Topological phase transitions in nonlinear driven-dissipative resonators
Prof. Oded Zilberberg (Universität Konstanz)

Topological classification of matter has become crucial for understanding (meta-)materials, with associated quantized bulk responses and robust topological boundary effects [1]. Topological phenomena have also recently garnered significant interest in nonlinear systems [2]. In particular, weak nonlinearities can result in parametric gain, leading to “non-Hermitian” metamaterials and the associated topological classification of open systems [3]. Here, we venture into this expanding frontier using an approach that moves away from quasilinear approximations around the closed system classification. We harness instead the topology of structural stability of vector flows, and thus propose a new topological graph invariant to characterize nonlinear out-of-equilibrium dynamical systems via their equations of motion. We exemplify our approach on the ubiquitous model of a dissipative bosonic Kerr cavity, subject both to one- and two-photon drives. Using our classification, we can identify the topological origin of phase transitions in the system, as well as explain the robustness of a multicritical point in the phase diagram. We, furthermore, identify that the invariant distinguishes population inversion transitions in the system in similitude to a Z2 index. Our approach is readily extendable to coupled nonlinear cavities by considering a tensorial graph index.

References
[1] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Rev. Mod. Phys. 91, 015006 (2019).
[2] A. Szameit, and M. C. Rechtsman, Nat. Phys. (2024).
[3] K. Ding, C. Fang, and G. Ma, Nat. Rev. Phys. 4, 745 (2022).