Quantum dynamics of time-dependent Hamiltonian models in finite and infinite Hilbert spaces

The knowledge of exact solutions of the dynamical problem of a two-level system subjected to classical time-dependent fields is exploited to exactly solve the quantum dynamics of more complex systems, such as interacting spin-qudits and interacting quantum oscillators. We start from the analysis of the symmetries possessed by the Hamiltonian operator of the system under scrutiny to identify invariant Hilbert subspaces. Within each of these subspaces we succeed in mapping the original dynamics into that of a Hamiltonian model expressed in terms of appropriate fictitious variables. For example, in the case of an invariant subspace characterized by a SU(2) symmetry, the original dynamical problem is traced back to that of a single spin J (whose value depends on the size of the subspace) subjected to external time-dependent fields. Thanks to the Group Theory, the time evolution operator of the spin J can be generated by the one relative to a single spin-1/2. This general approach is applied to two problems. The first deals with two interacting qubits subjected to local linearly varying fields. We show that Landau-Majorana-Stueckelberg-Zener transitions are possible, although a transverse constant field is absent, thanks to the presence of the coupling between the spins. The second one explores the quantum dynamics of a system composed of a quantum harmonic oscillator and an inverted quantum harmonic oscillator known as Glauber oscillator. In both cases our results are exact and stem from the approach discussed above.