Manifestations of flow topology in a quantum driven-dissipative system

Driven-dissipative bosonic systems exhibit a rich variety of nonequilibrium steady states (NESS) due to the interplay between coherent drives, interactions, and dissipation. In the semiclassical limit, the of phase-space dynamics governs the stability and structure of these dynamical phases. Topological transitions occur when the number, chirality, and connectivity of NESS changes, reflecting a global reorganization of the dynamical phase-space landscape. Here, we study the impact of these topological signatures in a quantum driven-dissipative Kerr oscillator. Employing a Lindblad master equation and quantum-trajectory methods, we reveal that quantum dynamics retain key topological features of the underlying semiclassical flows, with clear signatures accessible via quantum state tomography and linear response. In this manner, we predict flow-topological phases whose signatures arise from a reorganization of fluctuation modes and are not captured by Liouvillian-gap closing alone. We, thus, generalize the conventional criteria for diagnosing phase transitions. Our findings position phase-space flow topology as a powerful tool to identify and control robust quantum phases, enabling advances in error correction and sensing.