Valley Vortex States and Degeneracy Lifting via Photonic Higher-Band Excitation

We demonstrate valley-dependent vortex generation in photonic graphene. Without breaking inversion symmetry, the excitation of two valleys leads to the formation of an optical vortex upon Bragg reflection to the third equivalent valley, with its chirality determined by the valley degree of freedom. Vortex-antivortex pairs with valley-dependent topological charge flipping are also observed and corroborated by numerical simulations. Furthermore, we develop a three-band effective Hamiltonian model to describe the dynamics of the coupled valleys and find that the commonly used two-band model is not sufficient to explain the observed vortex degeneracy lifting. Such valley-polarized vortex states arise from high-band excitation without a synthetic-field-induced gap opening. Our results from a photonic setting may provide insight for the study of valley contrasting and Berry-phase-mediated topological phenomena in other systems.

Figure 1

Numerical simulation of pseudospin vortex states from valley-selective excitation. (a) The HCL composed of induced waveguides, where the inset illustrates sublattices $A$ and $B$ marked with red and blue dots, respectively. (b) The interfering probe beams used for lattice excitation, where the inset illustrates the excitation condition in real space. (c) Illustration of the excitation condition when the two $K$ (top row) or $K^{\prime }$ (bottom row) valleys are selectively excited in momentum space. (d) Output intensity pattern of the probe beams. (e) Corresponding spectrum, where blue dashed circle marks the Bragg-reflected component and white dashed hexagon depicts the first BZ. (f) Phase structure of the Bragg-reflected component.

Figure 2

Experimental demonstration corresponding to Fig. 1. (a) The HCL established by optical induction. (b) The $k$-space spectrum of the lattice beam, where $K$ and $K^{\prime }$ valleys are marked. (c)–(e) Vortex generation when two $K$ (top row) or $K^{\prime }$ (bottom row) valleys are selectively excited. (c) Output intensity patterns. (d) Input (top subpanel) and output (bottom subpanel) spectra, where blue dashed circles mark the newly generated spectral component at the third valley. (e) Interferograms from the Bragg-reflected components, where the vortex position or chirality is marked.

Figure 3

Demonstration of valley-dependent vortex pair generation. The excitation scheme is the same as in Fig. 1 for $K$ (top row) or $K^{\prime }$ (bottom row) valleys, except that the probe beam is shifted to the intersite position [see the inset in (a)]. (a) Output intensity patterns. (b) Far-field patterns of Bragg-reflected component from the third valley. (c) Interferograms of (b) with an inclined plane wave, where the vortex pair is marked. (d) Numerically calculated phase of the Bragg-reflected component showing the vortex pair of opposite chirality.

Figure 4

Numerical projection of Bragg-reflected field into two spinor components when two $K$ valleys are selectively excited. The top row corresponds to on-site excitation (Figs. 1 and 2) for single-vortex generation, and the bottom row corresponds to intersite excitation (Fig. 3) for vortex pair generation. (a),(b) Intensity and phase patterns of the sublattice ${\varphi }_A$ component; (c),(d) corresponding results of the sublattice ${\varphi }_B$ component. For excitation of two $K^{\prime }$ valleys, the topological charges for all vortices are reversed.

Figure 5

Calculated intensity and phase of the Bragg-reflected field into the third valley from the three-band model. (a)–(d) Shown are the results obtained by solving the effective propagation of Eqs. (3) and (4) using normalized parameters $\tilde{z}=0.5$ and $V=1$, with a width beam $w=0.8$. (a),(b) For on-site excitation to generate a single vortex (corresponding to Figs. 1 and 2). (c),(d) For intersite excitation to generate a vortex-antivortex pair (corresponding to Fig. 3).