The relevance and the importance of the study of time-dependent quantum models of interacting spin system stem from their applicability in several fields of Science.

Interacting spin systems, thanks to the versatility of the spin language, are useful to describe effectively interacting systems like quantum magnetic materials, interacting quantum dots, nitrogen-vacancy centers interactions, arrays of interacting superconducting qubits, ultracold atomic gas and trapped ions.

Recently, the interest in spin systems has grown exponentially also thanks to the birth and the fast development of the Quantum Information and Quantum Computation and Quantum Processing areas.

Time-dependent Hamiltonians are very useful, sometimes even necessarily required, in a lot of physical systems belonging to the scenarios previously mentioned. In general, quantum dynamical problems related to time dependent Hamiltonians are hardly solvable. In fact, e.g., the easiest problem of a simple two-level system or a spin 1/2 subjected to a time dependent magnetic field (SU(2) dynamical problem) cannot be solved in the general case.

In the first part of the seminar the exact quantum dynamics of a single spin-1/2 in a generic time-dependent classical magnetic field (generalized Rabi system) is investigated and compared with the quantum motion of a spin- 1/2 studied by Rabi. The notion of time-dependent resonance condition is introduced and carefully legitimated and analysed.

Several examples help to disclose analogies and departures of the quantum motion induced in a generalized Rabi system with respect to that exhibited by the spin-1/2 in a magnetic field precessing around the z-axis (standard Rabi scenario).

In the second part, a technique through which apply the solutions for a single spin problem to more complex spin systems is proposed. Three physically relevant time-dependent spin Hamiltonian models are considered as examples, bringing to light interesting physical properties and aspects.