aa r X i v : . [ qu a n t - ph ] J un Exact closed form analytical solutions for vibrating cavities

Pawe l W¸egrzyn

Marian Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Cracow, Poland ∗ For one-dimensional vibrating cavity systems appearing in the standard illustration of the dy-namical Casimir eﬀect, we propose an approach to the construction of exact closed-form solutions.As new results, we obtain solutions that are given for arbitrary frequencies, amplitudes and timeregions. In a broad range of parameters, a vibrating cavity model exhibits the general property ofexponential instability. Marginal behavior of the system manifests in a power-like growth of radiatedenergy.

PACS numbers: 42.50.Lc, 03.70.+k, 11.10.-z

I. INTRODUCTION

The dynamical variety of the Casimir eﬀect has been studied in numerous papers in the last two decades [1, 2].The most favorable model for theoretical considerations refers to the original Casimir’s setup [3] of a cavity composedof two perfectly conducting parallel plates. The dynamical modiﬁcation means that the distance between plateschanges with time. Then, we come across new fascinating phenomena in the context of quantum ﬁeld theory. Themost impressive manifestation of unusual non-classical properties of the theory of quantized ﬁelds is the eﬀect ofparticle production ”from nothing”. It is referred as dynamical Casimir eﬀect (DCE) or motion induced radiation(MIR). Most of theoretical papers explore so-called vibrating cavities, where oscillations of cavity walls are periodicin time. Attention of their authors is attracted by the instability of quantum ﬂuctuations due to the parametricresonance. They hope the resonance enhancement of particle production could lead to macroscopic eﬀects that wouldbe experimentally detected. Nowadays, the resonant instability of the vacuum that is followed by an explosive particleproduction in a vibrating cavity is believed to be accessible for measurements [4, 5, 6, 7, 8]. However, we are stillawaiting for conﬁrmation of successful experimental projects.In this paper, we consider the behavior of the electromagnetic ﬁeld in a one-dimensional vibrating cavity. Thisproblem was studied initially by Moore [9]. A cavity with one stationary wall and one moving wall with some prescribedtrajectory had been elaborated there. Moore did not solve any particularly interesting cavity model, but he made athorough study of a general theory of solving such models. His basic approach was generally used and developed in nextyears [10]-[72]. From theoretical point of view, this simpliﬁed model deals with several hard and important problems.We mean the task of solving a wave equation with time-dependent boundary conditions, the diﬃculties with analyticaldescription of physical systems under the parametric resonance conditions, the problem of quantization of ﬁelds inlimited regions with moving boundaries, the squeezing of quantum states or the problems of quantum entanglementand decoherence. One-dimensional vibrating cavities provide the simplest theoretical laboratories to study theseissues in quantum ﬁeld theory. The methods that are worked out there can be essentially adapted for more complexmodels. Some results and ideas are endorsed as well. There are advanced approaches to include non-perfect [55, 56]or partly transmitting [72] cavity walls, ﬁnite temperature eﬀects [32, 46, 47] or proceed to three-dimensional case[25, 31, 39, 40, 49, 51, 52, 62].In the case of one-dimensional cavities and ”scalar electrodynamics”, main achievements of numerous investigationswere obtained either in the framework of the eﬀective Hamiltonian approach [1, 23, 35] or using various numericalapproaches [24, 48, 54, 66, 67, 69, 70]. Analytical solutions obtained through the perturbation methods with eﬀectiveHamiltonians hold only for small amplitudes [28, 29, 37] or for particular time regimes, either short time [30] or longtime [20] limits. In many investigations, a frequency of cavity vibrations is assumed to match a resonance frequency.It is well known from classical mechanics [73] that parametric resonance occurs also for frequencies that are not ﬁnelytuned provided that respective amplitudes of oscillations are suﬃciently large. Oﬀ resonant behavior of vibratingcavities is usually studied in the limit of small detuning from resonance frequencies [38, 58].Our aim is to gather exact analytical and global solutions for vibrating cavities. In fact, there are few knownsolutions that can be described by closed form expressions. The ﬁrst closed form exact solution to describe a cavityvibrating at its resonance frequency was presented by Law [21]. Law’s solution corresponds to a cavity that oscillates ∗ Electronic address: [email protected] basically sinusoidally for small amplitudes. The frequency of oscillations is twice the lowest eigenfrequency of thecavity (so-called ”principal resonance” [1]). Law found travelling wave packets in the energy density of the ﬁeld. Henoted ”sub-Casimir” quantum ﬂuctuations far away from the wave packets. Next, Wu et al. [41] presented a family ofexact analytical solutions for all resonance frequencies. In the particular case of the second resonance frequency, theirsolution is matching Law’s solution. They described emerging of wave packets in the energy density, indicated sub-Casimir ﬂuctuations and emphasized the absence of wave packets in the ﬁrst resonant channel with the fundamentalcavity eigenfrequency (”semi-resonance” [1]). One can be puzzled that for any one of known exact solutions, thereappears a power-like resonance instability. Faithfully, the total energy of the ﬁeld increases quadratically with timethere. On the other hand, it is well recognized from other cavity models that the total radiated energy typicallygrows exponentially with time [24, 27, 37, 43, 44, 48]. For instance, one can refer to the asymptotic formulas found byDodonov et al. [1] for cavities that undergo harmonic oscillations. It was generally argued [24, 27] that an exponentialresonant instability is typical for vibrating cavities, while a power-like behavior constitutes a critical boundary betweenstability and instability regions deﬁned by domains of parameters [63]. In this paper, we ﬁnd solutions that revealexponential instability generally and exhibit a power-like law as a marginal eﬀect. Moreover, all previously knownexact solutions applied only to resonance frequencies. Here, we provide exact solutions that are adjustable for allfrequencies. This paper presents a rich class of exact and closed-form solutions, in addition all formerly presentedsolutions [1, 21, 41] are captured here as particular cases and examined in a more comprehensive way.Our paper is organized as follows. In Section II, we present our way of representing solutions to describe thequantum dynamics in a vibrating cavity. It relies on SL (2 , R ) symmetry of the algebraic structure that exists forthe quantized scalar ﬁeld in a static cavity [50, 64, 74]. Actually, we abandon Moore’s function that is awkward inuse. We put forward another object, called a fundamental map by us, that has remarkable analytical properties.These properties are collected in Section III, together with appropriate mathematical formulae for primary physicalfunctions, namely the vacuum expectation of the energy density of the ﬁeld inside a cavity and the total radiatedenergy. Finally, in Section IV we present a collection of exact closed-form solutions. The results are summarized inSection V. II. REPRESENTATION OF SOLUTIONS TO DESCRIBE THE QUANTUM DYNAMICS IN AVIBRATING CAVITY

In the standard physical setup, we have an electromagnetic resonator of length L composed of two perfectly reﬂectingwalls. Initially, the cavity is static. Then, it undergoes vibrations with a constant frequency ω . In literature, it isfrequently assumed that the cavity length L is related with the period of oscillations T = 2 π/ω . In this paper, wewill keep the parameters L and T independent. The parameters provide the characteristic physical length scales. Thestatic cavity length L deﬁnes the magnitude of Casimir interactions. In particular, it speciﬁes the scale of quantumﬂuctuations leading to the production of particles. The period T is the scale of parametric excitations of the systemcaused by some external force. It is very useful in numerical computations to put T = π . The parametric resonanceis expected when L and T are of the same order. Eventually, it depends also on an amplitude of vibrations. We arewilling to yield a phase diagram (Arnold’s diagram [73]) that exhibits stability and instability regions.The derivation of the simplest mathematical model leads to the quantization of free scalar ﬁeld A ( x, t ) with Dirichletboundary conditions imposed at the boundary walls x = 0 and x = L ( t ). The trajectory of the oscillating wall isperiodic: L ( t + T ) = L ( t ). It is important to assume that L ( t ) > | ˙ L ( t ) | ≤ v max < L ( t ) = L for t < A N ( t, x ) = i √ πN [exp ( − iω N R ( t + x )) − exp ( − iω N R ( t − x ))] . (1)The cavity eigenfrequencies ω N = N π/L are called resonance frequencies. We expect that the parametric resonanceoccurs at these frequencies for any amplitudes. However, the instability of the system may appear also for otherfrequencies provided that the amplitude of oscillations is suﬃciently large. Usually, it is a hard task to get the pictureof the asymptotic behavior of the system for any frequencies and amplitudes. Our knowledge of the system comes tous through the Moore’s function R given by the following equation: R ( t + L ( t )) − R ( t − L ( t )) = 2 L . (2)Usually, Moore’s function R is deﬁned as a dimensionless function (phase function). In this paper, we will prefer todeﬁne this function in dimensions of length. There is no general theory of solving Eq.(2). Before we present a big setof exact solutions of the above problem, it is worth to recall some useful symmetry of the static cavity system [50, 64].In the static region for t <

0, the quantized theory is invariant under the conformal transformations: t ± x → R min ( t ± x ) , (3)with the functions R min deﬁned by: R min ( τ ) = 2 ω arctan (cid:16) σ (tan ω τ (cid:17) , (4)where σ ( τ ) = ( Aτ + B ) / ( Cτ + D ) is any homography and ω is the lowest resonance frequency. Subsequent branchesof multivalued function arctan should be always chosen and linked together in such a way that a resulted function R min is continuous. It is described here a well-known SL (2 , R ) symmetry of free scalar ﬁelds quantized on a strip[74]. Surprisingly, this symmetry is rarely exploited in numerous papers on physical models of the quantum ﬁeld ina one-dimensional cavity. In particular, the symmetry helps to solve the puzzling problem why there is no resonantbehavior of the system for the fundamental resonance frequency ω = π/L .In this paper, we will be searching for exact solutions of Eq.(2) in the following form: R ( τ ) = 2 ω arctan (cid:16) ∆ n ( τ ) (tan ωτ (cid:17) + shif t . (5)In order to obtain closed-form solutions, we assume the range of arctan to be [ − π/ , π/

2] (principal branch) andappropriate shifts will be explicitly speciﬁed throughout. For instance, the linear Moore’s function, which describes astatic cavity, should be represented as: R static ( τ ) = τ − πω = 2 ω arctan (tan ωτ ⌊ ωτ π − ⌋ πω , (6)where we have used the standard notation for the ﬂoor function. The construction of the representation Eq.(5) istied up with the well-known idea from classical mechanics [73]. To explore the dynamics of periodic systems withparametric resonance, it is a handy way to deal with mappings for single periods. Here, we need a set of maps ∆ n numerated by the number n . Fortunately, the maps are not independent. We prove that it is enough to specify onlythe ﬁrst map ∆ . Henceforth, a function ∆ ( v ) is going to be called a fundamental map throughout this paper. Thismap deﬁnes the auxiliary function f : f ( τ ) = 2 ω arctan (cid:16) ∆ (tan ωτ (cid:17) + shif t , (7)which is a solution of a simpler problem than Eq.(2) (see equations for billiard functions in [63]): f ( t + L ( t )) = t − L ( t ) . (8)Since the cavity is static in the past, we have always that f ( τ ) = τ − L for τ < L . The subject is also simpliﬁed dueto the fact that the auxiliary function f fulﬁls the periodicity condition: f ( τ + T ) = f ( τ ) + T . (9)In general, the Moore’s function R ( τ ) is not subject to any periodic conditions. The reason lies in the lack of periodicityof the index n ( τ ), that assigns a map to a particular point.It is straightforward to prove that a fundamental map ∆ ( v ) designates unambiguously a Moore’s function R ( τ ).The solution of Moore’s equation (2) can be build according to the formula: R ( τ ) = f ◦ n ( τ ) ( τ ) + 2 L [ n ( τ ) − . (10)Looking at the representation Eq.(5), one can check easily: ∆ n = (∆ ) ◦ n . Throughout this paper, we use (∆ ) ◦ n tonote n -fold composition ∆ ◦ ∆ ◦ ... ◦ ∆ . It remains only to describe the step function n ( τ ) that appears in Eq.(5)and Eq.(10). As the function f ( τ ) is increasing, the region for τ ≥ L can be covered by intervals [ L n − , L n ), where L n ≡ ( f − ) ◦ n ( L ). The map number n ( τ ) equals n if the point τ lies inside [ L n − , L n ). Map markers L n will be calledmilestones throughout this papers. If τ ∈ [ L n − , L n ), then f ( τ ) ∈ [ L n − , L n − ). Thus, it is easy to ﬁnd the followingrecurrence relation, which is also very convenient for numerical purposes: n ( τ ) = (cid:26) τ < L n ( f ( τ )) τ ≥ L (11)In order to provide a glimpse to details of future calculations with the representation Eq.(5) or Eq.(7), we take a lookat the solution given by Eq.(48). This solution will be discussed later, but we glance over the borders of intervalsfor corresponding functions f and R there. They are depicted in Fig.1. Performing appropriate calculations, oneshould take into account that all variables and mappings are valid only in deﬁned domains. Typically, the regionsuitable for calculations of Bogoliubov coeﬃcients or total radiated energies is covered by two subsequent maps ∆ n .It makes evaluations of integrations and derivations of formulas more complex. This is the price we have paid forthe replacement of Moore’s equation by simpler relation Eq.(8). However, we will convince ourselves that this wayis eﬀective as a method for obtaining analytical results. Later, the details of calculations will be always skipped, sothat the pattern in Fig.1 is the only commentary on practical calculations.

0 5 10 15 20 25 30 35 ωτ /2n=0 n=1 n=2n=0 n=1 n=2periods of motionbranches of arctanborders of maps Figure 1: The borders of diﬀerent change intervals for the Moore’s function R deﬁned by Eq.(5) with Eq.(48).

It is diﬃcult to derive the function R ( τ ) from Eq.(2) for some prescribed trajectory L ( t ). A great number ofnumerical approaches and approximate solutions were presented in other papers, but only few exact solutions areknown. One way to obtain exact solutions is to specify the function f ( τ ), and then the trajectory L ( t ) can be givenin a parametric form: (cid:26) t = [ τ + f ( τ )] / L ( t ) = [ τ − f ( τ )] / f ( τ ) represents an admissible physical trajectory provide that it fulﬁls the following require-ments [48]: ( i ) f ( τ ) = τ − L for τ < L ( ii ) − v max v max ≤ ˙ f ( τ ) ≤ v max − v max ( iii ) f ( τ ) < τ (13)In this paper, we will be exploiting the representation Eq.(5) to describe solutions of equations for the electromagneticﬁeld in an oscillating one-dimensional cavity. Before we start with the construction of solutions, we describe generalproperties of fundamental maps ∆ extracted from proper solutions. III. GENERAL PROPERTIES OF FUNDAMENTAL MAPS ∆ Knowledge of Moore’s function enables us to draw out all information about the vibrating cavity system. The mostimportant object to calculate is the vacuum expectation value of the energy density: h T ( t, x ) i = ̺ ( t + x ) + ̺ ( t − x ) . (14)Using appropriate formulas given in [64] and our representation Eq.(5), we can easily calculate: ̺ ( τ ) = − ω π + ω − ω π " v n ( τ ) ( v ) ∆ ′ n ( τ ) ( v ) − ω π (1 + v ) S [∆ n ( τ ) ]( v ) , (15)where v = tan ( ωτ /

2) and S [∆ n ( τ ) ]( v ) denotes the Schwartz derivative of ∆ n ( τ ) with respect to v . The total quantumenergy radiated from the cavity can be calculated from: E ( t ) = Z L ( t )0 dx h T ( t, x ) i = Z t + L ( t ) t − L ( t ) dτ ̺ ( τ ) = 2 ω Z dv v ̺ ( v ) . (16)The most useful is the last formula which enables us to calculate the total energy by integration with respect to v .However, we should remember from the comment in the previous section on the pattern in Fig.1 that the replacement v = tan ( ωτ /

2) is valid only for a single period of cavity motion. The interval of integration [ t − L ( t ) , t + L ( t )] is tobe divided into parts representing separate periods of motion. The map number n ( τ ) may change at most once perperiod.Let us remind the relation ∆ n = (∆ ) ◦ n , and we need only to specify the fundamental map ∆ . The knowledge ofthis map makes it possible to predict the evolution of the system and describe the resonance behavior. Henceforth,our exploration of a quantum ﬁeld theory system is quite similar to examination of classical mechanics models underthe parametric resonance [73]. We need only to analyze the asymptotic behavior of iterations of the mapping ruledby ∆ , which is known from the ﬁrst period of motion.We are going to make a list of general properties of fundamental maps ∆ . First, we include information that thecavity is assumed to be static in the past, i.e. for times t <

0. It follows that a solution for t > t = 0. In our context, there is no need to demand that the sewing is perfectly smooth.For instance, we can accept that a force which causes cavity motion may be suddenly switched on. Such a solutionmay lead to some Dirac delta terms in its function for energy density, but from physical point of view the solutionis acceptable and useful for applications, so that it is deﬁnitely worth saving them. Henceforth, let us propose someminimal set of requirements for sewing. We put forward three sewing conditions at the initial time t = 0. Thetrajectory of the cavity wall and its velocity should be continuous: L ( t = 0) = L and ˙ L ( t = 0) = 0. Moreover, thereshould be no sudden local growth of energy: h T ( t = 0 , x ) i = − π/ (24 L ), i.e. the local energy density matches theCasimir energy density of vacuum ﬂuctuations at the initial time. It is now straightforward to gather a full set ofinitial conditions for the fundamental map ∆ ( v ):∆ ( v ) = − v ∆ ′ ( v ) = 1 S [∆ ]( v ) = 0 ; v ≡ tan ωL (cid:18) π LT (cid:19) = tan (cid:18) π ωω (cid:19) . (17)The last condition implies that the construction of a fundamental map is yet a non-linear problem. Next, we imposethe requirement that the velocity of cavity wall should never exceed v max . The maximal velocity is a parameter ofthe cavity model and the only limitation is that v max <

1. From Eq.(13)(ii) we obtain:1 − v max v max ≤ v ( v ) ∆ ′ ( v ) ≤ v max − v max . (18)This is a strong constraint on possible maps. One immediate consequence is that our function is increasing: ∆ ′ ( v ) > ( v ) is singular at some v s :lim v → v s ∓ ∆ ( v ) = ± ∞ , (19)then it is easy to prove that the following limit is ﬁnite and diﬀerent from zero:lim v → v s ∆ ′ ( v )∆ ( v ) = − lim v → v s v − v s )∆ ( v ) . (20)Henceforth, it follows the function ∆ ( v ) may have only poles of order one:∆ ( v ) = h ( v )( v − v )( v − v ) ... ( v − v s ) , (21)where the numerator h ( v ) is an analytical function. Taking Eq.(18) together with Eq.(21), we note that for largevalues of v the function h ( v ) shows the following asymptotic: h ( v ) ∼ | v | k , k ∈ { s − , s, s + 1 } . (22)Finally, we look at the representation Eq.(7) and the periodicity condition Eq.(9). We conclude that the number ofsingularities s in the map ∆ for the representation Eq.(7) is at most one. Actually, we could replace ω with sω in therepresentation Eq.(7) and allow for more complex form deﬁned by Eq.(21). The same performance as that in SectionIV might give new exact closed form solutions, but we will not examine this idea here.In general, for large arguments either the function ∆ is unbounded or it takes a ﬁnite limit. Therefore, therespective continuity condition corresponds to one of two choices:∆ ( ±∞ ) = ±∞ or ∆ ( −∞ ) = ∆ (+ ∞ ) = ﬁnite value . (23)Let us summarize this section. The basic set of solutions Eq.(1) for a quantum cavity system can be fully speciﬁedby Moore’s function Eq.(5). In turn, this function is to be reconstructed from the fundamental map ∆ . Thefundamental map is associated with the ﬁrst period of motion. Some basic physical requirements lead to strongmathematical conditions on the application of function ∆ to cavity models which are admissible from physical pointof view. This includes suitable sewing conditions Eq.(17) at some distinguished point v , the inequalities Eq.(18)introduced by a limitation v max on a cavity wall velocity and continuity condition Eq.(23). Moreover, the function∆ may have at most one singularity (only a simple pole) and it behaves for large arguments according to Eq.(22) ( s is a number of singularities, i.e. 0 or 1 here).For some given fundamental map ∆ that fulﬁls all required mathematical conditions, it may be still diﬃcult toderive trajectory L ( t ) from Eq.(7) and Eq.(12) or map ranges L n and index function n ( τ ) from Eq.(11). However,it is possible for rational functions. In the following section, we will discuss such solutions. They form a big andinteresting family of exactly solvable cavity models. In particular, they include all examples of exact closed formsolutions on vibrating cavities known from other papers [21, 41]. IV. EXACT CLOSED FORM ANALYTICAL SOLUTIONS

We use the considerations of the previous sections to ﬁnd a family of exactly solvable quantum models of vibratingcavities. The static cavity length L is ﬁxed and it characterizes a physical scale. According to the naive understandingof parametric resonance, the frequency of vibrations ω should be close to one of the resonance frequencies ω N . Itmeans that L is close to N T /

2. However, it should be conﬁrmed in a speciﬁc cavity model whether this naive criterionof resonance is justiﬁed. Moreover, it turns out there is a more subtle situation when L is an odd multiplicity of T / v in Eq.(17) is inﬁnite. Such cases should be analyzed in our treatment separately. A. Linear fundamental maps ∆ We begin by addressing the case when a fundamental map ∆ is a polynomial. The condition Eq.(18) that velocitiesare not approaching the speed of light is very restrictive here. It allows only for a linear function. First, we will examinethe case when v is ﬁnite.

1. Finite values of v Our method of proceeding follows closely on the formalism presented in the previous sections. Inserting a linearfunction into conditions (17), we pick out:∆ ( v ) = v − v ; v ≡ tan ωτ . (24)It is easy to verify that the above function fulﬁls all physical requirements Eq.(17), Eq.(18) and Eq.(23). Let us deﬁnethe natural number M and the angle parameter θ by: L = ( M + θπ ) T ; M = 1 , , , ... , | θ | < π . (25)The parameter M can be interpreted as the order of the resonance. We will go through this subsection and see thatthe parameter M is better to characterize the resonance channel than N . The auxiliary function f for τ ≥ L fromEq.(7) and its inverse function f − for τ ≥ − L yield: f ( τ ) = ω arctan (tan ωτ − v ) + (cid:0) ⌊ τT + ⌋ − M (cid:1) T ,f − ( τ ) = ω arctan (tan ωτ + 2 v ) + (cid:0) ⌊ τT + ⌋ + 2 M (cid:1) T ,v = tan ωL = tan θ . (26)The corresponding trajectory of the cavity wall for t ≥ L ( t ) = L + 1 ω arcsin (sin θ cos ( ωt )) − θω . (27)For small parameters θ , the oscillations of the cavity wall are close to a sinusoidal wave (see Fig.2). With increasing θ , -1 0 1 2 3 4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 leftwallx=0 rightwallx=L(t)t/T x/L -1 0 1 2 3 4-0.2 0 0.2 0.4 0.6 0.8 1 1.2 leftwallx=0 rightwallx=L(t)t/T x/L Figure 2: The trajectories of the cavity walls for the cavity motion Eq.(27) with M = 2 and θ = π/ they are nearer to a triangle wave. The wall oscillations take place between M T and L . The amplitude of vibrationsis ∆ L = 2 | θ | /ω , and the maximal velocity yields: v max = | sin θ | . (28)The Moore’s function for τ ≥ L can be calculated from Eq.(10): R ( τ ) = 2 ω arctan (cid:16) tan ωτ − n ( τ ) tan θ (cid:17) + shif t . (29) -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60-2-1 0 1 0 1 2 3 60 61 62 63 55 56 57 58-10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 -10 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 ωτ/2ω R (τ)/2 Figure 3: Moore’s function for the cavity motion Eq.(27) with M = 2 and θ = π/ The representation Eq.(29) can be eﬀectively used if we are able to assign appropriate maps. It is nice that we arein a position to calculate the milestones L n and the map number n ( τ ) from Eq.(11) exactly: L n = ω arctan (2 n + 1) v + (2 n + 1) M T ,n ( τ ) = n ( τ ) − τ − L n ( τ ) − ) + Θ( τ − L n ( τ ) ) ,n ( τ ) = ⌊ τ / (2 M T ) + 1 / ⌋ , (30)where the Heaviside step function is deﬁned with Θ(0) = 1. The Moore’s function R ( τ ) from Eq.(29) for some speciﬁcmotion of type Eq.(27) is shown in Fig.3. This function is always a small deviation from the linear function Eq.(6) thatdescribes the static case. Disturbances caused by the cavity motion are magniﬁed in Fig.3 for small and large functionarguments. With increasing arguments they approach a well-known staircase shape (”devil’s staircase”). However,the steps are hardly regular. If we look at the asymptotic behavior of the Moore’s function, then we are convinced thatit is not a good object for practical calculations, both analytical (perturbation methods) and numerical. Thus, thetransformations for phase functions like Eq.(5) are necessary to get a feasible way to perform mathematical analysisof vibrating cavities in quantum ﬁeld theory.The shape function for the energy density Eq.(15) for the solution Eq.(24) reads: ρ ( τ ) = − ω π + ω − ω π (cid:20) v v − n ( τ ) tan θ ) (cid:21) . (31)A snapshot of the energy density is displayed in Fig.4. In general, there are M wave packets travelling left and M wave packets travelling right. Their localization and their evolution can be easily derived and it is in full agreementwith results of procedures described in [63] and generalized in [68]. One can successfully derive periodic optical pathsand calculate cumulative Doppler factors, cumulative conformal anomaly contributions and other quantities. Here, weskip such details. Far from the narrow packets, in the so-called sub-Casimir region [21] the energy density is constantand its asymptotic value is: T out ( τ ) ∼ = − ω π = − (cid:18) M + 2 θπ (cid:19) ρ , for large τ , (32) -20.5-20-19.5-19-18.5-18-17.5-17 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T / ρ x/L Figure 4: The energy density of the cavity Eq.(27) with M = 2 and θ = π/ ρ = π/ (24 L ). where ρ = π/ (24 L ) is the magnitude of Casimir energy density for a static cavity of length L . Most of the energyis concentrated in narrow wave packets. The heights of peaks are proportional to t , and their widths shrinks like t − . It suggests that the total energy grows with time like t . It is true, and one can calculate from Eq.(16) an exactformula. Here, we give only an asymptotic formula for large times: E ( t ) ∼ = ω ( ω − ω )24 M π (tan θ ) t , t ≫ . (33)As usual, there is no resonant behavior for the the lowest resonance frequency. However, the resonance emerges forall frequencies above this threshold: ω > ω (or equivalently: for L > T / ω N , either the cavity is static or the motionis singular (a triangular wave trajectory). Therefore, we should learn that the resonance frequencies are auxiliaryobjects, and real behavior of any physical cavity system depends on its individual features. A speciﬁc feature of thesolution Eq.(27) is that for some ﬁxed initial cavity length L , the resonance appear for almost all frequencies abovesome threshold. However, there is no possibility to adjust the amplitude of vibrations. There exists an exact relationbetween the amplitude and the frequency: ∆ LL = (cid:12)(cid:12)(cid:12)(cid:12) − ω ω ⌊ ω ω + 12 ⌋ (cid:12)(cid:12)(cid:12)(cid:12) . (34)The most important problem for any linear dynamical system that exhibits a parametric resonance phenomenon isto ﬁnd stable and unstable regimes for periodically excited parameters. Usually, the parametric resonance domainsare depending on three crucial parameters: frequency and amplitude of periodic excitation and damping coeﬃcient.The relevant Fig.5 exhibits the phase diagram for the cavity model described in this section. Since the frequencyand the amplitude are related by Eq.(34), the instable solutions Eq.(27) are represented by points on the curve tothis plot. The instability of solutions is quadratic according to Eq.(33). It was suggested in [63] that a cavity modelwith a power-like instability appears as some boundary limit. If we extended the model Eq.(27) to possess more freeparameters, then other points in Fig.5 would represent cavity motions when the amplitude and the frequency do notmatch Eq.(34). Below the border curve, the cavity system would be stable, and for states represented by points thatare placed above the curve we would observe the resonance with the exponential growth of the total radiated energy.The last example of solution considered in this paper will justify such predictions. However, we are not able to prove0 ∆ L/L ω / ω ∆ L/L ω / ω Figure 5: The phase diagram for the cavity model Eq.(27): the relative amplitude of cavity oscillations versus the frequencyas the multiplicity of the fundamental frequency. that the statement is generally true. Note famous Arnold’s tongue structure [73] in Fig.5. However, the tongues arerather broad. In the classical theories [73], Arnold’s tongue has usually a narrow knife shape.

2. Inﬁnite values of v We get inﬁnite values of sewing points in Eq.(17) if the cavity oscillates at odd resonance frequencies ω = ω M − , L = ( M −

12 ) T ; M = 1 , , , ... . (35)The conditions Eq.(17), Eq.(18) and Eq.(23) are satisﬁed by a linear map with an arbitrary intercept parameterizedby θ (warning: θ has diﬀerent meaning that the same parameter in the previous subsection. For convenience, we haveredeﬁned this parameter here in such a way that numerous formulae match those of the previous section):∆ ( v ) = v − θ ; | θ | < π . (36)The trajectory of the cavity wall L ( t ) is the same as in Eq.(27). But now, the physical situation is diﬀerent. Inthe previous section, we were almost free to adjust the frequency of the oscillations. If the frequency were ﬁxed,the amplitude would be given by Eq.(34). Here, the frequency is not arbitrary, but the amplitude of the oscillations2 | θ | /ω is adjustable. These solutions are already known and they were presented ﬁrst in [41] (they correspond to thesolutions numbered by m ≡ N − f ( τ ) = ω arctan (tan ωτ + 2 tan θ ) + (cid:0) ⌊ τT + ⌋ − M + 1 (cid:1) T ,f − ( τ ) = ω arctan (tan ωτ − θ ) + (cid:0) ⌊ τT + ⌋ + 2 M − (cid:1) T . (37)Again v max = | sin θ | . The milestones L n and the map number n ( τ ) are given by much simpler formulae: L n = (2 n + 1) L,n ( τ ) = ⌊ τ L + ⌋ . (38)1The Moore’s function is given by the following formula: R ( τ ) = 2 ω arctan (cid:16) tan ωτ − n ( τ ) tan θ (cid:17) + ⌊ τT + 12 ⌋ T − L . (39)The proﬁle function for the energy density is given by: ρ ( τ ) = − (2 M − π L + M ( M − π L (cid:20) v v + 2 n ( τ ) tan θ ) (cid:21) . (40)Now, it is much more easier to work out the integral Eq.(16). Actually, it is interesting to look into an exact andclosed form formula for the total energy produced inside the cavity: E ( t ) = M ( M − π tan θ L t ++ M ( M −

1) tan θ M − L (cid:20) π sign θ − arctan (cid:18) ωt/

2) 1+ √ ωt tan θ − ( t/L +1 − α ( t )) sin ωt tan θ − √ ωt tan θ +( t/L +1 − α ( t )) sin ωt tan θ (cid:19)(cid:21) ( t + (1 − α ( t )) L ) + − (2 M − π L L ( t ) + M ( M − π L + M ( M − π tan θ L α ( t )(1 − α ( t ))+ M ( M −

1) tan θ M − L t/ (2 L ) − α ( t )) tan θ +( t/L +1 − α ( t )) tan ( ω ( t + L ( t )) /

2) tan θ ω ( t + L ( t )) / t/L +2 − α ( t )) tan θ ) , (41)where α ( t ) ≡ t L − ⌊ t L ⌋ . (42)Similarly to the solution presented in the previous subsection, the total energy of the system grows quadraticallywith time. We have extracted the leading term. However, the next to leading terms that are linear in time play animportant role as well. They cause that the energy is not irradiated continuously but rather in sudden jumps. Toverify that, we should take two leading terms from Eq.(41) and make the approximation for large values of t . As aresult we obtain: E ( t ) ∼ = M ( M − π tan θ M − L (cid:18) ⌊ tT ⌋ + Θ( θ ) (cid:19) (43)The presence of Heaviside function means that the problem is not analytical in the parameter θ , i.e. with respectto the change of direction of oscillations. We see that impulses of energy growth occur every period. For smallamplitudes, the energy is proportional to the square of the amplitude. This is in a agreement with a non-relativisticlimit of small velocities. It is amusing to consider a quasi-classical analogue of the model. Suppose, that at the initialstate we have an uniform distribution of energy of classical ﬁelds. The value of the energy density equals the absolutevalue of the static Casimir energy: ρ = π/ L . It corresponds to the classical potential A ( t, x ) = ϕ ( t + x ) + ϕ ( t − x )with ϕ ( τ ) = πτ / L . The classical energy is given by [63]: E cl t ) = Z L ( t )0 dx T ( t, x ) = Z t + L ( t ) t − L ( t ) dτ ˙ ϕ ( τ ) . (44)Next, we allow for the classical evolution of the electromagnetic system. From classical equations of motion we get: ϕ ( τ ) = ϕ ( f ( τ )). Using initial conditions, we obtain a classical global solution: ϕ ( τ ) = π L R ( τ ) . (45)We have encountered almost the same asymptotic formula for the energy as in the quantum case. The only exceptionis the coeﬃcient: E cl ( t ) ∼ = π tan θ M − L (cid:18) ⌊ tT ⌋ + Θ( θ ) (cid:19) (46)The results are confronted in Fig.6. We plot coeﬃcients of asymptotic energy formulae for the ﬁrst eight resonancechannels. To make plots more readable, we have used continuous lines. Classically, the strongest growth is for thefundamental resonance frequency, next resonances are less eﬀective. For the quantum case, the situation reverses.Due to the eﬀect of quantum anomaly, there is no resonance in the ﬁrst channel. Next, there is a rapid saturationfor higher resonance channels. In units of π tan θ/ L , the sum of a classical coeﬃcient and a quantum coeﬃcient isjust a unity.2 order of resonancequantum coefficientclassical coefficient order of resonancequantum coefficientclassical coefficient Figure 6: Coeﬃcients for quantum Eq.(43) and classical Eq.(46) asymptotic energy formulae. The coeﬃcients are in units of π tan θ/ L . B. Homographic fundamental maps ∆ We now turn the discussion to the case of maps ∆ that are rational functions with single poles:∆ ( v ) = h ( v ) v − v . (47)From Eq.(18) we ﬁnd that h ( v ) is at most a quadratic function. The periodicity condition Eq.(9) allows only for onesingularity of ∆ per period. Therefore, we can only consider homographic maps. It is convenient for us to start thediscussion with inversions, and then we will be looking at a general case.

1. Inversion map

We evaluate that Eq.(17) and other necessary conditions are satisﬁed by maps:∆ ( v ) = − v v , v = tan ωL θ . (48)There are no solutions for singular v , so that ω = ω N − and | θ | < π/

2. Moreover, we are forced to assume θ = 0and this way all resonance frequencies are excluded here: ω = ω N . The auxiliary function f for τ ≥ L from Eq.(7)and its inverse function f − for τ ≥ − L yield: f ( τ ) = ω arctan ( v / tan ωτ ) + (cid:0) ⌊ τT ⌋ − M + Θ( − θ ) (cid:1) T ,f − ( τ ) = ω arctan ( v / tan ωτ ) + (cid:0) ⌊ τT ⌋ + 2 M + Θ( θ ) (cid:1) T ,L = ( M + θπ ) T . (49)The trajectory of the cavity wall for t > L ( t ) = L − θω + sign θω h π − arcsin (cos 2 θ cos ( ωt )) i . (50)3Evidently, the maximal velocity is now: v max = | cos 2 θ | . (51)In the limit ω → ω N , we encounter a triangle wave trajectory. For ω = ( ω N + ω N − ) /

2, our solution degenerates to astatic one. The oscillations do always take place between L and L + sign θ ( π − | θ | ) /ω . The corresponding milestonesfor our representation of Moore’s function are given by: L n = ( − n L + ⌊ n + 12 ⌋ (4 M + sign θ ) T . (52)We make the energy density explicit: ρ ( τ ) = − ω π for τ ∈ [ L k − , L k ) − ω π + ω − ω π v (1+ v ) v + v for τ ∈ [ L k , L k +1 ) (53)There are wave packets in the energy density, but there is no unbounded growth of the total energy. The quantumcavity system is stable and its total accumulated energy oscillates with the period (4 M + sign θ ) T .The solution is also well-deﬁned for ω < ω ( L < T / M = 0 and θ >

2. Homographic map

As a ﬁnal and the most interesting application of our ideas, we consider a solution with a fundamental map beinga homographic function. So then, upon confrontation with initial conditions Eq.(17), we set:∆ ( v ) = − v v + v ( v − v ) v − v , (54)where v = tan ( ωL/

2) and v is an arbitrary parameter. It is straightforward to check that physical solutions existon condition that v = v . In passing, we note that the solutions that have been described in the previous subsectionare reproduced for v = 0.The evaluation of relevant auxiliary functions f and f − ends with the results: f ( τ ) = ω arctan ∆ ( v ) + (cid:0) ⌊ τT + ⌋ − M + Θ( v − v ) − Θ( v − v ) (cid:1) T ,f − ( τ ) = ω arctan ∆ − ( v ) + (cid:0) ⌊ τT + ⌋ + 2 M + Θ( v + v ) − Θ( v − v ) (cid:1) T ,L = ( M + π arctan v ) T . (55)The milestones are given by: L n = 2 ω arctan ∆ − n ( v ) + " (2 n + 1) M + n − X k =0 Θ(∆ − k ( v ) + v ) − n Θ( v − v ) T . (56)The angle parameter θ may be introduced here by using the following formula:tan θ = 1 + v v − v . (57)With the above deﬁnition, the derivation of the trajectory of the cavity wall from Eq.(12) gives us:sin ( ωL ( t ) + θ ) = sin ( ωL + θ ) cos ωt , (58)and it can be disentangled successfully: L ( t ) = L + 1 ω [arcsin (sin ( ωL + θ ) cos ( ωt )) − arcsin (sin ( ωL + θ ))] . (59)4We have assumed throughout this paper that the functions arcsin and arctan have their ranges restricted to[ − π/ , π/ v max = | sin ( ωL + θ ) | , (60)while the amplitude of oscillations is given by ∆ L = (2 /ω ) arctan v max .It is important to point some special cases of Eq.(59). For M = 1, v = 0 and v = 1 / (2 tan θ ), we get a cavity modelinvestigated by Law in [21]. It was the ﬁrst exact closed form solution presented in literature. The generalization ofthis solution for any M is leading to a second set of solutions described in the paper by Wu et al. [41] (the solutionswith m = 2 N in their notation). All solutions correspond to cavity vibrations with resonance frequencies ω = ω N . Itwas established for Law’s and Wu’s solutions that there appears resonant instability with a power-like behavior. Weare not going to discuss these solutions here and send the reader back to the original papers. We wish only to notethat there is one more class of solutions with a power-like instability. We obtain these solutions if we put v = 2 v inEq.(54). Let us describe them very brieﬂy. The frequency of oscillation is a free parameter and may be tuned to anyvalue greater than the fundamental frequency ω . But the amplitude of oscillations is uniquely deﬁned by the choiceof frequency.The solutions with exponential instability are obtained for v = 0 and v = 2 v . The derivation of the maps ∆ n forthe representation of Moore’s function Eq.(5) requires the calculation of n -fold composition of the fundamental mapgiven by Eq.(54). It is easy for homographies, so that the result is:∆ n ( v ) = ∆ ◦ n ( v ) = ( λ − λ ) 2( λ n + λ n ) v + ( λ − λ )( λ n − λ n )4( λ n − λ n ) v + 2( λ − λ )( λ n + λ n ) , (61)where: λ , = − v ± p v (2 v − v ) . (62)The proﬁle function for the energy density is then given by: ρ ( τ ) = − ω π + ω − ω π ( v − v ) n ( τ ) " v ( λ n + λ n v + ( λ − λ )( λ n − λ n )4 ) + ( λ n − λ n λ − λ v + λ n + λ n ) . (63)We restrict ourselves to calculate only the approximate value of the total energy for large times. Therefore, the ﬁrstterm in Eq.(63) can be omitted, while the second term integrated over one period gives: Z v =+ ∞ v = −∞ dτ ρ ( τ ) = ω − ω π T r ( H T H ) det ( H ) , (64)where H is a matrix composed of coeﬃcients of the homography ∆ n in Eq.(61). It can be easily calculated that: T r ( H T H ) det ( H ) = 14 (cid:18) v + 1 v (cid:19) "(cid:18) λ λ (cid:19) n ( τ ) + (cid:18) λ λ (cid:19) n ( τ ) − (cid:18) v − v (cid:19) . (65)For simplicity, we have assumed here that the number n ( τ ) do not change in the interval of integration. Further, weassume small amplitudes: 2 L ( t ) ≈ L ≈ M T and obtain the approximate formula: E ( t ) ∼ = 4 M − (cid:18) v + 1 v (cid:19) cosh " v − p v (2 v − v ) v + p v (2 v − v ) t L . (66)In the above brief calculation, we have demonstrated that it is rather easy and safely in our treatment to performapproximate calculations and skip insigniﬁcant details. In fact, the treatment described in Section II and III is welladapted for perturbative methods. However, the relation for the minimal amplitude of oscillations ∆ L min enough totrigger exponential instability of the cavity system vibrating at some ﬁxed frequency ω can be derived exactly:∆ L min L = (cid:12)(cid:12)(cid:12) ωω − ⌊ ωω + ⌋ (cid:12)(cid:12)(cid:12) ωω . (67)As the velocity of cavity wall cannot reach a speed of light, we obtain also the upper limit for amplitudes of oscillations5 ∆ L/L ω / ω Figure 7: The phase diagram for stability and instability regions for the cavity model Eq.(50). ∆ L min at given frequency ω : ∆ L max L = ω ω . (68)The above relation allows us to set up the phase diagram of stability and instability regions. A black area in Fig.7correspond to the instability region of the vibrating cavity model Eq.(50). Below this area, we have deﬁned frequenciesand amplitudes the cavity model is stable. A marginal behavior appears when the energy grows quadratically withtime. It is observed for v = 0 (resonance frequencies, boundaries between adjacent Arnold’s tongues) and v = 2 v (boundary points between stability and instability regions). Other parts of the diagram correspond to points wherethe cavity model is not well deﬁned (physical assumptions about L ( t ) at the beginning of Section II are violated). V. CONCLUSIONS

We have presented a rich class of exact and closed form analytical solutions for the quantum vacuum ﬁeld in aone-dimensional cavity vibrating under the parametric resonance conditions. The solutions are valid for all times,frequencies of cavity oscillations and/or their amplitudes are free parameters. For small amplitudes, cavity oscillationsare close to sinusoidal ones. In view of these properties, we can expect our solutions to yield all generic features knownfrom other investigations on vibrating cavity models in a single dimension.The representation of solutions Eq.(5) that appears in our treatment is based on SL (2 , R ) symmetry of scalar ﬁeldsquantized in a static cavity. We have introduced the notion of fundamental maps that are more convenient to proceedthan Moore’s phase functions. There is a direct mathematical relationship between iterations of fundamental mapsand the mechanism of parametric resonance. This is the way we can get insight into the regions of stability andinstability of the model (see Fig.7). One can calculate the rate of increase of the energy and the Lyapunov exponents.The stability-instability transition points and points between adjacent Arnold’s tongues correspond to cavity modelswith a power-like instability. Thus, the most crucial questions can be tackled. If we insist on detailed calculationsor exact formulas, then we have to set up how regions in space are covered by our maps (see Fig.1). Summarizingtechnical aspects, we can tackle with any solution successfully and completely provided that we know its fundamentalmap ∆ Eq.(7) and respective ranges of maps L n together with n ( τ ) Eq.(11). In this paper, general properties offundamental maps for any physically reasonable solutions have been described. This setup is also a good start pointfor perturbative calculations.6To the best of our knowledge, both exact closed form solutions for oﬀ resonant frequencies and exact closed formsolutions with exponential instability for vibrating cavities were not presented before. It refers also to Arnold’s phasediagrams of stability-instability regions for solutions of vibrating cavity models. It is also important to state that themechanism of parametric resonance in the quantum ﬁeld theory shares many common features with its analogue inthe classical theory. We believe that this similarity could be maintained also for three-dimensional vibrating cavities.The same should be true for the relevance of the symmetry of the quantized cavity model. VI. REFERENCES [1] V.V.Dodonov, Modern Nonlinear Optics, Part 1, in: M.W.Evans (Ed.), Adv. in Chem. Phys., vol. 119, Wiley, New York,309 (2001).[2] K. A. Milton, J.Phys. A37, R209 (2004).[3] H.B.G. Casimir, Proc. Kon. Ned. Akad. Wetensch. , 793 (1948).[4] C.Braggio et al., Europhys.Lett. , 754 (2005).[5] M. Brown-Hayes et al., J. Phys. A: Math Gen. , 6195(2006).[6] W.J.Kim, J.H.Brownell, R.Onofrio, Phys.Rev.Lett. , 200402 (2006).[7] E.Arbel-Segev et al., preprint http://xxx.lanl.gov/abs/quant-ph/0606099v3 (2006).[8] A.V.Dodonov, S.S.Mizrahi, V.V.Dodonov, preprint http://xxx.lanl.gov/abs/quant-ph/0612067v2 (2007).[9] G.T. Moore. J. Math. Phys. , 2679 (1970).[10] S.A.Fulling,P.C.W.Davies, Proc.Roy.Soc.London A348 , 79 (1976).[11] P.C.W.Davies, S.A.Fulling, Proc.Roy.Soc.London

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