Quantum Walks in artificial electric and gravitational Fields

QWs are simple formal analogues of classical random walks. They have been first considered by Feynman [1] as possible discretizations of the free Dirac dynamics in flat space-time. They have been introduced in the physics literature by [2] and [3] and the continuous-time version first appeared in [4] . They have been realized experimentally in [5–12] and are important in many fields, ranging from fundamental quantum physics [12, 13] to quantum algorithmics [14, 15], solid state phsyics [16–19] and biophysics [20, 21]. Following Feynman’s idea, several authors have studied the continuous limit of various QWs. The first publications [1, 22–28] only addressed QWs with constant coefficients and recent work has extended the discussion to QWs with timeand space-dependent coefficients [29–32], in both (1 + 1) and (1+2) space-time dimensions. In particular, a new method was developed in [29–32] to investigate the continuous limit of QWs with non constant coefficients. This method delivers interesting results, not only for standard QWs, but also for ‘derived’ QWs obtained from original QWs by keeping only one time-step out of two [32] . So far, this new method has only been applied to particular families of walks. This article presents the systematic application of this method to all QWs in (1 + 1) space-time dimensions. The main conclusions are: (i) all families of walks do not admit a continuous limit (ii) when the limit exists, it coincides, in all cases but one, with the dynamics of a Dirac fermion coupled to an artificial electric field and/or relativistic gravitational field. These theoretical conclusions are illustrated by numerical simulations. Connections with previous results as well as other topics like transport in graphene are also discussed.