SU(2) and SU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states
C. Brif
TL;DR
Brif introduces algebra eigenstates (AES) as eigenstates of complex linear combinations of Lie-algebra generators to unify coherent-state constructions for dynamical groups. The paper develops a general AES formalism with analytic representations and applies it to $SU(2)$ and $SU(1,1)$, solving the AES eigenvalue problem exactly and deriving closed-form expressions for quantum statistics. It shows how Perelomov generalized CS and intelligent states arise as AES subsets, including Barut-Girardello representations, and provides explicit expansions in the standard bases. The results offer a unified, analytic toolkit for squeezing, interferometry, and quantum-optical applications across compact and noncompact groups.
Abstract
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.