Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation

We present a representative set of analytic stationary-state solutions of the nonlinear Schr\"odinger equation for a symmetric double-square-well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry-preserving even and odd states, nonlinearity introduces quite exotic symmetry-breaking solutions---among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry-breaking localized solutions to form macroscopic quantum superposition states and explore a simple model for the exponentially small tunneling splitting.