A typical description of Bose-Einstein condensate goes along the line of "multiple atoms in the same ground state can be described by the same wave function". But hold on. Atoms are not elementary particles. They consists of electrons and quarks, and according to a usual way to present Standard Model and QED/QCD, different elementary particles correspond to wave solutions in different quantum fields. Those fields are orthogonal and cannot interact with each other directly. They interact by exchanging virtual force particles. Thus, there is no and cannot be a "quantum field of atom" where "wave function of atom" is an excitation.
I suppose that just as in the case of cooper pairs, when authors talk about "wave function of atom" what they really mean is rather a process where different quantum fields interact via virtual particles, and result of this process behaves as if this was a quantum wave solution for a particle, even though really it is not.
Is this correct? Is there any publication that explicitly construct an equation compatible with QED that gives a solution which looks-like-and-behaves-like a "wave function of an atom" even though there is no special field for its value? What is the value domain of such a wave function if it cannot be any one quantum field? Or is there a completely different method to assign meaning to a "wave function of atom"?