## Saturday, September 06, 2014

### What is the Casimir effect?

This is another in an occasional series of posts where I try to explain some physical phenomena and concepts in a comparatively accessible way.  I'm going to try hard to lean toward a lay audience here, with the very real possibility that this will fail.

You may have heard of the Casimir effect, or the Casimir force - it's usually presented in language that refers to "quantum fluctuations of the electromagnetic field", and phrases like "zero point energy" waft around.  The traditional idea is that two electrically neutral, perfectly conducting plates, parallel to each other, will experience an attractive force per unit area given by $\hbar c \pi^{2}/(240 a^{4})$, where $a$ is the distance between the plates.  For realistic conductors (and even dielectrics) it is possible to derive analogous expressions.  For a recent, serious scientific review, see here (though I think it's behind a paywall).

To get some sense of where these forces come from, we need to think about van der Waals forces.  It turns out that there is an attractive force between neutral atoms, say helium atoms for simplicity.  We are taught to think about the electrons in helium as "looking" like puffy, spherical clouds - that's one way to visualize the electron's quantum wave function, related to the probability of finding the electron in a given spot if you decided to look through some experimental means.  If you imagine using some scattering experiment to "take a snapshot" of the helium atom, you'd find the two electrons located at particular locations, probably away from the nucleus.  In that sense, the helium atom would have an "instantaneous electric dipole moment".  To use an analogy with magnetic dipoles, imagine that there are little bar magnets pointing from the nucleus to each electron.  The influence (electric field in the real atom; magnetic field from the bar magnet analogy) of those dipoles drops off in distance like $1/r^{3}$.  Now, if there was a second nearby atom, its electrons would experience the fields from the first atom.  This would tend to influence its own dipole (in the magnet analogy, instead of the bar magnets pointing on average in all directions, they would tend to align with the field from the first atom, rather like how a compass needle is influenced by a nearby bar magnet).   The result would be an attractive force, proportional to $1/r^{6}$.

In this description, we ignored that it takes time for the fields from the first atom to propagate to the second atom.  This is called retardation, and it's one key difference between the van der Waals interaction (when retardation is basically assumed to be unimportant) and so-called Casimir-Polder forces.

Now we can ask, what about having more than two atoms?  What happens to the forces then?  Is it enough just to think of them as a bunch of pairs and add up the contributions?  The short answer is, no, you can't just think about pair-wise interactions (interference effects and retardation make it necessary to treat extended objects carefully).

What about exotic quantum vacuum fluctuations, you might ask.  Well, in some sense, you can think about those fluctuations and interactions with them as helping to set the randomized flipping dipole orientations in the first place, though that's not necessary.  It has been shown that you can do full, relativistic, retarded calculations of these fluctuating dipole effects and you can reproduce the Casimir results (and with greater generality) without saying much of anything about zero point stuff.  That is why while it is fun to speculate about zero point energy and so forth (see here for an entertaining and informative article - again, sorry about the paywall), there really doesn't seem to be any way to get net energy "out of the vacuum".

DaveC said...

It seems uncertain whether the Casimir force, between two solids in vacuum, has ever really been measured. I know someone who collaborated with Steve Lamoreaux on it who is very skeptical. That person says none of the teams who tried allowed properly for electrostatic effects, and they don't trust each others' results. Anyone have more information?

Douglas Natelson said...

DaveC, a college classmate of mine (and fellow postdoc at Bell back in the day) worked on this. His most well known paper on this is here. I've always found it pretty convincing, to the extent that I have included it as an example in my forthcoming nano textbook. Figure 2 in that paper shows the 1/d dependence of the electrostatic forces that has to be nulled away. Figure 3 shows the distance dependence of the force when the electrostatic contribution has been nulled, and the residuals when comparing to a fit to the expected Casimir form. Error bars get a bit wonky below 100 nm separation, but it looks pretty good to me. If anyone has more detailed information on a controversy about such measurements, please feel free to post.

Anonymous said...

Well, Doug, I was all set to go out and buy the textbook, before you undermined all of your credibility with the following:
" it's electrons would experience the fields from the first atom."

Really? An apostrophe in a possessive for "it"? I guess Princeton's standards just aren't what they once were.

:)

Douglas Natelson said...

Autocorrect. That's my excuse and I'm sticking to it.