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Photonics Research
Vol. 7, Issue 6, 630 (2019)

Nonreciprocal unconventional photon blockade in a spinning optomechanical system

Baijun Li¹, Ran Huang¹, Xunwei Xu², Adam Miranowicz^3、4、5, and Hui Jing^1、6

Author Affiliations

¹Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, 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

³Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan

⁴Faculty of Physics, Adam Mickiewicz University, 61-614 Poznan, Poland

⁵e-mail: miran@amu.edu.pl

⁶e-mail: jinghui73@gmail.com

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DOI: 10.1364/PRJ.7.000630 Cite this Article Set citation alerts

Baijun Li, Ran Huang, Xunwei Xu, Adam Miranowicz, Hui Jing. Nonreciprocal unconventional photon blockade in a spinning optomechanical system[J]. Photonics Research, 2019, 7(6): 630 Copy Citation Text

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Abstract

We propose how to achieve quantum nonreciprocity via unconventional photon blockade (UPB) in a compound device consisting of an optical harmonic resonator and a spinning optomechanical resonator. We show that, even with very weak single-photon nonlinearity, nonreciprocal UPB can emerge in this system, i.e., strong photon antibunching can emerge only by driving the device from one side but not from the other side. This nonreciprocity results from the Fizeau drag, leading to different splitting of the resonance frequencies for the optical counter-circulating modes. Such quantum nonreciprocal devices can be particularly useful in achieving back-action-free quantum sensing or chiral photonic communications.

Δ_{F} = \pm \frac{n r Ω ω_{R}}{c} (1 - \frac{1}{n^{2}} - \frac{λ}{n} \frac{d n}{d λ}) = \pm η Ω,

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H = ℏ Δ_{L} a_{L}^{†} a_{L} + ℏ (Δ_{R} + Δ_{F}) a_{R}^{†} a_{R} + ℏ ω_{m} b^{†} b + ℏ J (a_{L}^{†} a_{R} + a_{R}^{†} a_{L}) + ℏ g a_{R}^{†} a_{R} (b^{†} + b) + i ℏ ϵ_{d} (a_{L}^{†} - a_{L}),

(2)

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\frac{d}{d t} q = ω_{m} p, \frac{d}{d t} p = - ω_{m} q - g_{b} a_{R}^{†} a_{R} - \frac{γ_{m}}{2} p + ξ, \frac{d}{d t} a_{L} = - (\frac{κ_{L}}{2} + i Δ_{L}) a_{L} - i J a_{R} + ϵ_{d} + \sqrt{κ_{L}} a_{L, in}, \frac{d}{d t} a_{R} = - (\frac{κ_{R}}{2} + i Δ_{R}^{'}) a_{R} - i J a_{L} - i g_{b} q a_{R} + \sqrt{κ_{L}} a_{R, in},

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⟨ ξ (t) ξ (t^{'}) ⟩ = \frac{1}{2 π} \int d ω e^{- i ω (t - t^{'})} Γ_{m} (ω),

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Γ_{m} (ω) = \frac{ω γ_{m}}{2 ω_{m}} [1 + \coth (\frac{ℏ ω}{2 k_{B} T})],

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⟨ a_{K, in}^{†} (t) a_{K, in} (t^{'}) ⟩ = 0, ⟨ a_{K, in} (t) a_{K, in}^{†} (t^{'}) ⟩ = δ (t - t^{'}),

(6)

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\frac{d}{d t} δ q = ω_{m} δ p, \frac{d}{d t} δ p = - ω_{m} δ q - g_{b} (β^{*} δ a_{R} + β δ a_{R}^{†}) - \frac{γ_{m}}{2} δ p + ξ, \frac{d}{d t} δ a_{L} = - (\frac{κ_{L}}{2} + i Δ_{L}) δ a_{L} - i J δ a_{R} + \sqrt{κ_{L}} a_{L, in}, \frac{d}{d t} δ a_{R} = - (\frac{κ_{R}}{2} + i Δ_{R}^{'}) δ a_{R} - i J δ a_{L} - i g_{b} q_{s} δ a_{R} - i g_{b} β δ q + \sqrt{κ_{R}} a_{R, in} .

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δ a_{L} (ω) = E (ω) a_{L, in} (ω) + F (ω) a_{L, in}^{†} (ω) + G (ω) a_{R, in} (ω) + H (ω) a_{R, in}^{†} (ω) + Q (ω) ξ (ω),

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E (ω) = \sqrt{κ_{L}} \frac{A_{1} (ω)}{A_{5} (ω)}, F (ω) = - \sqrt{κ_{L}} \frac{A_{2} (ω)}{A_{5} (ω)}, G (ω) = \sqrt{κ_{R}} \frac{A_{3} (ω)}{A_{5} (ω)}, H (ω) = - \sqrt{κ_{R}} \frac{A_{4} (ω)}{A_{5} (ω)}, Q (ω) = - i \frac{g_{b} χ (ω)}{ω_{m} A_{5} (ω)} [β A_{3} (ω) + β^{*} A_{4} (ω)],

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A_{1} (ω) = [{(\frac{κ_{R}}{2} + i ω)}^{2} + Δ_{R}^{'' 2}] V_{1}^{-} (ω) - g_{b}^{4} {| β |}^{4} {[\frac{χ (ω)}{ω_{m}}]}^{2} V_{1}^{-} (ω) + J^{2} V_{2}^{+}, A_{2} (ω) = - i J^{2} g_{b}^{2} β^{2} \frac{χ (ω)}{ω_{m}}, A_{3} (ω) = - i J V_{1}^{-} (ω) V_{2}^{-} - i J^{3}, A_{4} (ω) = - J g_{b}^{2} β^{2} \frac{χ (ω)}{ω_{m}} V_{1}^{-} (ω), A_{5} (ω) = V_{1}^{+} A_{1} (ω) + i J A_{3} (ω),

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Δ_{R}^{''} = Δ_{R}^{'} + g_{b} q_{s} - g_{b}^{2} {| β |}^{2} χ (ω), χ (ω) = ω_{m}^{2} / (ω_{m}^{2} - ω^{2} + \frac{i ω γ_{m}}{2}), V_{1}^{\pm} (ω) = \frac{κ_{L}}{2} \pm i (Δ_{L} - ω), V_{2}^{\pm} (ω) = \frac{κ_{R}}{2} \pm i (Δ_{R} - ω) .

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g_{L}^{(2)} (0) = \frac{{| α |}^{4} + 4 {| α |}^{2} R_{1} + 2 Re [α^{* 2} R_{2}] + R_{3}}{{({| α |}^{2} + R_{1})}^{2}},

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⟨ δ a_{L}^{\pm} (t) δ a_{L} (t) ⟩ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} X_{a_{L}^{\pm} a_{L}} d ω,

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δ a_{L}^{+} (t) = δ a_{L}^{†} (t), δ a_{L}^{-} (t) = δ a_{L} (t), and X_{a_{L}^{†} a_{L}} = {| Q (- ω) |}^{2} Γ_{m} (- ω) + {| F (- ω) |}^{2} + {| H (- ω) |}^{2}, X_{a_{L} a_{L}} = Q (ω) Q (- ω) Γ_{m} (- ω) + E (ω) F (- ω) + G (ω) H (- ω) .

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\dot{ρ} = \frac{1}{i ℏ} [H, ρ] + \frac{κ_{L}}{2} L [a_{L}] (ρ) + \frac{κ_{R}}{2} L [a_{R}] (ρ) + \frac{γ_{m}}{2} ({\bar{n}}_{m} + 1) L [b] (ρ) + \frac{γ_{m}}{2} {\bar{n}}_{m} L [b^{†}] (ρ),

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| φ ⟩ = C_{00} | 0, 0 ⟩ + C_{10} | 1, 0 ⟩ + C_{01} | 0, 1 ⟩ + C_{20} | 2, 0 ⟩ + C_{11} | 1, 1 ⟩ + C_{02} | 0, 2 ⟩,

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Δ_{opt} \approx \frac{- a_{3} + sgn (E) \sqrt{λ_{1}} - \sqrt{λ_{2}}}{4 a_{4}}, g_{opt} = \sqrt{- \frac{ω_{m} [Δ_{opt} (4 Δ_{opt}^{2} + 5 κ^{2}) + Δ_{F} λ_{3}]}{2 (2 J^{2} - κ^{2}) + 2 Δ_{F} λ_{4}}},

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Baijun Li, Ran Huang, Xunwei Xu, Adam Miranowicz, Hui Jing. Nonreciprocal unconventional photon blockade in a spinning optomechanical system[J]. Photonics Research, 2019, 7(6): 630

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