Spin is the fundamental property that distinguishes the two types of elementary particles: fermions with half-integer spins and bosons with integer spins. Photons, which are the quanta of light, have been long recognized as spin-1 gauge bosons. The polarization of the light is commonly accepted as its “intrinsic” spin degree of freedom. However, in free space, only two transverse polarizations are allowed. Thus, the photon spin is always only connected to the two circular polarizations. To construct the full quantum spin operator of light, longitudinal polarized photon modes have to be introduced.
Left and right circular polarization and their associated angular momenta
When a light beam is circularly polarized, each of its photons carries a spin angular momentum (SAM) of , where is the reduced Planck constant and the sign is positive for left and negative for right circular polarizations (this is adopting the convention from the point of view of the receiver most commonly used in optics). This SAM is directed along the beam axis (parallel if positive, antiparallel if negative).
The above figure shows the instantaneous structure of the electric field of left () and right () circularly polarized light in space. The green arrows indicate the propagation direction.
The mathematical expressions reported under the figures give the three electric-field components of a circularly polarized plane wave propagating in the direction, in complex notation.
Then, one can verify that both and satisfy the canonical angular momentum commutation relations
and they commute with each other .
After the plane-wave expansion, the photon spin can be re-expressed in a simple and intuitive form in the wave-vector space
where the column-vector is the field operator of the photon in wave-vector space and the matrix
is the spin-1 operator of the photon with the SO(3) rotation generators
and the two unit vectors denote the two transverse polarizations of light in free space and unit vector denotes the longitudinal polarization.
Due to the longitudinal polarized photon and scalar photon have been involved, both and are not gauge invariant. To incorporate the gauge invariance into the photon angular momenta, a re-decomposition of the total QED angular momentum and the Lorenz gauge condition have to be enforced. Finally, the direct observable part of spin and orbital angular momenta of light are given by
We can define the annihilation operators for circularly polarized transverse photons:
with polarization unit vectors
Then, the transverse-field photon spin can be re-expressed as
For a single plane-wave photon, the spin can only have two values , which are eigenvalues of the spin operator . The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves: