The equilibrium properties of a weakly interacting atomic Bose gas across the Berezinskii-Kosterlitz-Thouless (BKT) and Bose-Einstein condensation (BEC) phase transitions are numerically investigated through a dimensionality crossover from two to three dimensions. The crossover is realized by confining the gas in an experimentally feasible hybridized trap which provides homogeneity along the planar
xy directions through a box potential in tandem with a harmonic transverse potential along the transverse
z direction. The dimensionality is modified by varying the frequency of the harmonic trap from tight to loose transverse trapping. Our findings, based on a stochastic (projected) Gross-Pitaevskii equation, showcase a continuous shift in the character of the phase transition from BKT to BEC, and a monotonic increase of the identified critical temperature as a function of dimensionality, with the strongest variation exhibited for small chemical potential values up to approximately twice the transverse confining potential.
In this thesis we study the dimensional crossover of a nonrelativistic Bose gas with pointlike repulsive interactions. Our focus is on the dimensional crossover from two to three dimensions, which is of particular relevance in ultracold experiments. Both the two-dimensional and the three-dimensional Bose gas feature a phase transition to a superfluid state. We investigate how the phase transition temperature changes as a function of the spatial extent of the transverse dimension, which is confined in a potential well with periodic boundary conditions. For these calculations we employ Functional Renormalization Group equations. Further, we relate the 3D and 2D s-wave scattering lengths and show how the lower-dimensional system inherits its properties from the higher-dimensional system during the renormalization group flow. In these calculations we introduce the crossover function, which is of central importance in the dimensional crossover. We also investigate other compactifications. We calculate the crossover function for a box confinement and introduce approximate flow equations for a confinement in a harmonic trap.
We present an improved many-body T-matrix theory for partially Bose-Einstein condensed atomic gases by treating the phase fluctuations exactly. The resulting mean-field theory is valid in arbitrary dimensions and able to describe the low-temperature crossover between three-, two-, and one-dimensional Bose gases. When applied to a degenerate two-dimensional atomic hydrogen gas, we obtain a reduction of the three-body recombination rate, which compares favorably with experiment. Supplementing the mean-field theory with a renormalization-group approach to treat the critical fluctuations, we also incorporate into the theory the Kosterlitz-Thouless transition that occurs in a homogeneous Bose gas in two dimensions. In particular, we calculate the critical conditions for the Kosterlitz-Thouless phase transition as a function of the microscopic parameters of the theory. The proposed theory is further applied to a trapped one-dimensional Bose gas, where we find good agreement with exact numerical results obtained by solving a nonlinear Langevin field equation.