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arXiv:1402.0706

Topics in the Mathematical Physics of Cold Bose Gases

Jakob Yngvason

Published Feb 4, 2014• 0 citations
arXivHTMLPDF

Abstract

In these notes of six lectures on selected topics in the theory of cold, dilute Bose gases, presented at the 5th Warsaw School of Statistical Physics in June 2013, the following topics are discussed: 1) The concept of BEC, 2) the ground state energy of a dilute Bose gas with short range interactions, 3) Gross-Pitaevskii theory and BEC in trapped gases, 4) Bose gases in rotating traps and quantized vortices, and 5) strongly correlated phases in the lowest Landau level generated by rapid rotation.

DOI
10.48550/arxiv.1402.0706

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In these notes of six lectures on selected topics in the theory of cold, dilute Bose gases, presented at the 5th Warsaw School of Statistical Physics in June 2013, the following topics are discussed: 1) The concept of BEC, 2) the ground state energy of a dilute Bose gas with short range interactions, 3) Gross-Pitaevskii theory and BEC in trapped gases, 4) Bose gases in rotating traps and quantized vortices, and 5) strongly correlated phases in the lowest Landau level generated by rapid rotation.
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We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes a discussion of Bose–Einstein condensation, the excitation spectrum for trapped gases and its relation to superfluidity, as well as the appearance of quantized vortices in rotating systems. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schrödinger equation.
The material presented here consists of lecture notes written for a five two-hour sections course given during the 2004 edition of the Brazilian School on Statistical Mechanics.The topics covered are 1.Bose-Einstein condensation of trapped ideal gases; 2. Effective two-body interactions and exact results in the Gross-Pitaevski limit; 3. Atomic structure effects, including "Feshbach resonances" and hybrid condensates; 4. Elementary excitations in the Bogoliubov approximation and beyond; and 5. Splitt traps, interference phenomena and condensate arrays.
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.
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