Estimation of temperature in micromaser-type systems

  • B. Farajollahi
  • M. Jafarzadeh
  • H. Rangani Jahromi
  • M. Amniat-Talab
Article
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Abstract

We address the estimation of the number of photons and temperature in a micromaser-type system with Fock state and thermal fields. We analyze the behavior of the quantum Fisher information (QFI) for both fields. In particular, we show that in the Fock state field model, the QFI for non-entangled initial state of the atoms increases monotonously with time, while for entangled initial state of the atoms, it shows oscillatory behavior, leading to non-Markovian dynamics. Moreover, it is observed that the QFI, entropy of entanglement and fidelity have collapse and revival behavior. Focusing on each period that the collapses and revivals occur, we see that the optimal points of the QFI and entanglement coincide. In addition, when one of the subsystems evolved state fidelity becomes maximum, the QFI also achieves its maximum. We also address the evolved fidelity versus the initial state as a good witness of non-Markovianity. Moreover, we interestingly find that the entropy of the composite system can be used as a witness of non-Markovian evolution of the subsystems. For the thermal field model, we similarly investigate the relation among the QFI associated with the temperature, von Neumann entropy, and fidelity. In particular, it is found that at the instants when the maximum values of the QFI are achieved, the entanglement between the two-qubit system and the environment is maximized while the entanglement between the probe and its environment is minimized. Moreover, we show that the thermometry may lead to optimal estimation of practical temperatures. Besides, extending our computation to the two-qubit system, we find that using a two-qubit probe generally leads to more effective estimation than the one-qubit scenario. Finally, we show that initial state entanglement plays a key role in the advent of non-Markovianity and determination of its strength in the composite system and its subsystems.

Keywords

Quantum Fisher information Quantum entanglement Thermal field Micromaser 

Notes

Acknowledgements

We wish to acknowledge the financial support of the MSRT of Iran, Urmia University and Jahrom University.

References

  1. 1.
    Paris, M.G.: Quantum estimation for quantum technology. Int. J. Quant. Inf. 7, 125–137 (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Giovannetti, V., Lloyd, S., Maccone, L.: Advances in quantum metrology. Nat. Photon. 5, 222–229 (2011)ADSCrossRefGoogle Scholar
  3. 3.
    Lu, X.-M., Wang, X., Sun, C.P.: Quantum Fisher information flow and non-Markovian processes of open systems. Phys. Rev. A 82(4), 042103 (2010)ADSCrossRefGoogle Scholar
  4. 4.
    Chang, L., Li, N., Luo, S., Song, H.: Optimal extraction of information from two spins. Phys. Rev. A 89(4), 042110 (2014)ADSCrossRefGoogle Scholar
  5. 5.
    Zhong, W., Sun, Z., Ma, J., Wang, X., Nori, F.: Fisher information under decoherence in Bloch representation. Phys. Rev. A 87(2), 022337 (2013)ADSCrossRefGoogle Scholar
  6. 6.
    Jiang, Z.: Quantum Fisher information for states in exponential form. Phys. Rev. A 89(3), 032128 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Ma, J., Wang, X.: Fisher information and spin squeezing in the Lipkin–Meshkov–Glick model. Phys. Rev. A 80(1), 012318 (2009)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Sun, Z., Ma, J., Lu, X.-M., Wang, X.: Fisher information in a quantum-critical environment. Phys. Rev. A 82(2), 022306 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    Yao, Y., Xiao, X., Ge, L., Wang, X.-G., Sun, C.-P.: Quantum Fisher information in noninertial frames. Phys. Rev. A 89(4), 042336 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Giovannetti, V., Lloyd, S., Maccone, L.: Quantum metrology. Phys. Rev. Lett. 96(1), 010401 (2006)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Berrada, K.: Non-Markovian effect on the precision of parameter estimation. Phys. Rev. A 88(3), 035806 (2013)ADSCrossRefGoogle Scholar
  12. 12.
    Ma, J., Huang, Y.-X., Wang, X., Sun, C.P.: Quantum Fisher information of the Greenberger–Horne–Zeilinger state in decoherence channels. Phys. Rev. A 84(2), 022302 (2011)ADSCrossRefGoogle Scholar
  13. 13.
    Rangani Jahromi, H., Amniat-Talab, M.: Geometric phase, entanglement, and quantum Fisher information near the saturation point. Ann. Phys. 355, 299–312 (2015)ADSMathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rangani Jahromi, H.: Relation between quantum probe and entanglement in n-qubit systems within Markovian and non-Markovian environments. J. Mod. Opt. 64(14), 1377–1385 (2017)ADSCrossRefGoogle Scholar
  15. 15.
    Helstrom, C.: Quantum Detection and Estimation Theory. Elsevier Science, Amsterdam (1976)MATHGoogle Scholar
  16. 16.
    Jozsa, R., Abrams, D.S., Dowling, J.P., Williams, C.P.: Quantum clock synchronization based on shared prior entanglement. Phys. Rev. Lett. 85(9), 2010–2013 (2000)ADSCrossRefGoogle Scholar
  17. 17.
    Peters, A., Chung, K.Y., Chu, S.: Measurement of gravitational acceleration by dropping atoms. Nature 400, 849 (1999)ADSCrossRefGoogle Scholar
  18. 18.
    Bollinger, J.J., Itano, W.M., Wineland, D.J., Heinzen, D.J.: Optimal frequency measurements with maximally correlated states. Phys. Rev. A 54(6), R4649–R4652 (1996)ADSCrossRefGoogle Scholar
  19. 19.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefMATHGoogle Scholar
  20. 20.
    Hayashi, M.: Quantum Information: An Introduction. Springer, Berlin (2006)MATHGoogle Scholar
  21. 21.
    Bengtsson, I., Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2007)MATHGoogle Scholar
  22. 22.
    Amniat-Talab, M., Rangani Jahromi, H.: On the entanglement and engineering phase gates without dynamical phases for a two-qubit system with Dzyaloshinski–Moriya interaction in magnetic field. Quant. Inf. Proc. 12(2), 1185–1199 (2013)CrossRefMATHGoogle Scholar
  23. 23.
    Rangani Jahromi, H., Amniat-Talab, M.: Noncyclic geometric quantum computation and preservation of entanglement for a two-qubit Ising model. Quant. Inf. Proc. 14(10), 3739–3755 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Bennett, C.H., Brassard, G., Crpeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels. Phys. Rev. Lett. 70(13), 1895–1899 (1993)ADSMathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Yin, Z.-Q., Li, H.-W., Chen, W., Han, Z.-F., Guo, G.-C.: Security of counterfactual quantum cryptography. Phys. Rev. A 82(4), 042335 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    Noh, T.-G.: Counterfactual quantum cryptography. Phys. Rev. Lett. 103(23), 230501 (2009)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Morimae, T.: Strong entanglement causes low gate fidelity in inaccurate one-way quantum computation. Phys. Rev. A 81(6), 060307 (2010)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    Schaffry, M., Gauger, E.M., Morton, J.J.L., Fitzsimons, J., Benjamin, S.C., Lovett, B.W.: Quantum metrology with molecular ensembles. Phys. Rev. A 82(4), 042114 (2010)ADSCrossRefGoogle Scholar
  29. 29.
    Demkowicz-Dobrzański, R., Maccone, L.: Using entanglement against noise in quantum metrology. Phys. Rev. Lett. 113(25), 250801 (2014)ADSCrossRefGoogle Scholar
  30. 30.
    Huelga, S.F., Macchiavello, C., Pellizzari, T., Ekert, A.K., Plenio, M.B., Cirac, J.I.: Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79(20), 3865–3868 (1997)ADSCrossRefGoogle Scholar
  31. 31.
    Kacprowicz, M., Demkowicz-Dobrzaski, R., Wasilewski, W., Banaszek, K., Walmsley, I.A.: Experimental quantum-enhanced estimation of a lossy phase shift. Nat. Photon. 4, 357 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    Chaves, R., Brask, J.B., Markiewicz, M., Kolodyński, J., Acín, A.: Noisy metrology beyond the standard quantum limit. Phys. Rev. Lett. 111(12), 120401 (2013)ADSCrossRefGoogle Scholar
  33. 33.
    Dinani, H.T., Berry, D.W.: Loss-resistant unambiguous phase measurement. Phys. Rev. A 90(2), 023856 (2014)ADSCrossRefGoogle Scholar
  34. 34.
    Pezzé, L., Smerzi, A.: Entanglement, nonlinear dynamics, and the Heisenberg limit. Phys. Rev. Lett. 102(10), 100401 (2009)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Li, N., Luo, S.: Entanglement detection via quantum Fisher information. Phys. Rev. A 88(1), 014301 (2013)ADSCrossRefGoogle Scholar
  36. 36.
    Hyllus, P., Gühne, O., Smerzi, A.: Not all pure entangled states are useful for sub-shot-noise interferometry. Phys. Rev. A 82(1), 012337 (2010)ADSCrossRefGoogle Scholar
  37. 37.
    Boixo, S., Datta, A., Davis, M.J., Flammia, S.T., Shaji, A., Caves, C.M.: Quantum metrology: dynamics versus entanglement. Phys. Rev. Lett. 101(4), 040403 (2008)ADSCrossRefGoogle Scholar
  38. 38.
    Tilma, T., Hamaji, S., Munro, W.J., Nemoto, K.: Entanglement is not a critical resource for quantum metrology. Phys. Rev. A 81(2), 022108 (2010)ADSCrossRefGoogle Scholar
  39. 39.
    Datta, A., Shaji, A.: Quantum metrology without quantum entanglement. Mod. Phys. Lett. B 26(18), 1230010 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Sahota, J., Quesada, N.: Quantum correlations in optical metrology: Heisenberg-limited phase estimation without mode entanglement. Phys. Rev. A 91(1), 013808 (2015)ADSCrossRefGoogle Scholar
  41. 41.
    Kosloff, R.: Quantum thermodynamics: a dynamical viewpoint. Entropy. 15(6), 2100 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kosloff, R., Levy, A.: Quantum heat engines and refrigerators: continuous devices. Annu. Rev. Phys. Chem. 65(1), 365–393 (2014)ADSCrossRefGoogle Scholar
  43. 43.
    Armour, A.D., Blencowe, M.P., Schwab, K.C.: Entanglement and Decoherence of a Micromechanical Resonator via Coupling to a Cooper-Pair Box. Phys. Rev. Lett. 88(14), 148301 (2002)ADSCrossRefGoogle Scholar
  44. 44.
    Kleckner, D., Bouwmeester, D.: Sub-kelvin optical cooling of a micromechanical resonator. Nature (London) 444, 75 (2006)ADSCrossRefGoogle Scholar
  45. 45.
    Rocheleau, T., Ndukum, T., Macklin, C., Hertzberg, J.B., Clerk, A.A., Schwab, K.C.: Preparation and detection of a mechanical resonator near the ground state of motion. Nature 463, 72–75 (2009)ADSCrossRefGoogle Scholar
  46. 46.
    Brunelli, M., Olivares, S., Paris, M.G.A.: Qubit thermometry for micromechanical resonators. Phys. Rev. A 84(3), 032105 (2011)ADSCrossRefGoogle Scholar
  47. 47.
    Boyd, R.W.: Nonlinear Optics. Elsevier Science, Amsterdam (2008)Google Scholar
  48. 48.
    Neumann, P., Jakobi, I., Dolde, F., Burk, C., Reuter, R., Waldherr, G., Honert, J., Wolf, T., Brunner, A., Shim, J.H., Suter, D., Sumiya, H., Isoya, J., Wrachtrup, J.: High-precision nanoscale temperature sensing using single defects in diamond. Nano Lett. 13(6), 2738–2742 (2013)ADSCrossRefGoogle Scholar
  49. 49.
    Kucsko, G., Maurer, P.C., Yao, N.Y., Kubo, M., Noh, H.J., Lo, P.K., Park, H., Lukin, M.D.: Nanometre-scale thermometry in a living cell. Nature 500, 54 (2013)ADSCrossRefGoogle Scholar
  50. 50.
    Toyli, D.M., de las Casas, C.F., Christle, D.J., Dobrovitski, V.V., Awschalom, D.D.: Fluorescence thermometry enhanced by the quantum coherence of single spins in diamond. Proc. Natl. Acad. Sci. USA 110(21), 8417–8421 (2013)ADSCrossRefGoogle Scholar
  51. 51.
    Brunelli, M., Olivares, S., Paternostro, M., Paris, M.G.A.: Qubit-assisted thermometry of a quantum harmonic oscillator. Phys. Rev. A 86(1), 012125 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    Higgins, K.D., Lovett, B.W., Gauger, E.M.: Quantum thermometry using the ac Stark shift within the Rabi model. Phys. Rev. B 88, 155409 (2013)ADSCrossRefGoogle Scholar
  53. 53.
    Raitz, C., Souza, A.M., Auccaise, R., Sarthour, R.S., Oliveira, I.S.: Experimental implementation of a nonthermalizing quantum thermometer. Quant. Inf. Proc. 14(1), 37–46 (2015)CrossRefGoogle Scholar
  54. 54.
    Correa, L.A., Mehboudi, M., Adesso, G., Sanpera, A.: Individual quantum probes for optimal thermometry. Phys. Rev. Lett. 114(22), 220405 (2015)ADSCrossRefGoogle Scholar
  55. 55.
    Guo, L.-S., Xu, B.-M., Zou, J., Shao, B.: Improved thermometry of low-temperature quantum systems by a ring-structure probe. Phys. Rev. A 92(5), 052112 (2015)ADSCrossRefGoogle Scholar
  56. 56.
    Stace, T.M.: Quantum limits of thermometry. Phys. Rev. A 82(1), 011611 (2010)ADSCrossRefGoogle Scholar
  57. 57.
    Jevtic, S., Newman, D., Rudolph, T., Stace, T.M.: Single-qubit thermometry. Phys. Rev. A 91(1), 012331 (2015)ADSCrossRefGoogle Scholar
  58. 58.
    Braunstein, S.L., Caves, C.M., Milburn, G.J.: Generalized uncertainty relations: theory, examples, and Lorentz invariance. Ann. Phys. 247(1), 135–173 (1996)ADSMathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Raimond, J.M., Brune, M., Haroche, S.: Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys. 73(3), 565–582 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Yan, X.-Q., Meng, K.: A comparison of quantum discord and entanglement in a micromaser-type system. Int. J. Theor. Phys. 53(8), 2746–2752 (2014)CrossRefMATHGoogle Scholar
  61. 61.
    Wang, J., Tian, Z., Jing, J., Fan, H.: Quantum metrology and estimation of Unruh effect. Sci. Rep. 4, 7195 (2014)ADSCrossRefGoogle Scholar
  62. 62.
    Latune, C.L., Sinayskiy, I., Petruccione, F.: Quantum force estimation in arbitrary non-Markovian–Gaussian baths. Phys. Rev. A 94(5), 052115 (2016)ADSCrossRefGoogle Scholar
  63. 63.
    Kish, S.P., Ralph, T.C.: Estimating spacetime parameters with a quantum probe in a lossy environment. Phys. Rev. D 93(10), 105013 (2016)ADSMathSciNetCrossRefGoogle Scholar
  64. 64.
    Paris, M.G.A.: Quantum probes for fractional Gaussian processes. Phys. A 413, 256–265 (2014)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Jahromi, H.R.: Parameter estimation in plasmonic QED. Opt. Commun. 411, 119–125 (2018)ADSCrossRefGoogle Scholar
  66. 66.
    Huang, C.Y., Ma, W., Wang, D., Ye, L.: How the relativistic motion affect quantum Fisher information and Bell non-locality for multipartite state. Sci. Rep. 7, 38456 (2017)ADSCrossRefGoogle Scholar
  67. 67.
    Rangani Jahromi, H., Amniat-Talab, M.: Precision of estimation and entropy as witnesses of non-Markovianity in the presence of random classical noises. Ann. Phys. 360, 446–461 (2015)MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Wootters, W.K.: Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80(10), 2245–2248 (1998)ADSCrossRefMATHGoogle Scholar
  69. 69.
    Holevo, A.S., Giovannetti, V.: Quantum channels and their entropic characteristics. Rep. Progr. Phys. 75(4), 046001 (2012)ADSMathSciNetCrossRefGoogle Scholar
  70. 70.
    Rivas, A., Huelga, S.F., Plenio, M.B.: Quantum non-Markovianity: characterization, quantification and detection. Rep. Prog. Phys. 77(9), 094001 (2014)ADSMathSciNetCrossRefGoogle Scholar
  71. 71.
    Chruściński, D., Kossakowski, A.: Markovianity criteria for quantum evolution. J. Phys. B 45(15), 154002 (2012)ADSCrossRefGoogle Scholar
  72. 72.
    Jozsa, R.: Fidelity for mixed quantum states. J. Mod. Opt. 41, 2315–2323 (1994)ADSMathSciNetCrossRefMATHGoogle Scholar
  73. 73.
    Aharonov, Y., Massar, S., Popescu, S.: Measuring energy, estimating Hamiltonians, and the time-energy uncertainty relation. Phys. Rev. A 66(5), 052107 (2002)ADSMathSciNetCrossRefGoogle Scholar
  74. 74.
    Chruściński, D., Kossakowski, A.: Witnessing non-Markovianity of quantum evolution. Eur. Phys. J. D 68(1), 7 (2014)ADSCrossRefGoogle Scholar
  75. 75.
    Yan, Y.-A., Zhou, Y.: Hermitian non-Markovian stochastic master equations for quantum dissipative dynamics. Phys. Rev. A 92(2), 022121 (2015)ADSMathSciNetCrossRefGoogle Scholar
  76. 76.
    Mazzola, L., Laine, E.M., Breuer, H.P., Maniscalco, S., Piilo, J.: Phenomenological memory-kernel master equations and time-dependent Markovian processes. Phys. Rev. A 81(6), 062120 (2010)ADSCrossRefGoogle Scholar
  77. 77.
    Chruściński, D., Kossakowski, A., Rivas, Á.: Measures of non-Markovianity: divisibility versus backflow of information. Phys. Rev. A 83(5), 052128 (2011)ADSCrossRefGoogle Scholar
  78. 78.
    Wilde, M.: Quantum Information Theory. Cambridge University Press, Cambridge (2013)CrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • B. Farajollahi
    • 1
  • M. Jafarzadeh
    • 1
  • H. Rangani Jahromi
    • 2
  • M. Amniat-Talab
    • 1
  1. 1.Physics Department, Faculty of SciencesUrmia UniversityUrmiaIran
  2. 2.Physics Department, Faculty of SciencesJahrom UniversityJahromIran

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