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量子参数估计在量子度量学中有着重要的应用, 量子Cramer-Rao下界表明量子参数估计精度极限与量子Fisher信息是直接相关的. 本文利用量子参数估计理论对光场与原子失谐很大(大失谐)的Jaynes-Cummings模型耦合常数进行估计. 制备探测初态为Qubit系统与光场的直积态, 光场分别为Fock态、热态和相干态, 分别计算了这三种探测态经大失谐Jaynes-Cummings模型哈密顿量演化后复合系统以及Qubit和光场系统的量子Fisher信息. 通过分析发现, 复合系统的量子Fisher信息随平均光子数单调递增, Qubit基态与激发态的等权叠加态为最优探测态, 此时量子Fisher信息达到最大值; 当探测态的光场为Fock态和热态时, 关于被估计参数的信息都包含于Qubit系统; 对于大失谐Jaynes-Cummings模型耦合常数的估计, 光场为热态或相干态时耦合常数的估计精度高于光场为Fock态时的精度.
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关键词:
- 量子Fisher信息 /
- 参数估计 /
- 量子Cramer-Rao下界 /
- Jaynes-Cummings模型
Quantum parameter estimation is one of the most important applications in quantum metrology. The basic theory of quantum parameter estimation－quantum Cramer-Rao bound－shows that the precision limit of quantum parameter estimation is directly related to quantum Fisher information. Therefore quantum Fisher information is extremely important in the quantum parameter estimation. In this paper we use quantum parameter estimation theory to estimate the coupling constant of the Jaynes-Cummings model with large detuning. The initial probing state is the direct product state of qubit and radiation field in which Fock state, thermal state and coherent state are taken into account respectively. We calculate the quantum Fisher information of the hybrid system as well as qubit and radiation field for each probing state after the parameter evolution under the Hamiltonian of the Jaynes-Cummings model with large detuning. The results show that the quantum Fisher information increases monotonically with the average photon number increasing. The optimal detection state is that when the qubit system is in the equal weight superposition of the ground and the excited state, at this time the quantum Fisher information always reaches a maximum value, When the radiation field of probing state is Fock state or the thermal state, the information about the estimated parameter is included only in the qubit. The estimation accuracy of the coupling constant with thermal state or coherent state is higher than that with Fock state.-
Keywords:
- quantum Fisher information /
- parameter estimation /
- quantum Cramer-Rao bound /
- Jaynes-Cummings model
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图 1 光场为Fock态时QFI随不同变量的变化(
${H_{{\rm{q}}1}}(\varOmega ) = {H_1}(\varOmega )$ ) (a)平均光子数$ \bar n_1 $ ; (b) θFig. 1. Variation of QFI with different variables when the radiation field is a Fock state (
${H_{{\rm{q}}1}}(\varOmega ) = {H_1}(\varOmega )$ ): (a) The average photon number,${\bar n_1}$ ; (b) θ.
图 2 光场为热态时QFI随不同变量的变化(
${H_{{\rm{q}}2}}(\varOmega ) = {H_2}(\varOmega )$ ) (a)平均光子数${\bar n_2}$ ; (b) θFig. 2. Variation of QFI with different variables when the radiation field is a thermal state (
${H_{{\rm{q}}2}}(\varOmega ) = {H_2}(\varOmega )$ ): (a) The average photon number,${\bar n_2}$ ; (b) θ.
图 3 腔场为相干态时QFI随不同变量的变化 (a)平均光子数
${\bar n_3}$ ; (b) θFig. 3. Variation of QFI with different variables when the radiation field is a coherent state: (a) The average photon number,
${\bar n_3}$ ; (b) the θ. -
[1] Pang S S, Brun T A 2014 Phys. Rev. A 90 022117
Google Scholar
[2] 井晓幸 2016 博士学位论文 (杭州: 浙江大学)
Jing X X 2016 Ph. D. Dissertation (Hangzhou: Zhejiang University) (in Chinese)
[3] Genoni M G, Giorda P, Matteo G A P 2008 Phys. Rev. A 78 032303
Google Scholar
[4] Brida G, Degiovanni I, Florio A, Genovese M, Giorda P, Meda A, Matteo G A P, Shurupov A 2010 Phys. Rev. Lett. 104 100501
Google Scholar
[5] Giovannetti V, Lloyd S, Maccone L 2006 Phys. Rev. Lett. 96 010401
Google Scholar
[6] Monras A 2006 Phys. Rev. A 73 0338821
[7] Genoni M G, Olivares S, Matteo G A P 2011 Phys. Rev. Lett. 106 153603
Google Scholar
[8] Monras A, Matteo G A P 2007 Phys. Rev. Lett. 98 160401
Google Scholar
[9] Genoni M G, Invernizzi C, Matteo G A P 2009 Phys. Rev. A 80 033842
Google Scholar
[10] Lu X M, Wang X G 2021 Phys. Rev. Lett. 126 120503
Google Scholar
[11] Matteo G A P 2008 Int. J. Quant. Inf. 0804 2981
[12] Helstrom C W 1967 Phys. Lett. A 25 101
Google Scholar
[13] Yuen H P, Lax M 1973 IEEE Trans. Inf. Th. 19 740
Google Scholar
[14] Helstrom C W, Kennedy R S 1974 IEEE Trans. Inf. Th. 20 16
Google Scholar
[15] Braunstein S, Caves C 1994 Phys. Rev. Lett. 72 3439
Google Scholar
[16] Braunstein S, Caves C, Milburn G 1996 Ann. Phys. 247 135
Google Scholar
[17] Fisher R A 1925 Proc. Camb. Phil. Soc. 22 700
Google Scholar
[18] Helstrom C W 1969 J. Stat. Phys. 1 231
Google Scholar
[19] Holevo A S 1982 Probabilistic and Statistical Aspects of Quantum Theory (North Holland Amsterdam) pp52–96, 160–168
[20] Holevo A S 2001 Statistical Structure of Quantum Theory (Berlin, Heidelberg: Springer) pp45–70
[21] Pezze L, Smerzi A, Oberthaler M K, Schmied R, Treutlein P 2018 Rev. Mod. Phys. 90 035005
Google Scholar
[22] Gabbrielli M 2018 arXiv: 1810.10537 [quant-ph]
[23] Pezze' L, Smerzi A 2014 arXiv: 1411.5164 [quant-ph]
[24] 任志红, 李岩, 李艳娜, 李卫东 2019 物理学报 68 040601
Google Scholar
Ren Z H, Li Y, LiY N, Li W D 2019 Acta Phys. Sin. 68 040601
Google Scholar
[25] Royfriened B 1998 Physics from Fisher Information (Cambridge: Cambridge University Press) pp22–62
[26] Liu J, Jing X X, Zhong W, Wang X G 2014 Commun. Theor. Phys. 61 45
Google Scholar
[27] Liu J, Yuan H D, Lu X M, Wang X G 2020 J. Phys. A:Math. Theor. 53 023001
Google Scholar
[28] Watanabe D Y 2014 Formulation of Uncertainty Relation Between Error and Disturbance in Quantum Measurement by Using Quantum Estimation Theory (Tokyo: Springer Theses) pp19–44
[29] Genoni M G, Tufarelli T 2019 Phys. A:Math. Theor. 52 434002
Google Scholar
[30] Monras A 2006 Phys. Rev. A 73 033821
Google Scholar
[31] Smirne A, Kolodynski J, Huelga S F, Dobrzanski R D 2016 Phys. Rev. Lett. 116 120801
Google Scholar
[32] Pirandola S, Lupo C 2017 Phys. Rev. Lett. 118 100502
Google Scholar
[33] Lupo C, Pirandola S 2016 Phys. Rev. Lett. 117 190802
Google Scholar
[34] Invernizzi C, Korbman M, Venuti L C, Matteo G A P 2008 Phys. Rev. A 78 042106
Google Scholar
[35] Schneiter F, Qvarfort S, Serafini A, Xuereb A, Braun D, Rätzel D, Bruschi D E, 2020 Phys. Rev. A. 101 033834
Google Scholar
[36] Sanavio C, Bernad J Z, Xuereb A 2020 Phys. Rev. A 102 013508
[37] Genoni M G, Invernizzi C 2012 Eur. Phys. J. Spec. 10 1140
[38] Gerry C, Knight P 2005 Introductory Quantum Optics (Cambridge: Cambridge University Press) pp25–27, 105–107
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