Is the instantaneous polarisation direction of circularly polarised light observable?

Question

In classical electromagnetism, a circularly polarised electromagnetic wave is described by an electric field $\mathbf E(\mathbf r, t)=\mathbf E_0 e^{i(\mathbf k\cdot\mathbf x-\omega t)}$, where $\mathbf E_0\in\mathbb C^3$ has the form (assuming without loss of generality that the propagation direction is $z$) $$\mathbf E_0=(E_0,i E_0,0),\qquad E_0\in\mathbb R.$$ Because only the real part of $\mathbf E$ is physical, this corresponds to the physical electric field $$\mathrm{Re}[\mathbf E(\mathbf r,t)]=E_0\big( \cos(kz-\omega t),\sin(kz-\omega t), 0 \big).$$ In other words, the direction of the electric field rotates in the $xy$ plane.

This should mean that, even if only in principle, it should be possible to measure the instantaneous direction of the polarisation of a coherent light beam. For example, I should be able to set up an experiment in which, by measuring the polarisation with an high enough frequency, I should observe the polarisation direction rotating in time.

In particular, I shold then also be able to find positions where to put linear polarisers in such a way to have the light always pass through the polarisers without attenuation (because if I know the frequency of the light, I also know the distance corresponding to a full rotation of the polarisation, and then if I put linear polarisers at these positions in a suitable direction, the light would always cross them without any attenuation).

What I find odd is that assuming the above is true, I'm confused as to how this situation would be described when switching to the formalism of quantum mechanics (QM). In QM, circular polarisation is a specific polarisation state, which I can write as $|0\rangle+i|1\rangle$, and there is no notion of a "rotation in time". This is at odds with the classical description, so how do we describe this classical situation from a QM point of view (if at all possible)?

Answer 1

Not sure I understand what exactly you mean by timing the measurement. If we reduce the frequency enough so that we are dealing with low frequency radio waves, then one can measure the direction of the field based on the force it would exert on a charged particle. However, for light, the frequency is too high to do that. The integration time of a photo detector is typically much longer than the period of a light field. So although one can argue about this on a theoretical level, one would not be able to perform such an experiment in practice.

In the quantum case, remember that one can always perform a unitary transformation to convert linear polarization basis into a circular polarization basis. In effect, the measurement setup performs such a unitary transformation to bring the state into the eigenbasis of the measurement setup.

Another issue with the measurement is the phase. (Perhaps this is the core of your problem.) At very low frequencies, we can measure the phase of the field, because the time it takes for the measurement is much shorter than the period of the wave. We cannot do this at light frequencies. Therefore, we can only measure phase as a relative concept using interference effects. It is for this reason that the quantum measurement becomes probabilistic. We simply don't know what the absolute phase is, because we cannot measure it.