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. The Horak paper shows that Bloch energies (i.e. eigenmodes for a particle in a periodic potential…not to be confused with the Bloch oscillation frequency) will be changed by the backaction. This is because the presence of atoms in the cavity shifts the cavity resonance and hence the depth of the intracavity lattice: the Bloch energies are dependent on the potential depth. This change in potential depth due to backaction is fully included in our treatment. However, a change in the depth of a periodic potential does not change the Bloch frequency because this depends on the lattice’s spatial period, not the depth. Because this key aspect of our treatment can clearly lead to confusion we have modified the following sentence in the conclusions on p19, and added the reference to Horak’s paper: “Chief among these is that the backaction does not alter the frequency of the Bloch oscillations. By contrast, in the harmonic oscillator case there is the so-called optical spring effect which gives a dependence of oscillator frequency on field amplitude and detuning. To be clear, other motional frequencies are altered: because the intracavity lattice depth is modulated by the backaction this will affect certain types of atomic motion, for example the oscillation frequency of an atom about the bottom of one of the potential minima \cite{Horak00}. Nevertheless, the Bloch oscillation frequency is robust against this depth modulation because it only depends on the lattice period, not its depth.” Concerning the change in wavelength (and hence change in lattice constant) due to the refractive index of the gas, this effect is tiny. We consider it in section 6, on p17 and show that the correction to the refractive index is of order 10^-9 for our parameters. This is to be distinguished from the effective change in the cavity length which, due to the resonance effect, is substantial and is precisely what lies behind the amplitude modulation of the lattice.
(3) The authors claim there should be no backaction of the field on the oscillation frequencies. Also the BEC will lead to an effective change of the cavity length and lattice constant, which should renormalize the wavelength and thus om_recoil. This is somehow in contradiction with the eigenfrequency calculations (see Fig.4) in Horak, P.,Physical Review A, 61(3), 033609, 2000.