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

Felix Thiel, David A. Kessler

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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.

Original languageEnglish
Article number012218
JournalPhysical Review A
Issue number1
StatePublished - Jul 2020

Bibliographical note

Funding Information:
F.T. is grateful to Deutsche Forschungsgemeinschaft (DFG, Germany) for support under Grants No. TH 2192/1-1 and No. TH 2192/2-1. Support from Israel Science Foundation through Grant No. 1898/17 is acknowledged.

Publisher Copyright:
© 2020 American Physical Society.


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