With the objective of simulating the physical behavior of electrons in a dynamic background, we investigate a cold atomic Bose-Fermi mixture confined in an optical lattice potential solely affecting the bosons. The bosons, residing in the deep superfluid regime, inherit the periodicity of the optical lattice, subsequently serving as a dynamic potential for the polarized fermions. Owing to the atom-phonon interaction between the fermions and the condensate, the coupled system exhibits a Berezinskii-Kosterlitz-Thouless transition from a Luttinger liquid to a Peierls phase. However, under sufficiently strong Bose-Fermi interaction, the Peierls phase loses stability, leading to either a collapsed or a separated phase. We find that the primary function of the optical lattice is to stabilize the Peierls phase. Furthermore, the presence of a confining harmonic trap induces a diverse physical behavior, surpassing what is observed for either bosons or fermions individually trapped. Notably, under attractive Bose-Fermi interaction, the insulating phase may adopt a fermionic wedding-cake-like configuration, reflecting the dynamic nature of the underlying lattice potential. Conversely, for repulsive interaction, the trap destabilizes the Peierls phase, causing the two species to separate.

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. 

We investigate a one-dimensional Rabi-Hubbard type of model, arranged such that a quantum dot is sandwiched between every cavity. The role of the quantum dot is twofold, to transmit photons between neighboring cavities and simultaneously act as an effective photon nonlinearity. We consider three-level quantum dots in the A configuration, where the left and right leg couple exclusively to the left or right cavity. This noncommuting interaction leads to two highly entangled incompressible phases, separated by a second-order quantum phase transition; the degrees of freedom of the quantum dots can be viewed as a dynamical lattice for the photons which spontaneously breaks Z(2) symmetry due to a bosonic Peierls instability, leading to a phase with dimerized order. Additionally, we find a normal insulating phase and a superfluid phase that acts as a quantum many-body superradiant phase. In the superradiant phase, a Z(2) symmetry is broken and the phase transition falls within the universality class of the transverse-field Ising model. Finally, we show that the model can be interpreted as a Z(2) lattice gauge theory in the absence of a dipolar field on the lower qutrit levels.

This licentiate thesis concerns topics in non-interacting and interacting quantum physics in one dimension. We present the notions of Wannier functions and tight-binding models. Quantum walks are discussed, quantum mechanical analogues to random walks. We demonstrate the ideas of Bloch oscillation and super-Bloch oscillation - revivals of quantum states for particles in a periodic lattice subject to a constant force. Next, the Rabi model of light-matter interaction is derived. The concept of quantum phase transitions is presented for the Dicke model of superradiance. The idea of adiabatic elimination is used to highlight the connectedness of the Dicke model. Finally, we present a one-dimensional interacting system of resonators and artificial atoms that could be built as a superconducting circuit. Using adiabatic elimination as well as matrix product states, we find the phase diagram of this model.

We identify a type of periodic evolution that appears in driven quantum systems. Provided that the instantaneous (adiabatic) energies are equidistant we show how such systems can be mapped to (time-dependent) tilted single-band lattice models. Having established this mapping, the dynamics can be understood in terms of Bloch oscillations in the instantaneous energy basis. In our lattice model the site-localized states are the adiabatic ones, and the Bloch oscillations manifest as a periodic repopulation among these states, or equivalently a periodic change in the system's instantaneous energy. Our predictions are confirmed by considering two different models: a driven harmonic oscillator and a Landau-Zener grid model. To strengthen the link between our energy Bloch oscillations and the original spatial Bloch oscillations we add a random disorder that breaks the translational invariance of the spectrum. This verifies that the oscillating evolution breaks down and instead turns into a diffusive spreading. Finally, we consider a trapped ion setup and demonstrate how the mechanism can be utilized to prepare motional cat state of the ion.

With the objective of simulating the physical behaviour of electrons moving in a dynamical background, we study a cold atomic Bose-Fermi mixture in an optical lattice potential felt only by the bosons. The bosons, assumed to be in the deep superfluid regime, inherit the periodicity of the optical lattice and subsequently act as a dynamical potential for the polarized fermions. Due to the atom-phonon interaction between the fermions and the condensate, the coupled system displays a Berezinskii-Kosterlitz-Thouless transition from a Luttinger liquid to a Peierls phase. For sufficiently strong Bose-Fermi interaction, however, the Peierls phase becomes unstable and is succeeded by either a collapsed or a separated phase. We find that the main role of the optical lattice amounts to stabilizing the Peierls phase. Furthermore, the presence of a confining harmonic trap leads to a rich physical behaviour beyond what is observed for either bosons or fermions separately trapped. In particular, for an attractive Bose-Fermi interaction, the insulating phase may develop a fermionic wedding-cake like configuration reflecting the dynamical nature of the underlying lattice potential. For repulsive interaction, on the other hand, we conclude that the trap destabilizes the Peierls phase and the two species separate.