The first detection time of a quantum state under random probing
- URL: http://arxiv.org/abs/2012.01763v1
- Date: Thu, 3 Dec 2020 08:51:21 GMT
- Title: The first detection time of a quantum state under random probing
- Authors: David A. Kessler, Eli Barkai, Klaus Ziegler
- Abstract summary: 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.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
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