Polarons in Ultracold Atomic Gases

The dynamics of an impurity coupled to a quantum system represents a conceptually simple, yet highly non-trivial physical system. For instance, an electron moving through an ionic lattice will locally distort the lattice (due to Coulomb repulsion). The combined system of the impurity plus the local distortion is conventionally referred to as a polaron. The polaron acts like an isolated particle in many ways, and has a well-defined energy and mass that can differ considerably from that of the isolated impurity.

Polarons play essential roles in a variety of Condensed Matter systems including superconductors, Kondo systems, and colossal magnetoresitance materials. Additionally, at the theoretical level, polarons are interesting in
their own right. Ror instance, one of the first non-trivial application of Feynman's path integral focused on computing the mass of a polaron.

Despite years of investigation dating back to the early work of Frohlich and Feynmann, many aspects of polarons are still not understood. For this reason, in the past five years, a large amount of experimental work employing gases of ultra-cold atoms focusing on polarons has been carried out. The ultra-cold systems offer a number of distinct advantages over their solid-state material counterparts. In particular, with ultra-cold atoms, clean realisations of minimal models, thought to describe solid-state polarons, can be achieved. Additionally, due to the high degree of tunability (using, for instance, Feshbach resonances) novel regimes of polaron physics can be explored with ultra cold gases.

This PhD project will investigate polarons in mixtures of ultra-cold gases. It has recently been shown that when the coupling between the atomic species is weak, the system can be described by the so-called Frohlich Hamiltonian [see, for instance, New Journal of Physics, 19, 103035, 2017]. The Frohlich Hamiltonian is a canonical model in condensed matter physics, which describes the coupling between particles and phonons. It is the
starting point for describing standard superconductors. However, when the coupling between the gases is not weak, the Frohlich model ceases to provide an accurate description of the system.

In this project, we will use an alternative starting point - the full microscopic Hamiltonian of the coupled system. Many of the basic relations for polarons (like their mass and energy) are phrased in the context of the Frohlich model. Therefore, our first task will be to phrase these in a more general context. Next, we will we will employ the so-called Gross-Pitaevskii mean-field theory to this system.

This mean-field theory captures the non-linearities that are absent in the Frohlich model. However, the Gross-Pitaevskii, as it is a mean-field theory, does not provide a full account of the quantum nature of the polaron. To capture such quantum effects, and to assess their importance, we will employ the so-called truncated Wigner expansion.

While recent experimental work utilising ultra-cold gases has shown excellent agreement with theory in the weak-coupling regime, there are major outstanding puzzles in the strong-coupling regime For instance, the experimentally measured mass of the polaron in the strong-coupled regime [see, for instance, PRA 85, 023623 (2012)] shows substantial disagreement with theoretical results to date. A major goal of this PhD project is to solve this puzzle.