# Can coherent superpositions of a neutron and antineutron exist?

In my recent post I learned that electric charge is always conserved in contrast to strangeness quantum number, which limits the types of Hadrons that can be build. Furthermore, also different masses in the superpositions are ... at least dubious.

Now there exist Kaons in the following form: $$|K_0^S\rangle = \frac{|d \bar{s}\rangle - |\bar{d} s\rangle }{\sqrt{2}}$$ $$|K_0^L\rangle = \frac{|d \bar{s}\rangle + |\bar{d} s\rangle }{\sqrt{2}}$$

which are coherent superpositions of different a particle with it's antiparticle.

I wonder, whether something like this could exist:

$$|N\rangle = \frac{|u d d \rangle + |\bar{u} \bar{d} \bar{d}\rangle }{\sqrt{2}}$$

If yes: y1) How would one observe that such a particle exists? y2) Has this been observed in any experiment?

If no: n1) What is the physical reason that it can not exist? (All the quantum numbers are the same).

• Do you mean: is there a particle which is a coherent superposition of a neutron and an antineutron? Or do you mean: is it theoretically possible to put a neutron and an antineutron into coherent superposition? These are two different questions which probably have two different answers. – Peter Shor Jun 11 '16 at 20:03
• arxiv.org/abs/0902.0834 – Count Iblis Jun 11 '16 at 20:04
• You could certainly (in theory) design an experiment in which a coherent superposition of a neutron and an antineutron traveled for some distance, after which it gets measured to check that it really was in superposition. But you can also (in theory) design an experiment in which a superposition of one photon and two photons travels for some distance, after which it gets measured ... . I certainly wouldn't call this a 1.5-photon-number particle. – Peter Shor Jun 12 '16 at 12:22
• In the kaons you mention, |d$\bar{\mathrm{s}}$⟩ turns into |$\bar{\mathrm{d}}$s⟩ on a reasonable time scale, and this happens in nature, so it makes sense to consider the superpositions as particles. If there are processes that don't preserve baryon number and turn a neutron into an anti-neutron, these happen at time scales much, much longer than the half-life of a neutron. – Peter Shor Jun 12 '16 at 12:33
• Baryon number is not a sacred quantum number. It is an experimental question whether it is violated or not, and the answers, to y1) and not so far to y2) are in the review. – Cosmas Zachos Jun 12 '16 at 16:42

## 1 Answer

So there's this funny rule whose provenance I can't recall, but whose essence is: everything that is not forbidden, eventually happens. This rule is particularly fecund in quantum mechanics. If the process you describe is allowed, then every neutron already is a superposition of neutron and antineutron, and the question is just whether the oscillations between neutron and antineutron can be observed.

The problem is that, in order to observe the oscillation, you have to let the neutron wavefunction evolve in such a way that there isn't any difference in energy between $n$ and $\bar n$ for a substantial fraction of the oscillation time. This means that every interaction with ordinary matter, or with a magnetic field, constitutes "a measurement" and resets the state to pure neutron. (This is one of the important distinctions from the kaon case: the neutron, unlike the kaon, carries angular momentum and a magnetic moment.)

The Particle Data Group give a lower limit of about $10^8\rm\,s \approx 3\,yr$ for $n\to\bar n$ oscillations, based (in the "free neutron" case) on no detections in an experiment from the early nineties. The idea is that you make a whole boatload of slow neutrons, pass them through a very long pipe where they don't touch the walls and where the magnetic field has been made very feeble, and catch them on a detector at the bottom. Ordinary neutrons will make a few-MeV signal in a detector; antineutrons will make 2000 MeV of fast pions, quite distinctive.

There is talk about doing a new $n\bar n$ oscillation search; here are some recent slides and a recent paper.

To address comments by Peter Shor about experimental accessibility: suppose the $n\leftrightarrow\bar n$ oscillation period is $T = 10^{12}\rm\,s = 2\pi/\omega_{n\bar n}$. (I don't know what $n\leftrightarrow\bar n$ time scale is competitive with proton decay in limiting grand unified theories, so I made up one that's bigger than the current limit but not infinite.) Simply ("simply," heh) get $N$ neutrons and put them in a bottle for one neutron lifetime, $\tau_n = 10^3\rm\,s$. Half of them have decayed. The other half now have a wavefunction $$\left|\psi\right> = \left|n\right> \cos\tau_n\omega_{n\bar n} + \left|\bar n\right> \sin\tau_n\omega _{n\bar n}$$ and the fraction you expect to find in the antineutron state is $\sin^2\tau_n\omega_{n\bar n} \approx 4\pi^2\times10^{-18}$. So if you want to see $40\pm6$ antineutrons, you need $10^{18}$ neutrons.

This is a lot of neutrons (it's a microgram!) but it's not inaccessible. I've been involved in experiments which have captured $\gtrsim 10^{18}$ neutrons in a year or so of beam time, to look at part-per-billion asymmetries with real statistical significance.

• en.wikipedia.org/wiki/Totalitarian_principle – Rococo Jun 13 '16 at 3:22
• But ... if the neutron wavefunction evolves in this way, it happens much, much, much, much, much slower than the lifetime of a neutron. So it doesn't matter whether there is any interaction with ordinary matter or not: the neutron decays on average in less than an hour, and they probably take billions of years or more to oscillate to an antineutron. – Peter Shor Jun 13 '16 at 19:32
• Indeed, the neutron mean lifetime is 880 s, while the neutron-antineutron oscillation time has so far been bounded to $\geq 10^8 s$, but of course these lifetimes are population parameters: not every neutron is dead and gone after 15 mins! This is the beauty of the cleverly sensitive searches detailed in the review article cited twice. – Cosmas Zachos Jun 13 '16 at 21:35
• @PeterShor There's a lot of interesting parameter space between the present limit $10^8\rm\,s$ and your estimate $10^{16}\rm\,s$; see update. – rob Jun 14 '16 at 1:26