Nonlinear Schrödinger equation for a -symmetric delta-function double well

Abstract

The time-independent nonlinear Schrödinger 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 parity–time-(-) symmetric optical waveguides, e.g., the coalescence of modes at an exceptional point at a critical value of the loss/gain parameter, and the breaking of 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 symmetric, but observe coalescence and bifurcation scenarios different from those known from linear -symmetric Hamiltonians.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Quantum physics with non-Hermitian operators’.