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arXiv:1609.01751 · Physical Review A

Quantum-fluctuation-induced time-of-flight correlations of an interacting trapped Bose gas

Izabella Lovas, Balázs Dóra, Eugene Demler, Gergely Zaránd

Published Feb 24, 2017• 8 citations
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Abstract

We investigate numerically the momentum correlations in a two-dimensional, harmonically trapped interacting Bose system at $T=0$ temperature, by using a particle number preserving Bogoliubov approximation. Interaction-induced quantum fluctuations of the quasicondensate lead to a large anticorrelation dip between particles of wave numbers $\mathbf{k}$ and $\ensuremath{-}\mathbf{k}$ for $|\mathbf{k}|\ensuremath{\sim}1/{R}_{c}$, with ${R}_{c}$ the typical size of the condensate. The anticorrelation dip found is a clear fingerprint of coherent quantum fluctuations of the condensate. In contrast, for larger wave numbers, $|\mathbf{k}|\ensuremath{\gg}1/{R}_{c}$, a weak positive correlation is found between particles of wave numbers $\mathbf{k}$ and $\ensuremath{-}\mathbf{k}$, in accordance with the Bogoliubov result for homogeneous interacting systems.

DOI
10.1103/physreva.95.023625

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We investigate numerically the momentum correlations in a two-dimensional, harmonically trapped interacting Bose system at T=0T=0T=0 temperature, by using a particle number preserving Bogoliubov approximation. Interaction-induced quantum fluctuations of the quasicondensate lead to a large anticorrelation dip between particles of wave numbers k\mathbf{k}k and \ensuremath−k\ensuremath{-}\mathbf{k}\ensuremath−k for ∣k∣\ensuremath∼1/Rc|\mathbf{k}|\ensuremath{\sim}1/{R}_{c}∣k∣\ensuremath∼1/Rc​, with Rc{R}_{c}Rc​ the typical size of the condensate. The anticorrelation dip found is a clear fingerprint of coherent quantum fluctuations of the condensate. In contrast, for larger wave numbers, ∣k∣\ensuremath≫1/Rc|\mathbf{k}|\ensuremath{\gg}1/{R}_{c}∣k∣\ensuremath≫1/Rc​, a weak positive correlation is found between particles of wave numbers k\mathbf{k}k and \ensuremath−k\ensuremath{-}\mathbf{k}\ensuremath−k, in accordance with the Bogoliubov result for homogeneous interacting systems.
We investigate the position- and momentum-space two-body correlations in a weakly interacting, harmonically trapped atomic Bose-Einstein condensed gas at low temperatures. The two-body correlations are computed within the Bogoliubov approximation and the consequences of the finite system size are highlighted in contrast to the spatially homogeneous case. In the position space, we recover the antibunching induced by the repulsive interatomic interaction in the condensed fraction localized around the trap center and the bunching in the outer thermal cloud. In the momentum space, bunching signatures appear for either equal or opposite values of the momentum and display peculiar features as a function of the momentum and the temperature. In analogy to the optical Hanbury-Brown and Twiss effect, the amplitude of the bunching signal at closeby momenta is fixed by the chaotic nature of the matter field state and its linewidth is shown to be set by the (inverse of the) finite spatial size of the associated in-trap momentum components. In contrast, the linewidth of the bunching signal at opposite momenta is only determined by the condensate size.
We analyze the two-body momentum correlation function for a uniform, weakly interacting, one-dimensional Bose gas. We show that the strong positive correlation between opposite momenta, expected in a Bose-Einstein condensate with a true long-range order, almost vanishes in a phase-fluctuating quasicondensate where the long-range order is destroyed. Using the Luttinger liquid approach, we derive an analytic expression for the momentum correlation function in the quasicondensate regime, showing (i) the reduction and broadening of the opposite-momentum correlations (compared to the singular behavior in a true condensate) and (ii) an emergence of anticorrelations at small momenta. We also numerically investigate the momentum correlations in the crossover between the quasicondensate and the ideal Bose-gas regimes using a classical field approach and show how the anticorrelations gradually disappear in the ideal-gas limit.
We predict the existence of a dip below unity in the second-order coherence function of a partially condensed ideal Bose gas in harmonic confinement, signaling the anticorrelation of density fluctuations in the sample. The dip in the second-order coherence function is revealed in a canonical-ensemble calculation, corresponding to a system with fixed total number of particles. In a grand-canonical ensemble description, this dip is obscured by the occupation-number fluctuation catastrophe of the ideal Bose gas. The anticorrelation is most pronounced in highly anisotropic trap geometries containing small particle numbers. We explain the fundamental physical mechanism which underlies this phenomenon, and its relevance to experiments on interacting Bose gases.
We study numerically the low-temperature behavior of a one-dimensional Bose gas trapped in an optical lattice. For a sufficient number of particles and weak repulsive interactions, we find a clear regime of temperatures where density fluctuations are negligible but phase fluctuations are considerable, i.e., a quasicondensate. In the weakly interacting limit, our results are in very good agreement with those obtained using a mean-field approximation. In coupling regimes beyond the validity of mean-field approaches, a phase-fluctuating condensate also appears, but the phase-correlation properties are qualitatively different. It is shown that quantum depletion plays an important role.
We present a Hartree-Fock-Bogoliubov (HFB) theoretical treatment of the two-dimensional trapped Bose gas and indicate how semiclassical approximations to this and other formalisms have lead to confusion. We numerically obtain results for the quantum-mechanical HFB theory within the Popov approximation and show that the presence of the trap stabilizes the condensate against long wavelength fluctuations. These results are used to show where phase fluctuations lead to the formation of a quasicondensate.
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.