Based on the conclusion that the generalized Bargmann representation of a two-mode Fock state is a two-variable Hermite polynomial function [Hong-Yi Fan and Jun-hua Chen, Phys. Lett. A303 (2002) 311] we derive the generalized Bargmann representation of the spin coherent state and some new relations in the generalized function space.
We reveal that the two-variable Hermite function h m , n , which is the generalized Bargmann representation of the two-mode Fock state, involves quantum entanglement of harmonic oscillator's wave functions. The Schmidt decomposition of h m , n is derived. It also turns out that h m , n can be generated by windowed Fourier transform of the single-variable Hermite functions. As an application, the wave function of the two-variable Hermite polynomial state S ( r ) H m , n (μ a 1 † ,μ a 2 † )|00⟩, which is the minimum uncertainty state for sum squeezing, in ⟨η| representation is calculated.
A new complex function space whose basis is the single-variable Hermite polynomial H2jξ*+τξ2τ is constructed, which is related to both entangled state representation and spin coherent state in Schwinger bosonic realization. New binomial theorem involving two-variable Hermite polynomial is derived, which helps to constitute the new complex function space. We also present a new integration transformation of the basis H2jξ*+τξ2τ with its reciprocal transformation which is useful to deriving some operator identities.
Following the lines of the recent paper of J.-P. Gazeau and F. H. Szafraniec [J. Phys. A: Math. Theor. 44, 495201 (2011)], we construct here three types of coherent states, related to the Hermite polynomials in a complex variable which are orthogonal with respect to a non-rotationally invariant measure. We investigate relations between these coherent states and obtain the relationship between them and the squeezed states of quantum optics. We also obtain a second realization of the canonical coherent states in the Bargmann space of analytic functions, in terms of a squeezed basis. All this is done in the flavor of the classical approach of V. Bargmann [Commun. Pure Appl. Math. 14, 187 (1961)].