Diffusive-ballistic crossover in 1D quantum walks

Details Diffusive-ballistic crossover in 1D quantum walks Diffusive-ballistic crossover in 1D quantum walks

2002-09-04

arxiv.org/abs/quant-ph/0209036v3

Phys. Rev. Lett. 90 (2003) 230602

Physics

Quantum Physics

Final version close to published; 4 pages, 3 figures, RevTeX 4

Scientific paper

10.1103/PhysRevLett.90.230602

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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