Okay, here's some Wikipedia links for the physics challenged:
Bose-Einstein Condensate (BEC)
Think about it.
Not all naturally-occurring neutronium would be in a BEC, but if it is possible, then some neutron stars could eventually cool enough to form a BEC. It would take a very long time, of course. Since there have been supernovae since the beginning of the universe (minus a few million years) there should be some neutron stars which have cooled enough to reach the critical temperature. There could even be one within our galaxy; being a few millionths of a degree above zero Kelvin, it would be basically invisible. It would have to be in a relatively matter-free area of space, because a neutron star has immense gravity.
I can't even get a grip on the implications right now, but it seems to me that the entire star could be treated as a quantum object--which could be really interesting. Put an entire neutron star into superposition with one state some distance away; then collapse the wave function with a bias for the distant position, and guess what? You've (probably) teleported a stellar mass.
Now my head hurts.
* * *
Looking at the link for neutronium led me to the more generic entry on degenerate matter, which has some of its own interesting bits and pieces. Quark matter--wow, that should be really interesting stuff, assuming you could survive studying it. And if preons exist, what kind of density is required to achieve that?
The thing that really blows my mind about all this is that these states should be able to exist outside of a black hole. A black hole is infinitely dense and physics can't cope with what goes on inside of one. (As far as I know, anyway.) But stop short of a black hole, and what do you have? What happens if a neutron star gets cold? Does it collapse further into a quark star, and from there into a preon star? What is the maximum density of matter? Is there one?
I also have to wonder what happens when one passes the electron degeneracy limit--the point at which ordinary matter turns into neutronium. The proton and electron combine to form a neutron; it seems to me that there has to be some kind of an energy release from this interaction, but I can't put my finger on the exact mechanism.
When a star blows up, the event starts with an implosion--the star collapses--and some of the matter is converted into neutronium. (A supernova usually leaves a neutron star behind.) Does any of the energy of the explosion come from the conversion of normal matter into neutronium? If so, how much?
...This is the kind of stuff I think about when I'm trying to get to sleep. That's what I get for being a physics geek.