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.
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.
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.
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.
Orbital angular momentum entanglement (OAM) is one of the very intriguing topics in quantum physics. In addition to discovering and exploring its underlying mechanics, recent studies have also demonstrated a progress towards expanding degree of its entanglement. In this paper, we explore OAM entanglement by applying the Heisenberg uncertainty principle to the quantum position correlation within the azimuthal region. In particular, we decompose the pump light into a set of pump cone states characterized by their radii. The OAM entanglement can then be manipulated by controlling the radius of the pump cone state, the length of the nonlinear crystal and also the OAM carried by the pump field. That is followed by a detailed discussion and analysis. Such an exploration not only bring us a deeper understanding of OAM entanglement, but also help us to implement the high-dimensional quantum information tasks based on OAM entanglement.
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.
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(iâθ) term, where â can take any integer value and θ 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.
Introduction Although the orbital angular momentum (OAM) of light is often quantified in terms of ℓℏ per photon, one should recognise that this result can be derived directly from Maxwell's equations and is most certainly not an exclusively quantum property. Following the recognition of optical OAM in 1992 [1], the immediate thrust of experimental work was to observe OAM transfer to microscopic objects. These exciting experiments by Rubinsztein- Dunlop [2, 3] and others established that the predicted angular momentum was truly mechanical, but did not provide insight on OAM at the level of single photons. Laguerre- Gaussian modes were identified as being a natural basis set for describing beams with OAM, where their phase terms describe the helical phasefronts corresponding to an OAM of ℓℏ per photon. The polarisation of light has played an important historical role in the experimental investigation of quantum entanglement, quantum gate operations and quantum cryptography. A potential advantage of OAM is that it resides in a higher-dimensional state space which, as follows from the theory, is infinite-dimensional. One of the experimental challenges is to access a large number of these dimensions. In quantum communication, the most obvious advantage of high n -dimensional systems is the increased bandwidth [4], but qunits also lead to stronger security [5, 6]. An increase in the number of entangled quantum modes has also been shown to improve measures of nonlocality [7], to allow quantum computation with better fault tolerance [8] and to simplify quantum logic structures [9].