Correlated dynamics of driven cascaded quantum optical cavities

The interaction between a light cavity mode and other quantum systems provides an excellent platform for a multitude of applications in quantum technologies. However, more striking features arise when considering more than one cavity coupled in a unidirectional manner, forming a so-called cascaded configuration.In this thesis, we first investigate the dynamics of a system composed of a pair of cavity light modes arranged in a cascade configuration, interacting with a thermally excited beam of phaseonium atoms serving as ancillas. We provide exact closed dynamics for the first cavity over arbitrarily long interaction times.We emphasize the role played by the characteristic coherence phase of phaseonium atoms in determining the steady states of both the cavity fields and the ancillas. Additionally, we demonstrate how the second cavity follows a non-Markovian evolution due to interactions with the "used" ancillary atoms, facilitating information exchange with the first cavity.By adjusting the parameters of the phaseonium atoms, we can determine the final stable temperature reached by the cavities, thereby enabling both heating and cooling processes.Subsequently, we focus on the interaction between a light mode and a mechanical harmonic oscillator via radiation pressure, studying the dynamics of a pair of optomechanical systems interacting dissipatively with a waveguide in a unidirectional manner.Focusing on the regime where the cavity modes can be adiabatically eliminated, we derive an effective coupling between the two mechanical modes and explore the classical and quantum correlations established between the modes in both the transient and stationary regimes.We highlight the asymmetric nature of these correlations due to the unidirectional coupling and find that a constant amount of steady correlations can exist at long times. Furthermore, we demonstrate that this unidirectional coupling establishes a temperature gradient between the mirrors, dependent on the frequencies' detuning.Additionally, we analyze the power spectrum of the output guide field and show how, thanks to the chiral coupling, it is possible to reconstruct the spectra of each single mirror from such spectrum.Finally, we consider the case of a chain of optomechanical systems, always coupled unidirectionally by means of a waveguide, and find that by adiabatically eliminating the cavity modes, the effective interaction between the mechanical modes exhibits a behavior that extends beyond simple chain-like interactions, forming a network-like structure. This has significant implications for engineering a quantum network.