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Birkhäuser Basel eBooks

Phase Fluctuations and Coherence of an Almost Ideal Condensate

Лев П. Питаевский, S. Stringari

Published Jan 1, 2003• 0 citations
Publisher

Abstract

The quantum description of a Bose-Einstein condensates in double-well traps connected by a Josephson junction is discussed. The quantization in terms of the number and phase canonically-conjugated operators gives correct energy levels, but results in finite phase fluctuations in the absence of interaction. This seems to contradict the perfect coherence of a system of non-interacting particles. The problem can be solved by calculating the observables that characterize the coherence properties of a system, as, for example, the momentum distribution. It occurs that the visibility of interference fringes depends both on the fluctuations of the phase and the number of atoms. The visibility is calculated in the limit of small interaction and the results exhibit the correct behavior. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-7907-1_62

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The quantum description of a Bose-Einstein condensates in double-well traps connected by a Josephson junction is discussed. The quantization in terms of the number and phase canonically-conjugated operators gives correct energy levels, but results in finite phase fluctuations in the absence of interaction. This seems to contradict the perfect coherence of a system of non-interacting particles. The problem can be solved by calculating the observables that characterize the coherence properties of a system, as, for example, the momentum distribution. It occurs that the visibility of interference fringes depends both on the fluctuations of the phase and the number of atoms. The visibility is calculated in the limit of small interaction and the results exhibit the correct behavior. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The quantum and thermal fluctuations of the phase are investigated in a cold Bose gas confined by a double well trap. The coherence of the system is discussed in terms of the visibility of interference fringes in both momentum and coordinate space. The visibility is calculated at zero as well as at finite temperature. The thermal fluctuations are shown to affect significantly the transition from the coherent to the incoherent regime even at very low temperatures. The coherence of an array of multiple condensates is also discussed.
Abstract This chapter discusses several phenomena related to the coherence of trapped Bose–Einstein condensates. These include the long-range behaviour of the one-body density matrix and the interference fringes exhibited by two expanding condensates overlapping in space. Interference patterns in the momentum distribution are also discussed. A special section is devoted to the Josephson oscillations in the double-well potential, including the plasma oscillation and self-trapping phenomena. The procedures required to quantize the Josephson Hamiltonian, the phenomena of decoherence and fluctuations of the phase, and the derivation of the Bose–Hubbard model in the problem of the double well, are also discussed.
We study many-body quantum coherence and interaction blockade in two Josephson-linked Bose-Einstein condensates. We introduce universal operators for characterizing many-body coherence without limitations on the system symmetry and total particle number N . We reproduce the results for both coherence fluctuations and number squeezing in symmetric systems of large N , and reveal several peculiar phenomena that may occur in asymmetric systems and systems of small N . For asymmetric systems, we show that, due to an interplay between asymmetry and inter-particle interaction, the coherence fluctuations are suppressed dramatically when | E C / E J |≪1, and both resonant tunneling and interaction blockade take place for large values of | E C / E J |, where E C and E J are the interaction and tunneling energies, respectively. We emphasize that the resonant tunneling and interaction blockade may allow creating single-atom devices with promising technology applications. We demonstrate that for the systems at finite temperatures the formation of self-trapped states causes an anomalous behavior.
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.
The effects of quantum and thermal fluctuations upon the fringe structure predicted to be observable in the momentum distribution of coupled Bose-Einstein condensates are studied by the effective-potential method. For a double-well trap, the coherence factor recently introduced by Pitaevskii and Stringari (e-print cond-mat/0104458) is calculated using the effective potential approach and is found in good agreement with their result. The calculations are extended to the case of a one-dimensional array of condensates, showing that quantum effects are essentially described through a simple renormalization of the energy scale in the classical analytical expression for the fringe structure. The consequences for the experimental observability are discussed.
Bose–Einstein condensates of particles behave in many ways like coherent radiation fields, and the realization of Bose–Einstein condensation in dilute gases has opened up the experimental study of interactions between coherent matter waves. In addition, the existence of these dilute trapped quantum gases has prompted a re-examination of a number of theoretical issues. In Sec. 13.1 we discuss Josephson tunnelling of a condensate between two wells and the role of fluctuations in particle number and phase. The number and phase variables play a key role in the description of the classic interference experiment, in which two clouds of atoms are allowed to expand and overlap (Sec. 13.2). Rather surprisingly, an interference pattern is produced even though initially the two clouds are completely isolated. Density fluctuations in a Bose gas are studied in Sec. 13.3, where we relate atomic clock shifts to the two-particle correlation function. The ability to manipulate gaseous Bose–Einstein condensates by lasers has made possible the study of coherent matter wave optics and in Sec. 13.4 we describe applications of these techniques to observe solitons, Bragg scattering, and non-linear mixing of matter waves. How to characterize Bose–Einstein condensation in terms of the density matrix is the subject of Sec. 13.5, where we also consider fragmented condensates.