Quantum total detection probability from repeated measurements I. The bright and dark states.

We investigate a form of quantum search, where a detector repeatedly probes some quantum particle with fixed rate $1/\tau$ until it is first successful. This is a quantum version of the first-passage problem. We focus on the total probability, $P_\text{det}$, that the particle is eventually detected in some state, for example on a node $r_\text{d}$ on a graph, after an arbitrary number of detection attempts. For finite graphs, and more generally for systems with a discrete spectrum, we provide an explicit formula for $P_\text{det}$ in terms of the energy eigenstates. We provide two derivations of this formula, one proceeding directly from the renewal equation for the generating function of the detection amplitude and a second based on a study of the so-called "dark states." The dark states are those that are never detected, and constitute a subspace of the Hilbert space. All states orthogonal to the dark space in finite systems are in fact "bright", with $P_\text{det}=1$. Dark states can arise either from degeneracies of the energy spectrum, or from energy levels that have no projection on the detection state. We demonstrate how breaking the degeneracy infinitesimally can restore $P_\text{det}=1$. For finite systems, it is found that $P_\text{det}$ is independent of the measurement frequency $1/\tau$, except for special resonant values. Our formula for $P_\text{det}$ fails for infinite systems, in which case the result is only an upper bound of $P_\text{det}$. We show how this breakdown occurs in the case of the infinite line. In two follow-ons to this paper, we unravel the relation between $P_\text{det}$ and the system's symmetry.