Quantum transport in a time-dependent random potential with a finite correlation time

Abstract

Using an earlier density matrix formalism in momentum space we study the motion of a particle in a time-dependent random potential with a finite correlation time τ, for 0 < t ≪ τ. Within this domain we consider two subdomains bounded by kinetic time scales (t _c ² = 2mℏ ^-1 c ², c ² = σ ², ξ ², σξ, with 2σ the width of an initial wavepacket and the correlation length of the gaussian potential fluctuations), where we obtain power law scaling laws for the effect of the random potential in the mean squared displacement 〈x ²〉 and in the mean kinetic energy 〈E _kin〉. At short times, ≪ min (t _σ ², 1/2t _ξ ²), 〈x ²〉 and 〈E _kin〉 scale classically as t ⁴ and t ², respectively. At intermediate times, t _σξ ≪ t ≪ 2t _σ ² and 1/2t _ξ ² ≪ t ≪ t _σξ, these quantities scale quantum mechanically as t ^3/2 and as √t, respectively. These results lie in the perspective of recent studies of the existence of (fractional) power law behavior of 〈x ²〉 and 〈E _kin〉 at intermediate times. We also briefly discuss the scaling laws for 〈x ²〉 and 〈E _kin〉 at short times in the case of spatially uncorrelated potential.