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Diffusive-ballistic crossover in 1D quantum walks

Daniel K. Wojcik·J. R. Dorfman
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Abstract

We show that particle transport in a uniform, quantum multi-baker map, is generically ballistic in the long time limit, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. Random matrix theory provides explicit analytical predictions for the mean square displacement of a particle in the system. These results exhibit a crossover from diffusive to ballistic motion, with crossover time from diffusive to ballistic motion on the order of the inverse of Planck's constant. We can argue, that for a large class of 1D quantum random walks, similar to the quantum multi-baker, a sufficient condition for diffusion in the semi-classical limit is classically chaotic dynamics in each cell. Using an initial equilibrium density matrix, we find that diffusive behavior is recovered in the semi-classical limit for such systems, without further interactions with the environment.

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Daniel K. Wojcik, J. R. Dorfman. 2003-09-10. Diffusive-ballistic crossover in 1D quantum walks. https://doi.org/10.1103/physrevlett.90.230602

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