Introduction

Although proposed by Einstein [Ein24,Ein25] for an ideal quantum gas a long time ago Bose-Einstein condensation (BEC) almost remained only a mathematical artifact. After many years of intense experimental activity in 1995 BEC was observed in alkali vapours in a remarkable series of experiments [AEM+95,DMA+95]. Since that time there has been an explosion of experimental and theoretical interest worldwide in the study of dilute Bose gases. The Bose condensate, a macroscopically occupied quantum wave, exhibits peculiar properties and often is referred to as a new state of matter.

One of the directions where very important achievements were made in the years passed from the first realization of the BEC in gases, is the development techniques of cooling quantum gases to extremely low temperatures and of trapping methods allowing for the realization of low-dimensional geometries (for example, [GVL+01,SKC+01,GBM+01,MSKE03,TOH+04,SMS+04]). This combination leads to highly non-trivial effects, like the fermionization of bosons which may happen in one-dimensional quantum system. The progress in cooling methods has led to the possibility of observing both Bose and Fermi systems at temperatures much smaller than the degeneration temperature. The development of the techniques of diagnostics allows to get a quantitative description of the system under investigation: the size of the cloud, release energy, the momentum distribution, structure factor and frequencies of collective excitations are available in many experiments.

The most widely used approach for the description of quantum degenerate bosonic system is the mean-field Gross-Pitaevskii (GP) theory[Gro61,Pit61]. In this approach all particles are considered to be in the same quantum state described by the condensate wave function, which evolves in time according to the Gross-Pitaevskii equation. The mean-field approach has proven very useful as it is mathematically much easier to solve the equation for one particle in an effective field of other particles, than to solve the full many body problem. The GP approach holds when the depletion of the condensate is negligible or, more generally, when the correlation length is much larger than the interparticle distance. One-dimensional gases in a regime of strong quantum correlations where the above condition fails, have already been realized.

The problem of solving the many-body Schrödinger equation and finding multidimensional averages integrating out $3N$ degrees of freedom is very complicated. The Monte Carlo methods are indispensable tools in the calculation of multidimensional integrals (see, for example[Cep95,CB95,MFS95]) and have been shown to be highly useful in the investigation of quantum systems (see, for example, [ABCG02,GGMB04]). We are most of all interested in the quantum properties of the system at zero temperature. The diffusion Monte Carlo method is the best suited for this type of study.

Figure 1: An example of realization of a quasi one dimensional bosonic system (taken from [MSKE03]). Two counterpropagating laser beams create a tight two dimensional optical lattice. In the transversal direction the gas is in the ground state of the confining potential. Excitation of the next levels is highly suppressed due to the low temperature $k_BT/\hbar \omega _\perp <6\cdot 10^{-3}$ and low value of the available one-dimensional energy $\mu /\hbar \omega _\perp -1<0.1$.

The confining potentials (magnetic trapping, optical trapping, etc.) can be well described by harmonic potentials. If the frequencies of the confinement are equal in all three directions (i.e. the trap is spherical) the sample of the gas inside is three-dimensional. If, instead, the trap is made tighter in two directions, the shape of the gas cloud becomes elongated, and in the limit $\omega_\perp\gg\omega_z$ the system becomes effectively one-dimensional (see Fig. 1). The crossover of a trapped gas from three- to one- dimensional behaviour is highly interesting and we have studied it using a Quantum Monte Carlo method[AG02].

The reduced dimensionality enhances the effect of interactions and the properties of a one-dimensional system can be very different from the ones of a three-dimensional gas. The phenomenon of Bose condensation is absent in a one-dimensional homogeneous system. Furthermore, the behavior of repulsive bosons is very peculiar in one dimensional system: in the limit of low density the particles get completely reflected in the process of two-body collisions (limit of impenetrable particles) and the interaction between particles plays a role of an effective Pauli principle. In this Tonks-Girardeu limit fermionization of bose particles happens. The wave function of strongly interacting bosons can be mapped onto a wave function of ideal spinless fermions[Gir60]. The system of bosons acquires many fermionic properties: the energy, pair distribution function, static structure factor, etc. are the same. In the low-density regime beyond mean-field effects are important and they can not be accounted for by the Gross-Pitaevskii approach, which is valid instead in the opposite regime of large densities. In the regime of intermediate densities (recently realized in experiments[TOH+04]) both methods are not applicable. The system with contact repulsive interaction (Lieb-Liniger gas) is exactly solvable. Many of its properties are known exactly: the ground-state energy [LL63], value of the pair distribution function at zero distance[GS03b], long- and short-range expansions of the OBDM[OD03]. Still the complete description of correlation functions in the Lieb-Liniger gas (also value at zero of the three-body correlation function which was measured in the experiment[TOH+04]) is unknown. The DMC is well suited to study this problem[AG03].

In a one-dimensional system with attractive contact interactions a two-body bound state appears for any strength of the $\delta$-potential While the Pauli exclusion principle prohibits fermions from occupying a state with the same quantum numbers, bosons are free to populate the lowest state. An exact result for a system of bosons[McG64] shows that the ground state is a soliton-like state with large negative energy. In a system of two-component fermions two particles with different spin can form a bound state (particles with parallel spins can not interact with contact potential), but other particles (or pairs) have to stay apart. The exact solution[Gau67,KO75] shows formation of dimers which form a gas-like state. In the dilute regime the internal structure of the composite dimer can be neglected and the system behaves as a Tonks-Girardeau gas of bosons with twice the mass of the atoms[ABGP04].

For a gas of 1D bosons we propose to obtain a large attractive interaction in this special way: start with a gas of repulsive bosons, increase the strength of the interactions using the Feshbach resonance up to Tonks-Girardeau regime ($a_{1D}\to-0$) and then change the sign of the interaction ($a_{1D}\to +0$). The new state (``super-Tonks'') will have correlations which are even stronger than in the Tonks-Girardeau gas. The super-Tonks gas is a metastable state which has analogies with a gas of hard-rods of size $a_{1D}$. The super-Tonks is a metastable state. A very important question is to find out if the super-Tonks gas is stable and, thus, can be realized in an experiment. We use the variational Monte Carlo method to investigate this problem[ABCG04a]. Interaction effects in quasi-one-dimensional systems can be studied in experiments by exciting ``breathing'' mode oscillations. The local density approximation can be used to obtain the density profile of a trapped system of bosons or fermions. We solve the local density approximation for a quite general class of equations of state analytically. Using the sum rule approach we extract the oscillation frequencies numerically for all densities and analytically in the limits where the expansion of the equation of state is known.

A peculiar property of a low temperature system is the possibility of being superfluid. One of the most important predictions of Landau theory of superfluidity is the existence of a finite critical velocity. If a body moves in a superfluid at $T=0$ with velocity $V$ less then $v_c$, the motion is dissipationless. At $V>v_c$ a drag force arises because elementary excitations are created. Recently, existence of a critical velocity for the superfluid motion in a Bose-Einstein condensed gas was confirmed in various experiments. For example, at MIT a trapped condensate was stirred by a laser beam and the dissipated energy was measured[RKO+99,ORV+00]. According to Landau if one imagines to move a small body through the system and there is no normal part, no dissipation will happen if the speed is smaller than the speed of sound. We calculate the effect of a small impurity moving through a condensate which is described by the Gross-Pitaevskii equation[AP02]. We want to find an answer to a question which is rather complicated. We know that in the large density mean-field regime that the system should be superfluid. On the other side, in the Tonks-Girardeau regime the system is mapped to the fermions, which are not superfluid.

An important question concerns effective interactions in 1D, i.e. how the one-dimensional effective coupling constant is related to the three dimensional $s$-wave scattering length. A solution for the problem of two-body scattering on a pseudopotential in a waveguide was found by Olshanii[Ols98] and shows a resonant behavior in the regime $a_{3D}\sim a_\perp$ due to virtual excitations of transverse modes confinement levels. Since in normal experimental conditions $a_{3D}\ll a_\perp$, a resonant scattering in the vicinity of a Feshbach resonance should be used in order to enter this regime. In experiments a possible way to fulfill this condition is to make use of the Feshbach resonance. An important question is to prove the existence of the confinement induced resonance in a many body system. We consider a resonant scattering on a smooth attractive potential of very small range and use Fixed-Node Monte Carlo to study the problem of quasi-one-dimensional Bose gases with large scattering length[ABGG04b,ABGG04a].

The use of Feshbach resonance allows one to vary the interaction strength in a controlled way and tune the scattering length essentially to any arbitrary value. Recent experiments on two-component ultracold atomic Fermi gases near a Feshbach resonance have opened the possibility of investigating the crossover from a Bose-Einstein condensate (BEC) to a Bardeen-Cooper-Schrieffer (BCS) superfluid. For positive values of the $s$-wave scattering length $a_{3D}$, atoms with different spins are observed to pair into bound molecules which, at low enough temperature, form a Bose condensate [JBA+03,GRJ03,ZSS+03]. The molecular BEC state is adiabatically converted into an ultracold Fermi gas with $a<0$ and $k_F\vert a\vert\ll 1$ [BAR+04a,BKC+04], where standard BCS theory is expected to apply. In the crossover region the value of $\vert a_{3D}\vert$ can be orders of magnitude larger than the inverse Fermi wave vector $k_F^{-1}$ and one enters a new strongly-correlated regime known as unitary limit [OHG+02,BAR+04b,BKC+04]. In dilute systems, for which the effective range of the interaction $R_0$ is much smaller than the mean interparticle distance, $k_FR_0\ll 1$, the unitary regime is believed to be universal. In this regime, the only relevant energy scale should be given by the energy of the noninteracting Fermi gas. The unitary regime presents a challenge for many-body theoretical approaches because there is not any obvious small parameter to construct a well-posed theory. Quantum Monte Carlo techniques are the best suited tools for treating strongly-correlated systems. We use Fixed-Node Monte Carlo method to obtain for the first time the equation of state covering all regimes (BEC, unitary, BCS)[ABCG04b]. The equation of state can be tested in experiments by measuring the frequencies of collective oscillations. We also investigate the behavior of correlation functions.

The structure of the Dissertation is as follows.

In the Chapter 1 we introduce the analytical approaches and approximations used in the subsequent Chapters. Chapter 2 explains in details the Quantum Monte Carlo methods used in the study. In Chapter 3 we consider a system of bosons in an anisotropic trap and study the transition from a three dimensional behaviour to a quasi one dimensional one as the trap is made very elongated. We study the properties of a quasi-one-dimensional Bose gas with resonant scattering in Chapter 4. The system of $\delta$-interacting bosons in the case of repulsive interactions (Lieb-Liniger gas) is investigated in Chapter 5 and in the case of attractive interactions in Chapter 6. The motion of an impurity as a test for superfluidity is considered in Chapter 7. In the next two chapters we consider systems of two component fermions in a quasi one dimensional system (Chapter 8) and in a three-dimensional uniform system (Chapter 9). Conclusions are drawn in the last Chapter (Chapter 9.4).