The monograph is devoted to the main aspects of theoretical description of quantum many-body systems with Bose-Einstein or Fermi-Dirac statistics. We provide a review of modern methods of theoretical research, both analytical and numerical, as well as experimental realizations of unique effects and phenomena in quantum systems. The main attention is given to the theoretical study of exotic states in degenerate gases of atoms and quasiparticles. We develop a methology to accurately determine chemical potentials and other thermodynamic characteristics of ideal Bose and Fermi gases. These allow us to describe the ultraslow-light phenomenon in ultracold atomic gases in the framework of the microscopic approach that accounts for the internal degrees of freedom of bound states. The formalism relies on the linear response theory applied for the case of weak perturbation by the external elecromagnetic field. We also study Bose-Einstein condensation of photons in ideal atomic gases in different regimes of degeneracy of atomic and photonic components. A special attention is given to the Bogoliubov method of quasiaverages and related theory of a weakly interacting Bose gas with condensate. We extend the Bogoliubov theory to examine the spinor condensate and its magnetic states, as well as the condensate of heteronuclear Fermi-Fermi molecules. We also study a spatially periodic state of Bose-Einstein condensate, which can serve as a model for supersolid state of matter and develop the hydrodynamic theory of the latter. We outline the main statements of the Fermi liquid theory including the Pomeranchuk stability conditions of the normal state. It is shown that the violation of stability conditions results in certain phase transitions associated with the breaking of translational symmetry and rotational symmetry in the momentum space. We review a progress in the theoretical description of ultracold Fermi gases consisting of neutral atoms in the presence of the optical lattice spatial modulation. In case of multicomponent systems, a number of unique magnetically-ordered states are theoretically studied and predicted. We also develop a theory of highly dispersive spinful excitons in cobaltite compounds that explains certain unusual magnetic properties and effects in the resonant inelastic X-ray scattering on these materials.
The monograph is devoted to the main aspects of theoretical description of quantum many-body systems with Bose-Einstein or Fermi-Dirac statistics. We provide a review of modern methods of theoretical research, both analytical and numerical, as well as experimental realizations of unique effects and phenomena in quantum systems. The main attention is given to the theoretical study of exotic states in degenerate gases of atoms and quasiparticles. We develop a methology to accurately determine chemical potentials and other thermodynamic characteristics of ideal Bose and Fermi gases. These allow us to describe the ultraslow-light phenomenon in ultracold atomic gases in the framework of the microscopic approach that accounts for the internal degrees of freedom of bound states. The formalism relies on the linear response theory applied for the case of weak perturbation by the external elecromagnetic field. We also study Bose-Einstein condensation of photons in ideal atomic gases in different regimes of degeneracy of atomic and photonic components. A special attention is given to the Bogoliubov method of quasiaverages and related theory of a weakly interacting Bose gas with condensate. We extend the Bogoliubov theory to examine the spinor condensate and its magnetic states, as well as the condensate of heteronuclear Fermi-Fermi molecules. We also study a spatially periodic state of Bose-Einstein condensate, which can serve as a model for supersolid state of matter and develop the hydrodynamic theory of the latter. We outline the main statements of the Fermi liquid theory including the Pomeranchuk stability conditions of the normal state. It is shown that the violation of stability conditions results in certain phase transitions associated with the breaking of translational symmetry and rotational symmetry in the momentum space. We review a progress in the theoretical description of ultracold Fermi gases consisting of neutral atoms in the presence of the optical lattice spatial modulation. In case of multicomponent systems, a number of unique magnetically-ordered states are theoretically studied and predicted. We also develop a theory of highly dispersive spinful excitons in cobaltite compounds that explains certain unusual magnetic properties and effects in the resonant inelastic X-ray scattering on these materials.
Motivated by rapid experimental progress in the fields of ultracold atoms and quantum optics, I present a series of theoretical studies which explore collective phenomena in quantum gases of atoms and photons. In Chapter 1, I highlight the major developments in the research field and identify the overarching themes and motivations. I also provide a roadmap for the rest of the thesis and summarize the main results. The remaining eight chapters contain original studies, organized along three broad motifs. In Chapters 2 through 5, I investigate how the nature of collective excitations and quasiparticles can be explored in modern experiments. More specifically, I model the dynamics of a spin impurity in a Bose lattice gas, develop a protocol for observing fractionalized excitations or anyons in an optical cavity, and characterize the collective dynamics of Bogoliubov quasiparticles and domain walls in a Fermi superfluid. In Chapters 6 and 7, I examine unconventional superfluid phases in spin-imbalanced Fermi gases. In particular, I propose a novel technique for engineering the long-sought-after Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase and study the relative stability of exotic phases across a dimensional crossover. Finally, Chapters 8 and 9 are devoted to studies of kinetics in out-of-equilibrium systems. I model the formation of a Bose-Einstein condensate in a dimple trap and characterize the approach to thermal equilibrium in quasi-one-dimensional geometries.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the s-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
The main topic of these lectures is the physics of quantum degenerate atomic Fermi gases both under uniform confinement and in harmonic traps. To begin with, a general introduction on ideal Fermi gases is proposed. This "academic" limit is relevant in experiments where ultracold fermionic atoms are all polarized in a single spin state: the low atom momenta together with the fermionic nature of the atoms forbid atomic s-wave scattering and the atoms can be considered as free. Interactions are then introduced for the case of a gas with two spin components. Interactions can be tuned via Feshbach resonances and control the microscopic features of the superfluid phase, ranging from the BCS limit (formation of Cooper pairs) to the BEC limit (Bose-Einstein condensation of dimers). Various theoretical approaches for the description of the BCS-BEC crossover and their predictions in comparison with the experimental results are discussed.
Recent theoretical and experimental progress on studying one-dimensional systems of bosonic, fermionic, and Bose-Fermi mixtures of a few ultracold atoms confined in traps is reviewed in the broad context of mesoscopic quantum physics. We pay special attention to limiting cases of very strong or very weak interactions and transitions between them. For bosonic mixtures, we describe the developments in systems of three and four atoms as well as different extensions to larger numbers of particles. We also briefly review progress in the case of spinor Bose gases of a few atoms. For fermionic mixtures, we discuss a special role of spin and present a detailed discussion of the two- and three-atom cases. We discuss the advantages and disadvantages of different computation methods applied to systems with intermediate interactions. In the case of very strong repulsion, close to the infinite limit, we discuss approaches based on effective spin chain descriptions. We also report on recent studies on higher-spin mixtures and inter-component attractive forces. For both statistics, we pay particular attention to impurity problems and mass imbalance cases. Finally, we describe the recent advances on trapped Bose-Fermi mixtures, which allow for a theoretical combination of previous concepts, well illustrating the importance of quantum statistics and inter-particle interactions. Lastly, we report on fundamental questions related to the subject which we believe will inspire further theoretical developments and experimental verification.
In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field B. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field Bc, asymptotic values of n0 (relative atom number in mF=0 state) at large B and the phase diagram of n0. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of n0. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature" Ts and the ``kinetic temperature" Tk with Ts≪Tk, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.
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.