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Journal of Physics A Mathematical and General

On the completeness of photon-added coherent states

J. M. Sixdeniers, K. A. Penson

Published Mar 26, 2001• 32 citations
Publisher

Abstract

We demonstrate explicitly the completeness of photon-added coherent states (PACSs), introduced by Agarwal and Tara (Agarwal G S and Tara K 1991 Phys. Rev. A 43 492) and defined, up to normalization, by (â † ) M | z ⟩, M = 0,1,2,...,, where â † is the boson creation operator and | z ⟩ are conventional Glauber-Klauder coherent states. We find the analytical form of the positive weight function in their resolution of unity by solving the associated Stieltjes power-moment problem. We furnish an example of generation of another set of PACSs which are complete.

DOI
10.1088/0305-4470/34/13/314

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We obtain and investigate the regular eigenfunctions of simple differential operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one. With the help of these eigenfunctions we construct a non-unitary analogue of boson displacement operator which will be acting on the vacuum. In this way we generate collective quantum states of the Fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly. These states are thus coherent states in the sense of Klauder. They span the truncated Fock space without first r lowest-lying basis states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in nature and are reminiscent of photon-added states at Agarwal et al.

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We demonstrate explicitly the completeness of photon-added coherent states (PACSs), introduced by Agarwal and Tara (Agarwal G S and Tara K 1991 Phys. Rev. A 43 492) and defined, up to normalization, by (â † ) M | z ⟩, M = 0,1,2,...,, where â † is the boson creation operator and | z ⟩ are conventional Glauber-Klauder coherent states. We find the analytical form of the positive weight function in their resolution of unity by solving the associated Stieltjes power-moment problem. We furnish an example of generation of another set of PACSs which are complete.
In the present work, we construct and investigate some properties of the photon-added spin coherent states (PA-SCSs). The Klauder’s minimal set of conditions required to obtain coherent states are discussed. We give the analytical form for the positive weight function in the resolution of unity. Finally, we examine the statistical properties of the PA-SCSs in terms of different parameters using the Mandel’s Q-parameter. All these quantities are expressed in terms of hypergeometric and Meijer G-functions, and so, the PA-SCSs are a new field of application for these functions.
We construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation dε(q,x)/dx=ε(q,qx) with 0⩽q⩽1 and ε(q,0)=1. We use this function in place of the usual exponential to generate new coherent states |q,z〉 from the vacuum, which are normalized and continuous in their label z. These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally-ordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is sub-Poissonian. For any q<1 the states |q,z〉 are squeezed; we obtain and discuss their signal to quantum noise ratio. The function ε(q,x) allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For q=1 all the above results reduce to those for conventional coherent states. Finally, we establish a link with q-deformed bosons.
The photon-added nonlinear coherent states, defined as adaggerm∣alpha,f>{a}^{\\dagger m}|\\alpha, f>adaggerm∣alpha,f> up to a normalization constant, are still nonlinear coherent states with the nonlinear function f(N−m)[1−m/(N+1)]f(N-m)[1-m/(N+1)]f(N−m)[1−m/(N+1)]. Here % |\\alpha, f> is the nonlinear coherent state with the nonlinear function f(N)f(N)f(N), NNN is the number operator, adaggera^{\\dagger}adagger is the boson creation operator and mmm is a non-negative integer. We introduce a new type of nonlinear coherent states with the nonlinear function f(N+m)[1+m/(N+1)]f(N+m)[1+m/(N+1)]f(N+m)[1+m/(N+1)] obtained from the photon-added nonlinear coherent states with negative values of m.m.m. The nonlinear coherent states corresponding to the positive and negative values of mmm are shown to be the result of nonunitarily deforming the number states ∣m>|m>∣m> and ∣0>|0>∣0>, respectively. As an example, we study the sub-Poissonian statistics and squeezing effects of the photon-added geometric states with negative values of mmm in detail.
We obtain and investigate the regular eigenfunctions of simple differential operators x^r d^{r+1}/dx^{r+1}, r=1, 2, ... with the eigenvalues equal to one. With the help of these eigenfunctions we construct a non-unitary analogue of boson displacement operator which will be acting on the vacuum. In this way we generate collective quantum states of the Fock space which are normalized and equipped with the resolution of unity with the positive weight functions that we obtain explicitly. These states are thus coherent states in the sense of Klauder. They span the truncated Fock space without first r lowest-lying basis states: |0>, |1>, ..., |r-1>. These states are squeezed, are sub-Poissonian in nature and are reminiscent of photon-added states at Agarwal et al.