We introduce a linear-scaling real-space methodology to compute time-resolved electrical responses of materials driven far from equilibrium, with energy relaxation and disorder treated on an equal footing. Applying this approach to gapped monolayer and AB-stacked (Bernal) bilayer graphene, when driven by a circularly polarized optical pulse, we observe the generation (suppression) of a finite Hall conductivity when the system is trivial (topological). This Hall signal oscillates during optical driving and remains sizable after the light is switched off before relaxing toward equilibrium. Remarkably, this dynamical Hall response is robust in the presence of realistic descriptions of disorder, suggesting that disorder and relaxation dynamics can be leveraged as design parameters rather than as limitations. More broadly, our new methodology enables the investigation of electrical responses in driven, complex disordered quantum materials and highlights how engineered energy-transfer pathways can enable ultrafast optoelectronic functionality.

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• Optoelectronics
• Photoinduced effect
• Quantum anomalous Hall effect
• Quantum transport
• Topological Hall effect
• Topological materials
• Transport phenomena
• Bilayer graphene
• Dirac semimetal
• Graphene
• Density matrix methods
• Methods in transport
• Numerical techniques
• Tight-binding model