Actually, we do not claim that oscillation frequencies in general will be unaffected by the backaction. Rather, we claim that one particular frequency, the Bloch frequency, will be unaffected.
(3) The authors claim there should be no backaction of the field on the oscillation frequencies. This is somehow in contradiction with the eigenfrequency calculations (see Fig.4) in Horak, P.,Physical Review A, 61(3), 033609, 2000.