Analytic solutions of density-matrix evolutions with the use of Racah tensorial decompositions

The problem of the dynamical evolution of an atomic system excited from state 1 to state $N$ in $N-1\mathrm{}$ steps by using one or more lasers is studied. The $N^2$-dimensional dynamical space is shown to factorize into $N$ independent subspaces under certain experimental conditions by using Racah tensorial representation. Different classes of multiple optical soliton and multilevel excitation solutions obtained previously are shown to arise from a unified scheme. A new class of multiphoton excitation solutions is also given. General equations of motion in the higher-dimensional subspaces are given but have not been solved.