First Detected Arrival of a Quantum Walker on an Infinite Line

Felix Thiel, Eli Barkai, David A. Kessler

Research output: Contribution to journalArticlepeer-review

50 Scopus citations

Abstract

The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and on the sampling time τ. Here, we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value that is distance independent.

Original languageEnglish
Article number040502
JournalPhysical Review Letters
Volume120
Issue number4
DOIs
StatePublished - 26 Jan 2018

Bibliographical note

Publisher Copyright:
© 2018 American Physical Society.

Funding

The authors acknowledge support of Israeli Science Foundation (ISF) under Grant No. 1898/17. F. T. is sustained by Deutsche Forschungsgemeinschaft (DFG) under Grant No. TH2192/1-1.

FundersFunder number
Deutsche ForschungsgemeinschaftTH2192/1-1
Israel Science Foundation1898/17

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