Hybrid quantum systems in the USC regime

The interaction between light and matter is one of the most fundamental phenomena in physics, and it has been extensively studied in various contexts, such as atomic physics, quantum optics, and solid-state physics. In particular, the regime of strong coupling, where the light-matter interaction energy exceeds the dissipation rates of the system, has attracted a lot of attention for its potential applications in quantum information processing, quantum metrology, and quantum simulation. However, in recent years, new regimes of light-matter interaction have emerged, where the coupling strength becomes comparable to or even larger than the bare frequencies of the subsystems. These regimes, known as the ultrastrong coupling regime and the deep-strong coupling regime, pose new challenges and opportunities for both theoretical and experimental investigations. In particular, it has been shown that as the coupling enters the ultrastrong coupling regime, many issues arise, like the violation of the gauge invariance principle, and the presence of virtual excitations in the ground state. In this thesis, I explore some novel aspects of the ultrastrong coupling (USC) and deepstrong coupling (DSC) regimes on hybrid systems. The focus will be on cavity quantum electrodynamics and I will use two prototypical models: the quantum Rabi model and the Hopfield model, which describe the interaction between a single-mode cavity field and a two-level system, or a multimode cavity field and a collection of two-level systems, respectively. I also consider the effects of pure dephasing, which originates from the non-dissipative information exchange between quantum systems and environments, and plays a key role in both spectroscopy and quantum information technology. Additionally, I will study the hopping mechanism in optomechanics. The main results of this thesis are as follows: I derive an effective Hamiltonian for a nonlinear oscillator coupled to a qubit using a polariton model. I show the importance of correctly applying the minimal coupling to satisfy the gauge invariance principle. I investigate the pure dephasing of light-matter systems in the USC and DSC regimes, using a dissipative quantum Rabi model and a Hopfield model. I find that the interaction can significantly affect the form of the stochastic perturbation describing the dephasing of a subsystem, depending on the adopted gauge. I study the optomechanical two-photon hopping in a system of two cavities separated by a vibrating two-sided perfect mirror. I show that this system displays photon-pair hopping between the two electromagnetic resonators, which is not due to tunnelling, but rather to higher-order resonant processes. This thesis is organized as follows: In Chapter 1, I introduce the basic concepts and tools of cavity quantum electrodynamics and the USC and DSC regimes. In Chapter 2, I present the model and the results for the nonlinear oscillator both uncoupled and coupled to a qubit. In Chapter 3, I present the model and the results for the pure dephasing of light-matter systems. In Chapter 4, I present the model and the results for the optomechanical two-photon hopping. In Chapter 5, I summarize the main conclusions and outlooks of this thesis.