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Communications in Theoretical Physics

Eigenstates of q -Deformed Two-Photon Annihilation Operators and Their Nonclassical Properties

Wei LianFu, Shun-Jin Wang, Xi Ding-ping

Published Oct 30, 2000• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/34/3/553

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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.
We introduce the inverse of the harmonic-oscillator creation and annihilation operators by their actions on the number states. We then show that three of the two-photon annihilation operators, a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^, a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$, and a^ 2^{2}2, possess eigenstates that are analogous to the often-used coherent and squeezed states. A family of the eigenstates of the operator a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$a^ is the customary squeezed vacuum S(\ensuremath{\sigma})\ensuremath{\Vert}0〉 while another family of the eigenstates of a^a^ $^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}\mathrm{\ensuremath{-}}1}$ is the state S(\ensuremath{\sigma})\ensuremath{\Vert}n=1〉. We find some interesting properties related to squeezing, bunching, and antibunching exhibited by these states. An alternative method of summing some series by using these states is also considered. It is hoped that these states will find applications in quantum optics and quantum mechanics in general.
Using the properties of the q-deformed boson creation and annihilation operators and their inversed operators, two kind of q-deformed pair coherent states are introduced. Antibunching effects and correlation properties between two modes in the states are investigated. It is shown that q-deformed pair coherent states exhibit antibunching effects and the photons of the two modes are correlated. These nonclassical effects are influenced by the parameter q. These effects increase when ｜lnq｜ increases.
A method for construction of the k orthonormalized quantum eigenstates of the higher powers ak{\mathit{a}}^{\mathit{k}}ak of the quantum-annihilation operator a is proposed, and their physical meaning is explored. Nonclassical properties of these quantum eigenstates are studied and the main results are as follows: (1) All of the eigenstates can be expressed as a linear superposition of k quantum-coherent states; (2) all of them are generalized quantum-coherent states; (3) for k\ensuremath{\ge}3, they have the Nth-order quantum squeezing for N=mk+k/2 (m=0,1,2, . . .) and even k, but no Nth-order quantum squeezing for other values of N and k; (4) for k\ensuremath{\ge}3, all of them exhibit the lowest- and higher-order quantum antibunching.
Inverses of the harmonic oscillator creation and annihilation operators by their actions on the number states are introduced. Three of the two photon annihilation operators, viz., a(sup +/-1)a, aa(sup +/-1), and a(sup 2), have normalizable right eigenstates with nonvanishing eigenvalues. The eigenvalue equation of these operators are discussed and their normalized eigenstates are obtained. The Fock state representation in each case separates into two sets of states, one involving only the even number states while the other involving only the odd number states. It is shown that the even set of eigenstates of the operator a(sup +/-1)a is the customary squeezed vacuum S(sigma) O greater than.
The antibunching effect for the eigenstates of the q deformed annihilation operator a k q(k ≥3) is investigated. Using the numerical method, we have studied the influences of the q parameter deformation on the effect in the case of k =3. The results show that the eigenstates of a k q exhibit antibunching effect when x=|z| 2, i.e. the intensity of the oscillator in q deformed coherent state, is in values of certain intervals, and the effect is evidently influenced by the values of parameter q . When the intensity of the q deformed light field is changed stronger gradually, the fluctuation of the photon number in the light field is changed between the classical (or quantum) and the quantum (or classical) properties alternatively.