Conditional no-jump dynamics of non-interacting quantum chains

We analyze the open dynamics of quantum systems conditioned on no jumps being detected. We first obtain general results relating the no-jump probability and the waiting-time distributions to the conditional evolution of specific system observables. These results are applied to single-qubit models, whose conditional dynamics is quite involved and shows a rich set of physical behaviors. Furthermore, we obtain general expressions for the no-jump dynamics of non-interacting fermionic-bosonic chains undergoing Gaussian-preserving dynamics. We show that the conditional dynamics is determined by a non-linear Riccati-type differential equation for the correlation matrix. Finally, we apply our results to chains of hopping particles under inhomogeneous jump rates and boundary driven systems in presence of pairing terms.