Research Article | Open Access
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. https://doi.org/10.1155/2011/393417
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 an application, photon number statistics related to coherent states under consideration are discussed.
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 of the Riemann sphere , and are of the from where is a fixed integer parameter and are number states of the field mode. Define , then the probability for the production of photons is given by the squared modulus of the projection of the BS onto the number state as The latter is recognized as the binomial probability density where are the probabilities of the two possible outcomes of a Bernoulli trial . Also, observe that the coefficients in the finite superposition of number states in (1.1) constitute an orthonormal basis of the null space of the second-order differential operator which constitutes (in suitable units and up to additive constant) a realization in of the SchrΓΆdinger operator with uniform magnetic field on , with a field strength proportional to (see ). The given orthonormal basis together with the reproducing kernel of the Hilbert space in (1.4) can be used to interpret the projection of the BS onto the number state mentioned in (1.2) by writing Note also that the space is nothing else than the eigenspace associated with the first eigenvalue of the spectrum of acting on , which consists of an infinite set of eigenvalues (spherical Landau levels) of the form with finite multiplicity; that is, the associated -eigenspace is of finite dimension equals to .
Here, we take the advantage of the fact that each of the eigenspaces in (1.8) admits an orthogonal basis, denoted , whose elements are expressed in terms of Jacobi polynomials , as well as a reproducing kernel in an explicit form (see ) to consider a set of coherent states by adopting a generalized coherent states form βΓ la Iwataβ  as where denotes the norm square of in . 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 photon counting probability distribution in the same way as for the space through the quantities The latter can be considered as a kind of generalized binomial probability distribution depending on an additional parameter . 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 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.  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 states expansion are allowed to have additional arbitrary phases), generalized binomial states (GBS) can be defined by The corresponding photon counting probability is given by which follows the binomial law with parameters and ; , . The connection with our notations in (1.1) and (1.2) can be made by setting , and . Note that in limits and the binomial state reduces to number states and , respectively. In a different limit of and with , the probability distribution (2.2) goes to the Poisson distribution which characterizes the coherent states of the harmonic oscillator. In fact, and as pointed out in , 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 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 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 . 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) .
3. An Orthonormal Basis of
Let denote the unit sphere with the standard metric of constant Gaussian curvature . We identify the sphere with the extended complex plane , called the Riemann sphere, via the stereographic coordinate ; . We shall work within a fixed coordinate neighborhood with coordinate obtained by deleting the βpoint at infinityβ . Near this point we use instead of the coordinate .
In the stereographic coordinate , the Hamiltonian operator of the Dirac monopole with charge reads [4, page 598] This operator acts on the sections of the -bundle with the first Chern class . We have denoted by the strength of the quantized magnetic field. We shall consider the Hamiltonian in (3.1) acting in the Hilbert space , being the Lebesgue measure on . Its spectrum consists of an infinite number of eigenvalues (spherical Landau levels) of the form with finite degeneracy (see [4, page 598]). In order to present expressions of the corresponding eigensections in the coordinate , we first mention that the shifted operator on is intertwined with the invariant Laplacian acting in the Hilbert space . Namely, we have and therefore any ket of is represented by As mentioned in the introduction, we denote by the eigenspace of in , corresponding to the eigenvalue given in (3.2). Then, by  together with (3.5) and the intertwining relation (3.4), we obtain the following orthogonal basis of this eigenspace: where is the polynomial function given by According to the Jacobi's formula  being the Gauss hypergeometric function, it follows that The latter can also be expressed in terms of Jacobi polynomials via the transformation [11, page 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 given in (3.6) as 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 :
Remark 3.1. In the case , the elements of the orthogonal basis reduce further to and the reproducing kernel reads
Remark 3.2. In the case of the -dimensional projective space 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 ) as have been obtained in .
4. Generalized Coherent States
We follow the formalism presented in  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 . Let be a functional Hilbert space with an orthonormal basis and a finite d-dimensional subspace of the Hilbert space , of square integrable functions on a given measured space , with an orthogonal basis . Then, is a reproducing kernel Hilbert space whose reproducing kernel is given by where we have set . Associated to the data of and , the generalized coherent states are the elements of defined by where stands for . Note that the choice of the Hilbert space defines a quantization of into by considering the inclusion map . Furthermore, it is straightforward to check that and to show that the corresponding coherent state transform (CST) on, defines an isometry from into . Thereby we have a resolution of the identity of , that is, we have the following integral representation:
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 with being the number states of the harmonic oscillator, which also knwon as Fock states.
We can now construct for each spherical Landau level given in (3.2) a set of generalized coherent states (GCS) according to formula (4.2) as with the following precisions:(i), being identified with .(ii) is the eigenspace of in with dimension .(iii) (in view of (3.13)).(iv) are the eigenfunctions given by (3.11) in terms of the Jacobi polynomials.(v) being the norm square of , given in (3.12).(vi) the space of polynomials of degree less than , which carries a unitary irreducible representation of the compact Lie group (see ). The scalar product in is written as (vii) is an orthonormal basis of , whose elements are given explicitly by
Definition 4.2. For given positive integers and , and fixed , the wave function of the GCS in (4.6) are expressed as for every .
Remark 4.3. Note that for , the expression above reduces to 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 onto the state , is given by This is in fact the probability of finding photons in the coherent state . More explicitly, in view of (3.11), the quantity in (5.1) reads As annonced in the introduction, we denote the expression in (5.2) by for , with or equivalently . Motivated by this quantum probability, we then state the following.
Definition 5.1. For fixed integers , the discrete random variable having the probability distribution with , and denoted by , , will be called the generalized binomial probability distribution associated to the weighted Hilbert space .
Remark 5.2. Note that for , the above expression in (5.3) reduces to which is the standard binomial distribution with parameters and .
A convenient way to summarize all the properties of a probability distribution is to explicit its characteristic function: where is a real number, is the imaginary unit, and denotes the expected value.
Proposition 5.3. For fixed , the characteristic function of is given by for every .
Proof. Recall first that for every given fixed nonnegative integer , the characteristic function in (5.5) can be written as The last equality follows using (5.3). Next, by making the change in (5.7), it follows that Instead of the Jacobi polynomials, it is convenient to consider the closely related function introduced in [14, page 270]. They can be defined through the formula [14, equation (1), page 288], with (i.e., ) and . We can then express the square of as follows: Therefore, (5.8) reduces further to The transition above holds using the fact that [14, page 288] While the last equality can be checked easily using the addition formula [14, equation (3), page 326] Here the involved complex angles , , and are given through equations (8), (8β²), and (8β²β²) in [14, page 270]. In our case, they yield the followings: for , so that Next, using the fact that which is a special case of (5.9), with , , and , we obtain Finally, by substituting (5.16) and (5.18) in (5.12), taking into account that and , we see that the characteristic function reads simply as where .
Now, the characteristic function contains important information about the random variable . For example, various moments may be obtained by repeated differentiation of in (5.6) with respect to the variable and evaluation at the origin as
Corollary 5.5. Let . The mean value and the variance of are given respectively by
Proof. Let recall first that for every fixed integer , we have Thus direct computation gives rise to To conclude, we have to use successively the facts that for , we have and , together with Thus, we have We have also and therefore
6. Photon Counting Statistics
For an arbitrary quantum state one may ask to what extent is βnonclassicalβ 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 , 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, characterizes Poissonian statistics. If , we have sub-Poissonian statistics, otherwise, statistics are super-Poissonian. In our context and for , 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 , we make use of the obtained statistical parameters and to calculate Mandel parameter corresponding to the random variable . The discussion with respect to the sign of this parameter gives rise to the following statement.
Proposition 6.1. Let and be nonnegative integers and set
Then, and the photon counting statistics are
(i) sub-Poissonian for points such that and ,
(ii) Poissonian for points such that or ,
(iii) super-Poissonian for such that ,
Proof. Assume that . Making use of (5.22), we see that the Mandel parameter can be written as follows , where we have set with . Hence, it is clear that , viewed as second-degree polynomials in , admits exactly two real solutions given by Now, assertions (i), (ii), and (iii) follow by discussing the sign of the parameter (i.e., the sign of ) with respect to the modulus of , keeping in mind that .
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