SL(2,C) MULTILEVEL DYNAMICS

In a finite ladder of states excited stepwise by a (laser) field with linearly increasing detunings described by the SU(2) group, loss rates are introduced that vary linearly with level number. The dynamical group of the model is identified to be R x SL(2,C), with R an overall exponential dissipation factor. By using the representation theory of the SL(2,C) group, the transition probability of flipping from one state to another is obtained in a simple formula generalizing the Majorana formula, which is valid for the coherent excitation only. For the case of the time-dependent driving field a SU(2), SU(1,1) coherent-state path-integral formalism is developed that leads to a Riccati equation for the matrix elements of the evolution operator.