Oppenheim Award Talk: Space-time symmetry and nonreciprocal parametric resonance in mechanical systems

By modulating the springs of a spring-mass network, i.e., by making their stiffnesses time-dependent, one can engineer an ‘active metamaterial’. These systems harness the power of parametric resonance to achieve exotic signal processing capabilities. A key example is the emergence of non-reciprocity, where perturbations generate waves that propagate unidirectionally.

This talk will present a theoretical framework for predicting the mechanical response of such active metamaterials. We will uncover the fundamental conditions for signal amplification and non-reciprocal transport by analyzing the system's inherent symmetries. Drawing parallels with open quantum systems, we employ non-Hermitian and Floquet analysis to characterize the internal symmetries dictated by the symplectic nature of classical mechanics. Furthermore, we generalize Floquet theory to account for combined spatio-temporal periodicities (a.k.a. space-time symmetry), thereby classifying the system's external symmetries.

Our work provides a significant advance in the theoretical understanding of space-time modulated structures, with direct implications for the design of active metamaterials. The developed framework for analyzing space-time symmetric systems also holds broad applicability in quantum condensed matter physics.