Theory of Degenerate Two-Dimensional Quantum Gases

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When we try to understand the world at very small length scales (e.g., 10,000 times smaller than the width of a human hair) or ultra-cold temperatures (e.g., nearly 273 degrees below zero Celsius), classical (Newtonian) physics reaches its limits, and we require a very different description of nature, known as quantum mechanics. Over the last two decades, experiments on ultra-cold atoms have allowed physicists an unprecedented opportunity to observe the bizarre world of quantum physics in a very controlled fashion. Typically, a gas of about 10,000-100,000 atoms (each atom is very small indeed) is cooled to ultra-cold temperatures, and trapped in a localized region of space. The atoms may then be exposed to additional spatial confinement (thereby lowering the dimensionality of the system), along with the possibility of tuning the way in which the atoms interact with each other. This amounts to having created in the laboratory, a tailor-made quantum system for physicists to study. Such systems are called quantum many-body systems, since they must be described by quantum mechanics, and consist of many particles. Cold atomic systems are inherently interesting, because they allow for a symbiotic, mutually inspiring dynamic between theory and experiment; that is, experiments help to verify theory, and theory provides the impetus for additional experiments. A quantum mechanical description of matter requires that we introduce another property to each atom, known as "spin". Spin is an internal degree of freedom (i.e., has nothing to do with the spatial properties of the atom) with no Newtonian analogue. The spin results in the atoms being classified as either fermions or bosons. Fermions are rather anti-social, meaning that they do not like to behave cooperatively. Bosons, on the other hand, prefer to behave in unison. The familiar laser is an excellent example of where the collective properties of bosons (in this case, photons) result in a coherent beam of light, which is not realizable for fermions owing to their uncooperative behaviour. My project is to investigate the quantum many-body problem through a theoretical investigation of the physical properties of low-dimensional ultra-cold atomic gases. Low-dimensional gases of fermions or bosons may have very different properties from their three-dimensional (3D) counterparts. The phenomenon known as Bose-Einstein condensation (BEC) is an archetypical example of where dimensionality has a profound influence on the collective properties of the quantum gas. In particular, a uniform 3D gas of bosons may have a BEC (a new state of matter in which all of the atoms may be described effectively as a single object), whereas a uniform 1D Bose gas is not permitted to have a BEC. Moreover, many of the theoretical tools used to address 3D systems are not applicable to lower dimensions, thereby requiring new formulations of the many-body problem which are not sensitive to the dimensionality of the system. Many of the paradigms for quantum computation, superconductivity and the physics of 2D sheets of graphene, rely on some of the special properties of low-dimensional quantum systems. In addition, the demand for electronic devices to be made smaller and smaller (i.e., the current carrying electrons must move in very small, restricted geometries) implies that new designs need to be considered, which necessitates a deep understanding of low-dimensional quantum systems. Therefore, my theoretical research in low-dimensional quantum systems will have an impact on the development of new technologies, new industries, and help stimulate future experimental and theoretical studies.