  A topological frequency converter represents a dynamical counterpart of the
integer quantum Hall effect, where a two-level system enacts a quantized
time-averaged power transfer between two driving modes of incommensurate
frequency. Here, we investigate as to what extent temporal coherence in the
quantum dynamics of the two-level system is important for the topological
quantization of the converter. To this end, we consider dissipative channels
corresponding to spontaneous decay and dephasing in the instantaneous
eigenbasis of the Hamiltonian as well as spontaneous decay in a fixed basis.
The dissipation is modelled using both a full Lindblad and an effective
non-Hermitian (NH) Hamiltonian description. For all three dissipation channels
we find a transition from the unperturbed dynamics to a quantum watchdog
effect, which destroys any power transfer in the strong coupling limit. This is
striking because the watchdog effect leads to perfectly adiabatic dynamics in
the instantaneous eigenbasis, at first glance similar to the unperturbed case.
Furthermore, it is found that dephasing immediately leads to an exponential
decay of the power transfer in time due to loss of polarisation in the mixed
quantum state. Finally, we discuss the appearance in the effective NH
trajectory description of non-adiabatic processes, which are suppressed in the
full Lindblad dynamics.
