Fig. 1. Experimental schemes. (a) Illustrative picture of the experimental setup. The focus lengths of the two lenses are both f=150mm. (b) The three-level Rb85 atomic configuration excited by the probe field E1 and the hexagonal coupling field E2. (c) and (d) Calculated spatial distribution of refractive index for Δ1=−80MHz (c) and Δ1=−190MHz (d) with Δ2=−100MHz, resulting in honeycomb and hexagonal lattices, respectively.

Fig. 2. Momentum space vortex generation in the honeycomb lattice with the two-photon detuning being 20 MHz. (a) Experimentally measured momentum space image interference with the reference beam. The dislocations in fringes correspond to the vortices (marked by white circles). (b) The corresponding phase pattern extracted from the interference image. Black arrows show the rotation direction.

Fig. 3. Numerical solution of the Schrödinger equation showing wave packet expansion in photonic graphene starting from the excitation of a single site. (a) The probability distribution in real space. (b) The probability distribution in momentum space. (c) The reciprocal space phase image of the WF corresponding to panel (b). The left and right vortices are marked with blue and red circles, respectively. (d) The reciprocal space WF interference with a plane wave. Green circles indicate the Dirac points. The snapshot shows that the WF has the C3v symmetry. Panels (e) and (f) show the results of tight-binding calculation and correspond to panels (a) and (d), respectively.

Fig. 4. Numerical solution of the Schrödinger equation showing wave packet expansion in photonic honeycomb and hexagonal lattices with various symmetries of the WFs. (a), (b) Honeycomb lattice, excitation of a single “benzene ring” (C6v): real space probability (a) and momentum space phase (b). (c), (d) The same for a honeycomb lattice, excitation between two sites (C2v symmetry). (e), (f) The same for a topologically trivial hexagonal lattice with C2v-excitation.

Fig. 5. Phase patterns in momentum space for alternative symmetries and lattices. Panels (a) and (b) demonstrate absence of vortices for the case of C6v symmetry signal in honeycomb lattice. The momentum space interference image shown in panel (a) contains no fork dislocations in fringe pattern and the extracted phase shown in panel (b) also demonstrates no vortex-like phase defects. Panels (c) and (d) show the interference pattern and extracted phase, respectively, for the C2v symmetry signal realized in the honeycomb lattice by excitation between two sites. Panels (e) and (f) are for the interference pattern and extracted phase, respectively, in the case of the C2v symmetry in hexagonal lattice realized experimentally by setting the two-photon detuning −90MHz.