Observation of cnoidal wave localization in nonlinear topolectric circuits

We observe a localized cnoidal (LCn) state in an electric circuit network. Its formation derives from the interplay of nonlinearity and the topology inherent to a Su-Schrieffer-Heeger (SSH) chain of inductors. Varicap diodes act as voltage-dependent capacitors, and create a nonlinear on-site potential. For a sinusoidal voltage excitation around midgap frequency, we show that the voltage response in the nonlinear SSH circuit follows the Korteweg-de Vries equation. The topological SSH boundary state, which relates to a midgap impedance peak in the linearized limit is distorted into the LCn state in the nonlinear regime, where the cnoidal eccentricity decreases from edge to bulk.

Figure 1

(a) Schematic of the nSSH circuit. Alternating inductances $L_1$ and $L_2$ connect the voltage nodes, reverse-biased varicap diodes to ground act as nonlinear capacitances and the parallel capacitor $C_{\text{p}}$ takes the parasitic influence of the measurement setup into account [19]. $\tilde{C}\gg C\left(V_{\text{offset}}\right)$ blocks the dc voltage offset. The gray rectangle depicts one unit cell. (b) Measurement of the voltage dependence of the four Siemens BB512 varicap diodes in parallel configuration and $C_{\text{p}}$. (c) One unit cell of the nSSH circuit on the PCB with [1] inductors of nominal values $L_{1,\text{nom}}=330\mu \mathrm{H}$ and $L_{2,\text{nom}}=180\mu \mathrm{H}$, and [2] varicap diodes. (d) (left) Theoretical (blue curve) and measured (gray crosses) dispersion relation for PBC in the linear limit at the operating point $V_{\text{offset}}=2.5\mathrm{V}$. (right) Small signal impedance analysis between nodes 1A, 1B, 2A, 3A and ground for OBC. An impedance peak at the midgap frequency $f_0=430\mathrm{k}\mathrm{Hz}$ indicates the localized SSH boundary state.

Figure 2

ac measurement of the LCn state at $V_{\text{offset}}=2.5\mathrm{V}$. (top) Sinusoidal input signal with frequency $f_0=430\mathrm{k}\mathrm{Hz}$ and amplitude $A_0=2.5\mathrm{V}$ fed into node 1A. (left) Steady-state voltage response at nodes 2A, 3A, and 4A starting at $t\approx 18\mu \mathrm{s}$. The sinusoidal input is deformed into the LCn state, where amplitude and eccentricity decrease from edge to bulk. (right) Phase space plot of one period of the voltage signal [indicated by the gray area in (a)], visualizing its deformation.

Figure 3

(a) Transformation of a segment of the nSSH circuit to a nonlinear LC resonator by treating the sublattice nodes B as virtual ground, due to their negligible amplitude in the LCn state. (b) Discrete Fourier transformation of the measured voltage response in the LCn state at first three sublattice A nodes (color encoded), normalized with respect to the peak to peak voltage $V_{\text{pp}}$. The crosses mark the evaluated values, guided by solid lines and with triangles at multiples of the base frequency $f_0$. Due to nonlinear effects, higher harmonics of the fundamental frequency $f_0=430\mathrm{k}\mathrm{Hz}$ are excited. The inset compares the Fourier components for different excitation amplitudes $A_0$ (marker encoded), and nodes 2A, 3A, and 4A, with the analytical solution in Eq. (3), depicted as gray lines. (c) Logarithmic plot of the normalized RMS voltage for the linear and nonlinear case with respective linear fits. The ratio $\mathrm{\Delta }$ between subsequent RMS values together with the theoretically expected value in gray is shown below. (d) Logarithmic plot of the eccentricity $m$ for different excitation amplitudes $A_0$ with linear fits.