Floquet topological system based on frequency-modulated classical coupled harmonic oscillators

TL;DR

This paper proposes a method to observe topological effects in classical coupled harmonic oscillators with frequency modulation, demonstrating the emergence of topologically protected states and band structures analogous to quantum Hall systems.

Contribution

It introduces a classical system model that simulates topological phenomena using Floquet theory and frequency modulation, connecting classical oscillators to quantum topological models.

Findings

Observation of topologically protected chiral edge states.

Mapping of the system to a Harper-Hofstadter model.

Extraction of Chern numbers from steady-state oscillations.

Abstract

We theoretically propose how to observe topological effects in a generic classical system of coupled harmonic oscillators, such as classical pendula or lumped-element electric circuits, whose oscillation frequency is modulated fast in time. Making use of Floquet theory in the high frequency limit, we identify a regime in which the system is accurately described by a Harper-Hofstadter model where the synthetic magnetic field can be externally tuned via the phase of the frequency-modulation of the different oscillators. We illustrate how the topologically-protected chiral edge states, as well as the Hofstadter butterfly of bulk bands, can be observed in the driven-dissipative steady state under a monochromatic drive. In analogy with the integer quantum Hall effect, we show how the topological Chern numbers of the bands can be extracted from the mean transverse shift of the steady-state oscillation amplitude distribution. Finally we discuss the regime where the analogy with the Harper-Hofstadter model breaks down.