Ghost Bifurcations in Coupled Parametric Oscillators

Nonlinear parametric resonator networks are promising candidates as Ising machines for annealing and optimization. An important question concerns the validity of this Ising analogy for this intrinsically nonlinear many-body systems. Here we address this question by considering a network of two parametric resonators. Even in the weak coupling regime, we find physics beyond the Ising paradigm with the occurrence of ghost bifurcations involving only unstable solutions. These bifurcations become highly relevant in the presence of noise as they determine the transition paths and the switching rates between stable solutions. We verify our findings with a simple table-top experiment involving capacitively coupled electrical resonators. The presented results demonstrate that the dynamics of these networks are not fully captured by the Ising analogy. Our work emphasizes the need for further exploration of many body effects beyond the Ising analogy for applications as optimization devices.