Dynamics beyond the adiabatic approximation

We explore the dynamics of mixed classical-quantum Hamiltonians, a class of systems that frequently appears in the study of physical systems characterized by both slow and fast modes. Traditionally such systems are studied within the adiabatic approximation and as a rule they display level crossings. We show that level crossings can have peculiar effects on the dynamics and, in the regime between the adiabatic and diabatic limits in both classical and quantum approaches, the full dynamics has a series of non-intuitive features.