Observation of Valley Landau-Zener-Bloch Oscillations and Pseudospin Imbalance in Photonic Graphene

We demonstrate intervalley Bloch oscillation (BO) and Landau-Zener tunneling (LZT) in an optically induced honeycomb lattice with a refractive-index gradient. Unlike previously observed BO in a gapped square lattice, we show nonadiabatic beam dynamics that are highly sensitive to the direction of the index gradient and the choice of the Dirac cones. In particular, a symmetry-preserving potential leads to nearly perfect LZT and coherent BO between the inequivalent valleys, whereas a symmetry-breaking potential generates asymmetric scattering, imperfect LZT, and valley-sensitive generation of vortices mediated by a pseudospin imbalance. This clearly indicates that, near the Dirac points, the transverse gradient does not always act as a simple scalar force, as commonly assumed, and the LZT probability is strongly affected by the sublattice symmetry as analyzed from an effective Landau-Zener Hamiltonian. Our results illustrate the anisotropic response of an otherwise isotropic Dirac platform to real-space potentials acting as strong driving fields, which may be useful for manipulation of pseudospin and valley degrees of freedom in graphenelike systems.

Figure 1

(a) Schematic of the experimental setup. The upper path is for optical induction of photonic graphene, and the bottom path is for probing through the lattice. The refractive-index ramp is induced by nonuniform white-light illumination. HWP, half-wave plate; PBS, polarizing beam splitter; RD, rotating diffuser; AM, amplitude mask; $P$, polarizer; SF, spatial filter; SBN, strontium barium niobate. (b) Output HCL intensity pattern. (Inset) Illustration of the $A/B$ sublattice structure. (c) Discrete diffraction of a Gaussian beam exiting the lattice. (d) Measured spectrum superimposed with schematic drawing of the first BZ, where symmetry points are marked by dots and paths for BO are depicted by dashed lines.

Figure 2

Symmetric BO under an $x$ index gradient. (a)–(e) Measured Fourier spectra at the lattice output for different input tilts of the probe beam (marked by the white circles). The arrow in (a) indicates the direction of the gradient. In (a), when the probe beam crosses the Dirac $K$ point, there is symmetric scattering to the other two equivalent $K$ points. In (d) and (e), the two scattered components cross the $K^{\prime }$ points, then predominantly return to the first BZ. (f)–(j) Corresponding results from numerical simulation. The path of this BO is illustrated by horizontal dashed lines in Fig. 1.

Figure 3

(Top two rows) Asymmetric BO under a $y$ index gradient. (a)–(e) Same as in Figs. 2 except that the index gradient is now in the $y$ direction. In (b)–(e), when the probe beam crosses the Dirac $K$ point, the scattering to the other two equivalent $K$ points is highly asymmetric, leading to imperfect tunneling into the second band. (f)–(j) Corresponding numerical results. The path of this BO is illustrated by a vertical dashed line in Fig. 1. (Bottom row) Experimental results (k)–(o) of BO through the $\mathrm{\Gamma }$ and $M$ symmetry points (no valley involved), corresponding to quasiadiabatic transport primarily in the first band.

Figure 4

Calculated real-space displacement of the beam’s “center of mass” for propagation beyond experimentally accessible distance for better comparison of the BO dynamics. (a) Coherent BO under an $x$ index gradient, displaying its persistence along the $x$ direction. (b) Asymmetric BO under a $y$ index gradient, showing faded BO due to imperfect LZT and loss of coherence of split beams upon traversing the Dirac point.

Figure 5

Valley interference after BO probed with a narrow input beam under different conditions. (a)–(d) Measured output intensity under (a) no gradient, (b) horizontal gradient, (c) vertical gradient for BO along $K$ point, and (d) vertical gradient for BO along $K^{\prime }$ point. The white circle indicates the input beam position. (e)–(h) Measured interferograms showing fringe forks (vortex positions) in the first BZ marked by dashed circles. (i),(j) Phase structures obtained from numerical simulations corresponding to (f),(g).