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Topological Transition in a Non-Hermitian Quantum Walk

February 22, 2010 - 12:30pm
Leonid Levitov

We analyze a quantum walk on a bipartite one-dimensional lattice, in which the particle can decay whenever it visits one of the two sublattices. The corresponding non-Hermitian tight-binding problem with complex potential for the decaying sites exhibits two distinct phases, distinguished by a winding number defined in terms of the Bloch eigenstates in the Brillouin zone [1]. We find that the mean displacement of a particle initially localized on one of the non-decaying sites is quantized as an integer, changing from zero to one at the critical point. By mapping this problem onto a Jaynes-Cummings-type model with decay, we find that the topological transition is relevant for a variety of experimental settings, in particular for superconducting qubits coupled to high quality resonators [2]. The quantized behavior stands in contrast with the smooth dependence expected for a classical random walk, and can serve as a hallmark of coherent quantum dynamics in ladder-like multilevel systems. A real-space implementation of the quantum walk may help to verify quantum coherence in vortex transport in Josephson arrays [3]. 

[1] M. S. Rudner, L. S. Levitov, Phys. Rev. Lett. 102, 065703 (2009). 
[2] A. Wallraff et al., Nature 431, 162-167 (2004). 
[3] A. van Oudenaarden, S. J. K. Vardy, and J. E. Mooij, Phys. Rev. Lett. 77, 4257 (1996).

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