Entanglement of two superconducting qubits induced by a thermal noise of a cavity with Kerr medium taking into account the atomic coherence

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The system consisting of two identical artificial atoms (qubits), resonantly interacting with the mode of quantum field of an ideal microwave cavity in the presence of Kerr nonlinearity, is considered. For the considered model, an exact solution of the quantum Liouville equation for the full density matrix of the system «two qubits + resonator field mode» is obtained. To solve the quantum equation of evolution, the representation of «dressed» states, that is, the eigenfunctions of the Hamiltonian, was used. A complete set of «dressed» states of the considered model is found. With its help, the solution of the evolution equation was initially found for coherent initial states of qubits and Fock states of the field, that is, states with a certain number of photons in the resonator mode. Then, the above solution was generalized to the case of the thermal state of the resonator field. A reduced density matrix of two qubits is found by averaging over the field variables. The two-qubit density matrix is used to calculate the parameter of qubit entanglement in the analytical form. Concurrence was chosen as a quantitative criterion for qubit entanglement. A numerical simulation of the time dependence of the consistency of qubits for various parameters of the model and the initial states of qubits was carried out. The most interesting result seems to be that taking into account the initial coherence of qubits in the model with Kerr nonlinearity leads to a significant increase in the maximum degree of entanglement of qubits induced by the thermal field, even in the case of high intensities of the resonator field.

About the authors

Evgeny K. Bashkirov

Samara National Research University

Author for correspondence.
Email: bash@samsu.ru


  1. Dutra S.M. Cavity Quantum Electrodynamics: The Strange Theory of Light in a Box. Hoboken: John Wiley & Sons, 2005, 389 p. DOI: https://doi.org/10.1002/0471713465
  2. Buluta I., Ashab S., Nori F. Neutral and artificial atoms for quantum computation. Rep. Prog. Phys., 2011, vol. 74, no. 10, p. 104401. DOI: https://doi.org/10.1088/0034-4885/74/10/104401
  3. Walther H. et al. Cavity quantum electrodynamics. Rep. Prog. Phys., 2006, vol. 69, pp. 1325–1382. DOI: https://doi.org/10.1088/0034-4885/69/5/R02
  4. Leibfried D. et al. Quantum dynamics of single trapped ions. Rev. Mod. Phys., 2003, vol. 75, pp. 281–324. DOI: https://doi.org/10.1103/RevModPhys.75.281
  5. Xiang Z.-L. et al. Hybrid quantum circuits: Superconducting circuits interacting with other quantum systems. Rev. Mod. Phys., 2013, vol. 85, pp. 623–653. DOI: https://doi.org/10.1103/RevModPhys.85.623
  6. Georgescu I.M., Ashhab S., Nori F. Quantum simulation. Rev. Mod. Phys., 2014, vol. 88, pp. 153–186. DOI: https://doi.org/10.1103/RevModPhys.86.153
  7. Gu X. et al. Microwave photonics with superconducting quantum circuits. Phys. Repts., 2017, vol. 718–719, pp. 1–102. DOI: https://doi.org/10.1016/j.physrep.2017.10.002
  8. Wendin G. Quantum information processing with superconducting circuits: a review. Rep. Prog. Phys., 2017, vol. 80, p. 106001. DOI: https://doi.org/10.1088/1361-6633/aa7e1a
  9. Li G.-Q., Pan X.-Y. Quantum information processing with nitrogen–vacancy centers in diamond. Chinese Physics, 2018, vol. 27, no. 2, p. 020304. DOI: https://doi.org/10.1088/1674-1056/27/2/020304
  10. Jaynes E.T., Cummings F.W. Comparison of quantum and semiclassical radiation theory with application to the beam maser. Proc. IEEE, 1963, vol. 51, no. 1, pp. 89–109. DOI: https://doi.org/10.1109/PROC.1963.1664
  11. Shore B.W., Knight P.L. The Jaynes–Cummings model. J. Mod. Opt., 1993, vol. 40, pp. 1195–1238. DOI: https://doi.org/10.1080/09500349314551321
  12. Larson J. Dynamics of the Jaynes–Cummings and Rabi models: Old wine in new bottles. Physica Scripta, 2007, vol. 76, pp. 146–160. DOI: https://doi.org/10.1088/0031-8949/76/2/007
  13. Yu H. et al. Quantum correlations between light and the kilogram-mass mirrors of LIGO. Nature, 2020, vol. 583, no. 12, pp. 43–47. DOI: https://doi.org/10.1038/s41586-020-2420-8
  14. Dai H., Fu S., Luo S. Atomic nonclassicality in the Jaynes-Cummings model. Phys. Lett. A, 2020, vol. 384, no. 12, p. 126371. DOI: https://doi.org/10.1016/j.physleta.2020.126371
  15. Puri S., Boutin S., Blais A. Engineering the quantum states of light in a Kerr-nonlinear resonator by two-photon driving. Quantum Information, 2017, vol. 3, no. 1, p. 18. DOI: https://doi.org/10.1038/s41534-017-0019-1
  16. Al Naim A.F. et al. Effects of Kerr medium in coupled cavities on quantum state transfer. J. Nonlin. Opt. Phys. Mater., 2018, vol. 27, no. 3, p. 1850035. DOI: https://doi.org/10.1142/S0218863518500352
  17. Al Naim A.F. et al. Effects of Kerr medium and Stark shift parameter on Wehrl entropy and the field puruty for two-photon Jaynes–Cummings model under dispersive approximation. J. Rus. Las. Res., 2019, vol. 40, no. 1, pp. 20‒29. DOI: https://doi.org/10.1007/s10946-019-09764-w
  18. Anwar S.J., Ramzan M., Khan M.K. Effect of Stark- and Kerr-like medium on the entanglement dynamics of two three-level atomic systems. Quant. Inform. Process, 2019, vol. 18, no. 6, p. 192. DOI: https://doi.org/10.1007/s11128-019-2277-7
  19. Adanmitonde A.J., Avosvu G.I.Yu., Dosa F.A. On the quantization of some generalized Jaynes–Cummings models in a Kerr-like environment. Teor. matem. fiz. Fiz.-mat. nauki, 2020, vol. 203, no. 3, pp. 451–466. DOI: https://doi.org/10.4213/tmf9835 (In Russ.)
  20. Aldaghfag S.A., Berrada K., Abdel-Khalek S. Entanglement and photon statistics of two dipole-dipole coupled superconducting qubits with Kerr-like nonlinearities. Results in Phys., 2020, vol. 16, p. 102978. DOI: https://doi.org/10.1016/j.rinp.2020.102978
  21. Bergeal N. et al. Phase-preserving amplification near the quantum limit with a Josephson ring modulator. Nature, 2010, vol. 465, no. 7294. pp. 64–68. DOI: https://doi.org/10.1038/nature09035
  22. Kirchmair G. et al. Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature. 2013. Vol. 495, no. 7440, pp. 205–209. DOI: https://doi.org/10.1038/nature11902
  23. Kim M.S. et al. Entanglement induced by a single-mode heat environment. Phys. Rev. A, 2002, vol. 65, no. 4, p. 040101. DOI: https://doi.org/10.1103/PhysRevA.65.040101
  24. Zhou L., Song H.S. Entanglement induced by a single-mode thermal field and criteria for entanglement. J. Opt. B, 2002, vol. 4, no. 6, pp. 425–429. DOI: https://doi.org/10.1088/1464-4266/4/6/310
  25. Bashkirov E.K. Entanglement induced by the two-mode thermal noise. Laser Phys. Lett., 2006, vol. 3, no. 3, pp. 145–150. DOI: https://doi.org/10.1002/lapl.200510081
  26. Bashkirov E.K., Stupatskaya M.P. The entanglement of two dipole-dipole coupled atoms induced by nondegenerate two-mode thermal noise. Laser Phys., 2009, vol. 19, no. 3, pp. 525–530. DOI: https://doi.org/10.1134/S1054660X09030281
  27. Bashkirov E.K., Mastyugin M.S. The influence of the dipole-dipole interaction and atomic coherence on the entanglement of two atoms with degenerate two-photon transitions. Opt. Spectr., 2014, vol. 116, no. 4, pp. 630–634. DOI: https://doi.org/10.1134/S0030400X14040067
  28. Zhang B. Entanglement between two qubits interacting with a slightly detuned thermal feld. Opt. Commun., 2010, vol. 283, no. 23, pp. 4676–4679. DOI: https://doi.org/10.1016/j.optcom.2010.06.094
  29. Hu Y.-H. et al. Coherence-enhanced entanglement between two atoms at high temperature. Chin. Phys. B, 2008, vol. 17, no. 5, pp. 1784–1790. DOI: https://doi.org/10.1088/1674-1056/17/5/039
  30. Bashkirov E.K., Mangulova E.G. Atomic entanglement induced by two-mode thermal noise in the presence of dipole-dipole interaction and atomic coherence. Vestn. Sam. gos. tehn. un-ta. Ser. Fiz.-mat. nauki, 2013, vol. 2, no. 31, pp. 177–184. DOI: https://doi.org/10.14498/vsgtu1160 (In Russ.)
  31. Evseev M.M., Bashkirov E.K. Thermal entanglement in Tavis–Cummings model with Kerr nonlinearity. 2020 International Conference on Information Technology and Nanotechnology (ITNT), 2020, p. 9253347. DOI: https://doi.org/10.1109/ITNT49337.2020.9253347
  32. Bashkirov E.K. Entanglement in Tavis–Cummings model with Kerr nonlinearity induced by a thermal noise. Proc. SPIE, 2021, vol. 11846, pp. 210–219. DOI: https://doi.org/10.1117/12.2588673
  33. Bashkirov E.K. Entanglement of two dipole-coupled qubits interacting with a detuned cavity thermal field. Proc. SPIE, 2019, vol. 11066, pp. 115–121. DOI: https://doi.org/10.1117/12.2522476

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