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Journal of Optics B Quantum and Semiclassical Optics

Polynomial intelligent states

Matthew M Milks, Hubert de Guise

Published Nov 16, 2005• 10 citations
Publisher

Abstract

The construction of su (2) intelligent states is simplified using a polynomial representation of su (2). The cornerstone of the new construction is the diagonalization of a 2 × 2 matrix. The method is sufficiently simple to be easily extended to su (3), where one is required to diagonalize a single 3 × 3 matrix. For two perfectly general su (3) operators, this diagonalization is technically possible but the procedure loses much of its simplicity owing to the algebraic form of the roots of a cubic equation. Simplified expressions can be obtained by specializing the choice of su (3) operators. This simpler construction will be discussed in detail.

DOI
10.1088/1464-4266/7/12/026

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The construction of su (2) intelligent states is simplified using a polynomial representation of su (2). The cornerstone of the new construction is the diagonalization of a 2 × 2 matrix. The method is sufficiently simple to be easily extended to su (3), where one is required to diagonalize a single 3 × 3 matrix. For two perfectly general su (3) operators, this diagonalization is technically possible but the procedure loses much of its simplicity owing to the algebraic form of the roots of a cubic equation. Simplified expressions can be obtained by specializing the choice of su (3) operators. This simpler construction will be discussed in detail.
We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to the SU(2) and SU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in the both cases, by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.