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Resonant-tunneling in discrete-time quantum walk

  • Kaname Matsue
  • Leo Matsuoka
  • Osamu Ogurisu
  • Etsuo SegawaEmail author
Regular Paper
  • 51 Downloads

Abstract

We show that discrete-time quantum walks on the line, \(\mathbb {Z}\), behave as “the quantum tunneling”. In particular, quantum walkers can tunnel through a double-well with the transmission probability 1 under a mild condition. This is a property of quantum walks which cannot be seen on classical random walks, and is different from both linear spreadings and localizations.

Keywords

Quantum walk Quantum mechanics Resonant-tunneling Stationary measures 

Notes

Acknowledgements

The authors would like to thank N. Konno for his kind discussion. This work was supported by JSPS KAKENHI Grant numbers JP17K14235, JP17H04978, JP24540208, JP16K05227, JP16K17637, and JP16K03939. KM was partially supported by Program for Promoting the reform of national universities (Kyushu University), Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan, World Premier International Research Center Initiative (WPI), MEXT, Japan.

References

  1. 1.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, emended edn. Dover Publications Inc, Mineola (2010)zbMATHGoogle Scholar
  2. 2.
    Konno, N.: Quantum random walks in one dimension. Quantum Inf. Process. 1(5), 345–354 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Strauch, F.W.: Discrete-time quantum walks: continuous limit and symmetries. J. Math. Phys. 48, 082102 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Molfetta, G.D., Debbasch, F.: Discrete-time quantum walks: continuous limit and symmetries. J. Math. Phys. 53, 123302 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arrighi, P., Nesme, V., Forets, M.: The Dirac equation as a quantum walk: higher dimensions, observational convergence. J. Phys. A Math. Theor. 47, 465302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shikano, Y.: From discrete time quantum walk to continuous time quantum walk in limit distribution. J. Comput. Theor. Nanosci. 10, 1558–1570 (2013)CrossRefGoogle Scholar
  7. 7.
    Messiah, A.: Quantum Mechanics, 1st edn. North-Holland, Amsterdam (1961)zbMATHGoogle Scholar
  8. 8.
    Tsu, R., Esaki, L.: Tunneling in a finite superlattice. Appl. Phys. Lett. 22, 562–564 (1973).  https://doi.org/10.1063/1.1654509 CrossRefGoogle Scholar
  9. 9.
    Chang, L.L., Esaki, L., Tsu, R.: Resonant tunneling in semiconductor double barriers. Appl. Phys. Lett. 24, 593–595 (1974).  https://doi.org/10.1063/1.1655067 CrossRefGoogle Scholar
  10. 10.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H., Exner, P.: Solvable Model in Quantum Mechanics. AMS Chelsea Publishing, Hartford (2004)CrossRefzbMATHGoogle Scholar
  11. 11.
    Exner, P., Seba, P.: Free quantum motion on a branching graph. Rep. Math. Phys. 28, 7–26 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gnutzmann, S., Smilansky, U.: Quantum graphs. Adv. Phys. 55, 527–625 (2006)CrossRefGoogle Scholar
  13. 13.
    Konno, N., Takei, M.: The non-uniform stationary measure for discrete-time quantum walks in one dimension. Quantum Inf. Comput. 15, 1060–1075 (2015)MathSciNetGoogle Scholar
  14. 14.
    Endo, T., Konno, N.: The stationary measure of a space-inhomogeneous quantum walk on the line. Yokohama Math. J. 60, 33–47 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Endo, T., Kawai, H., Konno, N.: Stationary Measures for the Three-State Grover Walk with One Defect in One Dimension (2016). arXiv:1608.07402
  16. 16.
    Kawai, H., Komatsu, T., Konno, N.: Stationary Measure for Two-State Space-Inhomogeneous Quantum Walk in One Dimension (2017). arXiv:1707.04040
  17. 17.
    Zhao, Z., Du, J., Li, H., Yang, T., Chen, Z.-B., Pan, J.-W.: Implement Quantum Random Walks with Linear Optics Elements (2002). arXiv:quant-ph/0212149
  18. 18.
    Higuchi, Yu., Konno, N., Sato, I., Segawa, E.: Quantum graph walk I: mapping to quantum walks. Yokohama Math. J. 59, 33–55 (2013)MathSciNetzbMATHGoogle Scholar

Copyright information

© Chapman University 2018

Authors and Affiliations

  1. 1.Institute of Mathematics for Industry, International Institute for Carbon-Neutral Energy Research (WPI-I2CNER)Kyushu UniversityFukuokaJapan
  2. 2.Graduate School of EngineeringHiroshima UniversityHigashihiroshimaJapan
  3. 3.Division of Mathematical and Physical SciencesKanazawa UniversityKanazawaJapan
  4. 4.Graduate School of Information SciencesTohoku UniversitySendaiJapan

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