Yes we agree that there will be extra noise due to the open nature of the cavity, and we already refer to this on p3 of the Introduction: “The disadvantage of working in a cavity is that quantum measurement backaction, in the form of random fluctuations in the cavity field due to photons spontaneously leaking out of the cavity, heats up the cold atoms and limits the coherence time of the measurement \cite{Ven13}.” where the cited paper is one of ours [Prasanna Venkatesh, B.; O’Dell, D. H. J.; Bloch oscillations of cold atoms in a cavity: Effects of quantum noise. Phys. Rev. A 2013, 88, 013848] where we directlycalculated the heating effect for Bloch oscillating atoms in a cavity within the Bogoliubov approximation.
Some rough estimate on the time scales should be added.