[Paper Review] Dirac Equation for Photons: Origin of Polarisation

Shinichi Saito|arXiv (Cornell University)|Mar 30, 2023

Experimental and Theoretical Physics Studies11 citations

TL;DR

The paper derives a 2D space-time Klein-Gordon framework for photons in a graded-index fibre, showing that confinement induces an effective mass and leads to a Dirac-like spin description that links photon spin to polarisation.

ABSTRACT

Spin is a fundamental degree of freedom, whose existence was proven by Dirac for an electron by imposing the relativity to quantum mechanics, leading to the triumph to derive the Dirac equation. Spin of a photon should be linked to polarisation, however, the similar argument for an electron was not applicable to Maxwell equations, which are already Lorentz invariant. Therefore, the origin of polarisation and its relationship with spin are not completely elucidated, yet. Here, we discuss propagation of coherent rays of photons in a graded-index optical fibre, which can be solved exactly using the Laguerre-Gauss or Hermite-Gauss modes in a cylindrical or a Cartesian coordinate. We found that the energy spectrum is massive with the effective mass as a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional ($1D$) non-relativistic Schrödinger equation, which is equivalent to the $2D$ 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 $2D$ Dirac equation with spin. We applied the Bardeen-Cooper-Schrieffer (BCS)-Bogoliubov theory of superconductivity to a coherent ray from a laser and identified a radiative Nambu-Anderson-Higgs-Goldstone mode for recovering the broken symmetry. The spin expectation value of a photon corresponds to the polarisation state in the Poincaré sphere, which is characterised by fixed phases after the onset of lasing due to the broken $SU(2)$ symmetry, and it is shown that its azimuthal angle is coming from the phase of the energy gap.

Motivation & Objective

• Demonstrate how a coherent photon beam in a GRIN fibre can be described by a 2D Klein-Gordon equation and a 1D Schrödinger equation for the propagation along the fibre.
• Show that the confinement-induced energy gap gives photons an effective mass and connects to spin via a Dirac-like formalism.
• Elucidate the relationship between photon polarisation and spin through SU(2) symmetry and its manifestation on the Poincaré sphere.
• Link the phase of the energy gap to the azimuthal angle of polarisation via a Bogoliubov/BCS-type framework for a coherent laser state.

Proposed method

• Derive the Helmholtz equation for a GRIN fibre with n(r)^2 = n0^2(1 - g^2 r^2) and obtain exact mode solutions (Hermite-Gauss and Laguerre-Gauss).
• Show the quadratic dispersion relation omega0^2 - 2 Delta omega0 - (v0 k)^2 = 0 and define Delta = hbar delta w0 (2n + |m| + 1).
• Define an effective mass m* from the dispersion via Delta = m* v0^2 and derive a 1D Schrödinger equation along z: iħ ∂t ψz^+ = -(ħ^2/2m*) ◻2 ψz^+ where ◻2 = ∂t^2/v0^2 - ∂z^2.
• Demonstrate that a unitary shift yields a 2D Klein-Gordon equation and discuss a 2D Klein-Gordon/BCS-Bogoliubov framework leading to a radiative Nambu–Anderson–Higgs–Goldstone mode.
• Discuss parity, time-reversal, and SU(2) spin structure to connect spin expectation values with Stokes parameters on the Poincaré sphere.
• Introduce a symmetry-broken, coherent laser state and interpret the azimuthal polarisation angle as the phase of the energy gap.

Experimental results

Research questions

• RQ1 How does GRIN fibre confinement modify photon dispersion and induce an effective mass?
• RQ2 Can a 2D Klein-Gordon/Dirac framework describe photon spin and polarisation in a guided mode?
• RQ3 What is the role of broken symmetry and the energy gap in relating polarisation phase to the energy gap phase?
• RQ4 How do parity and time-reversal operations affect spin, chirality, and propagation in GRIN fibres?

Key findings

• The mode solutions yield a quadratic dispersion that includes an energy gap Delta, making photons effectively massive in the GRIN fibre.
• A 2D space-time Klein-Gordon equation is obtained via a unitary transformation from the 1D Schrödinger equation along the propagation direction.
• The spin expectation value of a photon corresponds to the polarisation state on the Poincaré sphere, with the azimuthal angle linked to the energy-gap phase.
• A radiative Nambu–Anderson–Higgs–Goldstone mode is identified in the coherent laser state as a mechanism to recover broken symmetry.
• Parity and time-reversal symmetries produce distinct behavior for guided (gapped) versus radiative (gapless) modes, connecting spin, chirality, and propagation direction.