Conclusions

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, Bethe ansatz, local density approximation, etc.) the main tool of the study is the Quantum Monte Carlo method in different modifications (variational MC, diffusion MC, fixed-node MC). 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 main novel results can be summarized as follows.

We present exact Quantum Monte Carlo results of the ground-state energy and structure of a Bose gas confined in highly anisotropic harmonic traps. Starting from a 3D Hamiltonian, where interparticle interactions are modeled by a hard-sphere or a soft-sphere potentials, we show that the system exhibits striking features due to particle correlations. By reducing the anisotropy parameter $\lambda$, while the number of particles $N$ and the ratio $a/a_\perp$ of scattering to transverse oscillator length are kept fixed, the system crosses from a regime where Gross-Pitaevskii mean-field theory applies to a regime which is well described by the 1D Lieb-Liniger equation of state in local density approximation. In the cross-over region both theories fail and one must resort to exact methods to account properly for both finite size effects and residual 3D effects. For very small values of $\lambda$ we find clear evidence, both in the energy per particle and in the longitudinal size of the cloud, of the fermionization of the system in the Tonks-Girardeau regime.

We use different methods for studying the resonant scattering of a Bose gas in a highly elongated trap, when the system enters a quasi one dimensional regime. We make a fully three dimensional calculation of the lowest-lying gas-like state of the many body system using a microscopic Fixed-Node Monte Carlo method. In order to prove the presence of the confined induced resonance predicted by Olshanii in a many-body system we make a full microscopic one-dimensional calculation for contact interactions with renormalized coupling constant $g_{1D}$. The resulting energies are in excellent agreement. This agreement proves that a properly chosen many-body 1D Hamiltonian describes well 3D Bose gases in the quasi-one dimensional regime. We consider the Lieb-Liniger and the hard-rod equation of state of a 1D system treated within the local density approximation, which is expected to be correct for large number of particles. Our detailed microscopic studies suggest that these LDA treatments provide a good description of quasi-1D Bose gases. In particular, we suggest a simple treatment of 1D systems with negative $g_{1D}$ using the hard-rod equation of state. We address the question of stability of an inhomogeneous gas in this regime utilizing a variational many-body framework. We find that the lowest-lying gas-like state is stable for negative coupling constants, up to a minimum critical value of $\vert g_{1D}\vert$. Our numerical results suggest that the stability condition can be expressed as $n_{1D} a_{1D} \simeq0.35$.

Properties of the Lieb-Liniger gas are investigated in details. We calculate for the first time the behaviour of correlation functions in a wide range of the characteristic parameter $na_{1D}$ covering Gross-Pitaevskii and Tonks-Girardeau regimes. We obtain the one-body density matrix $g_1(z)$ and pair distribution function $g_2(z)$ for all densities. In particular we investigate the nontrivial regime $na_{1D}\approx 1$ which is relevant for current experiments. We study the dependence of the value at zero of the three-body correlation function $g_3(0)$ on the gas parameter $na_{1D}$ and compare it with experimental results obtained at NIST[TOH+04]. We find agreement between theory and experiment. We extract the momentum distribution $n(k)$ and static structure factor $S(k)$ for all densities. We discuss how the presence of a harmonic trapping modifies the properties of the system. Using the Haldane approach for one-dimensional liquids we calculate the asymptotic behaviour of the one-body density matrix, density-density correlation function, dynamic form factor. In particular a direct comparison with the DMC calculation shows that the accuracy of the obtained coefficient of the one-body density matrix decay is better than $0.3\%$ in the whole range of densities.

We propose a novel technique of creating a metastable gas-like state of attractive bosons by crossing a confinement induced resonance. Such a gas has correlations even stronger than in the Tonks-Girardeau regime where the coupling constant is very large $g_{1D}\to\infty$. We calculate the equation of state in this ``super-Tonks'' regime using the Variational Monte Carlo method and estimate the critical density for the onset of instability against cluster formation. The static structure factor and one-body density matrix are calculated exactly within the hard-rod model, which provides the correct description of the system for small values of the gas parameter. For harmonically trapped systems we provide explicit predictions for the frequency of the lowest compressional mode.

We have studied the motion of an impurity through the condensate at zero temperature by solving the Gross-Pitaevskii equation in a perturbative way. We calculated the energy of a slow impurity. We find that the $V=0$ energy agrees with Bogoliubov theory, the velocity contribution can be written as $m^*V^2/2$, where the effective mass $m^*$ contributes to the mass of the normal component. We find that the motion at small velocities is dissipationless in one-, two-, and three- dimensional systems, although motion with velocities larger than the speed of sound leads to a non-zero drag force due to Cherenkov radiation of phonons. The expressions for the drag force are calculated. We used results for the dynamic form factor of the exact Lieb-Liniger theory to investigate the velocity dependence of the drag force in a 1D system. The form factor is calculated with the help of the Haldane method[Hal81]. The drag force exists for arbitrarily small velocity of motion, but is very small in the mean-field limit.

We considered a quasi-one-dimensional system of two component Fermi gas with contact potential between fermions of different spins. We have investigated the cross-over from weak to strong coupling of harmonically trapped gases with both repulsive and attractive effective interactions. The frequency of the lowest breathing mode, which can provide an experimental signature of the cross-over, is calculated. We predict the existence of a stable molecular Tonks-Girardeau gas in the strongly attractive regime. We obtain description of trapped one- and three- dimensional gas in the local density approximation for a perturbative equation of state. Obtained predictions for the frequencies of the lowest breathing mode are compared with numerical solutions.

We have carried out a detailed study of the equation of state of a Fermi gas in the BEC-BCS crossover using Fixed Node Monte Carlo techniques. In the BCS regime and in the unitary limit our results are in agreement with known perturbation expansions and with previous Fixed Node Green Function MC calculations [CCPS03,CPCS04]. In the BEC regime, in our many body calculation we recover the equation of state of a gas of composite bosons with repulsive effective interactions which are well described by the molecule-molecule scattering length $a_m=0.6a$ recently calculated in Ref. [PSS04].

The results obtained in this dissertation are relevant for present and future experiments. We make direct comparison the three-body loss rate of a 1D Bose system measured in experiments[TOH+04] finding good agreement. The equation of state obtained here for the BEC-BCS crossover in a two component Fermi gase can be used to determine frequencies of collective modes, which have been recently measured in experiments[BAR+04a,KHG+04,KTT04]. It is important to note that the methods of obtaining quasi-one-dimensional cigar-shaped systems have been developed considerably in the last years and it is expected that many more experiments on low-dimensional systems will appear soon. Another important point is that the strength of interactions can be tuned in a controlled way through the application of an external magnetic field in the proximity of a Feshbach resonance. Strengths of interaction in quasi-one-dimensional systems can be controlled by means of confinement induced resonance. This allows to hope that many new properties of low-dimensional quantum systems will be measured soon and compared to theoretical predictions.