Luis Edgar Vicent

Finite two-dimensional oscillator: II. The radial model

A finite two-dimensional radial oscillator of (N + 1)2 points is proposed, with the dynamical Lie algebra so(4) = su(2)x⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain so(4)⊃so(3)⊃so(2). As before, there are a finite number of energies and angular momenta; the Casimir spectrum of the new chain provides the integer radii 0≤ρ≤N, and the 2ρ + 1 discrete angles on each circle ρ are obtained from the finite Fourier transform of angular momenta. The wavefunctions of the finite radial oscillator are so(3) Clebsch-Gordan coefficients. We define here the Hankel-Hahn transforms (with dual Hahn polynomials) as finite-N unitary approximations to Hankel integral transforms (with Bessel functions), obtained in the contraction limit N→∞.

Design of bright, fiber-coupled and fully factorable photon pair sources

From quantum computation to quantum key distribution, many quantum-enhanced applications rely on the ability to generate pure single photons. Even though the process of spontaneous parametric downconversion (SPDC) is widely used as the basis for photon-pair sources, the conditions for pure heralded single-photon generation, taking into account both spectral and spatial degrees of freedom, have not been fully described. We present an analysis of the spatio-temporal correlations present in photon pairs produced by type-I, non-collinear SPDC. We derive a set of conditions for full factorability in all degrees of freedom—required for the heralding of pure single photons—between the signal and idler modes. In this paper, we consider several possible approaches for the design of bright, fiber-coupled and factorable photon-pair sources. We show through numerical simulations of the exact equations that sources based on: (i) the suppression of spatio-temporal entanglement according to our derived conditions and (ii) a tightly focused pump beam together with optimized fiber-collection modes and spectral filtering of the signal and idler photon pairs, lead to a source brightness of the same order of magnitude. Likewise, we find that both of these sources lead to a drastically higher factorable photon-pair flux, compared to an unengineered source.

Generalized radiometry as a tool for the propagation of partially coherent fields

We use a recently introduced family of generalized radiances that are exactly conserved along rays for the numerical propagation of physically relevant properties of partially coherent fields. This exact generalized radiometric description of the wave fields makes the computations more efficient than those involving diffraction integrals. We concentrate on fields emitted by planar sources, and consider propagation both in two- and three-dimensional space.

Engineering an ideal indistinguishable photon-pair source for optical quantum information processing

The eventual success of optical quantum information (QI) processing depends critically on the available technologies. Photons produced via the nonlinear process of spontaneous parametric downconversion (SPDC) have now been used in a vast array of experiments in optical quantum computing and long-distance quantum communication. However, almost no source to date has been fully optimized. Here, we describe our progress in engineering an ideal photon-pair source for optical quantum information processing – polarization-entangled photons made indistinguishable by engineering spatio-spectral unentanglement at the source. We present solutions for several key challenges encountered in the development of the indistinguishable source. We expect such a source to directly translate into significantly enhanced QI protocols with higher rates, fidelities and efficiencies.

Unitary rotation of square-pixellated images

The unitary rotation of square-pixellated images is based on the finite su(2)-oscillator model, which describes systems whose values for position, momentum and energy, are discrete and finite. In a two-dimensional position space, this allows the construction of angular momentum states, orthonormal and complete, for which rotations are defined as multiplication by phases that carry the rotation angle. The decomposition of a digital square images in terms of these angular momentum states determines a unitary (hence invertible) rotation of the image, whose kernel can be computed as a four-dimensional array of real numbers.

The Fourier U(2) Group and Separation of Discrete Variables

The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4;<), whose maximal compact subgroup is the Fourier group U(2) F ; this includes isotropic and anisotropic Fourier transforms, screen rotations and gy- rations in the phase space of ray positions and optical momenta. Deforming classical optics into a Hamiltonian system whose positions and momenta range over a finite set of values, leads us to the finite oscillator model, which is ruled by the Lie algebra so(4). Two dis- tinct subalgebra chains are used to model arrays of N 2 points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N 2 -vectors in this space are digital (pixellated) images on either of these two grids, related by a unitary transformation. Here we examine the unitary action of the analogue Fourier group on such images, whose rotations are particularly visible.

Quantum phase space for the one-dimensional hydrogen atom on the hyperbola

We obtain the solution of the one-dimensional hydrogen atom on the hyperbola by transforming its Schrodinger equation into the modified Poschl-Teller equation. We write explicitly the wave functions both in configuration and momentum space, and check the contraction limit of the system to the flat space model. Finally, we find the closed form of the Wigner function for the states of this system.

COHERENT STATES FOR THE FINITE su(2)-OSCILLATOR MODEL

A finite oscillator is a system that has a discrete and finite number of spatial positions and energy levels accessible. The su(2)-oscillator model takes the generators of the angular momentum su(2) algebra and, giving to each one a definite physical interpretation, provides a consistent mathematical description of a finite oscillator. This work searches an example of identification of generalized coherent states, within the collection of states and wavefunctions of the finite su(2) oscillator model.

Covariant discretization of axis-symmetric linear optical systems

We propose a discretization strategy for systems with axial symmetry. This strategy replaces the continuous position coordinates by a discrete set of sensor points, on which the discrete wave fields transform covariantly with the group of 2 x 2 symplectic matrices. We examine polar arrays of sensors (i.e., numbered by radius and angle) and find the complete, orthonormal sets of discrete-waveguide Meixner functions; when the sensors come closer together, these tend to the Laguerre eigenmodes of the continuous waveguide. In particular, the fractional Hankel transforms are discretized in order to define the fractional Hankel-Meixner transforms and similarly for all axis-symmetric linear optical maps. Coherent states appear in the discrete cylindrical waveguide. Covariant discretization leads to the same Wigner phase-space function for both the discrete and the continuum cases. This reinforces a Lie-theoretical model for the phase space of discrete systems.

Design of bright, fiber-coupled and fully factorable photon pair sources for quantum information processing

We derive conditions for full spatio-temporal factorability in non-collinear type-I spontaneous parametric downconversion. We show that fulfilment of these conditions leads to a higher single-mode fiber-coupled brightness, compared to sources rendered factorable through spectral filtering.