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Communications in Theoretical Physics

Some Remarks on the Binomial States

Hong-Yi Fan

Published Dec 15, 1995• 3 citations
Publisher

Abstract

Using the Holstein–Primakoff boson realization of spin operators we can identify the binomial state with a particular spin coherent state. In so doing some properties of spin coherent states can be derived by simply using the binomial state. The displaced binomial state is introduced and its completeness relation is examined. That the concept of the binomial state can be extended to two-mode case is pointed out.

DOI
10.1088/0253-6102/24/4/483

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Using the Holstein–Primakoff boson realization of spin operators we can identify the binomial state with a particular spin coherent state. In so doing some properties of spin coherent states can be derived by simply using the binomial state. The displaced binomial state is introduced and its completeness relation is examined. That the concept of the binomial state can be extended to two-mode case is pointed out.
Using the q analog of the Holstein-Primakoff boson realization of the su(2) generators, we show that a type of q-deformed binomial state that corresponds to the Heine distribution can be identified as an su(2)q{)}_{\mathit{q}})q​ coherent state. This fact is a q extension of the fact that the ordinary binomial state is a particular su(2) coherent state when the Holstein-Primakoff transformation is employed.
Unitary operator coherent states, as defined by Klauder (1963), Perelomov (1972) and Gilmore (1974), are considered for the harmonic series irreducible representations of U(p,q). Various properties of these states are reviewed, including their reproducing kernel and measure explicit forms, as well as the representation they lead to for the U(p,q) generators. Starting from a contraction of the su(p,q) Lie algebra, both Dyson and Holstein-Primakoff boson realisations of u(p,q) are obtained in terms of pq pairs of boson creation, annihilation operators and of the generators of a U(p)*U(q) intrinsic group. Such boson realisations are applied to determine the matrix elements of the U(p,q) generators between discrete bases classified according to the chain U(p,q) contains/implies U(p)*U(q). Finally, the isomorphism between so(4,2) and su(2,2) is employed to derive the corresponding properties of the SO(4,2) unitary operator coherent states for application in atomic physics.