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arXiv:1412.4337

Landau levels for discrete-time quantum walks in artificial magnetic fields

Pablo Arnault, Fabrice Debbasch

Published Dec 14, 2014• 1 citation
arXivHTMLPDF

Abstract

A new family of 2D discrete-time quantum walks (DTQWs) is presented and shown to coincide, in the continuous limit, with the Dirac dynamics of a spin 1/2 fermion coupled to a constant and uniform magnetic field. Landau levels are constructed, not only in the continuous limit, but also for the DTQWs i.e. for finite non-vanishing values of the time- and position-step, by a perturbative approach in the step. Numerical simulations support the above results and suggest that the magnetic interpretation is valid beyond the scope of the continuous limit. The possibility of quantum simulation of condensed-matter systems by DTQWs is also discussed.

DOI
10.48550/arxiv.1412.4337

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A family of discrete-time quantum walks (DTQWs) on the line with an exact discrete U(N)\text{U}(N)U(N) gauge invariance is introduced. It is shown that the continuous limit of these DTQWs, when it exists, coincides with the dynamics of a Dirac fermion coupled to usual U(N)\text{U}(N)U(N) gauge fields in two-dimensional spacetime. A discrete generalization of the usual U(N)\text{U}(N)U(N) curvature is also constructed. An alternate interpretation of these results in terms of superimposed U(1)\text{U}(1)U(1) Maxwell fields and SU(N)\text{SU}(N)SU(N) gauge fields is discussed in the Appendix. Numerical simulations are also presented, which explore the convergence of the DTQWs towards their continuous limit and which also compare the DTQWs with classical (i.e., nonquantum) motions in classical SU(2)\text{SU}(2)SU(2) fields. The results presented in this paper constitute a first step towards quantum simulations of generic Yang-Mills gauge theories through DTQWs.
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