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

Abstract

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≡ℂ∪{∞}, and are of the from ||𝑧,𝐵⟩=1+|𝑧|²^−𝐵2𝐵_𝑗=0𝑗!((22𝐵𝐵−)!𝑗)!^1/2𝑧^𝑗||𝑗⟩,(1.1) where 𝐵∈ℤ₊ is a fixed integer parameter and |𝑗⟩ are number states of the field mode. Define 𝜇_𝑧∶=|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 ||⟨𝑗∣𝑧,𝐵⟩||²=(2𝐵)!𝑗!(2𝐵−𝑗)!𝜇^𝑗𝑧1−𝜇_𝑧^2𝐵−𝑗.(1.2) The latter is recognized as the binomial probability density ℬ(2𝐵,𝜇_𝑧) where {𝜇_𝑧,(1−𝜇_𝑧)} 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) ℎ^𝐵𝑗(𝑧)∶=1+|𝑧|²^−𝐵𝑗!((22𝐵𝐵−)!𝑗)!^1/2𝑧^𝑗,𝑗=0,1,2,…,2𝐵,(1.3) constitute an orthonormal basis of the null space 𝒜_𝐵𝕊²∶=𝜑∈𝐿²𝕊²,𝐻_𝐵[𝜑]=0(1.4) of the second-order differential operator 𝐻_𝐵∶=−1+|𝑧|²²𝜕²𝜕𝑧𝜕𝑧−𝐵1+|𝑧|²𝑧𝜕𝜕𝑧−𝑧𝜕𝜕𝑧+𝐵²|𝑧|²−𝐵,(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 ℎ^𝐵𝑗(𝑧) together with the reproducing kernel 𝐾_𝐵(𝑧,𝑤)=(2𝐵+1)1+𝑧𝑤^2𝐵1+|𝑧|²^−𝐵1+|𝑤|²^−𝐵(1.6) of the Hilbert space 𝒜_𝐵(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 ⟨𝑗∣𝑧,𝐵⟩=𝐾_𝐵(z,𝑧)^−(1/2)ℎ^𝐵𝑗(𝑧).(1.7) Note also that the space 𝒜_𝐵(𝕊²) is nothing else than the eigenspace associated with the first eigenvalue of the spectrum of 𝐻_𝐵 acting on 𝐿²(𝕊²), which consists of an infinite set of eigenvalues (spherical Landau levels) of the form 𝜖^𝐵𝑚∶=(2𝑚+1)𝐵+𝑚(𝑚+1),𝑚=0,1,2,…, with finite multiplicity; that is, the associated 𝐿²-eigenspace 𝒜_𝐵,𝑚𝕊²∶=𝜑∈𝐿²𝕊²,𝐻_𝐵[𝜑]=𝜖^𝐵𝑚𝜑(1.8) is of finite dimension equals to 𝑑_𝐵,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 ℎ^{𝑗𝐵,𝑚}(𝑧),𝑗=0,1,2,…,2𝐵+2𝑚, whose elements are expressed in terms of Jacobi polynomials 𝑃^{𝜂(𝜏,𝜍)}(⋅), as well as a reproducing kernel 𝐾_𝐵,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 ||𝑧,𝐵,𝑚⟩=𝐾_𝐵,𝑚(𝑧,𝑧)^{−(1/2)2𝐵+2𝑚}_𝑗=0ℎ^{𝑗𝐵,𝑚}(𝑧)√𝜌_𝐵,𝑚(𝑗)||𝑗⟩,(1.9) where 𝜌_𝐵,𝑚(𝑗) denotes the norm square of ℎ^{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 𝒜_𝐵,𝑚(𝕊²) a photon counting probability distribution in the same way as for the space 𝒜_𝐵(𝕊²)≡𝒜_𝐵,0(𝕊²) through the quantities 𝑝_𝑗2𝐵,𝜇_𝑧,𝑚=𝑚!(2𝐵+𝑚)!𝑗!(2𝐵+2𝑚−𝑗)!𝜇^{𝑧𝑗−𝑚}1−𝜇_𝑧^{2𝐵+𝑚−𝑗}𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}1−2𝜇_𝑧²𝑗=0,1,…,2𝐵+2𝑚.,(1.10) The latter can be considered as a kind of generalized binomial probability distribution 𝑋∼ℬ(2𝐵,𝜇_𝑧,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⟩=^𝑛_𝑗=0𝑗!(𝑛𝑛−!𝑗)!𝜇^𝑗(1−𝜇)^𝑛−𝑗^1/2𝑒^𝑖𝑗𝜃||𝑗⟩.(2.1) The corresponding photon counting probability is given by𝑝_𝑗(𝑛,𝜇)=𝑛!𝑗!(𝑛−𝑗)!𝜇^𝑗(1−𝜇)^𝑛−𝑗(2.2) which follows the binomial law 𝑌∼ℬ(𝑛,𝜇) with parameters 𝑛 and 𝜇; 𝑛∈ℤ₊, 0<𝜇<1. The connection with our notations in (1.1) and (1.2) can be made by setting 𝑛=2𝐵, 𝑧=|𝑧|𝑒^𝑖𝜃 and |𝑧|²=𝜇(1−𝜇)⁻¹. Note that in limits 𝜇→0 and 𝜇→1 the binomial state reduces to number states |0⟩ and |𝑛⟩, respectively. In a different limit of 𝑛→+∞ and 𝜇→0 with 𝑛𝜇→𝜆, the probability distribution (2.2) goes to the Poisson distribution 𝒫(𝜆)𝑝_𝑗(𝜆)=𝜆^𝑗𝑗!𝑒^−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∼ℬ(𝑛,𝜇) is given by 𝒞_𝑌(𝑡)=(1−𝜇)+𝜇𝑒^𝑖𝑡^𝑛(2.4) from which one obtains the mean value and the variance as 𝐸(𝑌)=𝑛𝜇,Var(𝑌)=𝑛𝜇(1−𝜇).(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 𝒜_𝐵,𝑚(𝕊²)

Let 𝕊²⊂ℝ³ denote the unit sphere with the standard metric of constant Gaussian curvature 𝜅=1. We identify the sphere 𝕊² with the extended complex plane ℂ∪{∞}, called the Riemann sphere, via the stereographic coordinate 𝑧=𝑥+𝑖𝑦; 𝑥,𝑦∈ℝ. 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 𝑧⁻¹.

In the stereographic coordinate 𝑧, the Hamiltonian operator of the Dirac monopole with charge 𝑞=2𝐵 reads [4, page 598]𝐻_𝐵∶=−1+|𝑧|²²𝜕²𝜕𝑧𝜕𝑧−𝐵𝑧1+|𝑧|²𝜕𝜕𝑧+𝐵𝑧1+|𝑧|²𝜕𝜕𝑧+𝐵²1+|𝑧|²−𝐵².(3.1) This operator acts on the sections of the 𝑈(1)-bundle with the first Chern class 𝑞. We have denoted by 𝐵∈ℤ₊ the strength of the quantized magnetic field. We shall consider the Hamiltonian 𝐻_𝐵 in (3.1) acting in the Hilbert space 𝐿²(𝕊²)∶=𝐿²(𝕊²,(1+|𝑧|²)⁻²𝑑𝜈(𝑧)), 𝑑𝜈(𝑧)=𝜋⁻¹𝑑𝑥𝑑𝑦 being the Lebesgue measure on ℂ≡ℝ². Its spectrum consists of an infinite number of eigenvalues (spherical Landau levels) of the form𝜖^𝐵𝑚∶=(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 𝐻_𝐵−𝐵 on 𝐿²(𝕊²) is intertwined with the invariant Laplacian Δ_2𝐵∶=−1+|𝑧|²²𝜕²𝜕𝑧𝜕𝑧+2𝐵𝑧1+|𝑧|²𝜕𝜕𝑧(3.3) acting in the Hilbert space 𝐿^2,𝐵(𝕊²)∶=𝐿²(𝕊²,(1+|𝑧|²)^−2−2𝐵𝑑𝜈(𝑧)). Namely, we have𝐻_𝐵−𝐵=1+|𝑧|²^−𝐵Δ_2𝐵1+|𝑧|²^𝐵,(3.4) and therefore any ket |𝜙⟩ of 𝐿^2,𝐵(𝕊²) is represented by1+|𝑧|²^−𝐵⟨𝑧∣𝜙⟩in𝐿²𝕊².(3.5) As mentioned in the introduction, we denote by 𝒜_𝐵,𝑚(𝕊²) the eigenspace of 𝐻_𝐵 in 𝐿²(𝕊²), corresponding to the eigenvalue 𝜖^𝐵𝑚 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: ℎ^{𝑗𝐵,𝑚}(𝑧)∶=1+|𝑧|²^−𝐵𝑧^𝑗𝑄_{𝐵,𝑚,𝑗}1+|𝑧||²𝑧|²,0≤𝑗≤2𝐵+2𝑚,(3.6) where 𝑄_{𝐵,𝑚,𝑗}(⋅) is the polynomial function given by𝑄_{𝐵,𝑚,𝑗}(𝑡)=𝑡^−𝑗(1−𝑡)^𝑗−2𝐵𝑑𝑑𝑡^𝑚𝑡^𝑗+𝑚(1−𝑡)^{2𝐵−𝑗+𝑚}.(3.7) According to the Jacobi's formula [11]𝑑𝑑𝑥^𝑚𝑥^{𝑐+𝑚−1}(1−𝑥)^𝑏−𝑐=Γ(Γ𝑐(+𝑐)𝑚)𝑥^𝑐−1(1−𝑥)^{2𝑏−𝑐−𝑚}𝐹₁(−𝑚,𝑏;𝑐;𝑥),(3.8)₂𝐹₁(𝑎,𝑏,𝑐;𝑥) being the Gauss hypergeometric function, it follows that 𝑄_{𝐵,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] ₂𝐹₁𝑘+𝜈+𝜚+1,−𝑘,1+𝜈;1−𝑡2=Γ𝑘(!𝑘Γ+(11++𝜈𝜈))𝑃^{𝑘(𝜈,𝜚)}(𝑡).(3.10) So that the orthogonal basis in (3.6) reads simply in terms of Jacobi polynomial as ℎ^{𝑗𝐵,𝑚}(𝑧)=𝑚!1+|𝑧|²^−𝐵𝑧^𝑗−𝑚𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}11+−||𝑧𝑧||²².(3.11) Also, one obtains the norm square of the eigenfunction ℎ^{𝑗𝐵,𝑚} given in (3.6) as 𝜌_𝐵,𝑚(𝑗)∶=‖‖ℎ^{𝑗𝐵,𝑚}‖‖^2𝐿_2(𝕊2)=𝑚!(𝑚+𝑗)!(2𝐵+𝑚−𝑗)!(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 𝒜_𝐵,𝑚(𝕊²): 𝐾_𝐵,𝑚(𝑧,𝑤)=(2𝐵+2𝑚+1)=1+𝑧𝑤^2𝐵1+|𝑧|²^𝐵1+|𝑤|²^𝐵×₂𝐹₁−𝑚,𝑚+2𝐵+1,1;1+|𝑧|𝑧|²−𝑤1|+²|𝑤|².(3.13)

Remark 3.1. In the case 𝑚=0, the elements of the orthogonal basis reduce further to ℎ^𝑗𝐵,0(𝑧)=(1+|𝑧|²)^−𝐵𝑧^𝑗 and the reproducing kernel reads 𝐾_𝐵,0(𝑧,𝑤)=(2𝐵+1)1+𝑧𝑤^2𝐵1+|𝑧|²^−𝐵1+|𝑤|²^−𝐵.(3.14)

Remark 3.2. In the case of the 𝑛-dimensional projective space ℙ(ℂ^𝑛)(=𝑆¹⧵𝑆^2𝑛+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 ℂ^𝑛) as 𝐻_𝐵∶=1+|𝑧|²^𝑛_𝑖_,𝑗=1𝛿_𝑖𝑗+𝑧_𝑖𝑧_𝑗𝜕²𝜕𝑧_𝑖𝜕𝑧_𝑗−𝐵^𝑛_𝑗=1𝑧^𝑗𝜕𝜕𝑧_𝑗−𝑧_𝑗𝜕𝜕𝑧_𝑗−𝐵²+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 (ℋ,⟨⋅,⋅⟩_ℋ) be a functional Hilbert space with an orthonormal basis {𝜙_𝑛}^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 {Φ_𝑛}^𝑑𝑛=1. Then, 𝒜² is a reproducing kernel Hilbert space whose reproducing kernel is given by 𝐾(𝑥,𝑦)∶=^𝑑_𝑛=1Φ_𝑛(𝑥)Φ_𝑛(𝑦)𝜌_𝑛;𝑥,𝑦∈Ω,(4.1) where we have set 𝜌_𝑛∶=‖Φ_𝑛‖^2𝐿_{2(Ω,𝑑𝑠)}. Associated to the data of (𝒜²,Φ_𝑛) and (ℋ,𝜙_𝑛), the generalized coherent states are the elements of ℋ defined by Φ_𝑥∶=𝜔_𝑑(𝑥)^𝑑−(1/2)_𝑛=1Φ_𝑛(𝑥)√𝜌_𝑛𝜙_𝑛;𝑥∈Ω,(4.2) where 𝜔_𝑑(𝑥) stands for 𝜔_𝑑(𝑥)∶=𝐾(𝑥,𝑥). Note that the choice of the Hilbert space ℋ defines a quantization of Ω into ℋ by considering the inclusion map 𝑥↦Φ_𝑥. Furthermore, it is straightforward to check that ⟨Φ_𝑥,Φ_𝑥⟩_ℋ=1 and to show that the corresponding coherent state transform (CST) 𝒲 onℋ, 𝒲[𝑓](𝑥)∶=𝜔_𝑑(𝑥)^1/2⟨Φ_𝑥,𝑓⟩_ℋ;𝑓∈ℋ,(4.3) defines an isometry from ℋ into 𝒜². Thereby we have a resolution of the identity of ℋ, that is, we have the following integral representation: 𝑓(⋅)=_Ω⟨Φ_𝑥,𝑓⟩_ℋΦ_𝑥(⋅)𝜔_𝑑(𝑥)𝑑𝑠(𝑥);𝑓∈ℋ.(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 ||𝜁⟩=𝑒^|𝜁|2^−(1/2)_𝑘≥0𝜁^𝑘√𝑘!||𝑘⟩(4.5) with 𝜙_𝑘∶=|𝑘⟩ being the number states of the harmonic oscillator, which also knwon as Fock states.

We can now construct for each spherical Landau level 𝜖^𝐵𝑚 given in (3.2) a set of generalized coherent states (GCS) according to formula (4.2) as𝜗_{𝑧,𝐵,𝑚}≡||𝑧,𝐵,𝑚⟩=𝐾_𝐵,𝑚(𝑧,𝑧)^{−(1/2)2𝐵+2𝑚}_𝑗=0ℎ^{𝑗𝐵,𝑚}(𝑧)√𝜌_𝐵,𝑚(𝑗)||𝜙^𝑗(4.6) with the following precisions:(i)(Ω,𝑑𝑠)∶=(𝕊²,(1+|𝑧|²)⁻²𝑑𝜈(𝑧)), 𝕊² being identified with ℂ∪{∞}.(ii)𝒜²∶=𝒜_𝐵,𝑚(𝕊²) is the eigenspace of 𝐻_𝐵 in 𝐿²(𝕊²) with dimension 𝑑_𝐵,𝑚=2𝐵+2𝑚+1.(iii)𝜔(𝑧)=𝐾_𝐵,𝑚(𝑧,𝑧)=2𝐵+2𝑚+1 (in view of (3.13)).(iv)ℎ^{𝑗𝐵,𝑚}(𝑧) are the eigenfunctions given by (3.11) in terms of the Jacobi polynomials.(v)𝜌_𝐵,𝑚(𝑗) being the norm square of ℎ^{𝑗𝐵,𝑚}, given in (3.12).(vi)ℋ∶=𝒫_𝐵+𝑚 the space of polynomials of degree less than 𝑑_𝐵,𝑚, which carries a unitary irreducible representation of the compact Lie group SU(2) (see [14]). The scalar product in 𝒫_𝐵+𝑚 is written as ⟨𝜓,𝜙⟩_{𝒫𝐵+𝑚}=𝑑_𝐵,𝑚_ℂ𝑑𝑠(𝜁)1+||𝜁||²^{−2(𝐵+𝑚)−2}𝜓(𝜁)𝜙(𝜁).(4.7)(vii){𝜙_𝑗;0≤𝑗≤2𝐵+2𝑚} is an orthonormal basis of 𝒫_𝐵+𝑚, whose elements are given explicitly by𝜙_𝑗(𝜉)∶=(2(𝐵+𝑚))!(2𝐵+𝑚−𝑗)!(𝑗+𝑚)!𝜉^𝑗+𝑚𝜁∈ℂ.(4.8)

Definition 4.2. For given positive integers 2𝐵 and 𝑚, and fixed 𝑧∈𝕊², the wave function of the GCS in (4.6) are expressed as 𝜗_{𝑧,𝐵,𝑚}(𝜉)=11+|𝑧|²^{−𝐵2𝐵+2𝑚}_𝑗=0√𝑚!(2𝐵+𝑚)!(2𝐵+2𝑚)!𝑗!(2𝐵+2𝑚−𝑗)!𝑧^𝑗−𝑚𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}11−+||𝑧𝑧||²²𝜉^𝑗,(4.9) for every 𝜁∈ℂ.

According to (4.4), the system of GCS |𝜗_{𝑧,𝐵,𝑚}⟩ solves then the unity of the Hilbert space 𝒫_𝐵+𝑚 as𝟏_{𝒫𝐵+𝑚}=𝑑_𝐵,𝑚_ℂ𝑑𝜈(𝑧)1+|𝑧|²⁻²||𝜗_{𝑧,𝐵,𝑚}𝜗_{𝑧,𝐵,𝑚}||.(4.10) They also admit a closed form [15], as 𝜗_{𝑧,𝐵,𝑚}(𝜉)=(2𝐵+2𝑚)!(2𝐵+𝑚)!𝑚!(11++𝜉|𝑧𝑧|²)²^𝐵𝜉−𝑧(1+𝜉𝑧)1+|𝑧|²^𝑚.(4.11)

Remark 4.3. Note that for 𝑚=0, the expression above reduces to ⟨𝜉∣𝑧,𝐵,0⟩=1+|𝑧|²^−𝐵(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 𝜗_{𝑧,𝐵,𝑚} onto the state 𝜙_𝑗, is given by ||𝜗_{𝑧,𝐵,𝑚},𝜙_𝑗_ℋ||²=1𝜌^{𝑗𝐵,𝑚}𝑑_𝐵,𝑚||ℎ^{𝑗𝐵,𝑚}(𝑧)||².(5.1) This is in fact the probability of finding 𝑗 photons in the coherent state 𝜗_{𝑧,𝐵,𝑚}. More explicitly, in view of (3.11), the quantity in (5.1) reads ||𝜗_{𝑧,𝐵,𝑚},𝜙_𝑗_ℋ||²=𝑚!(2𝐵+𝑚)!𝑗!(2𝐵+2𝑚−𝑗)!1+|𝑧|²^−2𝐵|𝑧|^{2(𝑗−𝑚)}𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}11+−||𝑧𝑧||²²².(5.2) As annonced in the introduction, we denote the expression in (5.2) by 𝑝_𝑗(2𝐵,𝜇_𝑧,𝑚) for 𝑗=0,1,2,…, with 𝜇_𝑧=|𝑧|²(1+|𝑧|²)⁻¹ or equivalently |𝑧|²=𝜇_𝑧(1−𝜇_𝑧)⁻¹. Motivated by this quantum probability, we then state the following.

Definition 5.1. For fixed integers 𝐵,𝑚∈ℤ₊, the discrete random variable 𝑋 having the probability distribution 𝑝_𝑗2𝐵,𝜇_𝑧,𝑚=𝑚!(2𝐵+𝑚)!𝑗!(2𝐵+2𝑚−𝑗)!𝜇^{𝑧(𝑗−𝑚)}1−𝜇_𝑧^{2𝐵+𝑚−𝑗}𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}1−2𝜇_𝑧²,(5.3) with 𝑗=0,1,2,…,2𝐵+2𝑚, and denoted by 𝑋∼ℬ(2𝐵,𝜇_𝑧,𝑚), 0<𝜇_𝑧<1, will be called the generalized binomial probability distribution associated to the weighted Hilbert space 𝒜_𝐵,𝑚(𝑆²).

Remark 5.2. Note that for 𝑚=0, the above expression in (5.3) reduces to 𝑝_𝑗2𝐵,𝜇_𝑧,0=(2𝐵)!𝑗!(2𝐵−𝑗)!𝜇^𝑗𝑧1−𝜇_𝑧^2𝐵−𝑗,𝑗=0,1,2,…,2𝐵,(5.4) which is the standard binomial distribution with parameters 2𝐵 and 0<𝜇_𝑧<1.

A convenient way to summarize all the properties of a probability distribution 𝑋 is to explicit its characteristic function: 𝒞_𝑋(𝑡)=∶𝐸𝑒^𝑖𝑡𝑋,(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∼ℬ(2𝐵,𝜇_𝑧,𝑚) is given by 𝒞_𝑚(𝑡)=𝑒^𝑖𝑚𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡^2𝐵𝑃^{𝑚(0,2𝐵)}1−4𝜇_𝑧1−𝜇_𝑧(1−cos(𝑡))(5.6) for every 𝑡∈ℝ.

Proof. Recall first that for every given fixed nonnegative integer 𝑚, the characteristic function 𝒞_𝑚(𝑡) in (5.5) can be written as 𝒞_𝑚(𝑡)=^2𝐵+2𝑚_𝑗=0=𝑒^𝑖𝑗𝑡𝑝_𝑗2𝐵,𝜇_𝑧,𝑚_2𝐵+2𝑚_𝑗=0𝑒^𝑖𝑗𝑡𝑚!(2𝐵+𝑚)!𝑗!(2𝐵+2𝑚−𝑗)!𝜇^{𝑧(𝑗−𝑚)}1−𝜇_𝑧^{2𝐵+𝑚−𝑗}𝑃^{𝑚(𝑗−𝑚,2𝐵+𝑚−𝑗)}1−2𝜇_𝑧².(5.7) The last equality follows using (5.3). Next, by making the change 𝑘=𝐵+𝑚−𝑗 in (5.7), it follows that 𝒞_𝑚(𝑡)=^𝐵+𝑚_𝑘=−_(𝐵+𝑚)𝑒^{𝑖(𝐵+𝑚−𝑘)𝑡}𝑚!(2𝐵+𝑚)!(𝐵+𝑚+𝑘)!(𝐵+𝑚−𝑘)!𝜇^{𝑧𝐵−𝑘}1−𝜇_𝑧^𝐵+𝑘𝑃^{𝑚(𝐵−𝑘,𝐵+𝑘)}1−2𝜇_𝑧².(5.8) Instead of the Jacobi polynomials, it is convenient to consider the closely related function 𝒫^{𝑙𝑟,𝑠}(𝑥) introduced in [14, page 270]. They can be defined through the formula [14, equation (1), page 288], 𝑃^{(𝑟−𝑠,𝑟+𝑠)𝑛−𝑟}(𝑥)=2^𝑟((𝑛𝑛−−𝑠𝑟))!!((𝑛𝑛++𝑠𝑟))!!^1/2(1−𝑥)^{(𝑠−𝑟)/2}(1+𝑥)^{−(𝑠+𝑟)/2}𝒫^{𝑛𝑟,𝑠}(𝑥),(5.9) with 𝑚=𝑛−𝐵 (i.e., 𝑛=𝐵+𝑚) and 𝑥=1−2𝜇_𝑧. We can then express the square of 𝑃^{𝑚(𝐵−𝑘,𝐵+𝑘)}(𝑥) as follows: 𝑃^{𝑚(𝐵−𝑘,𝐵+𝑘)}1−2𝜇_𝑧²=(𝐵+𝑚𝑚−!(𝑘2)𝐵!(+𝐵𝑚+)𝑚!+𝑘)!𝜇^{𝑧−𝐵+𝑘}1−𝜇_𝑧^{−𝐵−𝑘}𝒫^{𝐵+𝑚𝐵,𝑘}1−2𝜇_𝑧².(5.10) Therefore, (5.8) reduces further to 𝒞_𝑚(𝑡)=𝑒^{𝑖(𝐵+𝑚)𝑡𝐵+𝑚}_𝑘=−_(𝐵+𝑚)𝑒^{−𝑖𝑘𝑡}𝒫^{𝐵+𝑚𝐵,𝑘}1−2𝜇_𝑧²(5.11)_(⋆)=(−1)^𝐵𝑒^{𝑖(𝐵+𝑚)𝑡𝐵+𝑚}_𝑘=−_(𝐵+𝑚)=(−1)_𝐵𝑒^{−𝑖𝑘(𝑡−𝜋)}𝒫^{𝐵+𝑚𝐵,𝑘}1−2𝜇_𝑧𝒫^{𝐵+𝑚𝑘,𝐵}1−2𝜇_𝑧𝑒^{𝑖(𝐵+𝑚)𝑡}𝑒^{−𝑖𝐵(𝜑+𝜓)}𝒫^{𝐵+𝑚𝐵,𝐵}(cos(𝜃)).(5.12) The transition (⋆) above holds using the fact that [14, page 288] 𝒫^{𝑙𝑗,𝑘}(𝑥)=(−1)^𝑗+𝑘𝒫^{𝑙𝑘,𝑗}(𝑥).(5.13) While the last equality can be checked easily using the addition formula [14, equation (3), page 326] ^𝑠_𝑘=−𝑠𝑒^{−𝑖𝑘𝜏}𝒫^{𝑠𝑗,𝑘}cos𝜃₁𝒫^{𝑠𝑘,𝑙}cos𝜃₂=𝑒^{−𝑖(𝑗𝜑+𝑙𝜓)}𝒫^{𝑠𝑗,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: 𝑒cos(𝜃)=cos²(2𝛼)+sin²(2𝛼)cos(𝑡),_{𝑖((𝜑+𝜓)/2)}=−𝑖cos²(𝛼)+sin²(𝛼)𝑒^−𝑖𝑡𝑒^𝑖𝑡/2cos(𝜃/2)(5.15) for 𝜃₁=𝜃₂=2𝛼, so that 𝑒^{−𝑖𝐵(𝜑+𝜓)}=(−1)^𝐵cos𝜃2^−2𝐵cos²(𝛼)+sin²(𝛼)𝑒^𝑖𝑡^2𝐵𝑒^{−𝑖𝐵𝑡}.(5.16) Next, using the fact that 2^−𝑠(1+𝑥)^𝑠𝑃^{(0,2𝑠)𝑛−𝑠}(𝑥)=𝒫^{𝑛𝑠,𝑠}(𝑥),(5.17) which is a special case of (5.9), with 𝑠=𝐵, 𝑛−𝑠=𝑚, and 𝑥=cos(𝜃), we obtain 𝒫^{𝐵+𝑚𝐵,𝐵}(cos(𝜃))=cos𝜃2^2𝐵𝑃^{𝑚(0,2𝐵)}(cos(𝜃)).(5.18) Finally, by substituting (5.16) and (5.18) in (5.12), taking into account that sin²(𝛼)=𝜇_𝑧 and cos²(𝛼)=1−𝜇_𝑧, we see that the characteristic function 𝒞_𝑚(𝑡) reads simply as 𝒞_𝑚(𝑡)=𝑒^𝑖𝑚𝑡cos²(𝛼)+sin²(𝛼)𝑒^𝑖𝑡^2𝐵𝑃^{𝑚(0,2𝐵)}(cos(𝜃)),(5.19) where cos(𝜃)=1−4𝜇_𝑧(1−𝜇_z)(1−cos(𝑡)).

Remark 5.4. Note that by taking 𝑚=0 in (5.19), the characteristic function reduces to 𝐶_𝑌(𝑡)=cos²(𝛼)+sin²(𝛼)𝑒^𝑖𝑡^2𝐵=1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡^2𝐵,(5.20) which is the well-known characteristic function of the binomial random variable 𝑌∼ℬ(2𝐵,𝜇_𝑧) with parameters 𝑛=2𝐵∈ℤ₊ and 0<𝜇_𝑧<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 𝒞_𝑚(𝑡) in (5.6) with respect to the variable 𝑡 and evaluation at the origin as 𝐸𝑋^𝑘=1𝑖^𝑘𝜕^𝑘𝜕𝑡^𝑘𝐶_𝑚(𝑡)||||_𝑡=0.(5.21)

Corollary 5.5. Let 𝑚,2𝐵∈ℤ₊. The mean value and the variance of 𝑋∼ℬ(2𝐵,𝜇_𝑧,𝑚) are given respectively by Var(𝑋)=2𝐵𝜇_𝑧𝐸(𝑋)=𝑚+2𝐵𝜇_𝑧,1−𝜇_𝑧+2𝜇_𝑧1−𝜇_𝑧𝑚(2𝐵+𝑚+1).(5.22)

Proof. Let recall first that for every fixed integer 𝑚=0,1,2,…, we have 𝐸(𝑋)=𝜕𝒞^𝑚𝑖𝜕𝑡||||_𝑡=0Var(𝑋)=𝐸𝑋₂,−[𝐸(𝑋)]²=𝜕²𝒞_𝑚𝑖²𝜕𝑡²||||_𝑡=0−𝜕𝒞^𝑚𝑖𝜕𝑡||||_𝑡=0².(5.23) Thus direct computation gives rise to 𝜕𝒞_𝑚𝑖𝜕𝑡(𝑡)=⎡⎢⎢⎣𝑚+2𝐵𝜇^𝑧𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡−4𝜇^𝑧1−𝜇_𝑧sin(𝑡)⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝑖𝜕𝑥)||_{𝑥=cos(𝜃)}𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠⎤⎥⎥⎦𝒞^𝑚𝜕(𝑡),₂𝒞_𝑚𝑖²𝜕𝑡²=⎡⎢⎢⎣𝜕(𝑡)𝑖𝜕𝑡⎛⎜⎜⎝𝑚+2𝐵𝜇^𝑧𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡−4𝜇^𝑧1−𝜇_𝑧sin(𝑡)⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝑖𝜕𝑥)||_{𝑥=cos(𝜃)}×𝒞_𝑚𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠⎞⎟⎟⎠⎤⎥⎥⎦(𝑡)+⎡⎢⎢⎣𝑚+2𝐵𝜇^𝑧𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡−4𝜇^𝑧1−𝜇_𝑧sin(𝑡)⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝑖𝜕𝑥)||_{𝑥=cos(𝜃)}=⎡⎢⎢⎣2𝐵𝜇_𝑧𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠⎤⎥⎥⎦𝜕𝑖𝒞𝜕𝑡^𝑚(𝑡)𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡−2𝐵𝜇^2𝑧𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡²+4𝜇_𝑧1−𝜇_𝑧cos(𝑡)⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝜕𝑥)||_{𝑥=cos(𝜃)}×𝒞_𝑚𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠⎤⎥⎥⎦(𝑡)−4𝜇_𝑧1−𝜇_𝑧sin(𝑡)𝜕𝑖𝜕𝑡⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝑖𝜕𝑥)||_{𝑥=cos(𝜃)}𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠𝒞^𝑚+⎡⎢⎢⎣𝑚+2𝐵𝜇_𝑧(𝑡)𝑒^𝑖𝑡1−𝜇_𝑧+𝜇_𝑧𝑒^𝑖𝑡−4𝜇^𝑧1−𝜇_𝑧sin(𝑡)⎛⎜⎜⎝(𝜕𝑃^{𝑚(0,2𝐵)}(𝑥)/𝑖𝜕𝑥)||_{𝑥=cos(𝜃)}𝑃^{𝑚(0,2𝐵)}(cos(𝜃))⎞⎟⎟⎠⎤⎥⎥⎦𝜕𝑖𝒞𝜕𝑡^𝑚(𝑡).(5.24) To conclude, we have to use successively the facts that for 𝑡=0, we have cos(𝜃)=1 and 𝒞_𝑚(0)=1, together with 𝜕𝑃^{𝑚(𝑎,𝑏)}𝑖𝜕𝑥(𝑥)=𝑎+𝑏+2𝑚+1𝑃^{(𝑎+1,𝑏+1)𝑚−1}(𝑥),𝑃^{𝑚(𝑎,𝑏)}(1)=Γ(𝑎+𝑚+1)𝑚!Γ(𝑎+1).(5.25) Thus, we have 𝐸(𝑋)=𝜕𝒞^𝑚𝑖𝜕𝑡||||_𝑡=0=𝑚+2𝐵𝜇_𝑧𝒞_𝑚(0)=𝑚+2𝐵𝜇_𝑧.(5.26) We have also 𝜕²𝒞_𝑚𝑖²𝜕𝑡²|_𝑡=0=2𝐵𝜇_𝑧1−𝜇_𝑧+2𝜇_𝑧1−𝜇_𝑧(2𝐵+𝑚+1)𝑃^{(1,2𝐵+1)𝑚−1}(1)×𝒞_𝑚𝑃^{𝑚(0,2𝐵)}(1)(0)+𝑚+2𝐵𝜇_𝑧𝜕𝒞^𝑚𝑖𝜕𝑡|^𝑡=0,(5.27) and therefore Var(𝑋)=𝜕²𝒞_𝑚𝑖²𝜕𝑡²||||_𝑡=0−𝜕𝒞^𝑚𝑖𝜕𝑡||||_𝑡=0²=2𝐵𝜇_𝑧1−𝜇_𝑧+2𝜇_𝑧1−𝜇_𝑧𝑚(2𝐵+𝑚+1).(5.28)

Remark 5.6. Note that by taking 𝑚=0 in (5.22), we recover the standard values 𝐸(𝑌)=2𝐵𝜇_𝑧,Var(𝑌)=2𝐵𝜇_𝑧1−𝜇_𝑧(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 𝑚≠0, we make use of the obtained statistical parameters 𝐸(𝑋) and Var(𝑋) to calculate Mandel parameter corresponding to the random variable 𝑋∼ℬ(2𝐵,𝜇_𝑧,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_±(𝐵,𝑚)∶=1±1−1𝑚(2𝐵+𝑚)^1/2^1/2.(6.2) Then, 𝑟₋(𝐵,𝑚)≤1≤𝑟₊(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 𝑚≠0. Making use of (5.22), we see that the Mandel parameter can be written as follows 𝑄(𝑋)=𝑄_𝑚(𝜇_𝑧)=−𝑇_𝑚(𝜇_𝑧)/(2𝐵𝜇_𝑧+𝑚), where we have set 𝑇_𝑚𝜇_𝑧==2(𝜇𝐵+𝑚[2𝐵+𝑚+1])𝜇^2𝑧−2𝑚[2𝐵+𝑚+1]𝜇_𝑧+𝑚_𝑧−𝑚𝑑^𝐵,𝑚2𝐵+𝑚𝑑_𝐵,𝑚²−𝑚𝑑^𝐵,𝑚−12𝐵𝑚+𝑚²−14𝐵+𝑚𝑑_𝐵,𝑚²(6.3) with 𝑑_𝐵,𝑚∶=2𝐵+2𝑚+1. Hence, it is clear that 𝑇_𝑚(𝜇_𝑧)=0, viewed as second-degree polynomials in 𝜇_𝑧, admits exactly two real solutions given by 𝜇^±𝑧(𝐵,𝑚)∶=𝑚𝑑^𝐵,𝑚2𝐵+𝑚𝑑_𝐵,𝑚⎛⎜⎜⎝1±1−2𝐵+𝑚𝑑^𝐵,𝑚𝑚𝑑^2𝐵,𝑚^1/2⎞⎟⎟⎠.(6.4) Now, assertions (i), (ii), and (iii) follow by discussing the sign of the parameter 𝑄_𝑚(𝜇_𝑧) (i.e., the sign of −𝑇_𝑚(𝜇_𝑧)) with respect to the modulus of 𝑧∈ℂ∪{∞}, keeping in mind that |𝑧|²=𝜇_𝑧/(1−𝜇_𝑧).

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.