  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 Schroedinger 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.
