Geometry of symmetrized states

Abstract

A projection operator is defined in terms of coset representatives and "coset harmonics." This operator is then used to construct fully symmetrized states describing N identical two-level atoms. The properties and quantum numbers of the symmetrized states are fully discussed. Fully symmetrized and labeled states for N identical 3 (or r) level atoms are constructed by a natural extension of this procedure. The interaction Hamiltonian, which describes transitions in an r-level atom, is constructed from the shift operators in the algebra of SU(r). When each of the N identical atoms "sees" the same field dependence, transitions are only allowed between energy eigenstates occurring within the same SU(r) unitary irreducible representation. Physical dispersion of the particles out of the coherence region of the field leads to "leakage" out of an SU(r) representation. If each two-level atom evolves from the ground state under the same field dependence, the total system state is a coherent superposition of bases belonging to the fully symmetric representation {N = 2J}2. These states exist in 1-1 correspondence with points in the surface of a unit sphere. They are therefore called Bloch states. The properties of Bloch states are derived and are seen to resemble the properties of coherent photon states. The connection is made manifest by a group contraction process. All properties of Glauber states can be constructed by contraction from the corresponding properties of the Bloch states. The states corresponding to N-identical r-level particles evolving under a coherent field are constructed in the same way. These states exist in 1-1 correspondence with points in the coset space SU(r) U(r-1), which is a Riemannian-symmetric space. The eigenstate expansions, inner products, eigenvalue equations, and uncertainty relations for these states are derived. More generally, systems of N identical (p + q) level atoms, evolving from the lowest q levels under a coherent field, are described by states existing in 1 - 1 correspondence with Riemannian symmetric space SU(p + q) S[U(p)×U(q)]. The properties of these spaces are used to give a geometric analytic interpretation to these coherent atomic states. © 1972.