# Quantum mechanic description of optical polarization

The concept optical polarization describes how the electric field oscillates in the plane perpendicular to the propagation direction of the field. The electromagnetic field interacts with charged particles, like electrons, and forces these to oscillate too. If the ensuing oscillation make the electrons bob back and forth along a line the field’s polarization is said to be linear. However, in general the electric field will trace out an elliptic track, with a straight line and a circle as the extreme cases. The foundations of optical polarization was laid out already in the 1850’s. The theory is still holding up today. However, it only treats ordinary light and only consider the fields intensity itself and not intensity correlations.

Beginning in the 1980’s, groups have started to generate quantum light with radically different characteristics from ordinary light. (This development was enabled by the invention of lasers in the 1960’s and the development of non-linear optics in the 1970’s.) With such quantum light it is, e.g., possible to generate light pulses consisting of two photons in an equal superposition of the two opposite circular polarizations. Loosely speaking one can say that the photons rotate the electric field simultaneously in a right hand and a left hand direction. On the average the rotations cancel, and thus one could claim that such a light pulse lack polarization. However, if the polarization is measured, then with equal probability one would get either a left-hand, or a right-hand result. Should one make a more sophisticated measurement and measure the polarization correlation between the photons one would indeed see that the light is not unpolarized. This gives an example when the present, 160 years old theory is insufficient to characterize quantum light.

*Figure 1. The polarization variance in different directions on the Poincaré sphere for a certain two-photon state of light. The light is partially polarized. The theoretically expected result is to the left in (a), and the experimentally measured result in (b). To the right, a cut through the figures is displayed. The blue curve (solid) shows theory, the red curve (dashed) shows the experiment, and the black curve (dotted) shows the experimental result adjusted to the theory by a solid rotation (only). The reason the theoretical and experimental figures are misaligned is that the optical components used in the experiments are imperfect. For example, the so-called quarter- and half-wave plates are never exactly of the intended birefringence.*

These new phenomena are of fundamental character and may be seen as mere curiosities. However, photons, and in particular the polarization of photons, are information carriers in, e.g., quantum key distribution and quantum computing systems. The speed and ease with which one can send so called qubits through the polarization state of individual photons makes them ideal “communicators” between the different parts of a future quantum computer. Therefore, it is detrimental to have an accurate and all-compassing description of quantum polarization. The prevalent theory, invented long before quantum mechanics saw the light of day, was never intended to describe quantum phenomena.

To exemplify further, light oscillating in a random fashion in the plane perpendicular to the direction of propagation is called unpolarized. Light bulb and fluorescent light belong to this category. If one measures if the light is oscillating in a certain pattern, e.g., linearly, one gets the same answer as if we instead measured its circular oscillation. One gets an identical result if one measures one of two quantum entangled photons. Such a light pulse consists of two photons, each without individual polarization but the polarization of one photon (anti)correlated to the other photon. If one in a certain measurement finds that one of the photons is linearly polarized in the horizontal direction, then one will find that the other photon is polarized in the vertical direction. If instead one of the photons were right-hand circularly polarized, the other particle would be found to be left-hand circularly polarized. Since the individual photons are unpolarized, there is an equal probability to find any of the two photons in any polarization state. However, once one of the photon’s polarization state is determined, the state of the other photon’s polarization is also automatically determined. One cannot, a priori, know the polarization state of the composite system. The only thing that is for certain is that the composite system has anti-correlated polarization. This may seem paradoxical, but is well documented by numerous measurements.

Such polarization-entangled quantum states of light lack a classical description. This withstanding, they are very useful and versatile in quantum computing and quantum key distribution. Recently they have also attracted attention in biology and astronomy. This proves that the time has come to make a systematic theory of quantum light polarization. We have laid out a research program, including both theory and experiments, to characterize quantum light polarization. Our hypothesis is that by measuring polarization correlations, or more precisely increasingly higher order polarization moments, the full description of the light’s polarization properties will be reviled. The moments and their distribution of different orders can subsequently be used to assign a degree of quantum polarization the state. The purpose of the project is thus to update the present theory, dating from the 19th century to reflect the 21st century physics and technology in a better way.

### References:

- G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization”, arXiv:1005.3935, Opt. Comm. vol. 283, pp. 4440–4447, 2010.

- A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, Ch. Marquardt, G. Leuchs, L. L. Sanchez-Soto, “Assessing the Polarization of a Quantum Field from Stokes Fluctuation”, arXiv 1004.3283, Phys. Rev. Lett. vol. 105, pp. 153602-153605, 2010.

- G. Björk, J. Söderholm, Y.-S. Kim, Y.-S. Ra, H.-T. Lim, C. Kothe-Termén, Y.-H. Kim, L. L. Sánchez-Soto, A. B. Klimov, “Central-moment description of polarization for quantum states of light”, E-print arXiv: 1111.5512. Phys. Rev. A. vol. 85, pp. 053835, 2012.

Text by : Gunnar Björk