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Physical Review Letters

Collapses and Revivals of Bose-Einstein Condensates Formed in Small Atomic Samples

Ewan M. Wright, Daniel F. Walls, J. C. Garrison

Published Sep 9, 1996• 218 citations
Publisher

Abstract

The macroscopic wave function for atomic samples composed of a few thousand particles is shown to exhibit collapses and revivals on a few seconds time scale, while Bose-Einstein condensation remains in the form of off-diagonal long-range order in the one-particle reduced density matrix. A recently proposed measurement scheme which is sensitive to Bose-broken gauge symmetry, and hence to the macroscopic wave function, could be used to detect the collapses and revivals experimentally.

DOI
10.1103/physrevlett.77.2158

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The macroscopic wave function for atomic samples composed of a few thousand particles is shown to exhibit collapses and revivals on a few seconds time scale, while Bose-Einstein condensation remains in the form of off-diagonal long-range order in the one-particle reduced density matrix. A recently proposed measurement scheme which is sensitive to Bose-broken gauge symmetry, and hence to the macroscopic wave function, could be used to detect the collapses and revivals experimentally.
We investigate the quantum interference between two Bose-Einstein condensates formed in small atomic samples composed of a few thousand atoms both by imposing Bose broken gauge symmetry from the outset and also using an explicit model of atomic detection. In the former case we show that the macroscopic wave function collapses and revives in time, and we calculate the characteristic times for current experiments. Collapses and revivals are also predicted in the interference between two Bose-Einstein condensates which are initially in Fock states, a relative phase between the condensates being established via atomic detections corresponding to uncertainty in the number difference between them.
Quantum collapses and revivals are fascinating manifestations of interference. Of particular interest in recent years are macroscopic quantum interference effects in Bose-Einstein condensates. In this letter such effects will be studied for the two site Bose-Hubbard model that is a standard model for exploration of Bose-Einstein condensates. An analytic expression that is valid in the weak coupling limit for the difference in the occupation of the two sites is developed and tested numerically. It describes correctly the collapses and revivals. Moreover, it is demonstrated that a calculation to the first order in the interparticle interaction is required for the prediction of the collapse and revival times while the second order is required for the evaluation of the shape of the revival peaks. We believe that the result is relevant for a variety of situations where collapses and revivals are found.
We present a scheme for determining if the quantum state of a small trapped atomic Bose-Einstein condensate is a state with a well-defined number of atoms, a Fock state, or a state with a broken U(1) gauge symmetry, a coherent state. The proposal is based on the observation of Ramsey fringes. The population difference measured by a Ramsey fringe experiment will exhibit collapse and revivals due to the mean-field interactions. The collapse and revival times depend on the relative strengths of the mean-field interactions for the two components and the initial quantum state of the condensate.
In this paper, we have used a technique , when a weak link is created between two systems which have undergone some type of gauge symmetry breaking. In fact the Josephson effect has been observed in superfluid .Bose condensate atomic gases is generally expected. A system whose gauge symmetry has been spontaneously broken can be described by order parameter that behaves in much respect like a macroscopic function)(rΨ. For Bose Einstein Condensates of dilute alkali gases )(rΨis the wave function of the macroscopically occupied one atomic state