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 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 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
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) (gas parameter) where is the density of particles
and is the -wave scattering length,
b)
is the concentration of impurities
with a uniform random distribution,
c) where is the particle-impurity -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 becomes significantly smaller than the
condensate fraction .
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 in the
``weak'' disorder regime
is described correctly
by the prediction of the Bogoliubov model. We study the dependence
of the superfluid fraction and condensate fraction
on the density and disorder parameters and
. We find that in limit of dilute systems and weak disorder
both and are in agreement with analytical
predictions. The existence of scaling in , as
predicted by Bogoliubov model, is checked and is shown to be valid
over a large range of . The use of the DMC method enables us to
investigate the regime of strong disorder. At low density and large
values of 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 . We show that the
superfluid-insulator quantum transition is absent within our model
of non-overlapping impurities.
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