Evolution of wave fields and atom-field interactions

PHYSICAL REVIEW A

VOLUME 57, NUMBER 6

JUNE 1998

Evolution of wave fields and atom-field interactions in a cavity with one oscillating mirror Maciej Janowicz* Instytut Fizyki, Uniwersytet Mikołaja Kopernika, ulica Grudzia¸dzka 5/7, 87-100 Torun´, Poland ~Received 11 August 1997; revised manuscript received 26 January 1998! In this paper the dynamics of wave fields in a cavity with one sinusoidally moving mirror are investigated. The wave equation is solved approximately using the method of multiple time scales. Frequency shifts as well as amplitude and phase modulations of cavity modes are given. In all cases when it is possible, an approximate effective field Hamiltonian operator is derived. In the adiabatic case it is then used to analyze the interaction of radiation with a two-level atom passing through the cavity. Corrections to the Rabi oscillations due to the time-dependent frequency of the cavity photons are investigated. It is shown that adiabatic motion of a mirror can lead to suppression of the Rabi oscillations of atomic inversion. @S1050-2947~98!00106-1# PACS number~s!: 42.50.2p, 03.65.Ge, 02.30.Mv

I. INTRODUCTION

*Present address: Institute of Physics, Polish Academy of Sciences, Aleja Lotniko´w 32/46, 02-668 Warsaw, Poland. Electronic address: [email protected]

tonian approaches to very similar problems have been used by Razavy and Terning in @7#. Our procedure can be briefly described as follows. After rewriting the classical wave equations in convenient dimensionless variables, we get a linear partial differential equation with variable coefficients but with trivial boundary conditions. Then we investigate such a boundary-value problem with an additional assumption of the ‘‘smallness’’ of either the amplitude of the sinusoidal motion of the mirror or the frequency of these oscillations, or both. The dynamics of fields in the cavity are analyzed using many time scales that appear very naturally if several variables of the dimension of frequency and of different orders are present. It allows us to specify the most important physical processes that take place in various regimes of characteristic frequencies. It is well known that in the method of multiple scales, the zeroth-order approximation alone brings in many cases very satisfactory results, provided the amplitude and phase corrections are taken into account. We shall identify several cases when this zeroth-order approximation can be obtained by solving an effective classical Hamiltonian. In all these cases we naively quantize the problem by writing creation and annihilation operators instead of c-number complex amplitudes, commutators instead of canonical Poisson brackets, and effective quantum Hamiltonians instead of effective classical Hamiltonian functions. Then, in the adiabatic case, we add to this approximate quantum Hamiltonian a term representing the coupling with a two-level atom ~as well as the free atomic Hamiltonian! and try to solve the atom-radiation interaction problem in a cavity with oscillating walls. I would like to mention that there exist also exact solutions of the wave equations with periodically ~though not sinusoidally! oscillating walls @15#. Another solution of this type, based on the group-theoretical methods, will be presented elsewhere. The rest of the paper is organized as follows. In Sec. II we analyze what we call ~just for the sake of this paper! ‘‘the adiabatic case’’: it is assumed that the frequency of oscillations of the wall is much smaller than the fundamental frequency of the cavity; three interesting subcases are identified. In Sec. III the atom-field interactions are studied based on an approximate field Hamiltonian for adiabatic cases. Some final remarks are contained in Sec. IV. As not all of

1050-2947/98/57~6!/4784~7!/$15.00

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One of the most important branches of recent physical research involving the fundamental concept of quantum vacuum @1# has been cavity quantum electrodynamics ~CQED!. In CQED we disturb the ‘‘normal’’ electromagnetic vacuum in various ways; for instance, the properties of vacuum near one perfectly conducting mirror and between two such mirrors have been analyzed in much detail @2–4#. Configurations with moving mirrors @5# are particularly interesting since one should expect then the emission of photons @5–7# with nonclassical photon statistics @8–10# as well as the dynamical modification of the Casimir force @10–16#. Very detailed studies exist where the problem of emission of photons by a moving boundary between a material body and vacuum is investigated @17–21#. In the author’s opinion, detailed knowledge about the available classical solutions to the wave equation should be of principal importance for the fruitful study of quantum electrodynamical problems with moving boundaries. In this paper we use the method of multiple scales to obtain such solutions approximately for the case of sinusoidal motion of one mirror. Several different regimes of dynamics are identified depending on the presence of one or more small parameters in the system. It is to be noted that in another paper about the classical and quantum theory of fields in a cavity between two perfect moving mirrors @5# some approximate schemes to find the solutions to the wave equation have already been developed ~see also @11#!. I believe, however, that the method of multiple scales, applied here, has many advantages as it is universal, simple, and especially very well suited for studying systems with parametric resonance. It can also be used very efficiently to obtain approximate effective Hamiltonians as it is shown below. The last fact is especially important in the case of systems for which one cannot write an exact Hamiltonian for very fundamental reasons ~cf. @5#!. The concept of effective Hamiltonians has been introduced to CQED with moving mirrors by Law in @14#. Other Hamil-

57

© 1998 The American Physical Society

57

EVOLUTION OF WAVE FIELDS AND ATOM-FIELD . . .

the calculations are entirely trivial, the main body of the paper is accompanied by the Appendix.

where w mn 52n p

II. WAVE FIELDS IN A CAVITY WITH AN ADIABATICALLY MOVING MIRROR

Let us consider an idealized one-dimensional cavity with perfectly reflecting mirrors. We shall consider the electric and magnetic fields as depending on one spatial coordinate, say, z, only. Then one may introduce the potential A, also depending on time and the z coordinate, which satisfies the wave equation 1 c2

Att 5Axx .

~1!

]t

12x 2

g¨ ] u g˙ ] 2 u g˙ 2 ] 2 u 1 ] 2 u g˙ 2 ] u 2 2x 22x 1x 5 . g ]x g ]x]t g2 ]x g2 ]x2 g2 ]x2 ~2!

In this equation g˙ means dg/d t . Equation ~2! is to be completed by the boundary conditions. In the new variables they are simple: u50 at x50 and u50 at x51. Let us now expand A into a series that will guarantee the exact fulfillment of the boundary conditions:

A( `

u5

2 A ~ t ! sin~ n p x ! . L n51 n

~3!

The author’s computational skills have turned out to be insufficient to solve Eq. ~2! exactly. Instead, I have had to apply a very efficient approximate scheme, the method of multiple scales, provided at least one small parameter in Eq. ~2! is available. In some realistic situations it is indeed the case. Thus we look for approximate solutions to the wave equations that satisfy the boundary conditions exactly. From Eq. ~3! it follows that A¨ m 12

g˙ 2

g¨

g˙

( w mn A n 2 g (n w mn A n 22 g (n w mn A˙ n g2 n

2

g˙ 2 g2

(n

v mn A n 1

E

1

0

v mn 52 ~ n p ! 2

E

x sinm p x cosn p x dx, 1

0

x 2 sinm p x sinn p x dx.

Equation ~4! has been the starting point of analysis in several recent papers about QED in cavities with moving mirrors, e.g., @16,11#. Below we shall analyze Eq. ~4! in several different dynamical regimes. From now on it is assumed that g( t ) is a sinusoidal function g ~ t ! 511

The vectorial nature of A is irrelevant in the onedimensional case so that we consider it as a scalar field and denote it by u. Since on perfectly conducting mirrors the tangential components of the electric field must be zero, we have simple boundary conditions for u as well: u must be zero on both mirrors. Let us assume that one mirror is fixed at z50, whereas the second one oscillates periodically according to the law z5q(t)5L1A f (Vt)5Lg(Vt), where L is the size of the cavity at rest, A is now the amplitude of the oscillations, f is a periodic function of time, and g(Vt) is equal to 11(A/L) f (Vt). Let us introduce the following dimensionless variables: a new time variable t 5(c/L)t, a new spatial variable x5z/q(t), and a new ‘‘frequency’’ n 5(L/c)V. Upon substituting these variables into Eq. ~1!, we get

] 2u

4785

1 g

~ m p ! 2 A m 50, 2

A sin n t . L

A. Adiabatic case

In this subsection we shall consider the following situation. The amplitude of oscillations is small when compared to the size of the cavity at rest and the frequency of oscillations of one mirror is also small when compared to the fundamental frequency of the cavity v 1 5 p (c/L) ~such an adiabatic case has been investigated by different methods and from a different point of view in @22#!. Furthermore, we shall assume that these two ratios are of the same order, that is, we write A/L5 e and n 5 em , where e is much smaller than 1, while m is of the order of unity. Intuitively, one may expect that in such an adiabatic case there will be no coupling between various modes @each term in the expansion ~3! is called ‘‘a mode’’#, but there will be some shifts and modulations in phases and amplitudes. The quantitative analysis of the Appendix confirms these expectations. Let us now rewrite Eq. ~4! including only terms up to order e 2 : A¨ m 22 e 2 m cosemt

(n w mn A˙ n 1 ~ m p ! 2~ 122 e sinemt

13 e 2 sin2 emt ! A m 50.

~6!

(1) 2 We now write A m as a sum A m 5A (0) m 1 e A m 1 e A m (2) 1••• and consider all terms of this expansion as functions of several time scales ~see the Appendix!. This is the key point of the method of multiple scales @23#, which, together with the Krylov-Bogoliubov-Mitropolskii @24# method, is one of the most efficient analytical tools of nonlinear mechanics and theory of parametric resonance. Let us stress that, according to the spirit of the method of multiple scales, the zeroth-order terms A (0) m alone will contain corrections to the amplitudes and phases. Usually, to get satisfactory agreement with numerical or experimental results, it is enough to take into account only such a ‘‘corrected’’ zeroth-order term. From the calculations provided in the Appendix it follows that

*~ t ! , A m 'a m ~ t ! 1a m ~4!

~5!

where

~7!

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MACIEJ JANOWICZ

S

a m ~ t ! 5a m ~ 0 ! e 2im p t exp 2i e

F

3exp

S

mp ~ cosn t 21 ! n

e ~ 112w mm ! sinn t 2

e2 mp 1 2im p t 1i sin2 n t 2 2n

DG

D

order of 1. Using the results of the Appendix, we conclude that the approximation valid up to first order in e is given by Eq. ~7! with

F

a m ~ t ! 5a m ~ 0 ! exp 2im p t 2i e .

~8!

Note that in the method of multiple time scales, in order to obtain an expansion valid up to first order in e , one must calculate corrections to phases and amplitudes up to second order: This is seen in Eq. ~8! as well. Of course, if we had had e 50, we would get the exact solution for A m in the cavity at rest. There are several conclusions that can be drawn from Eq. ~8!. First, the motion of the cavity leads ~in the second-order of approximation! to some constant frequency shift, equal to m p e 2 . Second, already in the first order, there arises a modulation of the frequency. Third, in the second order we have to do with the modulation of the amplitude of oscillations in each mode. This effect can be called a ‘‘perodic’’ damping. Of course, if we return to our old variables (t,z), the time-dependent corrections to the zeroth-order approximation will be contained in the factor sin mpx as well. Let us now restrict ourselves to first-order corrections, which could be perfectly enough for very small e . This means that not only do we drop the terms proportional to e 2 in Eq. ~8!, but we should write sin mpx.sin mpz/L. Then we find that the approximate solution for A m is derivable from the effective Hamiltonian function Hef f5

57

(m v m~ 12 e sinVt ! a m a m*

~9!

@where v m 5(c/L)m p #, which defines the canonical trans* (0) to a m (t) and a m* (t), where formation from a m (0) and a m a m (t) is given by Eq. ~8!. If we now perform quite a naive quantization of the system by letting the complex amplitudes a and a * to acquire the destruction and creation operator status, we shall see that the main physical process connected with the adiabatic oscillations of the cavity wall is just the modulation of energy of photons in a particular mode. Neither the exchange of excitations between various modes nor any appreciable photon production can take place, as already pointed out in @22,11#. The same is true in the ‘‘subadiabatic’’ case considered below. B. Subadiabatic case

If the amplitudes of oscillations are much smaller than the size of the cavity A [ e !1 L and if the ratio of the frequencies V/ v 1 is still much smaller than e , one has to do with what I call the subadiabatic case. Here again v 1 5 p c/L is the fundamental frequency of the cavity. This case has some peculiarities, but it should be considered separately mainly because it is probably most realistic to access in laboratory practice. In order to take into account that V/ v 1 ! e , we write n 5 e 2 m , where m is of the

2im p

S

mp ~ cosn t 21 ! n

1 2 1 e2 e t2 sin2 n t 2 4 n

DG

~10!

.

It is clear from this expression that in the subadiabatic case we have to do with the frequency shift and the essential modulation of the phase only. On the other hand, the amplitude remains constant. Thus the evolution can be considered to be Hamiltonian up to second order. The Hamiltonian function is given by

Hef f5

(m v m

S

D

1 1 *. 12 e sinVt1 e 2 2 e 2 cos2Vt a m a m 2 2 ~11!

To complete this section let us finally consider the largeamplitude adiabatic case, in which one has A/L5O(1) while n [ e !1. In this case one can obtain A (0) m in the form A ~m0 ! 5 a m e 2im p T 2 1T 3 1c.c.,

~12!

where

T 25

S

E

T 3 5 w mm 1

dt , 11d sinn t

D

1 lnu 11d sinn t u , 2

and d5A/L. Formula ~12! shows that in the large-amplitude case one has to do with the very strong amplitude modulation already in first order in e and no simple Hamiltonian exists. The nonadiabatic resonant cases with n 5 p and n 52 p have been studied very carefully in recent papers @11–13#. In @13# important preliminary results about the atom-radiation interaction for n 52 p are contained. Thus the nonadiabatic cases will not be considered here. Let us mention only that the effective Hamiltonians can be also constructed using the method of multiple scales. They contain terms responsible for the strong coupling between various modes as well as, for n 52 p , the term analogous to the ~interaction-picture! Hamiltonian of the degenerate parametric amplifier. This means that the state of the electromagnetic field produced by the resonantly moving wall not only would be squeezed ~this is well known!, but would also exhibit antibunching of photons if the lowest-mode initial state were a coherent state @25#.

57

EVOLUTION OF WAVE FIELDS AND ATOM-FIELD . . .

S 12S 215 AN 21 a † s 12AN 21 s 21a5N 21 a † a s 11

III. INTERACTION OF A TWO-LEVEL ATOM WITH RADIATION IN A CAVITY WITH AN OSCILLATING MIRROR: ADIABATIC CASES

We shall now consider the interaction of a two-level atom with one mode of electromagnetic field in a cavity with a sinusoidally oscillating mirror. It is assumed that the atom is coupled to the lowest-frequency mode of the field. We shall use the free-field Hamiltonian ~9! with the complex ampli* replaced by creation and annihilation optudes a m and a m erators aˆ m and aˆ †m . In the following, however, carets over the operators will be dropped. Then the full Hamiltonian is given by (\51) H5 v 1 ~ 12 e sin Vt ! a † a1 vs 221G ~ s 21a1a † s 12! , ~13! where v is the energy difference between the lower and the upper atomic levels, s i j 5 u i &^ j u , u 1 & denotes the lower atomic state, u 2 & is the higher state, and G is the coupling constant. We have ignored the dependence of the coupling constant on time, which is justified in the adiabatic case. Also, we have written a instead of a 1 and a † instead of a †1 . In addition to the time scales associated with the field and, in particular, with the sinusoidal motion of one mirror, we now have to do with additional frequencies ~and therefore time scales! associated with the atom and its coupling with the field. The most important of them is the Rabi frequency ¯ , where N ¯ is the mean excitation number in the atom G AN plus field system ~we exclude here the case in which the atom enters the cavity in the ground state and the field is prepared in the vacuum state; it is obviously not interesting in the adiabatic regime!. The dynamical pattern strongly depends on the ratios of all these frequencies. We shall restrict ourselves to the analysis of some interesting cases defined by special relations among time scales appearing in the system. Upon writing the time-evolution operator U as U5WU 1 , where

S

W5exp 2i v 1 Nt2i

ev1 ~ cosVt21 ! a † a V

D

d U 5H 1 ~ t ! U 1 , dt 1

5N 21 ~ a † a1 s 22! s 115 s 115S 11 . It follows that H 1 (t) has a form perfectly analogous to that describing the two-level atom driven by an external laser field with the phase modulation studied, e.g., in @26–28#. In particular, in @27# it was discovered that the trapping of population is possible if one appropriately chooses the modulation frequency. We can easily show that the same phenomenon can occur in a cavity with a moving mirror. In fact, using the Fourier-series expansion of the exponents in Eq. ~15!, we get

i

dU 1 dt

F

¯ 5 G AN

AN

AN¯

¯ 12G AN

AN

ev1 V

~ S 21e i e v 1 /V 1S 12e 2i e v 1 /V !

`

~ S 21e i[ e v 1 /V2k ~ p /2! ]

1S 12e 2i[ e v 1 /V2k ~ p /2! ] ! J k

S D ev1 V

where the transformed Hamiltonian is H 1 ~ t ! 5DS 221G AN ~ S 21e 2i ~ e v 1 /V !~ cosVt21 !

where J l (y) denotes the lth-order Bessel function of the argument y. In the derivation of Eq. ~16! it has been assumed that D50 ~the atom and the lowest-frequency mode are per¯ is much smaller fectly tuned!. Let us now suppose that G AN than V. We have already assumed that V and e v 1 are of the same order. This means that, for a microwave cavity with a size of 1 cm, we should have, e.g., A;1 m m, V;105 Hz, ¯ ;103 2104 Hz. Under these assumptions, we can and GN easily obtain an approximate expression for U 1 ~using, e.g., the method of multiple time scales!, valid up to the first order ¯ /V, in G AN `

(

1S 12e 2i[ e v 1 /V2k ~ p /2! ] ! J k

S D

where D5 v 2 v 1 , S ii 5 s ii for i51,2, S 125 AN a s 12 , and S 215 AN 21 s 21a. It is an important fact that N commutes with all S i j ’s and that S 222S 11 , S 12 , and S 21 generate the same algebra ~in the sense of both Lie algebra and associative algebra! as s 222 s 11 , s 12 , and s 21 . For instance,

G

ev1 sinkVt V 1 , ~17! V

where 1 is the unit operator and

FS DAG FS DAG ev1 G Nt 2i ~ e i ~ e v 1 /V ! S 21 V

1e 2i ~ e v 1 /V ! S 12! sin J 0 ~15!

G

coskVt U 1 , ~16!

V 1 5cos J 0

21 †

S D

( AN¯ k51

F

~14!

1S 12e i ~ e v 1 /V !~ cos Vt21 ! ! ,

J0

1 G AN U 1 ' 122i ~ S e i[ e v 1 /V2k ~ p /2! ] V k51 k 21

and N5 s 221a † a is the excitation number operator ~being an operator constant of motion!, we find that U 1 satisfies the Schro¨dinger equation i

4787

ev1 G Nt . V

~18!

This approximation is valid even if J 0 ( e v 1 /V) is very small. As an immediate consequence we note that if the last condition holds, the Rabi oscillations of inversion will be suppressed and the population will be trapped in one of the ¯ . Inatomic states, at least for times of the order of 1/G AN deed, let J 0 ( e v 1 /V)50. Let the atom enter the cavity in the

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MACIEJ JANOWICZ

excited state and the field state be a ~for instance, it can be a coherent state!. Then we have for the population of the excited state

^ S 22~ t ! & 5 ^ 2; a u U †1 ~ t ! W † ~ t ! S 22~ 0 ! W ~ t ! U 1 ~ t ! u a ;2 & 5 ^ 2; a u U †1 ~ t ! S 22~ 0 ! U 1 ~ t ! u a ;2 & ' ^ 2; a u S 22~ 0 ! u a ;2 & 1

F K UA ( S D U L G 2iG 2; a V

3J k

NS 12

k

1 2i ~ e v /V ! 2k ~ p /2! 1 e k

ev1 sinkVt a ;2 1c.c 51. V

~19!

We see that the result does not actually depend on the initial state of the field. From the appropriate second-order calculations it follows that in fact the inversion oscillates very fast with the frequency V and its multiplicities, but these oscillations have very low amplitudes. The same is true about the mean photon number. Let us now consider the subadiabatic case. It is assumed that V! e v 1 . For instance, let the size of the cavity be '1 cm and the amplitude A of oscillations of one mirror '1 m m. Then e 51024 , e v 1 '106 Hz, and V'102 Hz. Let us ¯ '104 Hz, so that V!G AN ¯ ! e v 1. also assume that G AN Then we have another small parameter at our disposal. Let us call it d and define as d 5 AV/ e v 1 . Equation ~16! is still valid, but we can now use the asymptotic formula for the Bessel functions of large arguments to get finally U 1 '122i 3cos

A

S

2 G AN 1 2 ~ S 21e i[1/d 2k ~ p /2! ] 1H.c.! p Ae v 1 V ~ k odd! k

(

D

p p 2k 2 sin~ kVt ! . 2 4 d 1

2

~20!

It is clear from this above expression that the Rabi oscillation of atomic inversion and of the mean photon number will be again strongly suppressed. Also, we shall have to do with the elimination of some lines from the spectrum of the light spontaneously emitted by the atom and that transmitted by the cavity walls ~obviously, here we cannot treat such spectra consistently for the cavity has been assumed to be ideal with perfectly reflecting mirrors!. The problem of the spectra of radiation spontaneously emitted by the atom or transmitted through the walls ~one of which oscillates adiabatically! has been analyzed by Law and co-workers in @29#. They have considered a dynamical regime very different from that studied here: In their paper e v 1 ~in our notation! is much smaller than G and V52G. This has enabled them to ignore all the ‘‘Bessel sidebands’’ arising due to the Fourier expansion analogous to that in our Eq. ~16!. In their dynamical regime the condition J 0 ( e v 1 /V)50 cannot be fulfilled for reasonable values of v 1 and G. In our work e v 1 and V are both, in the adiabatic case, much larger than G and hence the Bessel sidebands would be of great importance in the spectra. Physically, the effect of suppression of the Rabi oscillations can be attributed to the interference of probability am-

57

plitudes ~cf. @27#!. However, we prefer another interpretation. The adiabatic motion of the walls introduces a dynamical modification of the coupling between the atom and the field. When the special relation between amplitudes and frequencies A v 1 5 p 0n LV (p 0n is the nth zero of the Bessel function J 0 ) is fulfilled, this modification actually becomes a cancellation. In a sense we then have to do with an internal resonance in the system. It does not ~of course! depend on the initial states of the atom plus field system. Let us also notice that oscillation of the wall does not bring any dynamical detuning of the field from the atom. To conclude this section let us note that an experiment with atoms in cavities with adiabatically or subadiabatically moving mirrors ~rather than with atoms in external laser fields! can be performed to verify that the trapping of population can take place in systems with sinusoidal phase modulation. In other words, one might measure the atomic energy before and after passing through the cavity with an oscillating mirror to find whether a particular relation between characteristic amplitudes and frequencies @ J 0 (A v 1 /LV)'0# does or does not hold. In this connection we propose an experiment with a cavity, one wall of which is made from a metallized piezoelectric element. Let an atom prepared in a Rydberg state pass through the cavity. To be more specific, let it be a rubidium atom with the transition 63p 3/2↔61d 3/2 , v '2 p 32.231010 Hz, tuned to the lowest-mode frequency of the cavity at rest. This means that the size of the cavity must be L'0.68 cm. We require that the piezoelectric effect makes one of the cavity walls oscillating with the frequency, say, V52 p 3106 Hz and amplitude A50.745 m m. To the author’s knowledge, piezoelements with such parameters can be performed; alternatively, one might think about using the effect of magnetostriction to make the wall oscillate @30#. After the atom, prepared in the excited state, passes through the cavity, one can measure its energy in the ionization chamber to verify our predictions as described above. IV. FINAL REMARKS

In this work the dynamics of classical wave fields in a cavity with a sinusoidally moving mirror have been analyzed. It has been found that the main physical processes involved depend on the relations between characteristic frequencies of the system and between the amplitude of oscillation of the cavity and its size. Therefore, several different dynamical regimes have been studied with the method of multiple time scales. It has been found that in adiabatic regime one should expect amplitude and phase modulation in each mode together with a constant shift of the frequencies of modes, but no exchange of energy between various modes and no appreciable photon creation. In the subadiabatic case one can construct a simple, effective, valid up to the second order, free-field Hamiltonian. After quantization, such a Hamiltonian describes photons with time-dependent energy. A similar procedure is possible in the adiabatic case, but only up to first order. In the large-amplitude adiabatic case no simple effective Hamiltonian exists: Already in first order one finds strong amplitude modulations. Interactions of a two-level atom with the cavity field in the case of adiabatic and subadiabatic motion of one mirror have been studied. It has been found that this motion can lead to suppression of

57

EVOLUTION OF WAVE FIELDS AND ATOM-FIELD . . .

the Rabi oscillations if a special condition of internal resonance is fulfilled. Under this condition the atom and the field become decoupled. This effect can be verified experimentally. It is the author’s hope to generalize the results to full electrodynamics in three dimensions.

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valid only for very small times. Therefore, we require that the right-hand side of Eq. ~A6! vanishes. This leads to the system of equations

] a m* *, 52im p sin~ m T 1 ! a m ]T1 ~A7!

]am 5im p sin~ m T 1 ! a m , ]T1

ACKNOWLEDGMENTS

It is a pleasure to thank Professor E. A. Hinds, Dr. G. Barton, Professor P. A. Maia Neto, and Professor J. Mostowski for many stimulating discussions.

with the obvious solution, a m 5 a m ~ T 2 ! e 2i ~ m p / m !~ cosm T 1 21 ! ,

* 5 a m ~ T 2 ! e i ~ m p / m !~ cosm T 1 21 ! . am

APPENDIX

In this appendix we derive the asymptotic formula for the approximations to A m satisfying Eq. ~4! using the method of multiple time scales. Let us consider Eq. ~6!. To obtain an approximate solution, let us introduce three different ‘‘times’’ T 0 5 t T 1 5 et T 2 5 ef ~ T 1 ! ,

~A1!

where f (T 1 ) is still to be specified. Then we have d ] ] ] 5 1e 1 e 2f 8~ T 1 ! 1••• dt ]T0 ]T1 ]T2

~A2!

Then, to get A (1) m we must only solve the same equation as Eq. ~A4! with the result

* ~ T 1 ,T 2 ! e im p T 0 . A ~m1 ! 5b m ~ T 1 ,T 2 ! e 2im p T 0 1b m

S

]2

S

D S

2

D

S

D

]2

1 ~ m p ! 2 A ~m0 ! 50, 2

]T0

with the solution ~A5!

* are still functions where the complex amplitudes a m and a m of T 1 and T 2 . In first order we get

S

D

S

] a m* im p T ]2 ]am 0 1 ~ m p ! 2 A ~m1 ! 52 im p 2im p e ]T0 ]T1 ]T1 12 ~ m p ! sinm T 1 ~ a m e 2

* e im p T 0 ! , 1a m

12 f 8 ~ T 1 ! 2

]T1

(n

D

]2 A~0! ] T 0] T 2 m w mn

] A ~n0 ! ]T0

23 ~ m p ! 2 sin2 ~ m T 1 ! A ~m0 ! .

~A4!

* e im p T 0 , A ~m0 ! 5a m e 2im p T 0 1a m

S

]2

12 m cos~ m T 1 !

Substituting these expressions and the expansion

into Eq. ~6! and equating terms standing at the same power of e , we get in zeroth order

D

2im p T 0

~A10!

This is an inhomogeneous linear equation for A (2) m . To avoid in the solution any terms proportional to powers of T 0 , the following relations must hold:

]bm 5im p sin~ m T 1 ! b m , ]T1

] b m* * 52im p sin~ m T 1 ! b m ]T1 ~A11!

* on T 1 is the same as that of ~thus the dependence of b m ,b m * ) and a m ,a m i f 8~ T 1 !

]am 1 5 @ i m cos~ m T 1 ! a m 12i m cos~ m T 1 ! w mm a m ]T2 2 12m p sin2 ~ m T 1 ! a m #

~A12!

plus the complex conjugate of Eq. ~A12!. Equation ~A12! has the solution

a m 5 ¯a m e 2iT 2 5 ¯a m e 2i ef ~ T 1 ! ,

~A6!

where m 5 n / e . The right-hand side of Eq. ~A6! contains terms proportional to e 2im p T 0 and e im p T 0 . This means that the special solution to the inhomogeneous equation ~A6! must contain terms proportional to T 0 , then the approximation would be

D

]2 ]2 1 ~ m p ! 2 A ~m2 ! 5 22 12 ~ m p ! 2 sinm T 1 A ~m1 ! ]T0 ] T 0] T 1

]2 ]2 ]2 2 5 12 e 1 e 1 f T 1•••. ! ~ 8 1 ] T 0] T 1 ]T1 ] T 0] T 2 ] t 2 ] T 20 ~A3!

A m 5A ~m0 ! 1 e A ~m1 ! 1 e 2 A ~m2 ! 1•••

~A9!

In second order with respect to e we find the following equation, which allows us to obtain a m (T 2 ), b m (T 1 ), and A (2) m :

and d2

~A8!

~A13!

where

f~ T1!5

i mp mp sin~ m T 1 !~ 112w mm ! 1 T 12 sin~ 2 m T 1 ! . 2 2 4m ~A14!

MACIEJ JANOWICZ

4790

At this point we may stop and consider ¯a and ¯a * as constants. Since the natural initial conditions include the relations A ~m0 ! ~ 0 ! 5A m0 ,

A ~m1 ! ~ 0 ! 50,

A ~m2 ! ~ 0 ! 50,

1 T 35 f~ e 2t !, e

T 45 c~ e 2t !.

f~ e 2t !5

mp @ cos~ e 2 mt ! 21 # . m ~A18!

In second order one finds i c 8~ e 2t !

]am 5m p sin2 ~ e 2 mt ! a m , ]T4

c ~ e 2 t ! 5m p

~A19!

@1# P. W. Milonni, The Quantum Vacuum ~Academic, New York, 1993!. @2# G. Barton, Proc. R. Soc. London, Ser. A 410, 141 ~1987!. @3# E. A. Hinds, Adv. At., Mol. Opt. Phys. 28, 237 ~1991!. @4# E. A. Hinds, Cavity Quantum Electrodynamics, edited by Paul R. Berman, Advances in Atomic, Molecular and Optical Physics, Suppl. 2 ~Academic, New York, 1995!. @5# G. T. Moore, J. Math. Phys. 11, 2679 ~1970!. @6# S. A. Fulling and P. C. W. Davies, Proc. R. Soc. London, Ser. A 348, 393 ~1976!. @7# M. Razavy and J. Terning, Phys. Rev. D 31, 307 ~1985!. @8# G. Barton and C. Eberlein, Ann. Phys. ~N.Y.! 227, 393 ~1993!. @9# V. V. Dodonov, A. B. Klimov, and V. I. Man’ko, Phys. Lett. A 149, 225 ~1990!. @10# V. V. Dodonov, A. B. Klimov, and D. E. Nikonov, J. Math. Phys. 34, 2742 ~1993!. @11# V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664 ~1996!. @12# V. V. Dodonov, Phys. Lett. A 213, 219 ~1996!. @13# V. V. Dodonov, Phys. Lett. A 207, 126 ~1995!. @14# C. K. Law, Phys. Rev. A 49, 433 ~1994!. @15# C. K. Law, Phys. Rev. Lett. 73, 1931 ~1994!. @16# C. K. Law, Phys. Rev. A 51, 2537 ~1995!.

F

G

1 2 1 e t2 sin~ 2 me 2 t ! . 2 4m ~A20!

From these expressions one obtains Eq. ~10!. In the large-amplitude adiabatic case we have the following equation for A m : d2 dt

A 52 e 2 m 2

d m d cos~ emt ! w mn A n ( 11dsin~ emt ! n dt ~ mp !2

@ 11d sin~ emt !# 2

Am ,

~A21!

where d5A/L is not small; e is defined as V/ v 1 . Upon the introduction of two time variables 1 T 2 5 f ~ et ! , e

~A17!

~the prime denotes the derivative of a function over its whole argument! with the solution a m 5 a m ~ T 4 ! e 2iT 3 ,

a m 5 ¯a m e 2iT 4 ,

~A16!

A m is assumed to depend on these three time scales independently. Then, proceeding in a fully analogous way to the adiabatic case, we find that in zeroth order the solution ~A5! holds, while in first order we get

]am i f 8~ e 2t ! 1sin~ e 2 mt ! a m 50 ]T3

with the solution

~A15!

* 50, ¯a m 5a m (0), and ¯a m* 5a m* . Hence we can write b m 5b m the approximate solution given by Eqs. ~7! and ~8! follows. Let us now consider the subadiabatic case. One of the mirrors of the cavity oscillates with the dimensionless frequency n 5 e 2 m , where m 5O(1). Let us introduce the time variables T 05 t ,

57

T 3 5 c ~ et !

~A22!

and using the same procedure as before, we find

* e im p T 2 1T 3 , A ~m0 ! 5 a m e 2im p T 2 1T 3 1 a m

~A23!

where T 25

S

E

T 3 5 w mm 1

dt , 11d sinn t

D

1 lnu 11d sinn t u . 2

@17# G. Barton and A. Calogeracos, Ann. Phys. ~N.Y.! 238, 227 ~1995!. @18# A. Calogeracos and G. Barton, Ann. Phys. ~N.Y.! 238, 268 ~1995!. @19# G. Barton, Ann. Phys. ~N.Y.! 245, 361 ~1995!. @20# G. Barton and C. A. North, Ann. Phys. ~N.Y.! 252, 72 ~1996!. @21# P. A. Maia Neto, J. Phys. A 27, 2167 ~1994!. @22# G. Calucci, J. Phys. A 25, 3873 ~1992!. @23# A. H. Nayfeh and D. Mook, Nonlinear Oscillations ~Wiley, New York, 1995!. @24# Yu. A. Mitropolskii, Problems of Asymptotic Theory of Nonstationary Oscillations ~Nauka, Moscow, 1964!. @25# D. Walls and G. Milburn, Quantum Optics ~Springer, New York, 1995!. @26# M. Janowicz, Phys. Rev. A 44, 3144 ~1991!. @27# G. S. Agarwal and W. Harshawardhan, Phys. Rev. A 50, R4465 ~1994!. @28# W. Harshawardhan and G. S. Agarwal, Phys. Rev. A 55, 2165 ~1997!. @29# C. K. Law, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A 52, 4095 ~1995!. @30# M. Jaworski and A. Sienkiewicz ~private communication!.