QM 101: Fun with Photons

Last time I explored the quantum spin of photons, which manifests as the polarization of light. (Note that all forms of light can be polarized. That includes radio waves, microwaves, IR, UV, x-rays, and gamma rays. Spin — polarization — is a fundamental property of photons.)

I left off with some simple experiments that demonstrated the basic behavior of polarized light. They were simple enough to be done at home with pairs of sunglasses, yet they demonstrate the counter-intuitive nature of quantum mechanics.

Here I’ll dig more into those and other experiments.

Let’s start by defining the notation:

\gamma_{in}\to[\theta]^{\rho}_{\eta}\to\gamma_{out}

The left γ (gamma) is a photon going into the filter(s); the right one is a photon leaving. Each filter consists of square brackets enclosing an angle, θ (theta), that indicates the filter’s linear polarization angle. (Note that the incoming photon has unknown and random polarization.)

Following each filter brackets is, upper, a probability, ρ (rho), and, lower, a total, η (eta). The probability is the likelihood of the photon passing this filter. The total is the overall probability the photon has passed all filters so far.

Let’s consider the simplest possible example:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to\gamma_{out}

The photon enters a single filter set at 0°. Because its polarization is random, it has, on average, a 50% chance of passing through the filter (the upper number). As there is only one filter, the total probability at this point is also 50%.

Thus, on average, 50% of the photons that enter this experiment also exit. For a linear polarizing filter receiving unpolarized light, that’s exactly what we’d expect. (“Unpolarized” light is light in which the photons — which each have definite polarizations — are randomly and evenly mixed, so the light is a blend.)

What we want to note, though, is that, regardless of the photon polarization before it entered the filter, now it has linear polarization at 0°. We confirm this with the first experiment I showed last time:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{0}^{\circ}]^{1.0}_{0.5}\to\gamma_{out}

Things are as just described for the first filter. We find that a second filter (or a third or fourth), if also set to 0°, still allows a total of 50% of the photons through the system. (The lower number of the last filter is the output probability.) Apparently the second (or third or fourth) filters are passing 100% of the photons they receive.

That only happens when all photons are polarized at the same angle as the filter. Recall that the probability of passing is the cosine-squared of the angle between the photon’s polarization and the filter. The first filter (we’re assuming) sets the photon’s polarization angle to 0° (from now on just referred to as photon angle).

With the second filter set at 0°, the difference is zero, and:

\rho=\cos({0}^{\circ}\!)^2=1.0

Theory says 100%, and all the photons do pass, experimentally. We can further show the first filter changes the photon angle with the second experiment from last time:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{90}^{\circ}]^{0.0}_{0.0}\to\gamma_{out}

Now no photons make it through the second filter. Assuming photon angle is set to 0° by the first filter, now the difference is 90° so:

\rho=\cos({90}^{\circ}\!)^2=0.0

Theory says 0% and, again, experiment matches. We’ve set the second filter to 0° and to 90° and gotten 100% and 0% respectively. What about halfway? What if we set it to 45°?

The difference is 45° so:

\rho=\cos({45}^{\circ}\!)^2=0.5

Theory says half of the photons with an angle of 0° pass a filter set to 45°, and experiment agrees:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{45}^{\circ}]^{0.5}_{0.25}\to\gamma_{out}

We see 25% of photons pass through both filters (50%×50%). Those that do have an angle of 45°. We could prove that with a third filter the same way we did with two:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{45}^{\circ}]^{0.5}_{0.25}\to[{45}^{\circ}]^{1.0}_{0.25}\to\gamma_{out}

We could also test differences of 90° and 45° as we did above, and we’d get results showing the output photon angle is 45°.

These experiments all demonstrate that photon polarization does change as a result of the photon interacting with the filter. (Of course it does. That’s the whole point of polarizing filters. To change the light.)

§

Things get more interesting with the third experiment I showed last time:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{45}^{\circ}]^{0.5}_{0.25}\to[{90}^{\circ}]^{0.5}_{0.125}\to\gamma_{out}

Although the [0°]>[90°] experiment allowed no photons to pass, adding a third filter between the two filters allows 12.5% of the input photons through. (And those that do have an angle of 90°.)

That should seem counter-intuitive. Adding an additional filter should, if anything, reduce the light even further, right? Isn’t that how filters normally work?

Not polarizing filters. They’re better seen as light changers, not light reducers. (It’s just that many of the changes involve reduction.) As it turns out, in this case, the more we add, the more light we can let through.

Let’s try adding two filters between the first and last:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[{30}^{\circ}]^{0.75}_{0.38}\to[{60}^{\circ}]^{0.75}_{0.28}\to[{90}^{\circ}]^{0.75}_{0.21}\to\gamma_{out}

This four-filter setup allows 21% of the light through. More filters, more light!

The notation gets crowded with lots of filters. Since, at least for now, each filter after the first has the same angle of difference to the previous, we can compress such multiple secondary filters like this:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to\!\left[3\!\times\![{30}^{\circ}]^{0.75}\right]^{0.42}_{0.21}\!\!\to\gamma_{out} 

Which is the same setup as the one just above: three secondary filters, each 30° different from the photon that strikes it. The inner upper number is still the probability of the photon passing one filter (here 75% for an angle of 30°). The outer upper number is the probability of a photon passing this bundle, and the lower number is still the total output probability for all filters so far.

The math is:

\eta={0.5}\times({0.75}^{3})={0.5}\times{0.421875}={0.2109375}

(Remember that the three secondary filters start with only 50% of the original photons because of the first filter.)

Let’s use a lot more filters. Rather than a 30° difference, let’s try a difference of 10° from filter to filter. It’ll take ten filters total to rotate the polarization from 0° to 90°. The first one to give us linear polarization at 0° and then nine more after that, set progressively at 10°, 20°, 30°… 90°. In compressed notation:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to\!\left[9\!\times\![{10}^{\circ}]^{0.97}\right]^{0.76}_{0.38}\!\!\to\gamma_{out}

A whopping 38% of the photons come through this 10-filter stack. The nine secondary filters pass almost 76% of the photons. (Keep in mind that the highest we could achieve would be 50% because the first filter only passes half the photons.)

Doubling the filters (shifting by 5°) gives us:

\eta={0.5}\cdot\!\left((\cos({5}^{\circ})^2)^{18}\right)={0.5}\cdot\!({0.9924}^{18})={0.5}\cdot{0.87175}={0.43588}

Now we get over 43% of the photons. Let’s take one more jump and use 90 secondary filters, each shifting just 1° (so 1°, 2°, 3°,… 90°):

\eta={0.5}\!\cdot\!\left((\cos({1}^{\circ})^2)^{90}\right)={0.5}\!\cdot\!({0.9997}^{90})={0.5}\!\cdot\!{0.97295}={0.48648}

Which is pretty surprising. If one filter allows 50%, and two, with the second at 90° allow none, how is it that 91 filters allow over 48%? (So close to the theoretical max that it’s almost as if those 90 filters aren’t there.)

An important note: This analysis assumes perfect polarizing filters. Those don’t exist in the real world, so these numbers are theoretical. Actual experimental results are close but not exact. For one thing, passing through 91 filters made of pretty much anything is going to absorb some photons. On the other hand, sometimes photons with the wrong angle pass through anyway. (Gate crashers!)

The real world is messy.

§ §

These experiments only begin to show some of the unusual results possible with photons. For instance, check out the Elitzur-Vaidman bomb tester. It and myriad other experiments leverage the ability of photons to interfere with themselves. (The LIGO gravity wave detector, for instance, depends on such interference.)

The season finale of this series (so to speak) is a discussion of Bell’s Inequality and the experiments that test it. Many of those experiments use polarized photons in setups not entirely unlike the ones shown above. More precisely, they involve gathering and analyzing data about when entangled pairs of photons pass (or don’t pass) through various filter angles given the assumption both photons initially have the same polarization state.

Part of understanding those experiments requires a good understanding of how photons interact with polarizing filters. We already know the basic theory:

\gamma_{in}\to[{0}^{\circ}]^{0.5}_{0.5}\to[\theta]^{\cos(\theta)^2}_{0.5\cos(\theta)^2}\to\gamma_{out}

There is always a first filter (or other method) to create linearly polarized photons in a known state. Here I’ve used a filter set at 0° knowing that any photons that pass through (on average 50% of those striking it) will have a linear polarization at 0° (plus a few gate crashers, and minus a few legit that got caught in the filter).

Given the following data points (as discussed above)…

  • At θ=0° the probability of passing is 100%
  • At θ=45° the probability of passing is 50%
  • At θ=90° the probability of passing is 0%

…it might be tempting to assume a linear relationship between the angle and the probability of passing through. It’s tempting to draw a straight line from 0% to 100%, since that line would pass through 50% just fine. It seems reasonable that, if a certain setting gives X% probability, doubling that setting should double the probability. That’s certainly what we see when we go from 45° to 90°.

Chart 1. Linear (black), cosine (blue), and cosine-squared  (red) curves.

But data points from the in between angles tell us that the probability of a photon passing is the cosine-squared of the angle between the photon’s angle and the filter’s angle.

Not at all a linear relationship. That’s very apparent in Chart 1, which shows both the cosine (blue) and cosine-squared (red) curves in comparison to the linear one (black).

It also shows how the red and black curves intersect at only three points, which leads to the sense of a linear relationship.

Indeed, in comparison to the blue cosine curve (which is 1/4 of a sine wave), the cosine-squared curve is much flatter and reasonably does approximate the linear curve. It’s the small differences in the high and low regions that distinguish quantum behavior from linear behavior.

Bell tests use filters in those regions since that’s where quantum behavior stands out. For instance, filters set at 22.5° (halfway between 0° and 45°) are a common setting.

§ §

More on this in the future!

(It could be a while. I got sidetracked by other things, science fiction and mystery books, mainly. And I got a bit overwhelmed by how much there is to learn with QM. First so much math to learn, and then the QM stuff. I had to take a break for a while, but I’m trying to get back into it now.)

Stay unpolarized, my friends! Go forth and spread beauty and light.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

8 responses to “QM 101: Fun with Photons

  • Wyrd Smythe

    It’s something to look back at the role of light in my life. My parents told me that, as an infant, my second word was “light” (the first was “star”), and that I’d never sleep in the car at night because I was always watching the lights.

    As a child my dad took me to the circus. After the big event, a trapeze act, my dad turned to me to see what I thought of it. I missed it because I was counting the spotlights. “Look!” I said, “There are 27 spotlights!” (My dad told this story many times. I’ve been a geek literally all my life.)

    In high school I discovered a whole new passion: theatre lighting. Duck to water. I’d already gotten into photography and, even earlier, astronomy. All subjects involving light. In general optics was always a topic of interest.

    In college it was TV and film production, and light is important in both of those. (My CS minor, OTOH, didn’t involve light, so there’s that.) And we used to spend a lot of time on the bluff or up in the Hollywood hills smoking weed and watching the city lights (which in Los Angeles are spectacular, at least when it wasn’t too smoggy).

    Over the years I’ve noticed how important light is to me personally. I seem to suffer, at least slightly, from SAD, and I vastly prefer sunny days to cloudy ones (well who doesn’t?). I get pretty bummed out by the short days of winter.

    There is also the metaphorical disinfecting properties of shining light on dark subjects. Illuminate the darkness; drive out the evil! Of course, actual light is healthy, too (but use sunblock).

    And now I’m studying actual photons. 😀

    • Wyrd Smythe

      There is also a memorable event from when I was in high school somewhere around 1971. I went to a local tech faire and not only saw this new-fangled thing called a “laser” but was actually able to stick my hand in the beam. Since most of what I’d read about lasers mentioned their ability to cut through things like steel, that was and experience.

      They’re so common now, but once upon a time lasers were a whole new thing, and I was there to see it!

  • Wyrd Smythe

    Two key points here:

    Firstly, as with measuring spin 1/2 particles on some axis, measuring photons with a polarizing filter changes the wavefunction of the particle. The interaction with the measuring device alters the particle — this is a form of wavefunction “collapse.” This is demonstrated by the experiments that make second or third measurements on the same particle. The information we gather in such an experiment has a different quality compared to single measurements on multiple particles we assume are essentially the same. (For instance, gathering statistics on the probability of a particle to pass through a filter requires many tests with many particles.)

    Secondly, also as with spin 1/2 particles, a measurement invalidates previous measurements. Spin direction and polarization are orthogonal properties, so the Heisenberg Uncertainty relationship obtains.

  • SelfAwarePatterns

    I can definitely understand being sidetracked with science fiction books (or fiction books in general). I’m a bit burnt out on non-fiction stuff at the moment. And work is consuming a lot of the limited bandwidth I do have for it. Reading about writing is consuming much of the rest.

    I’ve often wondered about the relationship of polarity and spin. I’ll need to swing back to these posts when I have capacity to process them properly.

    • Wyrd Smythe

      I was kinda surprised when I found out photon polarization and photon spin are the same thing! Spin is supposed to be so mysterious and non-physical. (And it still is. Photon polarization is a superposition of left- and right-handed spins of the same particle which just seems to emphasize how non-physical spin is.)

      I’ve noted your absence from my quantum posts — I expect it with math or baseball 😉 — but I know how demanding and draining work can be. Or how one needs to just mix it up once in a while.

  • Sai Sundar S

    Oh Lover of light , It is really interesting to know about polarization .And you formulated a filter for Malu’s Law . The regression curves were also informative Thank you for sharing

  • BB #75: Gloves and Shoes | Logos con carne

    […] Speaking of Bell tests, I’ve noticed that science writers often struggle to find a good metaphor that illustrates just what’s so weird about the correlation between entangled particles. Bell tests are complex, and because they squat in the middle of quantum weirdness, they’re hard to explain in any classical terms. […]

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