Theory of ultra-cold, two-dimensional trapped quantum gases

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There are two kinds of particles in nature: bosons and fermions.  Familiar examples of bosons and fermions are photons and electrons, respectively. However, even more complicated structures, such as different isotopes of Helium, are restricted to being either fermions or bosons. At ultra-low temperatures (e.g., one-one-millionth of a degree above absolute zero), the rules of quantum mechanics, as opposed to classical mechanics, tell us that bosons and fermions behave in very different ways.  In particular, very near absolute zero all bosons prefer to be in the same "quantum state''; that is, bosons are conformists and like to possess identical physical properties.  Fermions, on the other hand, are forbidden from occupying the same quantum state.  As a result, when large numbers of fermions or bosons are cooled to very low temperatures, the collective physical properties they exhibit can be dramatically different.     We are specifically interested in investigating ultra-cold gases of bosons and fermions when they are confined to effectively two dimensions. In a simplified picture, one can imagine this scenario by taking a box containing a gas, and then squeezing opposite sides of the container until the gas inside can only move freely in two independent directions.  These so-called two-dimensional quantum gases turn out to behave in ways that are simply not seen in three dimensions.  Understanding why the physics of these two-dimensional systems are so different from their three-dimensional counterparts is an important and challenging problem.   Using both traditional "pen-and-paper'' and sophisticated computational techniques, we will investigate the physical properties of these low-dimensional quantum gases. Some of the questions we will address are: "Can a gas of fermions ever behave like a gas of bosons?", "Does a two-dimensional gas of bosons always form a Bose-Einstein condensate?", and "What is the connection between superfluidity and the Bose-Einstein condensate?".   The answer to these questions, among others, will lead to important insights in diverse areas of physics, such as atom lasers, quantum computation, and the theory of superfluids.