Quantum phase space picture of Bose-Einstein Condensates in a double well: Proposals for creating macroscopic quantum superposition states and a study of quantum chaos

We present a quantum phase space model of Bose-Einstein condensate (BEC) in a double well potential. In a quantum two-mode approximation we examine the eigenvectors and eigenvalues and find that the energy correlation diagram indicates a transition from a delocalized to a fragmented regime. Phase space information is extracted from the stationary quantum states using the Husimi distribution function. We show that the mean-field phase space characteristics of a nonrigid physical pendulum arises from the exact quantum states, and that only 4 to 8 particles per well are needed to reach the semiclassical limit. For a driven double well BEC, we show that the classical chaotic dynamics is manifest in the dynamics of the quantum states. Phase space anal-ogy also suggests that a π phase displaced wavepacket put on the unstable fixed point on a separatrix bifurcates to create a superposition of two pendulum rotor states - a macroscopic superposition state of BEC. We show that the choice of initial barrier height and ramping, following a π phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.