= Condensation, matter-wave coherence and superfluidity
<Bose-Einstein condensation> concerns occupation of a <one-particle state>. <Matter-wave coherence> concerns field correlations between positions or times. <Superfluidity> concerns a flow response, including suppression of dissipation under appropriate conditions. A density peak diagnoses neither a relative <quantum phase> between two separately prepared clouds nor a critical flow speed. These observables require different experiments.
For example, in a single-mode fixed-number <Fock state>, $\langle\hat a\rangle=0$ but $\langle\hat a^\dagger\hat a\rangle=N$. Writing $\hat\psi(\mathbf r)=\phi(\mathbf r)\hat a$ for that occupied mode gives
$$
\langle\hat\psi^\dagger(\mathbf r)\hat\psi(\mathbf r')\rangle
=N\phi^*(\mathbf r)\phi(\mathbf r').
$$
Thus vanishing mean field does not imply vanishing first-order <matter-wave coherence>. For two independent fixed-number modes, $\langle\hat a^\dagger\hat b\rangle=0$: there is no ensemble mean interference term establishing a fixed relative <quantum phase>. Interpreting fringes in an individual detection record requires considering correlations and the measurement; assigning a pre-existing definite relative <quantum phase> is unnecessary.
<Wolfgang Ketterle>'s lecture describes a separate coherence test: two condensates expanded into one another and produced fringes. Selectively removing one condensate made the fringes disappear on alternating preparation cycles, testing against diffraction from the confinement. This compared two-cloud and one-cloud signals; it did not measure dissipationless flow.<Nobel lecture: When atoms behave as waves: Bose-Einstein condensation and the atom laser by Wolfgang Ketterle>[pp. 1143–1145, Fig. 15, condensate-elimination control]{ref}
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