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Photonics Research
Vol. 9, Issue 7, 1289 (2021)

Motional n-phonon bundle states of a trapped atom with clock transitions

Yuangang Deng^1、5、*, Tao Shi^2、3, and Su Yi^{2、3、4、6、*}

Author Affiliations

¹Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and Astronomy, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, China

²CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

³CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100049, China

⁴School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

⁵e-mail: dengyg3@mail.sysu.edu.cn

⁶e-mail: syi@itp.ac.cn

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DOI: 10.1364/PRJ.427062 Cite this Article Set citation alerts

Yuangang Deng, Tao Shi, Su Yi. Motional n-phonon bundle states of a trapped atom with clock transitions[J]. Photonics Research, 2021, 9(7): 1289 Copy Citation Text

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Abstract

Quantum manipulation of individual phonons could offer new resources for studying fundamental physics and creating an innovative platform in quantum information science. Here, we propose to generate quantum states of strongly correlated phonon bundles associated with the motion of a trapped atom. Our scheme operates in the atom–phonon resonance regime where the energy spectrum exhibits strong anharmonicity such that energy eigenstates with different phonon numbers can be well-resolved in the parameter space. Compared to earlier schemes operating in the far dispersive regime, the bundle states generated here contain a large steady-state phonon number. Therefore, the proposed system can be used as a high-quality multiphonon source. Our results open up the possibility of using long-lived motional phonons as quantum resources, which could provide a broad physics community for applications in quantum metrology.

H / ℏ = ω {\hat{a}}^{†} \hat{a} + \frac{Ω}{2} σ_{y} + δ σ_{z} + \frac{g_{x}}{2} ({\hat{a}}^{†} + \hat{a}) σ_{x} + \frac{i g_{k}}{2} ({\hat{a}}^{†} - \hat{a}) σ_{z},

(1)

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H^{'} / ℏ = ω {\hat{a}}^{†} \hat{a} + \frac{Ω}{2} σ_{z} - δ σ_{y} + \frac{g_{x}}{2} ({\hat{a}}^{†} + \hat{a}) σ_{x} - \frac{i g_{k}}{2} ({\hat{a}}^{†} - \hat{a}) σ_{y},

(2)

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\frac{d ρ}{d t} = - i [H^{'}, ρ] + \frac{κ_{e}}{2} D [σ_{-}] ρ + \frac{κ_{d}}{2} D [\hat{a}] ρ + γ_{d} D [{\hat{a}}^{†} \hat{a}] ρ,

(3)

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g_{n}^{(k)} (τ_{1}, \dots, τ_{n}) = \frac{⟨ \prod_{i = 1}^{k} {[{\hat{a}}^{†} (τ_{i})]}^{n} \prod_{i = 1}^{k} {[\hat{a} (τ_{i})]}^{n} ⟩}{\prod_{i = 1}^{k} ⟨ {[{\hat{a}}^{†} (τ_{i})]}^{n} {[\hat{a} (τ_{i})]}^{n} ⟩},

(4)

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H / ℏ = \frac{p^{2}}{2 M} + Ω_{l} (x) σ_{-} + Ω_{l}^{*} (x) σ_{+} + δ σ_{z} + V (x),

(A1)

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H / ℏ = \frac{{(p - A)}^{2}}{2 M} + (Ω_{0} x - i Ω / 2) σ_{-} + (Ω_{0} x + i Ω / 2) σ_{+} + V (x), = \frac{{(p - A)}^{2}}{2 M} + Ω_{0} x σ_{x} + \frac{Ω}{2} σ_{y} + δ σ_{z} + V (x), \equiv \frac{p^{2}}{2 M} + κ_{so} p_{x} σ_{z} + Ω_{0} x σ_{x} + \frac{Ω}{2} σ_{y} + δ σ_{z} + V (x),

(A2)

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H / ℏ = ω {\hat{a}}^{†} \hat{a} + \frac{Ω}{2} σ_{y} + δ σ_{z} + \frac{g_{x}}{2} ({\hat{a}}^{†} + \hat{a}) σ_{x} + \frac{i g_{k}}{2} ({\hat{a}}^{†} - \hat{a}) σ_{z},

(A3)

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H / ℏ = ω {\hat{a}}^{†} \hat{a} + \frac{Ω}{2} σ_{z} - δ σ_{y} + \frac{g_{x}}{2} ({\hat{a}}^{†} + \hat{a}) σ_{x} - \frac{i g_{k}}{2} ({\hat{a}}^{†} - \hat{a}) σ_{y} = ω {\hat{a}}^{†} \hat{a} + \frac{Ω}{2} σ_{z} - δ σ_{y} + λ_{+} ({\hat{a}}^{†} σ_{-} + \hat{a} σ_{+}) + λ_{-} ({\hat{a}}^{†} σ_{+} + \hat{a} σ_{-}),

(B1)

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| ψ ⟩ = \sum_{n = 0}^{\infty} C_{n, g} | n, g ⟩ + \sum_{n = 0}^{\infty} C_{n, e} | n, e ⟩,

(B2)

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i \frac{d | ψ ⟩}{d t} = \sum_{n = 0}^{\infty} ({\dot{C}}_{n, g} | n, g ⟩ + {\dot{C}}_{n, e} | n, e ⟩),

(B3)

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H | ψ ⟩ = \sum_{n = 0}^{\infty} [n ω C_{n, g} | n, g ⟩ + (n ω + Ω) C_{n, e} | n, e ⟩] = \sum_{n = 0}^{\infty} (i δ C_{n, e} | n, g ⟩ - i δ C_{n, g} | n, e ⟩) + \sum_{n = 0}^{\infty} (\sqrt{n} λ_{+} C_{n, g} | n - 1, e ⟩ + \sqrt{n + 1} λ_{+} C_{n, e} | n + 1, g ⟩) + \sum_{n = 0}^{\infty} (\sqrt{n + 1} λ_{-} C_{n, g} | n + 1, e ⟩ + \sqrt{n} λ_{-} C_{n, e} | n - 1, g ⟩),

(B4)

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{\dot{C}}_{n, g} = n ω C_{n, g} + i δ C_{n, e} + \sqrt{n} λ_{+} C_{n - 1, e} + \sqrt{n + 1} λ_{-} C_{n + 1, e},

(B5)

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{\dot{C}}_{n, e} = (n ω + Ω) C_{n, e} - i δ C_{n, g} + \sqrt{n + 1} λ_{+} C_{n + 1, g} + \sqrt{n} λ_{-} C_{n - 1, g} .

(B6)

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(\begin{matrix} ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋰ \\ \dots & n ω & i δ & 0 & \sqrt{n + 1} λ_{-} & 0 & 0 & \dots \\ \dots & - i δ & n ω + Ω & \sqrt{n + 1} λ_{+} & 0 & 0 & 0 & \dots \\ \dots & 0 & \sqrt{n + 1} λ_{+} & (n + 1) ω & i δ & 0 & \sqrt{n + 2} λ_{-} & \dots \\ \dots & \sqrt{n + 1} λ_{-} & 0 & - i δ & (n + 1) ω + Ω & \sqrt{n + 2} λ_{+} & 0 & \dots \\ \dots & 0 & 0 & 0 & \sqrt{n + 2} λ_{+} & (n + 2) ω & i δ & \dots \\ \dots & 0 & 0 & \sqrt{n + 2} λ_{-} & 0 & - i δ & (n + 2) ω + Ω & \dots \\ ⋰ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ \end{matrix}) .

(B7)

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Yuangang Deng, Tao Shi, Su Yi. Motional n-phonon bundle states of a trapped atom with clock transitions[J]. Photonics Research, 2021, 9(7): 1289

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