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

自然界中有两种粒子：玻色子和费米子。 我们熟悉的玻色子和费米子的例子分别是光子和电子。然而，甚至更复杂的结构，如氦的不同同位素，都被限制为费米子或玻色子。在超低温下（例如，相对于经典力学，量子力学的规则告诉我们玻色子和费米子的行为方式非常不同。 特别是，在非常接近绝对零度的情况下，所有玻色子都倾向于处于相同的“量子态”;也就是说，玻色子是一致的，并且喜欢拥有相同的物理性质。 另一方面，费米子被禁止占据相同的量子态。 因此，当大量的费米子或玻色子被冷却到非常低的温度时，它们所表现出的集体物理性质可能会显着不同。 我们对研究玻色子和费米子的超冷气体（当它们被有效地限制在二维时）特别感兴趣。在一个简化的图片中，人们可以想象这样的场景：拿一个装有气体的盒子，然后挤压容器的两侧，直到里面的气体只能在两个独立的方向上自由移动。 这些所谓的二维量子气体的行为方式在三维空间中是看不到的。 理解为什么这些二维系统的物理与三维系统的物理如此不同是一个重要而具有挑战性的问题。 使用传统的“纸笔”和复杂的计算技术，我们将研究这些低维量子气体的物理性质。我们将要解决的一些问题是：“费米子气体的行为是否与玻色子气体相似？一个二维玻色子气体总是形成玻色-爱因斯坦凝聚吗？超流性和玻色-爱因斯坦凝聚之间的联系是什么？". 这些问题的答案将在物理学的不同领域产生重要的见解，如原子激光器，量子计算和超流体理论。

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.