Researching | Pseudospin-2 in photonic chiral borophene

Journals >Photonics Research >Volume 11 >Issue 5 >Page 869 > Article

Photonics Research
Vol. 11, Issue 5, 869 (2023)

Pseudospin-2 in photonic chiral borophene | Editors' Pick

Philip Menz^1,*, Haissam Hanafi¹, Daniel Leykam², Jörg Imbrock¹, and Cornelia Denz^1,3

Author Affiliations

¹Institute of Applied Physics, University of Muenster, Muenster 48149, Germany

²Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore

³Physikalisch-Technische Bundesanstalt (PTB), 38116 Braunschweig, Germany

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DOI: 10.1364/PRJ.486916 Cite this Article Set citation alerts

Philip Menz, Haissam Hanafi, Daniel Leykam, Jörg Imbrock, Cornelia Denz, "Pseudospin-2 in photonic chiral borophene," Photonics Res. 11, 869 (2023) Copy Citation Text

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Abstract

Pseudospin is an angular momentum degree of freedom introduced in analogy to the real electron spin in the effective massless Dirac-like equation used to describe wave evolution at conical intersections such as the Dirac cones of graphene. Here, we study a photonic implementation of a chiral borophene allotrope hosting a pseudospin-2 conical intersection in its energy–momentum spectrum. The presence of this fivefold spectral degeneracy gives rise to quasiparticles with pseudospin up to

\pm 2

. We report on conical diffraction and pseudospin–orbit interaction of light in photonic chiral borophene, which, as a result of topological charge conversion, leads to the generation of highly charged optical phase vortices.

{\hat{H}}_{k} = t (\begin{matrix} 0 & 1 & e^{- i a_{1} k} & e^{- i a_{2} k} & e^{- i a_{2} k} & 1 \\ 1 & 0 & 1 & e^{- i a_{2} k} & e^{i a_{3} k} & e^{i a_{3} k} \\ e^{i a_{1} k} & 1 & 0 & 1 & e^{i a_{3} k} & e^{i a_{1} k} \\ e^{i a_{2} k} & e^{i a_{2} k} & 1 & 0 & 1 & e^{i a_{1} k} \\ e^{i a_{2} k} & e^{- i a_{3} k} & e^{- i a_{3} k} & 1 & 0 & 1 \\ 1 & e^{- i a_{3} k} & e^{- i a_{1} k} & e^{- i a_{1} k} & 1 & 0 \end{matrix}),

(1)

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| ψ_{- 2} ⟩ = \frac{1}{\sqrt{6}} {(+ 1 e^{i \frac{1}{3} π} e^{i \frac{2}{3} π} - 1 e^{i \frac{4}{3} π} e^{i \frac{5}{3} π})}^{T}, | ψ_{- 1} ⟩ = \frac{1}{\sqrt{6}} {(+ 1 e^{i \frac{2}{3} π} e^{i \frac{4}{3} π} + 1 e^{i \frac{2}{3} π} e^{i \frac{4}{3} π})}^{T}, | ψ_{0} ⟩ = \frac{1}{\sqrt{6}} {(+ 1 - 1 + 1 - 1 + 1 - 1)}^{T}, | ψ_{+ 1} ⟩ = \frac{1}{\sqrt{6}} {(+ 1 e^{i \frac{4}{3} π} e^{i \frac{2}{3} π} + 1 e^{i \frac{4}{3} π} e^{i \frac{2}{3} π})}^{T}, | ψ_{+ 2} ⟩ = \frac{1}{\sqrt{6}} {(+ 1 e^{i \frac{5}{3} π} e^{i \frac{4}{3} π} - 1 e^{i \frac{2}{3} π} e^{i \frac{1}{3} π})}^{T},

(2)

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l = m_{s}^{in} - m_{s}^{out} .

(3)

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i \frac{\partial}{\partial z} ψ (x, y, z) = [- \frac{1}{2 k_{0} n_{0}} \nabla_{⊥}^{2} - k_{0} Δ n (x, y)] ψ (x, y, z) = \hat{H} ψ (x, y, z) .

(A1)

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{\hat{H}}_{k} = t (\begin{matrix} 0 & 1 & 1 - i a_{1} k & 1 - i a_{2} k & 1 - i a_{2} k & 1 \\ 1 & 0 & 1 & 1 - i a_{2} k & 1 + i a_{3} k & 1 + i a_{3} k \\ 1 + i a_{1} k & 1 & 0 & 1 & 1 + i a_{3} k & 1 + i a_{1} k \\ 1 + i a_{2} k & 1 + i a_{2} k & 1 & 0 & 1 & 1 + i a_{1} k \\ 1 + i a_{2} k & 1 - i a_{3} k & 1 - i a_{3} k & 1 & 0 & 1 \\ 1 & 1 - i a_{3} k & 1 - i a_{1} k & 1 - i a_{1} k & 1 & 0 \end{matrix}),

(C1)

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U = \frac{1}{\sqrt{6}} (\begin{matrix} e^{i \frac{5}{3} π} & e^{i \frac{5}{3} π} & 1 & e^{i \frac{4}{3} π} & e^{i \frac{1}{3} π} & 1 \\ e^{i \frac{4}{3} π} & - 1 & - 1 & 1 & e^{i \frac{2}{3} π} & 1 \\ - 1 & e^{i \frac{1}{3} π} & 1 & e^{i \frac{2}{3} π} & - 1 & 1 \\ e^{i \frac{2}{3} π} & e^{i \frac{5}{3} π} & - 1 & e^{i \frac{4}{3} π} & e^{i \frac{4}{3} π} & 1 \\ e^{i \frac{π}{3}} & - 1 & 1 & 1 & e^{i \frac{5}{3} π} & 1 \\ 1 & e^{i \frac{π}{3}} & - 1 & e^{i \frac{2}{3} π} & 1 & 1 \end{matrix}) .

(C2)

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{\hat{H}}_{eff} = t (\begin{matrix} - 1 & \frac{1}{2} (k_{x} - i k_{y}) & 0 & 0 & 0 \\ \frac{1}{2} (k_{x} + i k_{y}) & - 1 & \frac{1}{2} (k_{x} - i k_{y}) & 0 & 0 \\ 0 & \frac{1}{2} (k_{x} + i k_{y}) & - 1 & \frac{1}{2} (k_{x} - i k_{y}) & 0 \\ 0 & 0 & \frac{1}{2} (k_{x} + i k_{y}) & - 1 & \frac{1}{2} (k_{x} - i k_{y}) \\ 0 & 0 & 0 & \frac{1}{2} (k_{x} + i k_{y}) & - 1 \end{matrix}),

(C3)

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{\hat{H}}_{eff} = t (\begin{matrix} - 1 & \frac{1}{2} k e^{- i θ} & 0 & 0 \\ \frac{1}{2} k e^{i θ} & - 1 & \frac{1}{2} k e^{- i θ} & 0 & 0 \\ 0 & \frac{1}{2} k e^{i θ} & - 1 & \frac{1}{2} k e^{- i θ} & 0 \\ 0 & 0 & \frac{1}{2} k e^{i θ} & - 1 & \frac{1}{2} k e^{- i θ} \\ 0 & 0 & 0 & \frac{1}{2} k e^{i θ} & - 1 \end{matrix}),

(C4)

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β_{1,2,3,4,5} = t (- 1 - \sqrt{3} k), t (- 1 - k), - t, t (- 1 + k), t (- 1 + \sqrt{3} k) .

(C5)

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S_{z} = (\begin{matrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & - 2 \end{matrix}) .

(C6)

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S_{x} = \frac{1}{2} (\begin{matrix} 0 & 2 & 0 & 0 & 0 \\ 2 & 0 & \sqrt{6} & 0 & 0 \\ 0 & \sqrt{6} & 0 & \sqrt{6} & 0 \\ 0 & 0 & \sqrt{6} & 0 & 2 \\ 0 & 0 & 0 & 2 & 0 \end{matrix}),

(C7)

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S_{y} = \frac{1}{2 i} (\begin{matrix} 0 & 2 & 0 & 0 & 0 \\ - 2 & 0 & \sqrt{6} & 0 & 0 \\ 0 & - \sqrt{6} & 0 & \sqrt{6} & 0 \\ 0 & 0 & - \sqrt{6} & 0 & 2 \\ 0 & 0 & 0 & - 2 & 0 \end{matrix}),

(C8)

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{\hat{H}}_{eff} (k) = c_{0} k \cdot S + c_{1} k \cdot {S, S_{z}^{2}} - t I_{5},

(C9)

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{\hat{H}}_{eff} (k, θ) = \frac{c_{0} k}{2} (e^{- i θ} S_{+} + e^{i θ} S_{-}) + \frac{c_{1} k}{2} \cdot (e^{- i θ} {S_{+}, S_{z}^{2}} + e^{i θ} {S_{-}, S_{z}^{2}}) - t I_{5} .

(C10)

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Philip Menz, Haissam Hanafi, Daniel Leykam, Jörg Imbrock, Cornelia Denz, "Pseudospin-2 in photonic chiral borophene," Photonics Res. 11, 869 (2023)

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