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

TL;DR

This paper models Bose-Einstein condensates in a double well using quantum phase space, revealing classical-quantum correspondence, phase space dynamics, and proposing methods to generate macroscopic quantum superpositions and entangled states.

Contribution

It introduces a quantum phase space framework for BEC in a double well, linking classical phase space features with quantum states, and proposes protocols for creating controlled macroscopic superposition states.

Findings

Quantum states localize on classical phase space orbits.

Higher states form Schrödinger cat-like superpositions.

Classical chaos manifests in quantum Husimi distributions.

Abstract

We present a quantum phase space model of Bose-Einstein condensate (BEC) in a double well potential. In a two-mode Fock-state analysis 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. It is shown that the quantum states are localized on the known classical phase space orbits of a nonrigid physical pendulum, and thus the novel phase space characteristics of a nonrigid physical pendulum such as the $\pi$ motions are seen to be a property of the exact quantum states. Low lying states are harmonic oscillator like libration states while the higher lying states are Schr\"odinger cat-like superpositions of two pendulum rotor states. To study the dynamics in phase space, a comparison is made between a displaced quantum wavepacket and the trajectories of a swarm of points in classical phase space. For a driven double well, it is shown that the classical chaotic dynamics is manifest in the dynamics of the quantum states pictured using the Husimi distribution. Phase space analogy also suggests that a $\pi$ phase displaced wavepacket put on the unstable fixed point on a separatrix will bifurcate to create a superposition of two pendulum rotor states - a Schr\"odinger cat state (number entangled state) for BEC. It is shown that the choice of initial barrier height and ramping, following a $\pi$ phase imprinting on the condensate, can be used to generate controlled entangled number states with tunable extremity and sharpness.