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Photon creation and excitation of a detector in a cavity with a resonantly vibrating wall
a __
__ f!!B
ELSEVIER
30 October 1995 PHYSICS
LETTERS
A
Physics Letters A 207 (1995) 126-132
Photon creation and excitation of a detector in a cavity with a resonantly vibrating wall V.V. Dodonov RN. L.ebedev Physics Institute, Lminsky Prospekt 53, 117924 Moscow, Russian Federation Moscow Institute of Physics and Technology, I6 Gagarin Street, 140160 Zhukovskiy, Moscow Region, Russian Federation
Received 15 July 1995; accepted for publication 2 September 1995 Communicated by V.M. Agranovich
Abstract The problem of photon generation inside an ideal 3D cavity with resonantly vibrating walls is studied. The possibility of creating from a vacuum up to lo4 photons in a cavity with a Q-factor of about 3 x 10” is predicted. Different responses of detectors modeled by a harmonic oscillator and by a two-level atom are demonstrated. Keywords: Cavity electrodynamics; Vibrating boundary; Parametric resonance; Pield-atom interaction
1. Intruduction The problem of photon creation from a vacuum due to the motion of boundaries has attracted the attention of many authors during the last 25 years. Due to
its complexity it was considered mainly in the framework of the one-dimensional model (scalar electrodynamics) [ l-121. The 3D situation was considered only in the specific case when the wall moved with a constant velocity [ 13,141. Although some interesting exact and approximate solutions were found in the mentioned papers, they did not discuss a possible experimental verification of the phenomenon in luboruroryconditions (I do not consider the papers devoted to particle creation from moving mirrors, which served to model the processes in the nonstationary universe or in black holes). Recently, the possibility of generating a fantastic amount of photons (up to 10”) in the visible region, for a wall performing periodic instantaneous jumps between two stable posi-
Blsevier Science B.V. SUN 0375-9601(95)00691-5
tions, was claimed in Ref. [ 151. However, that result seems erroneous, since it was obtained in the framework of the adiabatic approximation, which is obviously incompatible with the jump motion (moreover, the concrete numerical values of the parameters chosen in Ref. [ 151 required a superluminal velocity of the wall during the jumps). Evidently, no photons can be produced in the case of adiabatic motion, as was demonstrated explicitly, e.g., in Refs. [ 5,8]. A correct estimation of the photon production due to a very fast shift of an ideal boundary was given recently in Ref. [ 161. There it was proposed to use a mirror formed by an electron-hole plasma created in a thin semiconductor slab due to a powerful femtosecond laser pulse. The upper limit of the total amount of photons was predicted to be about 106/cm3, which is equivalent to l-100 photons from each cm2 of the surface, for a distance between mirrors of 10-6-10-4 cm. The aim of the present paper is to study the case of an oscillatingmotion of a wall in a three-dimensional cavity
V. V. LWoonuv/ Physics Lprters A 207 (199.5) 126-132
and to analyze the possibility of observing the effect at the available experimental level. Besides, we consider (for the first time) the responses of different detectors to the resonant wall oscillations.
way to satisfy explicitly the time-dependent boundary conditions is to look for the solutions in the same functional form as in Eq. ( 1 ), but with the time-dependent parameter L(t), &kY*t)
2. Free evolution in the resonant case
127
= ~2C~~(x,YIL(t).Lr)fimn(f). m,n (2)
Let us choose a rectangular cavity with dimensions LX, L,., L, (briefly designated by symbol {L}), and consider the modes whose vector potentials are directed along the z-axis. When the dimensions of the cavity do not depend on time, thefield operaior in the Heisenberg representation reads
In the stationary case the operators o,,,,,(t) coincide with the (coordinate) quadrature components of the field mode operators. Rutting (2) into the wave equation one can arrive after some algebra at an infinite set of coupled ordinary di~erenti~ equations for the Heisenberg operators om (t) (see also Refs. [ 10,121>, .. &?knf &, (t) f&n = 2A(t) CgkjQjn
x [&,exp(-iw,t)
-+-ciL,exp(iw,t)],
(1)
where anln and ii;, are the usual annihilation and creation photon operators, and hvl(X~Yl{~)) x
=w&,~,V
sin( mrx/L,)
sin(nrry/l?,).
The choice of coefficients in Rq. ( 1) corresponds to the standard form of the field ~~iltoni~,
“I”
Now suppose that L, varies in time according to the given function L(t) . Then the space-time dependence of the field operator differs from IQ. ( 1). To find it one has to take into account two conditions: (i) the field must satisfy the wave equation d2AZ/at2 - c*AA, = 0, (ii) the vector potential must become zero at the surfaces of the cavity it is parallel to, i.e. AZlx+-,= Azlx=urt = A&a = A&o =0 111. In the one-dimensional case Moore [ 1] proposed an elegant ansatz, when both the wave equation and the boundary conditions were satisfied explicitly. Different exact and approximate solutions of Moore’s equation were found in Refs. [ 4-7,9,11]. A similar ansatz hardly exists in more than one dimensions. A natural
j
3
(The carets over the operators are omitted.) Here = 7fC{[k/L(t)12 -t (n/L,)*}1/2, A(t) = i( t)/L( t), and the constant numerical coefficients gkj read @kn(t)
Due to the normalization of the functions &, and due to the zero boundary conditions at n = L, these coefficients tl.lm Out t0 be antisymmetrical: g&j= -gjk. For a cavity with a reasonable dimension of some centimeters, the frequencies Wk,,must belong at least to the GHz band. It is hardly possible to cause the wall to oscillate as a whole at such frequencies. So we suppose that the oscillations of the sur$uce of the wall at frequency oW are excited, e.g., due to a piezo-effect. Then the amplitude of these oscillations a is connected with the relative deformation S in a standing acoustic wave inside the wall by the formula 6 = mwa/vs, where us is the sound velocity inside the wall, v, N 5 x IO3 m/s. Since usual materials cannot bear deformations exceeding the value S,, N 10e2, the velocity of the boundary cannot exceed the value v,, N 6;naxvsno 50 m/s (independent on the frequency), so the maximal dimensionless displacement 7 = a/h
128
Y.V. Lbdonou/ PhysicsLettersA 207 (I 5951126-132
is j&a N ( u,/2m)Sm, - 10m8 for wW N wn TCllo. Since 7 turns out to be extremely small, the only hope to observe any effect due to the motion of the boundary may be related to the parametric resonance case, when the wall oscillates at twice the eigenfrequency of some unperturbed mode: L(t) = Lo [ 1 - 7 cos(2w,t) I. Then one can use the wellknown method in the theory of parametric systems [ 171 of slowly varying ~plitudes, i.e. to look for the solutions of Eq. (3) in the form @k&t) = Skn(W exp(-iw&) + rlkn(V) exp(iu&).
(4)
To obtain the equations for & and ?J& one should put expressions (4) into Eq. (3)) mul~ply both sides withexp(&,t) or exp( -iw&)$ respectively, and average over fast oscillations, neglecting terms of the order of y* and higher. In this way, we eliminate all the terms in the right-hand side of Rq. (3), Indeed, the first and the second sums in the right-had side do not contain functions f&,, due to the ~tisy~et~ of the coefficients &j, whereas the last sum is proportional to h2 N y2. Consequently, after multiplication by the proper exponenti~ functions, the right-hand side will consist of terms with exponential factors of the type exp[ i( *win f Wknrf 20,“) t] with j # k. After averaging, all these terms are zero, since the spectrum mjn is not equidistant. Thus the problem is reduced to that of a onedimensional parametric oscillator with the time dependence of the eigenfrequency in the form o(t)
=uo[l+2ycos(2wot)],
QN f% = WC/L+ (However, this set of simplified equations can be solved exactly [ 181.) The quantum oscillator with a time-dependent frequency was investigate in numerous papers, beginning with Ref. [ 191, A detailed review can be found in Ref. [20). All the characteristics of the quantum oscillator are determined completely by the complex solution of the cfassicaf oscillator equation of motion B+W*(t)e=o,
(6)
satisfying the normalization condition de* - d’a = 2i. The initial conditions read e(0) = l/,/Z&, k(O) = i&. If the function o(t) takes the constant value wo at t 6 0 and at t > tf > 0, then the quantum mechanical average number of photons created from the ground state is given by the formula
Following Ref. [ 171, we look for the solution of Eq. (6) in the form e(t) = &
[u(t) eioor+u( t) evioor] ,
(8)
u f t) and u(t) being ~f~~~yvarying unctions of time.
Averaging over fast oscillations one can easily obtain the first order differential equations for the amplitudes (provided /y] < l), &= -ioeyv,
ti = iwoyu.
(9)
Their solutions satisfying the initial conditions (up to the terms of the order of y) read
(5)
where 2y = y[ 1 + (n&JmL,)2]-1iz, and 00 is the unperturbed eigenfrequency of the resonant mode. It should be emphasized that a reduction to a singleoscillator problem is possible only under the resonance conditions. If the time dependence of the eigenfrequency differs from Eq. (S), then all the modes strongly interact with each other [ 7,9, lo] (excluding the trivial case of an adiabatic motion of the wall, oW < 00, when no photons can be created). Moreover, in the one-dimensiona model we get an infinite set of coupled equations even after averaging over fast oscillations, since the frequency spectrum is equidis-
u(t) = cosh(oeyt),
u(t) = isinh(~o~~).
(IO)
Due to E!qs. (7), (8) and ( lo), the average number of photons grows in time exponentially, In> = lu12= sinh*(oeyt).
(11)
It is well known that the initial vacuum state of the oscillator is transformed into the squeezed vacwm state, if the frequency depends on time (see, e.g., reviews 120,211 and references therein). Looking at Rq. ( 11) one can immediately recognize the combination weyt as the so-called squeezing parameter. Therefore the
V.V. Dodonov/
probability to register n photons exhibits typical oscillations, Pi,, _ iltanh(wayt) 12’& Vm>! cosh( woyt) (2”‘m!)2’ 71)2,7,+1 = 0.
129
PhysicsLettersA 207fI 99%12&-I32
following two-dimensional H~ltoni~ governing the evolution of the coupled system “field oscillator + detector”,
(12)
Similar formulas for the amount of photons created in a cavity filled with a medium with time-dependent dielectric permeability were found in Ref. [ 221. For the frequency wa/27r N 10 GHz, the maximal value of parameter p = ywat reads p,, N 6008, the time t being expressed in seconds. Even if the amplitude of the vibrations were 100 times less than the maximal possible value, in t = 1 s one could get about sinh*(6) M 4 x IO4 photons in an empty cavity! The necessary Q-factor of the order of 3 x 1O’Owas achieved in experiments already several years ago [ 231. consequently, if the problem of exciting wall vibrations with sufficiently large amplitude (about 10-t” cm) were solved, then the effect of photon creation from a vacuum due to the vibrations of the boundary could be observed in a laboratory. The corresponding static relative deformation of the order of 1O-4 would require an electric field of about 3 x lo” V/cm in piezoelectric materials like BaTiOs, which is far less than the dielectric breakdown threshold. It is an open question how to achieve such deformations at GHz frequencies. 3. Interaction with a detector One of the possible methods of detecting photons created due to the motion of the wall may consist in placing inside the cavity some probe object. We consider two simplified models of a detector: a harmonic oscillator and a two-level atom. Their responses are quite different. 3.1. Harmonic oscillator Let a harmonic oscillator tuned at the resonant frequency be placed at the point of the maximum of the amplitude mode function rfim,,(x, y 1{L}) . Assuming the interaction between the oscillator and the field to be described by the standard minimal coupling term, -(e/mc)p s A, and neglecting a small influence of the nonresonant modes, one can easily arrive at the
(13) P, Q are the quadrature components of the field oscillator, p, q are those of the detector. A small dimensionless coupling coefficient K may be assumed to be constant, since its variations of the order of 7~ can be neglected in comparison with the relative variations of the eigenfrequency, which have the order of 9. Suppose that the lowest cavity mode is resonant. One can easily evaluate the dimensionless coupling const~t as K N ( e2/2~~c2 L) I/*, Then the ratio Y/K cannot exceed the value S,,( mu~Lf8rre2) 1/Zm 0.05 for L - 1 cm and the mass of the electron (and K 2 x 10e7 for these parameters}. Consequently, in real conditions Y/K < 1. In the time-independent case, w(t) = const = we, the system described by means of the Hamiltonian (13) possesses two eigenfrequencies, wf = wo( 1 f K) (provided 1~1 < 1). Let us assume that the wall vibrates at exactly twice the lower frequency w_, i.e. w(t) = wo[l+ 2ycos(2w_t)]. Then the lower and upper modes practically do not interact. Suppose that the initial state was the ground state for both the field and the probe oscillators. Using the general theory of multidimensional quantum systems with arbitrary quadratic H~ltoni~s, first proposed in Ref. [ 241 and examined in detail, e.g., in Ref. ]20], we have obtained the following expression for the wave function of the coupled system “field t probe oscillator” at t > 0 (the details of the calculations will be given elsewhere),
sl/
--!cash 1~ T
x exp{-it - $[a(t)Q2 -tb(t)q2
- 2c(t)qQ]),
a(t) = 1 + itanhpe-2i’P-iKe’@[tanh,ue-‘~(1+4anh~sin@ei~)-sin~], b(f) = I - i tanh p ev2@-itce’@[tanh~e-“(
1 -tanhpsincPe’p)
+sinp],
130
V. V. Dodonov /Physics
Letters A 207 fI 995) 126-132
c(t) = tanh~e--2i~-
+ ipc(1 - cosqoe-‘@ fi t~h2~sin~ei(~-‘~)), 4b= (w+ + w-) t = 26&t,
p =
ywot,
qpi = wit,
A= 1 - cosh~cos@.
(14) In all the formulas the terms of the order of K~ were neglected, as well as the terms proportional to y (except, of course, the ~guments of the hyperbolic functions) . Eq. (14) shows that at t > 0 the coupled system turns out to be in a two-mode squeezed state. In the short-time limit ,U < 1 the squeezing effect is very smaI1, since the coefficients a and b are close to unity, while c is of the order of K. On the contrary, if p 2 1, then all the terms proportional to K can be neglected, so that a(t)
= 1 t ix,
b(t) = 1 - ix,
c(t)
= ,y = tanh p ePziP- .
It is clear that the denser matrix of the detector f which is obtained from the density matrix of the total system p(Q,q;Q’,q’) = $(Q,q)(cl*(Q’,q’> by putting Q = Q’ and integrating over Q) also has a Gaussian form. Its properties are determined completely by the reduced covariance matrix (gab = i (SS+&&)- (2) (a}, a,b =p,9) M=:
(n> = i(T-
1) = &sinh*p,
i.e., it is half that in the free mode case. The photon distribution function can be expressed in terms of the Legendre polynomials, on the basis of the general formulas of Ref. [ 251,
sinh ,U
(16)
“=J1+3coshzlr It exhibits no oscillations for /.t 2 1, in contradistinction to the free-mode case ( 12).
~~~ /I
II 1 - tanhpsin6) = ; cosh2 p II tanhpcos+
tanhpcos+ 1 + tanhpsind,
II (1%
3.2. Two-level detector
’
Here (b = 250_. A similar matrix for the field oscillator can be obtained from Eq. (15) by changing the sign of the parameter p. There exists a strong correlation between the field and the detector in the long-time limit. For instance, the correlation coefficient between the quadrature components reads
(qQ>
rqQ = ~~ =
As was shown in Ref. [ 25 ] r the photon statistics in the Gaussian one-mode states is dete~n~ completely by two invests of the covariance matrix, d = det M, T = Tr M. Evidently, T is twice the energy of the quantum fluctuations. The parameter d characterizes the degree of purity of the quantum state, due to the relation Tr$’ = 1/2v/;i, p being the statistical operator of the system. The degnze of squeezing, i.e., tbe minimal possible value of the variance of some quadrature component, normalized by its vacuum value l/200, is equal to [26] s = T - dm. In the case under study, both subsystems have identical invariants, T = 4d = cash* ,u, so for p 2 1 they appear in highly mixed quantum states. As to the degree of squeezing, it turns out to be rather moderate: s = e-N cash p = $(l -e -2”). The average number of quanta in each subsystem equals
sinh fi cos 4 111 + (sinhycos+)a
4 O(K).
Now let us consider a simple model of a twolevel detector. The most significant features can be described, in the rotating wave approximation, in the framework of the following generalization of the Jaynes-Cummings Hamiltonian,
H = a+ff + $@ +
$sin(o,t)[a* 2
+
K(UCT+
+ Ub-)
+ (u+)~].
(17)
The eigenfr~uency of the unperturbed mode is assumed to be we = 1, D is the energy level difference of the detector, K and y < K have the same meaning as above; ~,a+ and CT+,U_,(TZ are the standard
V. V. Dodonov / Physics Letters A 207 f 1995) 126432
photon and spin operators. The wave function of the system “field + detector” can be written as e(t) = C[ck-)(t)Ilt,-)+ci+)(t)In,+)],withaclearmeaning of the symbols. If y = 0, then the known solution of the JC model reads [ 271
II 2 0,
),
c;-+ t) = c,
(18)
c,(,+)( t) = a, sin 9, exp( -itE,f ) +~,~cos~~exp(-irE~),
n > 0,
(19)
E:=n+i&h,,
tan 8, =
2h, - 1 + f2 “2 2A,+ 1 -R > *
(
We suppose that initially the system was in the ground state with the only nonzero coefficient &)(O) = 1, and that the frequency of the wail vibrations is close to twice the frequency of the un~~ur~ mode: ow = 2 - Y. Looking for the solution at y # 0 in the same form as ( IS), ( 19), but with time-dependent coefficients, and neglecting the rapidly oscillating terms containing exp( 2it), we get the following equation for the coefficient c( t), 6= ~v?${bl
sin31 exp[it(ill
tector in an excited state PC+) is always less than $. All the probabilities, in contradistinction to the first example, are periodically oscillating functions of time, Pt = P(+) =cos*fii sin*(&), P2 = sin2 61 sin*{cut).
c:Si (t) = a, cos tYnexp( -item) - 6, sin 6, exp( -itE;
131
It is interesting that the upper level of the detector can never be populated with 100% probability, since 6, > 0 for all values of the parameters. For n = 1, 6, = $, and Pt = P;! = PC+) = 1 sin*( $yt), A large detuning, 1 - n >> K, results in an increasing PAZ, since 41 + 0. However, in such a case LY-+ 0, as well, and the applicability of the JC model to the description of the interaction between the detector and the field becomes questionable. Due to the quite different responses of the oscillator and the two-level system to the field created by resonantly vibrating walls, it would’ be interesting to consider more realistic models of the detector. A suitable candidate could be some Rydberg atom excited to the N 100th level (then the transition frequencies between the neigh~uring levels belong just to the GHz band). This problem will be considered elsewhere.
Acknowledgement
I thank SM. Chum~ov, A.B. Klimov, VI. Man’ko, and H. Walther for discussions.
- v)]
-alcosfirexp[-it(hl+V)]}. Assuming Y = At = [ i( 1 - Q)* + 2~~1 ‘I2 and neglecting the terms oscillating with frequencies of the order of K, one can check that the infinite system of equations for a,, and b, is reduced to the following nyo equations, i‘= av’$sin29,
bl,
bj = -$Jiysin6t
Cons~uently, the only nonzero ~plitud~ onant case are (-) CO
c. in the res-
= cos( at),
c2(-)
= sin 41 sin(at) exp( -itE,‘-‘), t+1 = -cos6r sin(art)exp(-itE~-l), c1
where LY= 4 47 sin 61. Not more than two photons can be created, and the probability of finding the de-
References [ 1] G.T. Moore, J. Math. Phys. I I ( 1970) 2679. [2] S.A. Fulling and PCW. Davies, Proc. R. Sot. A 348 ( 1976) 393; N.D. Birrell and PCW. Davies, Quantum fields in curved space (Cambridge Univ. Press, Cambridge, 1982). [ 31 M. Razavy and J. Teming, Phys. Rev. D 3 I (1985) 307. [4] M. Castagnino and R. Ferraro, Ann. Phys. (NY) 154 (1984) 1. [ 51 S. Sarkar, in: Photons and quantum fluctuations, eds. E.R. Pike and H. Wahher (Hilger, Bristol, 1988) p. 151. [ 61 V.V. Dodonov, A.B. Klimov and V.I. Man’ko, Phys. Lett. A 149 (1990) 225. [7] V.V. Dodonov and A.B. Klimov, Phys. I&t. A 167 ( 1992) 309. [S] G. Calucci, J. Phys. A 25 ( 1992) 3873. [9] V.V. Dodonov, A.B. Klimov and D.E. Nikonov, J. Math. Phys. 34 (1993) 2742.
132
V,V. Dodonov/
Physics Letters A 207 (1995) 126-132
[IO] C.K. Law, Phys. Rev. A 49 (1994) 433.
[it] C.K. Law, Phys. Rev. Lett. 73 (1994) 1931. [ 121C.K. Law, Phys. Rev. A 51 ( 1995) 2537. [ 131 M. Bordag, G. Petrov and D. Robaschik, Yad. Fiz. 39 ( 1984) 1315 [ Sov.5. Nucl. Phys. 39 (1984) 8281; M. Bordag, F-M. Dittes and D. Robaschik, Yad. Fiz. 43 ( 1986) 1606 [Sov. J. Nucl. Phys. 43 (1986) 1034 f . [ 141 S. Hacyan, R. Jauregui and C. Villarreal, in: NASA Conference Publication 3286. Proc. Second lnt. Workshop on Harmonic oscillators, eds. D. Han and K.B. Wolf (NASA, Greenbelt, 1995) p. 61. [ 151 E. Sassaroli, Y.N. Srivastava and A. Widom, Phys. Rev. A 50 (1994) 1027. [ 161 Yu.E. Lozovik, V.G. Tsvetus and E.A. Vinogradov, Pis’ma Zh. Eksp. Teor. Fiz. 61 (1995) 711. [ 171 L.D. Landau and E.M. Lifshitz, Mechanics (Petgamon, Oxford, 1969). [ 181 V.V. Dodonov and A.B. Klimov, to be published. 1191 K. Husimi, Progr. Theor. Phys. 9 (1953) 381.
1201 V.V. Dodonov and V.I. Man’ko, in: Proceedings of Lebedev Physics Institute, Vol. 183. Invariants and the evolution of nonstationary quantum systems, ed. M.A. Markov (Nova Science, Commack, NY, 1989) p. 263. [21] V.V. Dodonov and V.I. Man’ko, in: Advances in chemical physics, Vol. LXXXV. Modem nonlinear optics, Part 3, eds. M. Evans and S. Kielich (Wiley, New York, 1994) p. 499. [22] V.V. Dodonov, A.B. Klimov and DE. Nikonov, Phys. Rev. A 47 (1993) 4422. [23] G. Rempe, F. Schmidt-Kaler and H. Walther, Phys. Rev. Len. 64 (1990) 2783. [24] V.V. Dodonov, LA. Malkin and V.I. Man’ko, Int. J. Theor. Phys. 14 (1975) 37. (251 V.V. Dodonov,O.V. Man’ko and V.I. Man’ko, Phys. Rev. A 49 (1994) 2993. [26] V.V. Dodonov, V.I. Man’ko and PG. Polynkin, Phys. L&t. A 188 (1994) 232. [27] W.H. Louisell, Radiation and noise in quantum electronics (McGraw-Hill, New York, 1964).
__ f!!B
ELSEVIER
30 October 1995 PHYSICS
LETTERS
A
Physics Letters A 207 (1995) 126-132
Photon creation and excitation of a detector in a cavity with a resonantly vibrating wall V.V. Dodonov RN. L.ebedev Physics Institute, Lminsky Prospekt 53, 117924 Moscow, Russian Federation Moscow Institute of Physics and Technology, I6 Gagarin Street, 140160 Zhukovskiy, Moscow Region, Russian Federation
Received 15 July 1995; accepted for publication 2 September 1995 Communicated by V.M. Agranovich
Abstract The problem of photon generation inside an ideal 3D cavity with resonantly vibrating walls is studied. The possibility of creating from a vacuum up to lo4 photons in a cavity with a Q-factor of about 3 x 10” is predicted. Different responses of detectors modeled by a harmonic oscillator and by a two-level atom are demonstrated. Keywords: Cavity electrodynamics; Vibrating boundary; Parametric resonance; Pield-atom interaction
1. Intruduction The problem of photon creation from a vacuum due to the motion of boundaries has attracted the attention of many authors during the last 25 years. Due to
its complexity it was considered mainly in the framework of the one-dimensional model (scalar electrodynamics) [ l-121. The 3D situation was considered only in the specific case when the wall moved with a constant velocity [ 13,141. Although some interesting exact and approximate solutions were found in the mentioned papers, they did not discuss a possible experimental verification of the phenomenon in luboruroryconditions (I do not consider the papers devoted to particle creation from moving mirrors, which served to model the processes in the nonstationary universe or in black holes). Recently, the possibility of generating a fantastic amount of photons (up to 10”) in the visible region, for a wall performing periodic instantaneous jumps between two stable posi-
Blsevier Science B.V. SUN 0375-9601(95)00691-5
tions, was claimed in Ref. [ 151. However, that result seems erroneous, since it was obtained in the framework of the adiabatic approximation, which is obviously incompatible with the jump motion (moreover, the concrete numerical values of the parameters chosen in Ref. [ 151 required a superluminal velocity of the wall during the jumps). Evidently, no photons can be produced in the case of adiabatic motion, as was demonstrated explicitly, e.g., in Refs. [ 5,8]. A correct estimation of the photon production due to a very fast shift of an ideal boundary was given recently in Ref. [ 161. There it was proposed to use a mirror formed by an electron-hole plasma created in a thin semiconductor slab due to a powerful femtosecond laser pulse. The upper limit of the total amount of photons was predicted to be about 106/cm3, which is equivalent to l-100 photons from each cm2 of the surface, for a distance between mirrors of 10-6-10-4 cm. The aim of the present paper is to study the case of an oscillatingmotion of a wall in a three-dimensional cavity
V. V. LWoonuv/ Physics Lprters A 207 (199.5) 126-132
and to analyze the possibility of observing the effect at the available experimental level. Besides, we consider (for the first time) the responses of different detectors to the resonant wall oscillations.
way to satisfy explicitly the time-dependent boundary conditions is to look for the solutions in the same functional form as in Eq. ( 1 ), but with the time-dependent parameter L(t), &kY*t)
2. Free evolution in the resonant case
127
= ~2C~~(x,YIL(t).Lr)fimn(f). m,n (2)
Let us choose a rectangular cavity with dimensions LX, L,., L, (briefly designated by symbol {L}), and consider the modes whose vector potentials are directed along the z-axis. When the dimensions of the cavity do not depend on time, thefield operaior in the Heisenberg representation reads
In the stationary case the operators o,,,,,(t) coincide with the (coordinate) quadrature components of the field mode operators. Rutting (2) into the wave equation one can arrive after some algebra at an infinite set of coupled ordinary di~erenti~ equations for the Heisenberg operators om (t) (see also Refs. [ 10,121>, .. &?knf &, (t) f&n = 2A(t) CgkjQjn
x [&,exp(-iw,t)
-+-ciL,exp(iw,t)],
(1)
where anln and ii;, are the usual annihilation and creation photon operators, and hvl(X~Yl{~)) x
=w&,~,V
sin( mrx/L,)
sin(nrry/l?,).
The choice of coefficients in Rq. ( 1) corresponds to the standard form of the field ~~iltoni~,
“I”
Now suppose that L, varies in time according to the given function L(t) . Then the space-time dependence of the field operator differs from IQ. ( 1). To find it one has to take into account two conditions: (i) the field must satisfy the wave equation d2AZ/at2 - c*AA, = 0, (ii) the vector potential must become zero at the surfaces of the cavity it is parallel to, i.e. AZlx+-,= Azlx=urt = A&a = A&o =0 111. In the one-dimensional case Moore [ 1] proposed an elegant ansatz, when both the wave equation and the boundary conditions were satisfied explicitly. Different exact and approximate solutions of Moore’s equation were found in Refs. [ 4-7,9,11]. A similar ansatz hardly exists in more than one dimensions. A natural
j
3
(The carets over the operators are omitted.) Here = 7fC{[k/L(t)12 -t (n/L,)*}1/2, A(t) = i( t)/L( t), and the constant numerical coefficients gkj read @kn(t)
Due to the normalization of the functions &, and due to the zero boundary conditions at n = L, these coefficients tl.lm Out t0 be antisymmetrical: g&j= -gjk. For a cavity with a reasonable dimension of some centimeters, the frequencies Wk,,must belong at least to the GHz band. It is hardly possible to cause the wall to oscillate as a whole at such frequencies. So we suppose that the oscillations of the sur$uce of the wall at frequency oW are excited, e.g., due to a piezo-effect. Then the amplitude of these oscillations a is connected with the relative deformation S in a standing acoustic wave inside the wall by the formula 6 = mwa/vs, where us is the sound velocity inside the wall, v, N 5 x IO3 m/s. Since usual materials cannot bear deformations exceeding the value S,, N 10e2, the velocity of the boundary cannot exceed the value v,, N 6;naxvsno 50 m/s (independent on the frequency), so the maximal dimensionless displacement 7 = a/h
128
Y.V. Lbdonou/ PhysicsLettersA 207 (I 5951126-132
is j&a N ( u,/2m)Sm, - 10m8 for wW N wn TCllo. Since 7 turns out to be extremely small, the only hope to observe any effect due to the motion of the boundary may be related to the parametric resonance case, when the wall oscillates at twice the eigenfrequency of some unperturbed mode: L(t) = Lo [ 1 - 7 cos(2w,t) I. Then one can use the wellknown method in the theory of parametric systems [ 171 of slowly varying ~plitudes, i.e. to look for the solutions of Eq. (3) in the form @k&t) = Skn(W exp(-iw&) + rlkn(V) exp(iu&).
(4)
To obtain the equations for & and ?J& one should put expressions (4) into Eq. (3)) mul~ply both sides withexp(&,t) or exp( -iw&)$ respectively, and average over fast oscillations, neglecting terms of the order of y* and higher. In this way, we eliminate all the terms in the right-hand side of Rq. (3), Indeed, the first and the second sums in the right-had side do not contain functions f&,, due to the ~tisy~et~ of the coefficients &j, whereas the last sum is proportional to h2 N y2. Consequently, after multiplication by the proper exponenti~ functions, the right-hand side will consist of terms with exponential factors of the type exp[ i( *win f Wknrf 20,“) t] with j # k. After averaging, all these terms are zero, since the spectrum mjn is not equidistant. Thus the problem is reduced to that of a onedimensional parametric oscillator with the time dependence of the eigenfrequency in the form o(t)
=uo[l+2ycos(2wot)],
QN f% = WC/L+ (However, this set of simplified equations can be solved exactly [ 181.) The quantum oscillator with a time-dependent frequency was investigate in numerous papers, beginning with Ref. [ 191, A detailed review can be found in Ref. [20). All the characteristics of the quantum oscillator are determined completely by the complex solution of the cfassicaf oscillator equation of motion B+W*(t)e=o,
(6)
satisfying the normalization condition de* - d’a = 2i. The initial conditions read e(0) = l/,/Z&, k(O) = i&. If the function o(t) takes the constant value wo at t 6 0 and at t > tf > 0, then the quantum mechanical average number of photons created from the ground state is given by the formula
Following Ref. [ 171, we look for the solution of Eq. (6) in the form e(t) = &
[u(t) eioor+u( t) evioor] ,
(8)
u f t) and u(t) being ~f~~~yvarying unctions of time.
Averaging over fast oscillations one can easily obtain the first order differential equations for the amplitudes (provided /y] < l), &= -ioeyv,
ti = iwoyu.
(9)
Their solutions satisfying the initial conditions (up to the terms of the order of y) read
(5)
where 2y = y[ 1 + (n&JmL,)2]-1iz, and 00 is the unperturbed eigenfrequency of the resonant mode. It should be emphasized that a reduction to a singleoscillator problem is possible only under the resonance conditions. If the time dependence of the eigenfrequency differs from Eq. (S), then all the modes strongly interact with each other [ 7,9, lo] (excluding the trivial case of an adiabatic motion of the wall, oW < 00, when no photons can be created). Moreover, in the one-dimensiona model we get an infinite set of coupled equations even after averaging over fast oscillations, since the frequency spectrum is equidis-
u(t) = cosh(oeyt),
u(t) = isinh(~o~~).
(IO)
Due to E!qs. (7), (8) and ( lo), the average number of photons grows in time exponentially, In> = lu12= sinh*(oeyt).
(11)
It is well known that the initial vacuum state of the oscillator is transformed into the squeezed vacwm state, if the frequency depends on time (see, e.g., reviews 120,211 and references therein). Looking at Rq. ( 11) one can immediately recognize the combination weyt as the so-called squeezing parameter. Therefore the
V.V. Dodonov/
probability to register n photons exhibits typical oscillations, Pi,, _ iltanh(wayt) 12’& Vm>! cosh( woyt) (2”‘m!)2’ 71)2,7,+1 = 0.
129
PhysicsLettersA 207fI 99%12&-I32
following two-dimensional H~ltoni~ governing the evolution of the coupled system “field oscillator + detector”,
(12)
Similar formulas for the amount of photons created in a cavity filled with a medium with time-dependent dielectric permeability were found in Ref. [ 221. For the frequency wa/27r N 10 GHz, the maximal value of parameter p = ywat reads p,, N 6008, the time t being expressed in seconds. Even if the amplitude of the vibrations were 100 times less than the maximal possible value, in t = 1 s one could get about sinh*(6) M 4 x IO4 photons in an empty cavity! The necessary Q-factor of the order of 3 x 1O’Owas achieved in experiments already several years ago [ 231. consequently, if the problem of exciting wall vibrations with sufficiently large amplitude (about 10-t” cm) were solved, then the effect of photon creation from a vacuum due to the vibrations of the boundary could be observed in a laboratory. The corresponding static relative deformation of the order of 1O-4 would require an electric field of about 3 x lo” V/cm in piezoelectric materials like BaTiOs, which is far less than the dielectric breakdown threshold. It is an open question how to achieve such deformations at GHz frequencies. 3. Interaction with a detector One of the possible methods of detecting photons created due to the motion of the wall may consist in placing inside the cavity some probe object. We consider two simplified models of a detector: a harmonic oscillator and a two-level atom. Their responses are quite different. 3.1. Harmonic oscillator Let a harmonic oscillator tuned at the resonant frequency be placed at the point of the maximum of the amplitude mode function rfim,,(x, y 1{L}) . Assuming the interaction between the oscillator and the field to be described by the standard minimal coupling term, -(e/mc)p s A, and neglecting a small influence of the nonresonant modes, one can easily arrive at the
(13) P, Q are the quadrature components of the field oscillator, p, q are those of the detector. A small dimensionless coupling coefficient K may be assumed to be constant, since its variations of the order of 7~ can be neglected in comparison with the relative variations of the eigenfrequency, which have the order of 9. Suppose that the lowest cavity mode is resonant. One can easily evaluate the dimensionless coupling const~t as K N ( e2/2~~c2 L) I/*, Then the ratio Y/K cannot exceed the value S,,( mu~Lf8rre2) 1/Zm 0.05 for L - 1 cm and the mass of the electron (and K 2 x 10e7 for these parameters}. Consequently, in real conditions Y/K < 1. In the time-independent case, w(t) = const = we, the system described by means of the Hamiltonian (13) possesses two eigenfrequencies, wf = wo( 1 f K) (provided 1~1 < 1). Let us assume that the wall vibrates at exactly twice the lower frequency w_, i.e. w(t) = wo[l+ 2ycos(2w_t)]. Then the lower and upper modes practically do not interact. Suppose that the initial state was the ground state for both the field and the probe oscillators. Using the general theory of multidimensional quantum systems with arbitrary quadratic H~ltoni~s, first proposed in Ref. [ 241 and examined in detail, e.g., in Ref. ]20], we have obtained the following expression for the wave function of the coupled system “field t probe oscillator” at t > 0 (the details of the calculations will be given elsewhere),
sl/
--!cash 1~ T
x exp{-it - $[a(t)Q2 -tb(t)q2
- 2c(t)qQ]),
a(t) = 1 + itanhpe-2i’P-iKe’@[tanh,ue-‘~(1+4anh~sin@ei~)-sin~], b(f) = I - i tanh p ev2@-itce’@[tanh~e-“(
1 -tanhpsincPe’p)
+sinp],
130
V. V. Dodonov /Physics
Letters A 207 fI 995) 126-132
c(t) = tanh~e--2i~-
+ ipc(1 - cosqoe-‘@ fi t~h2~sin~ei(~-‘~)), 4b= (w+ + w-) t = 26&t,
p =
ywot,
qpi = wit,
A= 1 - cosh~cos@.
(14) In all the formulas the terms of the order of K~ were neglected, as well as the terms proportional to y (except, of course, the ~guments of the hyperbolic functions) . Eq. (14) shows that at t > 0 the coupled system turns out to be in a two-mode squeezed state. In the short-time limit ,U < 1 the squeezing effect is very smaI1, since the coefficients a and b are close to unity, while c is of the order of K. On the contrary, if p 2 1, then all the terms proportional to K can be neglected, so that a(t)
= 1 t ix,
b(t) = 1 - ix,
c(t)
= ,y = tanh p ePziP- .
It is clear that the denser matrix of the detector f which is obtained from the density matrix of the total system p(Q,q;Q’,q’) = $(Q,q)(cl*(Q’,q’> by putting Q = Q’ and integrating over Q) also has a Gaussian form. Its properties are determined completely by the reduced covariance matrix (gab = i (SS+&&)- (2) (a}, a,b =p,9) M=:
(n> = i(T-
1) = &sinh*p,
i.e., it is half that in the free mode case. The photon distribution function can be expressed in terms of the Legendre polynomials, on the basis of the general formulas of Ref. [ 251,
sinh ,U
(16)
“=J1+3coshzlr It exhibits no oscillations for /.t 2 1, in contradistinction to the free-mode case ( 12).
~~~ /I
II 1 - tanhpsin6) = ; cosh2 p II tanhpcos+
tanhpcos+ 1 + tanhpsind,
II (1%
3.2. Two-level detector
’
Here (b = 250_. A similar matrix for the field oscillator can be obtained from Eq. (15) by changing the sign of the parameter p. There exists a strong correlation between the field and the detector in the long-time limit. For instance, the correlation coefficient between the quadrature components reads
(qQ>
rqQ = ~~ =
As was shown in Ref. [ 25 ] r the photon statistics in the Gaussian one-mode states is dete~n~ completely by two invests of the covariance matrix, d = det M, T = Tr M. Evidently, T is twice the energy of the quantum fluctuations. The parameter d characterizes the degree of purity of the quantum state, due to the relation Tr$’ = 1/2v/;i, p being the statistical operator of the system. The degnze of squeezing, i.e., tbe minimal possible value of the variance of some quadrature component, normalized by its vacuum value l/200, is equal to [26] s = T - dm. In the case under study, both subsystems have identical invariants, T = 4d = cash* ,u, so for p 2 1 they appear in highly mixed quantum states. As to the degree of squeezing, it turns out to be rather moderate: s = e-N cash p = $(l -e -2”). The average number of quanta in each subsystem equals
sinh fi cos 4 111 + (sinhycos+)a
4 O(K).
Now let us consider a simple model of a twolevel detector. The most significant features can be described, in the rotating wave approximation, in the framework of the following generalization of the Jaynes-Cummings Hamiltonian,
H = a+ff + $@ +
$sin(o,t)[a* 2
+
K(UCT+
+ Ub-)
+ (u+)~].
(17)
The eigenfr~uency of the unperturbed mode is assumed to be we = 1, D is the energy level difference of the detector, K and y < K have the same meaning as above; ~,a+ and CT+,U_,(TZ are the standard
V. V. Dodonov / Physics Letters A 207 f 1995) 126432
photon and spin operators. The wave function of the system “field + detector” can be written as e(t) = C[ck-)(t)Ilt,-)+ci+)(t)In,+)],withaclearmeaning of the symbols. If y = 0, then the known solution of the JC model reads [ 271
II 2 0,
),
c;-+ t) = c,
(18)
c,(,+)( t) = a, sin 9, exp( -itE,f ) +~,~cos~~exp(-irE~),
n > 0,
(19)
E:=n+i&h,,
tan 8, =
2h, - 1 + f2 “2 2A,+ 1 -R > *
(
We suppose that initially the system was in the ground state with the only nonzero coefficient &)(O) = 1, and that the frequency of the wail vibrations is close to twice the frequency of the un~~ur~ mode: ow = 2 - Y. Looking for the solution at y # 0 in the same form as ( IS), ( 19), but with time-dependent coefficients, and neglecting the rapidly oscillating terms containing exp( 2it), we get the following equation for the coefficient c( t), 6= ~v?${bl
sin31 exp[it(ill
tector in an excited state PC+) is always less than $. All the probabilities, in contradistinction to the first example, are periodically oscillating functions of time, Pt = P(+) =cos*fii sin*(&), P2 = sin2 61 sin*{cut).
c:Si (t) = a, cos tYnexp( -item) - 6, sin 6, exp( -itE;
131
It is interesting that the upper level of the detector can never be populated with 100% probability, since 6, > 0 for all values of the parameters. For n = 1, 6, = $, and Pt = P;! = PC+) = 1 sin*( $yt), A large detuning, 1 - n >> K, results in an increasing PAZ, since 41 + 0. However, in such a case LY-+ 0, as well, and the applicability of the JC model to the description of the interaction between the detector and the field becomes questionable. Due to the quite different responses of the oscillator and the two-level system to the field created by resonantly vibrating walls, it would’ be interesting to consider more realistic models of the detector. A suitable candidate could be some Rydberg atom excited to the N 100th level (then the transition frequencies between the neigh~uring levels belong just to the GHz band). This problem will be considered elsewhere.
Acknowledgement
I thank SM. Chum~ov, A.B. Klimov, VI. Man’ko, and H. Walther for discussions.
- v)]
-alcosfirexp[-it(hl+V)]}. Assuming Y = At = [ i( 1 - Q)* + 2~~1 ‘I2 and neglecting the terms oscillating with frequencies of the order of K, one can check that the infinite system of equations for a,, and b, is reduced to the following nyo equations, i‘= av’$sin29,
bl,
bj = -$Jiysin6t
Cons~uently, the only nonzero ~plitud~ onant case are (-) CO
c. in the res-
= cos( at),
c2(-)
= sin 41 sin(at) exp( -itE,‘-‘), t+1 = -cos6r sin(art)exp(-itE~-l), c1
where LY= 4 47 sin 61. Not more than two photons can be created, and the probability of finding the de-
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