View source for Spin angular momentum of light

{{Short description|Angular momentum deriving from photon spin}}

The '''spin angular momentum of light''' ('''SAM''') is the component of [[angular momentum of light]] that is associated with the [[Spin (physics)|quantum spin]] and the rotation between the [[polarization (waves)|polarization]] degrees of freedom of the photon.

== Introduction ==
Spin is the fundamental property that distinguishes the two types of elementary particles: [[Fermion|fermions]], with half-integer spins; and [[Boson|bosons]], with integer spins. [[Photon|Photons]], which are the [[Quantum|quanta]] of [[Electromagnetic radiation|light]], have been long recognized as spin-1 [[Gauge boson|gauge bosons]]. The polarization of the light is commonly accepted as its “intrinsic” spin [[Degrees of freedom (physics and chemistry)|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.

[[File:Sam v1.png|thumb|right|400px|Left and right circular polarization and their associated angular momenta]]
An [[electromagnetic wave]] is said to have [[circular polarization]] when its [[electric]] and [[magnetic field]]s rotate continuously around the beam axis during propagation. The [[circular polarization]] is left (<math>\mathrm{L}</math>) or right (<math>\mathrm{R}</math>) depending on the field rotation direction and, according to the convention used: either from the point of view of the source, or the receiver. Both conventions are used in science, depending on the context.

When a light beam is circularly polarized, each of its [[photon]]s carries a spin angular momentum (SAM) of <math>\pm\hbar</math>, where <math>\hbar</math> is the [[reduced Planck constant]] and the <math>\pm</math> 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 (<math>\mathrm{L}</math>) and right (<math>\mathrm{R}</math>) circularly polarized light in space. The green arrows indicate the [[Wave propagation|propagation]] direction.

The mathematical expressions reported under the figures give the three electric-field components of a circularly polarized plane wave propagating in the <math>z</math> direction, in [[complex number|complex]] notation.

== Mathematical expression ==
<!--In the following the main formula used for the spin angular momentum of light are given. — ??? -->

The general expression for the spin angular momentum is<ref>{{cite arXiv| last1=Yang|first1=L.-P.| last2=Khosravi|first2=F. |last3=Jacob|first3=Z. |year=2020 |title=Quantum spin operator of the photon |eprint=2004.03771|class=quant-ph}}</ref>

<math display="block">\mathbf{S} = \frac{1}{c}\int d^{3}x\mathbf{\pi}\times\mathbf{A},</math>

where  <math>c</math> is the speed of light in free space and <math>\mathbf{\pi}</math> is the [[Canonical momenta|conjugate canonical momentum]] of the [[magnetic vector potential|vector potential]] <math>\mathbf{A}</math>. The general expression for the orbital angular momentum of light is

<math display="block">\mathbf{L} = \frac{1}{c}\int d^{3}x\pi^{\mu}\mathbf{x}\times\mathbf{\nabla}A_{\mu},</math>

where <math>\mu=\{0,1,2,3\}</math> denotes four indices of the [[spacetime]] and [[Einstein notation|Einstein's summation convention]] has been applied. To quantize light, the basic equal-time commutation relations have to be postulated,<ref>{{Cite book|last1=Greiner|first1=W.|title=Field Quantization|last2=Reinhardt|first2=J.|isbn=9783642614859|chapter=Chap. 7|date=29 June 2013}}</ref>

<math display="block">[A^{\mu}(\mathbf{x},t),\pi^{\nu}(\mathbf{x}',t)]  =i\hbar cg^{\mu\nu}\delta^{3}(\mathbf{x}-\mathbf{x}'),</math>

<math display="block">[A^{\mu}(\mathbf{x},t),A^{\nu}(\mathbf{x}',t)]  =[\pi^{\mu}(\mathbf{x},t),\pi^{\nu}(\mathbf{x}',t)]=0,</math>

where <math>\hbar</math> is the [[Reduced Planck Constant|reduced Planck constant]] and <math>g^{\mu\nu}=\rm{diag}\{1,-1,-1,-1\}</math> is the metric tensor of the [[Minkowski space]].

Then, one can verify that both <math>\mathbf{S}</math> and <math>\mathbf{L}</math> satisfy the canonical angular momentum commutation relations

<math display="block">[S_{i},S_{j}] =i\hbar\epsilon_{ijk}S_{k},</math>

<math display="block">[L_{i},L_{j}] =i\hbar\epsilon_{ijk}L_{k},</math>

and they commute with each other <math>[S_{i},L_{j}] =0</math>.

After the plane-wave expansion, the photon spin can be re-expressed in a simple and intuitive form in the wave-vector space

<math display="block">\mathbf{S}=\hbar\int d^{3}k \hat{\phi}^{\dagger}_{\mathbf{k}}\mathbf{\hat{s}}\hat{\phi}_{\mathbf{k}}</math>

where the column-vector <math>\hat{\phi}_{\mathbf{k}}=[\hat{a}_{\mathbf{k},1},\hat{a}_{\mathbf{k},2},\hat{a}_{\mathbf{k},3}]^{T}</math> is the field operator of the photon in wave-vector space and the <math>3\times 3</math> matrix

<math display="block">\mathbf{\hat{s}}=\sum_{\lambda=1}^3\hat{s}_{\lambda}\mathbf{\epsilon}(\mathbf{k},\lambda)</math>

is the spin-1 operator of the photon with the SO(3) rotation generators

<math display="block">\hat{s}_{1}= \begin{bmatrix}
0 & 0 & 0\\

0 & 0 & -i\\

0 & i & 0
\end{bmatrix}, \qquad \hat{s}_{2}=\begin{bmatrix} 
0 & 0 & i\\

0 & 0 & 0\\

-i & 0 & 0
\end{bmatrix}, \qquad \hat{s}_{3}=\begin{bmatrix} 
0 & -i & 0\\

i & 0 & 0\\

0 & 0 & 0
\end{bmatrix},</math>

and the two unit vectors <math>\boldsymbol{\epsilon}(\mathbf{k},1)\cdot\mathbf{k} = \boldsymbol{\epsilon}(\mathbf{k},2) \cdot \mathbf{k} = 0</math> denote the two transverse polarizations of light in free space and unit vector <math>\boldsymbol{\epsilon}(\mathbf{k},3) = \mathbf{k}/|\mathbf{k}|</math> denotes the longitudinal polarization.

Due to the longitudinal polarized photon and scalar photon have been involved, both  <math>\mathbf{S}</math> and <math>\mathbf{L}</math> are not gauge invariant. To incorporate the gauge invariance into the photon angular momenta, a re-decomposition of the total [[Quantum electrodynamics|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

<math display="block">\mathbf{S}^{\rm obs}=i\hbar\int d^3 k(\hat{a}_{\mathbf{k},2}^{\dagger}\hat{a}_{\mathbf{k},1}-\hat{a}_{\mathbf{k},1}^{\dagger}\hat{a}_{\mathbf{k},2})\frac{\mathbf{k}}{|\mathbf{k}|}=\varepsilon_{0} \int d^{3}x\mathbf{E}_{\perp}\times \mathbf{A}_{\perp},</math>
and
<math display="block">\mathbf{L}^{\rm obs}_{M}=\varepsilon_{0}\int d^{3}x E_{\perp}^{j}\mathbf{x}\times\mathbf{\nabla}A_{\perp}^{j}</math>

which recover the angular momenta of classical transverse light.<ref>{{Cite book|last1=Cohen-Tannoudji|first1=C.|title=Photons and Atoms-Introduction to Quantum Electrodynamics|last2=Dupont-Roc|first2=J.|last3=Grynberg|first3=G.|publisher=Wiley-VCH|year=1997|isbn=9780471184331|chapter=Chap. 1}}</ref> Here, <math>\mathbf{E}_{\perp}</math>(<math>\mathbf{A}_{\perp}</math>) is the transverse part of the [[electric field]] ([[vector potential]]), <math>\varepsilon_0</math> is the [[vacuum permittivity]], and we are using [[International System of Units|SI units]].

We can define the annihilation operators for circularly polarized transverse photons:
<math display="block">\hat{a}_{\mathbf{k},\mathrm{L}} =\frac{1}{\sqrt{2}}\left(\hat{a}_{\mathbf{k},1}-i\hat{a}_{\mathbf{k},2}\right),</math>
<math display="block">\hat{a}_{\mathbf{k},\mathrm{R}} =\frac{1}{\sqrt{2}}\left(\hat{a}_{\mathbf{k},1}+i\hat{a}_{\mathbf{k},2}\right),</math>
with polarization unit vectors
<math display="block">\mathbf{e}(\mathbf{k},\mathrm{L}) =\frac{1}{\sqrt{2}}\left[\mathbf{e}(\mathbf{k},1)+i\mathbf{e}(\mathbf{k},2)\right],
</math>
<math display="block">\mathbf{e}(\mathbf{k},\mathrm{R}) =\frac{1}{\sqrt{2}}\left[\mathbf{e}(\mathbf{k},1)-i\mathbf{e}(\mathbf{k},2)\right].
</math>

Then, the transverse-field photon spin can be re-expressed as
<math display="block">\mathbf{S}^{\rm obs} = \int d^3k\hbar \left(\hat{a}^\dagger_{\mathbf{k},L} \hat{a}_{\mathbf{k},L} - \hat{a}^\dagger_{\mathbf{k},R} \hat{a}_{\mathbf{k},R}\right)\frac{\mathbf{k}}{|\mathbf{k}|},</math>

For a single plane-wave [[photon]], the spin can only have two values <math>\pm\hbar</math>, which are [[eigenvalues]] of the spin operator <math>\hat{s}_3</math>. The corresponding eigenfunctions describing photons with well defined values of SAM are described as circularly polarized waves:
<math display="block">|\pm\rangle = \begin{pmatrix} 1 \\ \pm i \\0\end{pmatrix}.</math>

==See also==
* [[Helmholtz equation]]
* [[Orbital angular momentum of light]]
* [[Polarization (physics)]]
* [[Photon polarization]]
* [[Spin polarization]]

== References ==
{{Reflist}}

==Further reading==
{{refbegin}}
*{{Cite book |last1=Born |first1=M. |name-list-style=amp |last2=Wolf |first2=E. |title=[[Principles of Optics|Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light]]|edition=7th |publisher=Cambridge University Press |location=Cambridge |year=1999 |isbn=978-0-521-64222-4 }}
*{{Cite book |last1=Allen |first1=L. |name-list-style=amp |last2=Barnnet |first2=Stephen M.|last3=Padgett |first3=Miles J. |title=Optical Angular Momentum |publisher=Institute of Physics |location=Bristol |year=2003 |isbn=978-0-7503-0901-1 }}
*{{Cite book |last1= Torres |first1=Juan P. |name-list-style=amp |last2=Torner |first2= Lluis|title=Twisted Photons: Applications of Light with Orbital Angular Momentum |publisher= Wiley-VCH |location=Bristol |year=2011 |isbn=978-3-527-40907-5 }}
{{refend}}

<!--- Categories --->
[[Category:Angular momentum of light]]
[[Category:Light]]
[[Category:Physical quantities]]