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

摘要

We present a quantum phase space model of Bose-Einstein condensate (BEC) in adouble well potential. In a two-mode Fock-state analysis we examine theeigenvectors and eigenvalues and find that the energy correlation diagramindicates a transition from a delocalized to a fragmented regime. Phase spaceinformation is extracted from the stationary quantum states using the Husimidistribution function. It is shown that the quantum states are localized on theknown classical phase space orbits of a nonrigid physical pendulum, and thusthe novel phase space characteristics of a nonrigid physical pendulum such asthe $\pi$ motions are seen to be a property of the exact quantum states. Lowlying states are harmonic oscillator like libration states while the higherlying states are Schr\"odinger cat-like superpositions of two pendulum rotorstates. To study the dynamics in phase space, a comparison is made between adisplaced quantum wavepacket and the trajectories of a swarm of points inclassical phase space. For a driven double well, it is shown that the classicalchaotic dynamics is manifest in the dynamics of the quantum states picturedusing the Husimi distribution. Phase space analogy also suggests that a $\pi$phase displaced wavepacket put on the unstable fixed point on a separatrix willbifurcate to create a superposition of two pendulum rotor states - aSchr\"odinger cat state (number entangled state) for BEC. It is shown that thechoice of initial barrier height and ramping, following a $\pi$ phaseimprinting on the condensate, can be used to generate controlled entanglednumber states with tunable extremity and sharpness.