We consider a quantum system that is initially localized at ${\mathit{x}}_{\text{in}}$ and that is repeatedly projectively probed with a fixed period $\tau$ at position ${\mathit{x}}_{\mathrm{d}}$. We ask for the probability $F_n$ that the system is detected at ${\mathit{x}}_{\mathrm{d}}$ for the very first time, where $n$ is the number of detection attempts. We relate the asymptotic decay and oscillations of $F_n$ with the system's energy spectrum, which is assumed to be absolutely continuous. In particular, $F_n$ is determined by the Hamiltonian's measurement spectral density of states (MSDOS) $f\left(E\right)$ that is closely related to the density of energy states (DOS). We find that $F_n$ decays like a power law whose exponent is determined by the power-law exponent $d_S$ of $f\left(E\right)$ around its singularities $E^{*}$. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of $F_n$ is accompanied by oscillations with frequencies that are determined by the singularities $E^{*}$. This gives rise to critical detection periods ${\tau }_c$ at which the oscillations disappear. In the ordinary case $d_S$ can be identified with the spectral dimension associated with the DOS. Furthermore, the singularities $E^{*}$ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of $F_n$ depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first-detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulas for the first-detection probability in these models.

• Received 15 February 2018

DOI:https://doi.org/10.1103/PhysRevA.97.062105