Quantum field theory for coherent photons: isomorphism between Stokes parameters and spin expectation values

Abstract

Stokes parameters (S) on the Poincaré sphere are very useful values to describe the polarisation state of photons. However, the fundamental principle on the nature of polarisation is not completely understood, yet, because we have no concrete consensus on how to describe spin of photons, quantum-mechanically. Here, we have considered a monochromatic coherent ray of photons, described by a many-body coherent state, and established a fundamental basis to describe the spin state of photons, in connection with a classical description based on Stokes parameters. We show that a spinor description of the coherent state is equivalent to Jones vector for polarisation states, and obtain the spin operators (Ŝ) of all components based on rotators in an SU(2) group theory. Polarisation controllers such as phase-shifters and rotators are also obtained as quantum-mechanical field operators to change the phase of the wavefunction for polarisation states. We show that the Stokes parameters are quantum-mechanical average of the spin operators, S=⟨Ŝ⟩.

Dirac equation for photons in a fibre: Origin of polarisation

“…Note that the wavefunction of a photon and the fundamental equation to describe the wavefunction have not been frequently discussed, as compared with the Schrödinger equation for an electron, except for the pioneering review article of Bialynicki-Birula [40] . A photon is an elementary particle, such that the probability of finding a photon at a certain position ( r ) is described by a wavefunction, , which satisfies the Helmholtz equation [10] , [11] , [46] , [47] , [32] where , t is time, is the vacuum permeability, and is the profile of the dielectric constant of a material. The values of permeability for most of the materials used in photonics are almost the same as those in a vacuum.…”

“…On the other hand, in a material with the refractive index profile of , a photon tends to propagate in a region where the refractive index is large, to minimise the optical path length, following Fermat's principle and the Eikonal equation [11] , [46] , [47] , [32] . The dispersion relationship for the wavenumber, k , in a material must be obtained by solving the Helmholtz equation.…”

“…The wavefunction of the 2 D Dirac equation is described by the 2-component spinor representation as where the spin up and down components correspond to the left- and right-circular polarisation states [46] , respectively, and the spinor components are and . Consequently, by assuming the proper factorisation of the corresponding 2 D Dirac equation, we have derived the spin of a photon from the Helmholtz equation, which is assigned for the polarisation degree of freedom.…”

“…We also show how the spin of a photon is linked to the origin of polarisation, as a manifestation of macroscopic coherence for the light from a laser source. To make the argument specific, we restrict our analysis to the propagation of a coherent ray of photons in a GRIN fibre [45] , [11] , [46] , [47] , [32] , but the extension to the other waveguide will be straightforward. The advantage to employing the GRIN fibre is the availability of the exact solution of the Helmholtz equation, and therefore we can theoretically treat it exactly.…”

“…For the coherent photons from a laser source, the many-body state is described by a coherent state [26] , [48] , [49] , which is a superposition state made of states with different number of photons. Because of the uncertainty in the number of photons, the phase is coherently fixed, while a macroscopic number of photons occupies the same state, which can be regarded as Bose-Einstein condensation [26] , [48] , [49] , [50] , [51] , [52] , [46] , [47] , [32] . We introduce the concept of the broken symmetry [53] to photons, which was originally established as the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity [54] , [55] , [56] , [57] , [50] , [51] , [52] .…”