Dimensional Crossover and Cold Atomic Gases*

Properties of correlated quantum systems depend drastically on the dimension: the lower the dimension, the more difficult it is for particles to avoid each other, thereby reinforcing the effects of the interactions. For example, a three-dimensional superfluid has a superfluid order parameter, while in two dimensions the order parameter is zero and the superfluid undergoes a topological Berezinski-Kosterlitz-Thouless transition. In one dimension a system of quantum bosons behaves as a Tomonaga-Luttinger liquid (TLL) with quasi long-range order only at zero temperature, and fermionization of the bosons.

While in most systems the dimension is well fixed, it is now more and more possible to realize, in particular in cold atomic gases, systems which are intermediate between two integer dimensions. This if for example the case when there are many one-dimensional tubes coupled by tunnelling of particles. Such systems are intermediate between three- or two-dimensional systems and one-dimensional ones. This situation leads to a set of novel physical properties known as dimensional crossover.

I will review in this talk the properties due to a dimensional crossover between one dimensional interacting quantum systems (TTL) and higher dimensional ones [1]. I will in particular discuss the case of a two-dimensional bosonic gas subjected to a periodic potential in one direction that can slice it into independent tubes and show how the measurements of the correlation functions such as the single particle correlation function, measured in time of flight, can reveal the mixed dimensionality of the system [2,3].

I will also connect these theoretical studies [2,3] with recent experiments that directly measured the dimensional crossover in a gas of caesium atoms [4]. Among the unexpected properties of such systems is the possibility to cool by reducing the dimensionality of the systems, as well as to use the correlation functions as a very precise thermometer [5].