# Attosecond spectral singularities in solid-state high-harmonic generation

## Abstract

Strong-field-driven electric currents in condensed-matter systems are opening new frontiers in petahertz electronics. In this regime, new challenges are arising as the roles of band structure and coherent electron–hole dynamics have yet to be resolved. Here, by using high-harmonic generation spectroscopy, we reveal the underlying attosecond dynamics that dictates the temporal evolution of carriers in multi-band solid-state systems. We demonstrate that when the electron–hole relative velocity approaches zero, enhanced constructive interference leads to the appearance of spectral caustics in the high-harmonic generation spectrum. We introduce the role of the dynamical joint density of states and identify its mapping into the spectrum, which exhibits singularities at the spectral caustics. By studying these singularities, we probe the structure of multiple unpopulated high conduction bands.

## Access options

from\$8.99

All prices are NET prices.

## Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

1. 1.

Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2011).

2. 2.

Vampa, G. et al. All-optical reconstruction of crystal band structure. Phys. Rev. Lett. 115, 193603 (2015).

3. 3.

Luu, T. T. et al. Extreme ultraviolet high-harmonic spectroscopy of solids. Nature 521, 498–502 (2015).

4. 4.

Garg, M. et al. Multi-petahertz electronic metrology. Nature 538, 359–363 (2016).

5. 5.

Liu, H. et al. High-harmonic generation from an atomically thin semiconductor. Nat. Phys. 13, 262–265 (2017).

6. 6.

Yoshikawa, N., Tamaya, T. & Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science 356, 736–738 (2017).

7. 7.

Hohenleutner, M. et al. Real-time observation of interfering crystal electrons in high-harmonic generation. Nature 523, 572–575 (2015).

8. 8.

Ndabashimiye, G. et al. Solid-state harmonics beyond the atomic limit. Nature 534, 520–523 (2016).

9. 9.

You, Y. S. et al. High-harmonic generation in amorphous solids. Nat. Commun. 8, 724 (2017).

10. 10.

You, Y. S. et al. Laser waveform control of extreme ultraviolet high harmonics from solids. Opt. Lett. 42, 1816–1819 (2017).

11. 11.

Hawkins, P. G., Ivanov, M. Y. & Yakovlev, V. S. Effect of multiple conduction bands on high-harmonic emission from dielectrics. Phys. Rev. A 91, 013405 (2015).

12. 12.

Silva, R., Blinov, I. V., Rubtsov, A. N., Smirnova, O. & Ivanov, M. High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems. Nat. Photon. 12, 266–270 (2018).

13. 13.

Vampa, G., McDonald, C., Orlando, G., Corkum, P. & Brabec, T. Semiclassical analysis of high harmonic generation in bulk crystals. Phys. Rev. B 91, 064302 (2015).

14. 14.

Ashcroft, N. W. & Mermin, N. D. in Solid State Physics 144–145 (Holt, Rinehart and Winston, 1976).

15. 15.

Van Hove, L. The occurrence of singularities in the elastic frequency distribution of a crystal. Phys. Rev. 89, 1189–1193 (1953).

16. 16.

Vampa, G. et al. Linking high harmonics from gases and solids. Nature 522, 462–464 (2015).

17. 17.

Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations. Nat. Photon. 8, 119–123 (2014).

18. 18.

Kemper, A., Moritz, B., Freericks, J. & Devereaux, T. Theoretical description of high-order harmonic generation in solids. New J. Phys. 15, 023003 (2013).

19. 19.

Higuchi, T., Stockman, M. I. & Hommelhoff, P. Strong-field perspective on high-harmonic radiation from bulk solids. Phys. Rev. Lett. 113, 213901 (2014).

20. 20.

Vampa, G. et al. Theoretical analysis of high-harmonic generation in solids. Phys. Rev. Lett. 113, 073901 (2014).

21. 21.

Golde, D., Meier, T. & Koch, S. High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations. Phys. Rev. B 77, 075330 (2008).

22. 22.

Tancogne-Dejean, N., Mücke, O. D., Kärtner, F. X. & Rubio, A. Impact of the electronic band structure in high-harmonic generation spectra of solids. Phys. Rev. Lett. 118, 087403 (2017).

23. 23.

Raz, O., Pedatzur, O., Bruner, B. D. & Dudovich, N. Spectral caustics in attosecond science. Nat. Photon. 6, 170–173 (2012).

24. 24.

Upstill, C. & Berry, M. IV Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Optics 18, 257–346 (1980).

25. 25.

You, Y. S., Reis, D. A. & Ghimire, S. Anisotropic high-harmonic generation in bulk crystals. Nat. Phys. 13, 345–349 (2017).

26. 26.

Wu, M. et al. Orientation dependence of temporal and spectral properties of high-order harmonics in solids. Phys. Rev. A 96, 063412 (2017).

27. 27.

Bruner, B. D. et al. Multidimensional high harmonic spectroscopy. J. Phys. B 48, 174006 (2015).

28. 28.

Li, J.-B. et al. Enhancement of the second plateau in solid high-order harmonic spectra by the two-color fields. Opt. Express 25, 18603–18613 (2017).

29. 29.

Ehrenreich, H. & Philipp, H. Optical properties of Ag and Cu. Phys. Rev. 128, 1622–1629 (1962).

30. 30.

Roessler, D. & Walker, W. Electronic spectrum and ultraviolet optical properties of crystalline MgO. Phys. Rev. 159, 733–738 (1967).

31. 31.

Faccialà, D. et al. Probe of multielectron dynamics in xenon by caustics in high-order harmonic generation. Phys. Rev. Lett. 117, 093902 (2016).

32. 32.

Silva, R., Martín, F. & Ivanov, M. High harmonic generation in crystals using maximally localized Wannier functions. Phys. Rev. B 100, 195201 (2019).

33. 33.

Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project for quantum simulations of materials. J. Phys. 21, 395502 (2009).

34. 34.

Levine, Z. H. & Allan, D. C. Linear optical response in silicon and germanium including self-energy effects. Phys. Rev. Lett. 63, 1719–1722 (1989).

35. 35.

Mostofi, A. A. et al. An updated version of Wannier90: a tool for obtaining maximally-localised Wannier functions. Comput. Phys. Commun. 185, 2309–2310 (2014).

## Acknowledgements

N.D. is the incumbent of the Robin Chemers Neustein Professorial Chair. N.D. acknowledges financial support from the Minerva Foundation, the Israeli Science Foundation, the Crown Center of Photonics and the European Research Council. M.K. acknowledges financial support from the Minerva Foundation and the Koshland Foundation.

## Author information

N.D., A.J.U. and G.O. conceived and planned the experiments. A.J.U., G.O., V.B., T.A.-P. and B.D.B. performed the measurements. A.J.U. and G.O. analysed the data. A.J.U., G.O., A.J.-G., C.M., R.E.F.S., N.D.K., M.K., A.N.R., O.S., M.I., B.Y. and T.B. developed the theoretical models and analysis. All authors discussed the results and contributed to writing the manuscript.

Correspondence to Nirit Dudovich.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–7 and Discussion.

## Rights and permissions

Reprints and Permissions