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Quantum and Semiclassical Optics Journal of the European Optical Society Part B

Boson inverse operators and associated coherent states

Anil Roy, C. L. Mehta

Published Oct 1, 1995• 12 citations

Abstract

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.

DOI: 10.1088/1355-5111/7/5/009

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In this paper, we introduce a new family of photon-added as well as photon-depleted q -deformed coherent states related to the inverse q -boson operators. These states are constructed via the generalized inverse q -boson operator actions on a newly introduced family of q -deformed coherent states (Quesne C 2002 J. Phys. A: Math. Gen. 35 9213 ) which are defined by slightly modifying the maths-type q -deformed coherent states. The quantum statistical properties of these photon-added and photon-depleted states, such as quadrature squeezing and photon-counting statistics, are discussed analytically and numerically in the context of both conventional (nondeformed) and deformed quantum optics.

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}

, 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.

A new kind of nonlinear coherent states, i.e. the k eigenstates of the powers B^k (k ≥ 3) of the generalized annihilation operator B^=a^†-1f(N^) of f-oscillators, are constructed using the inverse operators of the boson annihilation and creation operators. The set of these states may form a complete quantum mechanics representation. Using newly defined higher order squeezing and an antibunching, the properties of the Mth-order squeezing and the antibunching effect of the k states are investigated. The results show that the Mth-order [M = (n + 1/2)k; n = 0, 1, 2, …] squeezing effects exist in all of the k states when k is even and the antibunching effect exists in all of the k states.