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Photonics Research
Vol. 8, Issue 10, B15 (2020)

Disorder-protected quantum state transmission through helical coupled-resonator waveguides

JungYun Han^1、2, Andrey A. Sukhorukov³, and Daniel Leykam^1、2、*

Author Affiliations

¹Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, South Korea

²Basic Science Program, University of Science and Technology, Daejeon 34113, South Korea

³ARC Centre of Excellence for Transformative Meta-Optical Systems (TMOS), Nonlinear Physics Centre, Research School of Physics, The Australian National University, Canberra, ACT 2601, Australia

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DOI: 10.1364/PRJ.399919 Cite this Article Set citation alerts

JungYun Han, Andrey A. Sukhorukov, Daniel Leykam. Disorder-protected quantum state transmission through helical coupled-resonator waveguides[J]. Photonics Research, 2020, 8(10): B15 Copy Citation Text

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Abstract

We predict the preservation of temporal indistinguishability of photons propagating through helical coupled-resonator optical waveguides (H-CROWs). H-CROWs exhibit a pseudospin-momentum locked dispersion, which we show suppresses on-site disorder-induced backscattering and group velocity fluctuations. We simulate numerically the propagation of two-photon wave packets, demonstrating that they exhibit almost perfect Hong–Ou–Mandel dip visibility and then can preserve their quantum coherence even in the presence of moderate disorder, in contrast with regular CROWs, which are highly sensitive to disorder. As indistinguishability is the most fundamental resource of quantum information processing, H-CROWs may find applications for the implementation of robust optical links and delay lines in the emerging quantum photonic communication and computational platforms.

{\hat{H}}_{0, ccw} = \sum_{n} ({\hat{H}}_{a, ccw} + {\hat{H}}_{b, ccw} + {\hat{H}}_{a b, ccw} + {\hat{H}}_{a b, ccw}^{†}), {\hat{H}}_{a, ccw} = J {\hat{a}}_{n}^{†} (- i {\hat{a}}_{n - 1} + i {\hat{a}}_{n + 1}), {\hat{H}}_{b, ccw} = J {\hat{b}}_{n}^{†} (i {\hat{b}}_{n - 1} - i {\hat{b}}_{n + 1}), {\hat{H}}_{a b, ccw} = 2 J {\hat{a}}_{n}^{†} [{\hat{b}}_{n} + \frac{1}{2} ({\hat{b}}_{n - 1} + {\hat{b}}_{n + 1})],

(1)

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{\hat{H}}_{0, ccw} = \sum_{k} ψ_{k, ccw}^{†} [d (k) \cdot \hat{σ}] ψ_{k, ccw},

(2)

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H_{tot} (π + Δ k) \approx 2 J diag (Δ k, - Δ k, - Δ k, Δ k),

(3)

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{\hat{V}}_{ϵ} = \sum_{n} (V_{n, ϵ}^{(a)} {\hat{a}}_{n}^{†} {\hat{a}}_{n} + V_{n, ϵ}^{(b)} {\hat{b}}_{n}^{†} {\hat{b}}_{n}),

(4)

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i ω {\hat{ψ}}_{n, j} (ω) = i [{\hat{H}}_{0, j}, {\hat{ψ}}_{n, j} (ω)] - κ_{ex} {\hat{ψ}}_{n, j} (ω) (δ_{n, 1} + δ_{n, L}) - κ_{in} {\hat{ψ}}_{n, j} (ω) + \sqrt{2 κ_{ex}} {\hat{p}}_{in, j} (ω) δ_{n, 1},

(5)

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R_{1} (ω) = {| \frac{p_{in} (ω) - \sqrt{2 κ_{ex}} a_{1} (ω)}{p_{in} (ω)} |}^{2}, T_{1} (ω) = {| \frac{\sqrt{2 κ_{ex}} a_{L} (ω)}{p_{in} (ω)} |}^{2}, R_{2} (ω) = {| \frac{p_{in} (ω) - \sqrt{2 κ_{ex}} b_{1} (ω)}{p_{in} (ω)} |}^{2}, T_{2} (ω) = {| \frac{\sqrt{2 κ_{ex}} b_{L} (ω)}{p_{in} (ω)} |}^{2},

(6)

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{\hat{p}}_{out, 1} = - \sqrt{2 κ_{ex}} {\hat{a}}_{L}, {\hat{p}}_{out, 2} = - \sqrt{2 κ_{ex}} {\hat{b}}_{L} .

(7)

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τ_{j} = \frac{1}{i} \frac{d}{d ω} [\frac{p_{out, j} (ω)}{| p_{out, j} (ω) |}] (j = 1, 2),

(8)

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| out ⟩ = {\hat{ϕ}}_{a} {\hat{ϕ}}_{b} {| 00 ⟩}_{a b} = \int d ω \int d ω^{'} p_{out, 1} (ω) p_{out, 2} (ω^{'}) {\hat{a}}_{out}^{†} (ω) {\hat{b}}_{out}^{†} (ω^{'}) {| 00 ⟩}_{a b},

(9)

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(\begin{matrix} {\hat{c}}^{†} \\ {\hat{d}}^{†} \end{matrix}) = (\begin{matrix} t & i r \\ i r & t \end{matrix}) (\begin{matrix} {\hat{a}}_{out}^{†} \\ {\hat{b}}_{out}^{†} \end{matrix}),

(10)

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P_{coin} (τ_{c}) = \frac{1}{2} [1 - \frac{{| \int d ω p_{1}^{*} (ω) p_{2} (ω) e^{i ω τ_{c}} |}^{2}}{\int d ω {| p_{1} (ω) |}^{2} \int d ω^{'} {| p_{2} (ω^{'}) |}^{2}}] .

(11)

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{| 11 ⟩}_{a b} \to \frac{1}{\sqrt{2}} ({| 20 ⟩}_{a b} + {| 02 ⟩}_{a b}) .

(12)

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| out ⟩ = \frac{1}{\sqrt{2} \sqrt{N!}} ({\hat{ϕ}}_{a}^{N} + {\hat{ϕ}}_{b}^{N}) | 00 ⟩ = \frac{1}{\sqrt{2 N!}} [\int \prod_{i = 1}^{N} d ω_{i} p_{1} (ω_{i}) {\hat{a}}_{out}^{†} (ω_{i}) + \int \prod_{i = 1}^{N} d ω_{i} p_{2} (ω_{i}) {\hat{b}}_{out}^{†} (ω_{i})] | 00 ⟩,

(13)

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{| out (τ_{c}) ⟩}_{bef} = \frac{1}{2} \int d ω_{1} d ω_{2} [p_{1} (ω_{1}) p_{1} (ω_{2}) {\hat{a}}_{out}^{†} (ω_{1}) {\hat{a}}_{out}^{†} (ω_{2}) + p_{2} (ω_{1}) p_{2} (ω_{2}) {\hat{b}}_{out}^{†} (ω_{1}) {\hat{b}}_{out}^{†} (ω_{2}) e^{i (ω_{1} + ω_{2}) τ_{c}}] | 00 ⟩ .

(14)

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\frac{1}{\sqrt{2}} (| 20 ⟩ + e^{i θ} | 02 ⟩) \to \frac{1}{2 \sqrt{2}} [(1 - e^{i θ}) | 20 ⟩ + \sqrt{2} i (1 + e^{i θ}) | 11 ⟩ - (1 - e^{i θ}) | 02 ⟩] .

(15)

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P_{coin} = \frac{1 + \cos θ}{2} .

(16)

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| 20 ⟩ ⟨ 20 | = \frac{1}{2} \int d ω_{1} d ω_{2} {\hat{a}}_{out}^{†} (ω_{1}) {\hat{a}}_{out}^{†} (ω_{2}) | 00 ⟩ ⟨ 00 | {\hat{a}}_{out} (ω_{2}) {\hat{a}}_{out} (ω_{1}), | 02 ⟩ ⟨ 02 | = \frac{1}{2} \int d ω_{1} d ω_{2} {\hat{b}}_{out}^{†} (ω_{1}) {\hat{b}}_{out}^{†} (ω_{2}) | 00 ⟩ ⟨ 00 | {\hat{b}}_{out} (ω_{2}) {\hat{b}}_{out} (ω_{1}), | 11 ⟩ ⟨ 11 | = \int d ω_{1} d ω_{2} {\hat{a}}_{out}^{†} (ω_{1}) {\hat{b}}_{out}^{†} (ω_{2}) | 00 ⟩ ⟨ 00 | {\hat{b}}_{out} (ω_{2}) {\hat{a}}_{out} (ω_{1}),

(17)

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P_{coin} (τ_{c}) = \frac{1}{2} {1 + \frac{{[\int d ω p_{1}^{*} (ω) p_{2} (ω) e^{i ω τ_{c}}]}^{2} + c.c.}{[\int d ω {| p_{1} (ω) |}^{2}]^{2} + [\int d ω {| p_{2} (ω) |}^{2}]^{2}}} .

(18)

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Tr [ρ^{2} (τ_{c})] = 1 + 2 {\frac{({| \int d ω p_{1}^{*} p_{2} e^{i ω τ_{c}} |}^{2})^{2} - (\int d ω {| p_{1} |}^{2} \int d ω^{'} {| p_{2} |}^{2})^{2}}{[(\int d ω {| p_{1} |}^{2})^{2} + (\int d ω {| p_{2} |}^{2})^{2}]^{2}}},

(19)

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S_{a} : = - {Tr}_{a} [ρ_{a} \ln (ρ_{a})],

(20)

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S_{a} = - x \ln x - (1 - x) \ln (1 - x),

(21)

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x = \frac{[\int d ω {| p_{out, 1} (ω) |}^{2}]^{2}}{[\int d ω {| p_{out, 1} (ω) |}^{2}]^{2} + [\int d ω {| p_{out, 2} (ω) |}^{2}]^{2}} .

(22)

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{\hat{H}}_{tot} = {\hat{H}}_{sys} + {\hat{H}}_{env} + {\hat{H}}_{int},

(A1)

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{\hat{H}}_{env} = \int_{- \infty}^{\infty} d ω ω {\hat{p}}^{†} (ω, t) \hat{p} (ω, t), {\hat{H}}_{int} = i \int_{- \infty}^{\infty} d ω \sqrt{2 κ_{ex}} [{\hat{a}}^{†} (t) \hat{p} (ω, t) - h.c.],

(A2)

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\frac{d}{d t} \hat{p} (ω, t) = - i ω \hat{p} (ω, t) - \sqrt{2 κ_{ex}} \hat{a} (t) .

(A3)

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\hat{p} (ω, t) = e^{- i ω (t - t_{0})} \hat{p} (ω, t_{0}) - \sqrt{2 κ_{ex}} \int_{t_{0}}^{t} d t^{'} e^{- i ω (t - t^{'})} \hat{a} (t^{'}),

(A4)

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{\hat{p}}_{in} (t) : = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} d ω e^{- i ω (t - t_{0})} \hat{p} (ω, t_{0}) (t > t_{0}),

(A5)

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\int d ω \hat{p} (ω, t) = {\hat{p}}_{in} (t) - \sqrt{2 κ_{ex}} \int_{- \infty}^{\infty} d ω \int_{t_{0}}^{t} d t^{'} e^{- i ω (t - t^{'})} \hat{a} (t^{'}) = {\hat{p}}_{in} (t) - \frac{\sqrt{2 κ_{ex}}}{2} \hat{a} (t),

(A6)

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\hat{p} (ω, t) = e^{- i ω (t - t_{1})} \hat{p} (ω, t_{1}) - \sqrt{2 κ_{ex}} \int_{t_{1}}^{t} d t^{'} e^{- i ω (t - t^{'})} \hat{a} (t^{'}) .

(A7)

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{\hat{p}}_{out} (t) : = \frac{1}{\sqrt{2 π}} \int d ω e^{- i ω (t - t_{1})} \hat{p} (ω, t_{1}) (t < t_{1}),

(A8)

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\int d ω \hat{p} (ω, t) = {\hat{p}}_{out} + \sqrt{2 κ_{ex}} \int_{- \infty}^{\infty} d ω \int_{t}^{t_{1}} d t^{'} e^{- i ω (t - t^{'})} \hat{a} (t^{'}) = {\hat{p}}_{out} (t) + \frac{\sqrt{2 κ_{ex}}}{2} \hat{a} (t) .

(A9)

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{\hat{p}}_{in} (t) - {\hat{p}}_{out} (t) = \sqrt{2 κ_{ex}} \hat{a} (t),

(A10)

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{| 11 ⟩}_{a b} \to i r t \sqrt{2} {| 20 ⟩}_{a b} + i r t \sqrt{2} {| 02 ⟩}_{a b} + (t^{2} - r^{2}) {| 11 ⟩}_{a b},

(B1)

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{| 1001 ⟩}_{a_{1} b_{1} a_{2} b_{2}} \to i r t {| 1010 ⟩}_{a_{1} b_{1} a_{2} b_{2}} + i r t {| 0101 ⟩}_{a_{1} b_{1} a_{2} b_{2}} + t^{2} {| 1001 ⟩}_{a_{1} b_{1} a_{2} b_{2}} - r^{2} {| 0110 ⟩}_{a_{1} b_{1} a_{2} b_{2}},

(B2)

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\frac{1}{\sqrt{2}} ({| 20 ⟩}_{a b} + {| 02 ⟩}_{a b}) \to \frac{1}{\sqrt{2}} [(t^{2} - r^{2}) {| 20 ⟩}_{a b} - (t^{2} - r^{2}) {| 02 ⟩}_{a b} + 2 \sqrt{2} i r t {| 11 ⟩}_{a b}] .

(B3)

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\frac{1}{\sqrt{2}} ({| 1010 ⟩}_{a_{1} b_{1} a_{2} b_{2}} + {| 0101 ⟩}_{a_{1} b_{1} a_{2} b_{2}}) \to \frac{1}{\sqrt{2}} [(t^{2} - r^{2}) {| 1010 ⟩}_{a_{1} b_{1} a_{2} b_{2}} - (t^{2} - r^{2}) {| 0101 ⟩}_{a_{1} b_{1} a_{2} b_{2}} + 2 i r t {| 0110 ⟩}_{a_{1} b_{1} a_{2} b_{2}} + 2 i r t {| 1001 ⟩}_{a_{1} b_{1} a_{2} b_{2}}] .

(B4)

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ρ_{bef} (τ_{c}) = \frac{1}{A + B} (\begin{matrix} A & C (τ_{c}) \\ C^{*} (τ_{c}) & B \end{matrix}),

(C1)

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A = ⟨ 20 | ρ_{bef} (τ_{c}) | 20 ⟩ = : ρ_{bef, 2020} = {[\int d ω {| p_{out, 1} (ω) |}^{2}]}^{2}, B = ρ_{bef, 0202} = {[\int d ω {| p_{out, 2} (ω) |}^{2}]}^{2}, C (τ_{c}) = ρ_{bef, 2002} = {[\int d ω p_{out, 1}^{*} (ω) p_{out, 2} (ω) e^{i ω τ_{c}}]}^{2} .

(C2)

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ρ_{af, 2020} = ρ_{af, 0202} = - ρ_{af, 2002} = - ρ_{af, 0220} = \frac{1}{4 (A + B)} [A + B - C (τ_{c}) - C^{*} (τ_{c})], ρ_{af, 2011} = ρ_{af, 1120}^{*} = - ρ_{af, 0211} = - ρ_{af, 1102}^{*} = \frac{i}{2 \sqrt{2} (A + B)} [A - B + C (τ_{c}) - C^{*} (τ_{c})], ρ_{af, 1111} = \frac{1}{2 (A + B)} [A + B + C (τ_{c}) + C^{*} (τ_{c})],

(C3)

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U \otimes U : = U_{2} = \frac{1}{2} (\begin{matrix} 1 & i & i & - 1 \\ i & 1 & - 1 & i \\ i & - 1 & 1 & i \\ - 1 & i & i & 1 \end{matrix}) .

(C4)

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U_{2} = \frac{1}{2} (\begin{matrix} 1 & \sqrt{2} i & - 1 \\ \sqrt{2} i & 0 & \sqrt{2} i \\ - 1 & \sqrt{2} i & 1 \end{matrix}) .

(C5)

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Tr [ρ^{2} (τ_{c})] = \frac{A^{2} + B^{2} + 2 {| C (τ_{c}) |}^{2}}{{(A + B)}^{2}} = 1 + 2 \frac{{| C (τ_{c}) |}^{2} - A B}{{(A + B)}^{2}} .

(C6)

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S_{a} = - {Tr}_{a} [ρ_{a} \ln (ρ_{a})] = - \frac{A}{A + B} \ln \frac{A}{A + B} - \frac{B}{A + B} \ln \frac{B}{A + B} .

(C7)

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JungYun Han, Andrey A. Sukhorukov, Daniel Leykam. Disorder-protected quantum state transmission through helical coupled-resonator waveguides[J]. Photonics Research, 2020, 8(10): B15

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