IQIS Seminar: Triple slits and Born’s rule
Urbasi Sinha recently gave a seminar talk at IQIS about her test of the Born rule using a triple-slit setup, as well as an implementation of a single-qutrit system using the same setup. The bulk of the talk was based on an arXiv posting, and while it contains no earth-shattering results (Born was right, probably!), the motivation and experiment are quite interesting.
Born's rule is so ubiquitous in quantum mechanics that I had actually forgotten it had a special name and had to look it up before the talk; it's the rule which tells us that the probability distribution for a quantum system is given by |Ψ|². We know of course that this rule holds very well in a wide variety of situations, but then we know the same about Newton's laws, and look what happened there.
The fundamental principle being examined in the experiment is a rule for describing interference terms when combining probabilities, and was originally discussed by Rafael Sorkin in 1994. The original double-slit experiment demonstrates the wave nature of light (and other particles) by showing that the probability distribution resulting from a photon's passing through two slits is not equal to the sum of the probability distributions of having passed through one of the slits but not the other. The difference between these two possibilities is the interference term,
I(A,B) = P(A,B) - P(A) - P(B).
That is, the interference arising when slits A and B are both open is given by adding up the distributions when the slits are each open on their own, P(A) and P(B), and then subtracting the distribution arising when they are both open. If there is no interference, then I(A,B) vanishes. This is what we naïvely expect with our classical intuition for a single particle, and it is the fact that this is not the case (among other things) that gave rise to quantum mechanics.
What if there are three slits though? The corresponding rule is
I(A,B,C) = P(A,B,C) - P(A,B) - P(A,C) - P(B,C) + P(A) + P(B) + P(C).
If we assume that Born's rule is correct, then I(A,B,C) = 0. Running an experiment in which single quantum particles are sent through a three-slit apparatus can tell us what nature yields for I(A,B,C), and if it doesn't equal zero then something new and different is going on!
As alluded to earlier of course, no such violation was found. As far as they were able to determine, Born's rule does indeed hold. An interesting side note is that after the result appeared in Science last year (arXiv listing here), Max Born's son (in his nineties) wrote to the experimenters to thank them for their effort to prove his father's decades-old theoretical prediction.
Separating figures from a LaTeX document
Once again, I've run into something that shouldn't be an issue with a submission to PRA, but which nevertheless is. I generated some plots in Mathematica, but decided to label them with LaTeX. I thought this would be standard, given that it allows for a consistent font throughout the document, but the editors have said that using "commands in the manuscript source file to include axis labeling or other such content to your figures is a problem for us". Whatever the reasons for this, you can't argue with it if you want your manuscript published.
So, can you quickly extract each figure, complete with added labels, into a set of graphics files to be re-included in place of the original figures? Indeed you can! The first step is to use the preview package:
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
When you compile with this package active, every instance of the PreviewEnvironment will be rendered on its own page, and anything not in that environment will be suppressed. The tightpage option makes each page of the resulting file just big enough to contain the bounding box of its contents.
One warning: LaTeX labels get lost here, since they generally occur outside of the environment being extracted. This is important if you are, for example, placing a sub-figure label on the image. The first time you compile with preview active everything will look all right because the aux files will still remember the labels from the previous compile. If you start noticing `??' where you expect `(a)', just deactivate the preview package, recompile until things look right, then reactivate it.
The next step is to split the resulting file into a sequence of files, one per page in the original. My first attempt was to split either the postscript or PDF file using a command like
(Not the solution) pdf2ps allfigs.pdf fig%02d.ps
to split allfigs.ps into fig01.ps, fig02.ps, .... Unfortunately, this rasterizes the text, without even antialiasing it, and the result just doesn't look quite right. I was hoping to simply switch to pdf2eps or pdf2pdf for better luck, but at least on my Mac OS X machine these don't exist.
Instead, I found a page detailing a quick solution using Automator that spits out each page of a PDF into its own new PDF. The final step was simply to replace all of my lovely TikZ code with simple \includegraphics commands.
UPDATE: It turns out that the Automator solution trims the images a little too closely, and parts of the output PDFs are missing around the edges. I tried several other scripts and online PDF splitters, but ran into one of two problems with each. Either (a) the "split" PDFs contained only the first page, regardless of the output range asked for, or (b) the resulting PDFs were ever so slightly cropped, as with the Automator script. Finally I downloaded a program, PDFsam (PDF split and merge) which did the trick.
Finally, it turns out that I had to convert each of the individual PDF files to EPS format for submission to PRA. I accomplished this quickly and easily with a nice little script that runs Ghostscript.
Moustachioed Mayor Nenshi
OpenFile Calgary is looking for pictures of Calgary Mayor Naheed Nenshi sporting digitally added moustaches. Their goal is to draft him into growing a moustache for Movember, and I decided to try my hand at coming up with an entry.
I used a cc-licensed picture posted by flickr user 5of7, and have likewise licensed this minor derivative work, for whatever it's worth, under a Creative Commons Attribution-ShareAlike 2.0 Generic License.
You can find this and the other images people have submitted on the OpenFile Moustached Nenshi, a sampling page.