Topologically Enforced Bifurcations in Superconducting Circuits

The relationship of topological insulators and superconductors and the field of nonlinear dynamics is widely unexplored. To address this subject, we adopt the linear coupling geometry of the Su-Schrieffer-Heeger model, a paradigmatic example for a topological insulator, and render it nonlinearly in the context of superconducting circuits. As a consequence, the system exhibits topologically enforced bifurcations as a function of the topological control parameter, which finally gives rise to chaotic dynamics, separating phases that exhibit clear topological features.

Figure 1

(a) Sketch of the system. The coupling geometry resembles the one of the SSH model with alternating coupling strength. The system is subjected to external driving and dissipation (nonsketched). (b) Superconducting circuit giving rise to Eq. (1). (c) Nonlinear coupling potential of the nodes. (d) Effective coupling potential appearing due to a Fourier analysis of Eq. (1). (e) Topologically trivial limiting case $\alpha =t_0$, where the system consists of uncoupled dimers. (f) Topologically nontrivial limiting case $\alpha =-t_0$, where two uncoupled nodes at the boundary exist.

Figure 2

(a) Current $\mathrm{\Delta }{I}_1=I_1(t)-I_{\text{bulk}}(t)$ in units of $t_0/\mathcal{R}$ flowing from node $n=1$ through the resistance $R$ to the ground in the linear system $\delta =0$. Parameters are $\alpha =-0.4t_0$ (nontrivial, solid line) and $\alpha =0.2t_0$ (trivial, dashed line), $\mathcal{R}=0.02t_0\mathrm{\Omega }$, $I_{\text{ac}}=4t_0$, and $N=200$. $T_{\text{drive}}=2\pi /\mathrm{\Omega }$ denotes the driving period, where $\mathrm{\Omega }=\sqrt{2t_0}$. (b) As in (a) but for $\delta =0.95$. In this case, the system exhibits chaos. (c) $P_n(\mathrm{\Omega })$ in units of $(t_0/\mathcal{R}{)}^2$ in Eq. (5) for the time evolutions in (a) and (b). Throughout this Letter, we take $t_{\min }=500T_{\text{drive}}$ and $\tau =100T_{\text{drive}}$ to evaluate Eq. (5). (d) Dependence of $P_1(\mathrm{\Omega })$ and $P_{1,\text{tot}}$ in Eq. (6) as a function of $\delta$.

Figure 3

(a) Phase diagram for the order parameter $\chi$ defined in Eq. (7). Parameters are as in Fig. 2. Black lines depict the phase boundary obtained by the generalized force functionals Eq. (10). (b) Phase diagram of the topological order parameter $\mathrm{\Delta }$. (c) Level sets $G_1=0$ (solid line) and $G_2=0$ (dashed line) of Eq. (10) for fixed, but optimized $a_3$ for $I_{\text{ac}}=4t_0$. The corresponding parameters are marked in (a) by colored dots. The vanishing of the stable attractor at $\alpha \approx 0.25t_0$ triggers the chaotic dynamics observed in the central region of the phase diagram.