Fractal stabilization of Wannier-Stark resonances

– The quasienergy spectrum of a Bloch electron affected by dc-ac fields is known to have a fractal structure as function of the so-called electric matching ratio, which is the ratio of the ac field frequency and the Bloch frequency. This paper studies a manifestation of the fractal nature of the spectrum in the system “atom in a standing laser wave”, which is a quantum optical realization of a Bloch electron. It is shown that for an appropriate choice of the system parameters the atomic survival probability (a quantity measured in laboratory experiments) also develops a fractal structure as a function of the electric matching ratio. Numerical simulations under classically chaotic scattering conditions show good agreement with theoretical predictions based on random matrix theory. 1. – In this letter we study the spectral and dynamical properties of a Bloch particle affected by static and time-periodic forces: Ĥ = p̂/2 + cosx+ Fx+ Fω cos(ωt)x , p̂ = −h̄ d/dx , (1) where h̄ is the scaled Planck constant (see below). Originally this problem was formulated for a Bloch electron in dc-ac electric fields Ĥ = p̂/2m+V (x)+e[E+Eω cos(ωt)]x, V (x+a) = V (x) [1] and attracted much attention because of the similarity with the Hofstadter problem [2]. Indeed, the energy spectrum of a Bloch electron in a 2D lattice under the action of a constant magnetic fieldB depends on the magnetic matching ratio β = h/eBa (a is the lattice constant) and has a fractal structure as function of this parameter. Analogously, the quasienergy spectrum of system (1) depends on the electric matching ratio