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Journal of Modern Optics

Boson Inverse Operators and a New Family of Two-photon Annihilation Operators

Anil Roy, C. L. Mehta

Published Mar 1, 1995• 2 citations
Publisher

Abstract

We introduce the inverse of the Hermitian operator (ââ †) and express the Boson inverse operators â −1 and â †−1 in terms of the operators â, â † and (ââ †)−1. We show that these Boson inverse operators may be realized by Susskind-Glogower phase operators. In this way, we find a new two-photon annihilation operator and denote it as â 2(ââ †)−1. We show that the eigenstates of this operator have interesting non-classical properties. We find that the eigenstates of the operators (ââ †)−1 â 2, â(ââ †)−1 â and â 2(ââ †)−1 have many similar properties and thus they constitute a family of two-photon annihilation operators.

DOI
10.1080/09500349514550631

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We introduce the inverse of the Hermitian operator (ââ †) and express the Boson inverse operators â −1 and â †−1 in terms of the operators â, â † and (ââ †)−1. We show that these Boson inverse operators may be realized by Susskind-Glogower phase operators. In this way, we find a new two-photon annihilation operator and denote it as â 2(ââ †)−1. We show that the eigenstates of this operator have interesting non-classical properties. We find that the eigenstates of the operators (ââ †)−1 â 2, â(ââ †)−1 â and â 2(ââ †)−1 have many similar properties and thus they constitute a family of two-photon annihilation operators.
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.
Making use of the recently introduced inverses of boson creation and annihilation operators, the authors show that the 'photon-added coherent states' are eigenstates of the operator (a Dagger m aa Dagger -m ) and also of the operator (a-ma Dagger -1 ) with eigenvalue alpha . The authors discuss completeness of these states. They obtain another family of photon-added coherent states which are proportional to a -m mod alpha ), where m is a positive integer and mod alpha ) is a coherent state. They show that these are the eigenstates of the operator (a -m a m+1 ) with eigenvalue alpha . Analogously, the authors introduce 'photon-depleted coherent states', a Dagger -m mod alpha ), and show that these are the eigenstates of the operator (a+ma Dagger -1 ) with eigenvalue alpha . The conditions for squeezing and antibunching of photons have also been discussed.
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 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.
The inverse of paraboson annihilation and creation operators is introduced. The properties of these operators, their number state representation and commutation relations are discussed. Also introduced are the family of eigenstates of bilinears in paraboson operators and their inverses; and study the nonclassical properties, namely, squeezing and fluctuations in number of parabosons in these eigenstates. It is observed that a family of the eigenstates of a+−1a is the parasqueezed vacuum, S(σ)‖0,α〉. Similarly, a family of the eigenstates of aa+−1 is the parasqueezed first state, S(σ)‖n=1,α〉.