Splitting Probability in Monitored Quantum Dynamics

We consider a continuous-time quantum walk stroboscopically subjected to projective measurements on two distinct targets with sampling time period τ. We construct a general renewal framework for the first-detection amplitude at each target, from which we obtain the associated splitting probabilities. For Hamiltonians with reflection symmetry, we further show that each dual-target first-detection amplitude can be viewed as an interference effect between a pair of single-target first-detection amplitudes. This formulation yields exact expressions for the splitting probabilities and gives special sampling times at which they undergo pointwise discontinuous jumps due to a form of ergodicity breaking associated with dark states. As an illustrative example, we apply the framework to a one-dimensional tight-binding lattice chain with nearest-neighbor hopping, and study the splitting probabilities as functions of the initial position and sampling time. Below a critical sampling time, 0<τ< τ c where τ c = 2 π/ ΔE with ΔE being the energy bandwidth, the splitting probabilities for large system sizes become equal to 1/2, independent of the initial position, as long as this position does not coincide with target sites. Above the critical sampling time, τ at least τ c, the splitting probabilities deviate from 1/2 and develop oscillations as a function of initial position. The wavenumber associated with these oscillations is Q=2-1(rτ c/τ), where r is a positive integer. We also explore the effects of relative phase and spatial separation between different components of a coherent initial state on the splitting probabilities. Finally, we study how a finite number of measurements modifies the splitting probabilities and broadens the resonances at special sampling times.