Sub-laser cycle time scale of electronic response to strong laser fields enables attosecond dynamical imaging in atoms, molecules and solids. Optical tunneling and high harmonic generation are the hallmarks of attosecond imaging in optical domain, including imaging of phase transitions in solids. Topological phase transition yields a state of matter intimately linked with electron dynamics, as manifested via the chiral edge currents in topological insulators. Does topological state of matter leave its mark on optical tunneling and sub-cycle electronic response? We identify distinct topological effects on the directionality and the attosecond timing of currents arising during electron injection into conduction bands. We show that electrons tunnel across the band gap differently in trivial and topological phases, for the same band structure, and identify the key role of the Berry curvature in this process. These effects map onto topologically-dependent attosecond delays in high harmonic emission and the helicities of the emitted harmonics, which can record the phase diagram of the system and its topological invariants. Thus, the topological state of the system controls its attosecond, highly non-equilibrium electronic response to strong low-frequency laser fields, in bulk. Our findings create new roadmaps in studies of topological systems, building on ubiquitous properties of sub-laser cycle strong field response - a unique mark of attosecond science.
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