Advances in Mathematical Physics

Advances in Mathematical Physics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 393417 |

A. Ghanmi, A. Hafoud, Z. Mouayn, "Generalized Binomial Probability Distributions Attached to Landau Levels on the Riemann Sphere", Advances in Mathematical Physics, vol. 2011, Article ID 393417, 17 pages, 2011.

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

Academic Editor: Ali Mostafazadeh
Received08 Mar 2011
Accepted29 Mar 2011
Published19 Jun 2011


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 π•Š2≑ℂβˆͺ{∞}, and are of the from ||𝑧,𝐡⟩=1+|𝑧|2ξ€Έβˆ’π΅2𝐡𝑗=0ξ‚΅(2𝐡)!𝑗!(2π΅βˆ’π‘—)!1/2𝑧𝑗||π‘—βŸ©,(1.1) where π΅βˆˆβ„€+ is a fixed integer parameter and |π‘—βŸ© are number states of the field mode. Define πœ‡π‘§βˆΆ=|𝑧|2(1+|𝑧|2)βˆ’1, then the probability for the production of 𝑗 photons is given by the squared modulus of the projection of the BS |𝑧,𝐡⟩ onto the number state |π‘—βŸ© as ||||βŸ¨π‘—βˆ£π‘§,𝐡⟩2=(2𝐡)!πœ‡π‘—!(2π΅βˆ’π‘—)!𝑗𝑧1βˆ’πœ‡π‘§ξ€Έ2π΅βˆ’π‘—.(1.2) The latter is recognized as the binomial probability density ℬ(2𝐡,πœ‡π‘§) where {πœ‡π‘§,(1βˆ’πœ‡π‘§)} 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) β„Žπ΅π‘—ξ€·(𝑧)∢=1+|𝑧|2ξ€Έβˆ’π΅ξ‚΅(2𝐡)!𝑗!(2π΅βˆ’π‘—)!1/2𝑧𝑗,𝑗=0,1,2,…,2𝐡,(1.3) constitute an orthonormal basis of the null space π’œπ΅ξ€·π•Š2ξ€Έξ€½βˆΆ=πœ‘βˆˆπΏ2ξ€·π•Š2ξ€Έ,𝐻𝐡[πœ‘]ξ€Ύ=0(1.4) of the second-order differential operator π»π΅ξ€·βˆΆ=βˆ’1+|𝑧|2ξ€Έ2πœ•2πœ•π‘§πœ•π‘§ξ€·βˆ’π΅1+|𝑧|2ξ€Έξ‚΅π‘§πœ•βˆ’πœ•π‘§π‘§πœ•πœ•π‘§ξ‚Ά+𝐡2|𝑧|2βˆ’π΅,(1.5) which constitutes (in suitable units and up to additive constant) a realization in 𝐿2(π•Š2) of the SchrΓΆdinger operator with uniform magnetic field on π•Š2, with a field strength proportional to 𝐡 (see [4]). The given orthonormal basis β„Žπ΅π‘—(𝑧) together with the reproducing kernel 𝐾𝐡(𝑧,𝑀)=(2𝐡+1)1+𝑧𝑀2𝐡1+|𝑧|2ξ€Έβˆ’π΅ξ€·1+|𝑀|2ξ€Έβˆ’π΅(1.6) of the Hilbert space π’œπ΅(π•Š2) in (1.4) can be used to interpret the projection of the BS |𝑧,𝐡⟩ onto the number state |π‘—βŸ© mentioned in (1.2) by writing ξ€·πΎβŸ¨π‘—βˆ£π‘§,𝐡⟩=𝐡(z,𝑧)βˆ’(1/2)β„Žπ΅π‘—(𝑧).(1.7) Note also that the space π’œπ΅(π•Š2) is nothing else than the eigenspace associated with the first eigenvalue of the spectrum of 𝐻𝐡 acting on 𝐿2(π•Š2), which consists of an infinite set of eigenvalues (spherical Landau levels) of the form πœ–π΅π‘šβˆΆ=(2π‘š+1)𝐡+π‘š(π‘š+1),π‘š=0,1,2,…, with finite multiplicity; that is, the associated 𝐿2-eigenspace π’œπ΅,π‘šξ€·π•Š2ξ€Έξ€½βˆΆ=πœ‘βˆˆπΏ2ξ€·π•Š2ξ€Έ,𝐻𝐡[πœ‘]=πœ–π΅π‘šπœ‘ξ€Ύ(1.8) is of finite dimension equals to 𝑑𝐡,π‘šβˆΆ=2𝐡+2π‘š+1.

Here, we take the advantage of the fact that each of the eigenspaces in (1.8) admits an orthogonal basis, denoted β„Žπ‘—π΅,π‘š(𝑧),𝑗=0,1,2,…,2𝐡+2π‘š, whose elements are expressed in terms of Jacobi polynomials π‘ƒπœ‚(𝜏,𝜍)(β‹…), as well as a reproducing kernel 𝐾𝐡,π‘š(𝑧,𝑀) in an explicit form (see [5]) to consider a set of coherent states by adopting a generalized coherent states form β€œΓ  la Iwata” [6] as ||𝐾𝑧,𝐡,π‘šβŸ©=𝐡,π‘šξ€Έ(𝑧,𝑧)βˆ’(1/2)2𝐡+2π‘šξ“π‘—=0β„Žπ‘—π΅,π‘š(𝑧)√𝜌𝐡,π‘š||(𝑗)π‘—βŸ©,(1.9) where 𝜌𝐡,π‘š(𝑗) denotes the norm square of β„Žπ‘—π΅,π‘š(𝑧) in 𝐿2(π•Š2). 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 π’œπ΅,π‘š(π•Š2) a photon counting probability distribution in the same way as for the space π’œπ΅(π•Š2)β‰‘π’œπ΅,0(π•Š2) through the quantities 𝑝𝑗2𝐡,πœ‡π‘§ξ€Έ=,π‘šπ‘š!(2𝐡+π‘š)!πœ‡π‘—!(2𝐡+2π‘šβˆ’π‘—)!π‘§π‘—βˆ’π‘šξ€·1βˆ’πœ‡π‘§ξ€Έ2𝐡+π‘šβˆ’π‘—ξ‚€π‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2,𝑗=0,1,…,2𝐡+2π‘š.(1.10) The latter can be considered as a kind of generalized binomial probability distribution π‘‹βˆΌβ„¬(2𝐡,πœ‡π‘§,π‘š) depending on an additional parameter π‘š=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 SchrΓΆdinger 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,πœ‡,πœƒβŸ©=𝑛𝑗=0𝑛!πœ‡π‘—!(π‘›βˆ’π‘—)!𝑗(1βˆ’πœ‡)π‘›βˆ’π‘—ξ‚Ά1/2π‘’π‘–π‘—πœƒ||π‘—βŸ©.(2.1) The corresponding photon counting probability is given by𝑝𝑗(𝑛,πœ‡)=𝑛!πœ‡π‘—!(π‘›βˆ’π‘—)!𝑗(1βˆ’πœ‡)π‘›βˆ’π‘—(2.2) which follows the binomial law π‘ŒβˆΌβ„¬(𝑛,πœ‡) with parameters 𝑛 and πœ‡; π‘›βˆˆβ„€+, 0<πœ‡<1. The connection with our notations in (1.1) and (1.2) can be made by setting 𝑛=2𝐡, 𝑧=|𝑧|π‘’π‘–πœƒ and |𝑧|2=πœ‡(1βˆ’πœ‡)βˆ’1. Note that in limits πœ‡β†’0 and πœ‡β†’1 the binomial state reduces to number states |0⟩ and |π‘›βŸ©, respectively. In a different limit of 𝑛→+∞ and πœ‡β†’0 with π‘›πœ‡β†’πœ†, the probability distribution (2.2) goes to the Poisson distribution 𝒫(πœ†)π‘π‘—πœ†(πœ†)=𝑗𝑒𝑗!βˆ’πœ†,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 π‘ŒβˆΌβ„¬(𝑛,πœ‡) is given by π’žπ‘Œξ€·(𝑑)=(1βˆ’πœ‡)+πœ‡π‘’π‘–π‘‘ξ€Έπ‘›(2.4) from which one obtains the mean value and the variance as 𝐸(π‘Œ)=π‘›πœ‡,Var(π‘Œ)=π‘›πœ‡(1βˆ’πœ‡).(2.5) Therefore, the Mandel parameter, which measures deviation from the Poissonian distribution, 𝑄=Var(π‘Œ)𝐸(π‘Œ)βˆ’1=βˆ’πœ‡,(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 π’œπ΅,π‘š(π•Š2)

Let π•Š2βŠ‚β„3 denote the unit sphere with the standard metric of constant Gaussian curvature πœ…=1. We identify the sphere π•Š2 with the extended complex plane β„‚βˆͺ{∞}, called the Riemann sphere, via the stereographic coordinate 𝑧=π‘₯+𝑖𝑦; π‘₯,π‘¦βˆˆβ„. We shall work within a fixed coordinate neighborhood with coordinate 𝑧 obtained by deleting the β€œpoint at infinity” {∞}. Near this point we use instead of 𝑧 the coordinate π‘§βˆ’1.

In the stereographic coordinate 𝑧, the Hamiltonian operator of the Dirac monopole with charge π‘ž=2𝐡 reads [4, page 598]π»π΅ξ€·βˆΆ=βˆ’1+|𝑧|2ξ€Έ2πœ•2πœ•π‘§πœ•π‘§ξ€·βˆ’π΅π‘§1+|𝑧|2ξ€Έπœ•πœ•π‘§+𝐡𝑧1+|𝑧|2ξ€Έπœ•πœ•π‘§+𝐡2ξ€·1+|𝑧|2ξ€Έβˆ’π΅2.(3.1) This operator acts on the sections of the π‘ˆ(1)-bundle with the first Chern class π‘ž. We have denoted by π΅βˆˆβ„€+ the strength of the quantized magnetic field. We shall consider the Hamiltonian 𝐻𝐡 in (3.1) acting in the Hilbert space 𝐿2(π•Š2)∢=𝐿2(π•Š2,(1+|𝑧|2)βˆ’2π‘‘πœˆ(𝑧)), π‘‘πœˆ(𝑧)=πœ‹βˆ’1𝑑π‘₯𝑑𝑦 being the Lebesgue measure on ℂ≑ℝ2. Its spectrum consists of an infinite number of eigenvalues (spherical Landau levels) of the formπœ–π΅π‘šβˆΆ=(2π‘š+1)𝐡+π‘š(π‘š+1),π‘š=0,1,2,…,(3.2) with finite degeneracy 2𝐡+2π‘š+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 π»π΅βˆ’π΅ on 𝐿2(π•Š2) is intertwined with the invariant Laplacian Ξ”2π΅ξ€·βˆΆ=βˆ’1+|𝑧|2ξ€Έ2πœ•2πœ•π‘§πœ•π‘§+2𝐡𝑧1+|𝑧|2ξ€Έπœ•πœ•π‘§(3.3) acting in the Hilbert space 𝐿2,𝐡(π•Š2)∢=𝐿2(π•Š2,(1+|𝑧|2)βˆ’2βˆ’2π΅π‘‘πœˆ(𝑧)). Namely, we haveπ»π΅ξ€·βˆ’π΅=1+|𝑧|2ξ€Έβˆ’π΅Ξ”2𝐡1+|𝑧|2𝐡,(3.4) and therefore any ket |πœ™βŸ© of 𝐿2,𝐡(π•Š2) is represented byξ€·1+|𝑧|2ξ€Έβˆ’π΅βŸ¨π‘§βˆ£πœ™βŸ©in𝐿2ξ€·π•Š2ξ€Έ.(3.5) As mentioned in the introduction, we denote by π’œπ΅,π‘š(π•Š2) the eigenspace of 𝐻𝐡 in 𝐿2(π•Š2), corresponding to the eigenvalue πœ–π΅π‘š 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: β„Žπ‘—π΅,π‘šξ€·(𝑧)∢=1+|𝑧|2ξ€Έβˆ’π΅π‘§π‘—π‘„π΅,π‘š,𝑗|𝑧|21+|𝑧|2ξ‚Ά,0≀𝑗≀2𝐡+2π‘š,(3.6) where 𝑄𝐡,π‘š,𝑗(β‹…) is the polynomial function given by𝑄𝐡,π‘š,𝑗(𝑑)=π‘‘βˆ’π‘—(1βˆ’π‘‘)π‘—βˆ’2π΅ξ‚€π‘‘ξ‚π‘‘π‘‘π‘šξ€Ίπ‘‘π‘—+π‘š(1βˆ’π‘‘)2π΅βˆ’π‘—+π‘šξ€».(3.7) According to the Jacobi's formula [11]𝑑𝑑π‘₯π‘šξ€·π‘₯𝑐+π‘šβˆ’1(1βˆ’π‘₯)π‘βˆ’π‘ξ€Έ=Ξ“(𝑐+π‘š)π‘₯Ξ“(𝑐)π‘βˆ’1(1βˆ’π‘₯)2π‘βˆ’π‘βˆ’π‘šπΉ1(βˆ’π‘š,𝑏;𝑐;π‘₯),(3.8)2𝐹1(π‘Ž,𝑏,𝑐;π‘₯) being the Gauss hypergeometric function, it follows that 𝑄𝐡,π‘š,𝑗(𝑑)=(π‘š+𝑗)!𝑗!2𝐹1(βˆ’π‘š,2𝐡+π‘š+1,𝑗+1;𝑑).(3.9) The latter can also be expressed in terms of Jacobi polynomials via the transformation [11, page 283] 2𝐹1ξ‚€π‘˜+𝜈+𝜚+1,βˆ’π‘˜,1+𝜈;1βˆ’π‘‘2=π‘˜!Ξ“(1+𝜈)𝑃Γ(π‘˜+1+𝜈)π‘˜(𝜈,𝜚)(𝑑).(3.10) So that the orthogonal basis in (3.6) reads simply in terms of Jacobi polynomial as β„Žπ‘—π΅,π‘šξ€·(𝑧)=π‘š!1+|𝑧|2ξ€Έβˆ’π΅π‘§π‘—βˆ’π‘šπ‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ‚΅1βˆ’|𝑧|21+|𝑧|2ξ‚Ά.(3.11) Also, one obtains the norm square of the eigenfunction β„Žπ‘—π΅,π‘š given in (3.6) as 𝜌𝐡,π‘š(β€–β€–β„Žπ‘—)∢=𝑗𝐡,π‘šβ€–β€–2𝐿2(π•Š2)=π‘š!(π‘š+𝑗)!(2𝐡+π‘šβˆ’π‘—)!.(2𝐡+2π‘š+1)(2𝐡+π‘š)!(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 π’œπ΅,π‘š(π•Š2): 𝐾𝐡,π‘šξ€·(𝑧,𝑀)=(2𝐡+2π‘š+1)=1+𝑧𝑀2𝐡1+|𝑧|2𝐡1+|𝑀|2𝐡×2𝐹1ξƒ©βˆ’π‘š,π‘š+2𝐡+1,1;|π‘§βˆ’π‘€|2ξ€·1+|𝑧|2ξ€Έξ€·1+|𝑀|2ξ€Έξƒͺ.(3.13)

Remark 3.1. In the case π‘š=0, the elements of the orthogonal basis reduce further to β„Žπ‘—π΅,0(𝑧)=(1+|𝑧|2)βˆ’π΅π‘§π‘— and the reproducing kernel reads 𝐾𝐡,0ξ€·(𝑧,𝑀)=(2𝐡+1)1+𝑧𝑀2𝐡1+|𝑧|2ξ€Έβˆ’π΅ξ€·1+|𝑀|2ξ€Έβˆ’π΅.(3.14)

Remark 3.2. In the case of the 𝑛-dimensional projective space β„™(ℂ𝑛)(=𝑆1⧡𝑆2𝑛+1) equipped with the Fubini-Study metric, explicit formulae for the reproducing kernels of the eigenspaces associated with the SchrΓΆdinger operator with constant magnetic field written in the local coordinates (of the chart ℂ𝑛) as π»π΅ξ€·βˆΆ=1+|𝑧|2𝑛𝑖,𝑗=1𝛿𝑖𝑗+π‘§π‘–π‘§π‘—ξ€Έπœ•2πœ•π‘§π‘–πœ•π‘§π‘—βˆ’π΅π‘›ξ“π‘—=1ξƒ©π‘§π‘—πœ•πœ•π‘§π‘—βˆ’π‘§π‘—πœ•πœ•π‘§π‘—ξƒͺβˆ’π΅2ξƒ°+𝐡2(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 (β„‹,βŸ¨β‹…,β‹…βŸ©β„‹) be a functional Hilbert space with an orthonormal basis {πœ™π‘›}𝑑𝑛=1 and π’œ2 a finite d-dimensional subspace of the Hilbert space 𝐿2(Ξ©,𝑑𝑠), of square integrable functions on a given measured space (Ξ©,𝑑𝑠), with an orthogonal basis {Φ𝑛}𝑑𝑛=1. Then, π’œ2 is a reproducing kernel Hilbert space whose reproducing kernel is given by 𝐾(π‘₯,𝑦)∢=𝑑𝑛=1Φ𝑛(π‘₯)Φ𝑛(𝑦)πœŒπ‘›;π‘₯,π‘¦βˆˆΞ©,(4.1) where we have set πœŒπ‘›βˆΆ=‖Φ𝑛‖2𝐿2(Ξ©,𝑑𝑠). Associated to the data of (π’œ2,Φ𝑛) and (β„‹,πœ™π‘›), the generalized coherent states are the elements of β„‹ defined by Ξ¦π‘₯ξ€·πœ”βˆΆ=𝑑(π‘₯)π‘‘βˆ’(1/2)𝑛=1Φ𝑛(π‘₯)βˆšπœŒπ‘›πœ™π‘›;π‘₯∈Ω,(4.2) where πœ”π‘‘(π‘₯) stands for πœ”π‘‘(π‘₯)∢=𝐾(π‘₯,π‘₯). Note that the choice of the Hilbert space β„‹ defines a quantization of Ξ© into β„‹ by considering the inclusion map π‘₯↦Φπ‘₯. Furthermore, it is straightforward to check that ⟨Φπ‘₯,Ξ¦π‘₯βŸ©β„‹=1 and to show that the corresponding coherent state transform (CST) 𝒲 onβ„‹, 𝒲[𝑓]ξ€·πœ”(π‘₯)∢=𝑑(π‘₯)1/2⟨Φπ‘₯,π‘“βŸ©β„‹;π‘“βˆˆβ„‹,(4.3) defines an isometry from β„‹ into π’œ2. Thereby we have a resolution of the identity of β„‹, that is, we have the following integral representation: ξ€œπ‘“(β‹…)=Ω⟨Φπ‘₯,π‘“βŸ©β„‹Ξ¦π‘₯(β‹…)πœ”π‘‘(π‘₯)𝑑𝑠(π‘₯);π‘“βˆˆβ„‹.(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 ||ξ‚€π‘’πœβŸ©=|𝜁|2ξ‚βˆ’(1/2)ξ“π‘˜β‰₯0πœπ‘˜βˆš||π‘˜!π‘˜βŸ©(4.5) with πœ™π‘˜βˆΆ=|π‘˜βŸ© being the number states of the harmonic oscillator, which also knwon as Fock states.

We can now construct for each spherical Landau level πœ–π΅π‘š given in (3.2) a set of generalized coherent states (GCS) according to formula (4.2) asπœ—π‘§,𝐡,π‘šβ‰‘||𝐾𝑧,𝐡,π‘šβŸ©=𝐡,π‘šξ€Έ(𝑧,𝑧)βˆ’(1/2)2𝐡+2π‘šξ“π‘—=0β„Žπ‘—π΅,π‘š(𝑧)√𝜌𝐡,π‘š||πœ™(𝑗)𝑗(4.6) with the following precisions:(i)(Ξ©,𝑑𝑠)∢=(π•Š2,(1+|𝑧|2)βˆ’2π‘‘πœˆ(𝑧)), π•Š2 being identified with β„‚βˆͺ{∞}.(ii)π’œ2∢=π’œπ΅,π‘š(π•Š2) is the eigenspace of 𝐻𝐡 in 𝐿2(π•Š2) with dimension 𝑑𝐡,π‘š=2𝐡+2π‘š+1.(iii)πœ”(𝑧)=𝐾𝐡,π‘š(𝑧,𝑧)=2𝐡+2π‘š+1 (in view of (3.13)).(iv)β„Žπ‘—π΅,π‘š(𝑧) are the eigenfunctions given by (3.11) in terms of the Jacobi polynomials.(v)𝜌𝐡,π‘š(𝑗) being the norm square of β„Žπ‘—π΅,π‘š, given in (3.12).(vi)β„‹βˆΆ=𝒫𝐡+π‘š the space of polynomials of degree less than 𝑑𝐡,π‘š, which carries a unitary irreducible representation of the compact Lie group SU(2) (see [14]). The scalar product in 𝒫𝐡+π‘š is written as βŸ¨πœ“,πœ™βŸ©π’«π΅+π‘š=𝑑𝐡,π‘šξ€œβ„‚ξ‚€||𝜁||𝑑𝑠(𝜁)1+2ξ‚βˆ’2(𝐡+π‘š)βˆ’2πœ“(𝜁)πœ™(𝜁).(4.7)(vii){πœ™π‘—;0≀𝑗≀2𝐡+2π‘š} is an orthonormal basis of 𝒫𝐡+π‘š, whose elements are given explicitly byπœ™π‘—ξƒŽ(πœ‰)∢=(2(𝐡+π‘š))!(πœ‰2𝐡+π‘šβˆ’π‘—)!(𝑗+π‘š)!𝑗+π‘šπœβˆˆβ„‚.(4.8)

Definition 4.2. For given positive integers 2𝐡 and π‘š, and fixed π‘§βˆˆπ•Š2, the wave function of the GCS in (4.6) are expressed as πœ—π‘§,𝐡,π‘š1(πœ‰)=ξ€·1+|𝑧|2ξ€Έβˆ’π΅2𝐡+2π‘šξ“π‘—=0βˆšπ‘š!(2𝐡+π‘š)!(2𝐡+2π‘š)!𝑧𝑗!(2𝐡+2π‘šβˆ’π‘—)!π‘—βˆ’π‘šπ‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ‚΅1βˆ’|𝑧|21+|𝑧|2ξ‚Άπœ‰π‘—,(4.9) for every πœβˆˆβ„‚.

According to (4.4), the system of GCS |πœ—π‘§,𝐡,π‘šβŸ© solves then the unity of the Hilbert space 𝒫𝐡+π‘š asπŸπ’«π΅+π‘š=𝑑𝐡,π‘šξ€œβ„‚ξ€·π‘‘πœˆ(𝑧)1+|𝑧|2ξ€Έβˆ’2||πœ—π‘§,𝐡,π‘šπœ—ξ¬ξ«π‘§,𝐡,π‘š||.(4.10) They also admit a closed form [15], as πœ—π‘§,𝐡,π‘šξƒŽ(πœ‰)=(2𝐡+2π‘š)!ξ‚΅(2𝐡+π‘š)!π‘š!(1+πœ‰π‘§)21+|𝑧|2ξ‚Άπ΅ξƒ©ξ€·πœ‰βˆ’π‘§ξ€Έ(1+πœ‰π‘§)1+|𝑧|2ξƒͺπ‘š.(4.11)

Remark 4.3. Note that for π‘š=0, the expression above reduces to ξ€·βŸ¨πœ‰βˆ£π‘§,𝐡,0⟩=1+|𝑧|2ξ€Έβˆ’π΅(1+πœ‰π‘§)2𝐡,(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 πœ—π‘§,𝐡,π‘š onto the state πœ™π‘—, is given by ||ξ«πœ—π‘§,𝐡,π‘š,πœ™π‘—ξ¬β„‹||2=1πœŒπ‘—π΅,π‘šπ‘‘π΅,π‘š||β„Žπ‘—π΅,π‘š||(𝑧)2.(5.1) This is in fact the probability of finding 𝑗 photons in the coherent state πœ—π‘§,𝐡,π‘š. More explicitly, in view of (3.11), the quantity in (5.1) reads ||ξ«πœ—π‘§,𝐡,π‘š,πœ™π‘—ξ¬β„‹||2=π‘š!(2𝐡+π‘š)!𝑗!(2𝐡+2π‘šβˆ’π‘—)!1+|𝑧|2ξ€Έβˆ’2𝐡|𝑧|2(π‘—βˆ’π‘š)ξ‚΅π‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ‚΅1βˆ’|𝑧|21+|𝑧|2ξ‚Άξ‚Ά2.(5.2) As annonced in the introduction, we denote the expression in (5.2) by 𝑝𝑗(2𝐡,πœ‡π‘§,π‘š) for 𝑗=0,1,2,…, with πœ‡π‘§=|𝑧|2(1+|𝑧|2)βˆ’1 or equivalently |𝑧|2=πœ‡π‘§(1βˆ’πœ‡π‘§)βˆ’1. Motivated by this quantum probability, we then state the following.

Definition 5.1. For fixed integers 𝐡,π‘šβˆˆβ„€+, the discrete random variable 𝑋 having the probability distribution 𝑝𝑗2𝐡,πœ‡π‘§ξ€Έ=,π‘šπ‘š!(2𝐡+π‘š)!πœ‡π‘—!(2𝐡+2π‘šβˆ’π‘—)!𝑧(π‘—βˆ’π‘š)ξ€·1βˆ’πœ‡π‘§ξ€Έ2𝐡+π‘šβˆ’π‘—ξ‚€π‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2,(5.3) with 𝑗=0,1,2,…,2𝐡+2π‘š, and denoted by π‘‹βˆΌβ„¬(2𝐡,πœ‡π‘§,π‘š), 0<πœ‡π‘§<1, will be called the generalized binomial probability distribution associated to the weighted Hilbert space π’œπ΅,π‘š(𝑆2).

Remark 5.2. Note that for π‘š=0, the above expression in (5.3) reduces to 𝑝𝑗2𝐡,πœ‡π‘§ξ€Έ=,0(2𝐡)!πœ‡π‘—!(2π΅βˆ’π‘—)!𝑗𝑧1βˆ’πœ‡π‘§ξ€Έ2π΅βˆ’π‘—,𝑗=0,1,2,…,2𝐡,(5.4) which is the standard binomial distribution with parameters 2𝐡 and 0<πœ‡π‘§<1.

A convenient way to summarize all the properties of a probability distribution 𝑋 is to explicit its characteristic function: π’žπ‘‹ξ€·π‘’(𝑑)=βˆΆπΈπ‘–π‘‘π‘‹ξ€Έ,(5.5) where 𝑑 is a real number, βˆšπ‘–βˆΆ=βˆ’1 is the imaginary unit, and 𝐸 denotes the expected value.

Proposition 5.3. For fixed π‘š=0,1,2,…, the characteristic function of π‘‹βˆΌβ„¬(2𝐡,πœ‡π‘§,π‘š) is given by π’žπ‘š(𝑑)=π‘’π‘–π‘šπ‘‘ξ€·ξ€·1βˆ’πœ‡π‘§ξ€Έ+πœ‡π‘§π‘’π‘–π‘‘ξ€Έ2π΅π‘ƒπ‘š(0,2𝐡)ξ€·1βˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έξ€Έ(1βˆ’cos(𝑑))(5.6) for every π‘‘βˆˆβ„.

Proof. Recall first that for every given fixed nonnegative integer π‘š, the characteristic function π’žπ‘š(𝑑) in (5.5) can be written as π’žπ‘š(𝑑)=2𝐡+2π‘šξ“π‘—=0𝑒𝑖𝑗𝑑𝑝𝑗2𝐡,πœ‡π‘§ξ€Έ=,π‘š2𝐡+2π‘šξ“π‘—=0π‘’π‘–π‘—π‘‘π‘š!(2𝐡+π‘š)!πœ‡π‘—!(2𝐡+2π‘šβˆ’π‘—)!𝑧(π‘—βˆ’π‘š)ξ€·1βˆ’πœ‡π‘§ξ€Έ2𝐡+π‘šβˆ’π‘—ξ‚€π‘ƒπ‘š(π‘—βˆ’π‘š,2𝐡+π‘šβˆ’π‘—)ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2.(5.7) The last equality follows using (5.3). Next, by making the change π‘˜=𝐡+π‘šβˆ’π‘— in (5.7), it follows that π’žπ‘š(𝑑)=𝐡+π‘šξ“π‘˜=βˆ’(𝐡+π‘š)𝑒𝑖(𝐡+π‘šβˆ’π‘˜)π‘‘π‘š!(2𝐡+π‘š)!πœ‡(𝐡+π‘š+π‘˜)!(𝐡+π‘šβˆ’π‘˜)!π‘§π΅βˆ’π‘˜ξ€·1βˆ’πœ‡π‘§ξ€Έπ΅+π‘˜ξ‚€π‘ƒπ‘š(π΅βˆ’π‘˜,𝐡+π‘˜)ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2.(5.8) Instead of the Jacobi polynomials, it is convenient to consider the closely related function π’«π‘™π‘Ÿ,𝑠(π‘₯) introduced in [14, page 270]. They can be defined through the formula [14, equation (1), page 288], 𝑃(π‘Ÿβˆ’π‘ ,π‘Ÿ+𝑠)π‘›βˆ’π‘Ÿ(π‘₯)=2π‘Ÿξ‚΅(π‘›βˆ’π‘ )!(𝑛+𝑠)!(ξ‚Άπ‘›βˆ’π‘Ÿ)!(𝑛+π‘Ÿ)!1/2(1βˆ’π‘₯)(π‘ βˆ’π‘Ÿ)/2(1+π‘₯)βˆ’(𝑠+π‘Ÿ)/2π’«π‘›π‘Ÿ,𝑠(π‘₯),(5.9) with π‘š=π‘›βˆ’π΅ (i.e., 𝑛=𝐡+π‘š) and π‘₯=1βˆ’2πœ‡π‘§. We can then express the square of π‘ƒπ‘š(π΅βˆ’π‘˜,𝐡+π‘˜)(π‘₯) as follows: ξ‚€π‘ƒπ‘š(π΅βˆ’π‘˜,𝐡+π‘˜)ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2=(𝐡+π‘šβˆ’π‘˜)!(𝐡+π‘š+π‘˜)!πœ‡π‘š!(2𝐡+π‘š)!π‘§βˆ’π΅+π‘˜ξ€·1βˆ’πœ‡π‘§ξ€Έβˆ’π΅βˆ’π‘˜ξ‚€π’«π΅+π‘šπ΅,π‘˜ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2.(5.10) Therefore, (5.8) reduces further to π’žπ‘š(𝑑)=𝑒𝑖(𝐡+π‘š)𝑑𝐡+π‘šξ“π‘˜=βˆ’(𝐡+π‘š)π‘’βˆ’π‘–π‘˜π‘‘ξ‚€π’«π΅+π‘šπ΅,π‘˜ξ€·1βˆ’2πœ‡π‘§ξ€Έξ‚2(5.11)(⋆)=(βˆ’1)𝐡𝑒𝑖(𝐡+π‘š)𝑑𝐡+π‘šξ“π‘˜=βˆ’(𝐡+π‘š)π‘’βˆ’π‘–π‘˜(π‘‘βˆ’πœ‹)𝒫𝐡+π‘šπ΅,π‘˜ξ€·1βˆ’2πœ‡π‘§ξ€Έπ’«π΅+π‘šπ‘˜,𝐡1βˆ’2πœ‡π‘§ξ€Έ=(βˆ’1)𝐡𝑒𝑖(𝐡+π‘š)π‘‘π‘’βˆ’π‘–π΅(πœ‘+πœ“)𝒫𝐡+π‘šπ΅,𝐡(cos(πœƒ)).(5.12) The transition (⋆) above holds using the fact that [14, page 288] 𝒫𝑙𝑗,π‘˜(π‘₯)=(βˆ’1)𝑗+π‘˜π’«π‘™π‘˜,𝑗(π‘₯).(5.13) While the last equality can be checked easily using the addition formula [14, equation (3), page 326] π‘ ξ“π‘˜=βˆ’π‘ π‘’βˆ’π‘–π‘˜πœπ’«π‘ π‘—,π‘˜ξ€·ξ€·πœƒcos1π’«ξ€Έξ€Έπ‘ π‘˜,π‘™ξ€·ξ€·πœƒcos2ξ€Έξ€Έ=π‘’βˆ’π‘–(π‘—πœ‘+π‘™πœ“)𝒫𝑠𝑗,𝑙(cos(πœƒ)).(5.14) Here the involved complex angles πœ‘, πœ“, and πœƒ are given through equations (8), (8β€²), and (8β€²β€²) in [14, page 270]. In our case, they yield the followings: cos(πœƒ)=cos2(2𝛼)+sin2𝑒(2𝛼)cos(𝑑),𝑖((πœ‘+πœ“)/2)=ξ€·βˆ’π‘–cos2(𝛼)+sin2(𝛼)π‘’βˆ’π‘–π‘‘ξ€Έπ‘’π‘–π‘‘/2cos(πœƒ/2)(5.15) for πœƒ1=πœƒ2=2𝛼, so that π‘’βˆ’π‘–π΅(πœ‘+πœ“)=(βˆ’1)π΅ξ‚€ξ‚€πœƒcos2ξ‚ξ‚βˆ’2𝐡cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑑2π΅π‘’βˆ’π‘–π΅π‘‘.(5.16) Next, using the fact that 2βˆ’π‘ (1+π‘₯)𝑠𝑃(0,2𝑠)π‘›βˆ’π‘ (π‘₯)=𝒫𝑛𝑠,𝑠(π‘₯),(5.17) which is a special case of (5.9), with 𝑠=𝐡, π‘›βˆ’π‘ =π‘š, and π‘₯=cos(πœƒ), we obtain 𝒫𝐡+π‘šπ΅,π΅ξ‚€ξ‚€πœƒ(cos(πœƒ))=cos22π΅π‘ƒπ‘š(0,2𝐡)(cos(πœƒ)).(5.18) Finally, by substituting (5.16) and (5.18) in (5.12), taking into account that sin2(𝛼)=πœ‡π‘§ and cos2(𝛼)=1βˆ’πœ‡π‘§, we see that the characteristic function π’žπ‘š(𝑑) reads simply as π’žπ‘š(𝑑)=π‘’π‘–π‘šπ‘‘ξ€·cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑑2π΅π‘ƒπ‘š(0,2𝐡)(cos(πœƒ)),(5.19) where cos(πœƒ)=1βˆ’4πœ‡π‘§(1βˆ’πœ‡z)(1βˆ’cos(𝑑)).

Remark 5.4. Note that by taking π‘š=0 in (5.19), the characteristic function reduces to πΆπ‘Œξ€·(𝑑)=cos2(𝛼)+sin2(𝛼)𝑒𝑖𝑑2𝐡=ξ€·ξ€·1βˆ’πœ‡π‘§ξ€Έ+πœ‡π‘§π‘’π‘–π‘‘ξ€Έ2𝐡,(5.20) which is the well-known characteristic function of the binomial random variable π‘ŒβˆΌβ„¬(2𝐡,πœ‡π‘§) with parameters 𝑛=2π΅βˆˆβ„€+ and 0<πœ‡π‘§<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 π’žπ‘š(𝑑) in (5.6) with respect to the variable 𝑑 and evaluation at the origin as πΈξ€·π‘‹π‘˜ξ€Έ=1π‘–π‘˜πœ•π‘˜πœ•π‘‘π‘˜ξ€·πΆπ‘šξ€Έ||||(𝑑)𝑑=0.(5.21)

Corollary 5.5. Let π‘š,2π΅βˆˆβ„€+. The mean value and the variance of π‘‹βˆΌβ„¬(2𝐡,πœ‡π‘§,π‘š) are given respectively by 𝐸(𝑋)=π‘š+2π΅πœ‡π‘§,Var(𝑋)=2π΅πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έ+2πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έπ‘š(2𝐡+π‘š+1).(5.22)

Proof. Let recall first that for every fixed integer π‘š=0,1,2,…, we have 𝐸(𝑋)=πœ•π’žπ‘š||||π‘–πœ•π‘‘π‘‘=0,𝑋Var(𝑋)=𝐸2ξ€Έβˆ’[]𝐸(𝑋)2=πœ•2π’žπ‘šπ‘–2πœ•π‘‘2||||𝑑=0βˆ’ξƒ¬πœ•π’žπ‘š||||π‘–πœ•π‘‘π‘‘=0ξƒ­2.(5.23) Thus direct computation gives rise to πœ•π’žπ‘šβŽ‘βŽ’βŽ’βŽ£π‘–πœ•π‘‘(𝑑)=π‘š+2π΅πœ‡π‘§π‘’π‘–π‘‘ξ€·ξ€Ί1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έβˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€ΈβŽ›βŽœβŽœβŽsin(𝑑)(πœ•π‘ƒπ‘š(0,2𝐡)||(π‘₯)/π‘–πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)⎞⎟⎟⎠⎀βŽ₯βŽ₯βŽ¦π’ž(cos(πœƒ))π‘šπœ•(𝑑),2π’žπ‘šπ‘–2πœ•π‘‘2=βŽ‘βŽ’βŽ’βŽ£πœ•(𝑑)βŽ›βŽœβŽœβŽπ‘–πœ•π‘‘π‘š+2π΅πœ‡π‘§π‘’π‘–π‘‘ξ€·ξ€Ί1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έβˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€ΈβŽ›βŽœβŽœβŽsin(𝑑)(πœ•π‘ƒπ‘š(0,2𝐡)||(π‘₯)/π‘–πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)⎞⎟⎟⎠⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦(cos(πœƒ))Γ—π’žπ‘š(βŽ‘βŽ’βŽ’βŽ£π‘‘)+π‘š+2π΅πœ‡π‘§π‘’π‘–π‘‘ξ€·ξ€Ί1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έβˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€ΈβŽ›βŽœβŽœβŽsin(𝑑)(πœ•π‘ƒπ‘š(0,2𝐡)(||π‘₯)/π‘–πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦(cos(πœƒ))πœ•π’žπ‘š(=βŽ‘βŽ’βŽ’βŽ£π‘–πœ•π‘‘π‘‘)2π΅πœ‡π‘§π‘’π‘–π‘‘ξ€·ξ€Ί1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έβˆ’2π΅πœ‡2𝑧𝑒𝑖𝑑1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έ2+4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€ΈβŽ›βŽœβŽœβŽcos(𝑑)(πœ•π‘ƒπ‘š(0,2𝐡)||(π‘₯)/πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦(cos(πœƒ))Γ—π’žπ‘š(𝑑)βˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έπœ•sin(𝑑)βŽ›βŽœβŽœβŽπ‘–πœ•π‘‘(πœ•π‘ƒπ‘š(0,2𝐡)||(π‘₯)/π‘–πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)βŽžβŽŸβŽŸβŽ π’ž(cos(πœƒ))π‘š+⎑⎒⎒⎣(𝑑)π‘š+2π΅πœ‡π‘§π‘’π‘–π‘‘ξ€·ξ€Ί1βˆ’πœ‡π‘§ξ€»+πœ‡π‘§π‘’π‘–π‘‘ξ€Έβˆ’4πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€ΈβŽ›βŽœβŽœβŽsin(𝑑)(πœ•π‘ƒπ‘š(0,2𝐡)||(π‘₯)/π‘–πœ•π‘₯)π‘₯=cos(πœƒ)π‘ƒπ‘š(0,2𝐡)⎞⎟⎟⎠⎀βŽ₯βŽ₯⎦(cos(πœƒ))πœ•π’žπ‘šπ‘–πœ•π‘‘(𝑑).(5.24) To conclude, we have to use successively the facts that for 𝑑=0, we have cos(πœƒ)=1 and π’žπ‘š(0)=1, together with πœ•π‘ƒπ‘š(π‘Ž,𝑏)π‘–πœ•π‘₯(π‘₯)=π‘Ž+𝑏+π‘š+12𝑃(π‘Ž+1,𝑏+1)π‘šβˆ’1(π‘₯),π‘ƒπ‘š(π‘Ž,𝑏)(1)=Ξ“(π‘Ž+π‘š+1).π‘š!Ξ“(π‘Ž+1)(5.25) Thus, we have 𝐸(𝑋)=πœ•π’žπ‘š||||π‘–πœ•π‘‘π‘‘=0=ξ€·π‘š+2π΅πœ‡π‘§ξ€Έπ’žπ‘š(0)=π‘š+2π΅πœ‡π‘§.(5.26) We have also πœ•2π’žπ‘šπ‘–2πœ•π‘‘2|𝑑=0=2π΅πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έ+2πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έξƒ©π‘ƒ(2𝐡+π‘š+1)(1,2𝐡+1)π‘šβˆ’1(1)π‘ƒπ‘š(0,2𝐡)(1)ξƒͺξƒ­Γ—π’žπ‘š(ξ€Ί0)+π‘š+2π΅πœ‡π‘§ξ€»πœ•π’žπ‘š|π‘–πœ•π‘‘π‘‘=0,(5.27) and therefore πœ•Var(𝑋)=2π’žπ‘šπ‘–2πœ•π‘‘2||||𝑑=0βˆ’ξƒ©πœ•π’žπ‘š||||π‘–πœ•π‘‘π‘‘=0ξƒͺ2=2π΅πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έ+2πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έπ‘š(2𝐡+π‘š+1).(5.28)

Remark 5.6. Note that by taking π‘š=0 in (5.22), we recover the standard values 𝐸(π‘Œ)=2π΅πœ‡π‘§,Var(π‘Œ)=2π΅πœ‡π‘§ξ€·1βˆ’πœ‡π‘§ξ€Έ(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 β€œnonclassical” 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)βˆ’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 π‘š=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 π‘šβ‰ 0, we make use of the obtained statistical parameters 𝐸(𝑋) and Var(𝑋) to calculate Mandel parameter corresponding to the random variable π‘‹βˆΌβ„¬(2𝐡,πœ‡π‘§,π‘š). The discussion with respect to the sign of this parameter gives rise to the following statement.

Proposition 6.1. Let π‘š and 𝐡 be nonnegative integers and set π‘ŸΒ±ξƒ©ξ‚΅1(𝐡,π‘š)∢=1Β±1βˆ’π‘šξ‚Ά(2𝐡+π‘š)1/2ξƒͺ1/2.(6.2) Then, π‘Ÿβˆ’(𝐡,π‘š)≀1β‰€π‘Ÿ+(𝐡,π‘š) and the photon counting statistics are
(i) sub-Poissonian for points 𝑧 such that |𝑧|<π‘Ÿβˆ’(𝐡,π‘š) and |𝑧|>π‘Ÿ+(𝐡,π‘š),
(ii) Poissonian for points 𝑧 such that |𝑧|=π‘Ÿβˆ’(𝐡,π‘š) or |𝑧|=π‘Ÿ+(𝐡,π‘š),
(iii) super-Poissonian for 𝑧 such that π‘Ÿβˆ’(𝐡,π‘š)<|𝑧|<π‘Ÿ+(𝐡,π‘š),

Proof. Assume that π‘šβ‰ 0. Making use of (5.22), we see that the Mandel parameter can be written as follows 𝑄(𝑋)=π‘„π‘š(πœ‡π‘§)=βˆ’π‘‡π‘š(πœ‡π‘§)/(2π΅πœ‡π‘§+π‘š), where we have set π‘‡π‘šξ€·πœ‡π‘§ξ€Έ[]=2(𝐡+π‘š2𝐡+π‘š+1)πœ‡2𝑧[]πœ‡βˆ’2π‘š2𝐡+π‘š+1𝑧=ξƒ©πœ‡+π‘šπ‘§βˆ’π‘šπ‘‘π΅,π‘š2𝐡+π‘šπ‘‘π΅,π‘šξ€Έξƒͺ2βˆ’π‘šξ€·π‘‘π΅,π‘šβˆ’1ξ€Έξ€·2π΅π‘š+π‘š2ξ€Έβˆ’14𝐡+π‘šπ‘‘π΅,π‘šξ€Έ2(6.3) with 𝑑𝐡,π‘šβˆΆ=2𝐡+2π‘š+1. Hence, it is clear that π‘‡π‘š(πœ‡π‘§)=0, viewed as second-degree polynomials in πœ‡π‘§, admits exactly two real solutions given by πœ‡Β±π‘§(𝐡,π‘š)∢=π‘šπ‘‘π΅,π‘š2𝐡+π‘šπ‘‘π΅,π‘šξ€ΈβŽ›βŽœβŽœβŽξƒ©2ξ€·1Β±1βˆ’π΅+π‘šπ‘‘π΅,π‘šξ€Έπ‘šπ‘‘2𝐡,π‘šξƒͺ1/2⎞⎟⎟⎠.(6.4) Now, assertions (i), (ii), and (iii) follow by discussing the sign of the parameter π‘„π‘š(πœ‡π‘§) (i.e., the sign of βˆ’π‘‡π‘š(πœ‡π‘§)) with respect to the modulus of π‘§βˆˆβ„‚βˆͺ{∞}, keeping in mind that |𝑧|2=πœ‡π‘§/(1βˆ’πœ‡π‘§).

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 π‘ŸΒ±βˆΆ=π‘ŸΒ±(𝐡,π‘š) are as in (6.2).


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