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Acta Physica Sinica

Density-density correlation in quasi two-dimensional free expanding Bose-Einstein condensates

Yan Xu, Fan Wei, Bing Chen, Nan Xiang-Hong, Da Chen, Zhou Qiang, Zhang Lu-Yin

Published Jan 1, 2013• 0 citations
Publisher

Abstract

The effective Lagrangian density function and the quantum fluctuation of the wave function in the form of quantized operators are presented for a quasi two-dimensional Bose-Einstein condensate by means of Madelung transformation. This paper calculates the two-point density-density correlation function of the condensate during its free expansion after its confinement potential is removed. Results show that the two-point density-density correlation function in the long-wave limit is proportional to the wave number k and it tends to be a constant in the short-wave limit.

DOI
10.7498/aps.62.216701

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The effective Lagrangian density function and the quantum fluctuation of the wave function in the form of quantized operators are presented for a quasi two-dimensional Bose-Einstein condensate by means of Madelung transformation. This paper calculates the two-point density-density correlation function of the condensate during its free expansion after its confinement potential is removed. Results show that the two-point density-density correlation function in the long-wave limit is proportional to the wave number k and it tends to be a constant in the short-wave limit.
We study, within the framework of the classical-field approximation, the density correlations of a weakly interacting expanding Bose gas for the whole range of temperatures across the Bose-Einstein condensation threshold. We focus on elongated quasi-one-dimensional systems where there is a huge discrepancy between the existing theory and experimental results [A. Perrin et al., Nat. Phys. 8, 195 (2012)]. We find that the density correlation function is not reduced for temperatures below the critical one as it is predicted for the ideal gas or for a weakly interacting system within the Bogoliubov approximation. This behavior of the density correlations agrees with the above-mentioned experiment with the elongated system. Although the system was much larger than that studied here, we believe that the behavior of the density correlation function found there is quite generic. Our theoretical study indicates also large density fluctuations in the trap in the quasicondensate regime where only phase fluctuations were expected. We argue that the enhanced density fluctuations can originate in the presence of interactions in the system, or more precisely in the presence of spontaneous dark solitons in the elongated gas at thermal equilibrium.
We investigate theoretically the evolution of the two-point density correlation function of a low-dimensional ultracold Bose gas after release from a tight transverse confinement. In the course of expansion thermal and quantum fluctuations present in the trapped systems transform into density fluctuations. For the case of free ballistic expansion relevant to current experiments, we present simple analytical relations between the spectrum of ``density ripples'' and the correlation functions of the original confined systems. We analyze several physical regimes, including weakly and strongly interacting one-dimensional (1D) Bose gases and two-dimensional (2D) Bose gases below the Berezinskii-Kosterlitz-Thouless (BKT) transition. For weakly interacting 1D Bose gases, we obtain an explicit analytical expression for the spectrum of density ripples which can be used for thermometry. For 2D Bose gases below the BKT transition, we show that for sufficiently long expansion times the spectrum of the density ripples has a self-similar shape controlled only by the exponent of the first-order correlation function. This exponent can be extracted by analyzing the evolution of the spectrum of density ripples as a function of the expansion time.
We observe that a density-depleted vortex core evolves into concentric density ripples in a freely expanding quasi-two-dimensional Bose-Einstein condensate. The atomic density of the expanding condensate rapidly reduces due to the fast expansion along the tightly confining axial direction, and thus the transverse expansion of the condensate is not significantly affected by the atom-atom interactions. We find that the observed density profiles of the vortex cores are in good quantitative agreement with numerical simulations assuming no interactions. We analyze the defocused images of the vortex cores, where defocussing is caused by the free fall of the condensate during the expansion time. In the defocused images, the vortex core becomes magnified and appears to be filled after a certain expansion time which depends on the vortex charge number.
We observe that a density-depleted vortex core evolves into concentric density ripples in a freely-expanding quasi-two-dimensional Bose-Einstein condensate. The atomic density of the expanding condensate rapidly reduces due to the fast expansion along the tightly confined axial direction; thus, the transverse expansion of the condensate is not significantly affected by atom-atom interactions. We find that the observed density profiles of the vortex cores are in good quantitative agreement with numerical simulations assuming no interactions. We analyze the defocused images of the vortex cores, where defocussing is caused by the free fall of the condensate during the expansion time. In the defocused images, the vortex core becomes magnified and appears to be filled after a certain expansion time that depends on the vortex charge number.
We measure the two-point density correlation function of freely expanding quasicondensates in the weakly interacting quasi-one-dimensional (1D) regime. While initially suppressed in the trap, density fluctuations emerge gradually during expansion as a result of initial phase fluctuations present in the trapped quasicondensate. Asymptotically, they are governed by the thermal coherence length of the system. Our measurements take place in an intermediate regime where density correlations are related to near-field diffraction effects and anomalous correlations play an important role. Comparison with a recent theoretical approach described by Imambekov et al. yields good agreement with our experimental results and shows that density correlations can be used for thermometry of quasicondensates.
We measure the power spectrum of density fluctuations emerging in a freely expanding quasi-two-dimensional (2D) degenerate Bose gas, and we investigate the scaling behavior of the spectrum for the expansion time. The power spectrum shows an oscillatory shape for long expansion times, where the spectral peak positions are observed to be shifted to lower spatial frequencies than the theoretical prediction for a noninteracting expansion case. We find the spectral peak positions in good agreement with the recent numerical simulation presented by Mazets [Phys. Rev. A 86, 055603 (2012)], where the atom-atom interactions are taken into account. We present a mean-field description of the interaction effect in the expansion dynamics and quantitatively account for the observed spectral peak shifts. The spectral shift is intrinsic to the free expansion of a quasi-2D Bose gas due to finite axial confinement. Finally, we investigate the defocusing effect in the power spectrum measurement.