Bloch-Zener Oscillations in Binary Superlattices

Bloch-Zener oscillations, i.e., the coherent superposition of Bloch oscillations and Zener tunneling between minibands of a binary lattice, are experimentally demonstrated for light waves in curved femtosecond laser-written waveguide arrays. Visualization of double-periodicity breathing and oscillation modes is reported, and synchronous tunneling leading to wave reconstruction is demonstrated.

Figure 1

Schematic sketch of the system: (a) transverse potential of a binary lattice with additional linear gradient; (b) interleaved WS ladder spectrum; (c) dispersion curves of tight-binding minibands with equal waveguides (dashed curve) and detuned ($\sigma =\kappa$) waveguides (solid curve); (d) two-dimensional refractive index profile of the circularly-curved waveguide array. The inset in (d) is a microscope image showing cross sections of a few waveguides realized in fused silica glass.

Figure 2

Breathing modes in curved binary arrays for single waveguide excitation and for a Bloch period $z_B=37.5\mathrm{mm}$. (a) $\kappa =0.23{\mathrm{mm}}^{-1}$, no detuning; (b) $\kappa =0.23{\mathrm{mm}}^{-1}$, $\sigma /\kappa =0.4$, corresponding to synchronous ZT ($z_2=z_B$); (c) $\kappa =0.20{\mathrm{mm}}^{-1}$, $\sigma /\kappa =0.4$, corresponding to asynchronous ZT. In (a) and (b), $d=13\mu \mathrm{m}$ and $R=1.12\mathrm{m}$; in (c), $d=14\mu \mathrm{m}$ and $R=1.21\mathrm{m}$. Upper figures: measured fluorescence images, lower figures: corresponding simulations.

Figure 3

Same as Fig. 2, but for $d=11\mu \mathrm{m}$ and $R=1.9\mathrm{m}$, corresponding to $\kappa =0.37{\mathrm{mm}}^{-1}$ and $z_B=75\mathrm{mm}$. (a) no detuning; (b) $\sigma /\kappa =0.3$ (synchronous ZT); (c) $\sigma /\kappa =0.5$ (asynchronous ZT).

Figure 4

Oscillation modes in binary waveguide arrays with $z_B=75\mathrm{mm}$. Measured fluorescence images and estimated fractional band occupancies, with corresponding theoretical predictions, for (a) $\kappa =0.55{\mathrm{mm}}^{-1}$ no detuning; (b) $\kappa =0.55{\mathrm{mm}}^{-1}$, $\sigma /\kappa =0.22$; (c) $\kappa =0.538{\mathrm{mm}}^{-1}$, $\sigma /\kappa =0.41$.