Quantum stochastic resetting with long range hopping

The time needed by a quantum walker to reach a target site on a lattice can be minimized by implementing a resetting protocol, which lets the walker restart its motion at the initial site if it did not reach the target within a certain interval. This requires monitoring the target site by means of a detector, which clicks when the walker has reached the site. The optimal resetting rate is intimately related to the evolution of the probability that the detector clicks. We analyse the characteristic timescales of the monitored dynamics when the coupling between sites at distance d decays algebraically as d^α with α ∈ (0, ∞) and the dynamics induced by the detector is encompassed by an effective non-hermitian Hamiltonian. We show that the exponent α determines the equilibration time and controls the onset of spectral phase transitions. Our study allows us to determine the optimal resetting time as a function of α. It sheds light into the equilibration times in long-range models.