1. We show that for any finite-dimensional quantum systems, the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics.

2. An analytic approach to investigate the zero-temperature time evolution of the Jaynes-Cummings system with cavity losses is developed. With realistic coupling between the cavity and the environment assumed, a simple master equation is derived, leading to the explicit analytic solution for the resonant case. We examine the small and large detuning limits and discuss the condition where the widely used phenomenological treatment is justified. Explicit evaluations of the time evolutions for various initial states with finite detuning are also presented.

3. We propose to extend the adiabatic impulse approximation to multilevel systems to cope with non-adiabatic transitions seen in various fields of physics. This approximation method is shown to be equivalent to a series of unitary evolutions and facilitates to evaluate the dynamics numerically. In particular, we analyze the dynamics of the Landau–Zener grid model and the multilevel Landau–Zener–Stueckelberg–Majorana interference model, and confirm that the results are in good agreement with the exact dynamics evaluated numerically. We also derive the conditions for destructive interference to occur in the multilevel system.