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Multilevel Atomic Coherent States and Atomic Holomorphic Representation

Chang-qi Cao, Fritz Haake

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

The notion of atomic coherent states is extended to the case of multilevel atom collective. Based on atomic coherent states, a holomorphic representation for atom collective states and operators is defined. An example is given to illustrate its application.

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The notion of atomic coherent states is extended to the multilevel case. Since the representation based on coherent states is convenient in treating collective interactions of atoms with photons, and since many optical processes involve atoms of three or more levels, it is expected that this extension will play a role in the theory of such processes as cascade superfluorescence and superradiant lasing. Like their bosonic counterparts, atomic coherent states may be used to define a holomorphic representation for atomic states and operators. We discuss this in detail and give examples to illustrate applications.
"The possibility of describing noncommuting operators in quantum mechanics by classical type functions, and the associated expression of operator multiplication, is of considerable interest. The well known Wigner-Weyl-Moyal theory is an important example. Another is the diagonal representation of operators using standard coherent states. We develop general necessary and sufficient conditions for the existence of the diagonal representation in the context of a family of generalised coherent states associated with any unitary irreducible Lie group representation. Several examples illustrating these conditions, and interesting results in the Heisenberg-Weyl case, are presented".
The correlated states of an assembly of N multilevel atoms interacting with resonant radiation fields are determined. On the basis of the duality between the permutation group and the linear group, one can construct the internal states for an assembly of N atoms, with n levels each. The possible states are determined by various allowed Young diagrams, corresponding to various representations of the permutation group SN. The collective transition operators are obtained as the generators of the group SU(n).
The relationship between the atomic coherent-state representation of Arecchi et al. [Phys. Rev. A 6, 2211 (1972)] and the state multipoles is established. The state multipoles are used to develop a theory of generalized phase-space distributions for angular momentum (collective atomic) systems. The general theory for angular momentum systems is shown to have many features in common with the general theory for boson systems [Phys. Rev. D 2, 2161 (1970)]. These generalized phase-space distributions contain as a special case the coherent-state representation of Arecchi et al. The applications of the generalized phase-space distributions and state multipoles to the dynamical problems and to the calculation of multitime correlations are given. State-multipole techniques are used to give a brief discussion of the master equation describing cooperative resonance fluorescence.