Abstract This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its one-dimensional-like geometrical shape, and the deep one-dimensional regime where the motion is frozen in the radial direction. The cigar configuration is well suited to investigating the novel features exhibited by quantized vortices and solitons. In the deep one-dimensional regime Bose gases exhibits a very different behaviour with respect to usual three-dimensional BECs as a consequence of the quantum and thermal fluctuations of the phase. The Lieb–Liniger transition between the one-dimensional mean-field and the Tonks–Girardeau regimes is discussed, and important features concerning the frequency of the collective oscillations and the superfluid behaviour are pointed out. Finally, the chapter discusses some key features exhibited by one-dimensional Fermi gases.
Abstract This chapter discusses the behaviour of one-dimensional quantum gases. It discusses both the so-called cigar configurations, where the system is locally three-dimensional despite its one-dimensional-like geometrical shape, and the deep one-dimensional regime where the motion is frozen in the radial direction. The cigar configuration is well suited to investigating the novel features exhibited by quantized vortices and solitons. In the deep one-dimensional regime Bose gases exhibits a very different behaviour with respect to usual three-dimensional BECs as a consequence of the quantum and thermal fluctuations of the phase. The Lieb–Liniger transition between the one-dimensional mean-field and the Tonks–Girardeau regimes is discussed, and important features concerning the frequency of the collective oscillations and the superfluid behaviour are pointed out. Finally, the chapter discusses some key features exhibited by one-dimensional Fermi gases.
Abstract This chapter addresses the question of quantum gases in two dimensions. It investigates both the pancake and the deep two-dimensional regimes. In the former case the gas is locally three-dimensional along the axial direction. In contrast, in the deep two-dimensional regime the axial motion is frozen. The chapter investigates the new physical phenomena exhibited by these two-dimensional quantum gases exploring the behaviour of the equation of state and of the collective oscillations. It discusses the behaviour of a fast-rotating Bose gas in two-dimensions, and the structure of vortex lines during the transition between the two-dimensional Thomas–Fermi and the lowest Landau level regimes. In the deep two-dimensional regime quantum gases exhibit novel and important phenomena at finite temperatures, due to the absence of long-range order and the emergence of the Berezinskii–Kosterlitz–Thouless transition. The peculiar behaviour of the BCS–BEC transition in deeply two-dimensional Fermi gases is also discussed.
In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
We investigate the effects of dimensional reduction in Bose gases induced by a strong harmonic confinement in the transverse cylindric radial direction. By using a generalized Lieb-Liniger theory, based on a variational treatment of the transverse width of the Bose gas, we analyze the transition from a three-dimensional Bose-Einstein condensate to the one-dimensional Tonks-Girardeau gas. The sound velocity and the frequency of the lowest compressional mode give a clear signature of the regime involved. We also study the case of negative scattering length deriving the phase diagram of the Bose gas (uniform, single soliton, multisoliton, and collapsed) in toroidal confinement.
The loading of a Bose-Einstein condensate into a deep two-dimensional optical lattice provides a unique way to study one-dimensional Bose gases: the strong radial confinement freezes any motion in two dimensions, and for deep enough lattices, the system can be seen as an array of independent I D tubers. For our experimental parameters, the ID gas is predicted to be in an intermediate regime between the Tonks-Girardeau and the Thomas-Fermi regimes. We performed experiments showing that some long range phase coherence is present in this regime. We investigated correlation properties of these gases by studying their collective oscillations. In addition, we investigated the ID Mott transition by adiabatically loading the 1D gases into a 1D optical lattice.
The experimental achievement of Bose-Einstein condensation (BEC) in 1995 has aroused a large interest among the community of the physics of ultracold atomic gases.Recently new topics have attracted the attention of this very active field of research , in particular we could mention low dimensional systems, atoms in lattices, and classical and Fermi gases.As for the case of BEC, the study of the dynamical aspects of these systems provides a very valuable information.In particular time-of-flight experiments and measurements of collective excitations have been successfully realized providing a clear signature of the BEC and the regime of interaction.Following this line, in this thesis, we want to study some dynamical aspects of these new systems nowadays avaliable due to the remarkable development of the experimental techniques.In the first chapter we present the state of the art of the field, providing the historical path and the motivation for the work developed in the next chapters.In the second chapter, we discuss some general concepts in order to introduce the formalism and notation employed throughout the thesis, as well as those topics analyzed in more detail in the further chapters.In chapter three, we explain some general techniques which are useful to deal with one dimensional systems, and analyze in detail the expansion of a one dimensional Bose gas in a guide, which differs significantly from the expansion of a BEC.In particular we are able to predict the range of parameters for which the self-similarity of the expansion is violated.Chapter four is devoted to the physics of atoms in a periodic potential.We start, as in the previous chapter, introducing some general concepts and techniques employed further.First we discuss the physics of the superfluid case in a trap, showing that under certain conditions a three dimensional gas exhibits one dimensional properties.viii In particular, static properties and the lowest collective excitations differ significantly from those of a BEC.We extend the analysis to the insulating phase showing that correlation properties depend on the on-site one-dimensional regime.These correlations can be measured in current time of flight experiments.In the fifth chapter we analyze the dynamics of classical and Fermi gases.We show in detail how expansion and collective excitations depend on the interaction regime and the trap anisotropy.In particular, the expansion can be used to distinguish between a normal Fermi gas and a superfluid one.We summarize in the last chapter the new results presented in this thesis.