The topology of two-dimensional materials traditionally manifests itself through the quantization of the Hall conductance, which is revealed in transport measurements [1][2][3]. Recently, it was predicted that topology can also give rise to a quantized spectroscopic response upon subjecting a Chern insulator to a circular drive: Comparing the frequency-integrated depletion rates associated with drives of opposite orientations leads to a quantized response dictated by the topological Chern number of the populated Bloch band [4, 5]. Here we experimentally demonstrate this intriguing topological effect for the first time, using ultracold fermionic atoms in topological Floquet bands. In addition, our depletion-rate measurements also provide a first experimental estimation of the Wannier-spread functional, a fundamental geometric property of Bloch bands [6, 7]. Our results establish topological spectroscopic responses as a versatile probe, which could be applied to access the geometry and topology of many-body quantum systems, such as fractional Chern insulators [8].The discovery of topological states of matter has revolutionized band theory [1-3] by revealing the importance of the Bloch eigenstates and their geometric and topological properties, as captured by the Berry curvature [9] and topological Chern numbers [1-3]. These geometric band properties are associated with the adiabatic motion within a given Bloch band [9], and lead to striking effects such as the anomalous quantum Hall effect [10]. The topological invariant associated with Bloch bands (e.g. the Chern number) cannot be identified through the simple observation of the bulk energy bands, which can be accessed by spectroscopy. However, by evaluating not only the excitation frequencies, but also the excitation strengths [11], geometrical and topological properties become directly accessible via spectroscopy and lead to new topological phenomena [4, 5, 7]. In particular, subjecting a Chern insulator to a circular drive, and comparing the frequencyintegrated depletion rates Γ int ± = ∞ 0 Γ ± (ω)dω resulting from drives of opposite orientation (or chirality, ±), yields a quantized response [4]which is dictated by the Chern number C of the populated band. Here A cell is the area of the unit cell [12], E sp and quantized transport quantized depletion b a Energy Quas imom entum sp sp FIG. 1. Quantized responses in topological matter. a, In the quantum (anomalous) Hall effect, the Hall conductance relating the transverse current density j ⊥ to the applied electric field E follows a quantization law dictated by the Chern number C of the populated Bloch band [1, 9]. b, Our experiment reveals a distinct quantization law [4], which involves the depletion rates Γ ± of a Bloch band (inset) upon circular shaking, where (±) refer to the drive orientation. The differential integrated rate ∆Γ int ± also reveals the Chern number C, but is quadratic with respect to the driving strength E sp , reflecting its dissipative (interband) nature.ω are the strength and frequency of t...
