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Performance of an XXX Heisenberg model-based quantum heat engine and tripartite entanglement

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Abstract

We study the thermal entanglement between three qubits in a N-qubits isotropic spin \(\frac{1}{2}\) Heisenberg XXX chain, using the lower bound of concurrence. In particular, we show the dependence of the tripartite entanglement in terms of the magnetic field, temperature, the number of sites N in the chain and the lattice spacing between every three qubits. A N-qubit quantum heat engine is then constructed based on this multiqubit Heisenberg spin \(\frac{1}{2}\) XXX model, and the variation of different thermodynamic quantities (efficiency, work and heat released and absorbed) is studied with respect to the tripartite thermal entanglement in zero and nonzero magnetic field, as well as for odd and even N chains. The conditions for which the second law of thermodynamics is always preserved are established.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

References

  1. Schrödinger, E.: Discussion of probability relations between separated systems. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31, pp. 555–563. Cambridge University Press (1935)

  2. Murao, M., Jonathan, D., Plenio, M.B., Vedral, V.: Quantum telecloning and multiparticle entanglement. Phys. Rev. A 59(1), 156 (1999)

    ADS  Google Scholar 

  3. Nielsen, M.A., Chuang, I:. Quantum computation and quantum information (2002)

  4. Arnesen, M.C., Bose, S., Vedral, V.: Natural thermal and magnetic entanglement in the 1d Heisenberg model. Phys. Rev. Lett. 87(1), 017901 (2001)

    ADS  Google Scholar 

  5. Wang, Xiaoguang: Entanglement in the quantum Heisenberg xy model. Phys. Rev. A 64(1), 012313 (2001)

    ADS  Google Scholar 

  6. Kamta, G.L., Starace, A.F.: Anisotropy and magnetic field effects on the entanglement of a two qubit Heisenberg xy chain. Phys. Rev. Lett. 88(10), 107901 (2002)

    ADS  Google Scholar 

  7. Sun, Yang, Chen, Yuguang, Chen, Hong: Thermal entanglement in the two-qubit Heisenberg xy model under a nonuniform external magnetic field. Phys. Rev. A 68(4), 044301 (2003)

    ADS  Google Scholar 

  8. Wang, X., Fu, H., Solomon, A.I.: Thermal entanglement in three-qubit Heisenberg models. J. Phys. A Math. General 34(50), 11307 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  9. Wang, Xiaoguang, Zanardi, Paolo: Quantum entanglement and bell inequalities in Heisenberg spin chains. Phys. Lett. A 301(1–2), 1–6 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Coffman, V., Kundu, J., Wootters, W.K.: Distributed entanglement. Phys. Rev. A 61(5), 052306 (2000)

    ADS  Google Scholar 

  11. Mintert, Florian, Kuś, Marek, Buchleitner, Andreas: Concurrence of mixed bipartite quantum states in arbitrary dimensions. Phys. Rev. Lett. 92(16), 167902 (2004)

    ADS  Google Scholar 

  12. Li, Ming, Fei, Shao-Ming., Wang, Zhi-Xi.: A lower bound of concurrence for multipartite quantum states. J. Phys. A: Math. Theor. 42(14), 145303 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  13. Abaach, Sanaa, Faqir, Mustapha, El Baz, Morad: Long-range entanglement in quantum dots with Fermi-Hubbard physics. Phys. Rev. A 106, 022421 (2022)

    ADS  Google Scholar 

  14. Geva, E., Kosloff, R.: A quantum-mechanical heat engine operating in finite time. A model consisting of spin-1/2 systems as the working fluid. J. Chem. Phys. 96(4), 3054–3067 (1992)

    ADS  Google Scholar 

  15. He, Jizhou, Chen, Jincan, Hua, Ben: Quantum refrigeration cycles using spin-1 2 systems as the working substance. Phys. Rev. E 65(3), 036145 (2002)

    ADS  Google Scholar 

  16. Feng, Wu., Chen, Lingen, Sun, Fengrui, Chih, Wu., Li, Qing: Generalized model and optimum performance of an irreversible quantum Brayton engine with spin systems. Phys. Rev. E 73(1), 016103 (2006)

    ADS  Google Scholar 

  17. Feldmann, Tova, Kosloff, Ronnie: Performance of discrete heat engines and heat pumps in finite time. Phys. Rev. E 61(5), 4774 (2000)

    ADS  Google Scholar 

  18. Feldmann, Tova, Kosloff, Ronnie: Quantum four-stroke heat engine: thermodynamic observables in a model with intrinsic friction. Phys. Rev. E 68(1), 016101 (2003)

    ADS  Google Scholar 

  19. Feldmann, Tova, Kosloff, Ronnie: Characteristics of the limit cycle of a reciprocating quantum heat engine. Phys. Rev. E 70(4), 046110 (2004)

    ADS  Google Scholar 

  20. Feng, Wu., Chen, Lingen, Shuang, Wu., Sun, Fengrui, Chih, Wu.: Performance of an irreversible quantum Carnot engine with spin 1/ 2. J. Chem. Phys. 124(21), 214702 (2006)

    ADS  Google Scholar 

  21. Wang, JianHui, He, JiZhou, Mao, ZhiYuan: Performance of a quantum heat engine cycle working with harmonic oscillator systems. Sci. China, Ser. G 50(2), 163–176 (2007)

    MATH  Google Scholar 

  22. Chen, L.G., Liu, X.W., Ge, Y.L., Wu, F., Sun, F.R.: Ecological optimisation of irreversible harmonic oscillator Carnot refrigerator. J. Energy Inst. 86(2), 85–96 (2013)

    Google Scholar 

  23. Wang, Xinzhi, He, Yurong, Liu, Xing, Cheng, Gong, Zhu, Jiaqi: Solar steam generation through bio-inspired interface heating of broadband-absorbing plasmonic membranes. Appl. Energy 195, 414–425 (2017)

    Google Scholar 

  24. Geva, Eitan, Kosloff, Ronnie: On the classical limit of quantum thermodynamics in finite time. J. Chem. Phys. 97(6), 4398–4412 (1992)

    ADS  Google Scholar 

  25. Lin, Bihong, Chen, Jincan: Performance analysis of an irreversible quantum heat engine working with harmonic oscillators. Phys. Rev. E 67(4), 046105 (2003)

    ADS  Google Scholar 

  26. Lin, Bihong, Chen, Jincan: General performance characteristics of a quantum heat pump cycle using harmonic oscillators as the working substance. Phys. Scr. 71(1), 12 (2005)

    ADS  MATH  Google Scholar 

  27. Wang, Jianhui, He, Jizhou, Xin, Yong: Performance analysis of a spin quantum heat engine cycle with internal friction. Phys. Scr. 75(2), 227 (2007)

    ADS  MATH  Google Scholar 

  28. He, JiZhou, He, Xian, Tang, Wei: The performance characteristics of an irreversible quantum Otto harmonic refrigeration cycle. Sci. China, Ser. G 52(9), 1317–1323 (2009)

    MathSciNet  Google Scholar 

  29. Rezek, Yair, Kosloff, Ronnie: Irreversible performance of a quantum harmonic heat engine. New J. Phys. 8(5), 83 (2006)

    ADS  Google Scholar 

  30. Quan, H.T., Zhang, P., Sun, C.P.: Quantum-classical transition of photon-Carnot engine induced by quantum decoherence. Phys. Rev. E 73(3), 036122 (2006)

    ADS  Google Scholar 

  31. Scully, M.O., Zubairy, M.S., Agarwal, G.S., Walther, H.: Extracting work from a single heat bath via vanishing quantum coherence. Science 299(5608), 862–864 (2003)

    ADS  Google Scholar 

  32. Quan, Hai-Tao., Liu, Yu.-xi, Sun, Chang-Pu., Nori, Franco: Quantum thermodynamic cycles and quantum heat engines. Phys. Rev. E 76(3), 031105 (2007)

    ADS  MathSciNet  Google Scholar 

  33. Açıkkalp, Emin, Caner, Necmettin: Application of exergetic sustainable index to the quantum irreversible diesel refrigerator cycles for 1d box system. Eur. Phys. J. Plus 130(4), 1–8 (2015)

    ADS  Google Scholar 

  34. Scully, M.O.: Extracting work from a single thermal bath via quantum Negentropy. Phys. Rev. Lett. 87(22), 220601 (2001)

    ADS  Google Scholar 

  35. Henrich, M.J., Mahler, G., Michel, M.: Driven spin systems as quantum thermodynamic machines: fundamental limits. Phys. Rev. E 75(5), 051118 (2007)

    ADS  Google Scholar 

  36. Zhang, Ting, Liu, Wei-Tao., Chen, Ping-Xing., Li, Cheng-Zu.: Four-level entangled quantum heat engines. Phys. Rev. A 75(6), 062102 (2007)

    ADS  Google Scholar 

  37. Kieu, T.D.: Quantum heat engines, the second law and Maxwell’s daemon. Eur. Phys. J. D-Atomic Mol. Opt. Plasma Phys. 39(1), 115–128 (2006)

    ADS  Google Scholar 

  38. Zhang, G.-F.: Entangled quantum heat engines based on two two-spin systems with Dzyaloshinski–Moriya anisotropic antisymmetric interaction. Eur. Phys. J. D 49(1), 123–128 (2008)

    ADS  Google Scholar 

  39. de Oliveira, J.L.D., Rojas, M., Filgueiras, C.: Two coupled double quantum dots systems as an working substance for heat machines. arXiv preprint arXiv:2102.00908 (2021)

  40. Albayrak, Erhan: The entangled quantum heat engine in the various Heisenberg models for a two-qubit system. Int. J. Quantum Inf. 11(02), 1350021 (2013)

    MathSciNet  MATH  Google Scholar 

  41. He, Ji-Zhou., He, Xian, Zheng, Jie: Thermal entangled quantum heat engine working with a three-qubit Heisenberg xx model. Int. J. Theor. Phys. 51(7), 2066–2076 (2012)

    MATH  Google Scholar 

  42. Hill, S., Wootters, W.K.: Entanglement of a pair of quantum bits. Phys. Rev. Lett. 78(26), 5022 (1997)

    ADS  Google Scholar 

  43. Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245 (1998)

    ADS  MATH  Google Scholar 

  44. Horodecki, Ryszard, Horodecki, Paweł, Horodecki, Michał, Horodecki, Karol: Quantum entanglement. Rev. Modern Phys. 81(2), 865 (2009)

    ADS  MathSciNet  MATH  Google Scholar 

  45. An, N.B., Kim, J., Kim, K.: Nonperturbative analysis of entanglement dynamics and control for three qubits in a common Lossy cavity. Phys. Rev. A 82(3), 032316 (2010)

    ADS  MathSciNet  Google Scholar 

  46. Georgievskii, D.V., Shamolin, M.V.: Levi-Civita symbols, generalized vector products, and new integrable cases in mechanics of multidimensional bodies. J. Math. Sci. 187(3), 280–299 (2012)

    MathSciNet  MATH  Google Scholar 

  47. Jie, Ren, Shi-Qun, Zhu: Bipartite and tripartite entanglement in a three-qubit Heisenberg model. Commun. Theor. Phys. 46(6), 969 (2006)

    ADS  Google Scholar 

  48. Ahami, Nizar, El Baz, Morad: Thermal entanglement in a mixed spin Heisenberg xxx chain with DM interaction. Int. J. Quantum Inf. 19(05), 2150021 (2021)

    MathSciNet  MATH  Google Scholar 

  49. Mzaouali, Zakaria, Campbell, Steve, El Baz, Morad: Discrete and generalized phase space techniques in critical quantum spin chains. Phys. Lett. A 383(30), 125932 (2019)

    MathSciNet  MATH  Google Scholar 

  50. Mzaouali, Zakaria, El Baz, Morad: Long range quantum coherence, quantum classical correlations in Heisenberg xx chain. Physica A 518, 119–130 (2019)

    ADS  MathSciNet  MATH  Google Scholar 

  51. Fumani, F.K., Nemati, S., Mahdavifar, S., Darooneh, A.H.: Magnetic entanglement in spin-1/2 xy chains. Physica A Stat. Mech. Appl. 445, 256–263 (2016)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Wang, Xiaoguang: Threshold temperature for pairwise and many-particle thermal entanglement in the isotropic Heisenberg model. Phys. Rev. A 66(4), 044305 (2002)

    ADS  Google Scholar 

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Authors and Affiliations

  1. ESMaR, Faculty of Sciences, Mohammed V University in Rabat, Morocco, Av. Ibn Battouta, B.P. 1014, Agdal, Rabat, Morocco

    Khadija El Hawary & Morad El Baz

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  1. Khadija El Hawary
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  2. Morad El Baz
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Correspondence to Morad El Baz.

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Appendix: Additional figures

Appendix: Additional figures

In Figs. 10 and 11 the external magnetic field is chosen to be \(B = 3\) and \(B = 5\), respectively; all the curves are discontinuous, separated and the isolines of the work and the efficiency are both in the form of double closed loops. By increasing the magnetic field, the second law of thermodynamics is still preserved and the amount of heat absorbed and heat released still satisfy \( Q_1> -Q_2 > 0 \).

Fig. 10
figure 10

Variation of the Q1 (a), Q2 (b), W (c), and \(\eta \) (d) versus \(\textrm{LBC} _1\) and \(\textrm{LBC} _2\) for \(N=3\) and \(B =3\)

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Fig. 11
figure 11

Variation of the \(Q_1\) (a), \(Q_2\) (b), W (c), and \(\eta \) (d) with \(\textrm{LBC}_1\) and \(\textrm{LBC}_2\) for \(N=3\) and \(B =5\)

Full size image

For an anti-ferromagnetic chain with \(N=4\), \(N=5\) and \(N=6\), we contour plot the thermodynamic quantities with respect to the tripartite entanglement of each spin, for an external magnetic field \(B = 1\), and for \(T _1 = 2 \) and \( T _2 = 1\) (Figs. 12, 13, 14).

Fig. 12
figure 12

Variation of a \(Q_1\), b \(Q_2\), c W and d \(\eta \) with \(\textrm{LBC} _1\) and \(\textrm{LBC} _2\) for 3 nearest spins \(N=4\) for \(B =1\)

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Fig. 13
figure 13

Variation of a \(Q_1\), b \(Q_2\), c W and d \(\eta \) with \(\textrm{LBC} _1\) and \(\textrm{LBC} _2\) for 3 nearest spins \(N=5\) and \(B =1\)

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Fig. 14
figure 14

Variation of a \(Q_1\), b \(Q_2\), c W and d \(\eta \) with \(\textrm{LBC} _1\) and \(\textrm{LBC} _2\) for 3 nearest spins \(N=6\) and \(B =1\)

Full size image

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El Hawary, K., El Baz, M. Performance of an XXX Heisenberg model-based quantum heat engine and tripartite entanglement. Quantum Inf Process 22, 190 (2023). https://doi.org/10.1007/s11128-023-03911-8

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