A model of spontaneous emission for randomly distributed fluorescence centers

Volume 133, number6 PHYSICS LETTERS A 2! November 1988 A MODEL OF SPONTANEOUS EMISSION FOR RANDOMLY DISTRIBUTED FLUORESCENCE CENTERS Robert ALICKI, SlawomirRUDNICK1 and Slawomir SADOWSKI Institute of Theoretical Physics and Astrophysics, University of Gdahsk, PL-80-952 Gdajisk, Poland Received 17 August 1988; accepted for publication 13 September 1988 Communicated by A.R. Bishop Spontaneous emission of randomly distributed fluorescence centers is studied using a simple model ofharmonic oscillators or localized fermionic states and applying the theory of random matrices. A nonexponential decay law with time-dependent decay rate suppressed by collective effects is obtained. Spontaneous emission of N two-level atoms placed at fixed positions r~,k= 1, 2, N, is described by the following markovian master equation for the atomic density matrix [I]: ..., N ~ [S~,p,]—i d! I ~ Qk,[S~Si,pl] k,~/=I N + ~ Yki ([Sip,, S~] + [S~ ,p,S7J) (U . Here w is the renormalized atomic frequency. 12k/ describes a van der Waals dipole—dipole interaction decreasing like Irk r, and f= (Yk/) is a positively defined symmetric Nx N matrix. The matrix elements Yki are functions of rk r, and the orientation of the dipole moments. Their space dependence is oscillatory with leading terms I r,. r, I x sin(k Irk —r 11), k=w/c. One should mention that a qualitatively similar behaviour of the matrix F describing cooperative dissipation might be obtained for different types of centers (atoms, molecules, localized electron states, localized spins, etc.) and different kinds of interactions (interaction with photons, phonons, magnons, etc.). The simplest model describing collective effects in fluorescence is a mean-field version of (I) with QkI= Q, Yvi= y. This approximation is valid for a rather unphysical small sample case (max Irk r 1 ~ 1/k) or gives qualitative predictions for a pen — — -~ — —‘ — 272 cu-shaped sample [2]. The mean-field model predicts different cooperative phenomena like superradiance [2], subradiance [2,3] or limited thermalization (for the case of a thermal field [41) but the physical conditions of its validity are very restrictive. We would like to study a different case of a macroscopic sample with randomly distributed fluorescence centers. The density of centers is moderate so that we may putand Qk~ 0 and the values of Yki, k~ random in sign magnitude. In order to maketare this model solvable we replace two-level atoms by harmonic oscillators or localized states which may be occupied by fermions. As a consequence that spin-i operators S~,S~and S~ are replaced by a~,a~and a~a~.respectively where a~<and a~ satisfy canonical commutation or anticommutation relations. Moreover the matrix F is now a random matrix which is assumed to be given by +A ~ ‘2 ~— ~ Here y is the decay rate for an isolated fluorescence center,A = (ak/) is a random matrix described by the gaussian orthogonal ensemble [51, i.e. ak,=a/k are nonnally distributed with 2/N,0~J~ (3) <‘k~>°, (a~~>=J The mean-field scaling in (3) and the restriction 0 ~ J~~ are necessary to recover the positive defi. niteness of V (with probability one) in the limit 0375-9601/88/s 03.50 © Elsevier Science Publishers B.V. (North Holland Physics Publishing Division) Volume 133, number 6 PHYSICS LETTERS A 21 November 1988 N—~x.Assume now that the configuration ofcenters is fixed and hence V is specified. Then the master equation takes the form Now we may explicitly calculate the mean occupation number n ( r), the light intensity (per center) 1(r) = (w/y) dn(r)/dr and the time-dependent d decay rate #c(r)= T 2JI—y[dn(r)/dr]/n(r), (r~yt), n(0)e 2itJ2 \/(2J)—2e~dA N J [a~ak,pf] +1 ~ k,I=I — —2J (4) ykl{[akp,,afl+[ak,p,a,]}. (11) [61 Introducing a “single particle density matrix” a=(o~,),cJkI=tr(pa/ ak) one obtains ~~[I 1(2Jr)_2JI2(2Jr)] , (12) 1(r) =wn(0) (5) The mean occupation number ~V(t) is given by / ) N .~V(t)=tr( p, ~ a~a~ = \ tr ~ 2J12(2JT)), ‘I (2Jr) (13) where ‘I, 12 are modified Bessel functions. For Jr>> 1 we have k=I (14) 2(Jr)312 =tr(e”cr0) (6) In the case of a macroscopic sample and reasonable initial statesp 0, X(t) is an extensive quantity and hence in the limit possible configurations will be experiencedN~ooall (the analogy with spin glass may be noticed [7]). Therefore the mean occupation number per center n(t) is given by the average: . n(t)=e_~lim~<tr(o.oe_A>~)) . n(r)~n(0)2&” OJ— 1 \ J(r)~n(O)W e2(Jt)312(1_2J+3 / 1 l6Jr )‘ y2~ (15) __________ ~c(r)~~((l_2J)+J 16~3) 24 \ (16) . (7) The probability distribution for the gaussian orthogonal ensemble is invariant with respect to orthogot, WW~=W~W=l nal [5]. transformations Hence for any realA—~WAW and normalized vector ~= (~, ac(r) _~0~ 3-0 IN ~ \k./~I (e~)~ 1~)=~<tr(e~T)> (8) is independent of the choice of I~Therefore 4T)>=tr(a) <tr(~e ~<tr(e~T)>. 0.5 (9) The probability distribution p~) of eigenvalues of A is given in the limit N—~xby Wigner’s famous semicircular law [5] p(A)= = ~~(2J)2_~2 0 , , I~I~2J, I A I > 2J. (10) 3-1/2 0.0 Fig. 1. We obse~ea nonexponential decay law with the timedependent decay rate ic(r) suppressed by cooperative quantum interference effects. 273 Volume 133, number 6 PHYSICS LETTERS A This work is partially supported by the Polish Ministry of Education, project CPBP 01.03.3.3. References [1] G.S. Agarwal, Quantum optics, in: Springer tracts in modern physics, Vol. 70 (Springer, Berlin, 1974). [2] M. Gross and S. Haroche, Phys. Rep. 93 (1982) 301, and references therein. 274 21 November 1988 [3] A. Crulyellier, S. Liberman, D. Pavolini and P. Pillet, J. Phys. B 18(1985)3811. [4] R. Alicki, PhysicaA 150 (1988) 455. [5] M. L. Mehta, Random matrices (Academic Press, London, 1967). [6] R. Alicki and K. Lendi, Lectur notes in physics, Vol. 286. Quantum dynamical semigroups and application (Springer, Berlin, 1987). [7] D.C. Mattis, Springer series in solid state science, Vol. 55. The theory of magnetism II (Springer, Berlin, 1985).