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.

Keywords:

• coherent excitation
• Morris–Shore transformation
• dark states
• lambda linkage
• tripod linkage

Acknowledgements

Over the years I have enjoyed many instructive conversations with colleagues; those particularly relevant to the MS transformation and this article include Jim Morris, Carlos Stroud, Peter Knight, Klaas Bergmann, Razmik Unanyan, Leonid Yatsenko, Nikolay Vitanov, Andon Rangelov, Zsolt Kis and Michael Fleischhaur.

Notes

1. The two-state behavior of the MS-transformation directly involves all states in some way, unlike various approximations that replace the -state dynamics with a single two-state system [167, 168]; see Section 3.3.

2. Other common interactions include magnetic dipole, electric quadrupole and induced dipole (see [1] Section 9 and [3]).

3. The RWA requires that Rabi frequencies be much smaller than carrier frequencies.

4. Such processes require treatment by a density matrix [1–3].

5. By definition the Rabi frequency is the frequency of not .

6. The Shore–Cook approach discussed here leads, in some situations with odd-integer , to three-state behavior.

7. Thermal light responsible for thermodynamic equilibrium has a uniform distribution of propagation directions, and hence three possible axes for linear polarization, in contrast to the two transverse axes that are possible for a beam.

8. The definition using the arctan function should, in practice, be replaced by definitions using arcsin and arccos.