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Journals >Photonics Research >Volume 9 >Issue 5 >Page 879 > Article

Photonics Research
Vol. 9, Issue 5, 879 (2021)

Nonreciprocal transition between two nondegenerate energy levels

Xunwei Xu^1、2、*, Yanjun Zhao³, Hui Wang⁴, Aixi Chen^5、8, and Yu-Xi Liu^6、7

Author Affiliations

¹Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Key Laboratory for Matter Microstructure and Function of Hunan Province, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China

²Department of Applied Physics, East China Jiaotong University, Nanchang 330013, China

³Key Laboratory of Opto-electronic Technology, Ministry of Education, Beijing University of Technology, Beijing 100124, China

⁴Center for Emergent Matter Science (CEMS), RIKEN, Wako, Saitama 351-0198, Japan

⁵Department of Physics, Zhejiang Sci-Tech University, Hangzhou 310018, China

⁶Institute of Microelectronics, Tsinghua University, Beijing 100084, China

⁷Frontier Science Center for Quantum Information, Beijing 100084, China

⁸e-mail: aixichen@zstu.edu.cn

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DOI: 10.1364/PRJ.412904 Cite this Article Set citation alerts

Xunwei Xu, Yanjun Zhao, Hui Wang, Aixi Chen, Yu-Xi Liu. Nonreciprocal transition between two nondegenerate energy levels[J]. Photonics Research, 2021, 9(5): 879 Copy Citation Text

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Abstract

Stimulated emission and absorption are two fundamental processes of light–matter interaction, and the coefficients of the two processes should be equal. However, we will describe a generic method to realize the significant difference between the stimulated emission and absorption coefficients of two nondegenerate energy levels, which we refer to as a nonreciprocal transition. As a simple implementation, a cyclic three-level atom system, comprising two nondegenerate energy levels and one auxiliary energy level, is employed to show a nonreciprocal transition via a combination of synthetic magnetism and reservoir engineering. Moreover, a single-photon nonreciprocal transporter is proposed using two one-dimensional semi-infinite coupled-resonator waveguides connected by an atom with nonreciprocal transition effect. Our work opens up a route to design atom-mediated nonreciprocal devices in a wide range of physical systems.

H_{coh + dis} = (Ω - i γ) | a ⟩ ⟨ b | + (Ω^{*} - i γ) | b ⟩ ⟨ a | .

(1)

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H = (Δ_{a b} - i γ_{a}) | a ⟩ ⟨ a | - i γ_{b} | b ⟩ ⟨ b | + (Δ_{c b} - i γ_{c}) | c ⟩ ⟨ c | + (Ω_{a b} e^{i Φ} | a ⟩ ⟨ b | + Ω_{c b} | c ⟩ ⟨ b | + Ω_{c a} | c ⟩ ⟨ a | + H.c.),

(2)

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H_{eff} = (Δ_{a} - i Γ_{a}) | a ⟩ ⟨ a | + (Δ_{b} - i Γ_{b}) | b ⟩ ⟨ b | + J_{a b} | a ⟩ ⟨ b | + J_{b a} | b ⟩ ⟨ a |,

(3)

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J_{a b} \equiv Ω_{a b} e^{i Φ} - i \frac{Ω_{c a} Ω_{c b} (γ_{c} - i Δ_{c b})}{γ_{c}^{2} + Δ_{c b}^{2}},

(4)

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J_{b a} \equiv Ω_{a b} e^{- i Φ} - i \frac{Ω_{c a} Ω_{c b} (γ_{c} - i Δ_{c b})}{γ_{c}^{2} + Δ_{c b}^{2}} .

(5)

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H_{l} = Δ_{l} \sum_{j = 0}^{+ \infty} l_{j}^{†} l_{j} - ξ_{l} \sum_{j = 0}^{+ \infty} (l_{j}^{†} l_{j + 1} + H.c.),

(6)

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{\tilde{H}}_{eff} = J_{a b} | a ⟩ ⟨ b | + J_{b a} | b ⟩ ⟨ a | - i Γ_{a} | a ⟩ ⟨ a | - i Γ_{b} | b ⟩ ⟨ b |,

(7)

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H_{int} = g_{a} a_{0} | a ⟩ ⟨ g | + g_{b} b_{0} | b ⟩ ⟨ g | + g_{a} a_{0}^{†} | g ⟩ ⟨ a | + g_{b} b_{0}^{†} | g ⟩ ⟨ b | .

(8)

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Δ k \equiv π - 2 k_{half} .

(9)

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\tilde{H} = (ω_{a b} - i γ_{a}) | a ⟩ ⟨ a | - i γ_{b} | b ⟩ ⟨ b | + (ω_{c b} - i γ_{c}) | c ⟩ ⟨ c | + (Ω_{a b} e^{i ϕ_{a b}} e^{- i ν_{a b} t} | a ⟩ ⟨ b | + Ω_{c b} e^{i ϕ_{c b}} e^{- i ν_{c b} t} | c ⟩ ⟨ b | + Ω_{c a} e^{i ϕ_{c a}} e^{- i ν_{c a} t} | c ⟩ ⟨ a | + H.c.),

(A1)

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H = W^{†} \tilde{H} W + i \frac{d W^{†}}{d t} W = (Δ_{a b} - i γ_{a}) | a ⟩ ⟨ a | - i γ_{b} | b ⟩ ⟨ b | + (Δ_{c b} - i γ_{c}) | c ⟩ ⟨ c | + Ω_{a b} e^{i ϕ_{a b}} | a ⟩ ⟨ b | + Ω_{c b} e^{i ϕ_{c b}} | c ⟩ ⟨ b | + Ω_{c a} e^{i ϕ_{c a}} | c ⟩ ⟨ a | + H.c.,

(A2)

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| ψ ⟩ = A (t) | a ⟩ + B (t) | b ⟩ + C (t) | c ⟩ .

(B1)

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\dot{A} (t) = (- i Δ_{a b} - γ_{a}) A (t) - i Ω_{a b} e^{i Φ} B (t) - i Ω_{c a} C (t),

(B2)

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\dot{B} (t) = - γ_{b} B (t) - i Ω_{a b} e^{- i Φ} A (t) - i Ω_{c b} C (t),

(B3)

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\dot{C} (t) = (- i Δ_{c b} - γ_{c}) C (t) - i Ω_{c a} A (t) - i Ω_{c b} B (t) .

(B4)

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C (t) = \frac{- i Ω_{c a}}{γ_{c} + i Δ_{c b}} A (t) - \frac{i Ω_{c b}}{γ_{c} + i Δ_{c b}} B (t) .

(B5)

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\dot{A} (t) = - [i (Δ_{a b} - \frac{Ω_{c a}^{2} Δ_{c b}}{γ_{c}^{2} + Δ_{c b}^{2}}) + (γ_{a} + \frac{Ω_{c a}^{2} γ_{c}}{γ_{c}^{2} + Δ_{c b}^{2}})] A (t) - [i Ω_{a b} e^{i Φ} + \frac{Ω_{c a} Ω_{c b} (γ_{c} - i Δ_{c b})}{γ_{c}^{2} + Δ_{c b}^{2}}] B (t),

(B6)

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\dot{B} (t) = - [- i \frac{Ω_{c b}^{2} Δ_{c b}}{γ_{c}^{2} + Δ_{c b}^{2}} + (γ_{b} + \frac{Ω_{c b}^{2} γ_{c}}{γ_{c}^{2} + Δ_{c b}^{2}})] B (t) - [i Ω_{a b} e^{- i Φ} + \frac{Ω_{c a} Ω_{c b} (γ_{c} - i Δ_{c b})}{γ_{c}^{2} + Δ_{c b}^{2}}] A (t) .

(B7)

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H_{tot} = \sum_{l = a, b} H_{l} + {\tilde{H}}_{eff} + H_{int},

(C1)

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| E ⟩ = \sum_{j = 0}^{+ \infty} [u_{a} (j) a_{j}^{†} + u_{b} (j) b_{j}^{†}] | g, 0 ⟩ + A | a, 0 ⟩ + B | b, 0 ⟩,

(C2)

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Δ_{a} u_{a} (0) - ξ_{a} u_{a} (1) + g_{a} A = E u_{a} (0),

(C3)

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Δ_{b} u_{b} (0) - ξ_{b} u_{b} (1) + g_{b} B = E u_{b} (0),

(C4)

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- i Γ_{a} A + g_{a} u_{a} (0) + J_{a b} B = E A,

(C5)

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- i Γ_{b} B + g_{b} u_{b} (0) + J_{b a} A = E B,

(C6)

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Δ_{l} u_{l} (j) - ξ_{l} [u_{l} (j + 1) + u_{l} (j - 1)] = E u_{l} (j),

(C7)

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u_{l} (j) = e^{- i k_{l} j} + s_{l l} e^{i k_{l} j},

(C8)

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u_{l^{'}} (j) = s_{l^{'} l} e^{i k_{l^{'}} j},

(C9)

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E = Δ_{l} - 2 ξ_{l} \cos k_{l}, 0 < k_{l} < π,

(C10)

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A = \frac{(E + i Γ_{b}) g_{a} u_{a} (0) + J_{a b} g_{b} u_{b} (0)}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}},

(C11)

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B = \frac{(E + i Γ_{a}) g_{b} u_{b} (0) + J_{b a} g_{a} u_{a} (0)}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}} .

(C12)

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(Δ_{a} - E + {\bar{Δ}}_{a}) u_{a} (0) + J_{a b}^{'} u_{b} (0) = ξ_{a} u_{a} (1),

(C13)

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(Δ_{b} - E + {\bar{Δ}}_{b}) u_{b} (0) + J_{b a}^{'} u_{a} (0) = ξ_{b} u_{b} (1),

(C14)

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J_{a b}^{'} = \frac{J_{a b} g_{a} g_{b}}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}},

(C15)

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J_{b a}^{'} = \frac{J_{b a} g_{a} g_{b}}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}},

(C16)

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{\bar{Δ}}_{a} = \frac{(E + i Γ_{b}) g_{a}^{2}}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}},

(C17)

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{\bar{Δ}}_{b} = \frac{(E + i Γ_{a}) g_{b}^{2}}{(E + i Γ_{a}) (E + i Γ_{b}) - J_{b a} J_{a b}} .

(C18)

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(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) s_{a a} + J_{a b}^{'} s_{b a} = - ξ_{a} e^{i k_{a}} - {\bar{Δ}}_{a},

(C19)

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J_{b a}^{'} s_{a a} + (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) s_{b a} = - J_{b a}^{'} .

(C20)

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(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) s_{a b} + J_{a b}^{'} s_{b b} = - J_{a b}^{'},

(C21)

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J_{b a}^{'} s_{a b} + (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) s_{b b} = - ξ_{b} e^{i k_{b}} - {\bar{Δ}}_{b} .

(C22)

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L S = R,

(C23)

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S = (\begin{matrix} s_{a a} & s_{a b} \\ s_{b a} & s_{b b} \end{matrix})

(C24)

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L = (\begin{matrix} ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a} & J_{a b}^{'} \\ J_{b a}^{'} & ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b} \end{matrix}),

(C25)

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R = - (\begin{matrix} ξ_{a} e^{i k_{a}} + {\bar{Δ}}_{a} & J_{a b}^{'} \\ J_{b a}^{'} & ξ_{b} e^{i k_{b}} + {\bar{Δ}}_{b} \end{matrix}) .

(C26)

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s_{a a} = \frac{J_{a b}^{'} J_{b a}^{'} - (ξ_{a} e^{i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b})}{(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) - J_{a b}^{'} J_{b a}^{'}},

(C27)

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s_{b a} = \frac{2 i ξ_{a} J_{b a}^{'} \sin k_{a}}{(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) - J_{a b}^{'} J_{b a}^{'}},

(C28)

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s_{a b} = \frac{2 i ξ_{b} J_{a b}^{'} \sin k_{b}}{(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) - J_{a b}^{'} J_{b a}^{'}},

(C29)

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s_{b b} = \frac{J_{a b}^{'} J_{b a}^{'} - (ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{i k_{b}} + {\bar{Δ}}_{b})}{(ξ_{a} e^{- i k_{a}} + {\bar{Δ}}_{a}) (ξ_{b} e^{- i k_{b}} + {\bar{Δ}}_{b}) - J_{a b}^{'} J_{b a}^{'}} .

(C30)

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I_{l^{'} l} \equiv {| s_{l^{'} l} |}^{2} \frac{ξ_{l^{'}} \sin k_{l^{'}}}{ξ_{l} \sin k_{l}},

(C31)

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I_{b a} = {| s_{b a} |}^{2} = {| \frac{2 J_{b a} g^{2} ξ \sin k}{{[ξ e^{- i k} (- 2 ξ \cos k + i Γ) + g^{2}]}^{2}} |}^{2} .

(D1)

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| \sin k | = \frac{(| J_{b a} | - Γ) g^{2} \pm \sqrt{Θ}}{4 (g^{2} - ξ^{2}) ξ},

(D2)

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Θ = {(| J_{b a} | - Γ)}^{2} g^{4} - 4 (g^{2} - ξ^{2}) [4 (ξ^{2} - g^{2}) ξ^{2} + Γ^{2} ξ^{2} + g^{4}] .

(D3)

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| J_{b a} | = \frac{{(g^{2} + Γ ξ)}^{2}}{2 g^{2} ξ} .

(D4)

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| J_{b a} | = 2 Γ .

(D5)

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I_{b a} = {| s_{b a} |}^{2} = {| \frac{2 J_{b a} g^{2} ξ \sin k_{half}}{{[ξ e^{- i k} (- 2 ξ \cos k_{half} + i Γ) + g^{2}]}^{2}} |}^{2} = \frac{1}{2} .

(E1)

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2 (Γ - ξ) ξ {| \sin k_{half} |}^{2} + (1 - 2 \sqrt{2}) Γ^{2} | \sin k_{half} | + 2 (ξ - Γ) ξ + Γ^{2} = 0 .

(E2)

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| \sin k_{half} | = \frac{- (1 - 2 \sqrt{2}) \pm \sqrt{{(1 - 2 \sqrt{2})}^{2} - ζ (2 - ζ)}}{ζ} .

(E3)

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\frac{d}{d η} | \sin k_{half} | = \frac{d | \sin k_{half} |}{d ζ} \frac{d ζ}{d η} = 0,

(E4)

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η = \frac{1}{2} \Rightarrow ξ = \frac{Γ}{2} .

(E5)

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| \sin k_{half} | = 2 \sqrt{2} - 1 - 2 \sqrt{2 - \sqrt{2}},

(E6)

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Δ k_{\max} \equiv π - 2 \arcsin (2 \sqrt{2} - 1 - 2 \sqrt{2 - \sqrt{2}}) \approx 0.81 π .

(E7)

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Xunwei Xu, Yanjun Zhao, Hui Wang, Aixi Chen, Yu-Xi Liu. Nonreciprocal transition between two nondegenerate energy levels[J]. Photonics Research, 2021, 9(5): 879

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