼ Gðt1;x1; t2;x2Þ, ϕ ¼ ϕðt ;xÞ for shortness. In this the Appendix A and discuss miscellaneous calculation i i i notation “þ” and “−” signs mean that ϕ-operators come details in Appendixes B and C. from U† (i.e., from the anti-time-ordered part of the full propagator) or from U (i.e., from the time-ordered part), – † A. Schwinger Keldysh diagrammatic technique respectively. Similarly, “þ” and “−” vertices come from U λ λ In this paper we study quantum loop corrections to the and U and ascribe to the diagram factors i 4 and −i 4, scalar DCE which can be described by the following action respectively. For example, the one-loop correction to the ℏ 1 þ− (we set ¼ c ¼ ): G12 propagator is as follows: Z 1 1 λ 2 ∂ ϕ 2 − ϕ2 − ϕ4 1 S ¼ d x 2 ð μ ðt; xÞÞ 2 Vðt; xÞ ðt; xÞ 4 ðt; xÞ ; If the interaction is turned on and switched off adiabatically and jini is the true vacuum state of the free theory, then the ð1:2Þ interaction cannot disturb the jini state, i.e., jini and jouti states coincide. In this case in-in expectation value (1.3) can be reduced to the in-out expectation value, and Schwinger–Keldysh tech- where potential Vðt; xÞ determines the interaction of the nique reproduces the standard Feynman technique (see [37] for free field and the mirror. In Sec. II we consider the case of the details).

065005-2 DYNAMICAL CASIMIR EFFECT IN NONLINEAR VIBRATING … PHYS. REV. D 103, 065005 (2021) Z ∞ þ− þþ þþ 2 þ− The most interesting case of nonzero loop corrections is ΔG12 ¼ 3iλ dt3G13 ðG33 Þ G32 the case of so-called secular growth when n and/or κ t0Z pq pq ∞ t1þt2 → ∞ Δ − þ− −− 2 −− indefinitely grow in the limit T ¼ 2 , t ¼ t1 − 3iλ dt3G13 ðG33 Þ G32 : ð1:6Þ t2 const (t1 and t2 are external times of the exact Keldysh t0 ¼ propagator). Such a growth means that at some moment Note that we excluded disconnected diagrams because in (T ∼ 1=λ) loop corrections exceed tree-level values even for Schwinger–Keldysh technique such diagrams always can- an infinitesimal coupling constant. After this moment the cel each other. Also we took into account symmetry factor perturbation theory becomes inapplicable, so some kind (in this case it is 12) and included the imaginary unit i into of resummation must be performed in order to estimate the definition of propagators. the exact correlation functions. Such a resummation also It is easy to see that propagators (1.5) obey the relation allows to find the correct final state of the system after Gþþ þ G−− ¼ Gþ− þ G−þ. Hence, it is convenient to interactions are turned off [34–39,42,43]. perform the Keldysh rotation [50–52] and introduce the Besides that, the Keldysh propagator is closely con- Keldysh, retarded and advanced propagators: nected with the stress-energy flux. In a gaussian theory the relation is 1 1 K þ− −þ G G G ϕ1; ϕ2 ; 12 ¼ 2 ð 12 þ 12 Þ¼2 hf gi ∂ ∂ K hTtxi¼ t1 x2 G12jt1¼t2;x1¼x2 : ð1:9Þ R=A −− ∓ G12 ¼ G12 − G12 ¼θð t1 ∓ t2Þh½ϕ1; ϕ2i: ð1:7Þ On the tree level nonzero flux appears due to the ampli- These propagators have a simple physical interpretation. fication of zero-mode fluctuations, i.e., due to the fact that On the one hand, retarded and advanced propagators mode functions gpðt; xÞ do not coincide with a simple describe the propagation of some localized perturbations exponential function. This effect was observed and dis- (e.g., particle or quasiparticle). Hence, at the tree level they cussed in many seminal papers, e.g., [2–5].However,we do not depend on the state of the system (note that this would like to emphasize that quantum loop corrections also agrees with the definition (1.7), because commutator is a can make a significant contribution to the stress-energy c-number). On the other hand, Keldysh propagator contains flux. Indeed, the unlimited time growth of npq and κpq in the information about the state of the system: the decomposition (1.8) inevitably generates a valid con- Z Z tribution to the stress-energy flux (1.9). Moreover, for large 1 K dp dq † times loop corrections start to dominate. This is the other G ¼ δ þha a i g ðt1;x1Þg ðt2;x2Þ 12 2π 2π 2 pq p q p q reason to study the secular growth of quantum averages. Due to these reasons in this paper we calculate loop þhapaqigpðt1;x1Þgqðt2;x2ÞþH:c: ; ð1:8Þ corrections in the limit T ≫ Δt. In addition, we consider small coupling constants, λ → 0, λT ¼ const, in order to † single out the leading quantum corrections to the tree-level where δ ≡ δ p − q , a and a are bosonic creation pq ð Þ p p propagators. This limit has two important implications. and annihilation operators, and gpðt; xÞ is the mode First, loop corrections to the retarded and advanced ϕ Rfunction in the free field decomposition ðt; xÞ¼ propagators are negligible, because they do not grow in dp † 2π ½apgpðt; xÞþapgpðt; xÞ. Usually expectation values the limit T → ∞. For example, the first loop correction to † npq ≡ hapaqi and κpq ≡ hapaqi are diagonal in momenta, the retarded propagator is as follows: n ¼ n δ , κ ¼ κ δ − . In this case quantities n and Z pq p pq pq p p; q p ∞ κ are called energy level density and anomalous quantum R R K R p ΔG12 ¼ −3iλ dt3G13G33G32 average, respectively. In the case when mode functions gp Zt0 are not pure exponentials, these definitions should also take t1 −3 λ R K R ∼ O λ 0 into account the change in the modes (see the discussion in ¼ i dt3G13G33G32 ð T Þ: ð1:10Þ t2 the Sec. IV). However, we will apply the same terminology κ to the quantities npq and pq as well. For the second identity we used the causal structure of the Note that for a vacuum in-state, apjini¼0, both retarded propagator, i.e., took into account theta-functions npq ¼ 0 and κpq ¼ 0. At the tree-level they remain zero in the definition (1.7). Obviously, due to the specific limits during time evolution. Therefore, particle production is of integration this expression cannot grow in the limit related only to the change in the modes.However,in T ≫ Δt for a fixed Δt. The growth as Δt → ∞ may affect nonstationary situation these vacuum expectation values properties of quasiparticles, but it does not affect the state can also receive nonzero loop corrections. This would of the entire system or the stress-energy flux [39]. indicate the change in the state and refute the semiclassical Moreover, it is easy to check that loop corrections do approximation. not change the causal structure of retarded and advanced

065005-3 LIANNA A. AKOPYAN and DMITRII A. TRUNIN PHYS. REV. D 103, 065005 (2021) propagators. Therefore, higher-loop corrections possess the we obtain the following equation on the transverse com- same behavior [37,51–55]. Hence, we can neglect them in ponent of the vector potential Azðt; xÞ [56,57]: the limit in question. Second, in the limit in question leading loop corrections ∂2A ðt; xÞ ∂2A ðt; xÞ ω2 ðt; xÞ z − c2 z þ pe A ðt; xÞ¼0; to the energy level density and anomalous quantum average ∂t2 ∂x2 γðt; xÞ z in the decomposition (1.8) depend only on the average 2 2 4πe nðt; xÞ time: ωpeðt; xÞ¼ : ð1:14Þ me † † npq ¼hU ð∞;t0ÞT ½apðt1Þaqðt2ÞUð∞;t0Þi Here c is the speed of light, ω is Langmuir frequency, e † † pe ≈ hU ð∞;t0ÞT ½a ðTÞa ðTÞUð∞;t0Þi þ ; p q and me are electron charge and mass, nðt; xÞ is electron † density distribution and γðt; xÞ is the Lorentz factor of κpq ¼hU ð∞;t0ÞT ½apðt1Þaqðt2ÞUð∞;t0Þi the plasma sheet. We assume that plasma sheet moves ≈ † ∞ T ∞ hU ð ;t0Þ ½apðTÞaqðTÞUð ;t0Þi þ ; ð1:11Þ along the X-axis, so that function x ¼ xðtÞ describes its position at the moment t. Approximating the electron where ellipsis denote the subleading contributions in the λ → 0 ≫ Δ ∼ 1 λ density distribution by Dirac delta-function, nðt; xÞ¼ limit , T t, T = . Hence, expressions npqðTÞ δ − κ n0l ½x xðtÞ, where n0 is the average electron density and pqðTÞ can be interpreted as the exact energy level and l is the thickness of the electron layer, we obtain the density and anomalous quantum average at the moment T. equation on a two-dimensional massless scalar field: Thus, in this paper we calculate loop corrections to the tree-level quantum averages in the theory (1.2) and show ∂2 ∂2 α − 2 − δ − ϕ 0 that they possess secular growth. We start with the two-loop 2 c 2 ½x xðtÞ ðt; xÞ¼ ; ð1:15Þ ∂t ∂x γðt; xÞ (“sunset”) corrections, which are the simplest nontrivial ffiffiffiffiffiffi loop corrections. It is straightforward to show that in the ϕ p α 4π 2 λ → 0 ≫ Δ ∼ 1 λ where ðt; xÞ¼Azðt; xÞ S⊥, ¼ e n0l=me and S⊥ is limit , T t, T = these corrections are given the area of mirrors (this factor does not affect equations by the following formulas: of motion, but it is necessary for the correct dimensional Z reduction). As a rough approximation we can set l ∼ c=ω , 2 2 2 pe n T ≈ 2λ d x1d x2θ T − t1 θ T − t2 pqð Þ ð Þ ð Þ which yields α ∼ cωpe. A typical metal mirror has Z 16 −1 2 5 −1 3 ωpe ∼ 10 s , i.e., α=c ∼ 10 cm . Practically such dk × gp;1gq;2 gk;1gk;2 ; ð1:12Þ mirror also can be implemented by a breaking plasma 2π 17 −3 Z wakewave with typical parameters n0 ∼ 10 cm and −2 2 4 −1 2 2 2 l ∼ 10 cm; in this case α=c ∼ 10 cm [56,58].Yet κ T ≈−2λ d x1d x2θ T − t1 θ t1 − t2 pqð Þ ð Þ ð Þ another way to implement rapidly moving mirror involves Z 3 superconducting circuits which approximately reproduce dk 2 −1 g g 1 α ∼ 10 – × ½ p;1gq;2 þ gp;2gq;1 2π k; gk;2 ; ð1:13Þ the model (1.15) with =c cm [26 29]. Note that in the limit α → ∞ Eq. (1.15) reproduces the case of the ideal mirror: where we denoted gp;i ¼ gpðti;xiÞ for shortness. In the following sections we determine the exact modes for ∂2 ∂2 several types of the potential Vðt; xÞ in the theory (1.2), − 2 ϕ 0 ϕ 0 2 c 2 ðt; xÞ¼ ; ½t; xðtÞ ¼ : ð1:16Þ substitute them into the identities (1.12) and (1.13) and ∂t ∂x show that both n ðTÞ ∼ ðλTÞ2 and κ ðTÞ ∼ ðλTÞ2.We pq pq Moreover, the transmission coefficient of a nonideal mirror emphasize that these functions should be considered as is proportional to ω=α if ω ≪ α, where ω is the energy parts of the Keldysh propagator (1.8) which has a more of an incident wave. A derivation of this statement can fundamental meaning. be found in Sec. III A. Therefore, at such energy scales we can use a simplified Eq. (1.16) instead of more accurate B. Physical picture Eq. (1.15). It is useful to keep in mind the following physical picture Finally, recall that we would like to consider quantum related to the scalar DCE. Consider electromagnetic field corrections to free scalar theories (1.15) and (1.16).We 4 interacting with a thin layer of cold electronic plasma. choose λϕ as the simplest example of a non-Gaussian Fixing the Lorentz gauge and ignoring transverse effects2 theory (see Sec. IA). On the one hand, such interaction can be interpreted as a toy model of low-energy effective 2I.e., neglecting interactions between transverse and longitudinal QED [59,60], although there is no direct correspondence modes and setting transverse momentum k⊥ ¼ 0. The opposite case between these theories. On the other hand, in the context of 2 2 4 reproduces a massive scalar field with mass m ¼ k⊥. circuit QED λϕ theory describes a nonlinear waveguide

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[27,61]. In particular, such a nonlinear self-interacting term to the mirrors. In Secs. II C and II D we consider two of can be simulated by embedding a SQUID in the trans- the most natural adjustment options. In Sec. II E we also mission line [62–65]. In this case the dimensionless discuss periodically oscillating mirror trajectories which coefficient of nonlinearity (the ratio of the Kerr coefficient model a resonant cavity. and characteristic energy) can be estimated as K ∼ 10−6.In 2 our terminology this coefficient corresponds to K ∼ λ=ω0 ∼ A. Geometrical method of calculating modes 2 λΛ where ω0 is the characteristic energy of oscillations In this subsection we discuss the quantization of a and Λ is the characteristic size of the system. We expect massless scalar field on the background of two perfectly that such nonlinear effects can be combined with the reflective mirrors: reflecting boundary conditions and implemented using μ superconducting circuits similarly to the tree-level mea- ∂μ∂ ϕ ¼ 0; ϕ½t; LðtÞ ¼ ϕ½t; RðtÞ ¼ 0; ð2:1Þ surements [29,30]. In the remainder of this paper we use units ℏ ¼ c ¼ 1 where LðtÞ and RðtÞ denote the position of the left and unless otherwise specified. Also we assume the ðþ; −Þ right mirror at the moment t, respectively. We assume signature for the metric tensor. that mirrors are at rest before the moment t ¼ 0, i.e., Lðt<0Þ¼0 and Rðt<0Þ¼Λ. As was shown in [1,2,5], II. TWO IDEAL MIRRORS the quantized field is represented by the following mode decomposition: First of all, note that the theory of a massless scalar field with a single mirror considered in [48] does not have a X∞ † natural IR cutoff. This makes it difficult to distinguish ϕðt; xÞ¼ ½angnðt; xÞþangnðt; xÞ; ð2:2Þ between the true secular growth and standard IR divergen- n¼1 ces of loop integrals. For example, such artifacts arise in a † δ 0 pure stationary theory at small evolution times (see where ½am;an¼ mn. In the stationary case (LðtÞ¼ , Appendix A). In this section we study this issue by adding RðtÞ¼Λ for all t) the n ¼ 1 mode corresponds to a 3 ω π the second mirror at a finite distance. For simplicity we standing wave with the frequency 1 ¼ Λ. This is the assume that both mirrors are perfectly reflecting. lowest energy excitation of the cavity. The mode functions We confirm that at small evolution times quantum for an arbitrary motion of the mirrors can be written in averages in the two-mirror problem exhibit an unphysical terms of two auxiliary functions GðzÞ and FðzÞ: quadratic growth which can be eliminated by the proper i regularization (Sec. II C 1). However, at large evolution ffiffiffiffiffiffiffiffi −iπnGðtþxÞ −iπnFðt−xÞ gnðt; xÞ¼p ½e − e ; ð2:3Þ times this unphysical growth is replaced by a physically 4πn meaningful secular growth which indicates a change in the ’ state of the theory. On the one hand, this growth is which satisfy the generalized Moore s equations: associated with the violation of conformal invariance by − − 0 the λϕ4 interaction term. On the other hand, it is closely G½t þ LðtÞ F½t LðtÞ ¼ ; related to the violation of the energy-conservation law on a G½t þ RðtÞ − F½t − RðtÞ ¼ 2: ð2:4Þ nonstationary background (i.e., to the pumping of energy by an external force). We also reproduce the results of [48] In a stationary case these functions can be easily found z in the limit of infinitely distant mirrors. to be GðzÞ¼FðzÞ¼Λ. For arbitrary trajectories LðtÞ and In this section we mainly work with “broken” mirror RðtÞ Eqs. (2.4) can be solved recursively by the geometrical trajectories. Physically, such a trajectory corresponds to a method proposed in [10,11] and extended in [12].We single sudden kick which results in a discontinuous change briefly discuss the method in this subsection and apply it to in the velocity of the mirror. We assume that both mirrors several types of motions in Secs. II C, II D and II E. Recall move along “broken” trajectories. Moreover, we would like that we assume both mirrors are at rest before the moment z to consider asymptotically stationary motions, so we need t ¼ 0. This implies Gðz ≤ ΛÞ¼Fðz ≤ 0Þ¼Λ. Therefore, to equate the final velocities of the mirrors. Note that we it is convenient to define G static region (t þ x ≤ Λ) and F also need to adjust the moments when the kicks are applied static region (t − x ≤ 0) where the corresponding functions tþx t−x are fixed, Gðt þ xÞ¼ Λ , Fðt − xÞ¼ Λ . 3Another possibility to impose a natural IR cutoff is to The general idea of the geometrical method is to trace consider fields with nonzero physical mass. This approach was back functions GðzÞ and FðzÞ along the sequence of null investigated in [66]. However, the equations of motion for a lines until a null line intersects the time axis in a static massive scalar field on a nonhomogeneously moving mirror background are very difficult to solve. Due to the same reason region. In other words, this method uses that functions − loop corrections to npq and κpq are very difficult to calculate Gðt þ xÞ and Fðt xÞ from the decomposition (2.3) are analytically in this case. constant on the null lines t þ x ¼ const and t − x ¼ const.

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This allows to trace functions G and F along a null line between the mirrors. At the same time, Eqs. (2.4) relate functions G and F on reflected null lines. This, in turn, allows us to proceed until a null line reaches a static region where G or F is known. Let us illustrate this idea on the function GðzÞ (see Fig. 1). First, we draw a null line from the point ðz; 0Þ until it intersects the right mirror at the point ðt1;Rðt1ÞÞ: z ¼ t1 þ Rðt1Þ; hence;GðzÞ¼Gðt1 þ Rðt1ÞÞ: ð2:5Þ

Then we relate functions G and F on the right mirror:

GðzÞ¼Gðt1 þ Rðt1ÞÞ ¼ Fðt1 − Rðt1ÞÞ þ 2; ð2:6Þ FIG. 1. Calculation of the tfinal for the F static region (to the left) and G static region (to the right). Bold solid lines denote and draw a null line from the point ðt1;Rðt1ÞÞ to ðt2;Lðt2ÞÞ: word-lines of the mirrors, dashed lines denote null lines, and light gray areas denote F or G static regions. t1 − Rðt1Þ¼t2 − Lðt2Þ; hence;

Fðt1 − Rðt1ÞÞ ¼ Fðt2 − Lðt2ÞÞ: ð2:7Þ Note that the value of GðzÞ increases by 2 every time the null line hits the right mirror. This process is terminated Finally, we switch back to the function G and find the next only when a null line enters a static region where FðzÞ or intersection of the null line and the right mirror: GðzÞ can be evaluated explicitly. There are two possible ways to enter such a region. First, a null line reflecting off the right mirror can enter the F static region. Second, a null t2 þ Lðt2Þ¼t3 þ Rðt3Þ; hence; line reflecting off the left mirror can enter the G static Fðt2 − Lðt2ÞÞ ¼ Gðt2 þ Lðt2ÞÞ ¼ Gðt3 þ Rðt3ÞÞ: ð2:8Þ region. In both cases function GðzÞ reduces to the following expression: This defines the step of the recursion: 2 tfinal GðzÞ¼ n þ Λ ; ð2:10Þ GðzÞ¼Gðt1 þ Rðt1ÞÞ

¼ Gðt3 þ Rðt3ÞÞ þ 2 where n is the number of reflections off the right mirror and t is the moment at which the last null line intersect the 4 final ¼ Gðt5 þ Rðt5ÞÞ þ ¼: ð2:9Þ time-axis in the static region (see Fig. 1):

 P P n n−1 z − 2½ 1 Rðt2i−1Þ − 1 Lðt2iÞ; for F static region; t ¼ Pi¼ Pi¼ ð2:11Þ final − 2 n − n z ½ i¼1 Rðt2i−1Þ i¼1 Lðt2iÞ; for G static region:

Note that for timelike mirror worldlines moments Finally, let us discuss the possible choices for the … t1;t2; ;tfinal always exist and are unique [2,10]. functions LðtÞ and RðtÞ. We would like to consider Hence, the identity (2.10) correctly restores the function asymptotically uniform trajectories, i.e., trajectories with GðzÞ for all values of z. Also note that function (2.10) is fixed velocities L_ ð∞Þ¼R_ ð∞Þ¼β , jβ j < 1. Such 2Λ continuous because function tfinalðzÞ decreases by trajectories describe a cavity that is approximately sta- at points where the number of reflections nðzÞ increases tionary in the infinite past and future but undergoes by one. expansion or contraction at intermediate times. For con- The function FðzÞ can be restored using the similar venience we choose a coordinate system where both procedure [12]. However, for relatively simple functions mirrors are at rest (β− ¼ 0) before the moment t ¼ 0. LðtÞ it may be more convenient to use the identity The simplest example of such a nonstationary motion is a F½t − LðtÞ ¼ G½t þ LðtÞ. In this case it is sufficient to combination of so-called “broken” trajectories each of find the inverse function of zðtÞ¼t − LðtÞ (provided that which describes a single discontinuous change in the GðzÞ is known). velocity of the mirror:

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xðtÞ¼xðtxÞþβðt − txÞθðt − txÞ; ð2:12Þ where x ¼ L, R and tx is the moment when the corresponding mirror experiences a sudden kick. We remind that LðtLÞ¼0 1 and RðtRÞ¼Λ. The case tL ¼ tR ¼ 0 and β ¼ 4 is depicted on the Fig. 1. This trajectory can be considered as an approximation to a finite-period motion with constant proper acceleration w: 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  > 1 1 2 − 2 − 1 0 − γβ < w þ w ðt txÞ ; for γβ :> 1 pffiffiffiffiffiffiffiffi1 − 1 þ βðt − t Þ; for t − t > ; w 1−β2 x x w or to an eternally accelerated motion with velocity ex- First, let us remind that two-loop corrections to the ponentially close to β: energy level density and anomalous quantum average in the λϕ4 theory are given by the following expressions: β − − − β − − 1 − wðt txÞ xðtÞ xðtxÞ¼ ðt txÞ ð e Þ; for t>tx: Z Z Z w T Rðt1Þ T ≈ 2λ2 ð2:14Þ npqðTÞ dt1 dx1 dt2 t0 Lðt1Þ t0 Z ∞ Both of these trajectories reproduce (2.12) in the limit Rðt2Þ X w → ∞ and smoothly connect asymptotically uniform × dx2 Ip;m;n;kðt1;x1ÞIq;m;n;kðt2;x2Þ; regions. For the tree-level calculations this difference may Lðt2Þ m;n;k¼1 be crucial, because discontinuity in the velocity generates a ð2:17Þ singular stress-energy flux (e.g., see [2,3,67]). At the same Z Z Z time, in the following subsection we will argue that loop- T Rðt1Þ t1 κ ≈−2λ2 level calculations coincide for all asymptotically uniform pqðTÞ dt1 dx1 dt2 β 1 t0 Lðt1Þ t0 mirror trajectories (except the case j j¼ which cannot be Z ∞ Rðt2Þ X considered in our approach). As a result, two-loop correc- c × dx2 ðI ðt1;x1Þ tions to n and κ quadratically grow with time for all such p;m;n;k pq pq Lðt2Þ m;n;k¼1 trajectories, although the prefactors of this growth depend ↔ on the intermediate motion. Hence, we can use a simple × Iq;m;n;kðt2;x2Þþðp qÞÞ; ð2:18Þ “broken” trajectory (2.12) to illustrate the key points of the calculation. where we introduced for shortness

B. Secular growth as a consequence Ip;m;n;kðt1;x1Þ¼gpðt1;x1Þgmðt1;x1Þgnðt1;x1Þgkðt1;x1Þ; of the conformal invariance violation c Ip;m;n;kðt1;x1Þ¼gpðt1;x1Þgmðt1;x1Þgnðt1;x1Þgkðt1;x1Þ: The solution (2.3) was inspired by the conformal ð2:19Þ invariance of the free massless scalar field [1,2]. In fact, it is easy to check that the following conformal trans- We would like to estimate these integrals using the formation: conformal transformation (2.15). As we explained in w þ s ¼ Gðt þ xÞ;w− s ¼ Fðt − xÞ; ð2:15Þ the Sec. IA, we keep only the leading contributions in the limit λ → 0, T → ∞. Integrands of (2.17) and (2.18) reduces the problem (2.1) to the stationary problem with consist of oscillating exponents, so we can estimate both mirrors at rest: 1 c 1 jIp;m;n;kðt; xÞj < π4 and jIp;m;n;kðt; xÞj < π4. This implies μ ∂μ∂ ϕ ¼ 0; ϕðw; s ¼ 0Þ¼ϕðw; s ¼ 1Þ¼0; ð2:16Þ that the integration over the regions with finite areas cannot provide growing contributions to (2.17) and (2.18). Such which immediately implies the solution (2.3). However, contributions are negligible in the limit in question; there- 4 the λϕ interaction breaks the conformal invariance down. fore, such integrations can be excluded. Due to the same In this subsection we will show that this breakdown reason we can set t0 ¼ 0 (the integrals cannot indefinitely manifests itself in the secular growth of loop corrections. grow in the limit t0 → −∞ because mirrors are stationary For simplicity we will assume that the velocities of the in the past, see Appendix A). This rationale allows us to mirrors coincide at large times, i.e., L_ ðtÞ¼R_ ðtÞ¼β after simplify the integrals and single out the leading, growing some time t. The reasons for this (physically meaningful) with time contributions to (2.17) and (2.18). This is the only requirement will be explained below. type of contributions which survive in the limit in question.

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After the conformal transformation (2.15) we obtain the X∞ 0 0 − − 2 iπny following integrals: g ðyÞ¼g ½y y nðyÞ ¼ gne ; Z n¼−∞ 1 2 Z Z 0 −iπny T Rðt1Þ gn ¼ g ðyÞe dy; 2 0 dt1 dx1Ip;m;n;kðt1;x1Þ ∞ 0 Lðt1Þ X Z Z 0 0 − − 2 iπny G½TþLðTÞ 1 f ðyÞ¼f ½y y nðyÞ ¼ fne ; ≈ 0 0 − −∞ dw dsg ðw þ sÞf ðw sÞIp;m;n;kðw; sÞ; Z n¼ 0 0 1 2 0 −iπny 2:20 fn ¼ f ðyÞe dy; ð2:22Þ ð Þ 2 0

where nðyÞ ∈ N and 0 integral over dv in (2.21) has contributions dz z¼wþs dz z¼w−s that do not depend on u: the conformal factor, Ip;m;n;kðw; sÞ¼hpðw; sÞhmðw; sÞ × − π pffiffiffiffiffiffii i nðwþsÞ − Z hnðw; sÞhkðw; sÞ and hpðw; sÞ¼ 4πp ½e uþ2 0 −iπðpþmþnþkÞv −iπnðw−sÞ c dvg ðvÞe e . The integrals of Ip;m;n;kðt; xÞ and Ip;m;n;kðt; xÞ u 0 0 − Z ∞ have the same structure. Note that g ðw þ sÞ and f ðw sÞ uþ2 X iπlv −iπðpþmþnþkÞv are positive if we consider spacelike mirror worldlines with ¼ dv gne e jL_ ðtÞj < 1 and jR_ ðtÞj < 1. u l¼−∞ 2 Finally, let us make yet another change and introduce the ¼ gpþmþnþk: ð2:23Þ coordinates u ¼ w − s, v ¼ w þ s: Therefore, the integral (2.21) indefinitely grows with time, ≠ 0 Z Z provided that gpþmþnþk and T>t: T Rðt1Þ dt1 dx2Ip;m;n;kðt1;x1Þ Z 0 G½TþLðTÞ Lðt1Þ 0 Z Z duf ðuÞgpþmþnþk G½TþLðTÞ uþ2 1 0 0 G½tþLðtÞ ≈ du dv g ðvÞf ðuÞIp;m;n;kðu; vÞ: 0 2 ∼ − − u gpþmþnþk½T LðTÞ t þ LðtÞ: ð2:24Þ ð2:21Þ This, in turn, immediately implies the secular growth of npq and κpq: For general trajectories LðtÞ and RðtÞ this integral is very complex, but in some physically meaningful cases − − 2 ≈ 2λ2 ½T LðTÞ t þ LðtÞ it is significantly simpler that the initial integrals (2.17) npqðTÞ 4pffiffiffiffiffiffi ð4πÞ pq and (2.18). X∞ Namely, assume that the velocities of the mirrors coincide gpþmþnþkgq m n k × þ þ þ ; ð2:25Þ _ _ β mnk for large times, i.e., LðtÞ¼RðtÞ¼ for t>t.Inthiscase m;n;k¼1 the geometrical picture of Sec. II A implies that functions Δ 2 GðzÞ and FðzÞ periodically grow, i.e., Gðz þ zGÞ¼ 2 ½T − LðTÞ − t þ Lðt Þ 2 Δ 2 κ ðTÞ ≈−λ pffiffiffiffiffiffi GðzÞþ , Fðz þ zFÞ¼FðzÞþ . In the geometrical pq ð4πÞ4 pq language of Sec. II A the increase of the argument by Δz ¼ G X∞ ↔ 2Λ 2Λ g−pþmþnþkgqþmþnþk þðp qÞ or Δz ¼ adds an additional light ray reflection × : 1−jβj F 1þjβj mnk Λ m;n;k¼1 cycle to the derivation of G or F, respectively; is the distance between the mirrors at the reference frame for t>t. ð2:26Þ Hence, starting from some moment y the inverse functions gðyÞ and fðyÞ also periodically grow, i.e., gðy þ 2Þ¼ We remind that we keep only the leading expressions in the λ → 0 ≫ Δ ∼ 1 λ gðyÞþΔzG, fðy þ 2Þ¼fðyÞþΔzF.Whatisevenmore limit , T t, T = . The oscillating contributions important, their derivatives are simply periodic: g0ðy þ 2Þ¼ and contributions of the form λ2Tα with α < 2 simply die out g0ðyÞ, f0ðy þ 2Þ¼f0ðyÞ. Therefore, they can be expanded in this limit. Note that the sums over the virtual momenta are ∼ L ≫ 1 into a Fourier series: convergent because gn n for n :

X∞ g g X∞ Λ2 pþmþnþk qþmþnþk ∼ < ∞: ð2:27Þ mnk p m n k q m n k mnk m;n;k¼1 m;n;k¼1 ð þ þ þ Þð þ þ þ Þ

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Please also note that for some trajectories Fourier coef- z − Λ 2Λk t1 ¼ ; …;t2 1 ¼ t1 − ; ficients may be zero for all possible virtual momenta. In this 1 þ β kþ 1 − β2 κ case npq and pq do not receive growing with time Λ − 1 2 … contributions. An example of such a nonstationary motion t2k ¼ t2k−1 1 − β ;k¼ ; ; ;n: ð2:29Þ can be found in Sec. II D. Let us emphasize that secular growth (2.25) and (2.26) has a clear physical origin. In a fully stationary situation The total number of reflections is easily found using the energy conservation law forbids any kinetics (see periodical reflection pattern: Appendix A); however, the nonstationarity of the back-   ground violates this law4 and allows usually forbidden 1 − β z − Λ n ¼ ; ð2:30Þ processes to occur even at large evolution times. This 2 Λ violation is reflected in nonzero high order Fourier coef- ficients (2.22). Moreover, there are always contributions to where dxe denotes the least integer greater than or equal to (2.17) and (2.18) that do not depend on the integration x. Finally, F and G static regions in this picture correspond times (or their transformed counterparts), because mode to the following values of z>Λ: functions contain both incident and reflected waves. This results in the quadratic growth of quantum averages. 2Λ 2Λ Thus, the problem of calculating loop corrections ∶ − 1 − Λ − Λ F static region ðn Þ 1 − β

C. Simultaneous kicks It is also convenient to introduce the new coordinate In this subsection we consider the case of two simulta- δ ≡ x − LðtÞ, 0 < δ < Λ, which measures the distance to neous kicks [t ¼ t ¼ 0 in the notation (2.12)]: L R the left mirror. In these notations functions Gðt þ xÞ and Fðt − xÞ are related by a simple shift: LðtÞ¼βtθðtÞ;RðtÞ¼Λ þ βtθðtÞ: ð2:28Þ 1 þ β This type of the mirror motion is depicted on the Fig. 1.For Fðt; δÞ¼F½ð1 − βÞt − δ¼G ð1 þ βÞt − δ ; certainty we consider positive final velocities, 0 < β < 1, 1 − β although the discussion of this subsection is equally Gðt; δÞ¼G½ð1 þ βÞt þ δ: ð2:33Þ applicable to the case −1 < β < 0. Note that the distance between the mirrors in the observational reference frame is − Λ In this picture it is straightforward to see that functions G always fixed to be RðtÞ LðtÞ¼ . At the same time, the 2Λ and F periodically grow with time, i.e., Gðt þ 2 ; δÞ¼ distance in thep comovingffiffiffiffiffiffiffiffiffiffiffiffiffi frame reduces from Λ at the past 1−β 2 δ 2 2Λ δ δ 2 infinity to Λ 1 − β at the future infinity. Gðt; Þþ and Fðt þ 1−β2 ; Þ¼Fðt; Þþ . When these Let us apply the geometrical method to find the functions are multiplied by −iπn and substituted into the modes for the trajectories (2.28). First, the identity exponent of ex, this shift yields a factor e−2iπn ¼ 1. Hence, F½t − LðtÞ ¼ G½t þ LðtÞ immediately implies FðzÞ¼ the mode functions (2.3) are simply periodic with the 1 β 2Λ þ period Δt ¼ 2. Gð1−β zÞ for z>0, so it is sufficient to consider only the 1−β construction of GðzÞ. Second, moments of reflections are as The explicit form of the mode functions straightfor- follows: wardly follows from the identities (2.31), (2.32) and (2.33). The corresponding expressions can be found in the Appendix B. These expressions are too bulky to integrate 4It is obvious that quantum scalar field and nonuniformly moving mirror form an open system. Therefore, external forces them explicitly for arbitrary evolution times. However, can pump in and out its energy. This is what we call a violation of it is still possible to estimate the quantum averages (2.17) the energy conservation law. and (2.18) in the limits T ≪ Λ and T ≫ Λ.

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1. Small evolution times and regularization Λ Adapting the expressions from Appendix B to the region t<1þjβj we obtain the following identities for the mode functions: 8 1 β − π tþx −iπn þ t−x > pffiffiffiffiffiffii ½e i n Λ − e 1−β Λ ; if βt 4πn − π tþx − π t−x g ðt; xÞ¼ pffiffiffiffiffiffii ½e i n Λ − e i n Λ ; if t 4πn > 1−β 2β : −iπnð tþxþ Þ − π t−x pffiffiffiffiffiffii ½e 1þβ Λ 1þβ − e i n Λ ; if Λ − t

Note that in the G static region these modes coincide with the modes for a single mirror (compare with [48]). T We would like to calculate integrals (2.17) and (2.18) in the limit T ≪ Λ. Since the Λ is a small parameter here, we can T expand integrals into the TaylorR seriesR in Λ and keep the leadingR order.R Keeping in mind the stationary situation T Λþβt T Λþβt c (Appendix A) we expect integrals 0 dt βt dx Ip;m;n;kðt; xÞ and 0 dt βt dx Ip;m;n;kðt; xÞ to grow linearly with T.At the same time, the areas of the leftmost (βt

stat The same conclusion applies to the integrals (2.17) and and κpq → 0 (see Appendix A). Hence, the regularization (2.18). Hence: does not obscure the true secular growth which indicates the change in the state of the system and results in nonzero stat 2 stat 2 npq ≈ npq ∼ ðλΛTÞ ; κpq ≈ κpq ∼−ðλΛTÞ : ð2:36Þ corrections to the stress-energy tensor. Note that the analysis of this subsection does not depend In the limit of infinitely distant mirrors (Λ → ∞ and on the form of trajectories LðtÞ and RðtÞ after t ¼ 0 because πp p ¼ Λ ¼ const) this behavior can persist for a long time. for T ≪ Λ signals from the mirrors affect only regions with However, this fake “secular growth” does not correspond to areas ∼T2. Hence, the suggested regularization scheme can the change in the state of the system; in fact, it is just an be applied to an arbitrary motion of the mirrors. artifact of the incorrect IR cut-off choice. This unphysical Finally, note that loop calculations of the Sec. III and divergence can be cured by subtracting the corresponding paper [48] implicitly assume exactly the same regulariza- quantities from the stationary theory: tion for the quantum averages. Due to this reason we believe that these calculations correctly predict the behavior reg ≡ − stat κreg ≡ κ − κstat npq npq npq ; pq pq pq : ð2:37Þ of npq and κpq in the single-mirror limit Λ → ∞. Essentially, such a subtraction extends the integration 2. Large evolution times intervals to large negative times, i.e., reproduces the calculations in the limit t0 → −∞ instead of t0 ¼ 0 Let us apply the machinery developed in the Sec. II B (compare with the calculations of two-loop corrections to estimate npq and κpq for large times, T ≫ Λ. First, in [48]). Obviously, this way of regularization does not we rewrite the expression (2.32) using Heaviside step stat affect large times, T ≫ Λ, because in this limit npq → 0 functions:

 X∞ z z 2ðn − 1Þ z 2n 1 − β z 2βn GðzÞ¼θðΛ − zÞ þ θ − 1 − − θ − þ Λ Λ 1 − β Λ 1 − β 1 þ β Λ 1 þ β n¼1  2 2 2β θ z − n − θ z − 1 − n z − n þ Λ 1 − β Λ 1 − β Λ 1 − β : ð2:38Þ

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Having this representation it is straightforward to find the inverse function:  X∞ 1 β 2β Λ θ 1 − Λ θ − 2 − 1 − θ − 2 þ Λ − n gðyÞ¼ ð yÞ y þ ½ ðy ð n ÞÞ ðy nÞ 1 − β y 1 − β n¼1  2β Λ θ − 2 − θ − 2 1 Λ n þ½ ðy nÞ ðy ð n þ ÞÞ y þ 1 − β ; ð2:39Þ

X∞ ð1 − ð−1ÞpþmþnþkÞð1 − ð−1ÞqþmþnþkÞ and its derivative: S ¼ : p;q 4mnk p m n k q m n k m;n;k¼1 ð þ þ þ Þð þ þ þ Þ 2βΛ X∞ 0 Λ θ − 2 − 1 − θ − 2 ð2:44Þ g ðyÞ¼ þ 1 − β ½ ðy ð n ÞÞ ðy nÞ: n¼1 Note that S ¼ 0 if p and q have different parity. ð2:40Þ p;q However, Spq has nonzero nondiagonal terms that corre- spond to p and q of the same parity. This means that n The corresponding Fourier coefficients are also easy to pq and κpq do not reduce to the diagonal form as it happens in calculate: the most cases (e.g., see [37,39,42,43]). Z 1 2 Thus, in the case of simultaneous kicks (2.28) quantum 0 −iπny averages quadratically grow both with small (T ≪ Λ) gn ¼ 2 g ðyÞe dy 0 and large (T ≫ Λ) evolution times. The regularization 2βΛ 1 − ð−1Þn (2.37) eliminates the meaningless growth at small times ¼ − ; for n ≠ 0; 1 − β 2iπn and does not affect the meaningful growth at large times. Λ In the intermediate region these asymptotics are connected g0 ¼ 1 − β : ð2:41Þ by a smooth function which can be calculated numeri- cally (Fig. 2). Λ → ∞ πp 0 Finally, note that in the limit and p ¼ Λ ¼ const Finally, using that in this case t ¼ we obtain the approximate expressions for the quantum averages: two-mirror problem qualitatively reproduces the single- mirror problem considered in [48]. In both these cases S quantum averages quadratically grow with evolution ≈ λβΛ 2 p;q npqðTÞ ð TÞ × 6pffiffiffiffiffiffi ; ð2:42Þ time, although the velocity-dependent prefactors of these 32π pq 2 β Spqffiffiffiffi 2 n ∼ 2 p λT growths are slightly different: pq ð1þβÞ pq ð Þ in the 2 S 2 S− þ S − one-mirror case and n ∼ β ppqffiffiffiffi λT in the two-mirror κ ≈− λβΛ 2 p;q p; q pq pq ð Þ pqðTÞ ð TÞ × 6pffiffiffiffiffiffi ; ð2:43Þ 64π pq case. For small final velocities, jβj ≪ 1, this difference is negligible, which extends the correspondence to a quanti- where we introduced a short notation for the sum: tative level.

(a) (b)

FIG. 2. Numerically estimated functions n11ðTÞ (a) and jκ11ðTÞj (b). Different colors correspond to different mirror velocities β. The time T is measured in units of Λ and quantum averages are measured in units of 2λ2. Note that the regularization (2.37) was not applied.

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D. Synchronized kicks We would like to estimate the large-time asymptotics of κ The other notable option to adjust mirror trajectories functions npqðTÞ and pqðTÞ which do not depend on the is to consider synchronized kicks5 connected by a null line low-time behavior of the mode functions. Hence, for our 2Λ [tL ¼ tR þ Λ in the notation (2.12)]: purposes it is sufficient to consider only the region t>1−β where the mode functions (2.3) acquire the following form: LðtÞ¼βðt − ΛÞθðt − ΛÞ;RðtÞ¼Λ þ βtθðtÞ: ð2:45Þ

1−β i − π 1−β t − π β − π δ π δ In this case F and G static regions coincide, which ffiffiffiffiffiffiffiffi i nð ÞΛ i n i n1þβΛ i nΛ gnðt; xÞ¼p e ½e − e ; ð2:47Þ significantly simplifies the derivation of GðzÞ: 4πn

1 − β z 2β GðzÞ¼ þ ; for z>Λ: ð2:46Þ δ ≡ x − L t 0 < δ < 1 β Λ t> 2Λ 1 þ β Λ 1 þ β where ð Þ and ð þ Þ for all 1−β. Substituting these functions into integrals (2.17) and z κ Furthermore, it is straightforward to show that FðzÞ¼Λ (2.18) we obtain that neither npq nor pq grows in the for all z. limit T → ∞:

Z Z X∞ T T t1 t2 2 −iπð1−βÞðpþmþnþkÞΛ iπð1−βÞðqþmþnþkÞΛ 2 4 npqðTÞ ≈ 2λ dt1 dt2 e e Cðp; q; m; n; kÞ ∼∼λ Λ ; ð2:48Þ 2Λ 2Λ 1−β 1−β m;n;k¼1 Z Z X∞ T t1 t1 t2 2 iπð1−βÞðpþmþnþkÞΛ iπð1−βÞðqþmþnþkÞΛ 2 4 κpqðTÞ ≈ 2λ dt1 dt2 e e Dðp; q; m; n; kÞ ∼∼λ Λ : ð2:49Þ 2Λ 2Λ 1−β 1−β m;n;k¼1

Here we have introduced functions C and D that depend on momenta but do not depend on times t1 and t2: Z Z Λ Λ δ δ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Cðp; q; m; n; kÞ¼ d 1 d 2gp;1gq;2gm;1gn;1gk;1gm;2gn;2gk;2; ð2:50Þ 0 0 Z Z Λ Λ δ δ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ Dðp; q; m; n; kÞ¼ d 1 d 2ðgp;1gq;2 þ gq;1gp;2Þgm;1gn;1gk;1gm;2gn;2gk;2; ð2:51Þ 0 0

1−β − π β −iπn δ π δ where g˜ pffiffiffiffiffiffii e i n e 1þβΛ − ei nΛ . For the latter identities in (2.48) and (2.49) we used that time integrals are p;n ¼ 4πn ½ bounded (there are no singular contributions because the arguments of the exponential functions are never zero) and the sums over the virtual momenta are convergent: 1 Λ2 jCðp; q; m; n; kÞj ∼ jDðp; q; m; n; kÞj ≲ pffiffiffiffiffiffi ; hence; π4 pq mnk X∞ 2 X∞ 4 Λ × ½C or D Λ ∼ < ∞: p m n k q m n k p m n k q m n k mnk m;n;k¼1 ð þ þ þ Þð þ þ þ Þ m;n;k¼1 ð þ þ þ Þð þ þ þ Þ

Note that the dimensionless coefficients of proportionality p, m, n, k. Hence, it is impossible to get nonzero in identities (2.48) and (2.49) include the velocity β. contributions to the sum over virtual momenta in (2.25) One can also come to the same conclusion using the and (2.26). This implies that npq and κpq do not receive approach of the Sec. II B. Indeed, in this case we have growing with time loop corrections. 1þβ Λ − 2βΛ 0 1þβ Λ 2 gðyÞ¼1−β y 1−β and g ðyÞ¼1−β for y>y ¼ . Note that the difference between synchronized Λ ≠ Λ Therefore, the only nonzero coefficient of the Fourier (tL ¼ tR ) and unsynchronized (tL tR , e.g., 1þβ tL ¼ tR) kicks appears already at the tree level [2]. expansion is g0 ¼ 1−β Λ. At the same time, in our case 0 Namely, it can be shown that in the case of synchronized all frequencies are positive, i.e., p þ m þ n þ k> for all kicks the tree-level stress-energy tensor receives nonzero contributions only at the intermediate times, 0

065005-12 DYNAMICAL CASIMIR EFFECT IN NONLINEAR VIBRATING … PHYS. REV. D 103, 065005 (2021) the signal and the radiation pulse from the first kick coefficients (2.22) are zero and integrals (2.25) and “misses” the second kick and bounces back and forth (2.26) cannot receive secularly growing loop corrections. between the mirrors indefinitely. In the paper [2] this tree- The same reasoning also works for complex synchronized level effect was interpreted as evidence of particle creation. motions, e.g., the case where the second kick is applied In the former case this process is temporary (the radiation after several reflections of the first kick. emitted during the first kick is completely absorbed during In other words, the fine-tuning of kicks ensures the the second kick), and in the latter case it is permanent. energy conservation law in the infinite future which, in It is noteworthy that this difference persists at the loop turn, leads to the zero collision integral (compare with level: if the kicks are synchronized, then both tree-level and Appendix A). Note that for a finite-duration nonstationary IR two-loop-level contributions to the stress-energy tensor motion synchronization must be performed during the are negligible, whereas in the opposite case both contri- entire motion period. At the same time, without any butions are significant. Recall that loop corrections to the synchronization this argumentation does not work, energy stress-energy tensor are derived from the quantum averages conservation is violated and secular growth is possible. using (1.9). This difference also implies that ground state of the system behaves differently in the cases of synchronized E. Resonant cavity and unsynchronized kicks. From the point of view of Secs. II A and II B the Finally, let us apply the approach of Sec. II B to a one- difference between the cases of synchronized and dimensional resonantly oscillating cavity: unsynchronized kicks can be easily explained as follows. π First, in the case of synchronized kicks the number of 0 ϵΛ s t Lðt> Þ¼ sin Λ ; reflections from the left and the right mirror always differ by one. Second, the synchronization requirement implies sπt − Λ 1 … − 1 Rðt>0Þ¼Λ þ ϵΛ sin þ φ − ϵΛ sin φ; ð2:52Þ that Rðt2i−1Þ Lðt2iÞ¼ for i ¼ ; ;n . Hence, only Λ the first reflection point makes a nontrivial contribution to the identity (2.10). Finally, recall that we consider asymp- where ϵ ≪ 1 is the small parameter, s ∈ N defines the _ totically stationary motions, i.e., we assume that LðtÞ¼ frequency of oscillations and φ is the dephasing angle. For _ β φ 0 RðtÞ¼ for t>t. Together these observations imply that illustrative purposes we set ¼ (this describes the cavity 0 2 function GðzÞ is purely linear for z>z and g ðyÞ is oscillating as a whole) and s ¼ . In this case functions constant for y>y. Therefore, the high order Fourier GðzÞ and FðzÞ have the following form [16,17]:

2π X∞ z − 2ϵ z θ z − 2 − 1 − θ z − 2 O ϵ2 GðzÞ¼Λ sin Λ Λ ð n Þ Λ n þ ð Þ; 1 n¼ 2π X∞ z 2ϵ z θ z − 2 − θ z − 2 1 O ϵ2 FðzÞ¼Λ þ sin Λ Λ n Λ ð n þ Þ þ ð Þ: ð2:53Þ n¼0

This solution is valid even for relatively large arguments,6 Λ=ϵ ≪ z ≪ Λ=ϵ2. Now one can see that functions GðzÞ and FðzÞ are approximately periodic. Hence, the approach of Sec. II B is applicable even though the mirrors motion is not uniform at large times. The inverse functions in this case are straightforwardly determined up to the same order in ϵ:

X∞ gðyÞ¼Λy þ 2ϵ sin ð2πyÞ ½θðy − ð2n − 1ÞÞ − θðy − 2nÞ þ Oðϵ2Þ; n¼1 X∞ fðyÞ¼Λy − 2ϵ sin ð2πyÞ ½θðy − 2nÞ − θðy − ð2n þ 1ÞÞ þ Oðϵ2Þ: ð2:54Þ n¼0

This approximation is valid for y ≪ 1=ϵ2. Now it is easy to see that high Fourier coefficients are nonzero:

6Note that the naive approach based on a perturbative expansion in ϵ is applicable only for z ≪ Λ=ϵ, because at larger arguments identities (2.53) receive secularly growing corrections of the form ϵntm. The leading corrections of the form ϵntn can be resummed using a renormalization group technique discussed in [16,17]. However, for the case s ¼ 2 this resummation does not result in new corrections to the naive formula, so it can be simply extended to Λ=ϵ ≪ z ≪ Λ=ϵ2.

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−4in 1 − ð−1Þn where function xðtÞ determines the position of the mirror at g ϵΛ ; for n ≠−2; 0; 2; n ¼ 2 − 4 2 β ≡ dxðtÞ n the moment t, ðtÞ dt is the velocity of the mirror Λ πϵΛ β 1 α g0 ¼ ; and g2 ¼ : ð2:55Þ (j ðtÞj < for all t) and is the coefficient that controls the “ideality” of the mirror (the mirror is perfectly reflective Therefore, quantum averages receive secularly growing when α → ∞ and perfectly transparent when α ¼ 0). The loop corrections which are significant at relatively large quantized field can be represented by the mode decom- times, Λ=ϵ ≪ t ≪ Λ=ϵ2: position: Z ∞ dp † 2 Sp;q ϕ n ðTÞ ≈ ðλϵΛTÞ pffiffiffiffiffiffi ; ð2:56Þ ðt; xÞ¼ ½apgpðt; xÞþapgpðt; xÞ: ð3:2Þ pq 8π4 pq −∞ 2π † Here ap and ap are creation and annihilation operators 2 S−p;q þ Sp;−q κ ðTÞ ≈−ðλϵΛTÞ pffiffiffiffiffiffi : ð2:57Þ † pq 16π4 pq that obey the standard commutation relations, ½ap;aq¼ 2πδðp − qÞ; mode functions gp solve the corresponding Here we neglected the subleading Oðϵ2Þ and oscillating equation of motion: contributions. Also we introduced a short notation for the − sum over virtual momenta: μ pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðtÞ ∂μ∂ þ αδ gpðt; xÞ¼0; ð3:3Þ 1 − β2ðtÞ X∞ ð1 − ð−1ÞpþmþnþkÞð1 − ð−1ÞqþmþnþkÞ Sp;q ¼ and satisfy orthonormality conditions: 4mnk m;n;k¼1 p þ m þ n þ k q þ m þ n þ k ðg ;g Þ¼δðp − qÞ; ðg ;gÞ¼0; ð3:4Þ × : p q p q ðp þ m þ n þ kÞ2 − 4 ðq þ m þ n þ kÞ2 − 4 with respect to the Klein-Gordon inner product [4,5]: ð2:58Þ Z ∞ ≡ − ∂ − ∂ Thus, we have shown that a self-interacting massless scalar ðf; hÞ i dx½fðt; xÞ th ðt; xÞ h ðt; xÞ tfðt; xÞ: −∞ field on the background of resonantly oscillating mirrors receives secularly growing loop corrections to the quantum ð3:5Þ averages. This indicates the change in the ground state of the system. Also this means that loop corrections may Note that relations (3.2) and (3.4) automatically imply ϕ affect particle production in a resonant cavity. However, we the canonical equal-time commutation relations ½ ðt; xÞ; ∂ ϕ δ − emphasize that the final conclusion about the destiny of the t ðt; yÞ ¼ i ðx yÞ. Also note that in the ideal-mirror α → ∞ ground state and stress-energy flux can be made only after case ( ) the right-hand side (rhs) of this identity the resummation of the leading corrections from all loops. contains additional boundary terms because in this case modes have improper UV behavior (e.g., see [48,66]). As III. SINGLE NONIDEAL MIRROR we will see below, on the nonideal mirror background reflected waves are negligible in the UV limit, i.e., high- In this section we consider loop corrections to the DCE energy modes behave as simple plane waves. Hence, this on a single nonideal mirror background. First of all, we theory does not suffer from the problems of the ideal mirror discuss the quantization of a free two-dimensional massless theory. scalar field on such a background. We model the mirror In the paper [24] it was shown that mode functions with the delta-functional potential. Using the established satisfying these conditions can be represented as the sum of mode decomposition we calculate loop corrections to the reflected and transmitted waves: energy level density and anomalous quantum average. For simplicity we assume that mirror moves along a “broken” −iωu −iωfðvÞ e L e g ðt; xÞ¼θðpÞ θðxðtÞ − xÞ pffiffiffiffiffiffi − RωðvÞ pffiffiffiffiffiffi trajectory (2.12). p 2ω 2ω −iωu R e A. Free field quantization þ θðx − xðtÞÞTωðuÞ pffiffiffiffiffiffi 2ω Consider a free two-dimensional massless scalar field −iωv −iωgðuÞ e R e with the delta-potential background: þ θð−pÞ θðx − xðtÞÞ pffiffiffiffiffiffi − RωðuÞ pffiffiffiffiffiffi 2ω 2ω Z 1 α x − xðtÞ −iωv 2 ∂ ϕ 2 − δ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ2 L e S ¼ d x ð μ Þ ; ð3:1Þ þ θðxðtÞ − xÞTωðvÞ pffiffiffiffiffiffi ; ð3:6Þ 2 2 1 − β2ðtÞ 2ω

065005-14 DYNAMICAL CASIMIR EFFECT IN NONLINEAR VIBRATING … PHYS. REV. D 103, 065005 (2021) where ω ≡ jpj, u ≡ t − x and v ¼ t þ x are light-cone trajectory xðtÞ in the integrals (3.7) can be approximated by coordinates, and functions f, g are chosen such that 0 τ − τ0 ≪ αv0ðτÞ ≈ α a line at times < v00 τ γ3 ̈ . Hence, for a identities fðvÞ¼u, gðuÞ¼v are satisfied when the point ð Þ ðtÞjxðtÞj trajectory with relatively small proper acceleration, ðu; vÞ moves along the trajectory of the mirror (i.e., when jwðtÞj ¼ γ3ðtÞjẍðtÞj ≪ α, reflection coefficients can be u ¼ t − xðtÞ, v ¼ t þ xðtÞ). In these notations positive approximated by (3.8): (negative)-momentum modes correspond to the right (left)-moving waves. The reflection and transmission coef- α 2 R;L = w ω τ ≈ O ficients on the mirror are fixed by the stitching conditions R ð Þ þ ; where α=2 − iωDβ ðτÞ α which imply the following expressions (here we addition- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ally assume that the velocity of the mirror is uniform in the 1 βðτÞ infinite past): DðτÞ¼ : ð3:9Þ β 1 ∓ βðτÞ Z α τ α R 0 0 0 RωðτÞ¼ dτ exp iωðvðτÞ − vðτ ÞÞ − ðτ − τ Þ ; As was discussed in the Sec. IB, a realistic coefficient is 2 −∞ 2 α ∼ 101÷5 cm−1. This gives an estimated threshold accel- L R TωðτÞ¼1 − RωðτÞ; ∼ 2α ∼ 1020÷24 2 Z eration w c m=s . Therefore, for the most α τ α practical situations corrections to (3.8) can be neglected L 0 0 0 RωðτÞ¼ dτ exp iωðuðτÞ − uðτ ÞÞ − ðτ − τ Þ ; (although some experiments with plasma acceleration can 2 −∞ 2 achieve almost as large w [68]). R L TωðτÞ¼1 − RωðτÞ; ð3:7Þ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B. Loop corrections t 2 where τðtÞ¼τ0 þ dt 1 − β ðtÞ is the proper time of t0 In this subsection we use mode functions (3.6) with the the mirror and functions uðτÞ, vðτÞ denote the correspond- approximate reflection and transmission coefficients (3.9) ing coordinates on the mirror. Also we treated coefficients to calculate loop corrections to the energy level density and R and T as functions of τ via u ¼ uðτÞ, v ¼ vðτÞ. Note that anomalous quantum average. We consider corrections τ τ 4 uð Þ and vð Þ are invertible functions if the trajectory of the generated by the λϕ nonlinearity: τ mirror is timelike. Hence, we can define proper times u Z and τ such that uðτ Þ¼u and vðτ Þ¼v. These are the 1 α − λ v u v 2 ∂ ϕ 2 − δ pxffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixðtÞ ϕ2 − ϕ4 proper times of projections of the point ðu; vÞ onto the S ¼ d x ð μ Þ 2 : 2 2 1 − β ðtÞ 4 mirror along the lines u ¼ const and v ¼ const, respec- tively. Using this notation we can restore coefficients R ð3:10Þ L L and T for an arbitrary space-time point: RωðvÞ¼RωðτvÞ, R R L L R R We would like to single out only the leading quantum RωðuÞ¼RωðτuÞ, TωðvÞ¼TωðτvÞ, TωðuÞ¼TωðτuÞ. Note that in the limit ω ≫ α reflection coefficients tend corrections, so we work in the limit of small coupling λ → 0 → ∞ to zero due to fast oscillations of integrands in (3.7). Hence, constants and large evolution times, , T , λ in the UV region the reflected waves are negligible, i.e., T ¼ const. We remind that in this limit the leading κ modes (3.6) reduce to simple plane waves. This means that corrections to npq and pq are given by Eqs. (1.12) and in the UV region theory with a nonideal mirror reduces to a (1.13), respectively. standard 2D theory in empty space, so the UV renormal- Similarly to the case of two ideal mirrors (Sec. II B), izations can be performed in a standard way. energy level density (1.12) can be represented as the For an arbitrary trajectory coefficients (3.7) are very hard product of two integrals: to find explicitly, so we need to make an approximation to Z dk1dk2dk3 ≈ 2λ2 keep the calculations feasible. First, note that for a uniform npq 3 IpðTÞIqðTÞ; ð3:11Þ trajectory xðtÞ¼βt integrals in (3.7) can be explicitly ð2πÞ taken: where introduced a short notation for the integral: sffiffiffiffiffiffiffiffiffiffiffiffi Z Z α 2 1 β T ∞ R;L = Rω ðτÞ¼ ; where Dβ ¼ : ð3:8Þ I T dt1 dx1g 1g 1g 1g 1; α 2 − ω 1 ∓ β pð Þ¼ p; k1; k2; k3; = i Dβ t0 −∞ where g ≡ g ðt ;x Þ: ð3:12Þ p;n p n n Here Dβ are the Doppler factors for the right and left incident waves, so this formula has a transparent interpre- We would like to single out the secularly growing parts of tation in terms of Doppler shifts. Second, integrals (3.7) are the integral IpðTÞ. Such secular growth appears only when 0 1 predominantly gained on the interval 0 < τ − τ < α due to the integrand reduces to a product of the functions which the exponential decay of the integrand. Now note that depend on the same argument and has the same support.

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In the opposite case, i.e., when the integrand depends on both u and v, the resulting integral oscillates or decays when T → ∞. Therefore, such terms do not contribute to the secular growth and can be neglected. Also recall that Oðw=αÞ terms are practically negligible. In the mentioned approximation we straightforwardly obtain that IpðTÞ linearly grows with time:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iT IpðTÞ¼ gðp; k1;k2;k3Þ: ð3:13Þ 4 jpk1k2k3jðjpjþjk1jþjk2jþjk3jÞ

Here we have separated the universal prefactor and the variable term which consists of 8 different products of the mode parts: gðp; k1;k2;k3Þ¼½θðpÞθðk1Þθðk2Þθðk3Þþθð−pÞθð−k1Þθð−k2Þθð−k3Þ × T T T T R R R R − T−cT−c T−c T−c R−cR−c R−c R−c Dc 2 e−isðT−cxðTÞÞ jpj jk3j jk2j jk1j þ jpj jk3j jk2j jk1j ð jpj jk3j jk2j jk1j þ jpj jk3j jk2j jk1j½ β Þ Z T−cxðTÞ d −isu T−c u T−c u T−c u T−c u e−isu þ ½ jpjð Þ jk3jð Þ jk2jð Þ jk1jð Þ Z0 TþcxðTÞ X − − − − − 2 − fpg d −isv R c v R c v R c v R c v e isð tv vÞ J ; 3:14 þ ½ p ð Þ k3 ð Þ k2 ð Þ k1 ð Þ þ u;v ð Þ 0 j j j j j j j j fpg where c ¼ sgnðpÞ, s ¼jpjþjk1jþjk2jþjk3j, times tu and tv solve the equations u ¼ tu − xðtuÞ and v ¼ tv þ xðtvÞ. Also 2iω we have introduced a short notation for the transmission and reflection coefficients of the stationary mirror: Tω ¼ 2iω−α, 2 ω c α c i DβðuÞ c α 0 0 Rω ¼ 2 ω−α, and moving mirror: TωðuÞ¼2 ω c −α, RωðuÞ¼2 ω c −α. These coefficients come from the t< and t> i i DβðuÞ i DβðuÞ parts of the mirror trajectory, respectively. For brevity we have presentedP only four secularly growing terms which fpg correspond to a single combination of the mode parts; the others are denoted as fpg Ju;v and have the same structure. The explicit form of these terms can be found in the Appendix C. Analytic calculation of the integrals in (3.14) cannot be performed for an arbitrary mirror trajectory. However, in the limit of a “broken” trajectory (2.12) oscillating expressions reduce each other. Roughly speaking, these oscillating parts are nonzero only at the space-time regions that are causally connected with the segment of the accelerated motion of the mirror. For a broken trajectory this segment degenerates into a single dot, so the leading order term of gðp; k1;k2;k3Þ is simplified: gðp; k1;k2;k3Þ¼½θðpÞθðk1Þθðk2Þθðk3Þþθð−pÞθð−k1Þθð−k2Þθð−k3Þ X × T T T T R R R R − T−cT−c T−c T−c − R−cR−c R−c R−c Dc 2 Jfpg: ½ jpj jk3j jk2j jk1j þ jpj jk3j jk2j jk1j jpj jk3j jk2j jk1j jpj jk3j jk2j jk1jð βÞ þ u;v fpg ð3:15Þ

fpg Here once again Ju;v is the sum of all terms proportional to the mixed products of transmission and reflection coefficients. Substituting the calculated expression for the integral IpðTÞ into the energy level density (3.11), we obtain: Z λ 2 ðpTffiffiffiffiffiffiffiffiffiÞ dk1dk2dk3 gðp; k1;k2;k3Þg ðq; k1;k2;k3Þ O λ2 npq ¼ 3 þ ð TÞ: ð3:16Þ 8 jpqj ð2πÞ jk1k2k3j ðjpjþjk1jþjk2jþjk3jÞðjqjþjk1jþjk2jþjk3jÞ

Note that the integral over the virtual momenta is convergent. On the one hand, at large momenta the mirror is effectively transparent, so the integral reduces to the integral over the standard empty space modes. On the other hand, at small 1 momenta we can introduce an IR cutoff p ∼ Λ with a clear physical meaning (see Sec. II). Also one can check that this integral is not zero if the motion of the mirror is nonuniform. The calculations for the anomalous quantum average (1.13) are essentially the same. Indeed, the leading approximation to the κpq can be expressed in the following form: Z Z Z T þ∞ 2 dk1dk2dk3 κ ¼ −2λ dt1 dx1g 1g 1g 1½g I ðt1Þþg I ðt1Þ: ð3:17Þ pq 2π 3 k1; k2; k3; p;1 q q;1 p ð Þ t0 −∞

065005-16 DYNAMICAL CASIMIR EFFECT IN NONLINEAR VIBRATING … PHYS. REV. D 103, 065005 (2021) 8 P in in Thus, the final expression for the anomalous quantum < ½an gn ðt; xÞþH:c:; when t → −∞; 2 n average is also proportional to T : ϕðt; xÞ¼ P : out out ½an gn ðt; xÞþH:c:; when t → þ∞; Z n λ 2 κ − ðpTffiffiffiffiffiffiffiffiffiÞ dk1dk2dk3 ð4:2Þ pq ¼ 3 hðp; q; k1;k2;k3Þ 32 jpqj ð2πÞ jk1k2k3j in 0 O λ2T ; 3:18 and defined in- and out-states as an jini¼ and þ ð Þ ð Þ out an jouti¼0. In the notations of Sec. IA quantity N m corresponds to the diagonal part7 of the tree-level energy where the function hðp; q; k1;k2;k3Þ can be restored after level density. However, in Secs. II and III we have shown the calculation of integrals (3.17). This integral is con- that for sufficiently large evolution times loop corrections vergent and nonzero due to the same reasons as the integral to npq are of the same order as δpq. Note that these (3.16) for npq. corrections multiply the answer (4.1). Moreover, loop Thus, the DCE with a single nonideal mirror is essen- corrections also affect the nondiagonal parts of npq and tially equal to the case of an ideal mirror considered in [48]. anomalous quantum average κpq. Hence, the semiclassical The only difference is the presence of a dimensionful estimate (4.1) is incomplete. α parameter which determines the natural UV scale of the We expect the same reasoning to apply to other models theory. At the same time, we are mainly interested in the of the DCE, including four-dimensional setups. In fact, it is secular growth of loop corrections which is essentially an believed that in some approximation the modes of the IR effect. Therefore, it is not surprising that such a resonant cavity decouple [32,69–72]. In this approximation modification of the theory does not affect the behavior particle creation is qualitatively described by a quantum of the loop integrals (1.12) and (1.13). harmonic oscillator with time-dependent frequency. However, this qualitative model already posses a kind of IV. DISCUSSION AND CONCLUSION secular growth when anharmonic terms are included in the Hamiltonian [73]. This indicates that loop corrections play In this paper we have analyzed the role of nonlinearities an important role even in such simple models of the DCE. in the DCE, i.e., calculated quantum loop corrections to the It is noteworthy that both the tree-level and loop-level correlation functions of a self-interacting scalar field on stress-energy fluxes are associated with the violation of the the background of nonuniformly moving mirrors. We have conformal invariance. At the tree-level this invariance is considered the cases of two ideal mirrors and single violated by non-trivial boundary conditions [2]. In addition, nonideal mirror. We have shown that in both cases two- the secular growth of the loops reflects the conformal non- loop corrections to the Keldysh propagator (1.8) quadrati- invariance of the λϕ4 interaction term. This noninvariance cally grow with the evolution time. This implies that the directly brings the conformal factor into loop integrals stress-energy flux (1.9), energy level density (1.12) and (Sec. II B). anomalous quantum average (1.13) also receive secularly However, note that loop corrections to the quantum growing loop corrections. Hence, for large evolution times averages and stress-energy flux do not grow with time if ∼ 1 λΛ λ → 0 (T = ) loop corrections are significant even if . trajectories of mirrors are synchronized. Roughly speaking, This indicates a breakdown of the semiclassical approxi- synchronization forces the mode functions to “forget” the mation. Also this means that for large times particle periods of nonstationary motion (see the discussion at the creation in the DCE should be reconsidered. end of Sec. II D). The tree-level stress-energy flux on such a We would like to underline several important points background is also zero [2]. We find it remarkable that concerning our analysis. First of all, once again we the absence of particle creation is observed both at the emphasize that semiclassical approach to the particle tree-level and in loops. creation cannot be applied to an interacting theory. In fact, Finally, we emphasize that the definitive conclusion the number of created particles in this approach is calcu- about the destiny of the vacuum state and the stress-energy lated in terms of the Bogoliubov coefficients [5]: flux in the DCE can be made only after the resummation of X the leading corrections from all loops. To perform such a – N ¼hinjðaoutÞ†aoutjini¼ jβ j2; where resummation one needs to solve a system of the Dyson m m m nm – n Schwinger equations [34 36,55]. As a result, the tree-level out in quantum averages in (1.8) would be replaced by their βmn ¼ −ðgm ; ðgn Þ Þð4:1Þ P 7 out α in − β in † The relation an ¼P m ½ nmam nmðamÞ straightfor- and ·; · denotes the Klein-Gordon inner product. Here we N β β δ ð Þ wardly implies m ¼ n;k mn mkð nk þnnkÞ, where nnk ¼ in † in have introduced the field decomposition in the asymptotic hinjðan Þ ak jini. As we have discussedP in the Sec. IA, in the 0 N β 2 past and future: free theory nnk ¼ , hence, m ¼ n j nmj .

065005-17 LIANNA A. AKOPYAN and DMITRII A. TRUNIN PHYS. REV. D 103, 065005 (2021) renormalized values. Furthermore, in some special cases X∞ ϕ † the system of Dyson–Schwinger equations reduce to a ðt; xÞ¼ ½angnðt; xÞþangnðt; xÞ; 1 Boltzman kinetic equation which suggests a simple physi- n¼ − πn cally meaningful solution. Examples of such cases can be e i Λ t πnx g ðt; xÞ¼pffiffiffiffiffiffi sin ; ðA2Þ found in [37–39,42,43]. n πn Λ Unfortunately, it is still unclear how to perform such a † resummation for the DCE. On the one hand, we expect where ½am;an¼δmn. Now let us remind that the two-loop that higher loop corrections to the Keldysh propagator correction to npq reduces to the product of two integrals will not be suppressed in the limit λ → 0, T → ∞, (we can set t0 ¼ 0 due to the time translation invariance): λT ¼ const. In particular, the analysis similar to the X Sec. II B implies that one-loop corrections to the vertexes 2λ2 npqðTÞ¼ Ip;m;n;kðTÞIq;m;n;kðTÞ; quadratically grow in this limit. On the other hand, Z m;n;kZ the quadratic growth of the two-loop correction to the T Λ propagator is itself unconventional. Due to these reasons Ip;m;n;kðTÞ¼ dt dxgpðt; xÞgmðt; xÞgnðt; xÞgkðt; xÞ 0 0 Dyson–Schwinger equations of the theory (1.2) do not reduce to the kinetic equation.8 Therefore, one needs to ðA3Þ develop a completely new method to solve these equa- tions. We hope that the analysis of this paper will help to Substituting the modes (A2) into these integrals we obtain: deduce such a method. Ip;m;n;kðTÞ Z Z −iπt T Λ Λ ðpþmþnþkÞ ACKNOWLEDGMENTS e ffiffiffiffiffiffiffiffiffiffiffiffi ¼ dt dx 2p 0 0 8π pmnk We would like to thank Emil Akhmedov for proposing X π to us this problem, valuable discussions and proof- x × σmσnσk cos ðp þ σmm þ σnn þ σkkÞ reading of the text. Also we would like to thank Λ σm;σn;σk¼1 Sergey Alexeev, Kirill Gubarev and Gheorghe Sorin −iπT 1 − e Λ ðpþmþnþkÞ 1 Paraoanu for useful comments. L. A. A. would like to ¼ pffiffiffiffiffiffiffiffiffiffiffiffi iπ ðp þ m þ n þ kÞ 8π2 pmnk thank Hermann Nicolai and Stefan Theisen for the Λ X hospitality at the Albert Einstein Institute, Golm, where × σ σ σ Λδ σ σ σ 0: ðA4Þ the work on this project was partly done. The work m n k pþ mmþ nnþ kk; σ ;σ ;σ ¼1 of D. A. T. was supported by the Russian Ministry of m n k education and science and by the grant from the Now it is straightforward to see that at small times, T ≪ Λ, Foundation for the Advancement of Theoretical the integral linearly grows with time, I ðTÞ ∼ ΛT. “ ” p;m;n;k Physics and Mathematics BASIS . Hence, at such times the energy level density also 2 grows secularly, npqðTÞ ∼ ðλΛTÞ . However, at large times, T ≫ Λ, the time-dependent part reduces to the APPENDIX A: FALSE SECULAR GROWTH IN Dirac delta-function whose argument is never zero, THE STATIONARY THEORY −iπT 1−e Λ ðpþmþnþkÞ → δ πðpþmþnþkÞ 0 iπ ð Λ Þ¼ . Therefore, in this limit Let us show that a stationary theory may possess a fake ΛðpþmþnþkÞ “secular growth” at evolution times much less than the the correction to the energy level density is approximately → 0 → ∞ → zero, npq , as it should be. Similarly one can show that natural IR cutoff, although in the limit T both npq 2 κpq ∼−ðλΛTÞ for T ≪ Λ and κpq → 0 for T ≫ Λ. 0 and κpq → 0. Consider two ideal mirrors located at points x ¼ 0 and x ¼ Λ. This setup corresponds to the following This behavior of the quantum averages has a clear equation of motion for a massless scalar field: physical interpretation. At large evolution times the energy conservation and momentum conservation9 laws exclude any energy exchange; this means that the collision integral μ ∂μ∂ ϕ ¼ 0; ϕðt; 0Þ¼ϕðt; ΛÞ¼0; ðA1Þ is zero and quantum averages cannot receive nonzero loop corrections. However, at small evolution times the energy conservation law may be violated. Hence, usually forbid- which imply the following mode decomposition: den processes are allowed, so quantum averages may temporarily grow. Since this growth does not persist for 8 This also means that the interpretation of npq and κpq as the energy level density and anomalous quantum average may be 9Note that in this case the “momentum conservation” means misleading. It is safer to find the exact Keldysh propagator (1.8) the momentum conservation in the Brillouin zone, i.e., the total 2πn first and analyze its structure afterwards. momentum can change by Λ , n ∈ Z.

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τ 2Λ long evolution times, it reflects mere vacuum fluctuations functions increase by 2 when t increases by ¼ 1−β2,it rather than the change in the state of the system. Thus, we is convenient to introduce the number k: need to distinguish between this effect and the true secular   growth. − Λ ≡ t k τ : ðB1Þ APPENDIX B: MODE FUNCTIONS FOR THE CASE OF TWO KICKS Then it is straightforward to show that G½t>0; δ and In this Appendix we present the explicit expressions F½t>0; δ are given by the following piecewise linear for the functions G½t; δ¼G½ð1 þ βÞt þ δ, F½t; δ¼ functions (we assume that 0 < β < 1; the expressions for F½ð1 − βÞt − δ on the background (2.28). Since these the negative β can be obtained in the similar way):

8 ( 2 − 1þβ t − 1−β 0 δ Λ > 1−β 1−β 2β k 2 < τ ð Þt δ k ; > Λ þ 1þβ Λ þ 1þβ if 1−β δ 2 > k − < t 2 τ Λ 1−β τ > ( 1−β < k − < t t 1−β 0 δ 1 − 2 t − > k<τ ( > t 1−β 1 − 2 t − δ 1 > k<τ ð1−βÞt 1−β δ 2βðkþ1Þ Λ þ 1 β Λ þ 1 β ; if 2 þ þ 1−β t 1−β δ k þ 2 < τ

8 2 > ð1−βÞt − δ 2βðk−1Þ − 1þβ t − 1−β 2ðt=τ−kþ1Þ δ 1 > Λ Λ þ 1þβ ; if k 2 < τ ( > 1 β2 1−β δ 2ðt=τ−kÞ 2 > k − þ < t ð1þβÞt 1þβ δ 2βk 2 τ 2 Λ 1þβ 1þβ > Λ − 1−β Λ þ 1−β ; if > − 1−β t 1 − 2 − t δ 1 > k 2 < τ > 8 1 β2 1−β 2 − τ > > k − þ < t <> 2 τ 2 Λ 1þβ 1−β 2β 1−β Fðt; δÞ¼ ð Þt − δ þ k ; if k − < t Λ Λ 1þβ > 2 τ Λ 1−β τ > : 2 > t 1−β 2 t − δ 1 > k<τ þ > ( 1−β δ 2 > k 1 β 1 β 2β 1 þ 2 Λ 1þβ ðτ Þ > ð þ Þt − þ δ þ ðkþ Þ ; if > Λ 1−β Λ 1−β 1−β 1−β2 δ 2ðkþ1−t=τÞ 2 > < t − k< ; 1 < þ < ; > 2 τ 2 Λ 1þβ 1þβ > 2 : ð1−βÞt − δ 2βðkþ1Þ 1−β t 1−β 0 δ 1 − 2ðkþ1−t=τÞ Λ Λ þ 1þβ ; if k þ 2 < τ

Graphs of these functions for a fixed δ look like a At large β and fixed t these functions possess the following periodically growing saw. Namely, they are glued from behavior. At the largest part of the interval (of the length 1−β 1 − β Λ the alternating slow-growing (with the slope Λ ) and fast- ð Þ ) they look like a slowly oscillating exponential − π δ φ 1þβ ∼ i nΛþ 1 φ growing (with the slope Λ ) parts. For large β the teeth of function e , where the phase 1 does not depend the saw are very sharp; on the contrary, for small β graphs on δ. At the remaining part they oscillate much more − π 1þβ δ φ are almost smooth. The change of δ simply shifts the graphs rapidly, ∼e i n1−βΛþ 2 . In this region functions G and F up or down. increase by roughly 1, so the difference of phases to the left The mode functions are given by the formula (2.3) as and to the right from the fast-oscillation region is roughly usual: e−iπn. In other words, in even modes left and right slowly oscillating exponents simply continue each other, whereas i ffiffiffiffiffiffiffiffi −iπnGðt;δÞ −iπnFðt;δÞ in odd modes these exponents change the sign at the gnðt; δÞ¼p ½e − e : ðB4Þ 4πn gluing point.

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APPENDIX C: LOOP CORRECTIONS P fpg The full result for the function fpg Ju;v is presented in this section. Let us introduce two auxiliary integrals:

Int1 ¼½θð−pÞθðk1Þθðk2Þθðk3ÞþθðpÞθð−k1Þθð−k2Þθð−k3Þ

× RjpjTjk3jTjk2jTjk1j þ TjpjRjk3jRjk2jRjk1j s − ðRc T−c T−c T−c þ Tc R−c R−c R−c ½D−c2Þe−iðjk1jþjk2jþjk3jÞðT−cxðTÞÞ−ijpjðTþcxðTÞÞ 2jpj jpj jk3j jk2j jk1j jpj jk3j jk2j jk1j β s þ 1 β Z þc T−cxðTÞ d −isu Rc u T−c u T−c u T−c u e−iðjk1jþjk2jþjk3jÞu−ijpjvðuÞ þ ½ jpjð Þ jk3jð Þ jk2jð Þ jk1jð Þ Z0 TþcxðTÞ d −isv Tc v R−c v R−c v R−c v e−iðjk1jþjk2jþjk3jÞuðvÞ−ijpjv ; C1 þ ½ p ð Þ k3 ð Þ k2 ð Þ k1 ð Þ ð Þ 0 j j j j j j j j

Int2 ¼½θðpÞθðk1Þθð−k2Þθð−k3Þþθð−pÞθð−k1Þθðk2Þθðk3Þ

× RjpjTjk3jTjk2jRjk1j þ TjpjRjk3jRjk2jTjk1j s c −c −c c c −c −c c −c 2 −iðjk2jþjk3jÞðT−cxðTÞÞ−iðjpjþjk1jÞðTþcxðTÞÞ − ðR T T R þ T R R T ½Dβ Þe 2ðjpjþjk1jÞ jpj jk3j jk2j jk1j jpj jk3j jk2j jk1j s þ 1 β Z þc T−cxðTÞ d −isu Rc u T−c u T−c u Rc u e−iðjk2jþjk3jÞu−iðjk1jþjpjÞvðuÞ þ ½ jpjð Þ jk3jð Þ jk2jð Þ jk1jð Þ Z0 TþcxðTÞ d −isv Tc v R−c v R−c v T−c v e−iðjk2jþjk3jÞuðvÞ−iðjk1jþjpjÞv : C2 þ ½ p ð Þ k3 ð Þ k2 ð Þ k1 ð Þ ð Þ 0 j j j j j j j j where c ¼ sgnðpÞ, s ¼jpjþjk1jþjk2jþjk3j, vðuÞ¼tu þ xðtuÞ, uðvÞ¼tv − xðtvÞ. Then the final result for the function J can be written down in terms of the introduced integrals: X fpg Ju;v ¼ Int1 þ Int1ðp ↔ k1ÞþInt1ðp ↔ k2ÞþInt1ðp ↔ k3ÞþInt2 þ Int2ðk1 ↔ k2ÞþInt2ðk1 ↔ k3Þ: ðC3Þ fpg

[1] G. T. Moore, Quantum theory of the electromagnetic field in [7] W. G. Unruh, Notes on black hole evaporation, Phys. Rev. D a variable-length one-dimensional cavity, J. Math. Phys. 14, 870 (1976). (N.Y.) 11, 2679 (1970). [8] S. A. Fulling, Nonuniqueness of canonical field quantization [2] P. C. W. Davies and S. A. Fulling, Radiation from a moving in Riemannian space-time, Phys. Rev. D 7, 2850 (1973). mirror in two-dimensional space-time: conformal anomaly, [9] P. C. W. Davies, Scalar particle production in Schwarzschild Proc. R. Soc. A 348, 393 (1976). and Rindler metrics, J. Phys. A 8, 609 (1975). [3] P. C. W. Davies and S. A. Fulling, Radiation from moving [10] C. K. Cole and W. C. Schieve, Radiation modes of a cavity mirrors and from black holes, Proc. R. Soc. A 356, 237 with a moving boundary, Phys. Rev. A 52, 4405 (1995). (1977). [11] O. M´eplan and C. Gignoux, Exponential Growth of the [4] B. S. DeWitt, Quantum field theory in curved space-time, Energy of a Wave in a 1D Vibrating Cavity: Application to Phys. Rep. 19, 295 (1975). the Quantum Vacuum, Phys. Rev. Lett. 76, 408 (1996). [5] N. D. Birrell and P. C. W. Davies, Quantum Fields in [12] L. Li and B. Z. Li, Geometrical method for the generalized Curved Space (Cambridge University Press, Cambridge, Moore equations of a one-dimensional cavity with two England, 1984). moving mirrors, Chin. Phys. Lett. 19, 1061 (2002). [6] S. W. Hawking, Black hole explosions?, Nature (London) [13] C. K. Law, Resonance Response of the Quantum Vacuum to 248, 30 (1974). an Oscillating Boundary, Phys. Rev. Lett. 73, 1931 (1994).

065005-20 DYNAMICAL CASIMIR EFFECT IN NONLINEAR VIBRATING … PHYS. REV. D 103, 065005 (2021)

[14] C. K. Law, Interaction between a moving mirror and [35] E. T. Akhmedov and P. Burda, Solution of the Dyson– radiation pressure: A Hamiltonian formulation, Phys. Schwinger equation on de Sitter background in IR limit, Rev. A 51, 2537 (1995). Phys. Rev. D 86, 044031 (2012). [15] V. V. Dodonov and A. B. Klimov, Generation and detection [36] A. M. Polyakov, Infrared instability of the de Sitter space, of photons in a cavity with a resonantly oscillating boun- arXiv:1209.4135. dary, Phys. Rev. A 53, 2664 (1996). [37] E. T. Akhmedov, Lecture notes on interacting quantum [16] D. A. R. Dalvit and F. D. Mazzitelli, Renormalization group fields in de Sitter space, Int. J. Mod. Phys. D 23, approach to the dynamical Casimir effect, Phys. Rev. A 57, 1430001 (2014). 2113 (1998). [38] E. T. Akhmedov, F. K. Popov, and V. M. Slepukhin, Infrared [17] D. A. R. Dalvit and F. D. Mazzitelli, Creation of photons in dynamics of the massive ϕ4 theory on de Sitter space, Phys. an oscillating cavity with two moving mirrors, Phys. Rev. A Rev. D 88, 024021 (2013). 59, 3049 (1999). [39] E. T. Akhmedov, U. Moschella, K. E. Pavlenko, and F. K. [18] W. J. Kim, J. H. Brownell, and R. Onofrio, Detectability of Popov, Infrared dynamics of massive scalars from the Dissipative Motion in Quantum Vacuum via Superradiance, complementary series in de Sitter space, Phys. Rev. D Phys. Rev. Lett. 96, 200402 (2006). 96, 025002 (2017). [19] C. D. Fosco, F. C. Lombardo, and F. D. Mazzitelli, Quantum [40] E. T. Akhmedov, U. Moschella, and F. K. Popov, Characters dissipative effects in moving mirrors: A functional ap- of different secular effects in various patches of de Sitter proach, Phys. Rev. D 76, 085007 (2007). space, Phys. Rev. D 99, 086009 (2019). [20] Y. Nagatani and K. Shigetomi, Effective theoretical ap- [41] E. T. Akhmedov and E. T. Musaev, Comments on QED with proach to back reaction of the dynamical Casimir effect in background electric fields, New J. Phys. 11, 103048 (2009). (1 þ 1)-dimensions, Phys. Rev. A 62, 022117 (2000). [42] E. T. Akhmedov, N. Astrakhantsev, and F. K. Popov, [21] C. D. Fosco, A. Giraldo, and F. D. Mazzitelli, Dynamical Secularly growing loop corrections in strong electric fields, Casimir effect for semitransparent mirrors, Phys. Rev. D 96, J. High Energy Phys. 09 (2014) 071. 045004 (2017). [43] E. T. Akhmedov and F. K. Popov, A few more comments on [22] A. Lambrecht, M. T. Jaekel, and S. Reynaud, Motion secularly growing loop corrections in strong electric fields, Induced Radiation from a Vibrating Cavity, Phys. Rev. J. High Energy Phys. 09 (2015) 085. Lett. 77, 615 (1996). [44] E. T. Akhmedov, O. Diatlyk, and A. G. Semenov, Out of equi- [23] G. Barton and A. Calogeracos, On the quantum electrody- librium two-dimensional Yukawa theory in a strong scalar namics of a dispersive mirror. 1. Mass shifts, radiation, and wave background, Proc Steklov Inst Math / Trudy Matema- radiative reaction, Ann. Phys. (N.Y.) 238, 227 (1995). ticheskogo instituta imeni VA Steklova 309, 12 (2020). [24] N. Nicolaevici, Quantum radiation from a partially reflect- [45] E. T. Akhmedov, E. N. Lanina, and D. A. Trunin, Quantiza- ing moving mirror, Classical Quantum Gravity 18, 619 tion in background scalar fields, Phys. Rev. D 101, 025005 (2001). (2020). [25] N. Obadia and R. Parentani, Notes on moving mirrors, Phys. [46] E. T. Akhmedov and O. Diatlyk, Secularly growing loop Rev. D 64, 044019 (2001). corrections in scalar wave background, J. High Energy [26] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, Phys. 10 (2020) 027. Dynamical Casimir Effect in a Superconducting Coplanar [47] E. T. Akhmedov, H. Godazgar, and F. K. Popov, Hawking Waveguide, Phys. Rev. Lett. 103, 147003 (2009). radiation and secularly growing loop corrections, Phys. Rev. [27] J. R. Johansson, G. Johansson, C. M. Wilson, and F. Nori, D 93, 024029 (2016). Dynamical Casimir effect in superconducting microwave [48] E. T. Akhmedov and S. O. Alexeev, Dynamical Casimir circuits, Phys. Rev. A 82, 052509 (2010). effect and loop corrections, Phys. Rev. D 96, 065001 [28] P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, (2017); S. Alexeev, Secularly growing loop corrections to Stimulating Uncertainty: Amplifying the quantum vacuum the dynamical Casimir effect, arXiv:1707.02838. with superconducting circuits, Rev. Mod. Phys. 84, 1 (2012). [49] J. S. Schwinger, Brownian motion of a quantum oscillator, [29] C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. Math. Phys. (N.Y.) 2, 407 (1961). J. R. Johansson, T. Duty, F. Nori, and P. Delsing, Observa- [50] L. V. Keldysh, Diagram technique for nonequilibrium proc- tion of the dynamical Casimir effect in a superconducting esses, Zh. Eksp. Teor. Fiz. 47, 1515 (1964) [Sov. Phys. JETP circuit, Nature (London) 479, 376 (2011). 20, 1018 (1965)], http://www.jetp.ac.ru/cgi-bin/e/index/e/ [30] P. Lähteenmäki, G. S. Paraoanu, J. Hassel, and P. J. 20/4/p1018?a=list. Hakonen, Dynamical Casimir effect in a Josephson meta- [51] A. Kamenev, Many-Body Theory of Non-Equilibrium material, Proc. Natl. Acad. Sci. U.S.A. 110, 4234 (2013). Systems (Cambridge University Press, Cambridge, England, [31] V. V. Dodonov, Current status of the dynamical Casimir 2011). effect, Phys. Scr. 82, 038105 (2010). [52] J. Berges, Introduction to nonequilibrium quantum field [32] V. V. Dodonov, Fifty years of the dynamical Casimir effect, theory, AIP Conf. Proc. 739, 3 (2004). MDPI Phys. 2, 67 (2020). [53] J. Rammer, Quantum Field Theory of Non-Equilibrium [33] D. Krotov and A. M. Polyakov, Infrared sensitivity of States (Cambridge University Press, Cambridge, England, unstable vacua, Nucl. Phys. B849, 410 (2011). 2007). [34] E. T. Akhmedov, IR divergences and kinetic equation in de [54] E. A. Calzetta and B. L. B. Hu, Nonequilibrium Quantum Sitter space. Poincare patch: Principal series, J. High Energy Field Theory (Cambridge University Press, Cambridge, Phys. 01 (2012) 066. England, 2008).

065005-21 LIANNA A. AKOPYAN and DMITRII A. TRUNIN PHYS. REV. D 103, 065005 (2021)

[55] L. D. Landau and E. M. Lifshitz, Physical Kinetics on amplification and squeezing in Josephson parametric (Pergamon Press, Oxford, 1981), Vol. 10. amplifiers, Phys. Rev. Applied 8, 054030 (2017). [56] S. V. Bulanov, T. Zh. Esirkepov, M. Kando, A. S. Pirozhkov, [65] B. Yurke and E. Buks, Performance of cavity-parametric and N. N. Rosanov, Relativistic mirrors in plasmas. Novel amplifiers, employing Kerr nonlinearites, in the presence of results and perspectives, Phys. Usp. 56, 429 (2013). two-photon loss, J. Lightwave Technol. 24, 5054 (2006). [57] D. A. R. Dalvit, P. A. Maia Neto, and F. D. Mazzitelli, [66] L. Astrakhantsev and O. Diatlyk, Massive quantum scalar Fluctuations, dissipation and the dynamical Casimir effect, field theory in the presence of moving mirrors, Int. J. Mod. Lect. Notes Phys. 834, 419 (2011). Phys. A 33, 1850126 (2018). [58] S. V. Bulanov, T. Esirkepov, and T. Tajima, Light Intensi- [67] M. Castagnino and R. Ferraro, The radiation from moving fication towards the Schwinger Limit, Phys. Rev. Lett. 91, mirrors: The creation and absorption of particles, Ann. Phys. 085001 (2003); , Erratum, Phys. Rev. Lett. 92, 159901 (N.Y.) 154, 1 (1984). (2004). [68] W. P. Leemans, B. Nagler, A. J. Gonsalves, C. Toth, K. [59] W. Heisenberg and H. Euler, Folgerungen aus der Dirac- Nakamura, C. G. R. Geddes, E. Esarey, C. B. Schroeder, and schen theorie des positrons, Z. Phys. 98, 714 (1936). S. M. Hooker, GeV electron beams from a cm-scale acce- [60] J. S. Schwinger, On gauge invariance and vacuum polari- lerator, Nat. Phys. 2, 696 (2006). zation, Phys. Rev. 82, 664 (1951). [69] V. V. Dodonov and A. V. Dodonov, Quantum harmonic [61] B. Yurke and J. S. Denker, Quantum network theory, Phys. oscillator and nonstationary Casimir effect, J. Russ. Laser Rev. A 29, 1419 (1984). Res. 26, 445 (2005). [62] P. D. Nation, M. P. Blencowe, and E. Buks, Quantum [70] R. Schutzhold, G. Plunien, and G. Soff, Trembling cavities analysis of a nonlinear microwave cavity-embedded dc in the canonical approach, Phys. Rev. A 57, 2311 (1998). SQUID displacement detector, Phys. Rev. B 78, 104516 [71] V. V. Dodonov, A. B. Klimov, and V. I. Man’ko, Generation (2008). of squeezed states in a resonator with a moving wall, Phys. [63] J. Bourassa, F. Beaudoin, J. M. Gambetta, and A. Blais, Lett. A 149, 225 (1990). Josephson-junction-embedded transmission-line resonators: [72] V. V. Dodonov, Photon creation and excitation of a detector From Kerr medium to in-line transmon, Phys. Rev. A 86, in a cavity with a resonantly vibrating wall, Phys. Lett. A 013814 (2012). 207, 126 (1995). [64] S. Boutin, D. M. Toyli, A. V. Venkatramani, A. W. Eddins, I. [73] D. A. Trunin, Comments on the adiabatic theorem, Int. J. Siddiqi, and A. Blais, Effect of higher-order nonlinearities Mod. Phys. A 33, 1850140 (2018).

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