## 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 nth attempt. We calculate as well the mean and mean square, both of the number of the first successful detection attempt and the time until 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 until 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 interarrival time distribution from very narrow (δ peaked) to exponential, as resonances are wiped out by the randomness of the sampling.

Original language | English |
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Article number | 022222 |

Journal | Physical Review A |

Volume | 103 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2021 |

### Bibliographical note

Publisher Copyright:© 2021 American Physical Society.

### Funding

D.A.K. and E.B. acknowledge the support of the Israel Science Foundation, 1898/17. K.Z. acknowledges the support of the Julian Schwinger Foundation. D.A.K. acknowledges the assistance of Ruiqi Wang in the early stages of this work.

Funders | Funder number |
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Julian Schwinger Foundation for Physics Research | |

Israel Science Foundation | 1898/17 |