The first detection time of a quantum state under random probing

TL;DR

This paper derives formulas for the statistics of the first detection time of a quantum state under random measurement intervals, revealing how randomness affects detection efficiency and mean detection times.

Contribution

It provides explicit formulas and proofs for the detection probability, mean detection time, and the effect of different interval distributions in quantum systems.

Findings

Mean detection time equals mean attempt number times mean interval.

Total detection probability is unity for the return problem.

Detection time distribution changes qualitatively with interval randomness.

Abstract

We solve for the statistics of the first detection of a quantum system in a particular desired state, when the system is subject to a projective measurement at independent identically distributed random time intervals. We present formulas for the probability of detection in the $n$th attempt. We calculate as well the mean and mean square both of the number of the first successful detection attempt and the time till first detection. We present explicit results for a particle initially localized at a site on a ring of size $L$, probed at some arbitrary given site, in the case when the detection intervals are distributed exponentially. We prove that, for all interval distributions and finite-dimensional Hamiltonians, the mean detection time is equal to the mean attempt number times the mean time interval between attempts. We further prove that for the return problem when the initial and target state are identical, the total detection probability is unity and the mean attempts till detection is an integer, which is the size of the Hilbert space (symmetrized about the target state). We study an interpolation between the fixed time interval case to an exponential distribution of time intervals via the Gamma distribution with constant mean and varying width. The mean arrival time as a function of the mean interval changes qualitatively as we tune the inter-arrival time distribution from very narrow (delta peaked) to exponential, as resonances are wiped out by the randomness of the sampling.