second is the weighted contribution of new currents coming from the non-stationary modes of the original process. Finally, the third term is the current coming from the reset transitions. From the signs appearing in the last term, we see that there is a current loop from all the states to the reset state $C_0$ and then back from $C_0$ to all other states, so that $C_0$ acts both as a sink and source.

From this result, it is clear that the reset process will violate the condition of detailed balance, i.e., $J^\Gamma\neq 0$, if the original process satisfies the condition of detailed balance, i.e., $J=0$. In this case, it is known that the original process has a real spectrum, which implies from our results that the spectrum of the reset process must also be real, even though $J^\Gamma\neq 0$. This shows that currents are not necessarily associated with complex spectra of the dynamical generator. In fact, detailed balance is only a sufficient condition for the generator spectrum to be real, not a necessary condition.

In principle, it is also possible to have a reset process satisfying detailed balance ($J^\Gamma=0$) if the original process violates detailed balance ($J\neq 0$). However, this 