The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.
The refinement and the combination of laser cooling and evaporative cooling in dilute systems of alkali atoms lead to the achievement of Bose-Einstein condensation in 1995. Nowadays there is a fruitful interplay between theoretical and experimental results: in this context, the study of uniform Bose gases plays an important role in the comprehension of the static and dynamic properties of a bosonic system. Here we use quantum field theory, within the approach of functional integration, to study an homogeneous system of weakly-interacting bosonic cold atoms. After characterizing their thermodynamical properties, we derive - at a gaussian level - the number density and the superfluid density in three and in two spatial dimensions, expressed as functions of the condensate density and the temperature. Since the equations derived are valid for a generic interaction potential, we implement these formulas for bosons with a zero-range interaction, providing also an extension of these results with the inclusion of a finite-range interaction.
The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum fluctuations, which favour disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of its "marginal" behaviour. In an infinite uniform 2d fluid thermal fluctuations at any non-zero temperature are strong enough to destroy the fully ordered state associated with Bose--Einstein condensation, but are not strong enough to suppress superfluidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual "quasi-long-range" order at low temperatures leads to an interesting interplay between superfluidity and condensation in all experimentally relevant finite-size systems. In these notes we give an introduction to the physics of 2d Bose fluids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultra-cold atomic gases confined to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other fields of physics.
Abstract This volume introduces the basic concepts of Bose–Einstein condensation and superfluidity. It makes special reference to the physics of ultracold atomic gases; an area in which enormous experimental and theoretical progress has been achieved in the last twenty years. Various theoretical approaches to describing the physics of interacting bosons and of interacting Fermi gases, giving rise to bosonic pairs and hence to condensation, are discussed in detail, both in uniform and harmonically trapped configurations. Special focus is given to the comparison between theory and experiment, concerning various equilibrium, dynamic, thermodynamic, and superfluid properties of these novel systems. The volume also includes discussions of ultracold gases in dimensions, quantum mixtures, and long-range dipolar interactions.
This thesis is devoted to the rigorous study of dilute Bose gases with two-body and three-body interactions at zero temperature in two and three dimensions. Bose gases are many-body quantum systems composed of bosons, which notably do not obey the Pauli exclusion principle. As a consequence, at extremely low temperatures, a macroscopic fraction of the particles can occupy the same one-body quantum state. This phenomenon, known as Bose–Einstein condensation, was predicted in the 1920s by Bose and Einstein for noninteracting Bose gases, and first realised experimentally by the groups of Cornell and Wieman, and Ketterle, earning them the Nobel Prize for Physics in 2001. However, proving the occurrence of Bose–Einstein condensation in general interacting bosonic systems remains a major open problem and constitutes a central theme in this thesis. Another important property of bosonic matter is that it is generally more unstable than fermionic matter. In particular, for systems interacting via the Coulomb force, the Pauli exclusion principle is essential for explaining the stability of the second kind, which states that the energy and volume scale linearly with the number of particles. Nonetheless, bosonic systems with attractive but less singular interactions can also be stable, and the rigorous proof of this fact forms a second focus of this thesis. In three dimensions, we consider predominantly repulsive interactions. Firstly, we prove a leading order upper bound on the thermodynamic ground state energy of Bose gases with hard-core three-body interactions, which matches an existing lower bound. In addition, we obtain a simpler proof for integrable potentials. Secondly, we derive the leading order of the thermodynamic ground state energy of Bose gases with combined two-body and three-body interactions. More specifically, we show that, at the leading order, the ground state energy is entirely characterised by the scattering length of the two-body potential and the scattering hypervolume of the three-body potential. Thirdly, we study the time evolution of Bose–Einstein condensates with three-body interactions in the so-called Gross–Pitaevskii regime. We show that Bose–Einstein condensation is preserved under many-body evolution and that the condensate wavefunction evolves according to the quintic Gross–Pitaevskii equation. In particular, we show that the effective coupling constant is universal in the sense that it depends only on the scattering hypervolume of the interaction potential. As a by-product of our proof, we derive a stability of the second kind result for Bose gases with attractive two-body interactions and repulsive three-body interactions. In two dimensions, we consider trapped Bose gases with mean-field-type interactions, including the purely attractive case. We prove the stability of the second kind of the system and the convergence of the ground state energy per particle to that of a non-linear Schrödinger energy functional. As a consequence of the stability of the second kind, we also obtain Bose–Einstein condensation for the many-body ground states.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
The thermodynamic approach to density-functional theory (DFT) is used to derive a versatile theoretical framework for the treatment of finite-temperature (and in the limit, zero-temperature) Bose-Einstein condensates (BECs). The simplest application of this framework, using the overall density of bosons alone, would yield the DFT of Nunes [J. Phys. B 32, 4293 (1999)]. It is argued that a significant improvement in accuracy may be obtained by using additional density fields: the condensate amplitude and the anomalous density. Thus, two advanced schemes are suggested, one corresponding to a generalized two-fluid model of condensate systems, and another scheme, which explicitly accounts for anomalous density contributions and anomalous effective potentials. The latter reduces to the Hartree-Fock-Bogoliubov approach in the limit of weak interactions. For stronger interactions, a local-density approximation is suggested, but its implementation requires accurate data for the thermodynamic properties of uniform interacting BEC systems, including fictitious perturbed states of such systems. Provided that such data become available (e.g., from quantum Monte Carlo computation), DFT can be used to obtain high-accuracy theoretical results for the equilibrium states of BECs of various geometries and external potentials.
Shorttitle: The first experimental achievement of Bose Einstein condensation (BEC) in dilute alkali gases in 1995 has developed a renewed interest both theoretical and experimental. Lot of work has already been done over the last decade to study the different aspects of this condensation, mainly based on mean-field theories, hyperspherical approaches and quantum Monte Carlo methods. This thesis work presents an approximate many-body approach to investigate mainly the static (ground state) properties of the condensate, incorporating realistic inter-atomic interactions. Potential harmonic expansion method (PHEM), which includes two-body correlations only and disregards three or higher body correlations, has been adopted to analyze this condensate. This is ideally suited for BEC, which is required to be extremely dilute for its experimental realization. This treatment and analysis should be useful for the physics community especially working with weakly interacting ultra cold atoms, bosons and fermions as well.