SU(2) andSU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

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 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.

Analytic representations based on su(3) coherent states and Robertson intelligent states

“…The solutions of such general problem give the so-called in the literature algebra eigenstates or algebraic coherent states ( [13] and references therein). Taking specific constraints on the complex parameters occurring in this general eigenvalue equation, one can get various kind of coherent and squeezed states, in particular ones not discussed in this paper.…”

“…The states resulting from this minimization have different names in the literature such as correlated states [6][7][8] or Robertson intelligent states [9]. More recently, there has been much interest in such states for Lie algebras [9][10][11][12][13] as well as for quantum systems evolving in various potentials [14][15][16][17]. Robertson intelligent states for the quadrature components of Weyl generators of the algebras su(1, 1) and su (2) were constructed [9][10][11][12][13].…”

“…More recently, there has been much interest in such states for Lie algebras [9][10][11][12][13] as well as for quantum systems evolving in various potentials [14][15][16][17]. Robertson intelligent states for the quadrature components of Weyl generators of the algebras su(1, 1) and su (2) were constructed [9][10][11][12][13]. They were also defined for exactly solvable quantum systems as the eigenstates of complex combination of creation and annihilation operators [14][15][16][17].…”

Generalized coherent and squeezed states based on the h(1)⊕su(2) algebra

“…We follow ( §3.2) by the general construction of AES based on the algebra h(1) ⊕ su (2). These states are defined as eigenstates of an arbitrary linear combination of the generators of the considered algebra [8]. Then we consider special solutions to CS and SS for the so-called super-position and super-momentum operators ( §3.3).…”

“…AES [8] for this algebra are defined as eigenstates corresponding to a complex combination of the associated generators. A general hermitian operator A constructed from a combination of these generators is…”