We use an operational approach to discuss ways to measure the higher-order
cross-correlations between optical and matter-wave fields. We pay particular
attention to the fact that atomic fields actually consist of composite
particles that can easily be separated into their basic constituents by a
detection process such as photoionization. In the case of bosonic fields, that
we specifically consider here, this leads to the appearance in the detection
signal of exchange contributions due to both the composite bosonic field and
its individual fermionic constituents. We also show how time-gated counting
schemes allow to isolate specific contributions to the signal, in particular
involving different orderings of the Schr\"odinger and Maxwell fields.Comment: 11 pages, 2 figure
