Researching | Subnatural-linewidth fluorescent single photons

Journals >Photonics Research >Volume 12 >Issue 4 >Page 625 > Article

Photonics Research
Vol. 12, Issue 4, 625 (2024)

Subnatural-linewidth fluorescent single photons | Editors' Pick

He-Bin Zhang^1、2, Gao-Xiang Li^2、4、*, and Yong-Chun Liu^{1、3、5、*}

Author Affiliations

¹State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China

²Department of Physics, Huazhong Normal University, Wuhan 430079, China

³Frontier Science Center for Quantum Information, Beijing 100084, China

⁴e-mail: gaox@mail.ccnu.edu.cn

⁵e-mail: ycliu@tsinghua.edu.cn

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DOI: 10.1364/PRJ.514756 Cite this Article Set citation alerts

He-Bin Zhang, Gao-Xiang Li, Yong-Chun Liu. Subnatural-linewidth fluorescent single photons[J]. Photonics Research, 2024, 12(4): 625 Copy Citation Text

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Abstract

Subnatural-linewidth single photons are of vital importance in quantum optics and quantum information science. According to previous research, it appears difficult to utilize resonance fluorescence to generate single photons with subnatural linewidth. Here we propose a universally applicable approach to generate fluorescent single photons with subnatural linewidth, which can be implemented based on

Λ

-shape and similar energy structures. Further, the general condition to obtain fluorescent single photons with subnatural linewidth is revealed. The single-photon linewidth can be easily manipulated over a broad range by external fields, which can be several orders of magnitude smaller than the natural linewidth. Our study can be easily implemented in various physical platforms with current experimental techniques and will significantly facilitate the research on the quantum nature of resonance fluorescence and the technologies in quantum information science.

H = Δ_{e} σ_{e e} + Δ_{a} σ_{a a} + Δ_{s} s^{†} s + (Ω σ_{e g} + Ω_{r} σ_{g a} + g σ_{e a} s + H .c) .

(1)

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\dot{ρ} = - i [H, ρ] + γ_{1} D [σ_{g e}] ρ + γ_{2} D [σ_{a e}] ρ + κ D [s] ρ,

(2)

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L_{eff} ρ_{g} = γ_{1}^{*} D [σ_{g g}] ρ_{g} + γ_{2}^{*} D [σ_{a g}] ρ_{g} .

(3)

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γ_{i}^{*} = \frac{4 γ_{i} {| Ω |}^{2}}{{(γ_{1} + γ_{2})}^{2}},

(4)

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g_{s}^{(2)} (0) = \lim_{t \to \infty} \frac{⟨ s^{†} (t) s^{†} (t) s (t) s (t) ⟩}{{⟨ s^{†} (t) s (t) ⟩}^{2}} .

(5)

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I_{s} \equiv \frac{1}{Δ t} \leq Δ ω .

(6)

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I_{TLS} = Γ \frac{4 Ω_{T}^{2}}{Γ^{2} + 8 Ω_{T}^{2}},

(7)

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I_{TLS} = \frac{4 Ω_{T}^{2}}{Γ} ≪ Γ .

(8)

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I_{TLS} = \frac{Γ}{2},

(9)

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I_{N} = γ \frac{2 Ω^{2} Ω_{r}^{2}}{Ω^{4} + 2 Ω_{r}^{2} (2 γ^{2} + Ω^{2} + 2 Ω_{r}^{2})} .

(10)

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I_{N} = γ^{*} \frac{2 Ω_{r}^{2}}{γ^{* 2} + 4 Ω_{r}^{2}} .

(11)

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I_{N} = \frac{2 Ω_{r}^{2}}{γ^{*}} ≪ γ^{*} .

(12)

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I_{N} = \frac{γ^{*}}{2},

(13)

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{\dot{ρ}}_{σ} = - i [H_{A}, ρ_{σ}] + γ_{1} D [σ_{g e}] ρ_{σ} + γ_{2} D [σ_{a e}] ρ_{σ},

(A1)

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{\dot{ρ}}_{e e} = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) ρ_{e e} - i Ω ρ_{g e} + i ρ_{e g} Ω^{*}, {\dot{ρ}}_{e g} = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) ρ_{e g} + i Ω ρ_{e e} - i Ω ρ_{g g} + i ρ_{e a} Ω_{r}, {\dot{ρ}}_{e a} = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) ρ_{e a} - i Ω ρ_{g a} + i ρ_{e g} Ω_{r}^{*},

(A2)

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ρ_{g e} (t) \approx \frac{2 i Ω^{*}}{γ_{1} + γ_{2}} ρ_{g g} (t), ρ_{a e} (t) \approx \frac{2 i Ω^{*}}{γ_{1} + γ_{2}} ρ_{a g} (t), ρ_{e e} (t) \approx \frac{4 {| Ω |}^{2}}{{(γ_{1} + γ_{2})}^{2}} ρ_{g g} (t) .

(A3)

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{\dot{ρ}}_{g g} = γ_{1} ρ_{e e} + i Ω ρ_{g e} + i Ω_{r} ρ_{g a} - i Ω^{*} ρ_{e g} - i Ω_{r}^{*} ρ_{a g}, {\dot{ρ}}_{a a} = γ_{2} ρ_{e e} - i Ω_{r} ρ_{g a} + i Ω_{r}^{*} ρ_{a g}, {\dot{ρ}}_{g a} = - i Ω^{*} ρ_{e a} - i Ω_{r}^{*} ρ_{a a} + i Ω_{r}^{*} ρ_{g g} .

(A4)

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{\dot{ρ}}_{g} = - i [H_{T}, ρ_{g}] + L_{eff} ρ_{g},

(A5)

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{\tilde{ρ}}_{g g} = \frac{Ω_{r}^{2} (2 γ^{2} + Ω^{2} + 2 Ω_{r}^{2})}{M}, {\tilde{ρ}}_{a a} = \frac{Ω^{4} + Ω_{r}^{2} (2 γ^{2} - Ω^{2} + 2 Ω_{r}^{2})}{M}, {\tilde{ρ}}_{e e} = \frac{2 Ω^{2} Ω_{r}^{2}}{M}, {\tilde{ρ}}_{g e} = \frac{2 i γ Ω Ω_{r}^{2}}{M}, {\tilde{ρ}}_{a e} = \frac{- Ω^{3} Ω_{r} + 2 Ω Ω_{r}^{3}}{M}, {\tilde{ρ}}_{g a} = \frac{- i γ Ω^{2} Ω_{r}}{M},

(B1)

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S_{e a} (ω_{s}) = γ_{2} Re \int_{0}^{\infty} d τ \lim_{t \to \infty} ⟨ σ_{e a} (t) σ_{a e} (t + τ) ⟩ e^{i ω_{s} τ},

(C1)

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\partial_{τ} Y_{g g} (τ) = γ_{1} Y_{e e} (τ) + i Ω Y_{e g} (τ) - i Ω^{*} Y_{g e} (τ) + i Ω_{r} Y_{a g} (τ) - i Ω_{r}^{*} Y_{g a} (τ), \partial_{τ} Y_{a g} (τ) = - i Ω_{r}^{*} Y_{a a} (τ) - i Ω^{*} Y_{a e} (τ) + i Ω_{r}^{*} Y_{g g} (τ), \partial_{τ} Y_{e g} (τ) = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) Y_{e g} (τ) - i Ω^{*} Y_{e e} (τ) + i Ω^{*} Y_{g g} (τ) - i Ω_{r}^{*} Y_{e a} (τ), \partial_{τ} Y_{g a} (τ) = i Ω Y_{e a} (τ) + i Ω_{r} Y_{a a} (τ) - i Ω_{r} Y_{g g} (τ), \partial_{τ} Y_{a a} (τ) = γ_{2} Y_{e e} (τ) - i Ω_{r} Y_{a g} (τ) + i Ω_{r}^{*} Y_{g a} (τ), \partial_{τ} Y_{e a} (τ) = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) Y_{e a} (τ) + i Ω^{*} Y_{g a} (τ) - i Ω_{r} Y_{e g} (τ), \partial_{τ} Y_{g e} (τ) = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) Y_{g e} (τ) + i Ω Y_{e e} (τ) - i Ω Y_{g g} (τ) + i Ω_{r} Y_{a e} (τ), \partial_{τ} Y_{a e} (τ) = - (\frac{γ_{1}}{2} + \frac{γ_{2}}{2}) Y_{a e} (τ) - i Ω Y_{a g} (τ) + i Ω_{r}^{*} Y_{g e} (τ), \partial_{τ} Y_{e e} (τ) = - (γ_{1} + γ_{2}) Y_{e e} (τ) - i Ω Y_{e g} (τ) + i Ω^{*} Y_{g e} (τ) .

(C2)

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Y_{a g} (0) = {\tilde{ρ}}_{g e}, Y_{a a} (0) = {\tilde{ρ}}_{a e}, Y_{a e} (0) = {\tilde{ρ}}_{e e},

(C3)

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Y_{m n} (s) \equiv \int_{0}^{\infty} d τ Y_{m n} (τ) e^{- s τ},

(C4)

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S_{e a}^{e w} (ω_{s}) = S_{c}^{e w} (ω_{s}) + S_{n}^{e w} (ω_{s}) + S_{b}^{e w} (ω_{s}) .

(C5)

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S_{c}^{e w} (ω_{s}) = γ {| {\tilde{ρ}}_{a e} |}^{2} δ (ω_{s}) .

(C6)

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S_{n}^{e w} (ω_{s}) = \frac{- i {\tilde{ρ}}_{g e} Ω^{2} + {\tilde{ρ}}_{a e} γ Ω_{r}}{Ω} \frac{γ^{*}}{ω_{s}^{2} + γ^{* 2}} .

(C7)

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S_{b}^{e w} (ω_{s}) = - {\tilde{ρ}}_{a e} Ω Ω_{r} \frac{1}{ω_{s}^{2} + γ^{2}} - \frac{2 i {\tilde{ρ}}_{g e} Ω^{3}}{γ} \frac{γ^{2} - ω_{s}^{2}}{{(γ^{2} + ω_{s}^{2})}^{2}} .

(C8)

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\frac{\int_{\infty}^{\infty} d ω_{s} S_{b}^{e w}}{\int_{\infty}^{\infty} d ω_{s} S_{n}^{e w}} = \frac{Ω^{2}}{γ^{2}},

(C9)

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S_{e a}^{e w} (ω_{s}) = S_{c}^{e w} (ω_{s}) + S_{n}^{e w} (ω_{s}),

(C10)

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S_{e a}^{e s} (ω_{s}) = S_{c}^{e s} (ω_{s}) + S_{n}^{e s} (ω_{s}) + S_{b}^{e s} (ω_{s}) .

(C11)

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S_{c}^{e s} (ω_{s}) = γ {| {\tilde{ρ}}_{a e} |}^{2} δ (ω_{s}) .

(C12)

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S_{n}^{e s} (ω_{s}) = - \frac{i}{4} {\tilde{ρ}}_{g e} Ω (\frac{2 γ^{*}}{ω_{s}^{2} + γ^{* 2}} + \frac{γ^{*}}{{(ω_{s} - 2 Ω_{r})}^{2} + γ^{* 2}} + \frac{γ^{*}}{{(ω_{s} + 2 Ω_{r})}^{2} + γ^{* 2}}) .

(C13)

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S_{b}^{e s} (ω_{s}) = - {\tilde{ρ}}_{a e} Ω Ω_{r} \frac{1}{ω_{s}^{2} + γ^{2}} .

(C14)

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\frac{\int_{\infty}^{\infty} d ω_{s} S_{b}^{e s}}{\int_{\infty}^{\infty} d ω_{s} S_{n}^{e s}} = \frac{Ω^{2} - 2 Ω_{r}^{2}}{2 γ^{2}},

(C15)

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S_{e a}^{e s} (ω_{s}) = S_{c}^{e s} (ω_{s}) + S_{n}^{e s} (ω_{s}),

(C16)

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g_{e a}^{(2)} (τ) = 1 - \cos (2 Ω_{r} τ) e^{- γ^{*} τ} - \frac{Ω^{2} - 2 Ω_{r}}{2 γ Ω_{r}} \sin (2 Ω_{r} τ) e^{- γ^{*} τ},

(D1)

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H_{A} = Δ_{e} σ_{e e} + Ω σ_{e g} + Ω_{r} σ_{g a} + H .c .

(E1)

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{\dot{ρ}}_{g} = - i [{H^{'}}_{T}, ρ_{g}] + γ_{1 d}^{*} D [σ_{g g} ρ_{g}] + γ_{2 d}^{*} D [σ_{a g} ρ_{g}],

(E2)

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H_{T}^{'} = Δ_{eff} σ_{g g} + (Ω_{r} σ_{g a} + H.c) .

(E3)

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Δ_{eff} = - \frac{4 Δ_{e} {| Ω |}^{2}}{{(γ_{1} + γ_{2})}^{2} + 4 Δ_{e}^{2}}, γ_{1 d}^{*} = \frac{4 γ_{1} {| Ω |}^{2}}{{(γ_{1} + γ_{2})}^{2} + 4 Δ_{e}^{2}}, γ_{2 d}^{*} = \frac{4 γ_{2} {| Ω |}^{2}}{{(γ_{1} + γ_{2})}^{2} + 4 Δ_{e}^{2}},

(E4)

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Δ_{eff} = - \frac{Δ_{e} {| Ω |}^{2}}{γ^{2} + Δ_{e}^{2}}, γ_{d}^{*} = γ_{1 d}^{*} = γ_{2 d}^{*} = \frac{{| Ω |}^{2}}{γ^{2} + Δ_{e}^{2}},

(E5)

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{\dot{ρ}}_{r} = - i [H_{Rb}, ρ_{r}] + \sum_{i = - 1}^{1} Y_{i} D [| 1, i ⟩ ⟨ 0,0 |] ρ_{r} + κ D [s] ρ_{r} .

(F1)

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H_{Rb} = H_{0} + H_{I} + H_{B} + H_{S} .

(F2)

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H_{0} = Δ_{e} | F_{e}, m_{e} ⟩ ⟨ F_{e}, m_{e} | + H.c,

(F3)

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H_{I} = \sum_{g_{i}} V_{e g_{i}} | F_{e}, m_{e} ⟩ ⟨ F_{g}, m_{g_{i}} | + H.c .

(F4)

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V_{e g_{i}} = - ⟨ F_{e}, m_{e} | d | F_{g}, m_{g_{i}} ⟩ \cdot E = (- {1)}^{F_{e} - m_{e} + 1} (\begin{matrix} F_{e} & 1 & F_{g} \\ - m_{e} & q & m_{g_{i}} \end{matrix}) Ω_{L},

(F5)

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H_{B} = - μ_{B} g_{F} F \cdot B_{T} = μ_{B} g_{F} B_{T} \sum_{g_{i}, g_{j}} ⟨ F_{g}, m_{g_{i}} | F | F_{g}, m_{g_{j}} ⟩ \cdot e_{x} | F_{g}, m_{g_{i}} ⟩ ⟨ F_{g}, m_{g_{j}} |,

(F6)

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⟨ F_{g}, m_{g_{i}} | F | F_{g}, m_{g_{j}} ⟩ = ⟨ F ‖ F ‖ F ⟩ {(- 1)}^{F_{g} - m_{g_{i}}} (\begin{matrix} F_{g} & 1 & F_{g} \\ - m_{g_{i}} & q & m_{g_{j}} \end{matrix}) .

(F7)

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H_{S} = Δ_{s} s^{†} s + (g A_{d} s^{†} + H.c),

(F8)

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L_{eff} ρ_{r, g} = \sum_{i = - 1}^{1} Y_{i}^{*} D [| 1, i ⟩ ⟨ 1, - 1 |] ρ_{r, g},

(F9)

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He-Bin Zhang, Gao-Xiang Li, Yong-Chun Liu. Subnatural-linewidth fluorescent single photons[J]. Photonics Research, 2024, 12(4): 625

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