Cavity QED: unconventional phenomena and new perspectives on quantum technologies

Light can interact with matter in a variety of ways. When the electromagnetic field is confined in a cavity, larger light-matter coupling strengths can be achieved, which opens the possibility to discover unconventional phenomena that are usually impossible to observe. When the coupling strength is comparable to the frequency of the electromagnetic field, the so-called ultrastrong coupling (USC) regime is reached. In this regime, the rotating wave approximation (RWA) is no longer valid, and the interaction between light and matter must be treated in a non-perturbative way. Moreover, the number of total particles is no longer conserved, meaning that several processes (or virtual transitions behind them) can create particles (or virtual particles) from the vacuum. Nevertheless, in this regime, several theoretical issues arise. For example, the number of photons in the ground state acquires a non-negligible value, which means that, in principle, the cavity can emit photons even in the lowest energy state, which is unphysical. A breakdown of gauge invariance also occurs, suggesting that the standard approach must be treated properly. Furthermore, the standard way to describe open quantum systems in Markovian baths must be revised, since the subsystems in this regime are strongly correlated. In this Thesis, we explore the context of light-matter interaction, showing how these issues can be interpreted and fixed. We start by defining the basics of light-matter interaction. Starting from a Lagrangian approach, we derive the most two common gauges in cavity QED: the Coulomb gauge and the multipolar gauge (also known as dipole gauge in the case of a constant vector potential). We then pass in the Hamiltonian framework, deriving a full quantum treatment of light-matter interaction. Compared to the quantum harmonic oscillator, which has a linear energy spectrum, matter systems usually present non-linear energy spectra. Thus, a truncation of the Hilbert space to the lowest energy levels can simplify the treatment of the system, leading to the quantum Rabi model in the case of a two-level atom in interaction with a single-mode cavity field. However, this truncation process leads to the breakdown of gauge invariance. Specifically, the Coulomb gauge reproduces different results concerning those in the dipole gauge or even in the non-truncated system Hamiltonian. We show that this breakdown is due to the non-locality that the matter potential acquires when performing the projection into the lowest energy states. To overcome this issue, the correct Coulomb gauge can be obtained through a generalized minimal coupling replacement, which is introduced after the truncation and not before. The dipole gauge, however, is not always well defined. Indeed, in this frame, the minimal coupling replacement is performed in the photonic part of the Hamiltonian rather than in the matter part. Thus, the conjugate momentum of the vector potential is different from the standard one and contains the polarization vector of the matter system. Everything related to the electric field must be redefined in this gauge because the electric field is no longer the conjugate momentum of the vector potential. This includes photodetection, the interaction with an external environment, pure dephasing effects, and photon condensation. In this Thesis, we will focus on all these topics. First, we will show how to define the photodetection. Specifically, following Glauber's theory, it is defined as the product of the negative and positive frequency parts of the electric field. In the absence of any interaction with the matter, the negative and positive frequency parts correspond to the creation and destruction operator, respectively. However, when the interaction becomes relevant, this is no longer true. Moreover, we will derive a generalized master equation, which, compared to the standard one, is derived with a minimal amount of assumptions: the Born and Markov approximations. Using the standard master equation in the USC regime leads to a finite number of photons at the steady state, even at zero temperature, and with the correct photodetection operators. We then apply this solid approach to an incoherent pumping process, showing how thermal excitations of the atom can be detected from an emission spectrum of the electromagnetic field. We apply this framework also to coherent pumping, showing the peculiar effect of spontaneous scattering of Raman photons without vibrational degrees of freedom. As mentioned above, this process can be observed only when the number of particles is not conserved (i.e., the USC regime). We then study the effect of pure dephasing in the USC regime, showing a gauge-invariant treatment of this process, demonstrating also here that the standard approach is no longer valid when the light-matter coupling strength is comparable to the frequency of the electromagnetic field. Finally, we study the phenomenon of photon condensation (or superradiant phase transition), where the ground state acquires a macroscopic number of coherent photons. We prove that this phenomenon can not be achieved in the absence of a magnetic field, showing that previous works that claim the existence of this phenomenon used gauge-dependent approaches.