Particlelike Behavior of Topological Defects in Linear Wave Packets in Photonic Graphene

Topological defects, such as quantum vortices, determine the properties of quantum fluids. Their study has been at the center of activity in solid state and BEC communities. In parallel, the nontrivial behavior of linear wave packets with complex phase patterns was investigated by singular optics. Here, we study the formation, evolution, and interaction of optical vortices in wave packets at the Dirac point in photonic graphene. We show that while their exact behavior goes beyond the Dirac equation and requires a full account of the lattice properties, it can be still approximately described by an effective theory considering the phase singularities as “particles”. These particles are capable of mutual interaction, with their trajectory obeying the laws of dynamics.

Figure 1

(a) Experimental setup. ${\mathit{E}}_2$, ${\mathit{E}}_2^{\prime }$, and ${\mathit{E}}_2^{\prime \prime }$ are three coupling beams interfering to form a hexagonal optical lattice inside the rubidium vapor cell, which results in a honeycomb lattice for susceptibility due to the EIT effect. ${\mathit{E}}_1$ and ${\mathit{E}}_1^{\prime }$ are two probe beams from the same laser that construct a probe field featured by equally spaced vertical fringes. BS: $50/50$ beam splitter, CCD: charge coupled device camera. (b) The observed hexagonal coupling lattice. (c) The interference fringes formed by the two probe beams, both of which are set to carry an $\mathrm{OAM}=1$. (d) Spatial beam arrangement (before entering the cell) of the probe and coupling beams in the $XY$ plane. (e) The $\mathrm{\Lambda }$ three-level structure coupled by the probe and coupling fields. (f) The observed absorption (upper red curve, corresponding to the transition ${}^{85}\mathrm{Rb}$, $F=2\to F^{\prime }$) and EIT spectra (lower blue curve) versus the probe-field detuning. (g) The schematic for the generated hexagonal lattice. (h) and (i) The beam arrangements of the periodic probe field and the induced honeycomb lattice inside the medium for exciting A and B sublattices, respectively.

Figure 2

Wave packet in the Dirac equation. The two rows correspond to $4t/T=0.67(1.17)$. Intensity: (a), (d) $|{\psi }_A{|}^2$, (b), (e) $|{\psi }_B{|}^2$. Interference: (c), (f) $|{\psi }_A+{\psi }_B+e^{i{\mathbf{k}}_{\mathbf{r}}\mathbf{r}}{|}^2$, $k_{r}w=15$.

Figure 3

Wave packet evolution. The first two rows correspond to ${\mathrm{\Delta }}_1=45(56)\mathrm{MHz}$. (a), (c) the interference pattern with a reference beam; (b), (d) extracted phase [arrows in panel (b) show different contributions to the current]. The last row shows the vortex trajectory (e) $r(t)$ [black dots—experiment, red line—Eq. (4)] and (f) $y(x)$ [same as (e) + blue line—Eq. (5)].

Figure 4

Evolution of $L=+1$-wave packet. The two rows correspond to ${\mathrm{\Delta }}_1=80.4(96.4)\mathrm{MHz}$. Experiment: (a), (d) interference pattern, (b), (e) extracted phase. Vortex trajectories: (c) experiment (curves are guides for the eyes), (f) theory.

Figure 5

Evolution of $L=-1$-wave packet. The two rows correspond to ${\mathrm{\Delta }}_1=50(59.6)\mathrm{MHz}$. Experiment: (a), (d) interference patterns, (b), (e) extracted phase. Trajectories of the vortices: (c) experiment (smooth curves are guides for the eyes), (f) theory.