Some topics in the theory of ultracold atomic gases

Abstract

This thesis deals with a variety of topics that are relevant for the theory of ultracold atoms, with a focus on the many interaction regimes that can be obtained in these systems. The topics are presented in increasing order of complexity with regards to these interaction regimes. In the first chapter we are prompted by an experiment on a realization of an interacting Aubry-Andr´e Hamiltonian driven by a periodic modulation, where a localisation-delocalisation transition is observed. We will model an analogue non interacting system and show that it reproduces an equivalent phase diagram. Moreover we are able to provide a physical explanation for the critical amplitude and the critical frequency for the delocalisation transition. Next, we will consider a model of N atoms which are non interacting with the addition of a single light impurity that interacts with them. We will propose a simplified model based on the Born-Oppenheimer approximation and a polaron-like picture through which we are able to estimate the energy of the system and to address the question of the
existence of stable clusters bound by an impurity in the large N limit. In the rest of the thesis the pivot will be on the Andreev-Bashkin effect which describes the drag that each component of a mixture of two superfluids exerts on the other, as a result of their mutual interactions. We will first propose a microscopic theory based on linear response theory that describes the drag, and derive its implications on the nature of the excited states in a superfluid mixture. Then we will compute the effect of the drag on the spin speed of sound and the spin dipole mode which can in principle observed in experiments. Analytical results for the case of a weakly interacting mixture are presented as a benchmark for our methods. Finally we will focus on the Andreev-Bashkin effect in a Bose-Hubbard Hamiltonian in a one dimensional ring lattice. We will show that the effect is enhanced for attractive intraspecies interactions, more so close to the transition to paired superfluidity. A discussion on the correction brought by the drag in the low energy Luttinger theory for the model is also presented.

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Identifiers

ORCID for Donato Romito: orcid.org/0000-0001-6368-9675

ORCID for Carlos Lobo: orcid.org/0000-0001-7060-3905

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