Researching | Optimal initial states for quantum parameter estimation based on Jaynes–Cummings model [Invited]

Journals >Chinese Optics Letters >Volume 21 >Issue 10 >Page 102701 > Article

Chinese Optics Letters
Vol. 21, Issue 10, 102701 (2023)

Optimal initial states for quantum parameter estimation based on Jaynes–Cummings model [Invited]

Liwen Qiao¹, Jia-Xin Peng^1、2, Baiqiang Zhu^1、2, Weiping Zhang^{2、3、4、5}, and Keye Zhang^1、2、*

Author Affiliations

¹Quantum Institute for Light and Atoms, State Key Laboratory of Precision Spectroscopy, Department of Physics, School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China

²Shanghai Branch, Hefei National Laboratory, Shanghai 201315, China

³School of Physics and Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai 200240, China

⁴Shanghai Research Center for Quantum Sciences, Shanghai 201315, China

⁵Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China

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DOI: 10.3788/COL202321.102701 Cite this Article Set citation alerts

Liwen Qiao, Jia-Xin Peng, Baiqiang Zhu, Weiping Zhang, Keye Zhang. Optimal initial states for quantum parameter estimation based on Jaynes–Cummings model [Invited][J]. Chinese Optics Letters, 2023, 21(10): 102701 Copy Citation Text

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Abstract

Quantum parameter estimation is a crucial tool for inferring unknown parameters in physical models from experimental data. The Jaynes–Cummings model is a widely used model in quantum optics that describes the interaction between an atom and a single-mode quantum optical field. In this Letter, we systematically investigate the problem of estimating the atom-light coupling strength in this model and optimize the initial state in the full Hilbert space. We compare the precision limits achievable for different optical field quantum states, including coherent states, amplitude- and phase-squeezed states, and provide experimental suggestions with an easily prepared substitute for the optimal state. Our results provide valuable insights into optimizing quantum parameter estimation in the Jaynes–Cummings model and can have practical implications for quantum metrology with hybrid quantum systems.

Keywords

Jaynes–Cummings model parameter estimation theory quantum Fisher information

F_{g} = 4 (〈 \partial_{g} ψ_{g} | \partial_{g} ψ_{g} 〉 - {| 〈 ψ_{g} | \partial_{g} ψ_{g} 〉 |}^{2}),

(1)

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〈 Δ^{2} \hat{g} 〉 \geq 1 / ν F_{g} .

(2)

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{\hat{h}}_{g}^{H} = i (\partial_{g} {\hat{U}}_{g}^{†}) {\hat{U}}_{g},

(3)

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F_{g} = 4 [〈 ψ_{0} | ({\hat{h}}_{g}^{H})^{2} | ψ_{0} 〉 - {〈 ψ_{0} | {\hat{h}}_{g}^{H} | ψ_{0} 〉}^{2}] = 4 〈 ψ_{0} | (Δ {\hat{h}}_{g}^{H})^{2} | ψ_{0} 〉 .

(4)

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F_{g, Max} = [λ_{\max} ({\hat{h}}_{g}^{H}) - λ_{\min} ({\hat{h}}_{g}^{H} {)]}^{2},

(5)

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\hat{H} = \frac{ω_{0}}{2} {\hat{σ}}_{z} + ω {\hat{a}}^{†} \hat{a} + g ({\hat{σ}}_{+} \hat{a} + {\hat{a}}^{†} {\hat{σ}}_{-}),

(6)

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{\hat{H}}_{N} = [\begin{matrix} (N - \frac{1}{2}) ω + \frac{1}{2} Δ & g \sqrt{N} \\ g \sqrt{N} & \begin{matrix} (N - \frac{1}{2}) ω - \frac{1}{2} Δ \end{matrix} \end{matrix}] = g \sqrt{N} {\hat{σ}}_{1} + \frac{1}{2} Δ {\hat{σ}}_{3},

(7)

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{\hat{σ}}_{1} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], {\hat{σ}}_{2} = [\begin{matrix} 0 & - i \\ i & 0 \end{matrix}], {\hat{σ}}_{3} = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] .

(8)

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{\hat{h}}_{g}^{H, N} = c_{1} {\hat{σ}}_{1} + c_{2} {\hat{σ}}_{2} + c_{3} {\hat{σ}}_{3},

(9)

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c_{1} = - \frac{\sqrt{N} [4 N t g^{2} Ω + Δ^{2} \sin (t Ω)]}{Ω^{3}}, c_{2} = - \frac{Δ \sqrt{N} [1 - \cos (t Ω)]}{Ω^{2}}, c_{3} = - \frac{2 N Δ g [t Ω^{2} - Ω \sin (t Ω)]}{Ω^{4}} .

(10)

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{\hat{h}}_{g}^{H, N} \approx d_{1} {\hat{σ}}_{1} + d_{3} {\hat{σ}}_{3}

(11)

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d_{1} = - \frac{4 \sqrt{N^{3}} g^{2} t}{Ω^{2}}, d_{3} = - \frac{2 N Δ g t}{Ω^{2}} .

(12)

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{\hat{h}}_{g}^{H, N} = - \frac{2 N g t}{Ω} \vec{n} \cdot \vec{σ},

(13)

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| ψ_{0, Max}^{(N)} 〉 = \frac{1}{\sqrt{2}} (| e, N - 1 〉 + i | g, N 〉) .

(14)

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F_{g, Max}^{(N)} = \frac{16 N^{2} g^{2} t^{2}}{Ω^{2}} .

(15)

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| ψ_{0} 〉 = \oplus_{N = 0}^{\infty} C_{N} | ψ_{0}^{(N)} 〉,

(16)

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{\hat{h}}_{g}^{H} = \oplus_{N = 0}^{\infty} {\hat{h}}_{g}^{H, N} .

(17)

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F_{g} = 4 [\sum_{N = 0}^{\infty} P_{N} 〈 ψ_{0}^{(N)} | {({\hat{h}}_{g}^{H, N})}^{2} | ψ_{0}^{(N)} 〉] - 4 {(\sum_{N = 0}^{\infty} P_{N} 〈 ψ_{0}^{(N)} | {\hat{h}}_{g}^{H, N} | ψ_{0}^{(N)} 〉)}^{2} .

(18)

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F_{g}^{(N)} = 4 [P_{N} 〈 ψ_{0}^{(N)} | {({\hat{h}}_{g}^{H, N})}^{2} | ψ_{0}^{(N)} 〉] - 4 {(P_{N} 〈 ψ_{0}^{(N)} | {\hat{h}}_{g}^{H, N} | ψ_{0}^{(N)} 〉)}^{2} .

(19)

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F_{g}^{O} = \sum_{N = 0}^{\infty} P_{N} F_{g, Max}^{(N)},

(20)

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F_{g, Max}^{(N)} = \frac{16 N^{2} g^{2} t^{2}}{Δ^{2} + 4 N g^{2}} \approx 4 N t^{2} .

(21)

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F_{g, Max}^{(N)} \approx \frac{16 N^{2} g^{2} t^{2}}{Δ^{2}} .

(22)

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| ψ_{0}^{O, bs} 〉 = \sqrt{P_{0}} | g,0 〉 + \sqrt{P_{N}} \frac{| e, N - 1 〉 + i | g, N 〉}{\sqrt{2}} .

(23)

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F_{g}^{O, bs} = P_{0} F_{g, Max}^{(0)} + P_{N} F_{g, Max}^{(N)} = \frac{\bar{N}}{N} \frac{16 N^{2} g^{2} t^{2}}{Δ^{2} + 4 g^{2} N} .

(24)

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| ψ_{0}^{P} 〉 = | ψ_{o} 〉 \otimes | ψ_{a} 〉,

(25)

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| ψ_{o} 〉 = \sum_{n = 0}^{\infty} | c_{n} | e^{i ϕ n} | n 〉, | ψ_{a} 〉 = \frac{1}{\sqrt{2}} (| e 〉 + e^{- i χ} | g 〉),

(26)

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Liwen Qiao, Jia-Xin Peng, Baiqiang Zhu, Weiping Zhang, Keye Zhang. Optimal initial states for quantum parameter estimation based on Jaynes–Cummings model [Invited][J]. Chinese Optics Letters, 2023, 21(10): 102701

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