Ultrastrongly coupled light-matter systems: quantum phase transitions, spectral properties and effective models

The interaction of light and matter is a fundamental aspect in many fields of physics. In recent years, there has been a growing interest in the study of light-matter systems in the ultrastrong coupling (USC) regime, where the coupling strength between light and matter is comparable to the characteristic frequencies of the system, or even larger than them. This regime has enabled the prediction, and subsequently their experimental exploration, of many new physical phenomena, such as the presence of virtual photons in the system's ground state, the modification of the system's spectral properties, and processes that do not conserve the number of excitations, which are forbidden in the weak and strong coupling regimes. These phenomena have potential applications in many fields, including quantum information, quantum technologies, as well as in the study of fundamental aspects of quantum mechanics. Nevertheless, many questions remain open, both from a theoretical and experimental perspective. In this thesis, we investigate the properties of ultrastrongly coupled light-matter systems, focusing on quantum phase transitions, spectral properties, and effective models. We employ a combination of analytical and numerical techniques to study these systems, such as perturbative expansions, mean-field theory, and exact diagonalization. These results are relevant for understanding the behavior of light-matter systems in the USC regime, especially in cavity and circuit quantum electrodynamics (QED) settings. Indeed, many of the results presented here concern the spectral properties of these systems, either weakly or strongly coupled to an external bath. This thesis is structured as follows. In Chapter 1, we provide an overview of the current state of the art and introduce the necessary theoretical framework employed for describing light-matter interactions, with a particular focus on the USC regime and the models typically used to describe these systems. We also introduce the concept of quantum phase transitions (QPTs) and discuss their relevance in the context of light-matter systems. After presenting these introductory topics, in Chapter 2, we explore in detail one of the most fundamental models in quantum optics, the quantum Rabi model (QRM) and its generalizations, both in cavity and circuit QED implementations. This model describes the interaction between a two-level system and a single mode of the electromagnetic field, and is commonly used to effectively describe light-matter interactions in the USC regime. In particular, we analyze its spectral properties, such as coherent and incoherent emission spectra, in the USC regime and how they differ between various implementations. We then investigate the influence of higher energy levels of the matter subsystem, which are obviously neglected in a two-level description of the matter subsystem (or qubit, in circuit QED systems), on the system's behavior in the USC regime. This will lead us to the derivation of an effective Hamiltonian, the renormalized QRM (RQRM), that incorporates the effects of these higher-energy levels into the system's parameters while still retaining a two-level description of the matter component (qubit). Subsequently, in Chapter 3, this analysis is extended to the study of systems composed of multiple two-level emitters coupled to a single mode of the electromagnetic field, which can be described by an extended Dicke model and, in the thermodynamic limit, by the Hopfiled model. We explore the evolution of the system properties, such as energy levels and emission spectra, as a function of the number of emitters, and how the results expected in the thermodynamic limit are effectively recovered. Chapter 4 is dedicated to the study of QPTs in ultrastrongly coupled light-matter systems. In particular, we focus on the role of the electrostatic interactions between the emitters, which are often neglected in simpler models, and the impact they have on the system's behavior. We show that these interactions can significantly alter the occurrence and nature of QPTs. To this end, we investigate two realistic and specific arrangements of the emitters: a three-dimensional lattice of spatially separeted and highly localized dipoles and a two-dimensional layer of such dipoles embedded in a cavity. We analyze how the geometry of the emitter arrangement influences the system's properties, and we identify the conditions under which a QPT can occur. This results can be placed in the context of a long-standing debate about the occurrence of a superradiant phase transition (SPT) in such systems, which has been the subject of much theoretical investigation in recent years. Finally, Chapter 5 is devoted to the study of the open Dicke model, and specifically to the impact of external baths on the occurrence of the SPT in equilibrium conditions. We will find that our results differ significantly from previous works studying the open Dicke model in effective driven-dissipative systems. Specifically, we analyze how the presence of these baths modifies the critical point, as well as the macroscopic occupations in the superradiant phase. We also investigate the role of different types of baths, both ohmic and non-ohmic, and their influence on the system's behavior. In the end, we present coherent emission spectra for the open Dicke model for these types of baths and analyze their features in relation to the usual interpretations for systems that do not exhibit QPTs.