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arXiv:2008.13039 · Physical Review A

Position- and momentum-space two-body correlations in a weakly interacting trapped condensate

Salvatore Butera, David Clément, Iacopo Carusotto

Published Jan 4, 2021• 6 citations

Abstract

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.

DOI: 10.1103/physreva.103.013302

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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 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

T=0

and $\ensuremath{-}\mathbf{k}$, in accordance with the Bogoliubov result for homogeneous interacting systems.

Analyzing the noise in the momentum profiles of single realizations of one-dimensional Bose gases, we present the experimental measurement of the full momentum-space density correlations ⟨δn_{p}δn_{p^{'}}⟩, which are related to the two-body momentum correlation function. Our data span the weakly interacting region of the phase diagram, going from the ideal Bose gas regime to the quasicondensate regime. We show experimentally that the bunching phenomenon, which manifests itself as super-Poissonian local fluctuations in momentum space, is present in all regimes. The quasicondensate regime is, however, characterized by the presence of negative correlations between different momenta, in contrast to the Bogolyubov theory for Bose condensates, predicting positive correlations between opposite momenta. Our data are in good agreement with ab initio calculations.

Continuous center-of-mass position measurements performed on an interacting harmonically trapped Bose-gas are considered. Using both a semianalytical mean-field approach and a completely quantum numerical technique based on the positive

\mathbf{k}

representation, it is demonstrated that the atomic delocalization due to the measurement backaction is smaller for a strongly interacting gas. The numerically calculated second-order correlation functions demonstrate the appearance of atomic bunching as a result of the center-of-mass measurement. Although rather small, the bunching is also present for a strongly interacting gas, which is in contrast with the case of an unperturbed gas. The performed analysis allows us to speculate that, for relatively strong interactions, the size of the atomic cloud determined with a single snapshot measurement can become smaller than the ground-state cloud size.

A peculiarity of trapped atomic gases is that Bose-Einstein condensation (BEC) shows up not only in momentum space, where it is usually discussed in traditional textbooks, but also in coordinate space. This is the consequence of the inhomogeneity of these systems, which makes it possible to separate the condensate from the thermal component also in coordinate space. Actually, most of measurements in trapped gases have been so far limited to the study of density profiles and of the effects of BEC in coordinate space. The possibility of a direct measurement of the momentum distribution, as emerged from recent experiments based on two-photon Bragg scattering, is highly appealing. In fact, despite the inhomogeneity of the gas, BEC shows up more deeply in momentum than in coordinate space. In current experiments on harmonically-confined Bose gases, the sizes of the condensate R and of the thermal cloud R_T are in fact typically comparable.

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.