Quantum walks: The mean first detected transition time
We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψ...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
American Physical Society
2020-07-01
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| Series: | Physical Review Research |
| Online Access: | http://doi.org/10.1103/PhysRevResearch.2.033113 |
| _version_ | 1797211342957969408 |
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| author | Q. Liu R. Yin K. Ziegler E. Barkai |
| author_facet | Q. Liu R. Yin K. Ziegler E. Barkai |
| author_sort | Q. Liu |
| collection | DOAJ |
| description | We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψ_{in}〉 of the walker is orthogonal to the detected state |ψ_{d}〉. We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameters of the model, we find simple expressions describing the blow-up of the mean transition time. Using previous results on the fluctuations of the return time, corresponding to |ψ_{in}〉=|ψ_{d}〉, we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation. |
| first_indexed | 2024-04-24T10:24:58Z |
| format | Article |
| id | doaj.art-870b96db728b45529303e681a0e91e22 |
| institution | Directory Open Access Journal |
| issn | 2643-1564 |
| language | English |
| last_indexed | 2024-04-24T10:24:58Z |
| publishDate | 2020-07-01 |
| publisher | American Physical Society |
| record_format | Article |
| series | Physical Review Research |
| spelling | doaj.art-870b96db728b45529303e681a0e91e222024-04-12T16:57:29ZengAmerican Physical SocietyPhysical Review Research2643-15642020-07-012303311310.1103/PhysRevResearch.2.033113Quantum walks: The mean first detected transition timeQ. LiuR. YinK. ZieglerE. BarkaiWe consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate 1/τ. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state |ψ_{in}〉 of the walker is orthogonal to the detected state |ψ_{d}〉. We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameters of the model, we find simple expressions describing the blow-up of the mean transition time. Using previous results on the fluctuations of the return time, corresponding to |ψ_{in}〉=|ψ_{d}〉, we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.http://doi.org/10.1103/PhysRevResearch.2.033113 |
| spellingShingle | Q. Liu R. Yin K. Ziegler E. Barkai Quantum walks: The mean first detected transition time Physical Review Research |
| title | Quantum walks: The mean first detected transition time |
| title_full | Quantum walks: The mean first detected transition time |
| title_fullStr | Quantum walks: The mean first detected transition time |
| title_full_unstemmed | Quantum walks: The mean first detected transition time |
| title_short | Quantum walks: The mean first detected transition time |
| title_sort | quantum walks the mean first detected transition time |
| url | http://doi.org/10.1103/PhysRevResearch.2.033113 |
| work_keys_str_mv | AT qliu quantumwalksthemeanfirstdetectedtransitiontime AT ryin quantumwalksthemeanfirstdetectedtransitiontime AT kziegler quantumwalksthemeanfirstdetectedtransitiontime AT ebarkai quantumwalksthemeanfirstdetectedtransitiontime |