Field and density coherence of matter-wave fields

In analogy to Glauber's analysis of optical coherence, we adopt an operational approach to introduce different classes of atomic coherence associated with different types of measurements. For the sake of concreteness we consider specifically fluorescence, nonresonant imaging, and ionization. We introduce definitions of coherence appropriate to them, which we call electronic, density, and field coherence, respectively. We illustrate these concepts in various descriptions of Bose-Einstein condensation, showing that each of these descriptions makes different implicit assumptions on the coherence of the system. We also study the impact of elastic collision on the field and density coherence properties of atom lasers.