Coherent State Quantization and Moment Problem

Berezin-Klauder-Toeplitz (“anti-Wick”) or “coherent state” quantization of the complex plane, viewed as the phase space of a particle moving on the line, is derived from the resolution of the unity provided by the standard (or gaussian) coherent states. The construction of these states and their attractive properties are essentially based on the energy spectrum of the harmonic oscillator, that is on natural numbers. We follow in this work the same path by considering sequences of non-negative numbers and their associated “non-linear” coherent states. We illustrate our approach with the 2-d motion of a charged particle in a uniform magnetic field. By solving the involved Stieltjes moment problem we construct a family of coherent states for this model. We then proceed with the corresponding coherent state quantization and we show that this procedure takes into account the circle topology of the classical motion.


Introduction
One of the most interesting properties of standard or Glauber coherent states |z [1,2,3,4] is the Bayesian duality [5,6] that they encode between the discrete Poisson probability distribution, n → e −|z| 2 |z| 2 /n!, of obtaining n quantum excitations ("photons" or "quanta") in a measurement through some counting device, and the continuous Gamma probability distribution measure |z| 2 → e −|z| 2 |z| 2 /n! on the classical phase space.For this latter distribution, |z| 2 is itself a random variable, denoting the average number of photons, given that n photons have been counted.Such a duality underlies the construction of all types of coherent state families, provided they satisfy a resolution of the unity condition.It turns out that this condition is equivalent to setting up a "positive operator valued measure" (POVM) [7,4] on the phase space.Such a measure, in turn, leads to the quantization of the classical phase space, which associates to each point z ≡ (q + ip)/ √ 2 the one dimensional projection operator P z , projecting onto to the subspace generated by the coherent state vector, and then for z = z , P z P z = P z P z ).This "Berezin-Klauder-Töplitz" quantization (or "anti-Wick") [1,8,9] turns out, in this case, to be equivalent to the canonical quantization procedure.Clearly, this non-commutative version of the complex plane is intrinsically based on the nonnegative integers (appearing in the n! term).We then follow a similar path by considering sequences of nonnegative numbers which are far or not from the natural numbers [10].The resulting quantizations will then be looked upon as generalizations of the one yielded by the standard coherent states.
We illustrate our approach with the elementary model of the 2-d motion of a charged particle in a uniform magnetic field [11,12].By using a solution to a version of the Stieltjes moment problem [13,14] we construct a family of coherent states for this model.We prove that these states form an overcomplete set that is normalized and resolves the unity.We then carry out the corresponding coherent state quantization and we examine the consequences in terms of its probabilistic, functional, and localization aspects.
This article is organized as follows.In Section 2, we briefly review the standard coherent states and the way they allow painless quantization of the complex plane viewed as a phase space.The so-called nonlinear coherent states built from arbitrary sequences of numbers are described in Section 3 and we show how the moment problem immediately emerges from the exigence of unity resolution.If the positive case, the corresponding quantization of the complex plane is described in Section 4. In Section 5 we apply our formalism to the motion of a charged particle in a uniform magnetic field.There exist two families of coherent states for such a model, namely the Malkin-Man'ko states [15], which are just tensor products of standard coherent states, and the Kowalski-Rembielinski states [16].By introducing a kind of squeezing parameter q = e λ > 1 we extend the definition of the latter and solve the corresponding Stieltjes moment problem.This allows us to proceed with the quantization of the physical quantities and illustrate our study with numerical investigation.
where P z = |z z| is the orthogonal projector on vector |z and the integral should be understood in the weak sense.The proof is straightforward and stems from the orthogonality of the Fourier exponentials and from the integral expression of the gamma function which solves the moment problem for the factorial n!
|e n e n | = I .
Berezin-Klauder-Toeplitz-"anti-Wick" quantization or "coherent state quantization" Property (v) allows to define 1. a normalized positive operator-valued measure (POVM) on the complex plane equipped with its Lebesgue measure d 2 z π and its σ−algebra F of Borel sets: where L(H) + is the cone of positive bounded operators on H.
2. a quantization of the complex plane, which means that to a function f (z, z) in the complex plane there corresponds the operator A f in H defined by z n zn provided that weak convergence holds.
For the simplest functions f (z) = z and f (z) = z we obtain â|e 0 = 0 , (lowering operator) These two basic operators obey the canonical commutation rule : [â, â † ] = I.The number operator N = â † â is such that its spectrum is exactly N with eigenvectors e n : N |e n = n|e n .The fact that the complex plane has become non-commutative is apparent from the quantization of the real and imaginary

Coherent states for generic sequences
Let X = {x n } n∈N be a strictly increasing sequence such that x 0 = 0 and lim n→∞ x n = ∞.Then its associated exponential has an infinite convergence radius.Associated "coherent states" ("non-linear CS" in Quantum Optics) read as elements of H, a separable Hilbert space with orthonormal basis {|e n , n ∈ N}: These vectors still enjoy some properties similar to the standard ones.
Poisson-like distribution with average number of occurrences equal to |z| 2 (discrete probabilistic content).Consider the discrete probability distribution with parameter t ≥ 0: The average of the random variable n → x n is x n = t.Contrariwise to the standard case X = N, the continuous (gammalike x n ! with parameter n is not a probability distribution with respect to the Lebesgue measure dt: Finding the right measure amounts to solve a usually intractable moment problem.So, the map x n ! is not a (Gamma-like) probability distribution (with respect to the square of the radial variable in the complex plane) with x n+1 as a shape parameter, and this is a serious setback for the Berezin-Toeplitz quantization program.Indeed there is no reason to get now the resolution of the unity: Here F is a diagonal operator determined by the sequence of integrals . These integrals form a sequence of Stieltjes moments for the measure dt E(t) .
If the moment problem has a solution.Suppose that the Stieltjes moment problem [13,14] has a solution for the sequence (x n !) n∈N , i.e. there exists a probability distribution t → w(t) on [0, +∞) with infinite support such that We know that a necessary and sufficient condition for this is that the two matrices have strictly positive determinants for all n.Then, a natural approach is just to modify the measure on C by including the weight w(|z| 2 ) E(|z| 2 ).We then obtain the resolution of the identity: If the moment problem is solved by a measure t → w(t), then the vectors |v z enjoy all needed properties for quantization: x n ! is a (Gamma-like) probability distribution (with respect to the square of the radial variable) with x n+1 as a shape parameter and with respect to the modified measure on the complex plane Note that we might face with (x n !) an indeterminate moment sequence, which means that there are several representing measures.Then to each such a measure there corresponds a probability distribution on the classical phase space to be interpreted in terms of statistical mechanics.If the moment problem has no (explicit) solution.See an alternative in [10].

CS quantization with sequence X
If the moment problem has an explicit solution, one can proceed with the corresponding CS quantization of the complex plane since the family of vectors |v z solves the unity : with We proceed with this quantization like in the standard case: to a function f (z, z) in the complex plane there corresponds the operator A f in H defined by provided that weak convergence holds.For the simplest functions f (z, z) = z and f (z, z) = z we obtain â |e 0 = 0 , (lowering operator) together with new quantum localization properties.
5 An example: charged particle in a magnetic field Consider a classical nonrelativistic particle, charge −e, moving in the plane x 1 , x 2 and interacting with a constant and uniform magnetic field of intensityB perpendicular to the plane, described by a vector potential A only (A 0 = 0).The Hamiltonian of the particle is where ε ij is the Levi-Civita symbol.

Kowalski & Rembielinski coherent states
Kowalski and Rembielinski [16] have proposed the construction of CS for a particle in a uniform magnetic field by using their coherent states for the circle [17].The latter are constructed from the angular momentum operator Ĵ and the unitary operator Û that represents the position of the particle on the unit circle.These operators obey the commutation relations The introduction of these coherent states permits to avoid the problem of the infinite degeneracy present in the approach followed by Man'ko and Malkin [15], and, in addition, takes into in account the momentum part of the phase space.Consequently, the so obtained CS offer a better way to compare the quantum behavior of the system with the classical trajectories in the phase space.
Let us introduce the centre-coordinate operators where ω = eB/μ is the cyclotron frequency.The xi Introduce now the operators They obey the commutation rules [r 0± , r± ] = 0 .
The "relative" angular momentum operator Ĵ is proportional to the Hamiltonian Due to the rules, Ĵ can be viewed as the generator of rotations about the axis passing through the classical point (x The K&R CS |z 0 , ζ are constructed in the Hilbert space spanned by the orthonormal basis as solution to the eigenvalue equation: where we recognize a generalized exponential with x n ≡ e 2n n.

Squeezing/deforming the K& R states
The introduction of a "squeezing" parameter λ allows us to generalize the previous CS of a charged particle in a uniform magnetic field as an eigenvector of the commuting operators r0+ and Ẑλ , Operator Ẑλ coincides with the K& R Ẑ for λ = 2, and with just r− for λ = 0, i.e., the case of Malkin-Man'ko CS, which are actually tensor products of standard CS.Operator Ẑλ controls the dispersion relations of the angular momentum Ĵ and of the "position operator" r− .The corresponding CS read: t n /x n ! and x n ≡ e nλ n.The complex numbers z 0 and ζ parameterize, respectively, the position of the centre of the circle and the classical phase space state of the circular motion.Some properties of these CS make them more suitable with regard to the semi-classical behavior of a charged particle in a magnetic field, in comparison with the Malkin-Man'ko CS.The generalization involving λ allows one to exhibit better these interesting characteristics.

Resolution of the moment problem
The λ-CS |z 0 , ζ are the tensor product of the states |z 0 and |ζ , where the first one is a standard CS.So, in order to perform the CS quantization, we concentrate only on the states |ζ .For convenience, we put μω/2h = 1, and so ζ = ζ.Then, in the Fock basis {|n }, They resolve the unity in the Fock space spanned by the kets |n ,  Kowalski and Rembielinski observed that the approximate equality Ĵ ζ l does not hold for arbitrary small l, being really acceptable for |l| > 1 only.
) (raising operator) .These two basic operators obey the commutation rule :[â, â † ] = x N +1 − x N def = Δ N .The operator x N is defined by x N = â † â and is such that its spectrum is exactly the sequence X with eigenvectors e n : x N |e n = x n |e n .The triple {â, â † , Δ N } equipped with the operator commutator [• , •] generates (generically) an infinite Lie algebra which replaces the Weyl-Heisenberg Lie algebra.The quantization of the real and imaginary parts of z = 1 √ 2 (q + ip) yields position and momentum operators corresponding to the sequence X ,