Generalized Binomial Probability Distributions attached to Landau levels on the Riemann sphere

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 application, photon number statistics related to coherent states under consideration are discussed.


INTRODUCTION
The binomial states (BS) are the field states that are superposition of the number states with appropriately chosen coefficients [11]. Precisely, these states are labeled by points z of the Riemann sphere S 2 ≡ C ∪ {∞}, and are of the form where B ∈ Z + is a fixed integer parameter and | j > are number states of the field mode.
Define µ z to be 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 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 [1]. Also, observe that the coefficients in the finite superposition of number states in (1. of the second-order differential operator which constitutes (in suitable units and up to additive constant) a realization in L 2 (S 2 ) of the Schrödinger operator with uniform magnetic field on S 2 , with a field strength proportional to B (see [2]). The given orthonormal basis h B j (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 (S 2 ) 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 B j (z). (1.7) The space A B (S 2 ) is nothing else than the eigenspace associated with the first eigenvalue of the spectrum of H B acting on L 2 (S 2 ), which consists of an infinite set of eigenvalues (spherical Landau levels) of the form: with finite multiplicity; i.e., the associated L 2 −eigenspace 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 (1.9) admits an orthogonal basis denoted h B,m j (z), j = 0, 1, 2, · · · , 2B + 2m, whose elements are expressed in terms of Jacobi polynomials P (τ,ς) η (.), as well as a reproducing kernel K B,m (z, w) in an explicit form (see [9]) to consider a set of coherent states by adopting a generalized coherent states technique " à la Iwata" [5] as: . The coherent states in (1.10) possess a form similar to (1.1) and will enables us, starting from the observation made in (1.7), to attach to each eigenspace A B,m S 2 a photon counting probability distribution in the same way as for the space A B (S 2 ) ≡ A B,0 (S 2 ) through the quantities 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 (1.11) 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 (1.10). 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.

THE BINOMIAL STATES
The binomial states in their first form were introduced by Stoler et al. [11] 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 the coefficients of its j states expansion are allowed to have additional arbitrary phases) a generalized binomial state can be defined by and has as a photon counting probability which follows the binomial law Y ∼ B (n, µ) with parameters n and µ; n ∈ Z + , 0 < 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) which characterize the coherent states of the harmonic oscillator. In fact, and as pointed out in [11] 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 from which one obtains the mean value and the variance as Therefore, the Mandel parameter, which measures deviation from the Poissonian distribution, is always negative. Thus photon statistics in the the binomial states is always sub-Poissonian.

Remark 2.1.
We should note that a binomial state also admits a ladder operator definition [3] which means that this state is an eigenstate of a proper combination of the number operator and the annihilation operator via the Holstein-Primakoff realization of the Lie algebra of the group SU(2).

AN ORTHONORMAL BASIS OF A B,m (S 2 )
Let S 2 ⊂ R 3 denotes the unit sphere with the standard metric of constant Gaussian curvature κ = 1. We shall identify the sphere S 2 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 [2, p.598]: 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 dν(z) = π −1 dxdy being the Lebesgue measure on C≡R 2 . Its spectrum consists on an infinite number of eigenvalues (spherical Landau levels) of the form with finite degeneracy 2B + 2m + 1 (see [2, p.598]). In order to present expressions of the corresponding eigensections in the coordinate z, we first mention that the shifted operator . Namely, we have and therefore any ket |φ > of L 2,B (S 2 ) is represented by given in (3.2). Then, by [9] together with (3.5) and the intertwining relation (3.4), we obtain the following orthogonal basis of A B,m (S 2 ): where Q B,m,j (·) is the polynomial function given by According to the Jacobi's formula ( [7]): being the Gauss hypergeometric function, it follows that The latter can also be expressed in terms of Jacobi polynomials via the transformation ([7, p.283]) So that the orthogonal basis in (3.6) reads simply in terms of Jacobi polynomial as Also, one obtains the norm square of the eigenfunction h B,m j given in (3.6) as Finally, by Theorem 1 of [9, p.231] and thank to (3.5), we obtain the following expression for the reproducing kernel of the Hilbert eigenspace A B,m (S 2 ): Remark 3.1. Note that in the case m = 0, elements of the orthogonal basis reduce further to h B,0 j (z) = (1 + |z| 2 ) −B z j and the reproducing kernel reads simply as Remark 3.2. Note that in higher dimension, i.e., in the case of the n-dimensional projective space P(C n )(= S 1 \S 2n+1 ) equipped with the Fubini-Study metric, an 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 have been obtained in [4].

GENERALIZED COHERENT STATES
Let (H, ·, · H ) be a finite d-dimensional functional Hilbert space with an orthonormal basis {φ n } d n=1 and A 2 a finite d-dimensional subspace of the Hilbert space L 2 (Ω, ds), of square integrable functions on a given measured space (Ω, ds), with an orthogonal basis {Φ n } d n=1 . Then, A 2 is a reproducing kernel Hilbert space whose the reproducing kernel is given by where we have set ρ n := Φ n 2 L 2 (Ω,ds) . Associated to the data of (A 2 , Φ n ) and (H, φ n ), we introduce the following Definition 4.1. We define the generalized coherent states to be the elements of H given by 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 on H, defines an isometry from H into A 2 . Thereby we have a resolution of the identity, i.e., we have the following integral representation Remark 4.2. 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 with φ k := |k > being the number states of the harmonic oscillator.
• h B,m j (z) are the eigenfunctions given by (3.11) in terms of the Jacobi polynomials. • ρ B,m (j) being the norm square of h B,m j , given in (3.12). • 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 [12]). The scalar product in P B+m is written as • φ j ; 0 ≤ j ≤ 2B + 2m is an orthonormal basis of P B+m , whose elements are give explicitly by: According to (4.4), the system of GCS | ϑ z,B,m > solves then the unity of the Hilbert space P B+m as They also admit a closed form [8], as (4.11)

Remark 4.4.
Note that for m = 0, the expression above reduces to which are wave functions of Perelomov's coherent states based on SU(2) (see [10, p.62]).

GENERALIZED BINOMIAL PROBABILITY DISTRIBUTIONS
According to (4.2), we see that the squared modulus of ϑ z,B,m , φ j H , the projection of coherent state ϑ z,B,m onto the state φ j , is given by 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 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 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 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 (S 2 ).

Remark 5.2.
Note that for m = 0, the above expression in (5.3) reduces to , 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 : where t is a real number, i := √ −1 is the imaginary unit and E denotes the expected value or the mean value. We precisely establish the following result.
Proof. Recall first that for every given fixed nonnegative integer m, the characteristic function C m (t) in (5.4) can be written as according to the expression of p j (2B, µ z , m) given through (5.3). Next, by making the change (5.7) Instead of the Jacobi polynomials, it is convenient to consider the closely related function P l r,s (x) introduced in [ with m = n − B (i.e., n = B + m) and x = 1 − 2µ z . We can then express the square of P Therefore, (5.7) reduces further to The transition ( ) above holds using the fact that ([12, p.288]) P l j,k (x) = (−1) j+k P l k,j (x). While the last equality can be checked easily using the addition formula ([12, Eq3, p.326]) s ∑ k=−s e −ikτ P s j,k (cos(θ 1 ))P s k,l (cos(θ 2 )) = e −i(jϕ+lψ) P s j,l (cos(θ)).

Remark 5.4.
Note that by taking m = 0 in (5.15), the characteristic function reduces to 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, as mentioned at the beginning of this section, 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 (5.5) with respect to the variable t and evaluation at the origin as Proof. Let recall first that for every fixed integer m = 0, 1, 2, · · · , we have Thus direct computation gives rise to and 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 Thus, we have We have also and therefore Remark 5.6. Note that by taking m = 0 in (5.16) and (5.17), we recover the standard values of the binomial probability distribution as given in (2.5).

PHOTON COUNTING STATISTICS
For an arbitrary quantum state one may ask to what extent is "non-classical" in a sense that its properties differ from those of coherent states? In other words, is there any parameter that may reflect the degree on non-classicality of a given quantum state? In general, to define a measure of non-classicality of a quantum states one can follow several different approach. An earlier attempt to shed some light on the non-classicality of a quantum state was pioneered by Mandel [6], who investigated radiation fields and introduced the parameter 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 (5.18), and thereby the binomial states obeys sub-Poissonian statistics.
For m = 0, we make use of the obtained statistical parameters E(X) and Var(X) to calculate Mandel parameter corresponding 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: 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 ).
The figure below illustrates the quantum photon counting statistics with respect to the location in the extended complex plane of the point z labeling the generalized coherent states ϑ z,B,m discussed in Proposition 6.1. Here r ± := r ± (B, m) are as in (6.2).