The first detection time of a quantum state under random probing

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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Cited by in corpus (16)

• Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain
• First detection probability in quantum resetting via random projective measurements
• Restart uncertainty relation for monitored quantum dynamics
• Instability in the quantum restart problem
• Quantum random walk and tight-binding model subject to projective measurements at random times
• Stochastic resets in the context of a tight-binding chain driven by an oscillating field
• Tight-binding model subject to conditional resets at random times
• Uncertainty relation between detection probability and energy fluctuations
• Resonances of recurrence time of monitored quantum walks
• Dynamically emergent correlations in Brownian particles subject to simultaneous non-Poissonian resetting protocols
• Quantized dynamics in closed quantum systems
• First-detection-time statistics in many-body quantum transport
• Repeated measurements and random scattering in quantum walks
• Experimental measurement of quantum first-passage-time distributions
• Quantum evolution with random phase scattering
• Optimal detection of quantum states via projective measurements