Optically Reconfigurable Spin-Valley Hall Effect of Light in Coupled Nonlinear Ring Resonator Lattice

Haofan Yang, Jing Xu, Zhongfei Xiong, Xinda Lu, Ruo-Yang Zhang, Hanghang Li, Yuntian Chen, and Shuang Zhang

Phys. Rev. Lett. 127, 043904 – Published 23 July 2021

Abstract

Scattering immune propagation of light in topological photonic systems may revolutionize the design of integrated photonic circuits for information processing and communications. In optics, various photonic topological circuits have been developed, which were based on classical emulation of either quantum spin Hall effect or quantum valley Hall effect. On the other hand, the combination of both the valley and spin degrees of freedom can lead to a new kind of topological transport phenomenon, dubbed spin-valley Hall effect (SVHE), which can further expand the number of topologically protected edge channels and would be useful for information multiplexing. However, it is challenging to realize SVHE in most known material platforms, due to the requirement of breaking both the (pseudo)fermionic time-reversal ( $T$ ) and parity symmetries ( $P$ ) individually, but leaving the combined symmetry $S \equiv T P$ intact. Here, we propose an experimentally feasible platform to realize SVHE for light, based on coupled ring resonators mediated by optical Kerr nonlinearity. Thanks to the inherent flexibility of cross-mode modulation, the coupling between the probe light can be engineered in a controllable way such that spin-dependent staggered sublattice potential emerges in the effective Hamiltonian. With delicate yet experimentally feasible pump conditions, we show the existence of spin-valley Hall-induced topological edge states. We further demonstrate that both degrees of freedom, i.e., spin and valley, can be manipulated simultaneously in a reconfigurable manner to realize spin-valley photonics, doubling the degrees of freedom for enhancing the information capacity in optical communication systems.

Received 8 February 2021
Accepted 17 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.043904

Physics Subject Headings (PhySH)

Symmetry protected topological states Topological effects in photonic systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Haofan Yang¹, Jing Xu^1,2,*, Zhongfei Xiong¹, Xinda Lu¹, Ruo-Yang Zhang³, Hanghang Li¹, Yuntian Chen^1,2,†, and Shuang Zhang^4,5,‡

¹School of Optical and Electronic Information, Huazhong University of Science and Technology, Wuhan 430074, China
²Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China
³Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
⁴Department of Physics, University of Hong Kong, Hong Kong, China
⁵Department of Electrical and Electronic Engineering, University of Hong Kong, Hong Kong, China

^*Corresponding author. jing_xu@hust.edu.cn
^†Corresponding author. yuntian@hust.edu.cn
^‡Corresponding author. shuzhang@hku.hk

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Issue

Vol. 127, Iss. 4 — 23 July 2021

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Images

Figure 1
A 2D honeycomb array of ring resonators and its bulk band structure. (a) A 2D honeycomb array of ring resonators. Each ring supports CW and CCW mode. The illustration represents the pump-probe configuration and the corresponding mode coupling between the nearest-neighbor ring. (b) The schematic diagram for spin up and spin down. (c) Band structure for the case of nonpump comprising a honeycomb lattice ( $P_{p} = 0$ ). Note the band crossings at the Dirac point (the red dot). (d) Band structure for the spin-valley Hall photonic topological insulator. Dirac points are to open a gap with magnitude $Δ = 2 {| m |}^{2} / ω$ , which corresponds to the band gap in a Floquet photonic topological insulator. (e) Calculated Berry curvature of the first band near the $K$ and $K^{'}$ valleys and corresponding valley Chern numbers for type $A$ and type $B$ for the $ψ_{↑}$ .
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Figure 2
Calculated band structure for spin up (a) and spin down (b) of an $A B$ -type (blue lines) domain wall and a $B A$ -type (red lines) domain wall. Frequency of the band structure corresponds to the frequency relative to that of the probe light. The shaded regions correspond to the bulk states. The illustration in (a) is the schematic of a straight domain wall (black dashed line) between the upper domain with type $A (B)$ and the lower domain with type $B (A)$ . One-way propagation valley edge states of $A B$ -type domain wall (c)–(f). The propagating direction of excited edge states on (c) $K$ valley of $ψ_{↑}$ , (d) $K^{'}$ valley of $ψ_{↑}$ , (e) $K$ valley of $ψ_{↓}$ , and (f) $K^{'}$ valley of $ψ_{↓}$ agree with prediction from the band structure. The illustration in (e) shows the schematic diagrams of the domain wall and corresponding edge states. The red pentagram represents the source whose phase matches with the edge states at the $K / K^{'}$ valley. The color indicates the amplitude of $E_{z}$ profiles.
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Figure 3
Topological valley edge states of the $ψ_{↑}$ and $ψ_{↑}$ and the spatial separation of the two spins. (a) Simulated field map of $E_{z}$ profiles in the $x y$ plane. The red pentagram indicates the location of the excitation source. The illustration represents the schematic of the zigzag domain wall (black solid line) between the upper domain with type $A$ and the lower domain with type $B$ . The corner angle $α = 60 °$ and $ω = 120 °$ . The color indicates the amplitude of $E_{z}$ profiles. (b) Through the coupling of optical waveguides to ring resonators, the two spins, i.e., $ψ_{↑}$ (the blue arrow) and $ψ_{↓}$ (the red arrow), are separated.
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