Fig. 3: Band structures, measured beam delay and refractions for integer moving speed v = 1.
a Floquet bands of l = 0 for positive (solid) and negative (dashed) minibands outside (blue) and inside (red) the barrier with scalar and vector potential distribution (φ1, A1) = (0, 0), (Δφ, ΔA) = (−π/2, −π/2), β = π/3, and W = 15. The incident, transmitted and refracted packets are denoted by blue and red circles. b Calculated (blue curve) and measured (red circles) beam delays versus incident Bloch momentum k1+, which reaches dmax = 2 W, dmin = −2 W/3 and d = 0 at k1+ = −0.5π, 0.5π, and −0.148π. c Required gauge-potential difference combination Δφ + vΔA to achieve a transparency condition for each targeted k1+. d–f Measured field evolutions for k1+ = −0.5π, 0 and 0.5π, corresponding to dmax = 2 W (d), d = −0.28 W (e) and dmin = −2 W/3 (f). g–i Measured field evolutions for k1+ = −0.148π, 0 and 0.25π under Δφ + vΔA = −π, −2π/3, and −0.27π, corresponding to a transparent moving potential for a leftward (g), frozen (h), and rightward (i) propagating packet. In (d–i), the green dashed lines denote the two moving boundaries of moving potential. The blue solid lines denote the incident and transmitted packets’ trajectories in the (n, m) plane.
