Abstract Ultracold Bose gases are many-body systems with well-defined particle interactions that may serve as models for interacting quantum fields. The impact of virtual excitations is studied in the spatial transition zone created by a soft confinement potential that separates a degenerate ideal gas from a dense quasi-condensate. We compute within Bogoliubov theory the contribution of quasi-particles to the surface energy at zero temperature.
Abstract Ultracold Bose gases are many-body systems with well-defined particle interactions that may serve as models for interacting quantum fields. The impact of virtual excitations is studied in the spatial transition zone created by a soft confinement potential that separates a degenerate ideal gas from a dense quasi-condensate. We compute within Bogoliubov theory the contribution of quasi-particles to the surface energy at zero temperature.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
In this thesis we present a detailed investigation of the role played by quantum and thermal fluctuations in ultracold Bose gases. We begin with a review of several important concepts and analytical tools within a functional integration formalism. We first focus on the so-called zero-range approximation for the interaction potential, by recovering the Bogoliubov results and the Landau two-fluid model from a field-theory perspective. In deriving the beyond-mean-field equation of state, we are going to show that a crucial point concerns the proper regularization of the divergent zero-point energy. Among the alternative approaches to investigate finite-temperatures Bose gases, we apply the kinetic theory to explain some recent results on the sound propagation in two-dimensional Bose gases. We then move to consider the eventual corrections to the thermodynamics of Bose gases due to the finite-range character of the interaction potential. The coupling constants of the finite-range model are related to measurable scattering parameters through an effective-field-theory approach. The role of finite-range corrections is considered not only in three spatial dimensions but also in systems with lower dimensionalities. Our analytical predictions are in good agreement with available Monte Carlo simulations and consistent with other theoretical frames, as the Lieb-Lininger theory for one-dimensional systems. In the third chapter, the relevance of fluctuations is investigated from an alternative point of view. Indeed, for a single-component Bose gas we have actually considered their effect as deviations of thermodynamic quantities from the mean-field and zero-range picture. In the case of collapsing Bose mixtures, we are going to show that zero-temperature fluctuations play a crucial stabilizing role against the collapse instability. Because of this peculiar mechanism, ultracold mixtures can display finite-density configurations also in free space. Inspired by recent experiments, we characterize this novel self-bound state by comparison with bright solitons, following a variational scheme. We also consider the case of binary mixtures where a coherent internal coupling is turned on. In the last chapter, we move to deal with dipolar condensates. In particular, we are interested in beyond-mean-field effects leading to the formation of inhomogeneous ground states. In order to provide a reliable answer to the open issue of superfluid properties of these structures, we present our recent numerical investigation on the phase diagram of dipolar bosons.
In this dissertation, we study the few-body ultracold Bose-Fermi mixture and quasi-1D scattering at ultracold temperature. Degenerate quantum gases have attracted enormous attentions during the past two decades. They have opened new platforms of quantum simulation, precision measurement and quantum chemistry. The scattering properties of degenerate quantum gases are the first things to study in order to gain insight into various novel phenomena at ultracold temperatures.\nTo address the role of quantum statistics at ultracold temperature, we study the spectrum and dynamics of a few-body Bose-Fermi mixtures. In particular, we focus the dynamical evolution of a few-body Bose-Fermi mixture and concentrate on its universal behavior at large inter-particle scattering length. We predict the molecule formation efficiency in many-body Bose-Fermi mixtures by mapping this critical observable in few-body calculations. We also propose that a the quantum beat experiment could be used to measure the energy of the lowest Efimov trimer at unitarity.\nTo address the role of dimensionality, we study confinement induced resonances and similar phenomena in general transverse confining potentials. Well-separated energy scales in different dimensions allow the creation of reduced dimensional systems at ultracold temperatures. We develop a general framework to regularize the low-energy quasi-1D scattering phase shift associated with a zero range interaction.\nLastly, we discuss future prospects for the study of an ultracold Bose-Fermi mixture, and scattering in reduced dimensional systems of a more general topology.
We consider an ultracold Bose gas in a half-open potential well, otherwise confined to a quasi-one-dimensional geometry. The Bogoliubov equations for its elementary excitations are solved in the continuous spectrum to explore the particle and energy content near the edge of the gas, beyond the often applied local density approximation. In particular, gradients in the condensate density enhance density-dominated excitations in the border region. We discuss excess (missing) particles and their energy by comparing to suitable reference solutions for quasi-homogeneous systems. The density profile near the edge shows no Friedel oscillations, but a dipolar feature from the spill-out of thermally excited particles. The calculations are performed in the grand-canonical ensemble and in the thermodynamic limit.
Considering the Casimir effect due to phononic excitations of a weakly interacting dilute Bose–Einstein condensate (BEC), we derive a renormalized expression for the zero-temperature Casimir energy of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.
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.