Nonlinear Optical Galton Board: thermalization and continuous limit

TL;DR

This paper explores the nonlinear optical Galton board and its continuous limit, revealing thermalization phenomena and deriving a nonlinear Dirac equation that models its long-term behavior.

Contribution

It introduces the nonlinear optical Galton board, derives its continuous limit as a nonlinear Dirac equation, and demonstrates thermalization in both models.

Findings

NLOGB exhibits complex evolution leading to thermalized states.

The continuous limit of NLOGB is a nonlinear Dirac equation (NLDE).

NLDE thermalizes toward states similar to NLOGB.

Abstract

The nonlinear optical Galton board (NLOGB), a quantum walk like (but nonlinear) discrete time quantum automaton, is shown to admit a complex evolution leading to long time thermalized states. The continuous limit of the Galton Board is derived and shown to be a nonlinear Dirac equation (NLDE). The (Galerkin truncated) NLDE evolution is shown to thermalize toward states qualitatively similar to those of the NLOGB. The NLDE conserved quantities are derived and used to construct a stochastic differential equation converging to grand canonical distributions that are shown to reproduce the (micro canonical) NLDE thermalized statistics. Both the NLOGB and the Galerkin-truncated NLDE are thus demonstrated to exhibit spontaneous thermalization.