Quantum walk as a simulator of nonlinear dynamics: Nonlinear Dirac equation and solitons

TL;DR

This paper demonstrates that nonlinear quantum walks can simulate nonlinear Dirac dynamics, including soliton formation, and can control quantum diffusion, expanding the potential of quantum walks as universal quantum simulators.

Contribution

It introduces a nonlinear quantum walk framework capable of simulating nonlinear Dirac equations and solitons using a measurement-based feedforward scheme.

Findings

Successfully simulated nonlinear Dirac particles exhibiting soliton behavior.

Observed control over ballistic diffusion through nonlinear quantum walk dynamics.

Confirmed the framework's ability to replicate characteristic features of nonlinear solitons.

Abstract

Quantum walk (QW) provides a versatile tool to study fundamental physics and also to make a variety of practical applications. We here start with the recent idea of {\it nonlinear} QW and show that introducing {\it nonlinearity} to QW can lead to a wealth of remarkable possibilities, e.g., simulating nonlinear quantum dynamics thus enhancing the applicability of QW above the existing level for a universal quantum simulator. As an illustration, we show that the dynamics of a nonlinear Dirac particle can be simulated on an optical nonlinear QW platform implemented with a measurement-based-feedforward scheme. The nonlinear evolution induced by the feed-forward introduces a self-coupling mechanism to (otherwise linear) Dirac particles, which accordingly behave as a \emph{soliton}. We particularly consider two kinds of nonlinear Dirac equations, one with a scalar-type self-coupling (Gross-Neveu model) and the other with a vector-type one (Thirring model), respectively. Using their known stationary solutions, we confirm that our nonlinear QW framework is capable of exhibiting characteristic features of a soliton. Furthermore, we show that the nonlinear QW enables us to observe and control an enhancement and suppression of the ballistic diffusion.