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We analyze the results of a recent experiment with bosonic rubidium atomsharmonically confined in a quasi-two-dimensional geometry. In this experiment awell defined critical point was identified, which separates thehigh-temperature normal state characterized by a single component densitydistribution, and the low-temperature state characterized by a bimodal densitydistribution and the emergence of high-contrast interference betweenindependent two-dimensional clouds. We first show that this transition cannotbe explained in terms of conventional Bose-Einstein condensation of the trappedideal Bose gas. Using the local density approximation, we then combine themean-field (MF) Hartree-Fock theory with the prediction for theBerezinskii-Kosterlitz-Thouless transition in an infinite uniform system. Ifthe gas is treated as a strictly 2D system, the MF predictions for the spatialdensity profiles significantly deviate from those of a recent QuantumMonte-Carlo (QMC) analysis. However when the residual thermal excitation of thestrongly confined degree of freedom is taken into account, an excellentagreement is reached between the MF and the QMC approaches. For the interactionstrength corresponding to the experiment, we predict a strong correction to thecritical atom number with respect to the ideal gas theory (factor $\sim 2$). Aquantitative agreement between theory and experiment is reached concerning thecritical atom number if the predicted density profiles are used for temperaturecalibration.

The trapped two-dimensional Bose gas: from Bose-Einstein condensation to Berezinskii-Kosterlitz-Thouless physics