 As the states
formula_29 and
formula_30 are degenerate with respect to formula_26 for all formula_32, it is enough to diagonalize
formula_33 in the subspaces formula_34.The matrix elements of formula_33 in this subspace, formula_36 read

For a given formula_32, the energy eigenvalues of formula_39 are

where formula_41 is the Rabi frequency for the specific detuning parameter.The eigenstates formula_42 associated with the energy eigenvalues are given by

where the angle formula_45 is defined through
formula_46

It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates.