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The typical model for unpolarized light consists of two linearly polarized waves at right angles to each other, where the phase difference changes randomly every so often. Instead of two linear polarizations you can also use two opposite circular polarizations, or two elliptical polarizations--the statistical properties will be the same, described by the correlation matrix

$$\langle E_a E_b^* \rangle \propto \delta_{ab}.$$

Maybe I'm overextending here, but this is strongly reminiscent of a quantum mixed state with density matrix proportional to the identity matrix: you can describe the mixture using any basis you want. My question therefore is: is there anything quantum about the EM field of natural, unpolarized light? Does it have an instantaneously well defined value and polarization, even if it changes very fast? Or is its polarization intrinsically undefined?

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Classically unpolarized light does have a well defined instantaneous polarization vector at any moment in time. The direction of that vector just changes on time scales much faster than the detection bandwidth of whatever detector is under consideration so the detector just averages out (in time) it’s measurement result.

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    $\begingroup$ Might be also worth noting that a 'detector' can be a physical system, such as an atom. If the polarization varies faster than its internal timescale, then it sees unpolarized light. $\endgroup$
    – eranreches
    Commented Nov 1, 2022 at 22:13

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