I formulate a mean-field theory of Bose-Einstein condensation based on number-conserving many-body theory. I use techniques commonly adopted in nuclear structure theories to derive the mean-field equations. I compare my formulation with the Bogoliubov approach and I emphasize the key role played by the zero-energy eigenmode when we contrast these two approaches. The differences with analogous previous formulation are also pointed out.
I formulate a mean-field theory of Bose-Einstein condensation based on number-conserving many-body theory. I use techniques commonly adopted in nuclear structure theories to derive the mean-field equations. I compare my formulation with the Bogoliubov approach and I emphasize the key role played by the zero-energy eigenmode when we contrast these two approaches. The differences with analogous previous formulation are also pointed out.
The theoretical description of trapped weakly-interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the ‘maze’ of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This Tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain ‘classes of approaches ’ identified by their physical content, rather than their particular mathematical implementation. This Tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions such as the occupation number representation of second quantization are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean-field, number-conserving and stochastic) according to their particular needs. In addition to its ‘training element’, we hope that this Tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
The theoretical description of trapped weakly interacting Bose-Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the 'maze' of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain 'classes of approaches' identified by their physical content, rather than their particular mathematical implementation. This tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions, such as the occupation number representation of second quantization, are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross-Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean field, number conserving and stochastic) according to their particular needs. In addition to its 'training element', we hope that this tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
The theoretical description of trapped weakly interacting Bose–Einstein condensates is characterized by a large number of seemingly very different approaches which have been developed over the course of time by researchers with very distinct backgrounds. Newcomers to this field, experimentalists and young researchers all face a considerable challenge in navigating through the 'maze' of abundant theoretical models, and simple correspondences between existing approaches are not always very transparent. This tutorial provides a generic introduction to such theories, in an attempt to single out common features and deficiencies of certain 'classes of approaches' identified by their physical content, rather than their particular mathematical implementation. This tutorial is structured in a manner accessible to a non-specialist with a good working knowledge of quantum mechanics. Although some familiarity with concepts of quantum field theory would be an advantage, key notions, such as the occupation number representation of second quantization, are nonetheless briefly reviewed. Following a general introduction, the complexity of models is gradually built up, starting from the basic zero-temperature formalism of the Gross–Pitaevskii equation. This structure enables readers to probe different levels of theoretical developments (mean field, number conserving and stochastic) according to their particular needs. In addition to its 'training element', we hope that this tutorial will prove useful to active researchers in this field, both in terms of the correspondences made between different theoretical models, and as a source of reference for existing and developing finite-temperature theoretical models.
In this tutorial we present an introduction to some theoretical methods of quantum field theory applied to the description of a trapped Bose–Einstein condensate. First of all, we give a brief account of the main characteristics of the phenomenon of condensation and present the many-body Hamiltonian of the system. We outline some of the most important approaches used in the characterization of a condensed Bose gas, including the mean-field theory and the Hartree–Fock–Bogoliubov method. Finally we illustrate the use of these techniques addressing some important issues in quantum atom optics. We characterize the quantum state of a Bose–Einstein condensate (BEC) at zero temperature. We also describe a process of Beliaev coupling between quasiparticles using a method that includes terms beyond the usual Bogoliubov approach.
We formulate a conserving gapless mean-field theory for Bose-Einstein condensates on the basis of a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with density n and s-wave scattering length a to clarify its fundamental thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature ΔTc from the ideal-gas result T0 is positive and given to the leading order by ΔTc=2.33an1/3T0, in agreement with a couple of previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.
Since the nowadays growing interest in Bose-Einstein condensates (BEC) due to the expanded experimental evidence on various atomic systems within optical lattices in weak and strong coupling regimes, its many-body quantum bases and equations are elementarily presented emphasizing on the order parameter, pseudo-potential and mean field approaches, while emphasizing the common places with the Density Functional Theory, like the integral normalization of density or the chemical potential, as the main vehicles of both theories of matter (Putz, 2011d, 2011e).