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Sci Rep. 2019 Feb 27;9(1):2989. doi: 10.1038/s41598-019-40059-x.

Quantum walk hydrodynamics.

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Aix-Marseille Université, CNRS, École Centrale de Marseille, Institut Fresnel UMR 7249, 13013, Marseille, France.
Aix-Marseille Université, Université de Toulon, CNRS, LIS, France, Natural Computation research group, Marseille, France.
Departamento de Física Teórica and IFIC, Universidad de Valencia-CSIC, Dr. Moliner 50, 46100, Burjassot, Spain.
LERMA, UMR 8112, UPMC and Observatoire de Paris, 61 Avenue de l'Observatoire, 75014, Paris, France.
Laboratoire de Physique Statistique, École Normale Supérieure, PSL Research University, UPMC Univ Paris 06, Sorbonne Universités, Université Paris Diderot, Sorbonne Paris-Cité, CNRS, 24 Rue Lhomond, 75005, Paris, France.


A simple Discrete-Time Quantum Walk (DTQW) on the line is revisited and given an hydrodynamic interpretation through a novel relativistic generalization of the Madelung transform. Numerical results show that suitable initial conditions indeed produce hydrodynamical shocks and that the coherence achieved in current experiments is robust enough to simulate quantum hydrodynamical phenomena through DTQWs. An analytical computation of the asymptotic quantum shock structure is presented. The non-relativistic limit is explored in the Supplementary Material (SM).

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