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Physical Review A

SU(m,n) coherent states in the bosonic representation and their generation in optical parametric processes

Ravinder Rupchand Puri

Published Dec 1, 1994• 27 citations
Publisher

Abstract

The coherent states of SU(m,n) transformations arising in the description of certain multimode nonlinear parametric processes bilinear in boson operators are constructed. The nonclassical properties of the coherent states of SU(m) and those of SU(m,1) are identified. The dynamics generated by SU(m,n) Hamiltonian is studied.
DOI
10.1103/physreva.50.5309

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The coherent state representations of the group G equals W1 (direct product) G0 [where G0 equals SU(2) or SU(1,1)] are used in computer simulations of the dynamic of single two- level atom [G0 equals SU(2)] interacting with a quantized photon cavity mode [Jaynes-Cummings Model (JCM) without the rotating wave approximation] and, in general, nonlinear in photon variables. The second case [hyperbolic Jaynes - Cummings Model (HJCM), G0 equals SU(1,1)] corresponds to the quantum dynamics of quadratic nonlinear coupled oscillators (the parametric resonance on double field frequency and a three-wave parametric processes of nonlinear optics). Quasiclassical dynamical equations for parameters of approximately factorizable coherent states for these models are derived and regimes of motion for 'atom' and field variables are analyzed.
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.