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Phase transition to Bose-Einstein condensation for a Bosonic gas confined in a combined trap

We present a study of phase transition to macroscopic superfluidity for an ultracold bosonic gas confined in a combined trap formed by a one-dimensional optical lattice and a harmonic potential, focusing on the critical temperature of this system and the interference patterns of the Bose gas released from the combined trap. Based on a semiclassical energy spectrum, we develop an analytic approximation for the critical temperature $T_{c}$, and compare the analytic results with that obtained by numerical computations. For finite temperatures below $T_{c}$, we calculate the interference patterns for both the normal gas and the superfluid gas. The total interference pattern shows a feature of ``peak-on-a-peak". As a comparison, we also present the experimentally observed interference patterns of $^{87}$Rb atoms released from a one-dimensional optical lattice system in accord with our theoretical model. Our observations are consistent with the theoretical results.