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Journals >Chinese Optics Letters >Volume 20 >Issue 6 >Page 062701 > Article

Chinese Optics Letters
Vol. 20, Issue 6, 062701 (2022)

Three methods for the single-photon transport in a chiral cavity quantum electrodynamics system

Jiang-Shan Tang^1、2, Lei Tang¹, and Keyu Xia^1、2、*

Author Affiliations

¹College of Engineering and Applied Sciences, National Laboratory of Solid State Microstructures, and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210023, China

²School of Physics, Nanjing University, Nanjing 210023, China

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DOI: 10.3788/COL202220.062701 Cite this Article Set citation alerts

Jiang-Shan Tang, Lei Tang, Keyu Xia. Three methods for the single-photon transport in a chiral cavity quantum electrodynamics system[J]. Chinese Optics Letters, 2022, 20(6): 062701 Copy Citation Text

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Abstract

We investigate the single-photon transport problem in the system of a whispering-gallery mode microresonator chirally coupled with a two-level quantum emitter (QE). Conventionally, this chiral QE-microresonator coupling system can be studied by the master equation and the single-photon transport methods. Here, we provide a new approach, based on the transfer matrix, to assess the single-photon transmission of such a system. Furthermore, we prove that these three methods are equivalent. The corresponding relations of parameters among these approaches are precisely deduced. The transfer matrix can be extended to a multiple-resonator system interacting with two-level QEs in a chiral way. Therefore, our work may provide a convenient and intuitive form for exploring more complex chiral cavity quantum electrodynamics systems.

Keywords

chiral cavity QED system master equation method single-photon transport theory transfer matrix method

H = - Δ_{1} a^{†} a - Δ_{2} σ^{+} σ^{-} - Δ_{1} b^{†} b + i \sqrt{2 κ_{ex}} α_{in} (a^{†} - a) + g (a^{†} σ^{-} + σ^{+} a) + h (a^{†} b + b^{†} a),

(1)

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\dot{ρ} = - i [H, ρ] + κ_{tol} (2 a ρ a^{†} - a^{†} a ρ - ρ a^{†} a) + κ_{tol} (2 b ρ b^{†} - b^{†} b ρ - ρ b^{†} b) + γ (2 σ^{-} ρ σ^{+} - σ^{+} σ^{-} ρ - ρ σ^{+} σ^{-}),

(2)

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\dot{a} = i {\tilde{Δ}}_{1} a + α_{in} \sqrt{2 κ_{ex}} - i g σ^{-} - i h b,

(3a)

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{\dot{σ}}^{-} = i {\tilde{Δ}}_{2} σ^{-} + i g σ_{z} a,

(3b)

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\dot{b} = i {\tilde{Δ}}_{1} b - i h a,

(3c)

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〈 a 〉 = \frac{i α_{in} \sqrt{2 κ_{ex}} {\tilde{Δ}}_{1} {\tilde{Δ}}_{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} + 〈 σ_{z} 〉 g^{2}) - {\tilde{Δ}}_{2} h^{2}},

(4)

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t_{ω} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} + σ_{z} g^{2}] - {\tilde{Δ}}_{2} h^{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} + σ_{z} g^{2}) - {\tilde{Δ}}_{2} h^{2}} .

(5)

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H = \int d x c_{F}^{†} (x) (ω_{0} - i v_{g} \frac{\partial}{\partial x}) c_{F} (x) + \int d x c_{B}^{†} (x) (ω_{0} + i v_{g} \frac{\partial}{\partial x}) c_{B} (x) + (Ω - i κ_{in}) a^{†} a + (Ω - i κ_{in}) b^{†} b + (Ω_{e} - i γ) a_{e}^{†} a_{e} + Ω_{g} a_{g}^{†} a_{g} + \int d x δ (x) [V_{a} c_{F}^{†} (x) a + V_{a}^{*} a^{†} c_{F} (x)] + \int d x δ (x) [V_{b} c_{B}^{†} (x) b + V_{b}^{*} b^{†} c_{B} (x)] + g a σ_{+} + g^{*} a^{†} σ_{-} + h b^{†} a + h^{*} a^{†} b,

(6)

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| ψ 〉 = \int d x [{\tilde{φ}}_{F} (x, t) c_{F}^{†} (x) + {\tilde{φ}}_{B} (x, t) c_{B}^{†} (x)] | \emptyset 〉 + [{\tilde{e}}_{a} (t) a^{†} + {\tilde{e}}_{b} (t) b^{†} + {\tilde{e}}_{qe} (t) σ^{+}] | \emptyset 〉,

(7)

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t_{ω} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - g^{2}] - {\tilde{Δ}}_{2} h^{2}}{{\tilde{Δ}}_{1} ({\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - g^{2}) - {\tilde{Δ}}_{2} h^{2}},

(8)

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{\begin{cases} a_{1} = t^{*} b_{1} - κ^{*} a_{0} \\ b_{0} = t a_{0} + κ b_{1} \end{cases}, {\begin{cases} c_{1} = t^{*} d_{1} - κ^{*} c_{0} \\ d_{0} = t c_{0} + κ d_{1} \end{cases},

(9)

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\begin{matrix} (\begin{array}{l} a_{0} \\ b_{0} \\ c_{0} \\ d_{0} \end{array}) \end{matrix} = \frac{1}{κ^{*}} (\begin{matrix} - 1 & t^{*} & 0 & 0 \\ - t & 1 & 0 & 0 \\ 0 & 0 & - 1 & t^{*} \\ 0 & 0 & - t & 1 \end{matrix}) \begin{matrix} (\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) \end{matrix} \equiv M_{cpl} (\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) .

(10)

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(\begin{array}{l} a_{1} \\ b_{1} \\ c_{1} \\ d_{1} \end{array}) = M_{pro} M_{x} (\begin{array}{l} a_{3} \\ b_{3} \\ c_{3} \\ d_{3} \end{array}),

(11a)

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b_{3} = α_{2} e^{i θ_{2}} a_{3},

(11b)

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d_{3} = α_{2} e^{i θ_{2}} c_{3},

(11c)

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M_{pro} = (\begin{matrix} α_{1}^{- 1} e^{- i θ_{1}} & 0 & 0 & 0 \\ 0 & α_{3} e^{i θ_{3}} & 0 & 0 \\ 0 & 0 & α_{3}^{- 1} e^{- i θ_{3}} & 0 \\ 0 & 0 & 0 & α_{1} e^{i θ_{1}} \end{matrix}),

(12)

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(\begin{array}{l} a_{0} \\ b_{0} \\ c_{0} \\ d_{0} \end{array}) = M_{cpl} M_{pro} M_{x} (\begin{array}{l} a_{3} \\ b_{3} \\ c_{3} \\ d_{3} \end{array}) .

(13)

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M_{x} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

(14)

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t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ}}{- 1 + α t^{*} e^{i θ}} .

(15)

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M_{x} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 / t_{s} & r_{s} / t_{s} & 0 \\ 0 & - r_{s} / t_{s} & 1 / t_{s} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),

(16)

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t_{s} = \cos ε \approx 1 - \frac{ε^{2}}{2}, r_{s} = i \sin ε \approx i ε .

(17)

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t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α t^{*} e^{i θ} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} .

(18)

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M_{x} = (\begin{matrix} t_{qe}^{- 1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}),

(19)

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t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} t_{qe}}{- 1 + α {t^{*}}^{} e^{i θ} t_{qe}} .

(20)

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M_{x} = (\begin{matrix} t_{qe}^{- 1} & 0 & 0 & 0 \\ 0 & 1 / t_{s} & r_{s} / t_{s} & 0 \\ 0 & - r_{s} / t_{s} & 1 / t_{s} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) .

(21)

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t_{ω} = \frac{b_{0}}{a_{0}} = \frac{- t + α e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α {t^{*}}^{} e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} .

(22)

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α = e^{- κ_{in} τ_{rt}} \approx 1 - κ_{in} τ_{rt},

(23a)

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t = e^{- κ_{ex} τ_{rt}} \approx 1 - κ_{ex} τ_{rt} .

(23b)

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t_{qe} = \frac{ω - ω_{qe} + i (γ - Γ)}{ω - ω_{qe} + i (γ + Γ)},

(24)

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t_{ω} = \frac{- t + α e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}}{- 1 + α t^{*} e^{i θ} t_{qe} \frac{t_{s} - t^{*} α e^{i θ}}{1 - t_{s} t^{*} α e^{i θ}}} \approx \frac{κ_{ex} τ_{rt} - 1 + (1 - κ_{in} τ_{rt} + i Δ_{1} τ_{rt}) [(1 - \frac{2 i Γ}{{\tilde{Δ}}_{2} + i Γ}) (1 + \frac{ε^{2}}{i {\tilde{Δ}}_{1} τ_{rt} + ε^{2} / 2})]}{- 1 + (1 + i {\tilde{Δ}}_{1} τ_{rt}) [(1 - \frac{2 i Γ}{{\tilde{Δ}}_{2} + i Γ}) (1 + \frac{ε^{2}}{i {\tilde{Δ}}_{1} τ_{rt} + ε^{2} / 2})]} \approx \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - Γ (2 / τ_{rt} - κ_{tol})] - {\tilde{Δ}}_{2} \frac{ε^{2}}{τ_{rt}^{2}}}{{\tilde{Δ}}_{1} [{\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - Γ (2 / τ_{rt} - κ_{tol})] - {\tilde{Δ}}_{2} \frac{ε^{2}}{τ_{rt}^{2}}} = \frac{{\tilde{Δ}}_{1} [(Δ_{1} + i κ_{in} - i κ_{ex}) {\tilde{Δ}}_{2} - Γ (2 F - κ_{tol})] - {\tilde{Δ}}_{2} {(ε \times F)}^{2}}{{\tilde{Δ}}_{1} [{\tilde{Δ}}_{1} {\tilde{Δ}}_{2} - Γ (2 F - κ_{tol})] - {\tilde{Δ}}_{2} {(ε \times F)}^{2}} .

(25)

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(2 F - κ_{tol}) Γ = g^{2}, ε \times F = h,

(26)

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t_{qe} = \exp (i φ_{pha}) \exp (- φ_{dis}),

(27)

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Ω_{eff} \approx Ω (1 - \frac{φ_{pha} Ω}{2 π m ω}),

(28)

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Jiang-Shan Tang, Lei Tang, Keyu Xia. Three methods for the single-photon transport in a chiral cavity quantum electrodynamics system[J]. Chinese Optics Letters, 2022, 20(6): 062701

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