Electric quantum walks in two dimensions

We study electric quantum walks in two dimensions considering Grover, alternate, Hadamard, and discrete Fourier transform (DFT) quantum walks. In the Grover walk, the behavior under an electric field is easy to summarize: when the field direction coincides with the x or y axes, it produces a transient trapping of the probability distribution along the direction of the field, while when it is directed along the diagonals, a perfect two-dimensional (2D) trapping is frustrated. The analysis of the alternate walk helps one to understand the behavior of the Grover walk, as both walks are partially equivalent; in particular, it helps one to understand the role played by the existence of conical intersections in the dispersion relations, as we show that when these are removed, a perfect 2D trapping can occur for suitable directions of the field. We complete our study with the electric DFT and Hadamard walks in 2D, showing that the latter can exhibit perfect 2D trapping.