International Journal of Theoretical Physics

, Volume 57, Issue 5, pp 1455–1470 | Cite as

Environmental Effects on Two-Qubit Correlation in the Dispersive Jaynes-Cummings Model

  • Masashi Ban


Total, classical and quantum correlations as well as entanglement are studied for a two-qubit system, where each qubit is placed in a micro cavity described by the dispersive Jaynes-Cummings model. Not only the loss of cavity photons but also the effect of the qubit-photon interaction on the loss is taken into account. The two-qubit system is initially prepared in a Bell diagonal state with a single mixing parameter and the cavity photon is either in a superposition of vacuum and single-photon states or in a weak coherent state. It is shown that more correlation of the two qubits can survive as an initial value of the cavity photon number is smaller. There is a threshold value of the cavity photon number, below which the stationary state becomes inseparable. Furthermore it is found that the external environment which causes the cavity loss has two effects; one brings about the decay of the correlations and the other suppresses the decay.


Decoherence Entanglement Quantum correlation Classical correlation Dispersive Jaynes-Cummings model 


  1. 1.
    Peres, A.: Quantum Theory: Concepts and Methods. Kluwer, Dordrecht (1993)zbMATHGoogle Scholar
  2. 2.
    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865–942 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987)zbMATHGoogle Scholar
  4. 4.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  5. 5.
    Modi, K., Brodutch, A., Cable, H., Paterek, T., Vedral, V.: The classical-quantum boundary for correlations: Discord and related measures. Rev. Mod. Phys. 84, 1655–1707 (2012)ADSCrossRefGoogle Scholar
  6. 6.
    Ollivier, H., Zurek, W. H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2002)ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Datta, A., Shaji, A., Caves, C.M.: Quantum discord and the power of one qubit. Phys. Rev. Lett. 100, 050502 (2008)ADSCrossRefGoogle Scholar
  8. 8.
    Lanyon, B.P., Barbieri, M., Almeida, M.P., White, A.G.: Experimental quantum computing without entanglement. Phys. Rev. Lett. 101, 20051 (2008)CrossRefGoogle Scholar
  9. 9.
    Cavalcanti, D., Aolita, L., Boixo, S., Modi, K., Piani, M., Winter, A.: Operational interpretations of quantum discord. Phys. Rev. A 83, 032324 (2011)ADSCrossRefzbMATHGoogle Scholar
  10. 10.
    Ali, M., Rau, A.R., Alber, G.: Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)ADSCrossRefGoogle Scholar
  11. 11.
    Fanchini, F.F., Werlang, T., Brasil, C.A., Arruda, L.G.E., Caldeira, A.O.: Non-Markovian dynamics of quantum discord. Phys. Rev. A 81, 052107 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    Chen, Q., Zhang, C., Yu, S., Yi, X.X., Oh, C.H.: Quantum discord of two-qubit X states. Phys. Rev. A 84, 042313 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Maldonade-Trapp, A., Hu, A., Roa, L.: Analytical solutions and criteria for the quantum discord of two-qubit X-states. Quant. Inf. Process. 14, 1947–1958 (2015)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Yurischev, M.A.: On quantum discord of general X states. Quant. Inf. Process. 14, 3399–3421 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Luo, S.: Quantum discord for two-qubit systems. Phys. Rev. A 77, 042303 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Huang, Y.: Quantum discord for two-qubit X states: Analytical formula with very small worst-case error. Phys. Rev. A 88, 014302 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Girolami, D., Adesso, G.: Quantum discord for general two-qubit states: Analytical progress. Phys. Rev. A 83, 052108 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    Lu, X., Ma, J., Xi, Z., Wang, X.: Optimal measurements to access classical correlations of two-qubit states. Phys. Rev. A 83, 012327 (2011)ADSCrossRefGoogle Scholar
  19. 19.
    Maziero, J., Celeri, L.C., Serra, R.M., Vedral, V.: Classical and quantum correlations under decoherence. Phys. Rev. A 80, 044102 (2009)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Werlang, T., Souza, S., Fanchini, F.F., Villas Boas, C.J.: Robustness of quantum discord to sudden death. Phys. Rev. A 80, 024103 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    Yu, T., Eberly, J.H.: Finite-time disentanglement via spontaneous emission. Phys. Rev. Lett. 93, 140404 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    Yu, T., Eberly, J.H.: Quantum open system theory: Bipartite aspects. Phys. Rev. Lett. 97, 140403 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    Almeida, M.P., de Melo, F., Hor-Meyll, M., Salles, A., Walborn, S.P., Rebeiro, P.H.S., Davidovich, L.: Environment-induced sudden death of entanglement. Science 316, 579–582 (2007)ADSCrossRefGoogle Scholar
  24. 24.
    Laurat, J., Choi, K.S., Deng, H., Chou, C.W., Kimble, H.J.: Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling. Phys. Rev. Lett. 99, 180504 (2007)ADSCrossRefGoogle Scholar
  25. 25.
    Mazzola, L., Piilo, J., Maniscalco, S.: Sudden transition between classical and quantum decoherence. Phys. Rev. Lett. 104, 200401 (2010)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    Chanda, T., Pal, A.K, Biswas, A., Sen, A., Sen, U.: Freezing of quantum correlations under local decoherence. Phys. Rev. A 91, 062119 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    Maziero, J., Werlang, T., Fanchini, F.F., Celeri, L.C., Serra, R.M.: System-reservoir dynamics of quantum and classical correlations. Phys. Rev. A 81, 022116 (2010)ADSCrossRefGoogle Scholar
  28. 28.
    Hua, M., Tian, D.: Preservation of the geometric quantum discord in noisy environments. Ann. Phys. (NY) 343, 132–140 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Haikka, P., Johnson, T.H., Maniscalco1, S.: Non-Markovianity of local dephasing channels and time-invariant discord. Phys. Rev. A 87, 010103 (2013)Google Scholar
  30. 30.
    Ma, W., Xu, S., Shi, J., Ye, L.: Quantum correlation versus Bell-inequality violation under the amplitude damping channel. Phys. Lett. A 379, 2802–2807 (2015)ADSCrossRefzbMATHGoogle Scholar
  31. 31.
    Ciccarello, F., Giovannetti, V.: Creating quantum correlations through local nonunitary memoryless channels. Phys. Rev. A 85, 010102 (2012)ADSCrossRefGoogle Scholar
  32. 32.
    Shi, J.D., Wang, D., Ma, W.C., Ye, L.: Enhancing quantum correlation in open-system dynamics by reliable quantum operations. Quant. Inf. Process. 14, 3569–3579 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Karmakar, S., Sen, A., Bhar, A., Sarkar, D.: Effect of local filtering on freezing phenomena of quantum correlation. Quant. Inf. Process. 14, 2517–2533 (2015)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Obada, A.S.F., Hessian, H.A., Mohamed, A.B.A., Hashem, M.: Stationary discord and non-local correlations via qubit damping. J. Mod. Opt. 62, 918–926 (2015)ADSCrossRefGoogle Scholar
  35. 35.
    Li, Y., Xiao, X.: Recovering quantum correlations from amplitude damping decoherence by weak measurement reversal. Quant. Inf. Process. 12, 3067–3077 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Wang, C., Li, C., Nie, L., Li, X., Li, J.: Classical correlation, quantum discord and entanglement for two-qubit system subject to heat bath. Opt. Commun. 284, 2393–2401 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    Dajka, J., Mierzejewski, M., Luczka, J., Blattmann, R., Hänggi, P.: Negativity and quantum discord in Davies environment. J. Phys. A 45, 485306 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kondo, I., Ban, M.: Quantum discord for a bipartite qubit system in the Bloch channel. Quant. Inf. Process. 16, 196 (2017)ADSCrossRefzbMATHGoogle Scholar
  39. 39.
    Zheng, S., Guo, G.: Efficient scheme for two-atom entanglement and quantum information processing in cavity QED. Phys. Rev. Lett. 85, 2392–3295 (2000)ADSCrossRefGoogle Scholar
  40. 40.
    Walls, D.F., Milburn, G.J.: Quantum Optics. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  41. 41.
    Faria, J.G.P, Nemes, M.C.: Dissipative dynamics of the Jaynes-Cummings model in the dispersive approximation: analytical results. Phys. Rev. A 59, 3918–3925 (1999)ADSCrossRefGoogle Scholar
  42. 42.
    Zhang, J., Abdel-Aty, M.: Two atoms in dissipative cavities in dispersive limit: entanglement sudden death and long-lived entanglement. J. Phys. B 43, 025501 (2010)ADSCrossRefGoogle Scholar
  43. 43.
    Ban, M.: Nonequilibrium dynamics of the dispersive Jaynes-Cummings model by non-Markovian quantum master equation. J. Phys. A 43, 335305 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ban, M.: Exact time-evolution of the dispersive Jaynes-Cummings model: effect of initial correlation and master equation approach. J. Mod. Opt. 58, 640–651 (2011)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Ban, M.: Quantum master equation with damping operator including interaction effect for the Raman-coupled model with cavity damping. Int. J. Theor. Phys. 51, 151–166 (2012)CrossRefzbMATHGoogle Scholar
  46. 46.
    Zidan, N.: Entanglement and quantum discord of two moving atoms. Appl. Math. 5, 2485–2492 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Yao-Hua, H., Jun-Qiang, W.: Quantum correlations between two non-interacting atoms under the influence of a thermal environment. Chin. Phys. B 21, 014203 (2012)ADSCrossRefGoogle Scholar
  48. 48.
    Mirzaee, M.: Quantum discord of photon interaction with two two-level atoms under damping. Armenian J. Phys. 8, 114–121 (2015)Google Scholar
  49. 49.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  50. 50.
    Ban, M.: SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics. J. Math. Phys. 33, 3213–2228 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Ban, M.: Lie-algebra methods in quantum optics: Liouville-space formulation. Phys. Rev. A 47, 5093–5119 (1993)ADSCrossRefGoogle Scholar
  52. 52.
    Ban, M.: Decomposition formulas for SU(1,1) and SU(2) Lie algebras and their applications in quantum optics. J. Opt. Soc. Am. B 10, 1347–1359 (1993)ADSCrossRefGoogle Scholar
  53. 53.
    Wooters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245–2248 (1998)ADSCrossRefzbMATHGoogle Scholar
  54. 54.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A 34, 6899–6905 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Humanities and SciencesOchanomizu UniversityTokyoJapan

Personalised recommendations