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Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Phase dynamics and Bose–broken symmetry in atomic Bose–Einstein condensates

Daniel F. Walls, Matthew J. Collett, T. Wong, S. M. Tan, Ewan M. Wright

Published Dec 15, 1997• 7 citations
Publisher

Abstract

Collisions lead to the collapse and revival of the interference fringes between two atomic Bose–Einstein condensates. We study the collapses and revivals of the relative phase between the condensates for two different initial states. One state invokes Bose–broken symmetry allowing us to write the wavefunction as a superposition of total number (of both condensates) states whereas the other does not.

DOI
10.1098/rsta.1997.0136

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Collisions lead to the collapse and revival of the interference fringes between two atomic Bose–Einstein condensates. We study the collapses and revivals of the relative phase between the condensates for two different initial states. One state invokes Bose–broken symmetry allowing us to write the wavefunction as a superposition of total number (of both condensates) states whereas the other does not.
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.
In this review we discuss the dynamics of the phase of trapped Bose–Einstein condensates. In particular we consider the phenomena of phase decoherence (termed also as phase collapse, or diffusion), and phase revival in systems of interacting atoms. We analyse the dependence of the collapse and revival times on the trap potential, dimensionality of the gas, atom number fluctuations, and on the coherent dynamics of the condensate. We show that in a class of experimentally relevant systems, the collapse time is relatively short, and in some cases vanishes in the limit of a large number of atoms, implying that the trapped Bose gas cannot sustain a well-defined quantum phase, and that the phase memory is lost on a relatively short time scale. Furthermore, we calculate the relative atom number fluctuations or a model of two interacting condensates, and show that the fluctuations are generically sub-Poissonian.
We show that, when two Bose-Einstein condensates are combined, a spatial interference pattern ensues even if the condensates are in number states with no spontaneously broken gauge symmetry. In our measurement-theoretical approach the interference pattern arises from correlations between the detected positions of the atoms. We also simulate numerically experiments intended to distinguish number-state condensates and spontaneous symmetry breaking, the latter being modelled by coherent-state condensates. To detect the difference in the pair correlation functions, the experiment has to be repeated about a thousand times even for condensates with as few as five atoms each.
We study theoretically the macroscopic interference of two independent Bose condensates released from a double potential trap. The observation of fringes could serve as a test for the paradigm of broken gauge symmetry. By numerical solution of the nonlinear Schr\"odinger equation in three dimensions, the consecutive stages of expansion, overlap, and interference are investigated in order to facilitate the design of future experiments. The phase-space dynamics of the condensates is analyzed by means of the Wigner function. It turns out that the distance of interference fringes grows linearly in time with a velocity inversely proportional to the initial distance of the two condensates. The collisional reduction of the fringe visibility is estimated.