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Probing an interacting Bose gas in a quasi-two-dimensional trap

Shih-Kuang Tung

Published Jan 1, 2010• 2 citations
OpenAlex

Abstract

A fluctuating regime associated with the Berenzinskii-Kosterlitz-Thouless (BKT) transition is studied in a two-dimensional system of near-degenerate bosons (87Rb atoms). Two imaging procedures, the in-situ imaging method and the focusing-imaging method, are implemented to image the coordinate-space and momentum-space density distributions. The scaled compressibility is extracted from the coordinate-space density distributions obtained from in-situ images. Comparing the measured compressibility to the prediction from our semiclassical mean-field model, an onset of a regime which is beyond the mean-field prediction is identified and first resolved at phase-space density nλ2 ≳ 3. Information about the coherence of the system is extracted from the momentum-space distributions obtained from the focusing images. The spatial extent of the coherence at a size of the high density region in the system only appears at a distinguishably higher phase-space density of nλ2 ≈8. Therefore, a very interesting regime that is beyond mean-field prediction, but not yet a condensate, is identified. This regime, which does not exist in three dimensions, is a product of the enhanced interactions associated with reduced dimensionality.

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A fluctuating regime associated with the Berenzinskii-Kosterlitz-Thouless (BKT) transition is studied in a two-dimensional system of near-degenerate bosons (87Rb atoms). Two imaging procedures, the in-situ imaging method and the focusing-imaging method, are implemented to image the coordinate-space and momentum-space density distributions. The scaled compressibility is extracted from the coordinate-space density distributions obtained from in-situ images. Comparing the measured compressibility to the prediction from our semiclassical mean-field model, an onset of a regime which is beyond the mean-field prediction is identified and first resolved at phase-space density nλ2 ≳ 3. Information about the coherence of the system is extracted from the momentum-space distributions obtained from the focusing images. The spatial extent of the coherence at a size of the high density region in the system only appears at a distinguishably higher phase-space density of nλ2 ≈8. Therefore, a very interesting regime that is beyond mean-field prediction, but not yet a condensate, is identified. This regime, which does not exist in three dimensions, is a product of the enhanced interactions associated with reduced dimensionality.
We simulate a trapped quasi-two-dimensional Bose gas using a classical field method. To interpret our results we identify the uniform Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKT{T}_{\text{BKT}}TBKT​ as where the system phase-space density satisfies a critical value. We observe that density fluctuations are suppressed in the system well above TBKT{T}_{\text{BKT}}TBKT​ when a quasicondensate forms as the first occurrence of degeneracy. At lower temperatures, but still above TBKT{T}_{\text{BKT}}TBKT​, we observe the development of appreciable coherence as a prominent finite-size effect, which manifests as bimodality in the momentum distribution of the system. At TBKT{T}_{\text{BKT}}TBKT​ algebraic decay of off-diagonal correlations occurs near the trap center with an exponent of 0.25, as expected for the uniform system. Our results characterize the low-temperature phase diagram for a trapped quasi-two-dimensional Bose gas and are consistent with observations made in recent experiments.
In complementary images of coordinate-space and momentum-space density in a trapped 2D Bose gas, we observe the emergence of presuperfluid behavior. As phase-space density $\ensuremath{\rho}$ increases toward degenerate values, we observe a gradual divergence of the compressibility $\ensuremath{\kappa}$ from the value predicted by a bare-atom model, ${\ensuremath{\kappa}}_{\mathrm{ba}}$. $\ensuremath{\kappa}/{\ensuremath{\kappa}}_{\mathrm{ba}}$ grows to 1.7 before $\ensuremath{\rho}$ reaches the value for which we observe the sudden emergence of a spike at p=0p=0p=0 in momentum space. Momentum-space images are acquired by means of a 2D focusing technique. Our data represent the first observation of non-mean-field physics in the presuperfluid but degenerate 2D Bose gas.
We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allow us to characterize the crossover from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.
We analyze the results of a recent experiment with bosonic rubidium atoms harmonically confined in a quasi-two-dimensional (2D) geometry. In this experiment a well-defined critical point was identified, which separates the high-temperature normal state characterized by a single component density distribution, and the low-temperature state characterized by a bimodal density distribution and the emergence of high-contrast interference between independent 2D clouds. We first show that this transition cannot be explained in terms of conventional Bose–Einstein condensation of the trapped ideal Bose gas. Using the local density approximation (LDA), we then combine the mean-field (MF) Hartree–Fock theory with the prediction for the Berezinskii–Kosterlitz–Thouless (BKT) transition in an infinite uniform system. If the gas is treated as a strictly 2D system, the MF predictions for the spatial density profiles significantly deviate from those of a recent quantum Monte Carlo (QMC) analysis. However, when the residual thermal excitation of the strongly confined degree of freedom is taken into account, excellent agreement is reached between the MF and the QMC approaches. For the interaction strength corresponding to the experiment, we predict a strong correction to the critical atom number with respect to the ideal gas theory (factor ∼2). Quantitative agreement between theory and experiment is reached concerning the critical atom number if the predicted density profiles are used for temperature calibration.
Recent experiments on ultracold Bose gases in two dimensions have provided evidence for the existence of the Berezinskii-Kosterlitz-Thouless (BKT) phase via analysis of the interference between two independent systems. In this work we study the two-dimensional quantum degenerate Bose gas at finite temperature using the projected Gross-Pitaevskii equation classical field method. Although this describes the highly occupied modes of the gas below a momentum cutoff, we have developed a method to incorporate the higher momentum states in our model. We concentrate on finite-sized homogeneous systems in order to simplify the analysis of the vortex pairing. We determine the dependence of the condensate fraction on temperature and compare this to the calculated superfluid fraction. By measuring the first order correlation function we determine the boundary of the Bose-Einstein condensate and BKT phases, and find it is consistent with the superfluid fraction decreasing to zero. We reveal the characteristic unbinding of vortex pairs above the BKT transition via a coarse-graining procedure. Finally, we model the procedure used in experiments to infer system correlations [Hadzibabic et al., Nature 441, 1118 (2006)], and quantify its level of agreement with directly calculated in situ correlation functions.
We measure the momentum distribution of a two-dimensional trapped Bose gas and observe the increase of the range of coherence around the Berezinskii-Kosterlitz-Thouless (BKT) transition. We quantitatively compare our observed profiles to both a Hartree-Fock mean-field theory and quantum Monte Carlo simulations. In the normal phase, the momentum distribution is observed to sharpen well before the phase transition. This behavior is partially captured in a mean-field approach, in contrast to the physics of the BKT transition.