Nonlinear Schr\"odinger equation for a PT symmetric delta-functions double well

TL;DR

This paper analyzes a nonlinear Schr"odinger equation with PT symmetric delta-function double wells, revealing how nonlinearity affects mode coalescence, symmetry breaking, and bifurcation phenomena in systems modeling optical waveguides and Bose-Einstein condensates.

Contribution

It extends PT symmetric quantum models to include nonlinearity, showing novel bifurcation behaviors and PT symmetry preservation in nonlinear eigenstates.

Findings

Nonlinear eigenstates can preserve PT symmetry.

Mode coalescence occurs at exceptional points.

Nonlinearity introduces new bifurcation scenarios.

Abstract

The time-independent nonlinear Schr\"odinger equation is solved for two attractive delta-function shaped potential wells where an imaginary loss term is added in one well, and a gain term of the same size but with opposite sign in the other. We show that for vanishing nonlinearity the model captures all the features known from studies of PT symmetric optical wave guides, e.g., the coalescence of modes in an exceptional point at a critical value of the loss/gain parameter, and the breaking of PT symmetry beyond. With the nonlinearity present, the equation is a model for a Bose-Einstein condensate with loss and gain in a double well potential. We find that the nonlinear Hamiltonian picks as stationary eigenstates exactly such solutions which render the nonlinear Hamiltonian itself PT symmetric, but observe coalescence and bifurcation scenarios different from those known from linear PT symmetric Hamiltonians.