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Modern Physics Letters A

ON THE INVERSE OF TWO-MODE BOSON OPERATORS (a-b†) AND (a†-b)

Hong-Yi Fan, Zhihu Sun, HUI ZOU

Published Dec 28, 1999• 1 citation
Publisher

Abstract

We study the properties of the inverse of two-mode boson operators (a-b†) and (a†-b), by constructing the charged bosonic orthogonal state |q, r>, where r denotes the radius. We show that in |q, r> representation (a†-b)-1 and (a-b†)-1 lowers and ascends the charge quantum number q, respectively, which resembles the behavior of single-mode inverse operator [Formula: see text] in Fock space.

DOI
10.1142/s0217732399002911

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We study the properties of the inverse of two-mode boson operators (a-b†) and (a†-b), by constructing the charged bosonic orthogonal state |q, r>, where r denotes the radius. We show that in |q, r> representation (a†-b)-1 and (a-b†)-1 lowers and ascends the charge quantum number q, respectively, which resembles the behavior of single-mode inverse operator [Formula: see text] in Fock space.
New formulae are derived for the coefficients of normally ordered operator expansions for a single mode, which depend on the matrix elements in the Fock state representation. The closely connected problem of the reconstruction of operators F by the set of normally ordered moments Tr(a Dagger k a l f), (k, l=0, 1, 2, . . .), is solved. For this purpose, a new set of operators a k,l that is orthonormal to the set of normally ordered powers of boson operators a Dagger k a l is introduced and investigated. Furthermore, a symmetric and a Hermitian scalar product are defined in the linear space of linear operators for the boson mode. The duality obtained in the expansion and reconstruction formulae of the operators can be understood more deeply by considering not only the usual Hilbert space but also a rigged Hilbert space of linear operators for the boson mode.
The generalized inverses of q -boson operators denoted by are introduced via their acting on the q -number states. The even and odd number eigenstates of two photon operators are constructed. It is shown that these states can show quantum-squeezing. However, their even number states always show quantum-antibunching, while their odd number states always show quantum-bunching. This result is different from that in recent paper (C.L. Mehta, A.K. Roy and G.M. Saxena, Phys. Rev. A46 (1992) 1565) where the odd number eigenstates of operators with q ≡ 1 show bunching for some parameter values and antibunching for other parameter values.