Electric-circuit simulation of the Schrödinger equation and non-Hermitian quantum walks

Abstract: Recent progress has witnessed that various topological physics can be simulated by electric circuits under alternating current. However, it is still a nontrivial problem if it is possible to simulate the dynamics subject to the Schr\"{o}dinger equation based on electric circuits. In this work, we reformulate the Kirchhoff law in one dimension in the form of the Schr\"{o}dinger equation. As a typical example, we investigate quantum walks in $LC$ circuits. We also investigate how quantum walks are different in topological and trivial phases by simulating the Su-Schrieffer-Heeger model in electric circuits. We then generalize them to include dissipation and nonreciprocity by introducing resistors, which produce non-Hermitian effects. We point out that the time evolution of one-dimensional quantum walks is exactly solvable with the use of the generating function made of the Bessel functions.