Critical Energies and Wigner Functions of the Stationary States of the Bose Einstein Condensates in a Double-Well Trap

The quartic double-well potential contains the essential ingredients to study many-body systems within a rich semiclassical phase-space that includes an unstable point associated to the critical energy. This critical energy in the quantum realm causes symmetry breaking of the wavefunctions and produces a logarithmic divergence in the density of states, leading to an excited-state quantum phase transition (ESQPT). On the other hand, the Bose–Einstein Condensates (BEC) represent a promising platform to observe quantum mechanical phenomena at a macroscopic level. In this work, the lowest stationary states of the BEC in the mean field approximation are obtained via the Gross–Pitaevskii equation in a double-well trap. The critical energy at which the corresponding wavefunctions experience the symmetry breaking is estimated. It is also found that this critical energy is shifted from the local maximum of the trap as the interaction between bosons increases. The Wigner function is used to obtain a phase space representation of the stationary states. It is observed that the state with closest mean energy to its critical value shows vestiges of the separatrix in the semiclassical phase-space. The trends of the entropy, the nonorthogonality of the stationary states, and the nonclassicality via the negativities of the Wigner function are also examined.