Non-Hermitian and Zeno limit of quantum systems under rapid measurements

We investigate two models of a quantum system under rapid measurement performed to detect whether the system is in a given state |ψd). In the first, the detection process is modeled via an imaginary potential 2iℏ|ψd)(ψd|/τ. In the second approach, repeated strong projective measurements are performed on the otherwise unitarily evolving system with a fixed high frequency 1/τ. We compare the statistics of the random time T of first successful detection for the two models, considering both its probability density function F(t) and the moments (Tm). We show, by a direct comparison of the two solutions, that both approaches yield the same results for F(t) (and so the moments) in the small-τ limit, also called the Zeno limit, as long as the initial state |ψin) is not parallel to the detection state, so that |(ψd|ψin)|<1. When this condition is violated, however, the low probability density to detect the state on timescales much larger than τ is precisely a factor of 4 smaller in the non-Hermitian model. We express the solution of the Zeno limit of both problems formally in terms of an electrostatic analogy. Our results are corroborated by numerical simulations.