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Communications in Theoretical Physics

On Common Eigenvector of Parametric Interaction Hamiltonian and Number-Difference Operator Derived by Virtue of Entangled State Representation

Fan Hong‐Yi, Weibo Gao

Published Jan 1, 2007• 0 citations
Publisher

Abstract

By virtue of the properties of bipartite entangled state representation we derive the common eigenvector of the parametric Hamiltonian and the two-mode number-difference operator. This eigenvector is superposition of some definite two-mode Fock states with the coefficients being proportional to hypergeometric functions. The Gauss contiguous relation of hypergeometric functions is used to confirm the formal solution.

DOI
10.1088/0253-6102/47/1/027

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By virtue of the properties of bipartite entangled state representation we derive the common eigenvector of the parametric Hamiltonian and the two-mode number-difference operator. This eigenvector is superposition of some definite two-mode Fock states with the coefficients being proportional to hypergeometric functions. The Gauss contiguous relation of hypergeometric functions is used to confirm the formal solution.
As a continuum work of Bhaumik et al.3 who derived the common eigenvector of the number-difference operator Q≡(a†a-b†b) and pair-annihilation operator ab, 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 Ref. 5 in which the common eigenkets of Q and pair creators a†b† are discussed.
Since the commutation relation [x1-x2,a1+a2]=0,their common eigenvector |β,x〉 is constructed by using the technique of integration within an ordered product(IWOP) of operators.This state exhibits the properties of the coherent and entangled states.Its entangled properties and the relations with the EPR entangled state can be confirmed by Schmidt decomposition and the unilateral integral for|β,x〉.The superposition of the states |β=β1+iβ2,x〉 along β1 axis yields the ideal two-mode entangled state.
We study the eigenstate problem of a two-coupled oscillator system. A new entangled state representation |γ composed of the common eigenvectors of operators (x1+p2) and (p1+x2) is established. Eigenvalues and eigenvectors of the Hamiltonian are obtained in |γ representations. The same problem is studied in the second-quantization representation. We find that the second-quantization representation can be used to derive the normally ordered product expression of eigenvector in Fock space. In particular, we find that the ground state of the Hamiltonian is a kind of generalized two-mode squeezed state.