Coherent states of finite-level systems

Abstract

A method for constructing coherent states (CS) of finite-level systems with a given angular momentum is proposed. To this end we generalize the known spin equation (SE) to an infinite-dimensional Fock space. The equation describes a special quadratic system in the latter space. Its projections on $d$-dimensional subspaces, represent analogs of SE for $d$-dimensional systems in an external electromagnetic field which describe $d$-dimensional systems with a given angular moment. Using a modification of Malkin-Manko method developed in our earlier work, we construct the corresponding CS for the total quadratic system. Projections of the later CS on finite-dimensional subspaces we call angular moment CS (AMCS) of finite-level systems. The AMCS have a clear physical meaning, they obey the Schr\"odinger for a $d$-dimensional system with a given angular moment $j=\left(d-1\right)/2$ in an external electromagnetic field. Their possible exact solutions are constructed via exact solutions of the SE in $2$-dimensional space. The latter solutions can be found analytically and are completely described in our earlier works. A one subset of AMCS can be related to Perelomov spinning CS (PSCS). This reflects the fact that the set of possible AMCS is wider than the set of PSCS. AMCS states in a constant magnetic field are constructed. Some of them coincide with the Bloch CS.