Generation and robustness of bipartite non-classical correlations in two nonlinear microcavities coupled by an optical fiber

A.-B. A. Mohamed; H. Eleuch

doi:10.1364/JOSAB.35.000047

Journal of the Optical Society of America B
Vol. 35,
Issue 1,
pp. 47-53
(2018)
•https://doi.org/10.1364/JOSAB.35.000047

Generation and robustness of bipartite non-classical correlations in two nonlinear microcavities coupled by an optical fiber

A.-B. A. Mohamed and H. Eleuch

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A.-B. A. Mohamed and H. Eleuch, "Generation and robustness of bipartite non-classical correlations in two nonlinear microcavities coupled by an optical fiber," J. Opt. Soc. Am. B 35, 47-53 (2018)

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Abstract

We explore the bipartite non-classical correlations of two quantum wells coupled to two spatially separated microcavities. The microcavities are filled by linear optical media and linked by a single-mode optical fiber. We prove that it is possible to generate a correlation from the initially uncorrelated state. With particular coupling constants of the cavity-exciton and the fiber-cavity coupling, the maximum correlations can be generated periodically, and sudden changes occur only in the local quantum uncertainty. The optical susceptibility and coupling constants control the regularity, amplitudes, and frequencies of the correlation functions. When the system initially starts with a maximally correlated state, we show that the correlation robustness of the Wigner-Yanase skew information and Bell’s inequality depends not only on the coupling strengths and optical susceptibility but also on the dissipation rates. Deterioration of the correlations is substantially associated with the dissipation.

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Fig. 1. Schematic representation of two spatially separated semiconductor QWs (

A

B

), where each QW is in a driven microcavity by a reservoir. Microcavities are linked by an optical fiber.

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Fig. 2. For the two excitons,

ρ^{E_{A} E_{B}} (t)

L (t)

(solid curves),

U (t)

(dash curves),

B_{\max} (t)

(dash-dot curves) and negativity (dashed curves) for

| 1_{e} ⟩ \otimes | 1_{c} ⟩ \otimes | 1_{f} ⟩

with

χ = 0.0

in (a) and

χ = 0.9

in (b). At fixed values:

(λ_{A}, λ_{B}, ν, ϵ) = (1.42, 1.42, 1.0, 10^{- 3})

with

κ_{i} = γ_{i} = 0

. (c) and (d) are the same as (a) and (b), but for

λ_{B} = 0.42

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Fig. 3. Same as Figs. 2(a) and 2(b) but for

ν = 0.5

in (a) and (b), and for

κ_{i} = γ_{i} = 0.5

in (c) and (d).

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Fig. 4. Same as Fig. 1 but for the correlated initial state:

{| ψ (0) ⟩}_{cor} = [| Ψ_{2} ⟩ \otimes | 2_{e} ⟩ + | Ψ_{3} ⟩ \otimes | 3_{e} ⟩]

with

x_{i} = \frac{1}{\sqrt{8}}

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Fig. 5. Same as Figs. 3(a) and 3(b) but for

(λ_{A}, λ_{B}, ν) = (0.2, 0.2, 1.45)

in (a) and (b), and for

κ_{i} = γ_{i} = 0.2

in (c) and (d).

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Fig. 6. Same as Fig. 1(a), but for the two cavities,

ρ^{C_{A} C_{B}} (t)

in (a) and for

(λ_{A}, λ_{B}, ν, ϵ) = (0.2, 0.2, 1.8, 0.001)

in (b). (c) and (d) are the same as (a) and (b), but for

κ_{i} = γ_{i} = 0.4

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Fig. 7. Same as Fig. 6 but for the correlated initial state:

{| ψ (0) ⟩}_{cor} = [| Ψ_{2} ⟩ | 2_{e} ⟩ + | Ψ_{3} ⟩ | 3_{e} ⟩]

and for

(λ_{A}, λ_{B}, ν) = (1.5, 1.5, 0.2)

for

χ = 0.0

in (a) and

χ = 0.9

in (b).

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Equations (18)

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H_{int} = \sum_{k = A, B} χ^{k} ({\hat{a}}_{k}^{†}) {\hat{a}}_{k} + λ_{k}^{'} ({\hat{a}}_{k}^{+} {\hat{b}}_{k} + {\hat{b}}_{k}^{+} {\hat{a}}_{k}) + ϵ_{k}^{'} ({\hat{a}}_{k}^{+} + {\hat{a}}_{k}) + ν [\hat{o} ({\hat{a}}_{A}^{†} + {\hat{a}}_{B}^{†})] + ν [{\hat{o}}^{†} ({\hat{a}}_{A} + {\hat{a}}_{B})],

\frac{\partial ρ}{\partial t} = i ℏ [H_{int}, ρ] + \sum_{i = A, B} κ_{i} (2 {\hat{a}}_{i} ρ {\hat{a}}_{i}^{†} - {\hat{a}}_{i}^{†} {\hat{a}}_{i} ρ - ρ {\hat{a}}_{i}^{†} {\hat{a}}_{i}) + \frac{γ_{i}}{2} (2 {\hat{b}}_{i} ρ {\hat{b}}_{i}^{†} - {\hat{b}}_{i}^{†} {\hat{b}}_{i} ρ - ρ {\hat{b}}_{i}^{†} {\hat{b}}_{i}) .

i \frac{d}{d t} ρ = {\hat{H}}_{eff} ρ - {(ρ {\hat{H}}_{eff})}^{†},

H_{eff} = H_{int} - i ℏ \sum_{i = A, B} κ_{i} {\hat{a}}_{i}^{†} {\hat{a}}_{i} + \frac{γ_{i}}{2} {\hat{b}}_{i}^{†} {\hat{b}}_{i} .

{(H_{eff})}^{†} = H_{int} + i ℏ \sum_{i = A, B} κ_{i} {\hat{a}}_{i}^{†} {\hat{a}}_{i} + \frac{γ_{i}}{2} {\hat{b}}_{i}^{†} {\hat{b}}_{i} .

i ℏ \frac{d}{d t} | ψ (t) ⟩ = {\hat{H}}_{eff} | ψ (t) ⟩,

| ψ (t) ⟩ = \sum_{l = 1}^{4} | Ψ_{l} ⟩ \otimes | l_{e} ⟩ .

| Ψ_{1} ⟩ = x_{1} | 1_{c} ⟩ \otimes | 0_{f} ⟩ + x_{5} | 1_{c} ⟩ \otimes | 1_{f} ⟩ + x_{9} | 2_{c} ⟩ \otimes | 0_{f} ⟩ + x_{11} | 2_{c} ⟩ \otimes | 1_{f} ⟩ + x_{13} | 3_{c} ⟩ \otimes 0_{f} ⟩ + x_{15} | 3_{c} ⟩ \otimes | 1_{f} ⟩ + x_{17} | 4_{c} ⟩ \otimes | 0_{f} ⟩ + x_{18} | 4_{c} ⟩ \otimes | 1_{f} ⟩, | Ψ_{2} ⟩ = x_{2} | 1_{c} ⟩ \otimes | 0_{f} ⟩ + x_{6} | 1_{c} ⟩ \otimes | 1_{f} ⟩ + x_{14} | 3_{c} ⟩ \otimes | 0_{f} ⟩ + x_{16} | 3_{c} ⟩ \otimes | 1_{f} ⟩, | Ψ_{3} ⟩ = x_{3} | 1_{c} ⟩ \otimes | 0_{f} ⟩ + x_{7} | 1_{c} ⟩ \otimes | 1_{f} ⟩ + x_{10} | 2_{c} ⟩ \otimes | 0_{f} ⟩ + x_{12} | 2_{c} ⟩ \otimes | 1_{f} ⟩, | Ψ_{4} ⟩ = x_{4} | 1_{c} ⟩ \otimes | 0_{f} ⟩ + x_{8} | 1_{c} ⟩ \otimes | 1_{f} ⟩ .

{\dot{x}}_{1} = ϵ_{A} x_{13} + ϵ_{B} x_{9}, {\dot{x}}_{2} = - i λ_{B} x_{9} - {\tilde{γ}}_{B} x_{2}, {\dot{x}}_{3} = - i λ_{A} x_{13} - {\tilde{γ}}_{A} x_{3} + ϵ_{A} x_{14} + ϵ_{B} x_{10}, {\dot{x}}_{4} = - i λ_{A} x_{14} - i λ_{B} x_{10} - ({\tilde{γ}}_{A} + {\tilde{γ}}_{B}) x_{4}, {\dot{x}}_{5} = - i ν x_{9} - i ν A_{13} + ϵ_{A} A_{15} + ϵ_{B} x_{11}, {\dot{x}}_{6} = - i λ_{B} x_{11} - i ν x_{14} - {\tilde{γ}}_{B} x_{6}, {\dot{x}}_{7} = - i λ_{A} x_{15} - i ν x_{10} - {\tilde{γ}}_{A} x_{7} + ϵ_{A} x_{16} + ϵ_{B} A_{12} {\dot{x}}_{8} = - i λ_{A} x_{16} - i λ_{B} x_{12} - ({\tilde{γ}}_{A} + {\tilde{γ}}_{B}) x_{8}, {\dot{x}}_{9} = - i λ_{B} x_{2} - i ν x_{5} - κ_{B} x_{9} - i χ^{B} x_{9} + ϵ_{A} x_{17} + ϵ_{B} x_{1}, {\dot{x}}_{10} = - i λ_{A} x_{17} - i λ_{B} x_{4} - i ν x_{7} - (κ_{B} + {\tilde{γ}}_{A}) x_{10} - i χ^{B} x_{10} + ϵ_{B} x_{3}, {\dot{x}}_{11} = - i λ_{B} x_{6} - i ν x_{17} - κ_{B} x_{11} - i χ^{B} x_{11} + ϵ_{A} x_{18} + ϵ_{B} x_{5}, {\dot{x}}_{12} = - i λ_{A} x_{18} - i λ_{B} x_{8} - (κ_{B} + {\tilde{γ}}_{A}) x_{12} - i χ^{B} x_{12} + ϵ_{B} x_{7}, {\dot{x}}_{13} = - i λ_{A} x_{3} - i ν x_{5} - κ_{A} x_{13} - i χ^{A} x_{13} + ϵ_{A} x_{1} + ϵ_{B} x_{17}, {\dot{x}}_{14} = - i λ_{A} x_{4} - i λ_{B} x_{17} - i ν x_{6} - (κ_{A} + {\tilde{γ}}_{B}) x_{14} - i χ^{A} x_{14} + ϵ_{A} x_{2}, {\dot{x}}_{15} = - i λ_{A} x_{7} - i ν x_{17} - κ_{A} x_{15} - i χ^{A} x_{15} + ϵ_{A} x_{5} + ϵ_{B} x_{18}, {\dot{x}}_{16} = - i λ_{A} x_{8} - i λ_{B} x_{18} - (κ_{A} + {\tilde{γ}}_{B}) x_{16} - i χ^{A} x_{16} + ϵ_{A} x_{6}, {\dot{x}}_{17} = - i λ_{A} x_{10} - i λ_{B} x_{14} - i ν x_{15} - (κ_{A} + κ_{B}) x_{17} - i (χ^{A} + χ^{B}) x_{17} + ϵ_{A} x_{9} + ϵ_{B} x_{13}, {\dot{x}}_{18} = - i λ_{A} z_{12} - i λ_{B} x_{16} - i ν x_{11} - (κ_{A} + κ_{B}) x_{18} - i (χ^{A} + χ^{B}) x_{18} + ϵ_{A} x_{11} + ϵ_{B} x_{15},

I (ρ^{A B}, K^{Λ}) = - \frac{1}{2} Tr {[\sqrt{ρ^{A B}}, K^{Λ}]}^{2},

L_{A} (ρ^{A B}) = \min_{K^{Λ}} {I (ρ, K^{Λ})},

L (t) = 1 - λ_{\max} (W_{A B}),

w_{i j} = Tr {\sqrt{ρ^{A B}} (σ_{i} \otimes I) \sqrt{ρ^{A B}} (σ_{j} \otimes I)},

U_{c} (ρ) = \max_{K^{C}} I (ρ, K^{C}),

U (t) = {\begin{matrix} 1 - λ_{\min} (W_{A B}), & \vec{r} = 0; \\ 1 - \frac{1}{‖ \vec{r} ‖} \vec{r} W_{A B} {\vec{r}}^{T}, & \vec{r} \neq 0, \end{matrix}

B_{\max} (t) = 2 \sqrt{λ + \tilde{λ}},

{\vec{d}}_{1} = [\begin{matrix} x_{1} & x_{5} & x_{9} & x_{11} & x_{13} & x_{15} & x_{17} & A_{18} \end{matrix}], {\vec{d}}_{2} = [\begin{matrix} x_{2} & x_{6} & 0 & 0 & x_{14} & x_{16} & 0 & 0 \end{matrix}], {\vec{d}}_{3} = [\begin{matrix} x_{3} & x_{7} & x_{10} & x_{12} & 0 & 0 & 0 & 0 \end{matrix}], {\vec{d}}_{4} = [\begin{matrix} x_{4} & x_{8} & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .

{\vec{w}}_{1} = [\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} & x_{8} \end{matrix}], {\vec{w}}_{2} = [\begin{matrix} x_{9} & 0 & x_{10} & 0 & x_{11} & 0 & x_{12} & 0 \end{matrix}], {\vec{w}}_{3} = [\begin{matrix} x_{13} & x_{14} & 0 & 0 & x_{15} & x_{15} & 0 & 0 \end{matrix}], {\vec{w}}_{4} = [\begin{matrix} x_{17} & 0 & 0 & 0 & x_{17} & 0 & 0 & 0 \end{matrix}] .

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Equations (18)

Journal of the Optical Society of America B