Dinamica effettiva di sistemi quantistici driven chiusi e aperti

In recent years, the enormous progress in the field of Quantum Technologies has allowed the development of tools capable of creating and controlling quantum states in physical systems, called NISQ (Noisy Intermediate-Scale Quantum) computers. They constitute an important intermediate stage towards the demonstration of a large-scale quantum computer. Huge efforts have been and are being made in developing and engineering quantum hardware, and controlling it in order to perform novel quantum tasks in computation, communication and sensing. When dealing with real-life quantum hardware, three sources of errors have to be taken into account. First, the effects of the interaction of the system with an environment of uncontrollable degrees of freedom which may determine the loss of coherence. Second, imperfections in the control may introduce errors in performing quantum gates. Third, leakage from the computational subspace may cause an abrupt stop of the whole computation or at least part of it. States not belonging to the computational subspace may have an impact even when they are not populated during the dynamics since they still affect the principal system via virtual processes. In this work we focus on deriving effective low-frequency descriptions of the dynamics for both closed and open quantum system. In other words, our aim is to determine a dynamical map which describes effectively the original one in a specific range of parameters and thus identifying and properly "deal with" the terms of the map which contribute less in the time evolution of the system. Although not necessarily, we expect that the effective representation turns out to be simpler than the exact one. In the context of closed systems we propose a derivation of an effective Hamiltonian which captures effectively the dynamics of the original one in the "relevant" (e.g. those which are populated) subspace, thus "eliminating" from the formulation the rest of the Hilbert space. Our proposal only leverages on a coarse graining procedure of the time evolution and provides a systematic scheme to improve the approximations. The method is based on the Magnus expansion which expresses the solution of a differential equation into an exponential form. The result of the technique is a coarse-grained effective Hamiltonian valid at low energies which is able to average out the fast dynamics and to capture the degrees of freedom which are more relevant for the dynamics at long times. We test the method on a three-level system in a Lambda configuration, on an ac-driven two level system and on a generalized Rabi model. The same coarse graining approach is then used to discuss the dynamics of a two-state system driven by an ac field beyond the rotating wave approximation. This is a well-known problem in quantum optics, where however the amplitude of the driving field may be several orders of magnitude smaller than the natural atomic frequencies. On the contrary, in solid-state artificial atoms the presence of counterrotating terms may lead to errors affecting high-fidelity quantum gates. Coarse graining by the Magnus expansion proves to be a versatile tool, showing that the counterrotating terms determine the usual Bloch-Siegert shifts off-resonance (besides the Stark shift due to the corotating field), while giving at resonance, besides the Bloch-Siegert, a renormalization of the amplitude of the corotating field. As a final application of coarse graining by the Magnus expansion, we study the generalized Rabi model describing N+1 artificial atoms and a quantized harmonic where at least for one atom the coupling is ultrastrong coupling. This is a necessary condition to perform a routing operation in a modular quantum architecture when scaling to a large number N of target qubits. We obtain an effective Hamiltonian that accounts for the counterrotating terms as effective all-to-all couplings between the two-level systems besides energy shifts depending both on the the cavity and qubits states. Thanks to this analytic result, we could seek for an ultrafast protocol for multipartite entanglement generation in the ultrastrong coupling regime. The protocol is inspired by stimulated Raman adiabatic passage (STIRAP) since we aim to use the mode as a virtual quantum bus. This allows to suppress leakage from the computational subspace due to the dynamical Casimir effect. We find that renormalized terms in the low-energy Hamiltonian degrade the fidelity of adiabatic passage, but we are able to make this error correctable. Indeed, we found a new solution for the control problem of adiabatic passage via a dark state in a fully-connected quantum network, which requires only an additional local modulation of the detunings. Enlarging the scope to the dynamics of driven open systems, we study a generalized operatorial form of the Bloch-Redfield Markovian master equation and apply it to a two-level system driven by a single-tone ac classical field. The most common strategy, used especially in quantum optics, is the phenomenological approach, i.e. based on the Gorini-Kossakowski-Sudarshan-Lindblad equation, where the dissipator describes the undriven system, the drive being accounted by just adding a control term to the system Hamiltonian. We call this procedure the "local approach". On the contrary we can adopt an analysis starting from a microscopic Hamiltonian including both the interaction with the environment and the control field, which are then treated on the same footing. In this way the effect of the control field is taken into account from the outset. For an ac-driven control under the rotating wave approximation, we derive a time-independent differential equation in the frame rotating with the drive frequency. The resulting rates and jump operators are now modified by the presence of the control field. We name this procedure the "global approach". We will compare in this framework both the secular approximation and the full equation containing non-secular terms.