Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices

Friederike Klauck; Matthias Heinrich; Alexander Szameit

doi:10.1364/PRJ.409005

Journals >Photonics Research >Volume 9 >Issue 1 >Page A1 > Article

Photonics Research
Vol. 9, Issue 1, A1 (2021)

Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices

Friederike Klauck, Matthias Heinrich, and Alexander Szameit^*

Author Affiliations

Institute for Physics, University of Rostock, 18059 Rostock, Germany

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DOI: 10.1364/PRJ.409005 Cite this Article Set citation alerts

Friederike Klauck, Matthias Heinrich, Alexander Szameit. Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices[J]. Photonics Research, 2021, 9(1): A1 Copy Citation Text

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Abstract

We report on the experimental demonstration of two-photon quantum walks at the edge of a photonic Su–Schrieffer–Heeger lattice and compare them to those observed when launching photons at the edge of a homogeneous lattice. Whereas at the topological edge, one of the photons primarily remains close to the edge, both photons penetrate freely from the trivial edge into the bulk. This behavior manifests also in the average inter-particle distance, which is significantly larger at the topological edge. Hence, for a given propagation length, the entangled two-photon state launched at the topological edge extends over a wider domain of the lattice.

H = (\begin{matrix} 0 & c_{1} & 0 & \dots & 0 \\ c_{1} & 0 & c_{2} & □ & ⋮ \\ 0 & c_{2} & 0 & c_{1} & 0 \\ ⋮ & □ & c_{1} & 0 & ⋮ \\ 0 & \dots & 0 & \dots & 0 \end{matrix}) .

(1)

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H (k) = (\begin{matrix} 0 & c_{1} + c_{2} e^{i k} \\ c_{1} + c_{2} e^{- i k} & 0 \end{matrix}),

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E (k) = \pm \sqrt{c_{1}^{2} + c_{2}^{2} + 2 c_{1} c_{2} \cos k}

(3)

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Z = i \int_{B Z} ψ^{*} (k) \frac{\partial}{\partial k} ψ (k) \partial k,

(4)

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H = \frac{ℏ c}{n_{0}} \sum_{k} [β_{0} ({\hat{a}}_{k}^{†} {\hat{a}}_{k} + \frac{1}{2}) + \sum_{l} C_{k, l} {\hat{a}}_{k}^{†} {\hat{a}}_{l}],

(5)

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i \frac{d}{d z} {\hat{a}}_{k}^{†} (z) + \sum_{l} C_{k, l} {\hat{a}}_{k}^{†} = 0,

(6)

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{\hat{a}}_{k}^{†} (z) = \sum_{l} U_{k, l} (z) {\hat{a}}_{l}^{†} (0),

(7)

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U_{k, l} (z) = {(e^{i z C})}_{k, l} .

(8)

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{\bar{n}}_{m} (z) = ⟨ {\hat{a}}_{m}^{†} (z) {\hat{a}}_{m} (z) ⟩ = {| U_{m, k} (z) |}^{2} .

(9)

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{\bar{n}}_{m} (z) = {| U_{m, k} (z) |}^{2} + {| U_{m, l} (z) |}^{2} .

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Γ_{m, n} (z) = ⟨ {\hat{a}}_{m}^{†} (z) {\hat{a}}_{n}^{†} (z) {\hat{a}}_{n} (z) {\hat{a}}_{m} (z) ⟩

(11)

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Γ_{m, n} (z) = {| U_{m, k} (z) U_{n, l} (z) + U_{m, l} (z) U_{n, k} (z) |}^{2} .

(12)

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g (Δ) = \sum_{m} Γ_{m, m + Δ} .

(13)

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Friederike Klauck, Matthias Heinrich, Alexander Szameit. Photonic two-particle quantum walks in Su–Schrieffer–Heeger lattices[J]. Photonics Research, 2021, 9(1): A1

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