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Effects of the Coherence on the Parameter Estimation in a Quantum Metrology Scheme with Driving Fields

Abstract

The effects of the initial-state coherence on the quantum Fisher information(QFI) is studied in a parameter estimation scheme where the endowed unknown phase is measured after a driven two-level probing atom interacts with its environment. It is shown that the relatively large initial-state coherence couldn’t be necessary to achieve the large QFI, but it is necessary for the driving field to enhance the QFI. It is also found that the coherence plays its roles in the QFI through two paths. Generally, the QFI could lie at a low level in one path, but maintains a relatively high level in another path, even though the two paths correspond to the same initial-state coherence. This provides a way to achieve the large QFI with the less initial-state coherence.

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Change history

  • 15 February 2020

    The author found a mistake in their published article. The author wants to correct the presentation of his name from “Yuq-iang Liu” to “Yu-qiang Liu”.

References

  1. 1.

    Horodecki, R., Horodecki, P., Horodecki, M., Horodecki, K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

  2. 2.

    Ye, T.Y.: Quantum dialogue without information leakage using a single quantum entangled state. Int. J. Theor. Phys. 53(11), 3719 (2014)

  3. 3.

    Chen, Y.H., Shi, Z.C., Song, J., Xia, Y., Zheng, S.B.: Accelerated and noise-resistant generation of high-fidelity steady-state entanglement with rydberg atoms. Phys. Rev. A 97, 032328 (2018)

  4. 4.

    Namitha, CV, Satyanarayana, SVM: Role of initial coherence on entanglement dynamics of two qubit x states. J. Phys. B: At. Mol. Opt. Phys. 51(4), 045506 (2018)

  5. 5.

    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)

  6. 6.

    Sun, Y., Mao, Y., Luo, S.: From quantum coherence to quantum correlations. Europhys. Lett. 118(6), 60007 (2017)

  7. 7.

    Gisin, N.: Quantum nonlocality: how does nature do it? Science 326(5958), 1357 (2009)

  8. 8.

    Allahverdyan, A.E., Danageozian, A.: Quantum non-locality co-exists with locality. Europhys. Lett. 122(4), 40005 (2018)

  9. 9.

    Zheng, S.B.: One-step synthesis of multiatom greenberger-horne-zeilinger states. Phys. Rev. Lett. 87, 230404 (2001)

  10. 10.

    Brandão, F.G.S.L., Gour, G.: Reversible framework for quantum resource theories. Phys. Rev. Lett. 115, 070503 (2015)

  11. 11.

    Zanardi, P., Paris, M.G.A., Campos Venuti, L.: Quantum criticality as a resource for quantum estimation. Phys. Rev. A 78, 042105 (2008)

  12. 12.

    Kollas, N.K.: Optimization-free measures of quantum resources. Phys. Rev. A 97, 062344 (2018)

  13. 13.

    Chen, Y.H., Xia, Y., Chen, Q.Q., Song, J.: Efficient shortcuts to adiabatic passage for fast population transfer in multiparticle systems. Phys. Rev. A 89, 033856 (2014)

  14. 14.

    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)

  15. 15.

    Streltsov, A, Kampermann, H, Wölk, S, Gessner, M, Bruß, D: Maximal coherence and the resource theory of purity. New J. Phys. 20(5), 053058 (2018)

  16. 16.

    Chitambar, E., Gour, G.: Critical examination of incoherent operations and a physically consistent resource theory of quantum coherence. Phys. Rev. Lett. 117, 030401 (2016)

  17. 17.

    Winter, A., Yang, D.: Operational resource theory of coherence. Phys. Rev. Lett. 116, 120404 (2016)

  18. 18.

    Romero, E., Novoderezhkin, V.I., Van, G.R.: Quantum design of photosynthesis for bio-inspired solar-energy conversion. Nature 543(7645), 355 (2017)

  19. 19.

    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 (2003)

  20. 20.

    Bennett, C.H., DiVincenzo, D.P.: Quantum information and computation. Nature 404(6775), 247 (2000)

  21. 21.

    Kang, Y.H., Chen, Y.H., Shi, Z.C., Huang, B.H., Song, J., Xia, Y.: Nonadiabatic holonomic quantum computation using rydberg blockade. Phys. Rev. A 97, 042336 (2018)

  22. 22.

    Saffman, M.: Quantum computing with atomic qubits and rydberg interactions: progress and challenges. J. Phys. B: At. Mol. Opt. Phys. 49(20), 202001 (2016)

  23. 23.

    Hutter, A., Wootton, J.R., Loss, D.: Efficient Markov chain Monte Carlo algorithm for the surface code. Phys. Rev. A 89, 022326 (2014)

  24. 24.

    Wang, Z., Wu, W., Cui, G., Wang, J.: Coherence enhanced quantum metrology in a nonequilibrium optical molecule. New J. Phys. 20(3), 033034 (2018)

  25. 25.

    Toth, G., Apellaniz, I.: Quantum metrology from a quantum information science perspective. J. Phys. A: Math. Theor. 47(42), 15 (2014)

  26. 26.

    Giorda, P., Allegra, M.: Coherence in quantum estimation. J. Phys. A: Math. Theor. 51(2), 025302 (2017)

  27. 27.

    Giovannetti, V., Lloyd, S., Maccone, L.: Quantum metrology. Phys. Rev. Lett. 96, 010401 (2006)

  28. 28.

    Zhu, H., Hayashi, M.: Universally fisher-symmetric informationally complete measurements. Phys. Rev. Lett. 120, 030404 (2018)

  29. 29.

    Tan, Q.S., Yuan, J.B., Jin, G.R., Kuang, L.M.: Near-heisenberg-limited parameter estimation precision by a dipolar-bose-gas reservoir engineering. Phys. Rev. A 96, 063614 (2017)

  30. 30.

    Šafránek, D.: Simple expression for the quantum Fisher information matrix. Phys. Rev. A 97, 042322 (2018)

  31. 31.

    Liu, W.F., Xiong, H.N., Ma, J., Wang, X.: Quantum fisher information in the generalized one-axis twisting model. Int. J. Theor. Phys. 49(5), 1073 (2010)

  32. 32.

    Sun, Z., Ma, J., Lu, X.M., Wang, X.: Fisher information in a quantum-critical environment. Phys. Rev. A 82, 022306 (2010)

  33. 33.

    Cramer, H.: Mathematical Methods of Statistics. Princeton University, Princeton (1946)

  34. 34.

    Helstrom, C.W.: Quantum detection and estimation theory. J. Stat. Phys. 1(2), 231 (1969)

  35. 35.

    Brandt, H.E.: Positive operator valued measure in quantum information processing. Am. J. Phys. 67(5), 434 (1999)

  36. 36.

    Wiseman, H.M., Milburn, G.J.: Quantum Measurement and Control. Cambridge University Press, Cambridge (2014)

  37. 37.

    SALGADO, R.B.: Some identities for the quantum measure and its generalization. Mod. Phys. Lett. A 17(12), 711 (2002)

  38. 38.

    Yuan, H., Fung, C.H.F.: Fidelity and Fisher information on quantum channels. New J. Phys. 19(11), 113039 (2017)

  39. 39.

    Braunstein, S.L., Caves, C.M.: Statistical distance and the geometry of quantum states. Phys. Rev Lett. 72, 3439 (1994)

  40. 40.

    Šafránek, D.: Discontinuities of the quantum Fisher information and the bures metric. Phys. Rev. A 95, 052320 (2017)

  41. 41.

    Luo, S., Zhang, Q.: Informational distance on quantum-state space. Phys. Rev. A 69, 032106 (2004)

  42. 42.

    Caves, C.M.: Quantum-mechanical noise in an interferometer. Phys. Rev. D 23, 1693 (1981)

  43. 43.

    Tan, Q.S., Huang, Y., Kuang, L.M., Wang, X.: Dephasing-assisted parameter estimation in the presence of dynamical decoupling. Phys. Rev. A 89, 063604 (2014)

  44. 44.

    Metwally, N., Hassan, S.S.: Estimation of pulsed driven qubit parameters via quantum Fisher information. Laser Phys. Lett. 14(11), 115204 (2017)

  45. 45.

    Ren, Y.K., Wang, X.L., Zeng, H.S.: Protection of quantum Fisher information for multiple phases in open quantum systems. Quantum Inf. Process. 17(1), 5 (2017)

  46. 46.

    Li, Y.L., Xiao, X., Yao, Y.: Classical-driving-enhanced parameter-estimation precision of a non-markovian dissipative two-state system. Phys. Rev. A 91, 052105 (2015)

  47. 47.

    xiong Wu, S., Zhang, Y., shui Yu, C.: Local quantum uncertainty guarantees the measurement precision for two coupled two-level systems in non-markovian environment. Ann. Phys. 390, 71 (2018)

  48. 48.

    Wu, S.X., Yu, C.S.: The precision of parameter estimation for dephasing model under squeezed reservoir. Int. J. Theor. Phys. 56(4), 1198 (2017)

  49. 49.

    Chen, Y., Zou, J., Long, Z.W., Shao, B.: Protecting quantum Fisher information of n-qubit ghz state by weak measurement with flips against dissipation. Sci. Rep. 7, 6160 (2017)

  50. 50.

    Joo, J., Munro, W.J., Spiller, T.P.: Quantum metrology with entangled coherent states. Phys. Rev. Lett. 107, 083601 (2011)

  51. 51.

    Bollinger, J.J., Itano, W.M., Wineland, D.J., Heinzen, D.J.: Optimal frequency measurements with maximally correlated states. Phys. Rev. A 54, R4649 (1996)

  52. 52.

    Barnum, H., Nielsen, M.A., Schumacher, B.: Information transmission through a noisy quantum channel. Phys. Rev. A 57, 4153 (1998)

  53. 53.

    Chen, Y.H., Xia, Y., Chen, Q.Q., Song, J.: Fast and noise-resistant implementation of quantum phase gates and creation of quantum entangled states. Phys. Rev. A 91, 012325 (2015)

  54. 54.

    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)

  55. 55.

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

  56. 56.

    Zhong, W., Sun, Z., Ma, J., Wang, X., Nori, F.: Fisher information under decoherence in bloch representation. Phys. Rev. A 87, 022337 (2013)

  57. 57.

    Zeng, Y.X., Gebremariam, T., Ding, M.S., Li, C.: The influence of non-markovian characters on quantum adiabatic evolution. Ann. Phys. (Berlin, Ger.) 531(1), 1800234 (2019)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China, under Grant No.11775040 and No. 11375036, the Xinghai Scholar Cultivation Plan, and the Fundamental Research Fund for the Central Universities under Grants No. DUT18LK45.

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Correspondence to Chang-shui Yu.

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The original version of this article was revised: The author wants to correct the presentation of his name from "Yuq-iang Liu" to "Yu-qiang Liu".

Appendix: The Dynamics of the System

Appendix: The Dynamics of the System

The total Hamiltonian of system and reservoir reads

$$ \begin{array}{@{}rcl@{}} H_{S}&=&\frac{1}{2}({\varDelta} \sigma_{z}+{\varOmega}_{R}\sigma_{x}), \end{array} $$
(19)
$$ \begin{array}{@{}rcl@{}} H_{I}&=&\sum\limits_{k}(e^{i\omega_{L}t}\sigma_{+}+e^{-i\omega_{L}t}\sigma_{-})\otimes (g_{k}b_{k}+g_{k}^{\ast }b_{k}^{\dagger })\\ &=&A\otimes B, \end{array} $$
(20)
$$ \begin{array}{@{}rcl@{}} H_{B}&=&\sum\limits_{k}\omega_{k}b_{k}^{\dagger }b_{k}, \end{array} $$
(21)

with A and B defined by

$$ \begin{array}{@{}rcl@{}} A&=&e^{i\omega_{L}t}\sigma_{+}+e^{-i\omega_{L}t}\sigma_{-}, \end{array} $$
(22)
$$ \begin{array}{@{}rcl@{}} B&=&\sum\limits_{k}(g_{k}b_{k}+g_{k}^{\ast }b_{k}^{\dagger }). \end{array} $$
(23)

HS can be diagonalized as

$$ H_{S}=\frac{\omega_{0}}{2}\left\vert 1\right\rangle \left\langle 1\right\vert -\frac{\omega_{0}}{2}\left\vert 2\right\rangle \left\langle 2\right\vert, $$
(24)

where \(\omega _{0}=\sqrt {{\varDelta }^{2}+{{\varOmega }_{R}^{2}}}\) and

$$ \begin{array}{@{}rcl@{}} \left\vert 1\right\rangle =\left( \begin{array}{l} \cos \frac{\theta }{2} \\ \sin \frac{\theta }{2} \end{array} \right), \end{array} $$
(25)
$$ \begin{array}{@{}rcl@{}} \left\vert 2\right\rangle =\left( \begin{array}{l} -\sin \frac{\theta }{2} \\ \cos \frac{\theta }{2} \end{array} \right),{\kern-7.5pt} \end{array} $$
(26)

with \(\theta =\arcsin ({\varOmega }_{R}/\sqrt {{\varDelta }^{2}+{{\varOmega }_{R}^{2}}})\).

Thus one can obtain the eigenoperators of HS belonging to the different frequencies as

$$ \begin{array}{@{}rcl@{}} A(\omega_{0})&=&(\cos^{2}\frac{\theta}{2}e^{-i\omega_{L}t}-\sin^{2} \frac{\theta}{2}e^{i\omega_{L}t}) \tilde{\sigma}_{-}, \end{array} $$
(27)
$$ \begin{array}{@{}rcl@{}} A(-\omega_{0})&=&[A(\omega_{0})]^{\dagger}, \end{array} $$
(28)
$$ \begin{array}{@{}rcl@{}} A(0)&=&\sin \theta \cos (\omega_{L}t)\tilde{\sigma}_{z}, \end{array} $$
(29)

with

$$ \begin{array}{@{}rcl@{}} \tilde{\sigma}_{j}&=&U\sigma_{j}U^{\dagger }\text{, }j=\pm,z, \end{array} $$
(30)
$$ \begin{array}{@{}rcl@{}} U&=&\left( \begin{array}{cc} \cos \frac{\theta }{2} & -\sin \frac{\theta }{2} \\ \sin \frac{\theta }{2} & \cos \frac{\theta }{2} \end{array} \right) . \end{array} $$
(31)

Thus the interaction Hamiltonian in the interaction picture reads

$$ H_{I}(t)=[e^{-i\omega_{0}t}A(\omega_{0})+e^{i\omega_{0}t}A(-\omega_{0})+A(0)]\otimes B(t), $$
(32)

with \(B(t)={\sum }_{k}(g_{k}b_{k}e^{-i\omega _{k}t}+g_{k}^{\ast }b_{k}^{\dagger }e^{i\omega _{k}t})\). Substituting the above Hamiltonian and the eigenoperators into the Markovian master equation [55]

$$ \frac{d\rho_{S}(t)}{dt}=-{\int}_{0}^{\infty }ds tr_{B}[H_{I}(t),[H_{I}(t-s),\rho_{S}(t)\otimes\rho_{B}]], $$
(33)

with the secular approximation, one can arrive at

$$ \begin{array}{@{}rcl@{}} \dot{\rho}_{S}(t)&=&{\int}_{0}^{\infty }ds tr_{B}\left\{\right. [\cos^{4}\frac{\theta}{2}e^{-i(\omega_{0}+\omega_{L})s}+\sin^{4}\frac{\theta}{2}e^{i(-\omega_{0}+\omega_{L})s}] \\ &&\times [\tilde{\sigma}_{+}B(t-s)\rho_{S}(t)\rho_{B}\tilde{\sigma}_{-}-\tilde{\sigma}_{-}B(t)\tilde{\sigma}_{+}B(t-s)\rho_{S}(t)\rho_{B}] \\ &&+[\cos^{4}\frac{\theta }{2}e^{i(\omega_{0}+\omega_{L})s}+\sin^{4} \frac{\theta }{2}e^{i(\omega_{0}-\omega_{L})s}] \\ &&\times [\tilde{\sigma}_{-}B(t-s)\rho_{S}(t)\rho_{B}\tilde{\sigma}_{+}-\tilde{\sigma}_{+}B(t)\tilde{\sigma}_{-}B(t-s)\rho_{S}(t)\rho_{B}] \\ &&+[\frac{\sin^{2}\theta }{4}(e^{(-i\omega_{L})s}+e^{(i\omega_{L})s})] \\ &&\times[\tilde{\sigma}_{z}B(t-s)\rho_{S}(t)\rho_{B}\tilde{\sigma}_{z}-\tilde{\sigma}_{z}B(t)\tilde{\sigma}_{z}B(t-s)\rho_{S}(t)\rho_{B}] \left.\right\}. \end{array} $$
(34)

Using the correlation function

$$ \begin{array}{@{}rcl@{}} {\varGamma} (\omega )&=&{\int}_{0}^{\infty }ds e^{i\omega s}tr_{B}[B(t)B(t-s)\rho_{B}], \end{array} $$
(35)
$$ \begin{array}{@{}rcl@{}} \gamma (\omega )&=&{\varGamma} (\omega )+{\varGamma}^{\ast }(\omega )={\int}_{-\infty}^{\infty }ds e^{i\omega s}tr_{B}[B(s)B(0)\rho_{B}], \end{array} $$
(36)

then the master equation can be simplified as

$$ \begin{array}{@{}rcl@{}} \frac{d\rho }{dt}&=&\frac{\gamma_{+}}{2}(2\tilde{\sigma}_{+}\rho \tilde{ \sigma}_{-}-\tilde{\sigma}_{-}\tilde{\sigma}_{+}\rho -\rho \tilde{\sigma}_{-}\tilde{\sigma}_{+}) \\ &&+\frac{\gamma_{-}}{2}(2\tilde{\sigma}_{-}\rho \tilde{\sigma}_{+}-\tilde{ \sigma}_{+}\tilde{\sigma}_{-}\rho -\rho \tilde{\sigma}_{+}\tilde{\sigma}_{-})\\ &&+\frac{\gamma_{z}}{2}(2\tilde{\sigma}_{z}\rho \tilde{\sigma}_{z}-\tilde{\sigma}_{z}\tilde{\sigma}_{z}\rho -\rho \tilde{\sigma}_{z}\tilde{\sigma}_{z}), \end{array} $$
(37)

where

$$ \begin{array}{@{}rcl@{}} \gamma_{+}&=&\cos^{4}\frac{\theta }{2}\gamma(-\omega_{0}-\omega_{L})+\sin^{4}\frac{\theta }{2}\gamma(-\omega_{0}+\omega_{L}), \end{array} $$
(38)
$$ \begin{array}{@{}rcl@{}} \gamma_{-}&=&\cos^{4}\frac{\theta}{2}\gamma(\omega_{0}+\omega_{L})+\sin^{4}\frac{\theta}{2}\gamma(\omega_{0}-\omega_{L}), \end{array} $$
(39)
$$ \begin{array}{@{}rcl@{}} \gamma_{z}&=&\frac{\sin^{2}\theta }{4}[\gamma (\omega_{L})+\gamma(-\omega_{L})], \end{array} $$
(40)

and γ(ωx) is defined by

$$ \begin{array}{@{}rcl@{}} \gamma (\omega_{x} )&=&{\int}_{-\infty }^{\infty }ds e^{is\omega_{x}}\langle e^{iH_{B}t}Be^{-iH_{B}t}B\rangle \\ &=&{\int}_{-\infty }^{\infty } ds e^{is\omega_{x}}{\sum}_{k}|g_{k}|^{2}(e^{-is\omega_{k}}\langle b_{k}b_{k}^{\dagger }\rangle+e^{is\omega_{k}}\langle b_{k}^{\dagger}b_{k}\rangle ) \\ &=&{\int}_{-\infty }^{\infty } ds \int e^{is\omega_{x}}d\omega J(\omega )\left\{n(\omega)e^{is\omega}+[n(\omega)+1]e^{-is\omega}\right\} \\ &=&{\int}_{-\infty }^{\infty } d\omega J(\omega)n(\omega)\int ds e^{is(\omega_{x}+\omega)} +{\int}_{-\infty }^{\infty } d\omega J(\omega)(n(\omega)+1)\int ds e^{is(\omega_{x}-\omega)}\\ &=&2\pi{\int}_{-\infty }^{\infty } d\omega J(\omega)n\delta(\omega_{x}+\omega) +2\pi{\int}_{-\infty }^{\infty } d\omega J(\omega)(n+1)\delta(\omega_{x}-\omega). \end{array} $$
(41)

Now let’s assume that the interaction between system and bath do not depend on mode of oscillator, i.e, spectrum function [57] became flat

$$ J(\omega )=\frac{\gamma_{0}}{2\pi}. $$
(42)

Thus one can find that when ωx > 0

$$ \gamma(\omega_{x})=\gamma_{0} [n(\omega_{x})+1], $$
(43)

when ωx < 0

$$ \gamma(\omega_{x})=\gamma_{0}n(-\omega_{x}). $$
(44)

Thus, the master equation can be finally given by

$$ \frac{d\rho }{dt}=-i[H_{S},\rho ]+{\sum}_{j=\pm,z}\frac{1}{2}\gamma_{j}[2\tilde{ \sigma}_{j}\rho \tilde{\sigma}_{j}^{\dagger }-\tilde{\sigma}_{j}^{\dagger } \tilde{\sigma}_{j}\rho -\rho \tilde{\sigma}_{j}^{\dagger }\tilde{\sigma}_{j}]. $$
(45)

Expanding the density matrix ρ(t) as

$$ \rho (t)=\frac{1}{2}\left( I+\sum\limits_{k=x,y,z}\tilde{\beta}_{k}(t)\tilde{\sigma}_{k}\right), $$
(46)

and substituting it into (45), one can find the equations for βk(t) as

$$ \begin{array}{@{}rcl@{}} \frac{d\tilde{\beta}_{x}}{dt}&=&-\frac{1}{2}(\gamma_{+}+\gamma_{-}+4\gamma_{z})\tilde{\beta}_{x}-\omega_{0}\tilde{\beta}_{y}, \end{array} $$
(47)
$$ \begin{array}{@{}rcl@{}} \frac{d\tilde{\beta}_{y}}{dt}&=&-\frac{1}{2}(\gamma_{+}+\gamma_{-}+4\gamma_{z})\tilde{\beta}_{y}+\omega_{0}\tilde{\beta}_{x}, \end{array} $$
(48)
$$ \begin{array}{@{}rcl@{}} \frac{d\tilde{\beta}_{z}}{dt}&=&-(\gamma_{+}+\gamma_{-})\tilde{\beta}_{z}+\gamma_{+}-\gamma_{-}. \end{array} $$
(49)

Solving them we will arrive at

$$ \begin{array}{@{}rcl@{}} \tilde{\beta}_{x}(t)&=&e^{-\alpha(t)}[\tilde{\beta}_{x}(0)\cos(\omega_{0}t)-\tilde{\beta}_{y}(0)\sin(\omega_{0}t)], \end{array} $$
(50)
$$ \begin{array}{@{}rcl@{}} \tilde{\beta}_{y}(t)&=&e^{-\alpha(t)}[\tilde{\beta}_{y}(0)\cos(\omega_{0}t)+\tilde{\beta}_{x}(0)\sin(\omega_{0}t)], \end{array} $$
(51)
$$ \begin{array}{@{}rcl@{}} \tilde{\beta}_{z}(t)&=&e^{-\zeta(t)}[\tilde{\beta}_{z}(0)+\eta(t)], \end{array} $$
(52)

where

$$ \begin{array}{@{}rcl@{}} \alpha(t)&=&\frac{1}{2}(\gamma_{+}+\gamma_{-}+4\gamma_{z})t, \end{array} $$
(53)
$$ \begin{array}{@{}rcl@{}} \zeta(t)&=&(\gamma_{+}+\gamma_{-})t, \end{array} $$
(54)
$$ \begin{array}{@{}rcl@{}} \eta(t)&=&\frac{\gamma_{+}-\gamma_{-}}{\gamma_{+}+\gamma_{-}} [e^{(\gamma_{+}+\gamma_{-})t}-1], \end{array} $$
(55)

and βk(0) is determined by the initial state. Therefore, the solution of density matrix is

$$ \rho(t) =\frac{1}{2}\left( \begin{array}{cc} 1+\beta_{z}(t) & \beta_{x}(t)-i\beta_{y}(t) \\ \beta_{x}(t)+i\beta_{y}(t) & 1-\beta_{z}(t) \end{array} \right), $$
(56)

with

$$ \begin{array}{@{}rcl@{}} \beta_{x}(t)&=&\tilde{\beta}_{z}(t)\sin \theta +\tilde{\beta}_{x}(t)\cos \theta, \end{array} $$
(57)
$$ \begin{array}{@{}rcl@{}} \beta_{y}(t)&=&\tilde{\beta}_{y}(t), \end{array} $$
(58)
$$ \begin{array}{@{}rcl@{}} \beta_{z}(t)&=&\tilde{\beta}_{z}(t)\cos \theta -\tilde{\beta}_{x}(t)\sin \theta . \end{array} $$
(59)

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Yu, D., Liu, Y. & Yu, C. Effects of the Coherence on the Parameter Estimation in a Quantum Metrology Scheme with Driving Fields. Int J Theor Phys (2019). https://doi.org/10.1007/s10773-019-04194-5

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Keywords

  • Quantum Fisher information
  • Quantum coherence
  • Quantum metrology