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arXiv:1806.05479 · New Journal of Physics

Unified treatment of the total angular momentum of single photons via generalized quantum observables

Mário Motta, Giacomo Guarnieri, L. Lanz, Beatrix C. Hiesmayr

Published Dec 12, 2018• 2 citations
arXivHTMLPDF

Abstract

In this paper, we provide a consistent framework to address the notorious difficult decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component.We discuss the canonical decomposition into SAM and OAM components, which are the generators of internal and spatial rotations in the space of physical states.We find that those operators are mutually compatible but unsharp quantum observables, therefore Positive Operator-Valued Measures describe their joint measurements.We present another decomposition of the TAM, which we denote as a non-canonical one.The operators resulting from this decomposition are mutually incompatible but sharp quantum observables, thus Projector-Valued Measurements.This fact reflects their consistency with the transversality condition of single-photon wavefunctions, thus explains the underlying physics from a quantum information theoretic view.Furthermore, we discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.The difference between the canonical and non-canonical momenta leads to observable differences in higher-order statistical moments.

DOI
10.1088/1367-2630/aaf841

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IEEE photonics journal•Jan 26, 2016•30 citations
In this paper, we provide a consistent framework to address the notorious difficult decomposition of the single-photon total angular momentum (TAM) into a spin (SAM) and an orbital (OAM) component.We discuss the canonical decomposition into SAM and OAM components, which are the generators of internal and spatial rotations in the space of physical states.We find that those operators are mutually compatible but unsharp quantum observables, therefore Positive Operator-Valued Measures describe their joint measurements.We present another decomposition of the TAM, which we denote as a non-canonical one.The operators resulting from this decomposition are mutually incompatible but sharp quantum observables, thus Projector-Valued Measurements.This fact reflects their consistency with the transversality condition of single-photon wavefunctions, thus explains the underlying physics from a quantum information theoretic view.Furthermore, we discuss the implementations on joint measurements for both decompositions and provide an explicit calculation of all these quantities for circularly polarized Gaussian single-photon states.The difference between the canonical and non-canonical momenta leads to observable differences in higher-order statistical moments.
We show that when an electron or photon propagates in a cylindrically symmetric waveguide, its spin angular momentum (SAM) and its orbital angular momentum (OAM) interact. Remarkably, we find that the dynamics resulting from this spin-orbit interaction are quantitatively described by a single expression applying to both electrons and photons. This leads to the prediction of several rotational effects: the spatial or time evolution of either particle's spin-polarization vector is controlled by its OAM quantum number or, conversely, its spatial wave function is controlled by its SAM. We show that the common origin of these effects in electrons and photons is a universal geometric phase. We demonstrate how these phenomena can be used to reversibly transfer entanglement between the SAM and OAM degrees of freedom of two-particle states.
Whether the total angular momentum of the photon can be separated into spin and orbital parts has been a long-standing problem due to the constraint of transversality condition on its vector wavefunction. A careful analysis shows that the situation arises from a misuse of the constraint in quantum mechanics. To use the constraint properly, we convert the vector representation into a two-component representation in which the wavefunction is not constrained by any conditions. Upon doing so, we not only separate the spin conceptually from the orbital angular momentum but also identify the Berry gauge in which the two-component wavefunction can be canonically quantized. A corollary is that only in one particular Berry gauge can a radiation field be canonically quantized in terms of the plane waves. The degree of freedom to fix the Berry gauge, which reflects a symmetry of the constraint, turns out to have observable quantum effects.
The angular momentum of light is a useful resource for many applications. In specific physical architectures it can be considered as the sum of two independent terms, the spin and the orbital components, in analogy to particle systems. The spin angular momentum is related to the polarization of the optical beam, that is the direction of the oscillating electric field, whereas the orbital angular momentum is associated with the spatial distribution of the field. Being independent, spin and orbital angular momenta have been discovered and explored in separate contexts for many years, while only recently it has been considered the possibility to address both quantities on the same beam (or individual photons). The interaction between these two quantities gives rise to complex structures of the electromagnetic field, or to the so called classical entanglement in the domain of single photons. The research presented in this work aimed to show that combining spin and orbital angular momenta in light beams or single photons may be a useful tool for a variety of applications, with particular interest to the case of architectures characterized by spin-orbit interaction. This concept was made concrete through the design and the realization of several experiments, in the framework of singular optics, foundations of quantum mechanics, quantum information theory and quantum simulation.
Measurements in mutually unbiased bases (MUBs) are integral to quantum science. An underlying principle of a set of MUBs is that a measurement in one such basis provides no information about a measurement in one of the others. This feature is especially relevant for quantum key distribution. We perform a complete set of mutually unbiased measurements for high-dimensional orbital angular momentum (OAM) states of light. For a two-photon entangled state we show that specific joint measurements provide the minimum number of experimental settings required to characterize the quantum state. From these measurements we reconstruct the density matrix describing the high-dimensional two-photon entangled state. The experimental procedure that we outline can be applied to many different multipartite and high-dimensional systems.
Weak measurements are a unique tool for accessing information about weakly interacting quantum systems with minimal back action. Joint weak measurements of single-particle operators with pointer states characterized by a two-dimensional Gaussian distribution can provide, in turn, key information about quantum correlations that can be relevant for quantum information applications. Here we demonstrate that by employing two-dimensional pointer states endowed with orbital angular momentum (OAM), it is possible to extract weak values of the higher order moments of single-particle operators, an inaccessible quantity with Gaussian pointer states only. We provide a specific example that illustrates the advantages of our method both in terms of signal enhancement and information retrieval.
To address key challenges for both quantum communication and quantum computing applications in a simultaneous manner, we propose to employ the photon angular momentum approach by invoking the well-known fact that photons carry both the spin angular momentum (SAM) and the orbital angular momentum (OAM). SAM is associated with polarization, while OAM is associated with azimuthal phase dependence of the complex electric field. Given that OAM eigenstates are mutually orthogonal, in principle, an arbitrary number of bits per single photon can be transmitted. The ability to generate/analyze states with different photon angular momentum, by using either holographic or interferometric methods, allows the realization of quantum states in multidimensional Hilbert space. Because OAM states provide an infinite basis state, while SAM states are 2-D only, the OAM can also be used to increase the security for quantum key distribution (QKD) applications and improve computational power for quantum computing applications. The goal of this paper is to describe photon angular momentum based deterministic universal quantum qudit gates, namely, {generalized-X, generalized-Z, generalized-CNOT} qudit gates, and different quantum modules of importance for various applications, including (fault-tolerant) quantum computing, teleportation, QKD, and quantum error correction. For instance, the basic quantum modules for quantum teleportation applications include the generalized-Bell-state generation module and the QFT-module. The basic quantum module for quantum error correction and fault-tolerant computing is the nonbinary syndrome calculator module. The basic module for entanglement assisted QKD is either the generalized-Bell-state generation module or the Weyloperator-module. The possibility of implementing all these modules in integrated optics is discussed as well. Finally, we provide security analysis of entanglement assisted multidimensional QKD protocols, employing the proposed qudit modules, by taking into account the imperfect generation of OAM modes.