Two-state behavior in N-state quantum systems: The Morris–Shore transformation reviewed

Abstract

Descriptions of coherent excitation of multi-state quantum systems model changes through coupled ordinary differential equations that become more cumbersome as the number of involved quantum states increases. It is always useful to find simplifications of the equations, ideally involving exact analytic solutions. The ultimate simplification is to a set of independent pairs of quantum states. Such sets of independent two-state linkages occur naturally when we describe the excitation between two sets of degenerate Zeeman sublevels by a field that is linearly or circularly polarized. The Morris–Shore (MS) transformation, a Hilbert-space coordinate change, generalizes this procedure to replace the description of an elaborate linkage pattern of an -state Hamiltonian in the rotating-wave approximation by a set of independent two-state systems – a basis of paired bright states and excited states supplemented with dark states and spectator states. The three-state lambda-linkage found in stimulated Raman transitions is a simple example. Both bright and dark states have found application in various procedures for manipulating quantum states, either to achieve population transfer between states or to create coherent superpositions; the dark and spectator states coincide with particular dressed (or adiabatic) states. The present review describes the MS transformation, noting its historical background, and discusses examples of its use.