This experiment is called the three-polarizer paradox. Even if 'paradox' is not the right word for it, this experiment will help us to understand how polarizers work and what happens when you put a 45 degree polarizer inside two polarizers that are 90 degrees apart.
In this section, we will learn what polarizers are and what polarization is. We will try out different combinations of polarizers in the lab section and then see how the maths can be applied to it. The maths will use a new kind of notation called Dirac's notation, which will let us work the maths visually. We will use this new notation throughout all 7 experiments.
Polarizers are devices that only let light pass if it has a certain orientation (called polarization) and block all light that has a different polarization. If you place two polarizers at 90 degrees from each other, no light will pass the second polarizer. If they are both in the same orientation, then all the light that exits the first polarizer will pass through the second one.
Polarizers are very useful in optical setups. They are called polarizers because they can select photons of a certain polarization, and they also rotate the polarization of photons so that they exit with the same polarization as the polarizer. To understand this, you need to know that a polarizer at 0° would block half the photons in a beam at 45°. The other half that passes, will become polarized at 0°.
Now, we can understand the 3-polarizer paradox. If you place two perpendicular polarizers, no light will pass through them. But, if you place a third polarizer at 45 degrees between the first two, light will be able to pass. At first, this seems strange, because the first polarizer blocks vertical light, the second one at 45 degrees blocks some of that vertical light, and the third one still doesn't allow any horizontal light. In fact, the outgoing photon from the first polarizer will be horizontal. The second polarizer will then block half the photon, because it is at 45 degrees. However, all the outgoing photons will rotate to be at 45 degrees. The 45 degree beam will then reach the vertical polarizer, and again, half the photons will pass through it.
Now that you know better how the experiment works, have a play with this setup and try the figure out what is happening in other scenarios. Note that we considere a beam with an initial diagonal polarization. You can change the number of polarizers in the control panel and change their orientation by dragging the pink dot around. The percentage at the right shows the amount of light that reach thedetector, compare to the one emits by the laser at 45 degrees. If it says 100%, it means no light gets block at all. If it is 0%, then it is complet darkness at the end.
The prove you that animation is not lying, we built the setup in a lab. Let's have a look at the graph we made. Remember the initial light beam(coming out of the laser) is considered polarized at 45 degrees.
Tis table was made using the data we collected in the lab. First, we have the intensity of the initial beam, without any polarizers. The number of photon is the photon counted in the middle of the detector (which gives the maximum intensity of the beam). The four next values are measured with 1 polarizer, placed at different angle compared to the initial beam.
(still need to finish this paragraph....)
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.
On the left side, the laser emits a beam of photons. We then place a polarizing beam splitter to rotate the polarization of the beam at 45 degrees. Depending of the orientation of the beam splitter, you might not get 45 degrees, so you can add a half-wave plate to rotate it in the orientation you want: 45 degrees for us. Then, a set of lenses can be use to concentrate the beam in the dectector. The beamis now set to pass through all the polarizers we want. We tested with 0, 1, 2 and 3 polarizers. Finally, on the right-side, the detector is placed at the center af the beam.
You can download all our data here:
The pattern is created by moving the detector one side of the laser beam, to the other. At each step of the detetor, we count the number of photons captured in the time allowed (generally 1 or 2 seconds).
All of the acquisition can be found in those csv. They all have 3 columns: the position of the detector, the angle of the polarizers and the number of photons detected.
In the next sections, we will explain all our results with maths. We choose to use Dirac notation for all the experiment because it is super visual and it does not need any maths background to understand it. The math will stay very simple and visual. You only need to know how to add and multiply to fallow the steps.
With Dirac notation, we will use the "ket" symbol the represent the polarization of a photon. We can have vertical polarization $\ket{\updownarrow}$, horizontal polarization $\ket{\leftrightarrow}$, diagonal polarization $\ket{\diag}$ or anti-diagonal polarization $\ket{\adiag}$. Because the photons move in a wave motion (going left to right to left to right, etc), we can use double-sided arrow to show the polarization. It means a left polarization is the exact same as a right polarization.
In quantum, the letter "psi" ($\Psi$) is usually use to describ a quantum state, just like polarization. So let's write a state like this:
Now, the state needs to evolve throughout the experiment. To do that, we can define a linear operator that will dictate how the state evolve. For polarizer, let's call the operator P and denote it's polarizasing direction with an arrow P$_\diag$.
First, if the polarizor and the polarization of the incident photons are in the same direction, nothing should happen to the initial state.
If the polarizor is perpendicular to the initial beam, there is no light that should pass. BUT, the photons does not magically dispear from existance, they get absorb or reflected by the polarizor. They still exist, just not in our experiement anymore. We will denote those lost photon as $\ket{lost}$.
Even if the "lost" term cannot be detected anymore, we need to keep the "lost" part in our equation so the state stays normalized.
Now, if we have an angle that is not vertical or horizontal, we can always write it as a linear combination of both state. It will be way much easier to apply the polarizor on the state. The only rule is you always need to have a normalized state.
The results givin in this table are state of the photons. But we don't get to measure the state, we can know if a photon is detected or not. We will talk more about the detected intensity, because we have multiple photons at the same time.
The have the intensity, we need to take the complex conjugate. The word itself seems very complex, but dont worry, in our notation, it only means you multiply the state by itself. You also need to flip all the $\bra{}$ into $\ket{}$ and vice-versa.
For exemple, the complex conjugate of the state $\psi = \frac{1}{\sqrt{2}} (\ket{\updownarrow} + \ket{\leftrightarrow})$ is $\psi = \frac{1}{\sqrt{2}} (\bra{\updownarrow} + \bra{\leftrightarrow})$
Whenever you have a "bra" in front of a "ket", it is called a "braket" and you need to multiply both together. Mutliplying to state is generally called a scalar product (or a dot product).
To do a scalr product, you can either take the cosine of the angle between the two vectors, or you can decompose both vector horizontally and vertically. Note that a scalar product give a number, not a state.
Here is a table with the results of the most common scalar product we will work with.
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Now, what is the 2 angles are not parallel nor perpendicualar? There is two methods. First, we can decompose an angle in a horizontal part and a vertical part, then apply what we already know. Second method, we can take the cosine between the angle.
First method: let's say we want to apply a horizontal polarizer on a diagonal state, we will need to transform the diagonal in horizontal and vertical. Diagonal hide that half the photons are vertical, half horizontale.
*Note: we need to put $\sqrt{2}$ in front of each term, because the state need to be normalized. We will get to it very soon.
Once the state is split, we can apply our polarizer on each:
Now, let's see the second method with the cosine,which will lead us to the same result. By the way, this method will be way more usefull if we have weird angles. ike 30 degrees. How would you separate it in vertical and horizontal? The calculs may be tricky.
In this method, we can say the result of applying a polarizer is taking the cosine between the incident photon angle and the polarizer angle. Let's keep it general: let's say the photon come with a polarization of 45 degrees and the polarizer as and angle of P$_1$
Full approach
1. Let's start with the initial state as a diagonal polarization
2. Let's add the first polarizer, which is at an angle of 0 degree.
3. We can now add a second polarizer, with an angle of 45 degrees.
4. Now, let's add the third polarizer, with an angle of 90 degree.
We now have found all of the state of the photon for each place in the setup. Their one final thing we haven't say about the Dirac notation. A state does not represent anything physically. It only represent aquantum property of the photon. When we want to have something that can be measured, we need to multiply the state by itself, with atiny twist. In quantum, they called it the complex conjuagte. It means to multiply ot by itself, but all the $\ket{}$ becomes $\bra{}$ and vice-versa and the result gives the intensity of the resulting beam (I).
5. Let's do it for the first state we found
6. Let's do the same thing with the second state
7. Again, same thing with the third state
These are the actual measure theorical measure we are supposed to find. With only the first polarizers, the intensity should be cut in half. The second one also cut it in half (intensity is now one quarter) and so does the third one (intensity of one eighth). It is not random that they all cut the intensity in half, because they are all at 45 degrees to one another. We could do the same calculs with arbitrary angle, and we would find something different, as it is shown in the animation at the top of the page.
If you are curious of the results for other angles than the three previous one, you can do the exact same steps with any angles. We can also keep the angles as variables, and find a general equaton that will work at any angles.
The result for that general equation goes as fallow, where P1 is the angle of the first polarizer, P2 the second one and P3 the third.
For the complet step-by-step solution to find this equation, please refere to this document. There is also more details on the Dirac notation and on the previous development.
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If you want to see the three polarizer paradox from your own eyes, here are some Youtube video we found to be accurate and insightfull!
