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Entanglement dynamics of two coupled spins interacting with an adjustable spin bath: effect of an exponential variable magnetic field

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Abstract

The present work is devoted to studying the entanglement dynamics of two central spins coupled in a spin environment and subjected, simultaneously, to an external magnetic field changing with time t as an exponential function \({\mathfrak {B}}\left( 1-\mathrm{e}^{-\lambda t}\right) \). We want to determine whether interaction among central spins with an external magnetic field as well as preparation of bath in an appropriate spin coherent state, \(|\beta \rangle _\mathrm{{bath}}\), is shown to affect the decoherence process in a qualitatively significant manner. We show that the dynamics of the entanglement depends on the initial state of the central spins as well as the bath, the coupling constants and the strength of a magnetic field, \({\mathfrak {B}}, \lambda \). Compared with some cases already discussed in the literature as magnetic fields of periodic \(\sin (\lambda t)\) and \(\cos (\lambda t)\) functions, we can see that a magnetic field of exponential function \(\mathrm{e}^{-\lambda t}\) plays a very crucial role in the entanglement generation between the two-spin qubits and its protection. To do this, we use an operator technique of the Holstein–Primakoff transformation, and the dynamics of the reduced density matrix of two coupled spin qubits is obtained in both finite and infinite numbers of bath spins. We also derive the concurrence measure to quantify the entanglement of the reduced density matrix of the two coupled central spins and look for conditions that provide information on whether this becomes robust against decoherence. It has been shown that the entanglement distribution can be both amplified, stabilized and protected with \({\mathfrak {B}}, \lambda \) and \(\beta \). These results motivate developments toward the implementation or simulation of the purely theoretical model employing exponential fields.

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References

  1. Schrödinger, E.: The current situation in quantum mechanics. Naturwissenschaften 23, 807 (1935)

    ADS  MATH  Google Scholar 

  2. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  3. Barenco, A., Deutsch, D., Ekert, A., Jozsa, R.: Conditional quantum dynamics and logic gates. Phys. Rev. Lett. 74, 4083 (1995)

    ADS  Google Scholar 

  4. DiVincenzo, D.P.: Quantum computation. Science 270, 255 (1995)

    ADS  MathSciNet  MATH  Google Scholar 

  5. Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  6. Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Geometry of quantum states: an introduction to quantum entanglement. Phys. Rev. Lett. 70, 1895 (1993)

    ADS  MathSciNet  MATH  Google Scholar 

  7. Li, X., Pan, Q., Jing, J., Zhang, J., Xie, C., Peng, K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 88, 047904-1 (2002)

    ADS  Google Scholar 

  8. Bennett, C.H., Wiesner, S.J.: Quantum dense coding exploiting a bright Einstein–Podolsky–Rosen beam. Phys. Rev. Lett 69, 2881 (1992)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Bennett, C.H.: Communication via one-and two-particle operators on Einstein–Podolsky–Rosen states. Phys. Rev. Lett. 68, 3121 (1992)

    ADS  MathSciNet  Google Scholar 

  10. Ekert, A.K.: Quantum cryptography using any two nonorthogonal states. Phys. Rev. Lett. 67, 661 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  11. Loss, D., DiVincenzo, D.P.: Quantum cryptography based on Bell’s theorem. Phys. Rev. A 57, 120 (1998)

    ADS  Google Scholar 

  12. Burkard, G., Loss, D., DiVincenzo, D.P.: Coupled quantum dots as quantum gates. Phys. Rev. B 59, 2070 (1999)

    ADS  Google Scholar 

  13. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford University Press, Oxford (2002)

    MATH  Google Scholar 

  14. Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715 (2003)

    ADS  MathSciNet  MATH  Google Scholar 

  15. DiVincenzo, D.P., Loss, D.: Quantum computers and quantum coherence. J. Magn. Magn. Mater. 200, 202 (2000)

    ADS  Google Scholar 

  16. Modi, K.: Preparation of states in open quantum mechanics. Open Syst. Inf. Dyn. 18, 253 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Zurek, W.H.: Decoherence and the transition from quantum to classical. Phys. Today 44, 36 (1991)

    Google Scholar 

  18. Paz, J.P., Zurek, W.H.: Environment-induced decoherence and the transition from quantum to classical. In: Kaiser, R., Westbrook, C., David, F. (eds.) Coherent Matter Waves. Les Houches Session LXXII, EDP Sciences, pp. 533–614. Springer, Berlin (2001)

    Google Scholar 

  19. Cucchietti, F.M., Paz, J.P., Zurek, W.H.: Decoherence from spin environments. Phys. Rev. A 72, 052113 (2005)

    ADS  Google Scholar 

  20. Abragam, A.: The Principles of Nuclear Magnetism. Clarendon Press, Oxford (1978)

    Google Scholar 

  21. Cheung, T.T.P.: Spin diffusion in NMR in solids. Phys. Rev. B 23, 1404 (1981)

    ADS  Google Scholar 

  22. Dobrovitski, V.V., De Raedt, H.A.: Efficient scheme for numerical simulations of the spin-bath decoherence. Phys. Rev. E 67, 056702 (2003)

    ADS  Google Scholar 

  23. Schliemann, J., Khaetskii, A.V., Loss, D.: Spin decay and quantum parallelism. Phys. Rev. B 66, 245303 (2002)

    ADS  Google Scholar 

  24. Yu, W.-J., Xu, B.-M., Li, J.Z., Li, H., Shao, B.: Influences of initial states on entanglement dynamics of two central spins in a spin environment. Int. J. Theor. Phys. 55, 1460 (2016)

    MATH  Google Scholar 

  25. Guo, Y., Deng, H.-L.: Entanglement dynamics of two spin qubits in a spin environment with nonuniform coupling. Int. J. Theor. Phys. 53, 1459 (2014)

    MATH  Google Scholar 

  26. Tchoffo, M., Fouokeng, G.C., Tendong, E., Fai, L.C.: Dzyaloshinskii-Moriya interaction effects on the entanglement dynamics of a two qubit XXZ spin system in non-Markovian environment. J. Magn. Magn. Mater. 407, 358 (2016)

    ADS  Google Scholar 

  27. Majeed, M., Chaudhry, A.Z.: Effect of initial system-environment correlations with spin environments. Eur. Phys. J. D 73, 16 (2019)

    ADS  Google Scholar 

  28. Xiao-Zhong, Y., Hsi-Sheng, G., Ka-Di, Z.: Non-Markovian reduced dynamics and entanglement evolution of two coupled spins in a quantum spin environment. Phys. Rev. B 75, 045331 (2007)

    ADS  Google Scholar 

  29. Xiao-Zhong, Y., Hsi-Sheng, G., Ka-Di, Z.: Dynamics of a central electron spin coupled to an anti-ferromagnetic spin bath driven by a variable magnetic field in the Landau-Zener scenario. New J. Phys. 9, 219 (2007)

    Google Scholar 

  30. Tchoffo, M., Fouokeng, G.C., Massou, S., Afuoti, N.E., Nsangou, I., Fai, L.C., Tchouadeu, A.G., Kenn, J.-P.: Effect of the variable B-field on the dynamic of a central electron spin coupled to an anti-ferromagnetic qubit bath. World J. Cond. Matter Phys. 2, 246 (2012)

    ADS  Google Scholar 

  31. Semenov, Y.G., Kim, K.W.: Effect of an external magnetic field on electron-spin dephasing induced by hyperfine interaction in quantum dots. Phys. Rev. B 67, 073301 (2003)

    ADS  Google Scholar 

  32. Kane, B.E.: A silicon-based nuclear spin quantum computer. Nature 393, 133 (1998)

    ADS  Google Scholar 

  33. Kamtaand, G.L., Starace, A.F.: Anisotropy and magnetic field effects on the entanglement of a two qubit Heisenberg XY chain. Phys. Rev. Lett. 88, 107901 (2002)

    ADS  Google Scholar 

  34. Sun, Y., Chen, Y.G., Chen, H.: Thermal entanglement in the two-qubit Heisenberg XY model under a nonuniform external magnetic field. Phys. Rev. A 68, 044301 (2001)

    ADS  Google Scholar 

  35. Li, D.C., Cao, Z.L.: Entanglement in the anisotropic Heisenberg XYZ model with different Dzyaloshinskii–Moriya interaction and inhomogeneous magnetic field. Eur. Phys. J. D 50, 207 (2008)

    ADS  MathSciNet  Google Scholar 

  36. Lucamarini, M., Paganelli, S., Mancini, S.: Two-qubit entanglement dynamics in a symmetry-broken environment. Phys. Rev. A 69, 062308 (2004)

    ADS  Google Scholar 

  37. Lages, J., et al.: Decoherence by a chaotic many-spin bath. Phys. Rev. E 72, 026225 (2005)

    ADS  Google Scholar 

  38. Zyczkowski, K., Horodecki, P., Horodecki, M., Horodecki, R.: Dynamics of quantum entanglement. Phys. Rev. A 65, 012101 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  39. Yu, T., Eberly, J.H.: Finite-time disentanglement via spontaneous emission. Phys. Rev. Lett. 93, 140404 (2004)

    ADS  Google Scholar 

  40. Yu, T., Eberly, J.H.: Sudden death of entanglement. Science 323, 598 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  41. Lopez, C., Romero, G., Lastra, F., Solano, E., Retamal, J.: Sudden birth versus sudden death of entanglement in multipartite systems. Phys. Rev. Lett. 101, 080503 (2008)

    ADS  Google Scholar 

  42. Almeida, M.P., de Melo, F., Hor-Meyll, M., Salles, A., Walborn, S., Ribeiro, P.S., Davidovich, L.: Environment-induced sudden death of entanglement. Science 316, 579 (2007)

    ADS  Google Scholar 

  43. Laurat, J., Choi, K., Deng, H., Chou, C., Kimble, H.: Heralded entanglement between atomic ensembles: preparation, decoherence, and scaling. Phys. Rev. Lett. 99, 180504 (2007)

    ADS  Google Scholar 

  44. Barbosa, F., Coelho, A., De Faria, A., Cassemiro, K., Villar, A., Nussenzveig, P., Martinelli, M.: Robustness of bipartite Gaussian entangled beams propagating in lossy channels. Nat. Photonics 4, 858 (2010)

    ADS  Google Scholar 

  45. Lo Franco, R., 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)

    ADS  MathSciNet  MATH  Google Scholar 

  46. Leggio, B., Lo Franco, R., Soares-Pinto, D.O., Horodecki, P., Compagno, G.: Distributed correlations and information flows within a hybrid multipartite quantum-classical system. Phys. Rev. A 92, 032311 (2015)

    ADS  Google Scholar 

  47. Caruso, F., Giovannetti, V., Lupo, C., Mancini, S.: Quantum channels and memory effects. Rev. Mod. Phys. 86, 1203 (2014)

    ADS  Google Scholar 

  48. Rivas, A., Huelga, S.F., Plenio, M.B.: Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 77, 094001 (2014)

    ADS  MathSciNet  Google Scholar 

  49. Agarwal, G.S.: Control of decoherence and relaxation by frequency modulation of a heat bath. Phys. Rev. A 61, 013809 (1999)

    ADS  Google Scholar 

  50. Man, Z.-X., Xia, Y.-J., Lo Franco, R.: Harnessing non-Markovian quantum memory by environmental coupling. Phys. Rev. A 92, 012315 (2015)

    ADS  Google Scholar 

  51. D’Arrigo, A., Lo Franco, R., Benenti, G., Paladino, E., Falci, G.: Recovering entanglement by local operations. Ann. Phys. 350, 211 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Lo Franco, R., Bellomo, B., Andersson, E., Compagno, G.: Revival of quantum correlations without system-environment back-action. Phys. Rev. A 85, 032318 (2012)

    ADS  Google Scholar 

  53. Lo Franco, R.: Nonlocality threshold for entanglement under general dephasing evolutions: a case study. Quantum Inf. Process. 15, 2393 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  54. Lo Franco, R., D’Arrigo, A., Falci, G., Compagno, G., Paladino, E.: Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling. Phys. Rev. B 90, 054304 (2014)

    ADS  Google Scholar 

  55. Breuer, H.-P., Laine, E.-M., Piilo, J., Vacchini, B.: Nonlocality threshold for entanglement under general dephasing evolutions: a case study. Rev. Mod. Phys. 88, 021002 (2016)

    ADS  Google Scholar 

  56. Huelga, S.F., Rivas, A., Plenio, M.B.: Preserving entanglement and nonlocality in solid-state qubits by dynamical decoupling. Phys. Rev. Lett. 108, 160402 (2012)

    ADS  Google Scholar 

  57. Liu, B.H., Li, L., Huang, Y.F., Li, C.F., Guo, G.C., Laine, E.M., Breuer, H.P., Piilo, J.: Colloquium: Non-Markovian dynamics in open quantum systems. Nat. Phys. 7, 931 (2011)

    Google Scholar 

  58. Paladino, E., Galperin, Y.M., Falci, G., Altshuler, B.L.: 1/f noise: implications for solid-state quantum information. Rev. Mod. Phys. 86, 361 (2014)

    ADS  Google Scholar 

  59. Dehghani, A., Mojaveri, B., Jafarzadeh Bahrbeig, R., Nosrati, F., Lo Franco, R.: Entanglement transfer in a noisy cavity network with parity-deformed fields. J. Opt. Soc. Am. B 36, 1858 (2019)

    ADS  Google Scholar 

  60. Huang, Z., Kais, S.: Entanglement evolution of one-dimensional spin systems in external magnetic fields. Phys. Rev. A 73, 022339 (2006)

    ADS  Google Scholar 

  61. Paganelli, S., de Pasquale, F., Giampaolo, S.M.: Decoherence slowing down in a symmetry-broken environment. Phys. Rev. A 66, 052317 (2002)

    ADS  Google Scholar 

  62. Lucamarini, M., Paganelli, S., Mancini, S.: Two-qubit entanglement dynamics in a symmetry-broken environment. Phys. Rev. A 69, 062308 (2004)

    ADS  Google Scholar 

  63. San Ma, X., Min Wang, A., Dong Yang, X., You, H.: Entanglement dynamics and decoherence of three-qubit system in a fermionic environment. J. Phys. A 38, 2761 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  64. Breuer, H.P.: Exact quantum jump approach to open systems in bosonic and spin baths. Phys. Rev. A 69, 022115 (2004)

    ADS  Google Scholar 

  65. Breuer, H.P., Burgarth, D., Petruccione, F.: Non-Markovian dynamics in a spin star system: exact solution and approximation techniques. Phys. Rev. B 70, 045323 (2004)

    ADS  Google Scholar 

  66. Frasca, M.: 1/N-Expansion for the Dicke model and the decoherence program. Ann. Phys. 313, 26 (2004)

    ADS  MATH  Google Scholar 

  67. Wootters, W.K.: Entanglement of formation and concurrence. Quantum Inf. Comput. 1, 27 (2001)

    MathSciNet  MATH  Google Scholar 

  68. Zurek, W.H., Habib, S., Paz, J.P.: Coherent states via decoherence. Phys. Rev. Lett. 70, 1187 (1993)

    ADS  Google Scholar 

  69. Kim, M.S., Lee, J., Ahn, D., Knight, P.L.: Entanglement induced by a single-mode heat environment. Phys. Rev. A 65, 040101(R) (2002)

    ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Payame Noor University, P.O. Box 19395-3697, Tehran, Islamic Republic of Iran

    A. Dehghani

  2. Department of Physics, Azarbaijan Shahid Madani University, P.O. Box 51745-406, Tabriz, Islamic Republic of Iran

    B. Mojaveri & M. Vaez

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  1. A. Dehghani
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  2. B. Mojaveri
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Dehghani, A., Mojaveri, B. & Vaez, M. Entanglement dynamics of two coupled spins interacting with an adjustable spin bath: effect of an exponential variable magnetic field. Quantum Inf Process 19, 306 (2020). https://doi.org/10.1007/s11128-020-02803-5

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