Constructing Dynamical Symmetries

Optical excitation in the dipole approximation and other probes couple two quantum states of an unaddressed finite quantum mechanical discrete system. Thereby the interaction of the system with the probe is bilinear in the coherence between the two states and in the time-dependent strength of the probe. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for a Lie algebra that can be represented as a direct product of individual two state systems each using the population and the coherence between two states. Dynamical symmetries can be used to advantage to describe the progress of such systems in time. They also offer a compact and efficient representation for a density matrix of maximal entropy that evolves in time. Using the factorization approach of Wei and Norman, (J. Wei and E. Norman, Lie Algebraic Solution of Linear Differential Equations, J. Math. Phys. 4, 575 (1963)), we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two state systems. Thereby one can propagate, to all orders in perturbation theory, both the wave function and the density matrix with special reference to dynamical symmetries.