Theory of Dynamical Casimir Effect in nonideal cavities with time-dependent parameters Victor V. Dodonov Instituto de Física, Universidade de Brasília, Brazil Supported by CNPq (Brazilian agency, Brasília, DF)

Padua experiment C. Braggio, G. Bressi, G. Carugno, C. Del Noce, G. Galeazzi, A. Lombardi, A. Palmieri, G. Ruoso, D. Zanello “A novel experimental approach for the detection of the dynamical Casimir effect”, Europhys. Lett. 70, 754 (2005) “Semiconductor microwave mirror for a measurement of the dynamical Casimir effect”, Rev. Sci. Instrum. 75, (2004)

Plan A model of quantum damped oscillator with arbitrary time-dependent frequency and damping Photon generation rate in the case of periodical variations of frequency and damping Choice of optimal parameters and evaluation of possible numbers of created photons Calculation of the complex frequency shift

Effective Hamiltonian approach C. K. Law, `` Effective Hamiltonian for the radiation in a cavity with moving mirror and a time-varying dielectric medium´´, Phys. Rev. A 49, (1994). R. Schützhold, G. Plunien and G. Soff, ``Trembling cavities in the canonical approach´´, Phys. Rev. A 57, (1998). If Maxwell’s equation in a medium with time-independent parameters and boundaries can be reduced to Then for {L(t)} and the Dirichlet boundary conditions:

Quantum nonstationary oscillator K. Husimi, ``Miscellanea in elementary quantum mechanics - II´´, Prog. Theor. Phys. 9, (1953). I. A. Malkin, V. I. Man’ko and D. A. Trifonov, ``Coherent states and transition probabilities in a time-dependent electromagnetic field´´, Phys. Rev. D 2, (1970). For initial thermal state with th quanta, the mean number of quanta for t  R and T - reflection and transmission coefficients from an effective ``potential barrier´´  2 (t)

Nonstationary quantum damped oscillator  (t) =  (t) - i  (t)  (0) = 1

If then

If

Preservation of positivity of density matrix for any initial state

Periodical variations of frequency and damping  2 (t)=  2 (t+T): equidistant ``barriers’’ with period T and amplitude reflection and transmission coefficients r and f -1

One has to calculate the sum

2 n « 1 The phase  of the single-barrier inverse transmission coefficient must be known with an accuracy better than

Complex frequency shift If

For very short pulses:

If

Z=0.3

For Z = 0.3; η = 0.75; Δ = 1/30; A 0 = 10 : φ/π = Δf res = 15 MHz δf max = 10 MHz

Arbitrary pulses Account of delay time Point source

All models lead to the same conclusion, which is very favorable for the experiment: the non-uniformity of illumination or temporal spread of laser pulses will not deteriorate the rate of photon generation, if the recombination time is in the interval ps.

Optimization of geometry

Possible results and limitations Rectangular cavity 8x9x1 cm; η=0.75; η 3 =0.42; (λ=12 cm) T=0: 10 4 phot pulses 10 3 phot pulses G = 17 T (in K) T=4K: 10 4 phot pulses 10 3 phot pulses T=14K: 10 4 phot pulses Periodicity of laser pulses (with duration less than 10 ps) T  T 0 /2  200 ps; the total duration < 0.4  s; Q-factor  The optimal energy of each pulse W ~5  J The total energy E tot ~ 0.7 J

Unsolved theoretical problems Influence of coupling with other modes Full quantization scheme