Triptych of quantum matter in one dimension : Playful constructions in theoretical and computational physics

Abstract: In this compilation thesis, three different quantum mechanical systems are presented. All three have in common that they are one-dimensional quantum systems which, each in their own way, can be seen as analogous to other quantum systems. Chapter 2 is an introduction which takes on the ambitious task to guide the reader from standard quantum mechanical theory to the theory of condensed matter physics. In chapter 3 I present a theory for creating co- herent oscillations in the adiabatic eigenstates of time-periodic quantum systems. These are directly analogous to so-called Bloch oscillations which appear for electrons in periodical potentials which are subjected to a constant force. Chapter 4 concerns how phonon physics, in particular the Peierls instability, can be mimicked in systems of interacting ultacold quantum mechanical gases. In chapter 5, I present quantum mechanical circuits (circuit QED). We propose a circuit where the low-energy physics is analogous to a strongly interacting generalized spin chain, where the so-called superradiance directly correspond to a polarized Ising phase.