  We study a two dimensional (2D) system of interacting quantum bosons,
subjected to a continuous periodic potential in one direction. The correlation
of such system exhibits a dimensional crossover between a canonical 2D behavior
with Berezinski-Kosterlitz-Thouless (BKT) properties and a one-dimensional (1D)
behavior when the potential is large and splits the system in essentially
independent tubes. The later is in the universality class of Tomonaga-Luttinger
liquids (TLL). Using a continuous quantum Monte Carlo method, we investigate
this dimensional crossover by computing longitudinal and transverse superfluid
fraction as well as the superfluid correlation as a function of temperature,
interactions and potential. Especially, we find the correlation function
evolves from BKT to TLL type, with special intermediate behaviors appearing at
the dimensional crossover. We discuss how the consequences of the dimensional
crossover can be investigated in cold atomic gases experiments.
