Double slit experiment with one circular polarizer in front of only one slit in the which-way perspective?

Question

If one puts in front of a double slit two quarter-wave circular polarizers (QWP) in front of each slit, say one with 45° and the other with -45° orientation (fast and slow axis - i.e. clock- and anti-clockwise circular polarization) and shines linear polarized light on the slits, one does not get the typical interference pattern but the Gaussian-shaped diffraction pattern. This is the result of the overlap of the fringe- and anti-fringe patterns which, summed up, furnish the typical bell-shaped figure (these result from the two QWP induced pi-phase shift which causes the interference pattern with the central peak to be shifted left and right, respectively, with maxima overlapping minima and vice-versa).

One can see this also from the perspective of the which-way question in quantum mechanics which states that by placing the two QWP in front of the slits we have 'marked' the photons and we could recover the which-way information which leads to the loss of the interference pattern.

However, what happens if one places only one QWP in front of only one slit? As I understand it, the interference pattern would reappear as only one fringe or anti-fringe (right? Well, maybe not... this is part of my question).

I'm in doubt, because if one would analyze this in the which-way perspective photons are still 'labeled', that is, photons going through the slit without the QWP would be linearly polarized, those going through the slit with the QWP are circularly polarized and one could in principle recover the which way information, i.e. one expects still only the diffraction without interference pattern. So, a contradiction arises that hints possibly at a mistake or misunderstanding from my side, but can't find out where it is. Can anyone help?

Answer 1

However, what happens if one places only one QWP in front of only one slit? [...] if one would analyze this in the which-way perspective photons are still 'labeled', that is, photons going through the slit without the QWP would be linearly polarized, those going through the slit with the QWP are circularly polarized and one could in principle recover the which way information

This is not quite the case. In this situation, you are indeed 'labelling' the light that comes from both slits, but the labels are partially ambiguous: it is possible to distinguish the two labels to some extent, but it is not possible to do so in full.

The easiest way to understand why this is the case is to consider the only way to unambiguously determine whether light is linearly polarized along the chosen axis: i.e., using a polarizing beam splitter along this axis. Here, if you come in with light which is linearly polarized along the orthogonal axis, it will be fully deflected, allowing you to distinguish it from the intended target. But what about circularly polarized light? It effectively consists of a 50:50 mix of the two linear polarizations, so the beam splitter will just split it in half and send half of the power through each of the output ports. Thus, you're able to partially distinguish the two (a photon coming out of the second port would certify that it's not from the linearly-polarized slit) but you cannot do so all the time (since a photon coming out of the first port could have come from either slit).

Thus, the situation is halfway between the full-interference case (with no which-way information) and the fully-decohered case (with full which-way information). There is partial which-way information, so there will be partial interference: the fringes will be present, but the contrast will be significantly reduced.

Adapting my code from this answer, this is what the interference patterns will look like, for (i) counter-rotating circular sources, (ii) both linearly polarized on the same axis, and (iii) one of each:

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As you can see, there's a fair amount of interference left. But it's also important to note that the amplitude of the fringes does is smaller, and that the bottom of the troughs at the center no longer reaches down to the axis, i.e., there is no longer any complete destructive interference anywhere on the pattern.

Answer 2

The polarization state of the light illuminating the two slits should be taken into account. Interference between the portions of light passing through the two slits, when the polarization of light passing through one of the slits is rotated, results in a spatial modulation of the polarization state of photons detected at the screen. The pattern of polarization modulation may be calculated simply by summing the amplitudes of light waves (including polarization) from the two slits.

Answer 3

One way to think of this is to use the "the photon wave function" concept. A polarizer effectively destroys the original photon and creates a new function so now the photons passing thru the slits are not coherent anymore, thus the loss of the diffraction pattern. It is like having 2 separate sources one in front of each slit. When one polarizer is removed a portion of the photons passing thru both slits are coherent again and the partial pattern is seen.