Volume 155, number 2,3 PHYSICS LET1~ERS A 6 May 1991 Level statistics transitions in the spin—boson model C.H. Lewenkopf Max-Planck Institul fur Kernphysik, W-6900 Heidelberg, Germany M.C. Nemes, V. Marvulle, M.P. Pato and W.F. Wreszinski Inst ituto de Fisica, Universidade de São Paulo, C.P. 20516, 01498 São Paulo, S.P., Brazil Received 13 February 1991; accepted for publication 7 March 1991 Communicated by A.P. Fordy We present the nearest neighbour distribution (NND) and 43 statistics for the spin—boson model. The results show a standard Poisson—Wigner (GOE) transition in the NND for large spin, but the most interesting feature is the existence of a transition between the two extreme types of fluctuation spectra: picket-fence or harmonic-oscillator type for S small to Poisson for S large. The spin—boson model has recently been revisited 1.0 ‘ I [1—5]because of its interest in several areas of phys- 0.8 ics. In this note, we study the model’s level statistics, specifically the nearest neighbour distribution 0.6 - (NND) and the Dyson—Mehta 43 statistics [61. The model, in our generalized version, is described by the 0.4 - Hamiltonian 0.2 H=w 0a~a+wS~+G(S~a+Sa~) 0.0 +G’(Sa+S~a~) , (1) 0 1 2 3 S where a (a ~) is the annihilation (creation) opera- 1.0 / I tor corresponding to one boson, with [a, a ~] = 1; S~, / 0.8 - Si,, S~ are spin operators corresponding to spin quan- tum number S, satisfying SU (2) commutation re- o.o - lations [Sr, S~] =iS~+cyclic permutations, and S~ = (b) - S~ ± iS~1,.When the boson is interpreted as one mode < 0.4 of the electromagnetic field interacting with a 0.2 (2S+ 1)-level atom [7], the term with coupling con- stant G is the resonant one, and, hence, setting G’ =0 ______________________________ corresponds to the rotating-wave approximation 0 10 20 30 (RWA) [7]. We therefore refer to the term in (1) L with coupling constant G’ as the “counter-rotating” Fig. 1. (a) The histogram corresponds to the NND statistics for term. In the nonrelativistic theory of interaction be- S=~. G= 1 and G’ = 0.0. The dashed line is the theoretical curve tween atoms and the cutoff radiation field it follows corresponding to a Poisson distribution and the full curve the one corresponding to a Wigner surmize. (b) The points corre- that G = G’ [71,but, because the RWA is so good in spond to the numerical calculation of the 43(L) statistics for the quantum optics [7], we make G’ a variable param- same parameter value as above. The dashed and full lines arethe eter. Another reason for this choice is that, if G’ = 0, Poisson and GOE prediction respectively. Elsevier Science Publishers B.V. (North-Holland) 113