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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 393417, 17 pages
Research Article

Generalized Binomial Probability Distributions Attached to Landau Levels on the Riemann Sphere

1Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. BOX 1014, Agdal, Rabat 10000, Morocco
2Department of Mathematics, Faculty of Technical Sciences, Sultan Moulay Slimane University, P.O. Box 523, Béni Mellal 23000, Morocco

Received 8 March 2011; Accepted 29 March 2011

Academic Editor: Ali Mostafazadeh

Copyright © 2011 A. Ghanmi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A family of generalized binomial probability distributions attached to Landau levels on the Riemann sphere is introduced by constructing a kind of generalized coherent states. Their main statistical parameters are obtained explicitly. As an application, photon number statistics related to coherent states under consideration are discussed.

1. Introduction

The binomial states (BS) are the field states that are defined as a finite linear superposition of field number states weighted by a binomial counting probability distribution [1, 2]. Precisely, these states are labeled by points 𝑧 of the Riemann sphere 𝕊2{}, and are of the from ||𝑧,𝐵=1+|𝑧|2𝐵2𝐵𝑗=0(2𝐵)!𝑗!(2𝐵𝑗)!1/2𝑧𝑗||𝑗,(1.1) where 𝐵+ is a fixed integer parameter and |𝑗 are number states of the field mode. Define 𝜇𝑧=|𝑧|2(1+|𝑧|2)1, then the probability for the production of 𝑗 photons is given by the squared modulus of the projection of the BS |𝑧,𝐵 onto the number state |𝑗 as ||||𝑗𝑧,𝐵2=(2𝐵)!𝜇𝑗!(2𝐵𝑗)!𝑗𝑧1𝜇𝑧2𝐵𝑗.(1.2) The latter is recognized as the binomial probability density (2𝐵,𝜇𝑧) where {𝜇𝑧,(1𝜇𝑧)} are the probabilities of the two possible outcomes of a Bernoulli trial [3]. Also, observe that the coefficients in the finite superposition of number states in (1.1) 𝐵𝑗(𝑧)=1+|𝑧|2𝐵(2𝐵)!𝑗!(2𝐵𝑗)!1/2𝑧𝑗,𝑗=0,1,2,,2𝐵,(1.3) constitute an orthonormal basis of the null space 𝒜𝐵𝕊2=𝜑𝐿2𝕊2,𝐻𝐵[𝜑]=0(1.4) of the second-order differential operator 𝐻𝐵=1+|𝑧|22𝜕2𝜕𝑧𝜕𝑧𝐵1+|𝑧|2𝑧𝜕𝜕𝑧𝑧𝜕𝜕𝑧+𝐵2|𝑧|2𝐵,(1.5) which constitutes (in suitable units and up to additive constant) a realization in 𝐿2(𝕊2) of the Schrödinger operator with uniform magnetic field on 𝕊2, with a field strength proportional to 𝐵 (see [4]). The given orthonormal basis 𝐵𝑗(𝑧) together with the reproducing kernel 𝐾𝐵(𝑧,𝑤)=(2𝐵+1)1+𝑧𝑤2𝐵1+|𝑧|2𝐵1+|𝑤|2𝐵(1.6) of the Hilbert space 𝒜𝐵(𝕊2) in (1.4) can be used to interpret the projection of the BS |𝑧,𝐵 onto the number state |𝑗 mentioned in (1.2) by writing 𝐾𝑗𝑧,𝐵=𝐵(z,𝑧)(1/2)𝐵𝑗(𝑧).(1.7) Note also that the space 𝒜𝐵(𝕊2) is nothing else than the eigenspace associated with the first eigenvalue of the spectrum of 𝐻𝐵 acting on 𝐿2(𝕊2), which consists of an infinite set of eigenvalues (spherical Landau levels) of the form 𝜖𝐵𝑚=(2𝑚+1)𝐵+𝑚(𝑚+1),𝑚=0,1,2,, with finite multiplicity; that is, the associated 𝐿2-eigenspace 𝒜𝐵,𝑚𝕊2=𝜑𝐿2𝕊2,𝐻𝐵[𝜑]=𝜖𝐵𝑚𝜑(1.8) is of finite dimension equals to 𝑑𝐵,𝑚=2𝐵+2𝑚+1.

Here, we take the advantage of the fact that each of the eigenspaces in (1.8) admits an orthogonal basis, denoted 𝑗𝐵,𝑚(𝑧),𝑗=0,1,2,,2𝐵+2𝑚, whose elements are expressed in terms of Jacobi polynomials 𝑃𝜂(𝜏,𝜍)(), as well as a reproducing kernel 𝐾𝐵,𝑚(𝑧,𝑤) in an explicit form (see [5]) to consider a set of coherent states by adopting a generalized coherent states form “à la Iwata” [6] as ||𝐾𝑧,𝐵,𝑚=𝐵,𝑚(𝑧,𝑧)(1/2)2𝐵+2𝑚𝑗=0𝑗𝐵,𝑚(𝑧)𝜌𝐵,𝑚||(𝑗)𝑗,(1.9) where 𝜌𝐵,𝑚(𝑗) denotes the norm square of 𝑗𝐵,𝑚(𝑧) in 𝐿2(𝕊2). The coherent states in (1.9) possess a form similar to (1.1) and will enable us, starting from the observation made in (1.7), to attach to each eigenspace 𝒜𝐵,𝑚(𝕊2) a photon counting probability distribution in the same way as for the space 𝒜𝐵(𝕊2)𝒜𝐵,0(𝕊2) through the quantities 𝑝𝑗2𝐵,𝜇𝑧=,𝑚𝑚!(2𝐵+𝑚)!𝜇𝑗!(2𝐵+2𝑚𝑗)!𝑧𝑗𝑚1𝜇𝑧2𝐵+𝑚𝑗𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)12𝜇𝑧2,𝑗=0,1,,2𝐵+2𝑚.(1.10) The latter can be considered as a kind of generalized binomial probability distribution 𝑋(2𝐵,𝜇𝑧,𝑚) depending on an additional parameter 𝑚=0,1,2,. Thus, we study the main properties of the family of probability distributions in (1.10) and we examine the quantum photon counting statistics with respect to the location in the Riemann sphere of the point 𝑧 labeling the generalized coherent states introduced formally in (1.9).

The paper is organized as follows. In Section 2, we recall briefly the principal statistical properties of the binomial states. Section 3 deals with some needed facts on the Schrödinger operator with uniform magnetic field on the Riemann sphere with an explicit description of the corresponding eigenspaces. Section 4 is devoted to a coherent states formalism. This formalism is applied so as to construct a set of generalized coherent states attached to each spherical Landau level. In Section 5, we introduce the announced generalized binomial probability distribution and we give its main parameters. In Section 6, we discuss the classicality/nonclassicality of the generalized coherent states with respect to the location of their labeling points belonging to the Riemann sphere.

2. The Binomial States

The binomial states in their first form were introduced by Stoler et al. [1] to define a pure state of a single mode of the electromagnetic field for which the photon number density is binomial. Like the generalized coherent states (whose coefficients of its 𝑗 states expansion are allowed to have additional arbitrary phases), generalized binomial states (GBS) can be defined by ||𝑛,𝜇,𝜃=𝑛𝑗=0𝑛!𝜇𝑗!(𝑛𝑗)!𝑗(1𝜇)𝑛𝑗1/2𝑒𝑖𝑗𝜃||𝑗.(2.1) The corresponding photon counting probability is given by𝑝𝑗(𝑛,𝜇)=𝑛!𝜇𝑗!(𝑛𝑗)!𝑗(1𝜇)𝑛𝑗(2.2) which follows the binomial law 𝑌(𝑛,𝜇) with parameters 𝑛 and 𝜇; 𝑛+, 0<𝜇<1. The connection with our notations in (1.1) and (1.2) can be made by setting 𝑛=2𝐵, 𝑧=|𝑧|𝑒𝑖𝜃 and |𝑧|2=𝜇(1𝜇)1. Note that in limits 𝜇0 and 𝜇1 the binomial state reduces to number states |0 and |𝑛, respectively. In a different limit of 𝑛+ and 𝜇0 with 𝑛𝜇𝜆, the probability distribution (2.2) goes to the Poisson distribution 𝒫(𝜆)𝑝𝑗𝜆(𝜆)=𝑗𝑒𝑗!𝜆,𝑗=0,1,2,,(2.3) which characterizes the coherent states of the harmonic oscillator. In fact, and as pointed out in [1], the binomial states interpolate between number states (nonclassical states) and coherent states (classical states). It partakes of the properties of both and reduces to each in different limits.

The characteristic function of the random variable 𝑌(𝑛,𝜇) is given by 𝒞𝑌(𝑡)=(1𝜇)+𝜇𝑒𝑖𝑡𝑛(2.4) from which one obtains the mean value and the variance as 𝐸(𝑌)=𝑛𝜇,Var(𝑌)=𝑛𝜇(1𝜇).(2.5) Therefore, the Mandel parameter, which measures deviation from the Poissonian distribution, 𝑄=Var(𝑌)𝐸(𝑌)1=𝜇,(2.6) is always negative. Thus photon statistics in the binomial states is always sub-Poissonian.

Remark 2.1. One of the peculiarities of the GBS in (2.1) is that they can be exploited as reference states within schemes devoted at measuring the canonical phase of quantum electromagnetic fields. Moreover, they are the electromagnetic correspondent of the well-known coherent atomic states [7]. For more information and applications involving these binomial states in the context of cavity quantum electrodynamic, see, for example, [8, 9]. We should note that they also admit also a ladder operator definition which means that they are eigenstate of a proper combination of the number operator and the annihilation operator via the Holstein-Primakoff realization of the Lie algebra su(2) [10].

3. An Orthonormal Basis of 𝒜𝐵,𝑚(𝕊2)

Let 𝕊23 denote the unit sphere with the standard metric of constant Gaussian curvature 𝜅=1. We identify the sphere 𝕊2 with the extended complex plane {}, called the Riemann sphere, via the stereographic coordinate 𝑧=𝑥+𝑖𝑦; 𝑥,𝑦. We shall work within a fixed coordinate neighborhood with coordinate 𝑧 obtained by deleting the “point at infinity” {}. Near this point we use instead of 𝑧 the coordinate 𝑧1.

In the stereographic coordinate 𝑧, the Hamiltonian operator of the Dirac monopole with charge 𝑞=2𝐵 reads [4, page 598]𝐻𝐵=1+|𝑧|22𝜕2𝜕𝑧𝜕𝑧𝐵𝑧1+|𝑧|2𝜕𝜕𝑧+𝐵𝑧1+|𝑧|2𝜕𝜕𝑧+𝐵21+|𝑧|2𝐵2.(3.1) This operator acts on the sections of the 𝑈(1)-bundle with the first Chern class 𝑞. We have denoted by 𝐵+ the strength of the quantized magnetic field. We shall consider the Hamiltonian 𝐻𝐵 in (3.1) acting in the Hilbert space 𝐿2(𝕊2)=𝐿2(𝕊2,(1+|𝑧|2)2𝑑𝜈(𝑧)), 𝑑𝜈(𝑧)=𝜋1𝑑𝑥𝑑𝑦 being the Lebesgue measure on 2. Its spectrum consists of an infinite number of eigenvalues (spherical Landau levels) of the form𝜖𝐵𝑚=(2𝑚+1)𝐵+𝑚(𝑚+1),𝑚=0,1,2,,(3.2) with finite degeneracy 2𝐵+2𝑚+1 (see [4, page 598]). In order to present expressions of the corresponding eigensections in the coordinate 𝑧, we first mention that the shifted operator 𝐻𝐵𝐵 on 𝐿2(𝕊2) is intertwined with the invariant Laplacian Δ2𝐵=1+|𝑧|22𝜕2𝜕𝑧𝜕𝑧+2𝐵𝑧1+|𝑧|2𝜕𝜕𝑧(3.3) acting in the Hilbert space 𝐿2,𝐵(𝕊2)=𝐿2(𝕊2,(1+|𝑧|2)22𝐵𝑑𝜈(𝑧)). Namely, we have𝐻𝐵𝐵=1+|𝑧|2𝐵Δ2𝐵1+|𝑧|2𝐵,(3.4) and therefore any ket |𝜙 of 𝐿2,𝐵(𝕊2) is represented by1+|𝑧|2𝐵𝑧𝜙in𝐿2𝕊2.(3.5) As mentioned in the introduction, we denote by 𝒜𝐵,𝑚(𝕊2) the eigenspace of 𝐻𝐵 in 𝐿2(𝕊2), corresponding to the eigenvalue 𝜖𝐵𝑚 given in (3.2). Then, by [5] together with (3.5) and the intertwining relation (3.4), we obtain the following orthogonal basis of this eigenspace: 𝑗𝐵,𝑚(𝑧)=1+|𝑧|2𝐵𝑧𝑗𝑄𝐵,𝑚,𝑗|𝑧|21+|𝑧|2,0𝑗2𝐵+2𝑚,(3.6) where 𝑄𝐵,𝑚,𝑗() is the polynomial function given by𝑄𝐵,𝑚,𝑗(𝑡)=𝑡𝑗(1𝑡)𝑗2𝐵𝑑𝑑𝑡𝑚𝑡𝑗+𝑚(1𝑡)2𝐵𝑗+𝑚.(3.7) According to the Jacobi's formula [11]𝑑𝑑𝑥𝑚𝑥𝑐+𝑚1(1𝑥)𝑏𝑐=Γ(𝑐+𝑚)𝑥Γ(𝑐)𝑐1(1𝑥)2𝑏𝑐𝑚𝐹1(𝑚,𝑏;𝑐;𝑥),(3.8)2𝐹1(𝑎,𝑏,𝑐;𝑥) being the Gauss hypergeometric function, it follows that 𝑄𝐵,𝑚,𝑗(𝑡)=(𝑚+𝑗)!𝑗!2𝐹1(𝑚,2𝐵+𝑚+1,𝑗+1;𝑡).(3.9) The latter can also be expressed in terms of Jacobi polynomials via the transformation [11, page 283] 2𝐹1𝑘+𝜈+𝜚+1,𝑘,1+𝜈;1𝑡2=𝑘!Γ(1+𝜈)𝑃Γ(𝑘+1+𝜈)𝑘(𝜈,𝜚)(𝑡).(3.10) So that the orthogonal basis in (3.6) reads simply in terms of Jacobi polynomial as 𝑗𝐵,𝑚(𝑧)=𝑚!1+|𝑧|2𝐵𝑧𝑗𝑚𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)1|𝑧|21+|𝑧|2.(3.11) Also, one obtains the norm square of the eigenfunction 𝑗𝐵,𝑚 given in (3.6) as 𝜌𝐵,𝑚(𝑗)=𝑗𝐵,𝑚2𝐿2(𝕊2)=𝑚!(𝑚+𝑗)!(2𝐵+𝑚𝑗)!.(2𝐵+2𝑚+1)(2𝐵+𝑚)!(3.12) Finally, by Theorem 1 of [5, page 231] and thanks to (3.5), we obtain the following expression for the reproducing kernel of the eigenspace 𝒜𝐵,𝑚(𝕊2): 𝐾𝐵,𝑚(𝑧,𝑤)=(2𝐵+2𝑚+1)=1+𝑧𝑤2𝐵1+|𝑧|2𝐵1+|𝑤|2𝐵×2𝐹1𝑚,𝑚+2𝐵+1,1;|𝑧𝑤|21+|𝑧|21+|𝑤|2.(3.13)

Remark 3.1. In the case 𝑚=0, the elements of the orthogonal basis reduce further to 𝑗𝐵,0(𝑧)=(1+|𝑧|2)𝐵𝑧𝑗 and the reproducing kernel reads 𝐾𝐵,0(𝑧,𝑤)=(2𝐵+1)1+𝑧𝑤2𝐵1+|𝑧|2𝐵1+|𝑤|2𝐵.(3.14)

Remark 3.2. In the case of the 𝑛-dimensional projective space (𝑛)(=𝑆1𝑆2𝑛+1) equipped with the Fubini-Study metric, explicit formulae for the reproducing kernels of the eigenspaces associated with the Schrödinger operator with constant magnetic field written in the local coordinates (of the chart 𝑛) as 𝐻𝐵=1+|𝑧|2𝑛𝑖,𝑗=1𝛿𝑖𝑗+𝑧𝑖𝑧𝑗𝜕2𝜕𝑧𝑖𝜕𝑧𝑗𝐵𝑛𝑗=1𝑧𝑗𝜕𝜕𝑧𝑗𝑧𝑗𝜕𝜕𝑧𝑗𝐵2+𝐵2(3.15) have been obtained in [12].

4. Generalized Coherent States

We follow the formalism presented in [13] for constructing generalized coherent states, which can be considered as a generalization of the canonical ones when written as series expansion in the Fock basis according to [6]. Let (,,) be a functional Hilbert space with an orthonormal basis {𝜙𝑛}𝑑𝑛=1 and 𝒜2 a finite d-dimensional subspace of the Hilbert space 𝐿2(Ω,𝑑𝑠), of square integrable functions on a given measured space (Ω,𝑑𝑠), with an orthogonal basis {Φ𝑛}𝑑𝑛=1. Then, 𝒜2 is a reproducing kernel Hilbert space whose reproducing kernel is given by 𝐾(𝑥,𝑦)=𝑑𝑛=1Φ𝑛(𝑥)Φ𝑛(𝑦)𝜌𝑛;𝑥,𝑦Ω,(4.1) where we have set 𝜌𝑛=Φ𝑛2𝐿2(Ω,𝑑𝑠). Associated to the data of (𝒜2,Φ𝑛) and (,𝜙𝑛), the generalized coherent states are the elements of defined by Φ𝑥𝜔=𝑑(𝑥)𝑑(1/2)𝑛=1Φ𝑛(𝑥)𝜌𝑛𝜙𝑛;𝑥Ω,(4.2) where 𝜔𝑑(𝑥) stands for 𝜔𝑑(𝑥)=𝐾(𝑥,𝑥). Note that the choice of the Hilbert space defines a quantization of Ω into by considering the inclusion map 𝑥Φ𝑥. Furthermore, it is straightforward to check that Φ𝑥,Φ𝑥=1 and to show that the corresponding coherent state transform (CST) 𝒲 on, 𝒲[𝑓]𝜔(𝑥)=𝑑(𝑥)1/2Φ𝑥,𝑓;𝑓,(4.3) defines an isometry from into 𝒜2. Thereby we have a resolution of the identity of , that is, we have the following integral representation: 𝑓()=ΩΦ𝑥,𝑓Φ𝑥()𝜔𝑑(𝑥)𝑑𝑠(𝑥);𝑓.(4.4)

Remark 4.1. Note that formula (4.2) can be considered as a generalization (in the finite dimensional case) of the series expansion of the well-known canonical coherent states ||𝑒𝜁=|𝜁|2(1/2)𝑘0𝜁𝑘||𝑘!𝑘(4.5) with 𝜙𝑘=|𝑘 being the number states of the harmonic oscillator, which also knwon as Fock states.

We can now construct for each spherical Landau level 𝜖𝐵𝑚 given in (3.2) a set of generalized coherent states (GCS) according to formula (4.2) as𝜗𝑧,𝐵,𝑚||𝐾𝑧,𝐵,𝑚=𝐵,𝑚(𝑧,𝑧)(1/2)2𝐵+2𝑚𝑗=0𝑗𝐵,𝑚(𝑧)𝜌𝐵,𝑚||𝜙(𝑗)𝑗(4.6) with the following precisions:(i)(Ω,𝑑𝑠)=(𝕊2,(1+|𝑧|2)2𝑑𝜈(𝑧)), 𝕊2 being identified with {}.(ii)𝒜2=𝒜𝐵,𝑚(𝕊2) is the eigenspace of 𝐻𝐵 in 𝐿2(𝕊2) with dimension 𝑑𝐵,𝑚=2𝐵+2𝑚+1.(iii)𝜔(𝑧)=𝐾𝐵,𝑚(𝑧,𝑧)=2𝐵+2𝑚+1 (in view of (3.13)).(iv)𝑗𝐵,𝑚(𝑧) are the eigenfunctions given by (3.11) in terms of the Jacobi polynomials.(v)𝜌𝐵,𝑚(𝑗) being the norm square of 𝑗𝐵,𝑚, given in (3.12).(vi)=𝒫𝐵+𝑚 the space of polynomials of degree less than 𝑑𝐵,𝑚, which carries a unitary irreducible representation of the compact Lie group SU(2) (see [14]). The scalar product in 𝒫𝐵+𝑚 is written as 𝜓,𝜙𝒫𝐵+𝑚=𝑑𝐵,𝑚||𝜁||𝑑𝑠(𝜁)1+22(𝐵+𝑚)2𝜓(𝜁)𝜙(𝜁).(4.7)(vii){𝜙𝑗;0𝑗2𝐵+2𝑚} is an orthonormal basis of 𝒫𝐵+𝑚, whose elements are given explicitly by𝜙𝑗(𝜉)=(2(𝐵+𝑚))!(𝜉2𝐵+𝑚𝑗)!(𝑗+𝑚)!𝑗+𝑚𝜁.(4.8)

Definition 4.2. For given positive integers 2𝐵 and 𝑚, and fixed 𝑧𝕊2, the wave function of the GCS in (4.6) are expressed as 𝜗𝑧,𝐵,𝑚1(𝜉)=1+|𝑧|2𝐵2𝐵+2𝑚𝑗=0𝑚!(2𝐵+𝑚)!(2𝐵+2𝑚)!𝑧𝑗!(2𝐵+2𝑚𝑗)!𝑗𝑚𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)1|𝑧|21+|𝑧|2𝜉𝑗,(4.9) for every 𝜁.

According to (4.4), the system of GCS |𝜗𝑧,𝐵,𝑚 solves then the unity of the Hilbert space 𝒫𝐵+𝑚 as𝟏𝒫𝐵+𝑚=𝑑𝐵,𝑚𝑑𝜈(𝑧)1+|𝑧|22||𝜗𝑧,𝐵,𝑚𝜗𝑧,𝐵,𝑚||.(4.10) They also admit a closed form [15], as 𝜗𝑧,𝐵,𝑚(𝜉)=(2𝐵+2𝑚)!(2𝐵+𝑚)!𝑚!(1+𝜉𝑧)21+|𝑧|2𝐵𝜉𝑧(1+𝜉𝑧)1+|𝑧|2𝑚.(4.11)

Remark 4.3. Note that for 𝑚=0, the expression above reduces to 𝜉𝑧,𝐵,0=1+|𝑧|2𝐵(1+𝜉𝑧)2𝐵,(4.12) which are wave functions of Perelomov's coherent states based on SU(2) (see [16, page 62]).

5. Generalized Binomial Probability Distributions

According to (4.6), we see that the squared modulus of the projection of coherent state 𝜗𝑧,𝐵,𝑚 onto the state 𝜙𝑗, is given by ||𝜗𝑧,𝐵,𝑚,𝜙𝑗||2=1𝜌𝑗𝐵,𝑚𝑑𝐵,𝑚||𝑗𝐵,𝑚||(𝑧)2.(5.1) This is in fact the probability of finding 𝑗 photons in the coherent state 𝜗𝑧,𝐵,𝑚. More explicitly, in view of (3.11), the quantity in (5.1) reads ||𝜗𝑧,𝐵,𝑚,𝜙𝑗||2=𝑚!(2𝐵+𝑚)!𝑗!(2𝐵+2𝑚𝑗)!1+|𝑧|22𝐵|𝑧|2(𝑗𝑚)𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)1|𝑧|21+|𝑧|22.(5.2) As annonced in the introduction, we denote the expression in (5.2) by 𝑝𝑗(2𝐵,𝜇𝑧,𝑚) for 𝑗=0,1,2,, with 𝜇𝑧=|𝑧|2(1+|𝑧|2)1 or equivalently |𝑧|2=𝜇𝑧(1𝜇𝑧)1. Motivated by this quantum probability, we then state the following.

Definition 5.1. For fixed integers 𝐵,𝑚+, the discrete random variable 𝑋 having the probability distribution 𝑝𝑗2𝐵,𝜇𝑧=,𝑚𝑚!(2𝐵+𝑚)!𝜇𝑗!(2𝐵+2𝑚𝑗)!𝑧(𝑗𝑚)1𝜇𝑧2𝐵+𝑚𝑗𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)12𝜇𝑧2,(5.3) with 𝑗=0,1,2,,2𝐵+2𝑚, and denoted by 𝑋(2𝐵,𝜇𝑧,𝑚), 0<𝜇𝑧<1, will be called the generalized binomial probability distribution associated to the weighted Hilbert space 𝒜𝐵,𝑚(𝑆2).

Remark 5.2. Note that for 𝑚=0, the above expression in (5.3) reduces to 𝑝𝑗2𝐵,𝜇𝑧=,0(2𝐵)!𝜇𝑗!(2𝐵𝑗)!𝑗𝑧1𝜇𝑧2𝐵𝑗,𝑗=0,1,2,,2𝐵,(5.4) which is the standard binomial distribution with parameters 2𝐵 and 0<𝜇𝑧<1.

A convenient way to summarize all the properties of a probability distribution 𝑋 is to explicit its characteristic function: 𝒞𝑋𝑒(𝑡)=𝐸𝑖𝑡𝑋,(5.5) where 𝑡 is a real number, 𝑖=1 is the imaginary unit, and 𝐸 denotes the expected value.

Proposition 5.3. For fixed 𝑚=0,1,2,, the characteristic function of 𝑋(2𝐵,𝜇𝑧,𝑚) is given by 𝒞𝑚(𝑡)=𝑒𝑖𝑚𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡2𝐵𝑃𝑚(0,2𝐵)14𝜇𝑧1𝜇𝑧(1cos(𝑡))(5.6) for every 𝑡.

Proof. Recall first that for every given fixed nonnegative integer 𝑚, the characteristic function 𝒞𝑚(𝑡) in (5.5) can be written as 𝒞𝑚(𝑡)=2𝐵+2𝑚𝑗=0𝑒𝑖𝑗𝑡𝑝𝑗2𝐵,𝜇𝑧=,𝑚2𝐵+2𝑚𝑗=0𝑒𝑖𝑗𝑡𝑚!(2𝐵+𝑚)!𝜇𝑗!(2𝐵+2𝑚𝑗)!𝑧(𝑗𝑚)1𝜇𝑧2𝐵+𝑚𝑗𝑃𝑚(𝑗𝑚,2𝐵+𝑚𝑗)12𝜇𝑧2.(5.7) The last equality follows using (5.3). Next, by making the change 𝑘=𝐵+𝑚𝑗 in (5.7), it follows that 𝒞𝑚(𝑡)=𝐵+𝑚𝑘=(𝐵+𝑚)𝑒𝑖(𝐵+𝑚𝑘)𝑡𝑚!(2𝐵+𝑚)!𝜇(𝐵+𝑚+𝑘)!(𝐵+𝑚𝑘)!𝑧𝐵𝑘1𝜇𝑧𝐵+𝑘𝑃𝑚(𝐵𝑘,𝐵+𝑘)12𝜇𝑧2.(5.8) Instead of the Jacobi polynomials, it is convenient to consider the closely related function 𝒫𝑙𝑟,𝑠(𝑥) introduced in [14, page 270]. They can be defined through the formula [14, equation (1), page 288], 𝑃(𝑟𝑠,𝑟+𝑠)𝑛𝑟(𝑥)=2𝑟(𝑛𝑠)!(𝑛+𝑠)!(𝑛𝑟)!(𝑛+𝑟)!1/2(1𝑥)(𝑠𝑟)/2(1+𝑥)(𝑠+𝑟)/2𝒫𝑛𝑟,𝑠(𝑥),(5.9) with 𝑚=𝑛𝐵 (i.e., 𝑛=𝐵+𝑚) and 𝑥=12𝜇𝑧. We can then express the square of 𝑃𝑚(𝐵𝑘,𝐵+𝑘)(𝑥) as follows: 𝑃𝑚(𝐵𝑘,𝐵+𝑘)12𝜇𝑧2=(𝐵+𝑚𝑘)!(𝐵+𝑚+𝑘)!𝜇𝑚!(2𝐵+𝑚)!𝑧𝐵+𝑘1𝜇𝑧𝐵𝑘𝒫𝐵+𝑚𝐵,𝑘12𝜇𝑧2.(5.10) Therefore, (5.8) reduces further to 𝒞𝑚(𝑡)=𝑒𝑖(𝐵+𝑚)𝑡𝐵+𝑚𝑘=(𝐵+𝑚)𝑒𝑖𝑘𝑡𝒫𝐵+𝑚𝐵,𝑘12𝜇𝑧2(5.11)()=(1)𝐵𝑒𝑖(𝐵+𝑚)𝑡𝐵+𝑚𝑘=(𝐵+𝑚)𝑒𝑖𝑘(𝑡𝜋)𝒫𝐵+𝑚𝐵,𝑘12𝜇𝑧𝒫𝐵+𝑚𝑘,𝐵12𝜇𝑧=(1)𝐵𝑒𝑖(𝐵+𝑚)𝑡𝑒𝑖𝐵(𝜑+𝜓)𝒫𝐵+𝑚𝐵,𝐵(cos(𝜃)).(5.12) The transition () above holds using the fact that [14, page 288] 𝒫𝑙𝑗,𝑘(𝑥)=(1)𝑗+𝑘𝒫𝑙𝑘,𝑗(𝑥).(5.13) While the last equality can be checked easily using the addition formula [14, equation (3), page 326] 𝑠𝑘=𝑠𝑒𝑖𝑘𝜏𝒫𝑠𝑗,𝑘𝜃cos1𝒫𝑠𝑘,𝑙𝜃cos2=𝑒𝑖(𝑗𝜑+𝑙𝜓)𝒫𝑠𝑗,𝑙(cos(𝜃)).(5.14) Here the involved complex angles 𝜑, 𝜓, and 𝜃 are given through equations (8), (8′), and (8′′) in [14, page 270]. In our case, they yield the followings: cos(𝜃)=cos2(2𝛼)+sin2𝑒(2𝛼)cos(𝑡),𝑖((𝜑+𝜓)/2)=𝑖cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑡𝑒𝑖𝑡/2cos(𝜃/2)(5.15) for 𝜃1=𝜃2=2𝛼, so that 𝑒𝑖𝐵(𝜑+𝜓)=(1)𝐵𝜃cos22𝐵cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑡2𝐵𝑒𝑖𝐵𝑡.(5.16) Next, using the fact that 2𝑠(1+𝑥)𝑠𝑃(0,2𝑠)𝑛𝑠(𝑥)=𝒫𝑛𝑠,𝑠(𝑥),(5.17) which is a special case of (5.9), with 𝑠=𝐵, 𝑛𝑠=𝑚, and 𝑥=cos(𝜃), we obtain 𝒫𝐵+𝑚𝐵,𝐵𝜃(cos(𝜃))=cos22𝐵𝑃𝑚(0,2𝐵)(cos(𝜃)).(5.18) Finally, by substituting (5.16) and (5.18) in (5.12), taking into account that sin2(𝛼)=𝜇𝑧 and cos2(𝛼)=1𝜇𝑧, we see that the characteristic function 𝒞𝑚(𝑡) reads simply as 𝒞𝑚(𝑡)=𝑒𝑖𝑚𝑡cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑡2𝐵𝑃𝑚(0,2𝐵)(cos(𝜃)),(5.19) where cos(𝜃)=14𝜇𝑧(1𝜇z)(1cos(𝑡)).

Remark 5.4. Note that by taking 𝑚=0 in (5.19), the characteristic function reduces to 𝐶𝑌(𝑡)=cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑡2𝐵=1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡2𝐵,(5.20) which is the well-known characteristic function of the binomial random variable 𝑌(2𝐵,𝜇𝑧) with parameters 𝑛=2𝐵+ and 0<𝜇𝑧<1 as in (2.4).

Now, the characteristic function contains important information about the random variable 𝑋. For example, various moments may be obtained by repeated differentiation of 𝒞𝑚(𝑡) in (5.6) with respect to the variable 𝑡 and evaluation at the origin as 𝐸𝑋𝑘=1𝑖𝑘𝜕𝑘𝜕𝑡𝑘𝐶𝑚||||(𝑡)𝑡=0.(5.21)

Corollary 5.5. Let 𝑚,2𝐵+. The mean value and the variance of 𝑋(2𝐵,𝜇𝑧,𝑚) are given respectively by 𝐸(𝑋)=𝑚+2𝐵𝜇𝑧,Var(𝑋)=2𝐵𝜇𝑧1𝜇𝑧+2𝜇𝑧1𝜇𝑧𝑚(2𝐵+𝑚+1).(5.22)

Proof. Let recall first that for every fixed integer 𝑚=0,1,2,, we have 𝐸(𝑋)=𝜕𝒞𝑚||||𝑖𝜕𝑡𝑡=0,𝑋Var(𝑋)=𝐸2[]𝐸(𝑋)2=𝜕2𝒞𝑚𝑖2𝜕𝑡2||||𝑡=0𝜕𝒞𝑚||||𝑖𝜕𝑡𝑡=02.(5.23) Thus direct computation gives rise to 𝜕𝒞𝑚𝑖𝜕𝑡(𝑡)=𝑚+2𝐵𝜇𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡4𝜇𝑧1𝜇𝑧sin(𝑡)(𝜕𝑃𝑚(0,2𝐵)||(𝑥)/𝑖𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)𝒞(cos(𝜃))𝑚𝜕(𝑡),2𝒞𝑚𝑖2𝜕𝑡2=𝜕(𝑡)𝑖𝜕𝑡𝑚+2𝐵𝜇𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡4𝜇𝑧1𝜇𝑧sin(𝑡)(𝜕𝑃𝑚(0,2𝐵)||(𝑥)/𝑖𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)(cos(𝜃))×𝒞𝑚(𝑡)+𝑚+2𝐵𝜇𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡4𝜇𝑧1𝜇𝑧sin(𝑡)(𝜕𝑃𝑚(0,2𝐵)(||𝑥)/𝑖𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)(cos(𝜃))𝜕𝒞𝑚(=𝑖𝜕𝑡𝑡)2𝐵𝜇𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡2𝐵𝜇2𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡2+4𝜇𝑧1𝜇𝑧cos(𝑡)(𝜕𝑃𝑚(0,2𝐵)||(𝑥)/𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)(cos(𝜃))×𝒞𝑚(𝑡)4𝜇𝑧1𝜇𝑧𝜕sin(𝑡)𝑖𝜕𝑡(𝜕𝑃𝑚(0,2𝐵)||(𝑥)/𝑖𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)𝒞(cos(𝜃))𝑚+(𝑡)𝑚+2𝐵𝜇𝑧𝑒𝑖𝑡1𝜇𝑧+𝜇𝑧𝑒𝑖𝑡4𝜇𝑧1𝜇𝑧sin(𝑡)(𝜕𝑃𝑚(0,2𝐵)||(𝑥)/𝑖𝜕𝑥)𝑥=cos(𝜃)𝑃𝑚(0,2𝐵)(cos(𝜃))𝜕𝒞𝑚𝑖𝜕𝑡(𝑡).(5.24) To conclude, we have to use successively the facts that for 𝑡=0, we have cos(𝜃)=1 and 𝒞𝑚(0)=1, together with 𝜕𝑃𝑚(𝑎,𝑏)𝑖𝜕𝑥(𝑥)=𝑎+𝑏+𝑚+12𝑃(𝑎+1,𝑏+1)𝑚1(𝑥),𝑃𝑚(𝑎,𝑏)(1)=Γ(𝑎+𝑚+1).𝑚!Γ(𝑎+1)(5.25) Thus, we have 𝐸(𝑋)=𝜕𝒞𝑚||||𝑖𝜕𝑡𝑡=0=𝑚+2𝐵𝜇𝑧𝒞𝑚(0)=𝑚+2𝐵𝜇𝑧.(5.26) We have also 𝜕2𝒞𝑚𝑖2𝜕𝑡2|𝑡=0=2𝐵𝜇𝑧1𝜇𝑧+2𝜇𝑧1𝜇𝑧𝑃(2𝐵+𝑚+1)(1,2𝐵+1)𝑚1(1)𝑃𝑚(0,2𝐵)(1)×𝒞𝑚(0)+𝑚+2𝐵𝜇𝑧𝜕𝒞𝑚|𝑖𝜕𝑡𝑡=0,(5.27) and therefore 𝜕Var(𝑋)=2𝒞𝑚𝑖2𝜕𝑡2||||𝑡=0𝜕𝒞𝑚||||𝑖𝜕𝑡𝑡=02=2𝐵𝜇𝑧1𝜇𝑧+2𝜇𝑧1𝜇𝑧𝑚(2𝐵+𝑚+1).(5.28)

Remark 5.6. Note that by taking 𝑚=0 in (5.22), we recover the standard values 𝐸(𝑌)=2𝐵𝜇𝑧,Var(𝑌)=2𝐵𝜇𝑧1𝜇𝑧(5.29) of the binomial probability distribution as given in (2.5).

6. Photon Counting Statistics

For an arbitrary quantum state one may ask to what extent is “nonclassical” in a sense that its properties differ from those of coherent states? In other words, is there any parameter that may reflect the degree of nonclassicality of a given quantum state? In general, to define a measure of nonclassicality of a quantum states one can follow several different approach. An earlier attempt to shed some light on the nonclassicality of a quantum state was pioneered by Mandel [17], who investigated radiation fields and introduced the parameter 𝑄=Var(𝑋)𝐸(𝑋)1,(6.1) to measure deviation of the photon number statistics from the Poisson distribution, characteristic of coherent states. Indeed, 𝑄=0 characterizes Poissonian statistics. If 𝑄<0, we have sub-Poissonian statistics, otherwise, statistics are super-Poissonian. In our context and for 𝑚=0, as mentioned in Section 2, the fact that the binomial probability distribution has a negative Mandel parameter, according to (5.29), and thereby the binomial states obey sub-Poissonian statistics. For 𝑚0, we make use of the obtained statistical parameters 𝐸(𝑋) and Var(𝑋) to calculate Mandel parameter corresponding to the random variable 𝑋(2𝐵,𝜇𝑧,𝑚). The discussion with respect to the sign of this parameter gives rise to the following statement.

Proposition 6.1. Let 𝑚 and 𝐵 be nonnegative integers and set 𝑟±1(𝐵,𝑚)=1±1𝑚(2𝐵+𝑚)1/21/2.(6.2) Then, 𝑟(𝐵,𝑚)1𝑟+(𝐵,𝑚) and the photon counting statistics are
(i) sub-Poissonian for points 𝑧 such that |𝑧|<𝑟(𝐵,𝑚) and |𝑧|>𝑟+(𝐵,𝑚),
(ii) Poissonian for points 𝑧 such that |𝑧|=𝑟(𝐵,𝑚) or |𝑧|=𝑟+(𝐵,𝑚),
(iii) super-Poissonian for 𝑧 such that 𝑟(𝐵,𝑚)<|𝑧|<𝑟+(𝐵,𝑚),

Proof. Assume that 𝑚0. Making use of (5.22), we see that the Mandel parameter can be written as follows 𝑄(𝑋)=𝑄𝑚(𝜇𝑧)=𝑇𝑚(𝜇𝑧)/(2𝐵𝜇𝑧+𝑚), where we have set 𝑇𝑚𝜇𝑧[]=2(𝐵+𝑚2𝐵+𝑚+1)𝜇2𝑧[]𝜇2𝑚2𝐵+𝑚+1𝑧=𝜇+𝑚𝑧𝑚𝑑𝐵,𝑚2𝐵+𝑚𝑑𝐵,𝑚2𝑚𝑑𝐵,𝑚12𝐵𝑚+𝑚214𝐵+𝑚𝑑𝐵,𝑚2(6.3) with 𝑑𝐵,𝑚=2𝐵+2𝑚+1. Hence, it is clear that 𝑇𝑚(𝜇𝑧)=0, viewed as second-degree polynomials in 𝜇𝑧, admits exactly two real solutions given by 𝜇±𝑧(𝐵,𝑚)=𝑚𝑑𝐵,𝑚2𝐵+𝑚𝑑𝐵,𝑚21±1𝐵+𝑚𝑑𝐵,𝑚𝑚𝑑2𝐵,𝑚1/2.(6.4) Now, assertions (i), (ii), and (iii) follow by discussing the sign of the parameter 𝑄𝑚(𝜇𝑧) (i.e., the sign of 𝑇𝑚(𝜇𝑧)) with respect to the modulus of 𝑧{}, keeping in mind that |𝑧|2=𝜇𝑧/(1𝜇𝑧).

Figure 1 illustrates the quantum photon counting statistics with respect to the location in the extended complex plane of the point 𝑧 as discussed in Proposition 6.1. Here 𝑟±=𝑟±(𝐵,𝑚) are as in (6.2).

Figure 1: Quantum photon counting statistics.


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