One of the chapters of the book-in-progress talks about neutrino detection, drawing heavily on a forthcoming book I was sent for blurb/review purposes (about which more later). One of the little quirks of the book is that the author regularly referred to physicists trying to "trap" neutrinos. It took me a while to realize that he just meant "detect"-- coming from the AMO community, I naturally assume that "trap" means "localize to a small-ish region of space for a long-ish period of time." That is, after all, what I spent my Ph.D. work doing-- trapping cold atoms.

SteelyKid had a rough morning today, so I'm not quite in the right frame of mind for editing this chapter (which is what I really ought to be doing), but I started to make the effort. And immediately got distracted thinking of the "trap" issue. In particular, I made a mention in the text of the several hundred relic neutrinos from the Big Bang believed to be in every cubic centimeter of the universe, phrased in a way that made it sound like they were just sitting there. Which got me wondering what it would take to get neutrinos just sitting still in some region of space.

Of course, a real answer to this question would require me to know a whole bunch of stuff about neutrino physics that I don't actually know. So in the spirit of students the world over confronted with an exam question they don't know how to answer, I decided to change the question to something I do know how to attack, namely an estimate of the size of the "trap" you would need to have a neutrino sitting more or less still.

This still seems like an impossible problem, but the key word there is "estimate." And as long as you don't want a hard number, I can draw on one of the famous equations that give this blog its name, the Heisenberg Uncertainty Principle:

$latex \Delta x \Delta p \geq \frac{\hbar}{2} $

This says that the product of the uncertainty in the momentum of a particle and the uncertainty in its position must be greater than or equal to a non-zero constant. Thus, it's impossible to know both of those to arbitrary precision.

The main importance of this is as a concept, rather than something to calculate with, but there is one sort of calculation it's frequently used for, which is to estimate the properties of a confined particle. If you know that some particle is confined to a region of width $latex \Delta x $, then you know that there must be some uncertainty in its momentum as well. That means you'll never be sure of finding a trapped particle just sitting still, but you can put a rough limit on the velocity it will have given a particular trapping region. And from that, you can say what the energy of the lowest trap state ought to be, give or take.

So, if we were to confine a neutrino to some region of space, "trapping" it in the AMO sense of the word, what would the velocity be? Because I'm lazy, we'll use the classical approximation for momentum as just mass times velocity (which isn't as bad as it might seem, since the goal is to have slow-moving neutrinos, here), and get

$latex (m \Delta v) \Delta x \geq \frac{\hbar}{2} $ $latex v_{min} \approx \frac{\hbar}{2 m \Delta x} $

So, the approximate speed of a trapped neutrino decreases with increasing mass and decreases as you increase the size of the trapping region. Of course, getting an actual number requires a value for the neutrino mass, which we don't know in an absolute sense. But this is a ballpark kind of calculation, anyway, so we can just pick a value. If we say that our trapped neutrino has a mass of 1 eV/c2 in the units that particle physicists use (a value that's probably way too big, but convenient), the various constants end up giving you a relationship between approximate velocity in m/s and the "trap" size in meters that's really simple:

$latex v_{min} \approx 30/\Delta x $

So, a 1eV/c2 neutrino trapped in a 1m box would be moving at an approximate minimum speed of 30m/s. that's really fast, actually-- an electron trapped in the same size box would have a minimum uncertainty-derived speed of about 60 micrometers per second, half a million times smaller.

(As a sanity check, you can ask what this would predict for something like a BEC of atoms, which would be around 100,000 times heavier than an electron (ballpark), in a trap a micron on a side (ballpark), which gets you a minimum speed of about 0.6 mm/s, which is the right general range.)

So, what would it take to get neutrinos "just sitting there?" Well, it depends on your definition. My original phrasing mentioned a volume of one cubic centimeter. If you took that as the trap volume, your neutrinos would be moving at roughly 3000 m/s. If you want them at speeds comparable to the trapped laser-cooled atoms I'm used to, say 0.1 m/s, you would need a trap 300m on a side.

Of course, what you would make the walls of the trap of, in order to confine neutrinos to that volume, I have no idea. Given that you need a 100-m scale tank full of water, like the SuperK detector shown above in the "featured image," just to have a prayer of detecting a minuscule fraction of the vast number of neutrinos created in the Sun, I don't think we'll be actually trapping neutrinos any time soon...