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arXiv:1809.10967 · Physical Review A

Signatures of a universal jump in the superfluid density of a two-dimensional Bose gas with a finite number of particles

Krzysztof Gawryluk, Mirosław Brewczyk

Published Mar 20, 2019• 13 citations
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Abstract

We study, within the classical fields approximation, a two-dimensional weakly interacting uniform Bose gas of a finite number of atoms. By using a grand canonical ensemble formalism we show that such systems exhibit, in addition to the Berezinskii-Kosterlitz-Thouless (BKT) and thermal phases, an intermediate region. This intermediate region is characterized by a decay of current-current correlations at low momenta and by an algebraic decay of the first-order correlations with an exponent being larger than the critical value predicted by the BKT theory. The density of the superfluid fraction at the temperature which separates the BKT phase from the intermediate region approaches the one found by Nelson and Kosterlitz for two-dimensional superfluids while the number of atoms is increased.

DOI
10.1103/physreva.99.033615

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We study, within the classical fields approximation, a two-dimensional weakly interacting uniform Bose gas of a finite number of atoms. By using a grand canonical ensemble formalism we show that such systems exhibit, in addition to the Berezinskii-Kosterlitz-Thouless (BKT) and thermal phases, an intermediate region. This intermediate region is characterized by a decay of current-current correlations at low momenta and by an algebraic decay of the first-order correlations with an exponent being larger than the critical value predicted by the BKT theory. The density of the superfluid fraction at the temperature which separates the BKT phase from the intermediate region approaches the one found by Nelson and Kosterlitz for two-dimensional superfluids while the number of atoms is increased.
The physics in two-dimensional (2D) systems is very different from what we observe in three-dimensional (3D) systems. Thermal fluctuations in 2D systems are enhanced, and they prevent the conventional Bose-Einstein condensation (BEC) at non-zero temperatures by destroying the long-range order. However, a phase transition to a superfluid phase is still expected to occur in a 2D system along with an emergence of a quasi-long-range order, explained by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism. Within the BKT mechanism, a universal jump of the superfluid density in a 2D Bosonic system was theoretically predicted by Nelson and Kosterlitz, and was first observed in 2D \textsuperscript{4}He films by Bishop and Reppy. Recent experiments in trapped ultracold 2D Bose gas systems have shown signatures of the BKT transition, and its superfluidity. However, the universal jump in the superfluid density was not observed in these systems. Here we report the first observation of the universal jump in the superfluid density using an optically trapped ultracold 2D Bose gas. The measured superfluid phase space density at the BKT transition agrees well with the predicted value within our measurement uncertainty. Additionally, we measure the phase fluctuations in our density profiles to show that the BKT transition occurs first, followed by the BEC transition.
We analyze the theoretical derivation of the beyond-mean-field equation of state for two-dimensional gas of dilute, ultracold alkali-metal atoms in the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensate (BEC) crossover. We show that at zero temperature our theory — considering Gaussian fluctuations on top of the mean-field equation of state — is in very good agreement with experimental data. Subsequently, we investigate the superfluid density at finite temperature and its renormalization due to the proliferation of vortex–antivortex pairs. By doing so, we determine the Berezinskii–Kosterlitz–Thouless (BKT) critical temperature — at which the renormalized superfluid density jumps to zero — as a function of the inter-atomic potential strength. We find that the Nelson–Kosterlitz criterion overestimates the BKT temperature with respect to the renormalization group equations, this effect being particularly relevant in the intermediate regime of the crossover.
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.
In this work we investigate the unique properties of ultracold Bose gases in one and two dimensions. In two dimensions, we present simulations of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the projected Gross-Pitaevskii equation formalism. We confirm the presence of the BKT phase by observing algebraic decay in the first-order correlation function, unbinding of vortex pairs, and behaviour of the superfluid fraction. In one dimension we use a transverse Gaussian ansatz and the Lagrangian approach to derive an effective equation of motion for the one dimensional Bose gas at zero temperature.
We experimentally investigate the first-order correlation function of a trapped Fermi gas in the two-dimensional BEC-BCS crossover. We observe a transition to a low-temperature superfluid phase with algebraically decaying correlations. We show that the spatial coherence of the entire trapped system can be characterized by a single temperature-dependent exponent. We find the exponent at the transition to be constant over a wide range of interaction strengths across the crossover. This suggests that the phase transitions in both the bosonic regime and the strongly interacting crossover regime are of Berezinskii-Kosterlitz-Thouless type and lie within the same universality class. On the bosonic side of the crossover, our data are well described by the quantum Monte Carlo calculations for a Bose gas. In contrast, in the strongly interacting regime, we observe a superfluid phase which is significantly influenced by the fermionic nature of the constituent particles.
We investigate the phase ordering dynamics of a uniform two-dimensional Bose gas quenched to a finite temperature in the superfluid phase. Starting from a defect-rich, far-from-equilibrium state, we model the subsequent evolution with the projected Gross-Pitaevskii equation, which conserves both energy and particle number. By tuning the initial energy, we control the effective post-quench temperature and examine its role in the equilibration dynamics. We find that the gas exhibits universal behaviour at all temperatures, evidenced by spatio-temporal scaling of correlation functions and power-law growth of the correlation length ∼t1/z\sim t^{1/z}∼t1/z, with zzz the dynamical critical exponent. We find zzz to be temperature dependent, with z≈1.5z \approx 1.5z≈1.5 for post-quench temperatures just below the Berezinskii-Kosterlitz-Thouless (BKT) transition, and z≈1.9z \approx 1.9z≈1.9 for quenches to near-zero temperature. Analysis of the Porod tail of the momentum distribution suggests a temperature-dependent competition between vortices and sound waves in the coarsening process. The two-time correlation function also exhibits universal scaling, decaying as ∼t−λ/z\sim t^{-λ/z}∼t−λ/z, with autocorrelation exponent λλλ. Near the BKT transition we obtain λ≈2λ\approx 2λ≈2, whereas λλλ is found to diverge as the effective temperature approaches zero.