Solitons in a photonic nonlinear quantum walk: lessons from the continuum

TL;DR

This paper investigates soliton solutions in a nonlinear quantum walk model, deriving a nonlinear Dirac equation in the continuum limit, and demonstrates their stability and controllability through numerical simulations and external perturbations.

Contribution

It introduces a nonlinear quantum walk model with a state-dependent coin operator, derives its continuum limit as a nonlinear Dirac equation, and explores soliton stability and dynamics.

Findings

Solitons are stable and controllable structures in the nonlinear quantum walk.

The continuum limit reveals a nonlinear Dirac equation governing the system.

External phase perturbations affect soliton stability and trajectories.

Abstract

We analyse a nonlinear QW model which can be experimentally implemented using the components of the electric field on an optical nonlinear Kerr medium, which translates into a rotation in the coin operator, with an angle which depends (in a nonlinear fashion) on the state of the walker. This simple dependence makes it easy to 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, under some approximations, to gain some insight into the nature of soliton structures, which is illustrated by our numerical calculations. These solitons are stable structures whose trajectories can be modulated by choosing the appropriate initial conditions. We have also studied the stability of solitons when they are subject to an additional phase that simulates an external electric field, and also explored if they are formed in higher dimensional spaces.