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Quantum walks on a circle with optomechanical systems

Quantum Information Processing volume 14, pages 3595–3611 (2015)Cite this article

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Abstract

We propose an implementation of a quantum walk on a circle in an optomechanical system by encoding the walker on the phase space of a radiation field and the coin on a two-level state of a mechanical resonator. The dynamics of the system is obtained by applying Suzuki–Trotter decomposition. We numerically show that the system displays typical behaviors of quantum walks, namely the probability distribution evolves ballistically and the standard deviation of the phase distribution is linearly proportional to the number of steps. We also analyze the effects of decoherence by using the phase-damping channel on the coin space, showing the possibility to implement the quantum walk with present-day technology.

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Notes

  1. Usually in Hamiltonian (1) a linearization procedure is performed by assuming an undepleted cavity field, where the field operator is given by \(a\approx \alpha +\Delta a\), being \(\alpha \) a coherent amplitude and \(\Delta a\) a quantum fluctuation. Had we considered this linearization we would have ended up with the interaction term \(-\hbar g_0 (\alpha \Delta a^{\dagger }+\alpha ^* \Delta a) (b^{\dagger } + b)\), which turns out to lead to a quantum walk on a line given the displacement operator for the field \((\Delta a^{\dagger }+ \Delta a)\) for \(\alpha \) real. In this paper, we do not consider this procedure.

References

  1. Aharonov, Y., Davidovich, L., Zagury, N.: Quantum random walks. Phys. Rev. A 48(2), 1687–1690 (1993)

    Article  ADS  Google Scholar 

  2. Vieira, R., Amorim, E.P.M., Rigolin, G.: Dynamically disordered quantum walk as a maximal entanglement generator. Phys. Rev. Lett. 111(18), 180503 (2013)

    Article  ADS  Google Scholar 

  3. Childs, A.M., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.A.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the thirty-fifth ACM symposium on Theory of computing—STOC ’03, p. 59 (2003)

  4. Childs, A., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 1–11 (2004)

    Article  MathSciNet  Google Scholar 

  5. Portugal, R.: Quantum walks and search algorithms. Springer, Berlin (2013)

    Book  MATH  Google Scholar 

  6. Manouchehri, K., Wang, J.: Physical implementation of quantum walks. Springer, Berlin, Heidelberg (2014)

    Book  MATH  Google Scholar 

  7. Bouwmeester, D., Marzoli, I., Karman, G., Schleich, W., Woerdman, J.: Optical galton board. Phys. Rev. A 61(1), 013410 (1999)

    Article  ADS  Google Scholar 

  8. Souto Ribeiro, P., Walborn, S., Raitz, C., Davidovich, L., Zagury, N.: Quantum random walks and wave-packet reshaping at thesingle-photon level. Phys. Rev. A 78(1), 012326 (2008)

    Article  ADS  Google Scholar 

  9. 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(15), 153602 (2010)

    Article  ADS  Google Scholar 

  10. Zhang, P., Liu, B.H., Liu, R.F., Li, H.R., Li, F.L., Guo, G.C.: Implementation of one-dimensional quantum walks on spin-orbital angular momentum space of photons. Phys. Rev. A 81(5), 052322 (2010)

    Article  ADS  Google Scholar 

  11. Peruzzo, A., Lobino, M., Matthews, J.C.F., Matsuda, N., Politi, A., Poulios, K., Zhou, X.Q., Lahini, Y., Ismail, N., Wörhoff, K., Bromberg, Y., Silberberg, Y., Thompson, M.G., OBrien, J.L.: Quantum walks of correlated photons. Science 329(5998), 1500–1503 (2010). (New York, NY)

    Article  ADS  Google Scholar 

  12. Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Mosley, P.J., Andersson, E., Jex, I., Silberhorn, C.: Photons walking the line: a quantum walk with adjustable coin operations. Phys. Rev. Lett. 104(5), 050502 (2010)

    Article  ADS  Google Scholar 

  13. Schreiber, A., Cassemiro, K.N., Potoček, V., Gábris, A., Jex, I., Silberhorn, C.: Decoherence and disorder in quantum walks: from ballistic spread to localization. Phys. Rev. Lett. 106(18), 180403 (2011)

    Article  ADS  Google Scholar 

  14. Goyal, S.K., Roux, F.S., Forbes, A., Konrad, T.: Implementing quantum walks using orbital angular momentum of classical light. Phys. Rev. Lett. 110(26), 263602 (2013)

    Article  ADS  Google Scholar 

  15. Xue, P., Sanders, B., Leibfried, D.: Quantum walk on a line for a trapped ion. Phys. Rev. Lett. 103(18), 183602 (2009)

    Article  ADS  Google Scholar 

  16. 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(5937), 174–177 (2009). (New York, NY)

    Article  ADS  Google Scholar 

  17. 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(10), 100503 (2010)

    Article  Google Scholar 

  18. 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(9), 090504 (2009)

    Article  ADS  Google Scholar 

  19. Dür, W., Raussendorf, R., Kendon, V., Briegel, H.J.: Quantum walks in optical lattices. Phys. Rev. A 66(5), 052319 (2002)

    Article  ADS  Google Scholar 

  20. Travaglione, B., Milburn, G.: Implementing the quantum random walk. Phys. Rev. A 65(3), 032310 (2002)

    Article  ADS  Google Scholar 

  21. Sanders, B.C., Bartlett, S.D., Tregenna, B., Knight, P.L.: Quantum quincunx in cavity quantum electrodynamics. Phys. Rev. A 67(4), 042305 (2003)

    Article  ADS  Google Scholar 

  22. Xue, P., Sanders, B.C.: Quantum quincunx for walk on circles in phase space with indirect coin flip. New J. Phys. 10(5), 053025 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  23. Xue, P., Sanders, B., Blais, A., Lalumière, K.: Quantum walks on circles in phase space via superconducting circuit quantum electrodynamics. Phys. Rev. A 78(4), 042334 (2008)

    Article  ADS  Google Scholar 

  24. Hardal, A.Ü., Xue, P., Shikano, Y., Müstecaplıoğlu, Ö.E., Sanders, B.C.: Discrete-time quantum walk with nitrogen-vacancy centers in diamond coupled to a superconducting flux qubit. Phys. Rev. A 88(2), 022303 (2013)

    Article  ADS  Google Scholar 

  25. Aspelmeyer, M., Kippenberg, T.J., Marquardt, F.: Cavity optomechanics. arXiv:1303.0733 (2013)

  26. Poot, M., van der Zant, H.S.: Mechanical systems in the quantum regime. Phys. Rep. 511(5), 273–335 (2012)

    Article  ADS  Google Scholar 

  27. O’Connell, A.D., Hofheinz, M., Ansmann, M., Bialczak, R.C., Lenander, M., Lucero, E., Neeley, M., Sank, D., Wang, H., Weides, M., et al.: Quantum ground state and single-phonon control of a mechanical resonator. Nature 464(7289), 697–703 (2010)

    Article  ADS  Google Scholar 

  28. Chan, J., Alegre, T.P.M., Safavi-Naeini, A.H., Hill, J.T., Krause, A., Gröblacher, S., Aspelmeyer, M., Painter, O.: Laser cooling of a nanomechanical oscillator into its quantum ground state. Nature 478(7367), 89–92 (2011)

    Article  ADS  Google Scholar 

  29. Teufel, J.D., Donner, T., Castellanos-Beltran, M.A., Harlow, J.W., Lehnert, K.W.: Nanomechanical motion measured with an imprecision below that at the standard quantum limit. Nat. Nanotechnol. 4(12), 820–823 (2009)

    Article  ADS  Google Scholar 

  30. Suzuki, M.: Decomposition formulas of exponential operators and Lie exponentials with some applications to quantum mechanics and statistical physics. J. Math. Phys. 26, 601 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Suzuki, M.: Generalized Trotter’s formula and systematic approximants of exponential operators and inner derivations with applications to many-body problems. Commun. Math. Phys. 51(2), 183–190 (1976)

    Article  ADS  MATH  Google Scholar 

  32. Loudon, R.: The quantum theory of light. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  33. Kempe, J.: Quantum random walks—an introductory overview. Contemp. Phys. 44(4), 302–327 (2003). (Quant-ph/0303081)

    Article  MathSciNet  ADS  Google Scholar 

  34. de Oliveira, M.C., Moussa, M.H.Y., Mizrahi, S.S.: Continuous pumping and control of a mesoscopic superposition state in a lossy QED cavity. Phys. Rev. A 61, 063809 (2000)

    Article  ADS  Google Scholar 

  35. Nielsen, M.A., Chuang, I.L.: Quantum computation and quantum information. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  36. Verhagen, E., Deleglise, S., Weis, S., Schliesser, A., Kippenberg, T.J.: Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode. Nature 482(7383), 63–67 (2012)

    Article  ADS  Google Scholar 

  37. Chandrashekar, C.M., Srikanth, R., Subhashish, Banerjee: Symmetries and noise in quantum walk. Phys Rev. A 76, 022316 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  38. Okamoto, H., Gourgout, A., Chang, C.Y., Onomitsu, K., Mahboob, I., Chang, E.Y., Yamaguchi, H.: Coherent phonon manipulation in coupled mechanical resonators. Nat. Phys. 9(8), 480–484 (2013)

    Article  Google Scholar 

  39. Faust, T., Rieger, J., Seitner, M., Kotthaus, J., Weig, E.: Coherent control of a classical nanomechanical two-level system. Nat. Phys. 9(8), 485–488 (2013)

    Article  Google Scholar 

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Acknowledgments

JKM acknowledges financial supports from CNPq, Grants PCI-DB 302866/2014-0 and PDJ 165941/2014-6. RP acknowledges financial support from CNPq and FAPERJ. MCO acknowledges support by FAPESP and CNPq through the National Institute for Science and Technology of Quantum Information (INCT-IQ) and the Research Center in Optics and Photonics (CePOF).

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Author notes

  1. Jalil Khatibi Moqadam

    Present address: Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP, Brazil

Authors and Affiliations

  1. Laboratório Nacional de Computação Científica (LNCC), Petrópolis, RJ, Brazil

    Jalil Khatibi Moqadam & Renato Portugal

  2. Instituto de Física “Gleb Wataghin”, Universidade Estadual de Campinas, Campinas, SP, Brazil

    Marcos Cesar de Oliveira

Authors
  1. Jalil Khatibi Moqadam
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  2. Renato Portugal
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  3. Marcos Cesar de Oliveira
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Correspondence to Renato Portugal.

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Moqadam, J.K., Portugal, R. & de Oliveira, M.C. Quantum walks on a circle with optomechanical systems. Quantum Inf Process 14, 3595–3611 (2015). https://doi.org/10.1007/s11128-015-1079-9

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  • DOI: https://doi.org/10.1007/s11128-015-1079-9

Keywords

  • Optomechanical and electromechanical resonators
  • Quantum walk
  • Simulation