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Interacting Two-Particle Discrete-Time Quantum Walk with Percolation

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Abstract

We investigate discrete-time quantum walk of two interacting particles on one-dimensional percolation lattice. The dynamical percolation can make the quantum walk diffuse classically while the static percolation can make it localize at the neighbor of the initial site. According to the symmetry for two bosons, fermions and classical indistinguishable particles, three kind of initial states are considered. The interaction between two particles leads to bosons anti-bunching and fermions bunching. We focus on the joint probability distributions, the average distance and the meeting probability to describe the global properties of quantum walk. The joint probability distributions of two particles spread more out over the walking region in the dynamical percolation, while they are concentrated on the neighbor of the initial position in the statical case. For fermions and bosons initial states, both the curves of the meeting probability and that of the average distance occur crossing at a certain value of percolation probability, which is a result induced by the interplay between the interaction and percolation.

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References

  1. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48, 1687–1690 (1993)

    Article  ADS  Google Scholar 

  2. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  3. Kempe, J.: Quantum random walks: an introductory overview. Contemp. Phys. 44, 307–327 (2003)

    Article  ADS  Google Scholar 

  4. Venegas-Andraca, S.E.: Quantum walks: a comprehensive review. Quantum Inf. Process. 11, 1015–1106 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Schmitz, H., Matjeschk, R., Schneider, C., Glueckert, J., Enderlein, M., Huber, T., Schaetz, T.: Quantum walk of a trapped ion in phase space. Phys. Rev. Lett. 103, 090504 (2009)

    Article  ADS  Google Scholar 

  6. Karski, M., Förster, L., Choi, J.M., Steffen, A., Alt, W., Meschede, D., Widera, A.: Quantum walk in position space with single optically trapped atoms. Science 325, 174–177 (2009)

    Article  ADS  Google Scholar 

  7. Zähringer, F., Kirchmair, G., Gerritsma, R., Solano, E., Blatt, R., Roos, C.F.: Realization of a quantum walk with one and two trapped ions. Phys. Rev. Lett. 104, 100503 (2010)

    Article  Google Scholar 

  8. Broome, M.A., Fedrizzi, A., Lanyon, B.P., Kassal, I., Aspuru-Guzik, A., White, A.G.: Discrete single-photon quantum walks with tunable decoherence. Phys. Rev. Lett. 104, 153602 (2010)

    Article  ADS  Google Scholar 

  9. Lahini, Y., Verbin, M., Huber, S.D., Bromberg, Y., Pugatch, R., Silberberg, Y.: Quantum walk of two interacting bosons. Phys. Rev. A 86, 011603 (2012)

    Article  ADS  Google Scholar 

  10. Preiss, P.M., Ma, R., Tai, M.E., Lukin, A., Rispoli, M., Zupancic, P., Lahini, Y., Islam, R., Greiner, M.: Strongly correlated quantum walks in optical lattices. Science 347, 1229–1233 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Shenvi, N., Kempe, J., Whaley, K.B.: Quantum random-walk search algorithm. Phys. Rev. A 67, 052307 (2003)

    Article  ADS  Google Scholar 

  12. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  13. Lovett, N.B., Cooper, S., Everitt, M., Trevers, M., Kendon, V.: Universal quantum computation using the discrete-time quantum walk. Phys. Rev. A 81, 042330 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  14. Jeong, H., Paternostro, M., Kim, M.S.: Simulation of quantum random walks using the interference of a classical field. Phys. Rev. A 69, 012310 (2004)

    Article  ADS  Google Scholar 

  15. Rohde, P.P., Schreiber, A., Štefaňák, M., Jex, I., Silberhorn, C.: Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation. New J. Phys. 13, 013001 (2011)

    Article  ADS  Google Scholar 

  16. Pathak, P.K., Agarwal, G.S.: Quantum random walk of two photons in separable and entangled states. Phys. Rev. A 75, 032351 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  17. Omar, Y., Paunković, N., Sheridan, L., Bose, S.: Quantum walk on a line with two entangled particles. Phys. Rev. A 74, 042304 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Štefaňák, M., Kiss, T., Jex, I., Mohring, B.: The meeting problem in the quantum walk. J. Phys. A 39, 14965–14983 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Gamble, J.K., Friesen, M., Zhou, D., Joynt, R., Coppersmith, S.N.: Two-particle quantum walks applied to the graph isomorphism problem. Phys. Rev. A 81, 052313 (2010)

    Article  ADS  Google Scholar 

  20. Wang, Q.H., Li, Z.J.: Repelling, binding, and oscillating of two-particle discrete-time quantum walks. Ann. Phys. 373, 1–9 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Berry, S.D., Wang, J.B.: Two-particle quantum walks: entanglement and graph isomorphism testing. Phys. Rev. A 83, 042317 (2011)

    Article  ADS  Google Scholar 

  22. Carson, G.R., Loke, T., Wang, J.B.: Entanglement dynamics of two-particle quantum walks. Quantum Inf. Process. 14, 3193–3210 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Mookerjee, A., Dasgupta, I., Saha, T.: Quantum percolation. Int. J. Mod. Phys. B 09, 2989–3024 (1995)

    Article  ADS  Google Scholar 

  24. Sen, A.K., Bardhan, K.K., Chkrabarti, B.K.: Quantum and Semi-classical Percolation and Breakdown in Disordered Solids. Springer, Berlin (2009)

    MATH  Google Scholar 

  25. Anderson, P.W.: Absence of diffusion in certain random lattices. Phys. Rev. 109, 1492–1505 (1958)

    Article  ADS  Google Scholar 

  26. Lee, P.A., Ramakrishnan, T.V.: Disordered electronic systems. Rev. Mod. Phys. 57, 287–337 (1985)

    Article  ADS  Google Scholar 

  27. Evers, F., Mirlin, A.D.: Anderson transitions. Rev. Mod. Phys. 80, 1355–1417 (2008)

    Article  ADS  Google Scholar 

  28. Romanelli, A., Siri, R., Abal, G., Auyuanet, A., Donangelo, R.: Decoherence in the quantum walk on the line. Phys. A 347, 137–152 (2005)

    Article  MathSciNet  Google Scholar 

  29. Leung, G., Knott, P., Bailey, J., Kendon, V.: Coined quantum walks on percolation graphs. New J. Phys. 12, 123018 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  30. Chandrashekar, C.M., Busch, T.: Quantum percolation and transition point of a directed discrete-time quantum walk. Sci. Rep. 4, 6583 (2014)

    Article  ADS  Google Scholar 

  31. Kollár, B., Novotný, J., Kiss, T., Jex, I.: Discrete time quantum walks on percolation graphs. Eur. Phys. J. Plus 129, 103 (2014)

    Article  Google Scholar 

  32. Rigovacca, L., Di Franco, C.: Two-walker discrete-time quantum walks on the line with percolation. Sci. Rep. 6, 22052 (2016)

    Article  ADS  Google Scholar 

  33. Kendon, V.: Decoherence in quantum walks - a review. Math. Struct. Comput. Sci. 17, 1169–1220 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Alberti, A., Alt, W., Werner, R., Meschede, D.: Decoherence models for discrete-time quantum walks and their application to neutral atom experiments. New J. Phys. 16, 123052 (2014)

    Article  ADS  Google Scholar 

  35. Ahlbrecht, A., Cedzich, C., Matjeschk, R., Scholz, V.B., Werner, A.H., Werner, R.: Asymptotic behavior of quantum walks with spatio-temporal coin fluctuations. Quantum Inf. Process. 11, 1219–1249 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by 1331KSC, Natural Science Foundation of Shanxi Province (201601D011009) and Shanxi Scholarship Council of China (2015-012).

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Authors and Affiliations

  1. Institute of Theoretical Physics, Shanxi University, Taiyuan, 030006, China

    Xiao-Yu Sun, Qing-Hao Wang & Zhi-Jian Li

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  1. Xiao-Yu Sun
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  2. Qing-Hao Wang
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Correspondence to Zhi-Jian Li.

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Sun, XY., Wang, QH. & Li, ZJ. Interacting Two-Particle Discrete-Time Quantum Walk with Percolation. Int J Theor Phys 57, 2485–2495 (2018). https://doi.org/10.1007/s10773-018-3770-y

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  • DOI: https://doi.org/10.1007/s10773-018-3770-y

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