Quantum Engineering in Open Quantum Systems

The huge technological advancement achieved in the last years has allowed for the emergence of a new field of physics dubbed \u201cquantum engineering\u201d: with this term people refer to a wide range of topics, from planning and building physical systems for specific tasks to developing algorithms to control those systems, from ways to create specific quantum states to new theoretical tools to describe and plan new physical systems. As the field of quantum engineering covers many topics in physics, this is reflected in the community interested in it, ranging from quantum optics theorists to solid state experimentalists. This also includes the possibility, and sometimes the necessity, for a scientist willing to enter the field to study very different problems, as it happened for the material in this thesis, where at least two main topics are covered. One of them is the study of open quantum systems, more specifically in the context of collisional model and cascade networks. The latter are networks of quantum systems interacting through the interaction with a common environment with unidirectional, i.e. chiral, propagation of the signal. Thanks to the chirality of the environment it is possible to obtain non symmetrical couplings between the quantum systems composing the network, opening the way to engineer the steady state of the system. The tool used to derive master equation describing dynamics and properties of such systems is the one of collisional models: these models are nowadays extensively used in a wide range of topics concerning open quantum systems, from the description of both Markovian and non Markovian dynamics, to quantum optics and quantum thermodynamics. In collisional models the environment is depicted as a collection of smaller systems, dubbed ancillas, which interact in a collisional fashion with the quantum system under examination. This way of describing open systems dynamics leads to a discrete master equation on which it is then possible to enforce a continuous time limit. Among the advantages provided by such an approach there is the simplicity with which is possible to switch from a Markovian to a non-Markovian dynamics and the possibility of keeping track of the environmental degrees of freedom. The last feature cited is the one exploited in this thesis when studying a quantum system thermalizing through the interaction with a thermal bath: having at disposal the environmental state at each discrete step of the thermalization process, it is possible to compute the thermodynamic functionals relative to the environment. Specifically, by computing the quantum mutual information between the system and the environment, it is possible to show that the final joint state reached by the system and the environment is a factorized state. The other part of this thesis focuses instead on quantum state engineering by potential engineering. By appropriately engineering a potential profile, it is possible to obtain a class of quantum states, dubbed stretchable, which have the property of having a flat wave function in some regions, somehow analogously to what happens in photonic metamaterials: in this materials, where either the permittivity or the permeability is zero, the temporal and spatial variation of the electric field are decoupled, leading to the possibility of having a stretched wave with both large frequency and large wavelength. Finally, in this thesis it is shown how, by properly engineering a spatially varying potential landscape, it is possible to attach a geometric phase to the quantum state of a traveling wave. More specifically, as the confining potential of a traveling wave varies along a closed loop in parameters space, it is possible to implement an operation, usually called holonomy, which attaches a geometric phase to the state, analogously to what happens in the Berry phase phenomenon for a time dependent Hamiltonian