Quantum polarization, Dirac notation and erasure – Quantum Basics 3 of 3

Quantum Basics 1 of 3
Quantum Basics 2 of 3
Quantum Basics 3 of 3

Quantum mechanics is easy to test for yourself in this day and age, even down to the strangeness of quantum states. While quantum states can be very complicated there are a few examples, that despite their weirdness are simple enough to explore. What is a quantum state and how might you explore them? – well in this third installment of quantum basics I tackle just that sort of problem.

Please note: if you are having trouble seeing the interference pattern at the end of the video you may need to view it in HD as the effect is very subtle, and just the sort of thing compression destroys.

Materials: At least one pair of “Reel 3D glasses” (you don’t even have to go to a movie I just asked for a few pairs), isopropyl alcohol, toy laser, a light meter (or red LED hooked to a volt meter) , graphite sticks (mechanical pencil refills), card, tape scissors.
Experimental steps: In this video I do two experiments: the first, which I’ll call “polarization states” is more quantitative where you can actually write out the predicted values for the quantum system and compare them to measurement, while the second, which I’ll call “Quantum erasure” is just barely on the edge of being visible with the equipment used and is much more about the qualitative relationships.

Polarization states

  1. First you will need to remove the lenses from the 3d glasses. They pop out pretty easily as shown in the video.
  2. The lenses are made out of two layers, one is completely clear and the other is sort of grayish. When you look from the side you can just make out the seam between them. You will need to peel these two pieces of plastic apart for the experiment to work properly. There are little corners on the lens to help the plastic stay in the frame, these are a good place to start peeling. Also using an Xacto or other small blade to get things started can be helpful.
  3. Once peeled apart the two pieces will have a sticky adhesive on one side of them. I tried using Acetone, which worked a little but isopropyl alcohol worked much better for loosening the stickum from the two polarizer pieces. I used cotton swabs to rub on the surface then my fingernail to actually scrape the adhesive off the two pieces of plastic. This worked well enough but if you can think of a better solvent go for it.
  4. Now you can probably get away with using just one lens out of the glasses, that is one clear piece (circular polarizer) and one grayish piece (linear polarizer) but it’s a good idea to peel the second lens apart as well.
  5. Once you have your polarization pieces removed from the reel 3d glasses, separated and peeled you are ready to set things up. First you should orient the linear polarizers (the gray ones) parallel to each other. By rotating one of the polarizers about its center, while keeping the two parallel, you should be able to get them to go completely dark.
  6. Once you figured out the correct arrangement make sure to fix them in place by taping them to the table top (or whatever method suits you) but be sure there are at least a few centimeters between the two linear polarizers.
  7. Next attempt to shine the laser through the darkened polarizers. You may get a tiny amount of light from either looking at an odd angle or imperfections in the polarizers but essentially no light will come through.
  8. While shining the laser light at the blackened polarizers take one of the clear looking circular polarizers and slide it in between the two linear polarizers. Look for light that now comes through.
  9. If you have just linear polarizers you can still get this effect by inserting a third linear polarizer at 45 degrees to the other two. Also the math for this one is a little simpler (no imaginary numbers) when multiplying the states with each other later.
  10. Next calculate the expected values for the probability of the passage of a photon at each step along its journey (see math below).
  11. Finally use your light meter (or led with volt meter) to check the amount of light coming through at each stage of the polarizer to compare with the calculation. Generally you should expect less light to have made it through at each stage than the theory predicts since polarizers are imperfect. If there appears to be more light than the theory suggests you are probably measuring ambient light.

Quantum Erasure

  1. First a warning, this experiment is hard to execute and requires an understanding of the double slit from part one of Quantum Basics and a comprehension of the previously mentioned Polarization States experiment.
  2. Start by cutting two steips off of one of your linear polarizers. Make sure to cut these at EXACTLY 90 degrees from each other. If they aren’t cut at 90 the experiment will not work. Each strip should be a long rectangle, if you lay the two pieces on top of each other so the long axis lines up they should block out the light completely.
  3. Next you will need to make a horizontal hole in a piece of card stock, or note card. this hole should be as tall as your polarizer pieces are wide.
  4. Place a single piece of graphite across the middle of the gap and tape it in place. So you have one horizontal rectangular hole bisected by a single piece of graphite.
  5. Next slide your two linear polarizer pieces into the hole on either side of the graphite. It is a good idea to let the polarizer pieces actually touch each other just behind the graphite. Be careful that the polarizers are aligned and tape in place.
  6. Next on top of the linear polarizers place two more graphite sticks on either side of the first, being sure to leave a small gap. Tape in place these as well. Add extra tape to insure that there only two paths a light ray could go. One path should require going through the vertical linear polarizer and the other should require going through the horizontal linear polarizer.
  7. Test your apparatus, under normal circumstances you should see a diffraction pattern when shining the laser light through a double slit (see quantum basics 1). However, if the apparatus was made right you should see no diffraction or interference. This is because each side has been marked, or measured by the polarizer and as such the wave nature of light is hidden. If you do see an interference pattern reexamine your aparatus and adjust as needed.
  8. Once satisfied that there are two paths for the light to travel on and they no longer produce interference you will need to set up a viewing area. You will need to keep the laser fixed on (I used a rubber band this time) and shine it at the double slit with the polarizers you just made. Next set up a viewing screen (piece of white paper does fine) that is over 1 meter away.
  9. Next, with another large piece of the linear polarizer hold it in the way of the light after the slits, this is easiest to do near the “screen”. First try holding the polarizer vertically, then horizontally. You should see two slightly different blobs when you do this, corresponding to the two slits.
  10. Finally tilt the linear polarizer to 45 degrees in relation to the slits and observe. If done carefully enough the interference pattern will return. However, if any of the three polarizer pieces is out of alignment you may not have enough clarity to see the interference.
  11. BONUS: try using one of the circular polarizer pieces used in the Polarization States experiment. I couldn’t get this to work but I think the math works out that it could restore the interference.

Data:

Polarization States, data from the impromptu light meter (LED with volt meter).

DataTable1

Okay so 35% off from the prediction is pretty bad, but I think the issues were with my shoddy light detection. If I do this again I will use a real light meter, or use an infrared LED which would have more sensitivity for the laser light.

Math: In the above video I introduce Dirac notation also called bra-ket notation and it gets pretty in-depth, so let’s get to it. First of all we need the basic objects for bra ket notation, which turns out is a ket.

psi-ket
This is usually called the state of the system but it just symbolizes the particular way the system is behaving at some particular time. Symbolically it “Represents a state vector in the multidimensional Hilbert space” but in real language it just means how the quantum thing is being. We’ll go over a very simple system where there only ever two polarization options for the light (system) to be in, but when you get into position and energy one needs to actually imagine an infinite set of possible ways the system can be. For now we’ll draw the kets that are involved in the calculation I did in the video.
horizontal ket
Horizontal

vertical ket
Vertical

Circular ket
Circular

These are the three states a photon will move through as it traverses the arrangement for the “Polarization States” experiment. Each one of these states has another representation as a matrix. In this case we’re lucky since each matrix only has two elements. Again this is a very simple system, but if one were to imagine a general system containing a position or a momentum the number of entries in the matrix would have to be infinite, which is where all the hard math comes in for much of quantum physics – so many infinities >_< but I digress. For us we can write down the states of the kets like this.

horizontal with matrix
Horizontal

vertical ket with matrix
Vertical

Circular ket with matrix
Circular

There many more states than this and in general they look more complicated (sines and cosines with both real and imaginary parts) but for the specific example I thought it would be best to just consider few special cases. Be that as it may but…well what does this all mean?

Well the matrix representation of the state at any particular point is the map of the particular arrangement a quantum system is in. When a quantum system moves from one state to another (by interacting with a polarizer for instance) we count the amount of overlapping of these maps to find the probability that the system will move from one states to another. Symbolically we can write it something like this.

horizontal ket with circ bra

The technical name for this arrangement is the probability amplitude. That is this is the amount that horizontal light overlaps in it’s map with circular polarized light. We can write this out in matrix notation like this.

Horiz anf circ prob amp matrix
Notice how the left hand side, the circular polarization matrix is written as a row now instead of a column as above, this is because it is in the bra part of the notation. Bras point to the left and kets to the right. When you change a ket into a bra in addition to changing how the matrix is written you need to do what is called complex conjugation. What that means is to simply replace every imaginary “i” with “-i” and vise versa. So that negative symbol should be there (though it doesn’t come up until later). Multiplying the above setup through by normal matrix multiplication rules you wind up with

1 over root 2

now remember THIS IS NOT THE PROBABILITY. This is the probability amplitude. To move from this amount to the actual probability one needs to multiply this by its complex conjugate again. However since this does not have any imaginary “i” in it you just end up squaring it.

1 over root 2 squared

Which of course equals one half. Very often this step of moving from Probability Amplitude to true probability is simply called squaring and for simple examples that’s not that big of a deal, but always remember that there could be i’s lurking around and changing the final answer. This 1/2 probability by the way it the probability that light will make it from the first linear horizontal polarizer to after the circular polarizer in the “polarization States” experiment. To move on to the next step of this probability calculation you need to take this.

circ to vert

converting this to its matrix form you get this.

circ to vert matrix prob amp

This time the “i” is not negative because it’s in the original ket form. The ket is what we start with and we then check the overlap with the vertical bra. Notice how the bra has again been written like a row but this time there is nor sign switch because there were no imaginary numbers. Again calculating through by the normal rules of matrix multiplication we get.

i over root 2

This time we do end up with an “i” in our probability amplitude. That means to get the actual probability we need to multiply this by it’s complex conjugate so we do this.

i times neg i over root 2

which again works out to one half. Now this is the chance for a photon to move from after the circular polarizer out through the vertical linear polarizer. If we take the chance to get from horizontal to circular and multiply it by the change to go from circular to vertical we get the final probability to move through all three polarizers.

half times a half is quarter

One quarter or 25% should get through. Although this is a maximum I got down at about 16% for mine, but still this should impress you because if you take the probability amplitude for just horizontal getting through a vertical you get this.

prob vert vs horiz

Which is equal to ZERO! this is bizarre, light can’t get though until you add something more to interact with?!? Well that is the fun of quantum mechanics.

Now As for the math for the double slit arrangement with the polarizer…that is way beyond the scope of this post but if you’re interested I found this paper really useful.

Linkography:

Polarization of light – quantum basics part 3

https://www.youtube.com/watch?v=o4s28XVfQaU

https://www.youtube.com/watch?v=E-nOXpVPAPs

https://www.youtube.com/watch?v=PDzkRmOmwfA

https://www.youtube.com/watch?v=ycY2mUZHS84

https://www.youtube.com/watch?v=R-6St1rDbzo

Quantum notation:

https://www.youtube.com/watch?v=tKbzlOU_ETk

https://www.youtube.com/watch?v=LBFBQr_xKEM

https://www.youtube.com/watch?v=tKbzlOU_ETk

https://www.youtube.com/watch?v=pBh7Xqbh5JQ

https://www.youtube.com/watch?v=zNMzUf5GZsQ

https://www.youtube.com/watch?v=SeYXM-XBTyk

http://jamesowenweatherall.com/SCPPRG/EllermanDavid2012Man_QuantumEraser2.pdf

Commons Graphics:

“Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View” by Dave3457 – Own work. Licensed under Public Domain via Commons – https://commons.wikimedia.org/wiki/File:Circular.Polarization.Circularly.Polarized.Light_Circular.Polarizer_Creating.Left.Handed.Helix.View.svg#/media/File:Circular.Polarization.Circularly.Polarized.Light_Circular.Polarizer_Creating.Left.Handed.Helix.View.svg

2 Comments

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s