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.
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 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 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 (OAM) is one of the most recently discovered properties of light, and it is only in the past decade its quantum properties have been the subject of experimental investigations and have found applications. Unlike polarization, which is only bidimensional, orbital angular momentum provides, with relative ease, unprecedented access to a theoretically unbounded discrete state space. The process of spontaneous parametric down-conversion has long been used as a source of two-photon states that can be entangled in several degrees of freedom, including OAM. In this thesis, the properties of the natural OAM spectrum associated with the entangled states produced by parametric down-conversion were investigated. Chapters 2 and 3 describe the production and detection of tunable high-dimensional OAM entanglement in a down-conversion system. By tuning the phase-matching conditions and improving the detection stage, a substantial increase in the half-width of the OAM correlation spectrum was observed. The conjugate variable of OAM, angular position, was also considered when examining high-dimensional states entangled in OAM. In order to efficiently determine their dimension, high-dimensional entangled states were probed by implementing a technique based on phase masks composed of multiple angular sectors, as opposed to narrow single-sector analysers. Presented in chapter 4, this technique allows the measurements of tight angular correlations while maintaining high optical throughput. The states so produced were then used for a number of applications centred around the concept of mutually unbiased bases. One can define sets of mutually unbiased bases for arbitrary subspaces of the OAM state space. Two bases are mutually unbiased if the measurement of a state in one basis provides no information about the state as described in the other basis. Complete measurements in mutually unbiased bases of high-dimensional OAM spaces are presented in chapter 5. Measurements in sets of mutually unbiased bases are integral to quantum science and can be used in a variety of protocols that fully exploit the large size of the OAM state space; we describe their use in efficient quantum state tomography, quantum key distribution and entanglement detection. Caution is however necessary when dealing with state spaces embedded in higher-dimensional spaces, such as that provided by OAM. Experimental tests of Bell-type inequalities allow us to rule out local hidden variable theories in the description of quantum correlations. Correlations inconsistent with the states observed, or even with quantum mechanics, known as super-quantum correlations, have however been recorded previously in experiments that fail to comply with the fair-sampling conditions. Chapter 6 describes an experiment that uses a particular choice of transverse spatial modes for which super-quantum correlations persist even if the detection is made perfectly efficient. The sets of modes carrying OAM allow a complete description of the transverse field. The ability to control and combine additional degrees of freedom provides the possibility for richer varieties of entanglement and can make quantum protocols more powerful and versatile. One such property of light, associated with transverse modes possessing radial nodes in the field distribution, can be accessed within the same type of experimental apparatus used for OAM. In chapter 7, the radial degree of freedom is explored, together with OAM, in the context of Hong-Ou-Mandel interference.
Entanglement of the properties of two separated particles constitutes a fundamental signature of quantum mechanics and is a key resource for quantum information science. We demonstrate strong Einstein, Podolsky, and Rosen correlations between the angular position and orbital angular momentum of two photons created by the nonlinear optical process of spontaneous parametric down-conversion. The discrete nature of orbital angular momentum and the continuous but periodic nature of angular position give rise to a special sort of entanglement between these two variables. The resulting correlations are found to be an order of magnitude stronger than those allowed by the uncertainty principle for independent (nonentangled) particles. Our results suggest that angular position and orbital angular momentum may find important applications in quantum information science.
Entanglement in higher dimensions is an attractive concept that is a chal- lenge to realise experimentally. To this end, the entanglement of the orbital angular momentum (OAM) of photons holds promise. The OAM state-space is discrete and theoretically unbounded. In the work that follows, we investi- gate various aspects of OAM entanglement. We show how the correlations in OAM and its conjugate variable, angular position, are determined by phase- matching and the shape of the pump beam in spontaneous parametric down- conversion. We implement tests of quantum mechanics which have been previously done for other variables. We show the Einstein-Podolsky-Rosen paradox for OAM and angle, supporting the incompatibility of quantum me- chanics with locality and realism. We demonstrate violations of Bell-type inequalities, thereby discounting local hidden variables for describing the correlations we observe. We show the Hardy paradox using OAM, again highlighting the nonlocal nature of quantum mechanics. We demonstrate violations of Leggett-type inequalities, thereby discounting nonlocal hidden variables for describing correlations. Lastly, we have looked into the entan- glement of topological vortex structures formed from a special superposition of OAM modes and show violations of Bell-type inequalities confined to a finite, isolated volume.
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.