Simultaneous eigenstates of the number-difference operator and a bilinear interaction Hamiltonian derived by solving a complex differential equation

TL;DR

This paper derives the simultaneous eigenstates of a number-difference operator and a bilinear Hamiltonian using complex differential equations, extending previous work on eigenvectors of related operators in quantum optics.

Contribution

It introduces a method to find eigenstates of a specific bilinear Hamiltonian and number-difference operator using complex differential equations and special functions.

Findings

Eigenstates expressed in terms of two-variable Hermite polynomials.

Solution involves hypergeometric functions.

Extends previous eigenvector derivations in quantum optics.

Abstract

As a continuum work of Bhaumik et al who derived the common eigenvector of the number-difference operator Q and pair-annihilation operator ab (J. Phys. A9 (1976) 1507) we search for the simultaneous eigenvector of Q and (ab-a^{+}b^{+}) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Mod. Phys. Lett. A 9 (1994) 1291 by Fan and Klauder, in which the common eigenkets of Q and pair creators are discussed.