Extremal states for photon number and quadratures as gauges for nonclassicality

Rotated quadratures carry the phase-dependent information of the electromagnetic field, so they are somehow conjugate to the photon number. We analyze this noncanonical pair, finding an exact uncertatinty relation, as well as a couple of weaker inequalities obtained by relaxing some restrictions of the problem. We also find the intelligent states saturating that relation and complete their characterization by considering extra constraints on the second-order moments of the variables involved. Using these moments, we construct performance measures tailored to diagnose photon-added and Schr\"odinger catlike states, among others.Comment: 6 pages, 4 color figures. Comments welcome

Unpolarized states and hidden polarization

We capitalize on a multipolar expansion of the polarisation density matrix, in which multipoles appear as successive moments of the Stokes variables. When all the multipoles up to a given order $K$ vanish, we can properly say that the state is $K$th-order unpolarized, as it lacks of polarization information to that order. First-order unpolarized states coincide with the corresponding classical ones, whereas unpolarized to any order tally with the quantum notion of fully invariant states. In between these two extreme cases, there is a rich variety of situations that are explored here. The existence of \textit{hidden} polarisation emerges in a natural way in this context.Comment: 7 pages, 3 eps-color figures. Submitted to PRA. Comments welcome

Orbital angular momentum from marginals of quadrature distributions

We set forth a method to analyze the orbital angular momentum of a light field. Instead of using the canonical formalism for the conjugate pair angle-angular momentum, we model this latter variable by the superposition of two independent harmonic oscillators along two orthogonal axes. By describing each oscillator by a standard Wigner function, we derive, via a consistent change of variables, a comprehensive picture of the orbital angular momentum. We compare with previous approaches and show how this method works in some relevant examples.Comment: 7 pages, 2 color figure

On the Relationship between the One-Corner Problem and the M−M-Corner Problem for the Vortex Filament Equation

In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular M-corner polygon as initial datum can be explained at infinitesimal times as the superposition of M one-corner initial data. This fact is mainly sustained with the calculation of the speed of the center of mass; in particular, we show that several conjectures made at the numerical level are in agreement with the theoretical expectations. Moreover, due to the spatial periodicity, the evolution of VFE at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner; and this interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the M-corner case; and the numerical experiments carried out provide new arguments that support the multifractal character of the trajectory defined by one of the corners of the initial polygon

The Vortex Filament Equation as a Pseudorandom Generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE), $$X_t = X_s \wedge X_{ss},$$ taking a planar regular polygon of M sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based

Vortex filament equation for a regular polygon

In this paper, we study the evolution of the vortex filament equation,$$X_t = X_s \wedge X_{ss},$$with $X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gau$\beta$ sum. We also study the fractal behaviour of $X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal

Lost and found: the radial quantum number of Laguerre-Gauss modes

We introduce an operator linked with the radial index in the Laguerre-Gauss modes of a two-dimensional harmonic oscillator in cylindrical coordinates. We discuss ladder operators for this variable, and confirm that they obey the commutation relations of the su(1,1) algebra. Using this fact, we examine how basic quantum optical concepts can be recast in terms of radial modes.Comment: Some minor typos fixed

Vortex Filament Equation for a regular polygon in the hyperbolic plane

The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case

On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion

In this paper, we consider the evolution of the Vortex Filament equa- tion (VFE): Xt = Xs ∧ Xss, taking M-sided regular polygons with nonzero torsion as initial data. Us- ing algebraic techniques, backed by numerical simulations, we show that the solutions are polygons at rational times, as in the zero-torsion case. However, unlike in that case, the evolution is not periodic in time; more- over, the multifractal trajectory of the point X(0,t) is not planar, and appears to be a helix for large times. These new solutions of VFE can be used to illustrate numerically that the smooth solutions of VFE given by helices and straight lines share the same instability as the one already established for circles. This is accomplished by showing the existence of variants of the so-called Rie- mann’s non-differentiable function that are as close to smooth curves as desired, when measured in the right topology. This topology is motivated by some recent results on the well-posedness of VFE, which prove that the selfsimilar solutions of VFE have finite renormalized energy

Unraveling beam self-healing

We show that, contrary to popular belief, non only diffraction-free beams may reconstruct themselves after hitting an opaque obstacle but also, for example, Gaussian beams. We unravel the mathematics and the physics underlying the self-reconstruction mechanism and we provide for a novel definition for the minimum reconstruction distance beyond geometric optics, which is in principle applicable to any optical beam that admits an angular spectrum representation. Moreover, we propose to quantify the self-reconstruction ability of a beam via a newly established degree of self-healing. This is defined via a comparison between the amplitudes, as opposite to intensities, of the original beam and the obstructed one. Such comparison is experimentally accomplished by tailoring an innovative experimental technique based upon Shack-Hartmann wave front reconstruction. We believe that these results can open new avenues in this field