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We study the thermodynamic properties of a 2D array of coupledone-dimensional Bose gases. The system is realized with ultracold bosonic atomsloaded in the potential tubes of a two-dimensional optical lattice. Fornegligible coupling strength, each tube is an independent weakly interacting 1DBose gas featuring Tomonaga Luttinger liquid behavior. By decreasing thelattice depth, we increase the coupling strength between the 1D gases and allowfor the phase transition into a 3D condensate. We extract the phase diagram forsuch a system and compare our results with theoretical predictions. Due to thehigh effective mass across the periodic potential and the increased 1Dinteraction strength, the phase transition is shifted to large positive valuesof the chemical potential. Our results are prototypical to a variety oflow-dimensional systems, where the coupling between the subsystems is realizedin a higher spatial dimension such as coupled spin chains in magneticinsulators.

Cross-dimensional phase transition from an array of 1D Luttinger liquids to a 3D Bose-Einstein condensate