We present a number of new physical systems that may be addressed using
methods of time dependent transformation. A recap of results available for
two-state systems is given, with particular emphasis on the AC stark effect. We
give some results that are not well known, including the full solution for a
two state system in a static electric field with arbitrary direction.
Connection with established theorems in time optimal quantum control is given,
and a full discussion outlines some advanced results in matrix calculus. In
particular, we derive a set of matrix gates relevant to quantum information
theory and computation using time optimal unitary operators, and define the
hyperbolic equivalent of the quantum brachistochrone problem.Comment: 11 page