## Posts Tagged 'quantum measurement'

### Quantum mechanics: a polarizing topic

These 3D glasses (from an exhibition on Egypt at the Virginia Museum of Fine Arts, which explains the motif) have polarizing filters in each eye. If you look closely, you can see that one eye is darker than the other; that’s because the light from the computer monitor is polarized in such a way that it passes through one filter but not the other. [Credit: moi]

Culturally, we have an odd relationship with quantum mechanics. Writers of popular accounts often like to emphasize the strangeness of quantum physics, talking about Schrödinger’s cat, entanglement, and the like. However, nearly every aspect of modern technology involves quantum physics on some level: computers (which are present in most cars to run the fuel injection system), smartphones, lasers, all types of television and computer screens, even fluorescent and LED lights. To put it another way, the technological applications of quantum physics are so ubiquitous that we often forget they’re quantum, saving that term for the odder and harder-to-understand aspects of the field.

However, it’s really important to remember that physicists and chemists do understand quantum mechanics pretty well. Sure, there are intricacies of interpretation, and entanglement particularly can lead to headaches even for experienced scientists. For that reason, I thought it would be interesting to look at one of the simplest systems we know to highlight both how quantum physics works and why it’s strange.

The specific system I want to discuss is the polarization of light, which shows up in a wide variety of contexts, from sunglasses to 3D movies to LCD monitors on computers.

### Polar opposites

Most explanations of polarization don’t get into quantum physics, since it’s fairly easy to describe the phenomenon using classical (non-quantum) electromagnetic theory. In this picture, light is purely a wave made up of oscillating electric and magnetic fields, perpendicular to the direction the light is propagating. The polarization is just the orientation of the electric field part of the wave; see the video below for a typical visualization.[1]

I modified VPython code from Rob Salgado (available here) to create this movie. The wave moves from left to right; the blue arrows represent the electric field, while the red stand in for the magnetic field. You can see that while the fields reverse direction, they remain along a single axis. That represents linear polarization, with the polarization direction defined by the axis of the electric field – in this case, vertical. (There’s a second type known as circular polarization, which is a topic for another day.)

In the classical view, unpolarized light—such as emitted by the Sun or a lightbulb—consists of a mixture of all possible polarization orientations. That’s because the atoms emitting the light act independently of each other. A polarizing filter typically consists of long polymer molecules arranged in parallel strands. Those strands absorb light with polarization perpendicular to the direction the molecules are aligned, eliminating all but one of the orientations of of the light waves. In an ideal filter (the kind that never really exists) half the light is lost in the process.[2]

For me to demonstrate that to you mathematically is pretty straightforward (though it does require using a bit of calculus). The problem with the pure wave view arises when we want to think about photons: individual particles of light. If a single photon from an unpolarized light source is fired at an ideal polarizing filter, it’s not possible in advance to know its probability of passing through, no matter which way the filter is oriented. However, if you send enough photons at the filter one at a time, about fifty percent of them will get through to a detector on the far side. So what’s going on?

### Quantum state of mind

Looking at things quantum-mechanically, photons (like any other particle type) are described by a quantum state. The state is a mathematical bundle of all the relevant physical properties, including position, momentum, energy (which for photons is equivalent to the frequency of light), and spin. In the case of photons, spin and polarization are the same thing, so the polarization of light is completely described by the quantum state.

Thankfully we’ll be able to ignore all the other properties in the state, and just look at polarization by itself.[3] Now is where quantum physics asserts itself. The polarization state of a photon from an unpolarized source is indeterminate: there is no way to know in advance how it is oriented.

Schematic of a photon from an unpolarized light source passing through a single polarizing filter. [Credit: moi]

For simplicity, let’s consider a polarizing filter oriented vertically (relative to the lab floor). Our photon from the unpolarized source is a mathematical mixture of horizontal and vertical polarizations, something known as a superposition. We don’t know in advance how much of each ingredient is in the mixture, and there’s no way to reconstruct it after the fact. That’s because, if it gets through the filter, it now is vertically polarized, and any subsequent experiment acting on it will produce results accordingly. For example, if you place a second filter oriented horizontally, no photon will get through both: any photon surviving the first filter will be vertically polarized, and therefore absorbed by the second one.

If you add a second polarizing filter with perpendicular orientation to the first, no photon can pass through both. [Credit: moi]

From a quantum point of view, the polarizing filter performs a “measurement” on the photon. What that really means is an interaction that selects a certain physical property from the total quantum state: no human intervention is necessary for measurement to occur. In this case, the filter selects the vertical polarization part of the photon’s quantum state; how large that component is determines the probability the photon gets through the filter. (See the next section for the math on how that works.) However, we can’t reproduce the probability experimentally from a single photon — we can only determine if the photon survived or not. Instead, we can infer probabilities statistically from a large sample of photons, allowing us to reconstruct the complete wave character of light.

But we’re not done having fun with polarization! Take our two filters—one vertical, the other horizontal—and slip in a third filter between them angled at 45°. Now we’re performing three measurements on the photon: the first acts on the vertical component of the quantum state as before, creating (if the photon survives!) vertically polarized light. The second filter, however, is neither horizontal nor vertical, so it interacts accordingly with the vertically polarized photon to select the part that’s aligned with it. A photon that manages to get through the first two filters is now polarized at a 45° angle, and therefore has a horizontal component when it reaches the third filter. The result: the addition of the 45° filter allows a photon that was once vertically polarized to pass through a horizontal filter. Some photons manage to get to the detector with those three filters in those configurations, when it wouldn’t be able to without the middle 45° filter in place. If you switched the order of the second and third filter, no light would get through.

Adding a third filter between the two in the previous diagram with orientation between the two allows some transmission of light. [Credit: moi]

In terms of measurement, each filter only acts on the photon as it is when they interact. Each filter cares not (in anthropomorphic language) for the history of the photon, and the photon itself doesn’t “remember” its prior state after each measurement. It’s less that each filter changes the photon than that it selects the bit commensurate with what it “measures” while the rest can’t get through. Each measurement is independent of what went before.

In fact, many natural processes perform measurements, something known as decoherence: a process that simultaneously disrupts the expected quantum behavior and makes things act more like pre-quantum physics predicts. Decoherence is why we don’t have to take quantum entanglement into account in every physics calculation: though many objects start out entangled with each other, decoherence gradually disentangles them until they behave like independent systems. I will follow up this post with another dealing with entanglement and decoherence (unless my schedule eats me again, as it is wont to do).

The 3D glasses in the image beginning the post have opposite polarization filters in them. The film they were designed to view consists of two sets of images, also with opposite polarizations, projected simultaneously onto a screen. Each eye in the glasses therefore lets in just the light from one of those sets of images, and the brain interprets the combination as a three-dimensional shape.

### Just the math, ma’am

The original impetus for this post was a discussion on Twitter in which a few people asked if I could help them understand the math behind quantum physics a little better. Obviously that’s a huge topic, but let’s see how some of the discussion of polarization translates into the way a physicist would calculate it.

First, the quantum state is represented by a “state vector”, which comes in two basic forms: a “bra” $\langle\psi|$ and a “ket” $|\psi\rangle$. They are named that way because in combination they make a “bracket” (if you ignore the missing “c”): $\langle \psi | \psi \rangle$. ($\psi$ is the Greek letter “psi”, properly pronounced “psee”, but usually pronounced “sigh” by Americans for perverse reasons.) A photon’s state vector can be split into horizontal $|h\rangle$ and vertical $|v\rangle$ polarization states. Here’s the ket version:
$|\psi\rangle = a |h\rangle + b |v\rangle$
and the bra version
$\langle\psi| = a^*\langle h| + b^*\langle v|$
where $a$ and $b$ are complex numbers, and the “*” denotes the complex conjugate. If the photons is from an unpolarized source, we don’t know what a and b are, which kinda sucks, but that’s quantum physics for you. However, we can say something about their combination.

The horizontal and vertical polarization states are orthogonal: the bracket of opposite states equals zero. Mathematically,
$\langle h| v \rangle = \langle v | h \rangle = 0$
while
$\langle h| h \rangle = \langle v | v \rangle = 1$
More generally, the bracket of a state vector with itself yields 1:
$\langle \psi | \psi \rangle = a^* a + b^* b = |a|^2 + |b|^2 = 1$
We interpret all of this using the language of probability: $|a|^2$ is the probability the photon has horizontal polarization, while $|b|^2$ is the probability it’s vertically polarized. These probabilities are expressed as numbers between 0 (“ain’t no chance nohow”) and 1 (“right on”); the photon has no other polarization state possibilities, so the total probability it’s to be found in some combination of horizontal and vertical polarization is 1.

So what does a polarizing filter do? The horizontal filter in our earlier example measures just the horizontal part of the photon’s state. We say the filter projects the state onto one of horizontal polarization, and we write that as a quantum operator:
$\hat{P}(h) = |h\rangle \langle h|$
When this operator acts on the quantum state, it selects just the horizontal component:
$\hat{P}(h) |\psi\rangle = |h \rangle \langle h| \psi \rangle = b |h \rangle$
Again, we can’t know what $b$ is, but if we hurl many photons through the horizontal filter, we find that half of them get through. (If anyone wants to know how that works, let me know. It’s not terribly hard, but it’s going to make this already too-long post even longer!)

Now if we add a vertical filter, its projection operator is easy to guess:
$\hat{P}(v) = |v\rangle \langle v|$
Immediately it’s obvious that if you take first a horizontal then a vertical measurement, no photons get through, because
$\hat{P}(h) \hat{P}(v) = |h\rangle \langle h| v\rangle \langle v| = 0$
No need to even look at the quantum state!

Finally, what about that 45° angle filter? The polarization state for that is
$|r\rangle = \cos(45^\circ) |h\rangle + \sin(45^\circ)|v\rangle$
(where I’m meaning “r” as “rotated”) so the projection operator is
$\hat{P}(r) = |r\rangle\langle r| = \cos^2(45^\circ) |h\rangle\langle h| + \sin^2(45^\circ) |v\rangle\langle h| + \cos(45^\circ) \sin(45^\circ) \left( |h\rangle\langle v| + |v\rangle\langle h|\right)$
That’s not as complicated as it looks: any photon successfully passing through the vertical filter is vertically polarized, so we only need to worry about the bra vectors containing “v”:
$\hat{P}(r) |v\rangle = \sin^2(45^\circ) | v\rangle + \cos(45^\circ)\sin(45^\circ) | h \rangle$
Even though we started off with no horizontal polarization, passing the photon through the 45° filter acts on the quantum state to give it some!

Now I realize that went by very quickly; a description in a textbook would necessarily be more thorough. Hopefully this gives you a flavor of the math involved in quantum physics, though.

### Notes

1. A mathematical aside for those interested: the electric field of light is a vector, meaning a mathematical object that points in a particular direction. However, because light is a wave, the electric field flips by 180°, so that’s pointing the opposite direction. That means polarization isn’t strictly a vector; instead, it defines an axis. To put it another way, if the electric field starts pointing along the +x-axis, after a little while it will be pointing along the -x-axis, so we say the polarization direction is just “x”.
2. Technically this is true only if the light strikes the filter directly head-on; if the beam comes in at an angle, the amounts change accordingly.
3. This is an idealization, of course: polarizing filters have finite thickness and are not perfectly pure. That means there can be extra scattering, reflection, and absorption that are not described by the simple stuff I’m discussing here.