Quantum Information Processing

, Volume 14, Issue 7, pp 2657–2672 | Cite as

Witnessing the boundary between Markovian and non-Markovian quantum dynamics: a Green’s function approach

  • Shibei Xue
  • Rebing Wu
  • Tzyh-Jong Tarn
  • Ian R. Petersen


This paper presents a Green’s function-based root locus method to investigate the boundary between Markovian and non-Markovian open quantum systems in the frequency domain. A Langevin equation for the boson-boson coupling system is derived, where we show that the structure of the Green’s function dominates the system dynamics. In addition, by increasing the coupling between the system and its environment, the system dynamics are driven from Markovian to non-Markovian dynamics, which results from the redistribution in the modes of the Green’s function in the frequency domain. Both a critical transition and a critical point condition under Lorentzian noise are graphically presented using a root locus method. Related results are verified using an example of a boson-boson coupling system.


Non-Markovian open quantum systems Markovian dynamics   Green’s function Root locus Critical transition 



This research was supported by the Australian Research Council under grant FL110100020. Re-Bing Wu acknowledges support from TNlist and Natural Science Foundation of China (Grant Nos. 61374091 and 61134008).


  1. 1.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)MATHGoogle Scholar
  2. 2.
    Divincenzo, D.P.: Quantum computation. Science 270(5234), 255–261 (1995)MATHMathSciNetADSCrossRefGoogle Scholar
  3. 3.
    Chirolli, L., Burkard, G.: Decoherence in solid-state qubits. Adv. Phys. 57(3), 225–285 (2008)ADSCrossRefGoogle Scholar
  4. 4.
    Zhou, D., Lang, A., Joynt, R.: Disentanglement and decoherence from classical non-Markovian noise: random telegraph noise. Quantum Inf. Process. 9(6), 727–747 (2010)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Gardiner, C., Zoller, P.: Quantum Noise. Springer, Berlin (2000)MATHCrossRefGoogle Scholar
  6. 6.
    Petersen, I.R.: Cascade cavity realization for a class of complex transfer functions arising in coherent quantum feedback control. Automatica 47(8), 1757–1763 (2011)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hassen, S.Z.S., Heurs, M., Huntington, E.H., Petersen, I.R., James, M.R.: Frequency locking of an optical cavity using linear-quadratic Gaussian integral control. J. Phys. B At. Mol. Phys. 42(17), 175501 (2009)ADSCrossRefGoogle Scholar
  8. 8.
    Xue, S., Wu, R.B., Zhang, W.M., Zhang, J., Li, C.W., Tarn, T.J.: Decoherence suppression via non-Markovian coherent feedback control. Phys. Rev. A 86, 052304 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Tu, M.W.Y., Zhang, W.M.: Non-Markovian decoherence theory for a double-dot charge qubit. Phys. Rev. B 78(23), 235311 (2008)ADSCrossRefGoogle Scholar
  10. 10.
    Tu, M.W.Y., Lee, M.T., Zhang, W.M.: Exact master equation and non-Markovian decoherence for quantum dot quantum computing. Quantum Inf. Process. 8(6), 631–646 (2009)MATHCrossRefGoogle Scholar
  11. 11.
    Wu, R.-B., Li, T.-F., Kofman, A., Zhang, J., Liu, Yx, Pashkin, Y., Tsai, J.-S., Nori, F.: Spectral analysis and identification of noises in quantum systems. Phys. Rev. A 87, 022324 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Longhi, S.: Non-Markovian decay and lasing condition in an optical microcavity coupled to a structured reservoir. Phys. Rev. A 74, 063826 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Tan, H.T., Zhang, W.M.: Non-Markovian dynamics of an open quantum system with initial system-reservoir correlations: A nanocavity coupled to a coupled-resonator optical waveguide. Phys. Rev. A 83(3), 032102 (2011)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Burkard, G.: Non-Markovian qubit dynamics in the presence of \(1/f\) noise. Phys. Rev. B 79(12), 125317 (2009)ADSCrossRefGoogle Scholar
  15. 15.
    Anastopoulos, C., Shresta, S., Hu, B.L.: Non-Markovian entanglement dynamics of two qubits interacting with a common electromagnetic field. Quantum Inf. Process. 8(6), 549–563 (2009)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Xue, S., Zhang, J., Wu, R.B., Li, C.W., Tarn, T.J.: Quantum operation for a one-qubit system under a non-Markovian environment. J. Phys. B At. Mol. Phys. 44(15), 154016 (2011)ADSCrossRefGoogle Scholar
  17. 17.
    Orieux, A., DArrigo, A., Ferranti, G., Franco, R.L., Benenti, G., Paladino, E., Falci, G., Sciarrino, F., Mataloni, P.: Experimental on-demand recovery of entanglement by local operations within non-Markovian dynamics. Sci. Rep. 5, 8575 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    Bellomo, B., Lo Franco, R., Maniscalco, S., Compagno, G.: Two-qubit entanglement dynamics for two different non-Markovian environments. Phys. Scr. T140, 014014 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    D’Arrigo, A., Lo Franco, R., Benenti, G., Paladino, E., Falci, G.: Recovering entanglement by local operations. Ann. Phys. 350, 211–224 (2014)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Bellomo, B., Lo Franco, R., Maniscalco, S., Compagno, G.: Entanglement trapping in structured environments. Phys. Rev. A 78, 060302 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    Xu, J.-S., Sun, K., Li, C.-F., Xu, X.-Y., Guo, G.-C.: Experimental recovery of quantum correlations in absence of system-environment back-action. Nat. Commun. 4, 2851 (2013)ADSGoogle Scholar
  22. 22.
    Franco, R.L., Bellomo, B., Maniscalco, S., Compagno, G.: Dynamics of quantum correlations in two-qubit systems within non-Markovian environments. Int. J. Mod. Phys. B 27, 1345053 (2013)MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Chruściński, D., Maniscalco, S.: Degree of non-Markovianity of quantum evolution. Phys. Rev. Lett. 112, 120404 (2014)ADSCrossRefGoogle Scholar
  24. 24.
    Bylicka, B., Chruściński, D., Maniscalco, S.: Non-Markovianity and reservoir memory of quantum channels: a quantum information theory perspective. Sci. Rep. 4, 5720 (2014)CrossRefGoogle Scholar
  25. 25.
    Zhang, W.M., Lo, P.Y., Xiong, H.N., Tu, M.W.Y., Franco, N.: General non-Markovian dynamics of open quantum systems. Phys. Rev. Lett. 109, 170402 (2012)ADSCrossRefGoogle Scholar
  26. 26.
    Lei, C.U., Zhang, W.M.: Decoherence suppression of open quantum systems through a strong coupling to non-Markovian reservoirs. Phys. Rev. A 84, 052116 (2011)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Xiong, H.N., Zhang, W.M., Tu, M.W.Y., Braun, D.: Dynamically stabilized decoherence-free states in non-Markovian open fermionic systems. Phys. Rev. A 86, 032107 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    Liu, B.H., Li, L., Huang, Y.F., Li, C.F., Guo, G.C., Laine, E.M., Breuer, H.P., Piilo, J.: Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems. Nat. Phys. 7(12), 931–934 (2011)CrossRefGoogle Scholar
  29. 29.
    Breuer, H.P., Laine, E.M., Piilo, J.: Measure for the degree of non-Markovian behavior of quantum processes in open systems. Phys. Rev. Lett. 103, 210401 (2009)MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    Ogata, K.: Modern Control Engineering. PrenticeHall, Englewood Cliffs (1996)Google Scholar
  31. 31.
    Zhao, X.Y., Hedemann, S.R., Yu, T.: Restoration of a quantum state in a dephasing channel via environment-assisted error correction. Phys. Rev. A 88, 022321 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    Benedetti, C., Paris, M.G.A., Maniscalco, S.: Non-Markovianity of colored noisy channels. Phys. Rev. A 89, 012114 (2014)ADSCrossRefGoogle Scholar
  33. 33.
    Xue, S., Wu, R.B., Tarn, T.J.: Modeling and analysis of non-Markovian open quantum systems for coherent feedback. In 3rd IFAC International Conference on Intelligent Control and Automation Science, 3, pp. 365–370, (2013)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Shibei Xue
    • 1
  • Rebing Wu
    • 2
  • Tzyh-Jong Tarn
    • 3
  • Ian R. Petersen
    • 1
  1. 1.School of Engineering and Information TechnologyUNSW Canberra at the Australian Defence Force AcademyCampbell ACTAustralia
  2. 2.Department of AutomationTsinghua UniversityBeijingPeople’s Republic of China
  3. 3.Department of Electrical and Systems EngineeringWashington University St. LouisSt. LouisUSA

Personalised recommendations