Accuracy of the Gross-Pitaevskii Equation in a Double-Well Potential

The Gross-Pitaevskii equation (GPE) in a double-well potential produces solutions that break the symmetry of the underlying non-interacting Hamiltonian, i.e., asymmetric solutions. The GPE is derived from the more general second-quantized Fock Schro<spacing diaeresis>\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\textrm{o}}$$\end{document}dinger equation (FSE). We investigate whether such solutions appear in the more general case or are artifacts of the GPE. We use two-mode analyses for a variational treatment of the GPE and to treat the Fock equation. An exact diagonalization of the FSE in dual condensates yields degenerate ground states that are very accurately fitted by phase-state representations of the degenerate asymmetric states found in the GPE. The superposition of degenerate asymmetrical states forms a cat state. An alternative form of cat state results from a change of the two-mode basis set.