A new approach is proposed, extending the concept of geometric phases to adiabatic open quantum systems described by density matrices (mixed states). This new approach is based on an analogy between open quantum systems and dissipative quantum systems which uses a C * -module structure. The gauge theory associated with these new geometric phases does not employ the usual principal bundle structure but a higher structure, a categorical principal bundle (so-called principal 2-bundle or non-abelian bundle gerbes) which is sometimes a non-abelian twisted bundle. The need to site the gauge theory in this higher structure is a geometrical manifestation of the decoherence induced by the environment on the quantum system.

A generalized adiabatic approximation is formulated for a two-state dissipative Hamiltonian which is valid beyond weak dissipation regimes. The history of the adiabatic passage is described by superadiabatic bases as in the nondissipative regime. The topology of the eigenvalue surfaces shows that the population transfer requires, in general, a strong coupling with respect to the dissipation rate. We present, furthermore, an extension of the Davis-Dykhne-Pechukas formula to the dissipative regime using the formalism of Stokes lines. Processes of population transfer by an external frequency-chirped pulse-shaped field are given as examples.

The nonadiabatic temporal evolution which is associated with inelastic collisions and photoreactive processes typically produces a final state distribution that differs markedly from the initial state distribution. Nevertheless an adiabatic formalism is often used in a zeroth-order description of the processes; for time-periodic perturbations the Floquet theory has been used within an adiabatic framework to provide a compact dynamical theory which requires a basis composed of only a small number of Floquet eigenstates. The use of the generalized Floquet theory or of the concept of a super-adiabatic basis allows the adiabatic approach to be further extended to handle systems with quasi-periodic Hamiltonians. The present work proposes a new approach, in which the time duration of the interaction is artificially prolonged and special absorbing boundary conditions are introduced asymptotically over the lengthened time interval in such a way as to force the adiabaticity of the process. The method involves what can be thought of as time-dependent optical potentials. Some trial applications to semiclassical inelastic collisions and to photodissociation effects have shown that the use of the new technique permits a description of the dynamical processes which is so economical that the use of a single generalized Floquet eigenvector will suffice. The main technical feature of this constrained adiabatic trajectory method is that it converts the problem of solving the TDSE with an explicitly timedependent potential into that of solving a static complex eigenvalue problem.