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Journal of the Optical Society of America A

Discrete normalized Bargmann transform through the gyrator transform

Kenan Uriostegui

Published Apr 21, 2020• 0 citations
Publisher

Abstract

Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.

DOI
10.1364/josaa.387945

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Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.
We define a discrete Bargmann transform for discrete and finite functions by means of the coherent states in the su(2) finite harmonic oscillator model. The transform space is over a corresponding finite square mesh in the complex plane. From there, the inverse discrete Bargmann transform reconstitutes the original function with an average error of 10−7 for functions of unit norm. Three defining properties of the Bargmann transform also hold in this discrete version: (1) the Bargmann transform of finite su(2) harmonic oscillator functions are complex power functions; (2) the discrete su(1,1) repulsive oscillator functions self-reproduce in the chosen finite interval with little error; and (3) the oscillator dynamics under the fractional Fourier transform in Bargmann’s complex space correspond, in this discrete transform, to the dynamics under the su(2) fractional Fourier–Kravchuk transform.
Based on the coherence property under the Fourier transform, we introduce two types of coherent states into the finite discrete oscillator model of s u (2) algebra, and show that they are stable under the fractional Fourier–Kravchuk transform. Using the set of discrete coherent states of this model as biorthonormal bases, we propose two discrete transforms. The first transform maps f ∈ C N functions to the unit circle, and the second transform maps the same functions to Bargmann space; both transforms have analytic inverses. These transformations establish a connection between Fourier and Bargmann expansions, respectively, with the basis of Kravchuk symmetric functions. They also provide a framework for the study of finite-dimensional systems in a periodic or complex continuous space. We give some examples of applications to qudit systems.
Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x̂ and p̂ to a pair η̂ and ζ̂ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η̂, ζ̂ can be realized in the Schrödinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square-integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well-known Bargmann transform is the special case when η̂ and iζ̂ are the harmonic oscillator raising and lowering operators. The Moshinsky-Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.
This paper concerns a relationship between the kernel of the Bargmann transform and the corresponding canonical transformation. We study this fact for a Bargmann transform introduced by Thomas and Wassell [J. Math. Phys. 36, 5480–5505 (1995)]—when the configuration space is the two-sphere S2 and for a Bargmann transform that we introduce for the three-sphere S3. It is shown that the kernel of the Bargmann transform is a power series in a function which is a generating function of the corresponding canonical transformation (a classical analog of the Bargmann transform). We show in each case that our canonical transformation is a composition of two other canonical transformations involving the complex null quadric in C3 or C4. We also describe quantizations of those two other canonical transformations by dealing with spaces of holomorphic functions on the aforementioned null quadrics. Some of these quantizations have been studied by Bargmann and Todorov [J. Math. Phys. 18, 1141–1148 (1977)] and the other quantizations are related to the work of Guillemin [Integ. Eq. Operator Theory 7, 145–205 (1984)]. Since suitable infinite linear combinations of powers of the generating functions are coherent states for L2(S2) or L2(S3), we show finally that the studied Bargmann transforms are actually coherent states transforms.
We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.
The one-dimensional finite oscillator model satisfies that under commutator brackets, all operators close into the (dynamical) Lie algebra su(2); therefore the eigenvalues of position, momentum and energy operators are discrete and the eigenvalue of a compact u(l) central operator fixes the dimension of the su(2) representation. In this work we review the one-dimensional finite oscillator model with u(2) as the dynamical algebra where the wave functions, which involve Kravchuk polynomials, are discrete functions of angular momentum theory.