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      Dirac equation for photons in a fibre: Origin of polarisation

      research-article
      Heliyon
      Elsevier
      Dirac equation, Klein-Gordon equation, Polarisation, Spin angular momentum, Coherent state, Broken symmetry, Graded index fibre

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          Abstract

          Spin is a fundamental degree of freedom, which was discovered by Dirac for an electron in his relativistic quantum mechanics, known as the Dirac equation. The origin of spin for a photon is unclear because Maxwell's equations in a vacuum are Lorentz invariant without introducing the concept of spin. Here, the propagation of coherent rays of photons in a graded-index optical fibre is considered to discuss the origin of polarisation for photons using exact solutions of the Laguerre-Gauss and Hermite-Gauss modes. The energy spectrum is massive, and the effective mass is a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional (1 D) non-relativistic Schrödinger equation, which is equivalent to the 2 D space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the 2 D Dirac equation with spin. The spin expectation values of photons correspond to the polarisation state on the Poincaré sphere. As an application of the theory, a polarisation interferometer is proposed, whose energy spectrum shows a Dirac cone in the Stokes parameter space.

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          The Band Theory of Graphite

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            Two-dimensional gas of massless Dirac fermions in graphene.

            Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrödinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective 'speed of light' c* approximately 10(6) m s(-1). Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m(c) of massless carriers in graphene is described by E = m(c)c*2. This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.
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              Weyl and Dirac semimetals in three-dimensional solids

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                Author and article information

                Contributors
                Journal
                Heliyon
                Heliyon
                Heliyon
                Elsevier
                2405-8440
                21 March 2024
                15 April 2024
                21 March 2024
                : 10
                : 7
                : e28367
                Affiliations
                [0010]Center for Exploratory Research Laboratory, Research & Development Group, Hitachi, Ltd., 1-280 Higashi-Koigakubo, Kokubunji, 185-8601, Tokyo, Japan
                Article
                S2405-8440(24)04398-6 e28367
                10.1016/j.heliyon.2024.e28367
                11004707
                38601593
                10dc1086-5531-4bbd-ae87-83fe1f9f2ad6
                © 2024 The Author(s)

                This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

                History
                : 9 November 2023
                : 29 February 2024
                : 18 March 2024
                Categories
                Research Article

                dirac equation,klein-gordon equation,polarisation,spin angular momentum,coherent state,broken symmetry,graded index fibre

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