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arXiv:quant-ph/9704036 · Physical Review A

Phase preparation by atom counting of Bose-Einstein condensates in mixed states

R. Graham, T. Wong, Matthew J. Collett, S. M. Tan, Daniel F. Walls

Published Jan 1, 1998• 11 citations
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Abstract

We study the buildup of quantum coherence between two Bose-Einstein condensates that are initially in mixed states. We consider in detail the two cases where each condensate is initially in a thermal or a Poisson distribution of atom number. Although initially there is no relative phase between the condensates, a sequence of spatial atom detections produces an interference pattern with arbitrary but fixed relative phase. The visibility of this interference pattern is close to one for the Poisson distribution of two condensates with equal counting rates but it becomes a stochastic variable in the thermal case, where the visibility will vary from run to run around an average visibility of $\ensuremath{\pi}/4.$ In both cases, the variance of the phase distribution is inversely proportional to the number of atom detections in the regime where this number is large compared to one but small compared with the total mean number of atoms in the condensates.

DOI
10.1103/physreva.57.493

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We study the buildup of quantum coherence between two Bose-Einstein condensates that are initially in mixed states. We consider in detail the two cases where each condensate is initially in a thermal or a Poisson distribution of atom number. Although initially there is no relative phase between the condensates, a sequence of spatial atom detections produces an interference pattern with arbitrary but fixed relative phase. The visibility of this interference pattern is close to one for the Poisson distribution of two condensates with equal counting rates but it becomes a stochastic variable in the thermal case, where the visibility will vary from run to run around an average visibility of \ensuremathπ/4.\ensuremath{\pi}/4.\ensuremathπ/4. In both cases, the variance of the phase distribution is inversely proportional to the number of atom detections in the regime where this number is large compared to one but small compared with the total mean number of atoms in the condensates.
We investigate the creation of a relative phase between two Bose-Einstein condensates, initially in number states, by detection of atoms and show how the system approaches a coherent state. Two very distinct time scales are found: one for the creation of the interference is of the order of the detection time for a few single atoms and another, for the preparation of coherent states, of the order of the detection time for a significant fraction of the total number of atoms. Approximate analytic solutions are derived and compared with exact numerical results.
We investigate the creation of a relative phase between two Bose-Einstein condensates, initially in number states, by detection of atoms and show how the system approaches a coherent state. Two very distinct time scales are found: one for the creation of the interference is of the order of the detection time for a few single atoms and another, for the preparation of coherent states, is of the order of the detection time for a significant fraction of the total number of atoms. Approximate analytic solutions are derived and compared with exact numerical results.
We study the interference of two Bose-Einstein condensates within an elementary model. The detection of the atoms is modeled by adapting the standard theory of photon detection. Even though the condensates are taken to be in number states with no phases whatsoever, our stochastic simulations of atom detection produce interference patterns as would also be predicted on the basis of the phases of the macroscopic wave functions describing the condensates. In statistical mechanics terms, we have devised a method to analyze spontaneous symmetry breaking for an arbitrary (not necessarily larger) number of particles.
We study theoretically the interference patterns produced by the overlap of an array of Bose-Einstein condensates that have no phase coherence among them. We show that density-density correlations at different quasimomenta, which play an important role in two-condensate interference, become negligible for large NNN, where NNN is the number of overlapping condensates. In order to understand the physics of this phenomenon, it is sufficient to consider the periodicity of the lattice and the statistical probability distribution of a random-walk problem. The average visibility of such interference patterns decreases as ${N}^{\ensuremath{-}1∕2}$ for large NNN.
We show that the visibility in interference experiments with Bose-Einstein condensates is directly related to the condensate fraction. The probability distribution of the contrast over many runs of an interference experiment thus gives the full counting statistics of the condensed atom number. For two-dimensional Bose gases, we discuss the universal behavior of the probability distribution in the superfluid regime and provide analytical expressions for the distributions for both homogeneous and harmonically trapped samples. They are non-Gaussian and unimodal with a variance that is directly related to the superfluid density. In general, the visibility is a self-averaging observable only in the presence of long-range phase coherence. Close to the transition temperature, the visibility distribution reflects the universal order-parameter distribution in the vicinity of the critical point.
We study an elementary model of two tunneling Bose-Einstein condensates with an atom detector in one of the traps. Stochastic simulations of atom detection in one trap show an oscillation of atom number between traps even though the condensates are initially in equal number Fock states. The oscillation has a random amplitude depending on the phase difference between the condensates, which coincides with that given by a spontaneously broken symmetry argument. \textcopyright{} 1996 The American Physical Society.