Lectures W. Zwerger 2021

The four Lectures are intended to provide an introduction to the basic theoretical concepts which underly the physics of ultracold Bose gases. Their fundamental properties are discussed along with some key experiments, covering both classic topics and recent developments like supersolids in dipolar gases.

The first Lecture deals with Bose-Einstein Condensation (BEC) and Superfluidity in gases and liquids. It starts with the direct observation of long-range phase coherence in a trapped gas and discusses why the ground state of any fluid phase of Bosons always exhibits BEC. Gaseous and liquid states turn out to be separated by a quantum tricritical point which appears at vanishing scattering length. In its vicinity, self-bound liquid droplets of Bosons are stabilized by three-body repulsion. The associated unbinding of N-body bound states at a geometric sequence of scattering lengths extends the Efimov effect at N=3 to arbitrary N.

The second Lecture discusses the requirements for realizing a supersolid phase where BEC coexists with a mass density wave. Based on the Leggett bound on the superfluid fraction in phases with a periodic density modulation and the well understood example of superfluidity in optical lattices, it is shown that supersolids require a finite density of defects in their ground state. Such a situation is generically present in the density wave along the axial direction of dipolar gases in cigar – shaped traps which appears for dipolar lengths exceeding the short range scattering length. The emerging supersolid is a superfluid version of a smectic liquid crystal. It exhibits a novel low energy mode due to wave like propagation of defects, as predicted by Andreev and Lifshitz.

The third Lecture focusses on universal properties of ultracold gases probed at short distance which are a consequence of the fact that the effective range of interactions can be taken to zero. The resulting set of exact relations between thermodynamic properties and the short distance behavior of various correlation functions due to Tan provide an example for the operator product expansion originally developed in statistical and high-energy physics. In the context of ultracold gases, these methods allow to understand the negative line shift observed in Bragg scattering at large momentum which arises from processes where the momentum is transferred to two or more atoms simultaneously.

The fourth Lecture is devoted to the consequences of scale and conformal invariance in ultracold gases. Following a brief discussion of elementary examples like electrodynamics, it is shown that scale and conformal invariance appears for ultracold gases in 1D and 3D with unitarity limited scattering and generically in 2D. The additional symmetries give rise to universal features in both equilibrium properties and in dynamics. An example is the observation of breathers in 2D Bose gases at Collège de France, whose theoretical understanding is still incomplete. Finally, the implications of scale and conformal invariance for quantum limited transport coefficients like viscosity are discussed briefly.

Basic Concepts and some current Directions in Ultracold Gases