In a radioactive Bose-Einstein condensate, would all the atoms disintegrate simultaneously? They're all supposed to do the same thing - so I suppose they would? Also, would the resulting half-life be the same as that of the individual atoms it is made up of?
 A: There are several ways to destroy a Bose-Einstein condensate. The most common is temperature, which is why BECs are all low-temperature phenomena. For instance, helium becomes superfluid when a large fraction of the atoms enter the same quantum state, which happens around $\mathrm{2\,K = \frac16\,meV}/k$, so apparently the first excited state in fluid helium is somewhere around 200 µeV. A beta decay typically releases a few MeV, mega-eV, of energy, a billion times more than a BEC-destroying phonon. Most of that energy is carried away by the electron and the neutrino, but about 0.01% of the decay energy goes into recoil of the decaying nucleus — more than enough to knock it out of the condensate.
An interesting question is whether you could produce a condensate that decayed as a unit. It could be that's what a nucleus is: you don't get to say "the neutron in the $g_{3/2}$ orbital decayed," but instead "the nucleus decayed." Certainly it'd have to be a medium where the excitation energies are comparable to or smaller than the energies involved in the nuclear decays. You can also produce a condensate that undergoes electromagnetic electronic transitions as an ensemble: we call it a laser.
Another handwavy way to think of a condensate is as an ensemble where the distance between atoms is much shorter than the de Broglie wavelength of a typical atom, so that you can no longer make a clear separation between one atom's wavefunction and its neighbors. Interactions which preserve the condensate must have a long enough length scale (or low enough energy scale) to involve the entire condensate at once. But the length scale for the weak interaction goes like $\hbar c / m_W c^2 = 0.002\,\mathrm{fm}$, many times smaller than a nucleon; that interaction can't be spread across the ensemble the way that a low-temperature phonon can.
A: I think the answer should be "no", as they are phenomena happening in two different sectors.
That is, Bose-Einstein condensation involves the center-of-mass degrees of freedom of each atom. On the other hand, radioactive decay pertains to the internal interactions among constituent subatomic particles.
A: All the atoms have the same state, but the state of an atom usually does not influence its chance to decay (electron capture excluded, obviously - highly ionized atoms are less likely to decay due to electron capture simply by lack of electrons to capture).
