In this talk, I will discuss how the
geometry of quantum states can be revealed using a universal scheme based on excitation-rate
measurements upon periodic driving [1,2,3]. When applied to Chern insulators or
Landau levels, this approach leads to a quantized circular dichroism phenomenon
[1,2], which can be interpreted as the dissipative counterpart of the quantum
Hall effect. Besides, we will present protocols allowing for the experimental
detection of the quantum metric tensor [3,4], which could be applied to detect
new forms of monopoles in higher dimensions . Finally, I will report on the
first experimental observation of quantized circular dichroism in an ultracold
Fermi gas , and describe how this scheme can be applied to reveal the
topological order of fractional Chern insulators .
 D. T. Tran, A. Dauphin, A. G.
Grushin, P. Zoller and N. Goldman, Science Advances 3, e1701207 (2017).
 D. T. Tran, N. R. Cooper, and N. Goldman, Phys. Rev. A 97, 061602(R) (2018).
 T. Ozawa and N. Goldman, Phys. Rev. B 97, 201117(R) (2018).
 M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, M. B. Plenio, F. Jelezko, T. Ozawa, N. Goldman, S. Zhang and J. Cai, arXiv:1811.12840 (2018)
 G. Palumbo and N. Goldman, Phys. Rev. Lett. 121, 170401 (2018).
 L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fläschner, K. Sengstock, N. Goldman and C. Weitenberg, arXiv:1805.11077 (2018).
 C. Repellin and N. Goldman, arXiv:1811.08523 (2018).