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Communications in Theoretical Physics

SU(2) Coherent State Description of Two-Mode Bose–Einstein Condensates

Ying Wu, Xiaoxue Yang

Published May 15, 2002• 9 citations
Publisher

Abstract

We show that the evolution equations for mean quantities such as atom numbers and the inter-mode correlation for two-mode Bose–Einstein condensates form a closed set of equations in the SU(2) coherent state description, and they are identical in form to the two-mode mean-field model with only a slightly reduced two-body interaction strength. The exact analytical solutions to the evolution equations are also presented.

DOI
10.1088/0253-6102/37/5/539

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We show that the evolution equations for mean quantities such as atom numbers and the inter-mode correlation for two-mode Bose–Einstein condensates form a closed set of equations in the SU(2) coherent state description, and they are identical in form to the two-mode mean-field model with only a slightly reduced two-body interaction strength. The exact analytical solutions to the evolution equations are also presented.
A formalism for describing the coherence and interference properties of two atomic clouds of Bose-Einstein condensates (BEC) is presented, which is applicable even in the opposite limits when the BEC clouds are initially coherent and when they are initially independent. First, we develop a mean-field theory wherein one mean-field mode is used, and then, for fragmented (i.e., independent) condensates, we use a mean-field theory with two modes. We then develop a full two-mode field theory, with a field operator composed of a sum of two terms containing matter wave mode functions phi1 and phi2, that multiply the destruction operators of the modes, a1 and a2. When atom-atom interactions are present and when the mode functions overlap, the matter wave mode functions phi1 and phi2 develop components moving to the right and left, and this results in interference fringes in the density. At the many-body level, another source of interference arises from expectation values of the form (a(i)+a(j)) with i double dagger j, which become nonzero due to tunneling and interactions. We detail how these two sources of interference affect the density profile and the density-density correlation functions of Bose-Einstein condensates in the coherent and in the fragmented regimes.
In terms of exact solutions of the time-dependent Schr\"odinger equation for an effective giant spin modeled from a coupled two-mode Bose-Einstein condensate (BEC) with adiabatic and cyclic time-varying Raman coupling between two hyperfine states of the BEC, we obtain analytic time-evolution formulas of the population imbalance and relative phase between two components with various initial states, especially the SU(2) coherent state. We find the Berry phase depending on the number parity of atoms, and particle number dependence of the collapse revival of population-imbalance oscillation. It is shown that self-trapping and phase locking can be achieved from initial SU(2) coherent states with proper parameters.