Parity-Time Synthetic Phononic Media

Classical systems containing cleverly devised combinations of loss and gain elements constitute extremely rich building units that can mimic non-Hermitian properties, which conventionally are attainable in quantum mechanics only. Parity-time ($\mathcal{P}\mathcal{T}$) symmetric media, also referred to as synthetic media, have been devised in many optical systems with the ground breaking potential to create nonreciprocal structures and one-way cloaks of invisibility. Here we demonstrate a feasible approach for the case of sound where the most important ingredients within synthetic materials, loss and gain, are achieved through electrically biased piezoelectric semiconductors. We study first how wave attenuation and amplification can be tuned, and when combined, can give rise to a phononic $\mathcal{P}\mathcal{T}$ synthetic media with unidirectional suppressed reflectance, a feature directly applicable to evading sonar detection.

Figure 1

Gain or attenuation of sound in ZnO at (a) various carrier concentrations $n/n_i$ and for (b) different frequencies. The vertical dashed-dotted line marks the Cherenkov threshold ($v_d=v_s$). In panel (b) $n=n_i$ with a plasma frequency at 9 MHz. (c) From (a) we take the example where $n={10}^3{n}_i$ corresponding to a plasma frequency at 280 MHz and plot the elastic displacements $u$ at the gain or attenuation peaks and where the electric bias is switched off, $E_0=0$.

Figure 2

$c_{33}^{\prime }$ is computed versus $E_0$ for ZnO with $n={10}^4{n}_i$ at $\omega =0.7{\omega }_p$. Top scale represents the data as a function of $v_d/v_s$. Left(right) axis represents the real(imaginary) part of $c_{33}^{\prime }$. At the Cherenkov threshold, $v_d=v_s$ (dash-dotted line at $E_0=131.8\mathrm{kV}/\mathrm{m}$), the effective stiffness $c_{33}^{\prime }$ is purely real.

Figure 3

Scattering parameters for various ZnO slab setups biased with a voltage source. In all cases, the overall slab thickness $h=4\mathrm{cm}$. Upper panel: Slab is biased in forward direction compared to the sound direction (left to right), $E_G=150.0\mathrm{kV}/\mathrm{m}$. Middle panel: Bias in a bidirectional fashion with $E_G=150.0\mathrm{kV}/\mathrm{m}$ and $E_{-G}=-E_G$. Lower panel: Balanced loss and gain through a voltage divider in forward direction, $E_G=150.0\mathrm{kV}/\mathrm{m}$ and $E_L=113.5\mathrm{kV}/\mathrm{m}$. In the simulations the background is impedance matched to ZnO, hence, ${\rho }^B=\rho$ and $c_{33}^B=c_{33}$.

Figure 4

(a) $\mathcal{P}\mathcal{T}$ symmetric binary system where the gain (red) and loss (blue) constituents are biased in an bidirectional fashion. $h=4\mathrm{cm}$ and the passive nonpiezoelectric separator region, with ${\rho }^B=\rho$ and $c_{33}^B=c_{33}$, has thickness $s=0.5\mathrm{mm}$. (b) Scattering parameters spectrum is plotted around the exceptional point (dashed line) as a function of $E_g$, whereas $E_L$ is locked. Same parameters are taken as in Fig. 2. (c) $\alpha$ is given for the $\mathcal{P}\mathcal{T}$ symmetry structure in (a) along the stacking direction $z$ when sound propagates forward, backwards, and after a round trip.