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arXiv:quant-ph/9907075 · Journal of Optics B Quantum and Semiclassical Optics

Coherent states of nonlinear algebras: applications to quantum optics

V. Sunilkumar, Bindu A. Bambah, R. Jagannathan, Prasanta Kumar Panigrahi, Venkatraman Srinivasan

Published Apr 1, 2000• 48 citations
arXivHTMLPDF

Abstract

We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.

DOI
10.1088/1464-4266/2/2/311

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We present a general unified approach for finding the coherent states (CSs) of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the noncompact cases, to obtain the annihilation operator eigenstates, by finding the canonical conjugates of these operators. Generalized CSs, in the Perelomov sense, also follow from this construction. This allows us to explicitly construct CSs associated with various quantum optical systems.
We present a general unified approach for finding the coherent states of polynomially deformed algebras such as the quadratic and Higgs algebras, which are relevant for various multiphoton processes in quantum optics. We give a general procedure to map these deformed algebras to appropriate Lie algebras. This is used, for the non compact cases, to obtain the annihilation operator coherent states, by finding the canonical conjugates of these operators. Generalized coherent states, in the Perelomov sense also follow from this construction. This allows us to explicitly construct coherent states associated with various quantum optical systems.
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In this work we make use of deformed operators to construct the coherent states of some nonlinear systems by generalization of two definitions: i) As eigenstates of a deformed annihilation operator and ii) by application of a deformed displacement operator to the vacuum state. We also construct the coherent states for the same systems using the ladder operators obtained by traditional methods with the knowledge of the eigenfunctions and eigenvalues of the corresponding Schrödinger equation. We show that both methods yield coherent states with identical algebraic structure.
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Abstract The multiphoton algebras for one-dimensional Hamiltonians with infinite discrete spectrum, and for their associated kth-order SUSY partners are studied. In both cases, such an algebra is generated by the multiphoton annihilation and creation operators, as well as by Hamiltonians which are functions of an appropriate number operator. The algebras obtained turn out to be polynomial deformations of the corresponding single-photon algebra previously studied in literature. The Barut-Girardello coherent states, which are eigenstates of the annihilation operator, are obtained and their uncertainty relations are explored by means of the associated quadratures.
We obtain a large class of multiphoton annihilation operator (F) eigenstates by constructing an operator ${\mathit{G}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$ such that [F,${\mathit{G}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1. We show that almost all known coherent states, including the squeezed states and other nonclassical states such as the cat and the kitten states follow from our approach. Further, we show that all of them can be expressed as an exponential operator acting on the vacuum of the operator F. The technique can be easily generalized to deformed bosons.