Bright and dark solitons in a photonic nonlinear quantum walk: lessons from the continuum

Abstract

We propose a nonlinear quantum walk model inspired in a photonic implementation in which the
polarization state of the light field plays the role of the coin-qubit. In particular, we take profit of the
nonlinear polarization rotation occurring in optical media with Kerr nonlinearity, which allows to
implement a nonlinear coin operator, one that depends on the state of the coin-qubit. We consider
the space-time continuum limit of the evolution equation, which takes the form of a nonlinear Dirac
equation. The analysis of this continuum limit allows us to gain some insight into the existence
of different solitonic structures, such as bright and dark solitons. We illustrate several properties
of these solitons with numerical calculations, including the effect on them of an additional phase
simulating an external electric field.