30 January 2013

An Excursion into Quantum Mechanics

I've always wanted to see a tiny fraction of how quantum physics works, and I found a wonderful lecture series by Leonard Susskind, one of the fathers of string theory on YouTube (link). You might have heard about the double-slit experiment, where two light sources create a complicated pattern on a screen which is immediately destroyed if we "look at" how the light got there; Schrodinger's cat, which is both dead and alive, and about how quantum mechanics gives rise to a new place of consciousness in the world. The idea that there are alternative realities, widely used in SF and in the more mainstream, comes directly from Hugh Everett's 1957 interpretation of quantum mechanics, the many-worlds.

Along the way, I discovered amazing things, but also learned to see the strange quantum-mechanical effects in a different light, in which they make much more sense. I also encountered a very simple experiment that I think shows the strange ways of modern physics, but which everybody can try at home. No lasers, laboratories, expensive equipment needed.

Only three (linear) polarisers. These are half-grey plastic foils, and used to be there in front of and behind all LCD screens in calculators, monochrome handheld games, and aeon-old digital watches. They have a direction, and they only let through light "polarised" in that particular direction. That's why they are not completely transparent - they only let through a portion of the light hitting them.

If you put two polarisers on top of each other, and start to rotate one, at one point they won't let any light through - the area where they overlap becomes black. This is when their directions are orthogonal: light that goes through one is exactly the type of light blocked by the other.

So far, so good, nothing too interesting has happened. But now put a third polariser between the two without moving the first two ones. By rotating the middle one, you should be able to make the blackness disappear, and let some light through! How is that possible? More polarisers should block even more light, shouldn't they?

What actually happens is that light polarised at a given angle has a given chance of getting through a polariser - the closer the match between the angles, the higher the chance. At 0 degrees, when the photon is polarised just in the direction of the polariser, the chance is 100%; at 90 degrees, the chance is 0. (By the way, the photon is indivisable. Either the whole goes through, or none of it.) However, after it emerges from the polariser, it is certain to be polarised along the direction of the polariser. That means that the polariser in the middle, which is at, say, 45 degrees compared to the first polariser, will let some photons through, but then they will be polarised at 45 degrees, not zero - and have some chance of going through the third polariser as well.

Actually, if you think about it, this makes a lot of sense. If a polariser would only let through light that is already polarised in its direction, it would be completely black, as there are so many directions light can be randomly polarised in. I think this is a wonderful example of how our intuition (evolved based on large masses moving slowly) betrays us, and how these strange phenomena can actually be more logical than what we originally expected to happen.

But polarisation is also very useful in other experiments, in one of which a pair of photons is created in a way that we know that they are perpendicularly polarised. Now if we put a polariser in the way of one of the photons, and it goes through (when we then know its polarisation is aligned with the direction of the polariser), then every time, without fail, the other photon also goes through a perpendicularly aligned polariser. Either both go through, or neither. It is as if the first photon could send a message to the other that it should also align its polarisation - only that this message can travel faster than light. Einstein called this "spooky action at a distance," and devised ingenious ways of trying to explain it away (he couldn't). Others called it entanglement.

What's even more astonishing, is that it has been verified experimentally almost completely ruling out any other explanation, that the photons did not decide in advance whether they should both go through the polarisers or not. It is done based on Bell's inequality.

So when the pair of photons is generated, we don't really know how they are polarised - they are in a superposition of possible polarisations. When the first photon goes through a polariser (it is measured, in a way: it either got through or it didn't), those possibilities collapse, and we either know that the first photon got through and so will the second one, or know that the first photon didn't get through, and neither will the other.

This is what is called wavefunction collapse, and is, for example, at the heart of destroying the pattern created in a two-slit experiment when the photon is measured at the slits.

But what happens during this collapse? And when does it happen? We only see the photon either going through or not going through the polariser, and so when we measure, there must be a collapse. But what is a measurement? A photon hitting a polariser, in reality, only interacts with its atoms (or, rather, electrons), which get excited, enter various states, emit photons, and so on. A camera, detecting a photon, is also made of atoms. And particles can quite happily hit other particles without causing a wavefunction collapse.

Some, as detailed in the articles linked above, suggested that it is us, or, in less obviously egocentric language, consciousness that causes a wavefunction collapse. I think it is quite clearly nonsense. We don't need anything but atoms to explain the mind and consciousness - and so by Occam's razor, consciousness also consists of atoms. There's a difference in quantity, not quality.

But I was intrigued by how the illusion of the collapse comes about, and the first step was to learn more about entanglement, the process that combines multiple particles (and, ultimately, us) in a wavefunction.


Without really knowing how, I took paper and pen, and tried to figure out how the two photons in the pair I mentioned above are supposed to be described. I failed and had to try again and again, but in the end managed to put together something that looked vaguely promising. Along the way I was also guided by Dr Lvovsky's lecture notes. What follows might very well be completely mistaken, but I thought I'd include my notes here as a souvenir from my journey - much like a photo from a vacation posted on Facebook. And, the results appear to agree with Bell's inequality to boot. Nevertheless, if you know where I've gone amiss and how this should really be done, please feel free to leave a comment and let me know.

And with that, onto the equations. (Unfortunately, I cannot introduce the concepts used here like Hilbert space and bra-ket notation. But if you're new to these, please watch the lecture series by Prof Susskind, linked above.)

I imagine the following experiment: suppose there is a source that emits two entangled photons along the z axis, A and B, both linearly polarized, but in an orthogonal direction. Photon A encounters a linear polariser aligned with the x and y axes and a detector behind it. Photon B encounters a linear polariser rotated by \(\phi\) and another detector behind it. There is some apparatus, C, connected to both detectors. I think after the experiment, the apparatus should be in a superposition of detecting one, the other, both, or neither photons:
\[ |C\rangle = x\left|C_{A\&B}\right> + y\left|C_A\right>+ z\left|C_B\right>+ w\left|C_0\right>. \]
I'm trying to find the coefficients (amplitudes) x, y, z and w. The order in which the photons interact with the polarisers and detectors should not matter.

In isolation, the polarization of photon A can be described in the 2-dimensional Hilbert space \(H_A\). Let the observable associated with the x,y-aligned polariser be P with the following eigenvectors:
\[ P = \begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
~~~|+p\rangle = \begin{pmatrix}1\\0\end{pmatrix}
~~~|-p\rangle = \begin{pmatrix}0\\1\end{pmatrix}. \]
The polarization of photon B can be described in a similar space \(H_B\). Let the observable associated with the \(\phi\)-rotated polariser be \(P_\phi\) with the following eigenvectors (using the equivalents of \(|+p\rangle\) and \(|-p\rangle\) as the basis):
\[ P_\phi = \begin{pmatrix}\cos2\phi& \sin2\phi\\ \sin2\phi& -\cos2\phi\end{pmatrix}
~~~|+\phi\rangle = \begin{pmatrix}\cos\phi\\ \sin\phi\end{pmatrix}
~~~|-\phi\rangle = \begin{pmatrix}-\sin\phi\\ \cos\phi\end{pmatrix}. \]

The (linear) polarization of the entangled photons, if A is polarized in the a direction, can be described in \(H_A\otimes H_B\) as
\[ |\Psi\rangle = \cos a |+p\rangle\otimes|-p\rangle + \sin a |-p\rangle\otimes|+p\rangle, \]
and, using the \(|\pm p\rangle \otimes|\pm p\rangle\) basis, this can be expressed as
\[ |\Psi\rangle = \begin{pmatrix}0\\ \cos a\\ \sin a\\ 0\end{pmatrix}. \]
Now in order to find the outcome of the experiment, I considered \(P\otimes P_\phi\) as an observable in \(H_A\otimes H_B\) (tensor product of Hermitians is Hermitian). Its eigenvectors are the tensor products of the eigenvectors of P and \(P_\phi\), which are:
\[ \begin{aligned}
|+p\rangle\otimes|+\phi\rangle =& \begin{pmatrix}\cos\phi\\ \sin\phi\\ 0\\ 0\end{pmatrix} \\
|+p\rangle\otimes|-\phi\rangle =& \begin{pmatrix}-\sin\phi\\ \cos\phi\\ 0\\ 0\end{pmatrix} \\
|-p\rangle\otimes|+\phi\rangle =& \begin{pmatrix}0\\ 0\\ \cos\phi\\ \sin\phi\end{pmatrix} \\
|-p\rangle\otimes|-\phi\rangle =& \begin{pmatrix}0\\ 0\\ -\sin\phi\\ \cos\phi\end{pmatrix}
\end{aligned} \]

And from these, the amplitudes appear to be:
\[ \begin{aligned}
\langle+p+\phi|\Psi\rangle = \sin\phi\cos a &= x \\
\langle+p-\phi|\Psi\rangle = \cos\phi\cos a &= y \\
\langle-p+\phi|\Psi\rangle = \cos\phi\sin a &= z \\
\langle-p-\phi|\Psi\rangle = -\sin\phi\sin a &= w,
\end{aligned} \]
as we expect both photons to go through the polarisers at \(\phi=\pi/2\) and \(a=0\).

This result also seems to be in accordance with the reasoning around Bell's theorem, which suggests that the average correlation between the two detectors, defined as
\[ \text{Cor} = \frac{
\left(\begin{matrix}\text{number of experiments}\\ \text{showing correlation}\end{matrix}\right)
- \left(\begin{matrix}\text{number of experiments}\\ \text{with no correlation}\end{matrix}\right) }
{\text{number of experiments}}. \]
is not linear, but sinusoid. Based on the above amplitudes, this is
\[ \begin{aligned} \text{Cor} = \sin^2\phi\cos^2a + \sin^2\phi\sin^2a - \cos^2\phi\cos^2a - \cos^2\phi\sin^2a &\\
= \sin^2\phi - \cos^2\phi = -\cos(2\phi)&, \end{aligned}\]
which seems to be OK as we expect complete correlation at \(\phi=\pi/2\) (when the polarisers are aligned orthogonally) and again at \(\phi=3\pi/2\).

In our original experiment, the two polarisers were arranged perpendicularly, that is, \(\phi=\pi/2\). In this case,
\[ \begin{aligned}
\langle+p+\phi|\Psi\rangle &= x = \cos a \\
\langle+p-\phi|\Psi\rangle &= y = 0 \\
\langle-p+\phi|\Psi\rangle &= z = 0 \\
\langle-p-\phi|\Psi\rangle &= w = \sin a.
\end{aligned} \]
That is, it is not possible for only one photon to go through the polariser in its way. Depending on the initial direction of polarisation, \(a\), either both go through with greater probability, or neither.


What have I learned from this? Well, first and foremost, a first-hand experience of what has been described in many places: that quantum entanglement, the "spooky action at a distance," is nothing else but statistical correlation mistaken for causation. That is, that we ourselves become entangled with the first photon when we measure it, and then will necessarily see the second photon as we do without there being any "action". This leads, admittedly, to Everett's many-world interpretation, but what happens to entanglements on a macroscopic scale is a completely different question. Chapter 2 of Dr Lvovsky's lecture notes contains some indications as to how this could be described. Usually, it seems, data loss is inevitable, which leads to the (observed) collapse of superpositions.

My second observation partly follows from this problem that arises with many particles, and partly from some interesting facts I encountered. For example, although a complete Hilbert space is used to describe states, only unit vectors represent physical states. And even of those, a single phase (basically, a single direction of the many that define a vector) is irrelevant. There might very well be completely consistent and good reasons for these limitations that I'll learn later. But at the point it seems that the mathematical model does not fit the physical reality perfectly. It does not seem to be beautiful, however successful quantum mechanics is at describing and predicting phenomena. (We shouldn't forget how successful classical physics was in its own day.)

And then there is our rapidly evolving view of the Universe. It seems to constantly surprise us. We now need dark matter, then dark energy to make ends (and the two sides of equations) meet.

Because of these, I acquired a feeling that this is not all of the story. Maybe it's string theory. Maybe Stephen Wolfram is right, who suggested that ultimately, we won't be able to create equations that will predict nature at any point in the future, effectively shortcutting the calculations nature makes to arrive at her results. Maybe we can only simulate these calculations, but not outwit them.

And if so, I think it is premature to ask the big questions: are there alternative worlds? What will happen to the Universe? How come the Self, the I, is such a singular point in space-time, while nothing physical makes it stand out from the rest of the collections of particles?

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