Quantum concepts in optical polarization

Aaron Z. Goldberg; Aaron Z. Goldberg; Pablo de la Hoz; Pablo de la Hoz; Gunnar Björk; Andrei B. Klimov; Markus Grassl; Markus Grassl; Gerd Leuchs; Gerd Leuchs; Luis L. Sánchez-Soto; Luis L. Sánchez-Soto

doi:10.1364/AOP.404175

Advances in Optics and Photonics
Vol. 13,
Issue 1,
pp. 1-73
(2021)
•https://doi.org/10.1364/AOP.404175

Quantum concepts in optical polarization

Aaron Z. Goldberg, Pablo de la Hoz, Gunnar Björk, Andrei B. Klimov, Markus Grassl, Gerd Leuchs, and Luis L. Sánchez-Soto

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Aaron Z. Goldberg, Pablo de la Hoz, Gunnar Björk, Andrei B. Klimov, Markus Grassl, Gerd Leuchs, and Luis L. Sánchez-Soto, "Quantum concepts in optical polarization," Adv. Opt. Photon. 13, 1-73 (2021)

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Table of Contents Category
- Quantum Optics and Quantum Information
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About this Article
History
- Original Manuscript: July 29, 2020
- Revised Manuscript: November 11, 2020
- Manuscript Accepted: November 12, 2020
- Published: March 1, 2021

Abstract

We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.

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$S$	$M$	State	Constellation
1	1	$Ψ_{0} = 1$	radial line
$\frac{3}{2}$	1	$Ψ_{\pm \frac{3}{2}} = \frac{1}{\sqrt{2}}$	equatorial triangle
2	2	$Ψ_{- 1} = \sqrt{\frac{2}{3}}, Ψ_{2} = \sqrt{\frac{1}{3}}$	tetrahedron
$\frac{5}{2}$	1	$Ψ_{\pm \frac{3}{2}} = \frac{1}{\sqrt{2}}$	equatorial triangle + poles
3	3	$Ψ_{\pm 2} = \frac{1}{\sqrt{2}}$	octahedron
$\frac{7}{2}$	2	$Ψ_{- \frac{5}{2}} = Ψ_{\frac{1}{2}} = \sqrt{\frac{7}{18}}, Ψ_{\frac{7}{2}} = \sqrt{\frac{2}{9}}$	two triangles + pole
4	3	$Ψ_{\pm 4} = \sqrt{\frac{5}{24}}, Ψ_{0} = \sqrt{\frac{7}{12}}$	cube
$\frac{9}{2}$	2	$Ψ_{\pm \frac{9}{2}} = \frac{1}{\sqrt{6}}, Ψ_{\pm \frac{3}{2}} = \frac{1}{\sqrt{3}}$	three triangles
5	3	$Ψ_{\pm 5} = \frac{1}{\sqrt{5}}, Ψ_{0} = \frac{3}{\sqrt{5}}$	pentagonal prism
$\frac{11}{2}$	3	$Ψ_{\pm \frac{11}{2}} = \frac{\sqrt{17}}{12}, Ψ_{\pm \frac{5}{2}} = i \frac{\sqrt{55}}{12}$	pentagon + two triangles
6	5	$Ψ_{\pm 5} = \pm \frac{\sqrt{7}}{5}, Ψ_{0} = - \frac{\sqrt{11}}{5}$	icosahedron
7	4	$Ψ_{\pm 6} = \sqrt{\frac{854}{3645}}, Ψ_{\pm 3} = \sqrt{\frac{637}{13420}} + i \sqrt{\frac{512603}{9783180}}$	three squares + poles
		$Ψ_{0} = \sqrt{\frac{12561757}{163053000}} - i \sqrt{\frac{512603}{2013000}}$
10	5	$Ψ_{\pm 10} = \sqrt{\frac{187}{1875}}, Ψ_{\pm 5} = \pm \sqrt{\frac{209}{625}}, Ψ_{0} = \sqrt{\frac{247}{1875}}$	deformed dodecahedron

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Advances in Optics and Photonics