# Delayed choice quantum eraser experiment involving light and polarization filters

I read a post about trying Young's double slit experiment at home, and the first comment suggested a DIY delayed choice quantum eraser experiment. In the video, they use two polarization filters on the slits, one at $$0^{\circ}$$ and one at $$90^{\circ}$$, to trace the path of the light and thus stop the light from interfering, and then a $$45^{\circ}$$ angle one to erase the information and obtain the interference pattern again. I have a few questions about this:

1. Isn't the reason for the light not interfering in the first part of the experiment the fact that the two polarization filters set the light in orthonormal states that don't interact? I imagine that if after we polarize the light, we use another filter to determine the polarization angle, then we would have information about the path and collapse the wave function, but without the second polarization filter, we still don't have the path information, from my understanding.
2. When light doesn't interfere, I assumed that we would see the particle-lie behavior of light; then why don't we see two separate dots/lines on the screen, corresponding to the two slits, but rather a diffuse blob of light?
3. If instead we used, say, a $$0^{\circ}$$ and a $$10^{\circ}$$ polarization filter, should we be able to see interference?

Long story short, I don't think this is a very good example of a quantum eraser experiment. Nothing is happening here that can't be explained via classical electromagnetism; there's no explicitly "quantum" effects involved. But along the way, there are several misconceptions in your question that should be corrected.

Isn't the reason for the light not interfering in the first part of the experiment the fact that the two polarization filters set the light in orthonormal states that don't interact?

"Interact" is definitely the wrong word here. Light doesn't interact with other light; if you cross two beams of light and examine them after the crossing point, you will not be able to detect that the two beams had crossed. What's happening here is, specifically, interference, which is not an interaction between two light beams; instead, it just derives from the simultaneous presence of two electromagnetic fields at the same point in space. These two electromagnetic fields just add their intensities, because electromagnetism is linear.

The non-interference of two electromagnetic waves that are linearly polarized in orthogonal states is a purely classical effect. In particular, the Fresnel-Arago laws describe interference between arbitrary electromagnetic plane waves. It turns out that you can prove that the total intensity at a point where two electromagnetic waves are present contains three terms: one term proportional to the square of the first wave's electric field, one term proportional to the square of the second wave's electric field, and a third term, which depends on the phase difference of the waves and the dot product of the electric field vectors at that point. If the two waves are linearly polarized in orthogonal states, the dot product of their electric field vectors is always zero, and this third term vanishes. Hence, this is a prediction of classical electromagnetism.

I imagine that if after we polarize the light, we use another filter to determine the polarization angle, then we would have information about the path and collapse the wave function, but without the second polarization filter, we still don't have the path information, from my understanding.

In the quantum-mechanical explanation, what matters is not whether we the experimenters have which-path information. What matters is whether there's some way, in principle, to distinguish the photons that came from one slit from the photons that came from the other slit. This distinguishability is the "which-path" information. If you pass both sets of photons through another polarizer, the two sets of photons coming out of the polarizer will be indistinguishable again, because the polarizer just changed them all to the same polarization state. This means that they can interfere again. Putting a second polarizer in erases the which-path information for measurements beyond that polarizer.

But again, this is not a fundamentally quantum phenomenon. Everything going on here is fully in compliance with classical electromagnetism. In particular, Malus's Law describes the intensity of a linearly polarized electromagnetic wave after it passes through a polarizer. The derivation is straightforward: if you decompose the polarization vector into components parallel and perpendicular to the polarizer, the polarizer blocks the perpendicular component and passes the parallel component. From this, you get three effects:

1) The intensity of the passed light is proportional to the cosine of the angle between the light's polarization orientation and the polarizer's orientation;

2) The light that passes through the polarizer is polarized in the direction of the polarizer; and

3) Unpolarized light that passes through the polarizer comes out polarized with half the intensity it started with.

So, if you combine Malus's Law and the Fresnel-Arago laws, you can completely explain this demonstration using classical electromagnetism. By the Fresnel-Arago laws, two sources of coherent unpolarized light interfere in the same way as two sources of coherent polarized light with parallel polarizations, so without any polarizers, you see interference on a screen behind the two slits. If you place the orthogonal polarizers over the slits, by Malus's Law the light coming out of one slit is polarized orthogonally to the light coming out of the second slit. By the Fresnel-Arago laws, these two orthogonally-polarized light sources cannot interfere, so you don't see interference on the screen. But if you put another polarizer between the slits and the screen, by Malus's Law the light coming out of that polarizer has the same polarization, no matter what its input polarization was. This means that, beyond this polarizer, the light from the two slits now has the same polarization, and so, by the Fresnel-Arago laws, they can now once again interfere, producing an interference pattern on the screen again.

When light doesn't interfere, I assumed that we would see the particle-lie behavior of light; then why don't we see two separate dots/lines on the screen, corresponding to the two slits, but rather a diffuse blob of light?

Because you don't have an ideal point source behind the slits. Any real light source has a finite size, so light traveling in a variety of different directions can enter the slit. Light that enters the slit from different directions will hit the screen at different locations, producing a blob rather than two dots/slits.

If instead we used, say, a $$0^\circ$$ and a $$10^\circ$$ polarization filter, should we be able to see interference?

The dot product between the two electric field orientation vectors is nonzero, so by the Fresnel-Arago laws, yes, you will see interference, but it will be weaker than the interference without the polarizers in place (in other words, the variations in intensity as a percentage of the total intensity will be smaller). When you place another polarization filter between the slits and the screen, then all of the light has the same polarization again, and the interference returns to its original strength. The orienation of this polarizer doesn't affect the strength of the interference as a percentage of the maximum intensity on the screen, but it does affect the overall maximum intensity of the light on the screen.

• I understand most of what you're saying, but here's where I get lost; you said: "In the quantum-mechanical explanation, what matters is not whether we the experimenters have which-path information. What matters is whether there's some way, in principle, to distinguish the photons that came from one slit from the photons that came from the other slit. This distinguishability is the "which-path" information." 1/2 May 1, 2020 at 18:28
• and regarding the 0 and 10 degree example: "The dot product between the two electric field orientation vectors is nonzero, so by the Fresnel-Arago laws, yes, you will see interference, but it will be weaker than the interference without the polarizers in place" If the paths are distinguishable by the polarization angle (0 or 10 degrees) why do we still see interference? 2/2 May 1, 2020 at 18:28
• @Anthill In the quantum-mechanical explanation, the 10-degree polarization state can be decomposed into the superposition of a 0-degree polarization state and a 90-degree polarization state (since any two orthogonal polarization states form a basis of the space of possible polarization states). So the 0-degree component of the state is indistinguishable from the 0-degree polarization state produced by the other slit, so it interferes, while the 90-degree component of the state is distinguishable from the state produced by the other slit, so it does not interfere. May 1, 2020 at 18:34
• @Anthill This means that the amount of interference you see is proportional to the quantum-mechanical "overlap" between the two states (in technical terms, the inner product of the two polarization states). May 1, 2020 at 18:35
• @Anthill Meanwhile, in the classical explanation, all this talk of "which-path" information doesn't matter at all; the behavior is just explained by the vector addition of electric fields. May 1, 2020 at 18:39

@Anthill, ik this is coming in late but just came across this. I get what you're saying and yes, going to your point #1, you do have which path information as the measurement by the polarizers collapses the wave function. That being the case, a photon either goes through the vertical polarizer or the horizontal one. This shows up as a single blob on the screen. However, as you point out in your point #2, over repeated runs you will indeed see 2 blobs as at each run, a photon has an even 50% chance of passing through either the horizontal or the vertical.

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