Topological strong-field physics on sub-laser-cycle timescale


The sub-laser-cycle timescale of the electronic response to strong fields enables attosecond dynamical imaging in atoms, molecules and solids1,2,3,4, with optical tunnelling and high-harmonic generation the hallmarks of attosecond optical spectroscopy2,5,6,7. Topological insulators are intimately linked with electron dynamics, as manifested via the chiral edge currents8, but it is unclear if and how topology leaves its mark on optical tunnelling and sub-cycle electronic response. Here, we identify distinct bulk topological effects on directionality and timing of currents arising during electron injection into conduction bands. We show that electrons tunnel differently in trivial and topological insulators, 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 and helicities of emitted harmonics that record the phase diagram of the system. Our findings create new roadmaps in studies of topological systems, building on the ubiquitous properties of the sub-laser-cycle strong-field response—a unique mark of attosecond science.

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Fig. 1: The Haldane system in two phases.
Fig. 2: Helicity of harmonics in topological insulators.
Fig. 3: Topologically dependent time delays in harmonic emission.
Fig. 4: Topological signal in injected currents.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.


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R.E.F.S. and M.I. acknowledge support from Engineering and Physical Sciences Research Council/Defence Science and Technology Laboratory (EPSRC/DSTL) Multidisciplinary University Research Initiative (MURI) grant EP/N018680/1. R.E.F.S. acknowledges support from the European Research Council Starting Grant (ERC-2016- STG714870). Á.J.-G. and M.I. acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) Quantum Dynamics in Tailored Intense Fields (QUTIF) grant IV 152/6-1. O.S. acknowledges support from the DFG Schwerpunktprogramm 1840 ‘Quantum Dynamics in Tailored Intense Fields’ project SM 292/5-1, and Molecular Electron Dynamics Investigated by Intense Fields and Attosecond Pulses (MEDEA) project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 641789. B.A. received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 706538.

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R.E.F.S., Á.J.-G. and M.I. developed the idea. R.E.F.S. developed the numerical code, Á.J.-G. performed the simulations and analysed the data. R.E.F.S., Á.J.-G. and O.S. developed the analytical analysis I. B.A., O.S. and M.I. developed the analytical analysis II. Á.J.-G. and M.I. wrote the main part of the manuscript, which was discussed by all authors.

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Correspondence to R. E. F. Silva or Á. Jiménez-Galán.

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Silva, R.E.F., Jiménez-Galán, Á., Amorim, B. et al. Topological strong-field physics on sub-laser-cycle timescale. Nat. Photonics 13, 849–854 (2019).

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