Fast wave transport in two-dimensional $\mathcal{PT}$-symmetric lattices

Abstract

We present a theoretical investigation of wave dynamics in two-dimensional non-Hermitian $\mathcal{PT}$-symmetric lattices, where onsite, as well as inter-site control couplings are employed. Our analysis shows that these couplings can be tuned to achieve a direction-sensitive group velocity enhancement beyond what is possible in the uncontrolled (Hermitian) counterpart, while ensuring that the wave packet evolution remains bounded and dynamically stable. We derive a dedicated relation between the control parameters, providing a systematic condition under which stability is guaranteed. We then study the topological properties of the non-Hermitian system at hand, and use an experimental-ready topoelecric metamaterial platform to demonstrate the non-Hermitian couplings realization, and the resulting wave dynamics. This framework paves the way to designing stable and fast wave transport in planar non-Hermitian media.

Figures (5)

• Figure 1: (a) The lattice model schematic. (b) The first Brillouin zone of the Hermitian (gray hexagon) and the non-Hermitian $\mathcal{PT}$-symmetric lattice (red parallelogram). (c)-(h) Dispersion surfaces: the nominal Hermitian case for $\eta=1$ and $\gamma=0$ (c), the unbalanced non-Hermitian case, real and imaginary ($\Omega=\Omega_R+\mathrm{i}\Omega_I$), with six triangular exceptional rings for $\eta=1$ and $\gamma=0.6$ (d), the marginal Hermitian case for $\eta=2$ and $\gamma=0$ (e), the gapped Hermitian case for $\eta=2.5$ and $\gamma=0$ (f), the unbalanced non-Hermitian case, real and imaginary, with four elliptic exceptional rings for $\eta=2.5$ and $\gamma=0.6$ (g), and the non-Hermitian $\mathcal{PT}$-symmetric case for $\eta=2.5$ and $\gamma=0.236$ (h). For (c), (e), (f), and (h) the imaginary spectrum is zero.
• Figure 2: Group velocity increase analysis. (a), (b) The dispersion relations and the associated group velocities along the $y$ axis, for the $\mathcal{PT}$-symmetric systems with $\gamma_{pt}=0$$(\eta=2)$, $\gamma_{pt}=1.74$$(\eta=7)$, and $\gamma_{pt}=2.90$$(\eta=12)$, and the corresponding $\gamma$ obtained from Eq. \ref{['eq:gamma_eta_balance']}, on top of the nominal Hermitian system with $\gamma=0$ and $\eta=1$. The red crosses indicate numerical data obtained from the spacetime simulations of Fig. \ref{['fig:simulations']}. (c), (d) The same along the $x$ axis. The black curves in panels (a) and (b) respectively represent $\Omega_2$ and $v_{gy2}$ from Eq. \ref{['eq:vgy']}.
• Figure 3: (a) Berry curvature distribution $F(k_x,k_y)$ according to Eq. \ref{['eq:topology_eqs']}, inside and outside the exceptional surface, corresponding to the dispersions shown in Figs. \ref{['fig:dispersion']}(f) and \ref{['fig:dispersion']}(g) in panel (b).
• Figure 4: Time domain numerical simulations and group velocity increase demonstration for the generic lattice model in Fig. \ref{['fig:dispersion']}(a). (a)-(d) Time domain wave packet propagation in the $y$ direction of the $\mathcal{PT}$-symmetric lattice (normalized to $\Omega_0=1$). The initial distribution (a) corresponding to $k=0.6\pi$, and the final response for the combinations $\gamma_{pt}=0$$(\eta=2)$, $\gamma_{pt}=1.74$$(\eta=7)$, and $\gamma_{pt}=2.90$$(\eta=12)$, depicted in (b), (c), and (d), respectively.
• Figure 5: Topoelectrical metamaterial realization. (a) The unit cell schematics. (b) An operational amplifier in a negative impedance converting setup implementing the onsite gain element. (c) Voltage inputs location across a metamaterial of 12X10 hexagonal cells, depicted by Gaussian modulation, according to Eq. \ref{['eq:V_source']} for $k=0.6\pi$. The circles represent the nodes $A$ and $B$. The color map in this panel is not to scale. (d) Snapshot of the space-time response of the non-Hermitian $\mathcal{PT}$-symmetric metamaterial for $\eta=7$ and $\gamma_{pt}$ according to Eq. \ref{['eq:gamma_eta_balance']}, for nominal values of the electric circuit elements $C_0=150$$\mathrm{nF}$ and $L_0=220$$\mathrm{\mu H}$, and no resistance other than the controlled $R_\gamma$. (e) The same as in panel (d), but with a disorder in $C_0$, and in $R_0=23$ Ohm, which is an inherent element resistance, according to the diagram in panel (f). (f) Disorder in the values of capacitance $C_0$ and $R_0$, normalized by the nominal values. The color bar is relevant to panels (d) and (e), and indicates Volts.