Time-optimal control of a 3-level quantum system and its generalization to an n-level system

We solve the problem of steering a three-level quantum system from one eigen-state to another in minimum time and study its possible extension to the time-optimal control problem for a general n-level quantum system. For the three- level system we find all optimal controls by finding two types of symmetry in the problem: Z
₂
x S
₃
discrete symmetry and S
¹
continuous symmetry, and exploiting them to solve the problem through discrete reduction and symplectic reduction. We then study the geometry, in the same framework, which occurs in the time-optimal control of a general n-level quantum system.