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Introduction

Although proposed by Einstein [1,2] for an ideal quantum gas a long time ago (1924) Bose-Einstein condensation (BEC) remained only as a mathematical artifact until London ``rediscovered'' it in 1938 to explain the superfluidity of liquid $^4$He [3]. Recently (1995), after many years of struggle, BEC was observed in alkali vapors in a remarkable series of experiments [4,5]. Since that time there has been an explosion of experimental and theoretical interest worldwide in the study of dilute Bose gases (for a review see [6]). In the last years great attention has been also devoted to the investigation of disordered Bose systems. The experimental realizations of these systems are liquid $^4$He adsorbed in various types of porous media such as vycor and aerogel. These systems exhibit many interesting properties, which have not yet been fully understood theoretically, such as the suppression of superfluidity [7], a rich variety of elementary excitations [8] and a critical behavior near the phase transition different from the bulk [9]. In this Thesis we study a Bose gas in the presence of quenched impurities. This model provides a reasonable description of liquid $^4$He adsorbed in porous media and can become relevant for Bose condensed gases in the presence of heavy impurities. At zero temperature the system is described by the following parameters:
a) $na^3$ (gas parameter) where $n$ is the density of particles $na^3$ and $a$ is the $s$-wave scattering length,
b) $\chi=n^{imp}/n$ is the concentration of impurities with a uniform random distribution,
c) $b/a$ where $b$ is the particle-impurity $s$-wave scattering length In the first part of the Thesis we investigate the dilute Bose gas by treating the random external potential as a perturbation. In this regime one can work out analytically, within the Bogoliubov model, the effect of disorder on the ground-state energy, superfluid behavior and condensate fraction. In the second part of the Thesis we approach the problem by resorting to the Diffusion Monte Carlo (DMC) method. This numerical method solves exactly the many-body Schrödinger equation for the ground-state of a system of bosons. This method is used for the investigation of the weak disorder regime and results of the simulations agree with the predictions of the Bogoliubov model. Also the DMC method is well suited to study the regime of strong disorder. In this regime we investigate the relation between superfluid behavior and Bose-Einstein condensation. At low densities, we find that the superfluid and condensate components of the system are equally suppressed by the disorder. However, for the very large concentration of impurities, we find that the superfluid fraction $\rho _s/\rho $ becomes significantly smaller than the condensate fraction $N_0/N$. The structure of this Thesis is as follows: In the first chapter the mean-field Gross-Pitaevskii theory and the beyond mean-field Bogoliubov theory of the dilute Bose gas are briefly reviewed. The Gross-Pitaevskii equation for the order parameter is derived and applied to the calculatation of the ground-state energy of the system. The elementary excitation energies are obtained by considering the small oscillations of the order parameter around the equilibrium solution. Beyond mean-field approximation we discuss the Bogoliubov effective Hamiltonian of a dilute Bose gas and calculate within this model the excitation spectrum and corrections to the the ground-state energy arising from quantum fluctuations. The results for the fraction of noncondensed particles and the one-body density matrix at zero temperature are also discussed. In the second chapter we discuss the theory of a dilute Bose gas in the presence of disorder. Within the Bogoliubov model we study the effects of the weak external random potential, modeled by the uniform random distribution of quenched impurities. The corrections to the ground-state energy and the condensate depletion due to the external random potential are calculated, as well as the behavior of the one-body density matrix. The second part of this chapter is devoted to the microscopic definition of the superfluid density. By using the Bogoliubov model we investigate the effect of the external random field on the superfluid density. The same result is also obtained in a new alternative way which makes use of the Gross-Pitaevskii equation. Chapter Three is devoted to the Quantum Monte Carlo method. The Diffusion Monte Carlo technique is briefly described and its main features are discussed. We also discuss the implementation of the parallel version of the algorithm. The Diffusion Monte Carlo (DMC) and Variational Monte Carlo (VMC) methods are applied to a hard-sphere homogeneous Bose gas, and a specific trial wave-function is constructed and tested. The techniques of calculating the ground-state energy and the one-body density matrix are presented. A formula for the calculation of the superfluid density within the DMC algorithm is derived and proved to be unbiased by the trial wavefunction. All types of systematic errors present in the DMC algorithm applied to the hard-sphere model are investigated. In the last chapter we apply the DMC method to investigate a Bose gas in the presence of hard-sphere quenched impurities. We show that the ground-state energy of a dilute system $na^3 \ll 1$ in the ``weak'' disorder regime $\chi(b/a)^2 \ll
1$ is described correctly by the prediction of the Bogoliubov model. We study the dependence of the superfluid fraction $\rho _s/\rho $ and condensate fraction $N_0/N$ on the density $na^3$ and disorder parameters $\chi $ and $b/a$. We find that in limit of dilute systems and weak disorder both $\rho _s/\rho $ and $N_0/N$ are in agreement with analytical predictions. The existence of scaling in $R = \chi (b/a)^2$, as predicted by Bogoliubov model, is checked and is shown to be valid over a large range of $R$. The use of the DMC method enables us to investigate the regime of strong disorder. At low density and large values of $R$ we find that the system enters a regime where the superfluid density is strongly suppressed, whereas the condensate fraction is still large. The space dependence of the one-body density matrix is calculated and is shown to agree with analytical predictions at small densities $na^3$. We show that the superfluid-insulator quantum transition is absent within our model of non-overlapping impurities.
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Next: Dilute Bose gas Up: Thesis Previous: Contents   Contents