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arXiv:cond-mat/0308135 · Physical Review A

Second-order correlation function of a phase fluctuating Bose-Einstein condensate

L. Cacciapuoti, D. Hellweg, M. Kottke, T. Schulte, Wolfgang A. Ertmer, J. Arlt, K. Sengstock, Luis Santos, Maciej Lewenstein

Published Nov 20, 2003• 33 citations
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Abstract

The coherence properties of phase fluctuating Bose-Einstein condensates are studied both theoretically and experimentally. We derive a general expression for the N-particle correlation function of a condensed Bose gas in a highly elongated trapping potential. The second-order correlation function is analyzed in detail, and an interferometric method to directly measure it is discussed and experimentally implemented. Using a Bragg diffraction interferometer, we measure intensity correlations in the interference pattern generated by two spatially displaced copies of a parent condensate. Our experiment demonstrates how to characterize the second-order correlation function of a highly elongated condensate and to measure its phase coherence length.

DOI
10.1103/physreva.68.053612

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The coherence properties of phase fluctuating Bose-Einstein condensates are studied both theoretically and experimentally. We derive a general expression for the N-particle correlation function of a condensed Bose gas in a highly elongated trapping potential. The second-order correlation function is analyzed in detail, and an interferometric method to directly measure it is discussed and experimentally implemented. Using a Bragg diffraction interferometer, we measure intensity correlations in the interference pattern generated by two spatially displaced copies of a parent condensate. Our experiment demonstrates how to characterize the second-order correlation function of a highly elongated condensate and to measure its phase coherence length.
We measure the intensity correlation function of two interfering spatially displaced copies of phase fluctuating Bose-Einstein condensates. It is shown that this corresponds to a measurement of the phase correlation properties of the initial condensate. Analogous to the method used in the stellar interferometer experiment of Hanbury Brown and Twiss, we use spatial intensity correlations to determine the phase coherence lengths of elongated condensates. We find good agreement with our prediction of the correlation function and confirm the expected coherence length.
We probe the spatial coherence properties of a magnetically trapped Bose gas. Two matter wave beams are extracted from two spatially separated regions of the trap and overlap outside the trapping region. The visibility of the resulting interference pattern measures the phase coherence between the regions of extraction. By varying the spatial separation between the two regions the first-order spatial correlation function of the trapped Bose gas can be measured. The location of the minima of the interference pattern is reproducible, which experimentally confirms that the trapped Bose-Einstein condensate is not fragmented into individual condensates.
We present experimental and theoretical studies of the coherence properties of a Bose-Einstein condensate (BEC) using an interference technique. Two optical standing wave pulses of duration 100 ns and separation $\ensuremath{\Delta}t$ are applied to a condensate. Each standing wave phase grating makes small copies of the condensate displaced in momentum space. The quantum mechanical amplitudes of each copy interfere, depending on $\ensuremath{\Delta}t$ and on spatial phase variations across the condensate. We find that the behavior of a trapped BEC is consistent with a uniform spatial phase. A released BEC, however, exhibits large phase variation across the condensate.
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.
We study first- and second-order coherence of trapped dilute Bose gases using appropriate correlation functions. Special attention is given to the discussion of second-order or density correlations. Except for a small region around the surface of a Bose-Einstein condensate the correlations can be accurately described as those of a locally homogeneous gas with a spatially varying chemical potential. The degrees of first- and second-order coherence are therefore functions of temperature, chemical potential, and position. The second-order correlation function is governed both by the tendency of bosonic atoms to cluster and by a strong repulsion at small distances due to atomic interactions. In present experiments both effects are of comparable magnitude. Below the critical temperature the range of the bosonic correlation is affected by the presence of collective quasiparticle excitations. The results of some recent experiments on second- and third-order coherence are discussed. It is shown that the relation between the measured quantities and the correlation functions is much weaker than previously assumed.
The Hanbury Brown-Twiss experiment has proved to be an effective means of measuring two-point correlation function of identical particles. We analyze experimental observation of stripes formation of a phase fluctuating Bose-Einstein condensates in a highly elongated 3D traps [Dettmer {\em et al.}, Phys. Rev. Lett. {\bf 87}, 160406 (2001)] by means of axial two-point correlation functions. We also predict that the stripes are present in quasi-1D Bose gas in the mean-field as well as in the hard-core bosons regimes. These stripes can be realized by measuring the axial two-point correlation functions by using the Bragg interferometric method which is similar to the original Hanbury Brown and Twiss experimental setup.