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Dynamical phase transitions across slow and fast regimes in a two-tone driven Duffing resonator

Soumya S. Kumar1,*, Javier del Pino1,2, Letizia Catalini3,4,5, Alexander Eichler3,4, and Oded Zilberberg1,†

  • 1Department of Physics, University of Konstanz, 78464 Konstanz, Germany
  • 2Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain
  • 3Laboratory for Solid State Physics, ETH Zürich, 8093 Zürich, Switzerland
  • 4Quantum Center, ETH Zürich, 8093 Zürich, Switzerland
  • 5Center for Nanophotonics, AMOLF, 1098XG Amsterdam, The Netherlands

  • *Contact author: soumya.kumar@uni-konstanz.de
  • †Contact author: oded.zilberberg@uni-konstanz.de
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Phys. Rev. Research 8, 023298 – Published 15 June, 2026

DOI: https://doi.org/10.1103/d3b6-lnw6

Abstract

The response of nonlinear resonators to multifrequency driving reveals rich dynamics beyond conventional single-tone theory. We study a Duffing resonator under bichromatic excitation and identify a competition between the two drives, governed by their detuning and relative amplitudes. In the slow-beating regime, where the tones are closely spaced, the secondary tone acts as a modulation that induces dynamical phase transitions between coexisting stationary states. We introduce the cycle-averaged amplitude as an order parameter and map the resulting phase diagram as a function of the drive detuning and amplitude ratio, capturing the pronounced asymmetry observed for blue versus red detuning between the drives in experiment. We link the onset of these transitions to the resonance properties around the nonlinear stationary mode of the system. We apply a two-tone ansatz to this system, for the first time, to provide a comprehensive map of stationary states. Our results provide a framework for controlling driven nonlinear systems, enabling state manipulation, and sensing in nanomechanical, optical, and superconducting circuit platforms.

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Phys. Rev. Research 8, 023298– Published 15 June, 2026
  • Received 17 February 2026
  • Accepted 29 April 2026
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