This Dissertation presents results of a thorough study of ultracold bosonic and fermionic gases in three-dimensional and quasi-one-dimensional systems. Although the analyses are carried out within various theoretical frameworks (Gross-Pitaevskii, Bet
he ansatz, local density approximation, etc.) the main tool of the study is the Quantum Monte Carlo method in different modifications (variational Monte Carlo, diffusion Monte Carlo, fixed-node Monte Carlo methods). We benchmark our Monte Carlo calculations by recovering known analytical results (perturbative theories in dilute limits, exactly solvable models, etc.) and extend calculations to regimes, where the results are so far unknown. In particular we calculate the equation of state and correlation functions for gases in various geometries and with various interatomic interactions.
The phenomenon of Bose-Einstein condensation and superfluidity in a Bose gas with disorder is investigated. Diffusion Monte Carlo (DMC) method is used to calculate superfluid and condensate fraction of the system as a function of density and strength
of disorder at zero temperature. The algorithm and implementation of the Diffusion Monte Carlo method is explained in details. Bogoliubov theory is developed for the analytical description of the problem. Ground state energy, superfluid fraction and condensate fraction are calculated. It is shown that same results for the superfluid fraction can be obtained in a perturbative manner from Gross-Pitaevskii equation. Ground state energy, obtained from DMC calculations, is compared to predictions of Bogoliubov theory, which are found to be valid in the regime, when the strength of disorder is small. It is shown that unusual situation, when the superfluid fraction is smaller than the condensate fraction, can be realized in this system.
The ground-state properties of one-dimensional 3He are studied using quantum Monte Carlo methods. The equation of state is calculated in a wide range of physically relevant densities and is well reproduced by a power-series fit. The Luttinger liquid
theory is found to describe the long-range properties of the correlation function. The density dependence of the Luttinger parameter is explicitly found and interestingly it shows a non-monotonic behavior. Depending on the density, the static structure factor can be a smooth function of the momentum or might contain a peak of a finite or infinite height. Although no phase transitions are present in the system, we identify a number of physically different regimes, including an ideal Fermi gas, a Bose-gas, a super-Tonks-Girardeau regime, and a quasi-crystal.
