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).
 A: 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.
