On stochastic dynamics of hydrogenic ion transport through solids

We investigate the problem of hydrogenic ion transport through solids. To this end, we consider a system made of a single electron in a given potential and driven by a stochastic force uniformly distributed in time [D.G. Arbò et al., Phys. Rev. A 60 (1999) 1091], and we derive the time evolution equation of the ensemble average density matrix. We show that the random phase approximation of this equation leads to the rate equation model used to describe ion transport [P. Nicolai et al., J. Phys. B 23 (1990) 3609; J.P. Rozet et al., J. Phys. B 22 (1988) 33]. With the help of the Wigner transform, we also demonstrate that both quantum and classical dynamics of an electron under a train of kicks obeys the same equation within the semiclassical approximation. We conclude that seemingly different approaches may be regarded as different approximations of the same quantum problem and we show with a numerical example where differences arise