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Coherent States on Riemann Surfaces as m-Photon States

Apostol Vourdas

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

Coherent states on the m-sheeted sphere (for the SU(2) group) are used to define analytic representations. The corresponding generators create and annihilate clusters of m-photons. Non-linear Hamiltonians that contain these generators are considered and their eigenvectors and eigenvalues are explicitly calculated. The Holstein-Primakoff and Schwinger formalisms in this context are also discussed.

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Coherent states on the m-sheeted sphere (for the SU(2) group) are used to define analytic representations. The corresponding generators create and annihilate clusters of m-photons. Non-linear Hamiltonians that contain these generators are considered and their eigenvectors and eigenvalues are explicitly calculated. The Holstein-Primakoff and Schwinger formalisms in this context are also discussed.
SU(2) coherent states on an m-sheeted covering of the sphere are introduced, and properties like overcompleteness and resolution of the identity are studied. Operators J+(m), J−(m), Jz(m) that obey the SU(2) algebra but shift the ‖jn〉 states by m steps are considered, and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the standard SU(2) coherent states with respect to the usual angular momentum operators J+, J−, Jz. An extended SU(2) Bargmann representation on an m-sheeted sphere is introduced in which the transformations of an m-sheeted covering of the SU(2) group are implemented as extended Mobius conformal mappings. The formalism is applied in the study of Hamiltonians that contain the operators J+(m), J−(m), Jz(m) and describe processes where the state of a particle is shifted by m steps. The method is also used in the context of both the Holstein–Primakoff and Schwinger representation of SU(2).
Coherent states on the m-sheeted complex plane are introduced and properties like overcompleteness and resolution of the identity are studied. Operators am+, am which create and annihilate clusters of m particles are considered and it is shown that the properties of our coherent states with respect to them are analogous to the properties of the ordinary (Glauber) coherent states with respect to a+, a. An extended Bargmann representation in the m-sheeted complex plane and P and Q representations with respect to our coherent states, are studied. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.
We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all (N-1) Casimirs of SU(N) in terms of these creation and annihilation operators. The SU(N) coherent states belonging to any irreducible representations of SU(N) are labeled by the eigenvalues of the Casimir operators and are characterized by (N-1) complex orthonormal vectors describing the SU(N) manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.