  We study the thermodynamic properties of a 2D array of coupled
one-dimensional Bose gases. The system is realized with ultracold bosonic atoms
loaded in the potential tubes of a two-dimensional optical lattice. For
negligible coupling strength, each tube is an independent weakly interacting 1D
Bose gas featuring Tomonaga Luttinger liquid behavior. By decreasing the
lattice depth, we increase the coupling strength between the 1D gases and allow
for the phase transition into a 3D condensate. We extract the phase diagram for
such a system and compare our results with theoretical predictions. Due to the
high effective mass across the periodic potential and the increased 1D
interaction strength, the phase transition is shifted to large positive values
of the chemical potential. Our results are prototypical to a variety of
low-dimensional systems, where the coupling between the subsystems is realized
in a higher spatial dimension such as coupled spin chains in magnetic
insulators.
