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Bulletin of the American Physical Society

Topological Transition in a Non-Hermitian Quantum Walk: a new test for quantumness in driven artificial atoms and Josephson arrays

Mark Spencer Rudner, Leonid S Levitov

Published Mar 18, 2009• 0 citations
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Non-Hermitian topological edge states have many intriguing properties, however, to date, they have mainly been discussed in terms of bulk-boundary correspondence. Here, we propose using a bulk property of diffusion coefficients for probing the topological states and exploring their dynamics. The diffusion coefficient was found to show unique features with the topological phase transitions driven by parity-time (PT)-symmetric non-Hermitian discrete-time quantum walks as well as by Hermitian ones, despite the fact that artificial boundaries are not constructed by an inhomogeneous quantum walk. For a Hermitian system, a turning point and abrupt change appears in the diffusion coefficient when the system is approaching the topological phase transition, while it remains stable in the trivial topological state. For a non-Hermitian system, except for the feature associated with the topological transition, the diffusion coefficient in the PT-symmetric-broken phase demonstrates an abrupt change with a peak structure. In addition, the Shannon entropy of the quantum walk is found to exhibit a direct correlation with the diffusion coefficient. The numerical results presented herein may open up a new avenue for studying the topological state in non-Hermitian quantum walk systems.
We extend the non-Hermitian one-dimensional quantum walk model [Phys. Rev. Lett. 102, 065703 (2009)] by taking the dephasing effect into account. We prove that the feature of topological transition does not change even when dephasing between the sites within units is present. The potential experimental observation of our theoretical results in the circuit QED system consisting of superconducting qubit coupled to a superconducting resonator mode is discussed and numerically simulated. The results clearly show a topological transition in quantum walk and display the robustness of such a system to the decay and dephasing of qubits. We also discuss how to extend this model to higher dimension in the circuit QED system.
Scale-free localization is a distinctive phenomenon in non-Hermitian systems, where specific eigenstates, known as scale-free localized states (LSs), possess localization lengths that scale linearly with the system size. Here, we show that a single non-Hermitian defect in one-dimensional quantum walks (QWs) induces scale-free localization. Driven by non-Hermiticity, the QW system undergoes a pseudo-Hermitian (PH) phase transition and the PH symmetry breaking is accompanied by the emergence of scale-free localization. As non-Hermiticity is further increased, the system undergoes a second phase transition, marked by the emergence of two pairs of bound states. Crucially, we identify the critical points of both transitions using localization properties. Furthermore, we extend our investigation to two-dimensional QWs, where a single non-Hermitian defect also induces scale-free localization, thereby demonstrating the universality of this phenomenon. Our work provides a valuable perspective for further exploring defect physics and a type of novel LS in non-Hermitian quantum systems.