Physics Open

Lower bound of constant product 0.2ℏ between the pair of periodically angular uncertainties in a set of numerous singular light beams and its utility

Hsiao-Chih Huang

Published Aug 11, 2022• 0 citations

Abstract

The angular lower bound does not correspond to the Robertson lower bound in the range of coordinate variable and the correlation form between the pair of uncertainties. In addition, the pair of angular uncertainties in a singular light beam is unevaluated. Moreover, these two uncertainties are independent of n-fold symmetry of azimuthal intensity that are with the two invariable types in a light beam. Herein, we report the smaller difference between mean OAM and the product of azimuthal phase-gradient of beam cross-section and ℏ, the larger azimuthal coordinate range of one periodic helical wavefront in a set of numerous singular light beams, each of which utilises the superposition comprising two fractional OAM light beams that have a difference of δ in the azimuthal phase-gradient. This is attributed to the angular uncertain principle as an uncertainty relation between the pair of periodically angular uncertainties that is defined with the sub-n-fold periodicity of a light beam by a constant bound between 0.184ℏ and 0.204ℏ. This periodically angular lower bound corresponds to the Robertson lower bound for the infinity range of the coordinate variable and the constant correlation between the pair of uncertainties; however, it is stronger by 2.45 times of magnitude at least. Moreover, we quantitatively estimated the unevaluated OAM uncertainty in a singular light beam by using the periodically angular relation. We demonstrate that the OAM uncertainty is a monotonic function of the azimuthal phase-gradient difference δ, 0 ≤ δ ≤ n, in this singular light with identically mean OAMs of 0 and nℏ/2.

DOI: 10.1016/j.physo.2022.100115

Related papers

Uncertainty principle for orbital angular momentum and angular position with infinity

Hsiao-Chih Huang

The knowingly angular uncertainty principle (UP) holds the orbital angular momentum (OAM) is precisely defined in an optical vortex with angular position (AP) of 2{\pi} azimuthal coordinate ({\phi}) range. However, the pair of their uncertainties is discretely selected, not correspondent to the unselected linear momentum and position by h/4{\pi}. Herein, we demonstrate the less difference between mean OAM and the product of azimuthal phase-gradient and h/2{\pi}, the larger {\phi} range of one periodically helical wavefront in a set of numerous singular lights for both their infinitesimal and infinity. This is another angular UP for any pair of unselected OAM and AP by a stronger constant product 0.186h/2{\pi} on their underlying uncertainties. Moreover, there is no difference of known-AP uncertainties between these lights for the infinitely proposing AP uncertainty of their set. This UP would enhance existing OAM researches by its infinitely and certainly proposing- and known-AP uncertainties and stronger constant correlation.

May 19, 20200 citations

Uncertainty principle for orbital angular momentum and infinite angular position

Hsiao-Chih Huang

Heisenberg uncertainty principle (UP) holds that the nearly unlimited precise observation of linear momentum is for any states, and in complementarity, states with nearly infinite position space and vice versa for a particle by an lower bound h/4{\pi} on the product of two their underlying uncertainties [1]. In a cylindrical system, however, only the states with integral charges mh/2{\pi} allow the unlimitedly precise observation of orbital angular momentum (OAM), and in complementarity, angular position (AP) states with 2{\pi} range for round and doughnut light beams [2]. Moreover, the correlation of OAM and AP of entangled photon pairs is one order of magnitude stronger that those with independent particles [3]. Here we propose a UP of demonstrating arbitrary states of unlimitedly precise mean OAM refer to the azimuthal phase-gradient (PG), and in complementarity, states with infinite range AP, observed in a set of numerous singular light beams with simultaneously helical wavefront and phase-shift-front associated with three natures of wave, discrete OAM eigenmode and linear optics. These arbitrary OAM states of arbitrary phase-jump are extended by discrete precisions that consist of not only the OAM eigenstates but also the states with the max phase-jump. Furthermore, we theoretically demonstrate the double resolution limit of OAM eigenmodes and the super-2{\pi} angle period for two overlap probabilities of n-sectional fractional OAM state with one and two PGs, respectively, which is used to illustrate the quantum nonlocality in the spontaneously parametric down-conversion (SPDC) process.

arXiv:2005.09259May 19, 20200 citations

Transfer of Information Using Helically Phased Modes

Miles J. Padgett, Graham M. Gibson, Johannes Courtial

In 1992, Les Allen and coworkers within Han Woerdman's group realized that light beams with an azimuthal phase dependence described by an additional exp(iâÎ¸) term, where â can take any integer value and Î¸ is the angular position within the beam, carry an orbital angular momentum. This orbital angular momentum is additional to the spin angular momentum associated with circular polarization. These azimuthally phased beams comprise â intertwined helical phase fronts, skewed with respect to the beam axis, meaning that at all positions within the beam cross section, the wavefront normal and associated momentum density has a well-defined azimuthal component. The insight of Allen and coworkers was that for all beams described in this way, when integrated over the beam cross section, the orbital angular momentum (OAM) is ââ per photon. Laguerre-Gaussian and high-order Bessel beams are both examples of mode families having these properties. A further feature of these beams is that the phase singularity on the beam axis precludes a nonzero on-axis intensity and therefore all such beams with â â 0 have an annular intensity cross section (see Fig. 4.1). The circulation of the optical momentum and energy around the singularity leads them also to be called optical vortices. The further recognition that beams carrying defined amounts of OAM could be created within the laboratory means that over the last 15 years they have formed the basis of many investigations, ranging from the transfer of OAM to microscopic particles (in optical spanners or optical wrenches) to studies of the fundamental properties of light in both the classical and quantum regimes. However, although OAM is an immensely useful concept, one must emphasize that all its properties are simply those predicted by standard electromagnetic theory. The purpose of this chapter is to consider how sets of helical laser modes carrying OAM can be used for information transfer.

SPIE eBooksMar 16, 20100 citations

Correlation coefficients and the Robertson-Schroedinger uncertainty relations

Gordon N. Fleming

Calling the quantity; 2delta(A)delta(B)/||, with non-zero denominator, the uncertainty product ratio or UPR for the pair of observables, (A, B), it is shown that any non-zero correlation coefficient between two observables raises, above unity, the lower bound of the UPR for each member of an infinite collection of pairs of incompatible observables. Conversely, any UPR is subject to lower bounds above unity determined by each of an infinite collection of correlation coefficients. This result generalizes the well known Schroedinger strengthening of the Robertson uncertainty relations (with the former expressed in terms of the correlation coefficient rather than the anticommutator) where the UPR and the correlation coefficient both involve the same pair of observables. Two, independent, derivations of the result are presented to clarify its origins and some examples of its use are examined.

arXiv:quant-ph/0703226Mar 23, 20070 citations

On some states minimizing uncertainty relations: A new look at these relations

K. Urbanowski

The Heisenberg–Robertson (HR) and Robertson–Schrödinger (RS) uncertainty relations are studied. Analyzing the RS uncertainty relation we found, that there can exist a large set of states of the quantum system under considerations, for which the lower bound of the product of the standard deviations of a pair of noncommuting observables, A and B, is zero, and which differ from those described in the literature for the HR uncertainty relation. These states are not eigenstates of either the observable A or B. The correlation function for these observables in such states is equal to zero. We have also shown that the so-called “sum uncertainty relations” also do not provide any information about lower bounds on the standard deviations calculated for these states. We additionally show that the uncertainty principle in its most general form has two faces: One is that it is a lower bound on the product of standard deviations, and the other is that the product of standard deviations is an upper bound on the modulus of the correlation function of a pair of the noncommuting observables in the state under consideration.

arXiv:2411.08131Modern Physics Letters AMar 25, 20260 citations

Uncertainty principle for angular position and angular momentum

Sonja Franke‐Arnold, Stephen Mark Barnett, Eric Yao +3 more

The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.

New Journal of PhysicsAug 9, 2004265 citations