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 vertical bar psi(d)>. In the first, the detection process is modeled via an imaginary potential 2ih vertical bar psi(d)>. 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-tau limit, also called the Zeno limit, as long as the initial state vertical bar psi(in)> is not parallel to the detection state, so that vertical bar vertical bar < 1. When this condition is violated, however, the low probability density to detect the state on timescales much larger than tau 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.