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From the way the question is worded, I would assume you can treat this system as a BEC wavepacket in a potential barrier, where the potential barrier is given by your harmonical trap. Since the trap is harmonic, this is in analogy to the well-known vibrational wavepackets. I would also recommend reading this paper for some more insight: ...

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It seems to me that you just cannot tell the difference between a Bose condensate and nothing in this case. What will change if you add some photons or phonons with zero energy to the system? No characteristics of the system will change. So it seems to me we have no criterion to decide if there is a Bose condensate in this case, and what's more important, it ...

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Let me try to formulate the question more precisely and then give my answer. The phase'' $\theta$ of a BEC is introduced as $\langle \psi_N|\hat{a}|\psi_{N+1}\rangle = |\psi|e^{i\theta}$, where $|\psi_N\rangle$ is the ground state being occupied by $N$ bosons. Because the occupation number of the condensed state $|\psi_N\rangle$ is of the order $N$, the ...

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