# Synchronous oscillations locked on classical energy levels by two cooperating drives

## TL;DR

This paper demonstrates that classical macroscopic oscillators can exhibit discrete energy levels through a novel synchronization process driven by two fields, challenging the intuition that classical energies are continuous.

## Contribution

It introduces a new synchronization mechanism that creates classical energy levels in macroscopic oscillators driven by two strong fields, a phenomenon previously unknown.

## Key findings

- Classical energy levels can be formed via synchronization in macroscopic oscillators.
- The energy levels are stable and robust against noise.
- The phenomenon occurs beyond a certain drive intensity threshold.

## Abstract

It is intuitively imagined that the energy of a classical object always takes continues values and can hardly be confined to discrete ones like the energy levels of microscopic systems. Here, we demonstrate that such classical energy levels against intuition can be created through a previously unknown synchronization process for nonlinearly coupled macroscopic oscillators driven by two equally strong fields. Given the properly matched frequencies of the two drive fields, the amplitude and phase of an oscillator will be frozen on one of a series of determined trajectories like energy levels, and the phenomenon exists for whatever drive intensity beyond a threshold. Interestingly, the oscillator's motion can be highly sensitive to its initial condition but, unlike the aperiodicity in chaotic motion, it will nonetheless end up on such fixed energy levels. Upon reaching the stability, however, the oscillations on the energy levels are robust against noisy perturbation.