Acoustic Analogue of Electronic Bloch Oscillations and Resonant Zener Tunneling in Ultrasonic Superlattices

We demonstrate the existence of Bloch oscillations of acoustic fields in sound propagation through a superlattice of water cavities and layers of methyl methacrylate. To obtain the acoustic equivalent of a Wannier-Stark ladder, we employ a set of cavities with different thicknesses. Bloch oscillations are observed as time-resolved oscillations of transmission in a direct analogy to electronic Bloch oscillations in biased semiconductor superlattices. Moreover, for a particular gradient of cavity thicknesses, an overlap of two acoustic minibands occurs, which results in resonant Zener-like transmission enhancement.

Figure 1

Transfer-matrix calculation of the intensity distribution ($\log |P(z){|}^2$) through a sonic superlattice made of eight coupled water cavities. (a) Flat band case, $\Delta (1/d_W)=0\%$, where two minibands (MB1 and MB2) are clearly seen. (b) Tilted band case, $\Delta (1/d_W)=6\%$, where Wannier-Stark–like ladders (WSL1 and WSL2) are also shown. The structure of coupled water cavities is schematically shown above each panel; dark regions represent Plexiglas layers. The right panels depict the transmission spectra across the total structure.

Figure 2

(a) Amplitude of transmission calculated by transfer matrix (TM) for the multilayer described in Fig. 1b. The acoustical analogue of the electronic Wannier-Stark ladder is seen as a series of almost equidistant transmission peaks. (b) The corresponding experimental spectrum obtained from time-domain measurements in transmission. (c) Spacing between peaks ($\delta {\nu }_B$) in the corresponding WSL as a function of the gradient $\Delta (1/d_W)$. The gray lines are guides for the eye and allow to distinguish the linear WSL regime.

Figure 3

Temporal response of the system for various values of the gradient. The top panel shows the unperturbed probe pulse without superlattice sample. Lower panels show the obtained time-domain oscillations, which are the acoustic Bloch oscillations (in MB1). The period, $T_B$, measured are $41.5$, $32.7$, and $29.3\mu \mathrm{s}$ for gradients 6%, 8%, and 9%, respectively.

Figure 4

(a) Transfer-matrix (TM) calculation of the transmission spectra of a tilted superlattice around the value of gradient, $\Delta (1/d_W)$, where the first anticrossing of the acoustical WSL and hence Zener-like resonant effect occurs. The amplitude is plotted as a function of the reduced frequency, where ${\omega }_0$ is the central frequency. The inset shows the strength (see text) of the resonant peak arising from the interaction between the two states located in WSL1 and WSL2, respectively [see Fig. 1b]. (b) The transmission spectra measured by time-resolved transmission experiments. The inset shows a comparison between the behavior obtained by TM simulations (full lines) and experiments (symbols with error bars).