This experience is known as the delayed-choice quantum eraser. At first, the experiemnt might seems like if a measurment made in the present can alters one made in the past, but in fact, considering the photon in a superposition of state makes it work. This experiment was first realised in 1998 and was done a second time in Brazil in 2008 in a slitly diffrent setup (see the litterature section), more similar to the one will are doing. On one of the path, we have a double slit. Without entangled photon, we would get interference. This experiment shows that when we have entengled photon, the detctor D1 will never get alone interference, even if photon passed through the double slit. That curve is the one in purple in the following animation. But, when we look at the coincidence between D1 and D2, we get different results depending on the angle of all polarizers. That pattern is drawn is pink.
You can change all polarizer's angle with the pink dots. you can also modify the slit parameters, remove the BBO or try the four preset with the most intersting parameters.
The prove you that animation is not lying, we built the setup in a lab and got those curves. We tested the 4 presets and here is what we got for the coincidences (the pink curve in the animation):
These are the data we got from our lab setup. In the following section, we will explain what we did in the lab, what the data shows, what is the explaination for all of this and the maths behind the theorical curves.
Curious of what and entanglement optical setup looks like? Here is a few picture of what we have:
Here is the representation of the setup you see in the picture. All of the equipement is shown, but the scaling is not representative.
The BBOs creates pairs of photons in the state: $\ket{\psi_1} = \frac{1}{\sqrt{2}}(\ket{\updownarrow}_1\ket{\updownarrow}_2ket{\leftrightarrow}_1\ket{\leftrightarrow}_2)$. The downer path, path 1, starts with a double slit (b=200mm, d=400mm). Each slit has a polarizer in front (PA=0deg, PB=90deg). A 500mm lens is place in the middle between the double slit and the detector, both at 500mm. The third polarizer, PA, is at 45 degree. Finally, there is a linear actuator that moves the detector, D1, with a 200mm slit right in front of it.
The upper path, path 2, also has a 500 mm lens, places at 500mm of the detector and a rotative polarizer P2.
The detectors we use are called single photon detector. Their mounts are not shown in this setup, but the detector are linked to an optical fiber, which need a collider and a filter also attached to it. The detectors count the number of photon they see for a determine time. We move the detector D1 to count the number of photon at different position to recreate the full pattern. D2 doesn't need to move because we shoot a straight beam to it, no diffraction or interference. All componants that arrives before the BBO are only there to insure the beam is as small as it can be, perfectly perpendicular to the BBO and in the right polarization and phase. For more detail, the experiment 4 explicit their usage.
You can download all our data here:
To recreate a pattern, we moved an actuator along the beam. At each step, we took several points to be able the have an average and a standard deviation. We also did a few sweep over all distances to average even more.
All those files has 7 columns. The first two are the angle of the polarizers, but are steady in this experiement. The third one is the distance the actuator has done in μm . The forth column is the sweep number. Whenever the sweep change, the distance is reset to the starting point. Finally, the last three columns are the counts ont the detctor, there D1 (on the double slit path), D2 and the last is the coincidence between D1and D2.
Let's take the time to analyse those graphics. First, if we compare the four plots on the left, we see they are all the same (minus some little experiemental error), which is completly normal since nothing is moving on that path. Even when P2 is at 0 or 90 degree, we will select photons that went through either one slit or the other. But the other photons still exist, so we are not suppose to see a distance between the diffrraction. When we look at the graphics on the right, the cincidence, now we see changes. At 45 and -45 degrees, we clearly see interference. Both graphics are complementary (the peak becomes crest and crest becomes peak, At 0 and 90 degrees, we only see diffraction, but we see a small shift between the to curves, explain by the fact the photnos has not use the same slit. The maximum on the diffraction curve are centered in front of each slit.
In the next sections, we will cover how did we achives the experiement, the explaination of the result and the maths to find the theorical curve.
In the BBO crystal, we generate a pair of entangled photon in the state $\ket{\psi_1} = \frac{1}{\sqrt{2}}(\ket{\uparrow}_1\ket{\uparrow}_2 + \ket{\leftrightarrow}_1\ket{\leftrightarrow}_2)$. They are both sent to two different path. The bottom path as a double slit whereeach slit are covered with a polarizer at 0 or 90 degrees. If we let the setup like that, we could never see interference since the photons from different slit cannot interact. So we place a third polarizer at 45 degree. On the upper path, we only place one polarizer, both a polarizer that can be rotated at any angle. That way, when looking at coincidance, we can chose with the upper path polarizer P2 in which slit the photon went. Let's say we place the poalrizer at 0 degre, we now for certain coincidance with photons that went to the 0 degre slit. We won't get interference, since it acts like a single slit. If P2 is placed at 45 degree, the photon could have passed through any slit, so the interference there is. The most intriging part of this experience is went looking not at coincidance, but at all event happening at detector D1. If we ignore D2, we are looking at a "normal" photon beam going through a double slit. The obvious result would be interference, but that is not what this experiment really shows.
Full approach
1. Let's start with the initial state
2. We transform it into the right Bell state depending on the crystal used.
3. We let the photons travel to reach either one of the slit. In both case, the distance it travels is l. We need to change the phase of the photon in both path of that distance.
4. We apply the two polarizors of 0 and 90 degrees
5. We add the distance needed to reach the point C. We need to consider that photons from both slit won't travel the same distance. Here, I cheat just a little bit, I'll add right away all the distance to reach the detector instead of adding it in two step. The reason is because it depends on the position of P1, but we could place it anywere and it won't change the result. So, let's say that the photons that went thourgh slit A travels a ditance a and the oneform slit B travel a distance b.
6. Let's apply polarizor P1 of 45 degrees.
7. Let's apply polarizor P2. We want to keep the angle as a variable so we don't have to redo all calculus want changing the angle. Let's call that angle γ (gamma).
8. Finaly, the photons will travel to the detectors.
9. The final step to solve get to pattern of the coincidence, we need to take the complex conjugate of this equation. To do a complex conjugate, you take the result of this last equation and you multiply it by itself, exept all |⟩ becomes ⟨| and vice-versa and all ei becomes e−i.
10. This last equation is our interference pattern. We see that if γ is either 0 or 90, the term with the exponantial will be null. Which means no interference. Anything else will have interference. To have the final equation for the pattern, we need to multiply by the envelop of the diffration pattern. Which give the final result:
If you want to check out some papers that did the same experiment, here are the two best ressources we have found.
If you want a more interactive ressource, here are some Youtube video we found to be accurate and insightfull.
