  The Morris-Shore (MS) transformation is a powerful tool for decomposition of
the dynamics of multistate quantum systems to a set of two-state systems and
uncoupled single states. It assumes two sets of states wherein any state in the
first set can be coupled to any state in the second set but the states within
each set are not coupled between themselves. Another important condition is the
degeneracy of the states in each set, although all couplings between the states
from different sets can be detuned from resonance by the same detuning. The
degeneracy condition limits the application of the MS transformation in various
physically interesting situations, e.g. in the presence of electric and/or
magnetic fields or light shifts, which lift the degeneracy in each set of
states, e.g. when these sets comprise the magnetic sublevels of levels with
nonzero angular momentum. This paper extends the MS transformation to such
situations, in which the states in each of the two sets are nondegenerate. To
this end, we develop an alternative way for the derivation of Morris-Shore
transformation, which can be applied to non-degenerate sets of states. We
present a generalized eigenvalue approach, by which, in the limit of small
detunings from degeneracy, we are able to generate an effective Hamiltonian
that is dynamically equivalent to the non-degenerate Hamiltonian. The effective
Hamiltonian can be mapped to the Morris-Shore basis with a two-step similarity
transformation. After the derivation of the general framework, we demonstrate
the application of this technique to the popular Lambda three-state system, and
the four-state tripod, double-Lambda and diamond systems. In all of these
systems, our formalism allows us to reduce their quantum dynamics to simpler
two-state systems even in the presence of various detunings, e.g. generated by
external fields of frequency drifts.
