Researching | On-chip generation of hybrid polarization-frequency entangled biphoton states

Journals >Photonics Research >Volume 11 >Issue 2 >Page 270 > Article

Photonics Research
Vol. 11, Issue 2, 270 (2023)

On-chip generation of hybrid polarization-frequency entangled biphoton states | Spotlight on Optics

S. Francesconi¹, A. Raymond¹, R. Duhamel¹, P. Filloux¹, A. Lemaître², P. Milman¹, M. I. Amanti¹, F. Baboux^1、*, and S. Ducci¹

Author Affiliations

¹Université Paris Cité, CNRS, Laboratoire Matériaux et Phénomènes Quantiques, 75013 Paris, France

²Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

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DOI: 10.1364/PRJ.459879 Cite this Article Set citation alerts

S. Francesconi, A. Raymond, R. Duhamel, P. Filloux, A. Lemaître, P. Milman, M. I. Amanti, F. Baboux, S. Ducci. On-chip generation of hybrid polarization-frequency entangled biphoton states[J]. Photonics Research, 2023, 11(2): 270 Copy Citation Text

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Abstract

We demonstrate a chip-integrated semiconductor source that combines polarization and frequency entanglement, allowing the generation of entangled biphoton states in a hybrid degree of freedom without post-manipulation. Our AlGaAs device is based on type-II spontaneous parametric downconversion in a counterpropagating phase-matching scheme in which the modal birefringence lifts the degeneracy between the two possible nonlinear interactions. This allows the direct generation of polarization–frequency entangled photons at room temperature and telecom wavelength, and in two distinct spatial modes, offering enhanced flexibility for quantum information protocols. The state entanglement is quantified by a combined measurement of the joint spectrum and Hong–Ou–Mandel interference (raw visibility

70.1 % \pm 1.1 %

) of the biphotons, allowing to reconstruct a restricted density matrix in the hybrid polarization–frequency space.

ω_{p} \sin (θ) = ω_{s} n_{H} (ω_{s}) - ω_{i} n_{V} (ω_{i}) [Inter . H V],

(1)

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ω_{p} \sin (θ) = ω_{i} n_{V} (ω_{s}) - ω_{i} n_{H} (ω_{i}) [Inter . V H],

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| Ψ ⟩ = \iint d ω d ω^{'} [ϕ_{H V} (ω, ω^{'}) {\hat{a}}_{s, H}^{†} (ω) {\hat{a}}_{i, V}^{†} (ω^{'}) + ϕ_{V H} (ω, ω^{'}) {\hat{a}}_{s, V}^{†} (ω) {\hat{a}}_{i, H}^{†} (ω^{'})] | 0,0 ⟩,

(3)

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{| Ψ ⟩}_{HPF} = \frac{1}{\sqrt{2}} ({| H ω_{1} ⟩}_{s} {| V ω_{2} ⟩}_{i} + {| V ω_{2} ⟩}_{s} {| H ω_{1} ⟩}_{i}) .

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P_{c} (τ) = \frac{1}{2} - Re [\iint d ω d ω^{'} ϕ_{H V} (ω, ω^{'}) ϕ_{V H}^{*} (ω^{'}, ω) e^{- i (ω - ω^{'}) τ}],

(5)

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ϕ_{H V}^{pm} (ω_{-}) = \sqrt{π} w_{z} e^{- {(ω_{-} - μ)}^{2} / 2 Δ ω_{-}^{2}}, ϕ_{V H}^{pm} (ω_{-}) = \sqrt{π} w_{z} e^{- {(ω_{-} + μ)}^{2} / 2 Δ ω_{-}^{2}},

(6)

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P_{c} (τ) = \frac{1}{2} - \frac{1}{2} \exp (- \frac{τ^{2}}{2 Δ τ^{2}}) \cos (μ τ) .

(7)

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P_{c} (τ) = \frac{1}{2} [1 - V \exp (- \frac{τ^{2}}{2 Δ τ^{2}}) \cos (μ τ)] + a τ + b,

(8)

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ρ = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & p & \frac{V}{2} e^{i ϕ} & 0 \\ 0 & \frac{V}{2} e^{- i ϕ} & 1 - p & 0 \\ 0 & 0 & 0 & 0 \end{matrix}),

(9)

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f_{r} (ω) = \frac{\sqrt{R (1 - R)} \exp (i \frac{3 ω n L}{2 c})}{1 - R \exp (i \frac{2 ω n L}{c})}, f_{t} (ω) = \frac{\sqrt{1 - R} \exp (i \frac{ω n L}{2 c})}{1 - R \exp (i \frac{2 ω n L}{c})},

(A1)

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{\hat{a}}_{s, σ}^{†} (ω) \to f_{t} (ω) {\hat{a}}_{R, σ}^{†} (ω) + f_{r} (ω) {\hat{a}}_{L, σ}^{†} (ω), {\hat{a}}_{i, σ}^{†} (ω) \to f_{t} (ω) {\hat{a}}_{L, σ}^{†} (ω) + f_{r} (ω) {\hat{a}}_{R, σ}^{†} (ω),

(A2)

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| Ψ ⟩ = \iint d ω_{1} d ω_{2} {ϕ_{H V} (ω_{1}, ω_{2}) [f_{t} (ω_{1}) {\hat{a}}_{R, H}^{⏧} (ω_{1}) e^{- i ω_{1} τ} + f_{r} (ω_{1}) {\hat{a}}_{L, H}^{⏧} (ω_{1})] \cdot [f_{t} (ω_{2}) {\hat{a}}_{L, V}^{⏧} (ω_{2}) + f_{r} (ω_{2}) {\hat{a}}_{R, V}^{⏧} (ω_{2}) e^{- i ω_{2} τ}] + ϕ_{V H} (ω_{1}, ω_{2}) [f_{t} (ω_{1}) {\hat{a}}_{R, V}^{⏧} (ω_{1}) e^{- i ω_{1} τ} + f_{r} (ω_{1}) {\hat{a}}_{L, V}^{⏧} (ω_{1})] \cdot [f_{t} (ω_{2}) {\hat{a}}_{L, H}^{⏧} (ω_{2}) + f_{r} (ω_{2}) {\hat{a}}_{R, H}^{⏧} (ω_{2}) e^{- i ω_{2} τ}]} | 0,0 ⟩ .

(A3)

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{\hat{a}}_{1}^{⏧} (ω) \to \frac{1}{\sqrt{2}} [{\hat{a}}_{3}^{⏧} (ω) + i {\hat{a}}_{4}^{⏧} (ω)], {\hat{a}}_{2}^{⏧} (ω) \to \frac{1}{\sqrt{2}} [{\hat{a}}_{4}^{⏧} (ω) + i {\hat{a}}_{3}^{⏧} (ω)],

(A4)

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| Ψ_{c} ⟩ = \frac{1}{2} \iint d ω_{1} d ω_{2} [A (ω_{1}, ω_{2}) {\hat{a}}_{3, H}^{⏧} (ω_{1}) {\hat{a}}_{4, V}^{⏧} (ω_{2}) + B (ω_{1}, ω_{2}) {\hat{a}}_{3, H}^{⏧} (ω_{2}) {\hat{a}}_{4, V}^{⏧} (ω_{1}) + C (ω_{1}, ω_{2}) {\hat{a}}_{3, V}^{⏧} (ω_{1}) {\hat{a}}_{4, H}^{⏧} (ω_{2}) + D (ω_{1}, ω_{2}) {\hat{a}}_{3, V}^{⏧} (ω_{2}) {\hat{a}}_{4, H}^{⏧} (ω_{1})] | 0,0 ⟩,

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A (ω_{1}, ω_{2}) = ϕ_{H V} (ω_{1}, ω_{2}) [- f_{r} (ω_{1}) f_{r} (ω_{2}) e^{- i ω_{2} τ} + i f_{t} (ω_{1}) f_{r} (ω_{2}) e^{- i (ω_{1} + ω_{2}) τ} + i f_{r} (ω_{1}) f_{t} (ω_{2}) + i f_{t} (ω_{1}) f_{t} (ω_{2}) e^{- i ω_{1} τ}],

(A6)

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B (ω_{1}, ω_{2}) = ϕ_{V H} (ω_{1}, ω_{2}) [f_{r} (ω_{1}) f_{r} (ω_{2}) e^{- i ω_{2} τ} + i f_{t} (ω_{1}) f_{r} (ω_{2}) e^{- i (ω_{1} + ω_{2}) τ} + i f_{r} (ω_{1}) f_{t} (ω_{2}) - i f_{t} (ω_{1}) f_{t} (ω_{2}) e^{- i ω_{1} τ}],

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C (ω_{1}, ω_{2}) = ϕ_{V H} (ω_{1}, ω_{2}) [- f_{r} (ω_{1}) f_{r} (ω_{2}) e^{- i ω_{2} τ} + i f_{t} (ω_{1}) f_{r} (ω_{2}) e^{- i (ω_{1} + ω_{2}) τ} + i f_{r} (ω_{1}) f_{t} (ω_{2}) + i f_{t} (ω_{1}) f_{t} (ω_{2}) e^{- i ω_{1} τ}],

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D (ω_{1}, ω_{2}) = ϕ_{H V} (ω_{1}, ω_{2}) [f_{r} (ω_{1}) f_{r} (ω_{2}) e^{- i ω_{2} τ} + i f_{t} (ω_{1}) f_{r} (ω_{2}) e^{- i (ω_{1} + ω_{2}) τ} + i f_{r} (ω_{1}) f_{t} (ω_{2}) - i f_{t} (ω_{1}) f_{t} (ω_{2}) e^{- i ω_{1} τ}] .

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{\hat{M}}_{H V} = \int d ω_{3} {\hat{a}}_{3, H}^{†} (ω_{3}) | 0 ⟩ ⟨ 0 | {\hat{a}}_{3, H} (ω_{3}) \cdot \int d ω_{4} {\hat{a}}_{4, V}^{†} (ω_{4}) | 0 ⟩ ⟨ 0 | {\hat{a}}_{4, V} (ω_{4}), {\hat{M}}_{V H} = \int d ω_{3} {\hat{a}}_{3, V}^{†} (ω_{3}) | 0 ⟩ ⟨ 0 | {\hat{a}}_{3, V} (ω_{3}) \cdot \int d ω_{4} {\hat{a}}_{4, H}^{†} (ω_{4}) | 0 ⟩ ⟨ 0 | {\hat{a}}_{4, H} (ω_{4}) .

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P_{H V} (τ) = ⟨ Ψ_{c} | {\hat{M}}_{H V} | Ψ_{c} ⟩ = \frac{1}{4} \iint d ω_{3} d ω_{4} [| A (ω_{3}, ω_{4} {) |}^{2} + | B (ω_{3}, ω_{4} {) |}^{2} + A^{*} (ω_{3}, ω_{4}) B (ω_{4}, ω_{3}) + A (ω_{4}, ω_{3}) B^{*} (ω_{3}, ω_{4})],

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P_{V H} (τ) = ⟨ Ψ_{c} | {\hat{M}}_{V H} | Ψ_{c} ⟩ = \frac{1}{4} \iint d ω_{3} d ω_{4} [| C (ω_{3}, ω_{4} {) |}^{2} + | D (ω_{3}, ω_{4} {) |}^{2} + C^{*} (ω_{3}, ω_{4}) D (ω_{4}, ω_{3}) + C (ω_{4}, ω_{3}) D^{*} (ω_{3}, ω_{4})],

(A12)

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S. Francesconi, A. Raymond, R. Duhamel, P. Filloux, A. Lemaître, P. Milman, M. I. Amanti, F. Baboux, S. Ducci. On-chip generation of hybrid polarization-frequency entangled biphoton states[J]. Photonics Research, 2023, 11(2): 270

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