Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength W , where a quantum to classical crossover at time tc ∝ W −2 transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.
We obtain numerically a scale-invariant distribution of the bandwidths S for the critical Harper model, which is closely described by a semi-Poisson P(S) = 4Sexp(-2S) curve. After a suitable unfolding of spectra, derived from different boundary conditions, a semi-Poisson level spacing distribution and a sub-Poisson linear number variance are deduced from the bandwidth distribution. The obtained results support possible universality of the critical spectral statistics and suggest its connection to spectral multifractality.
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