Quantum walks as massless Dirac fermions in curved space-time

A particular family of time- and space-dependent discrete-time quantum walks (QWs) is considered in one-dimensional physical space. The continuous limit of these walks is defined through a procedure discussed here and computed in full detail. In this limit, the walks coincide with the propagation of a massless Dirac fermion in an arbitrary gravitational field. A QW mimicking the radial propagation of a fermion outside and inside the event horizon of a Schwarzschild black hole is explicitly constructed and simulated numerically. Thus, the family of QWs considered in our manuscript provides an analog system to study experimentally coherent quantum propagation in curved spacetime.

Figure 1

Density of the QW vs null geodesics (solid curves) of the 2D Schwarzschild metric for various values of $\lambda$ [see text and Eq. (14)]. The singularity is represented by the dotted and dashed line on the left and the horizon (which is a null geodesic) is represented by the dashed line. (a), (b) The two branches of the QW which starts inside the horizon end up on the singularity. The (red) solid line represents the limit of the definition domain $\mathcal{D}$ of the QW. (c) One branch of a QW which starts on the horizon stays on the horizon while the other branch ends up on the singularity. (d) One branch of the QW which starts outside the horizon propagates away from the black hole, and the other branch ends up on the singularity.