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...

Machine learning techniques such as artificial neural networks are currently revolutionizing many technological areas and have also proven successful in quantum physics applications. Here we employ an artificial neural network and deep learning techniques to identify quantum phase transitions from single-shot experimental momentum-space density images of ultracold quantum gases and obtain results, which were not feasible with conventional methods. We map out the complete two-dimensional topological phase diagram of the Haldane model and provide an accurate characterization of the superfluid-to-Mott-insulator transition in an inhomogeneous Bose-Hubbard system. Our work points the way to unravel complex phase diagrams of general experimental systems, where the Hamiltonian and the order parameters might not be known.Ultracold quantum gases have established as a formidable experimental platform to study paradigmatic quantum many-body systems in a well-controlled environment (1, 2). Important break-throughs include the realization of paradigmatic condensed matter models such as the Mott insulator transition or topological quantum matter. While these systems offer complementary observables to solid state systems, finding proper observables for quantum phases remains a key challenge, in particular in exotic systems such as non-local topological order of many-body localization. Here we explore a new approach building on modern machine learning techniques (3). Inspired by the success of convolutional neural networks in image recognition, we feed such networks with single images of momentum-space density, which are a standard experimental output of quantum gas experiments. We train it on large data sets of labelled images taken far away from the phase transition and apply the trained network to test data across the phase transition. The network is able to identify the correct position of the phase transition in parameter space from single experimental images. This is crucial advance for optimizing parameters, because the phase can now be determined from single images for direct decisions in the laboratory, and points towards future fully automated quantum simulators. We expect these techniques to be valuable also for in-situ snapshots as captured by quantum gas microscopes (4, 5). Similar approaches were previously applied to numerical Monte Carlo simulations of various physical models (6)(7)(8)(9)(10)(11)(12)(13). Neural networks are also opening new avenues in other areas of quantum physics, such as the representation of quantum many-body states (14,15) or the optimization of complex systems (16)(17)(18).We demonstrate the power of artificial neural networks on two physical examples, namely the topological phase transition in the Haldane model and the superfluid-to-Mott-insulator transition in the Bose-Hubbard model, both realized for cold atoms in optical lattices. We show that we can perform tasks, which were not possible with conventional techniques, such as the determination of non-local topological order from a sing...

We report an experimental investigation of the facilitated excitation dynamics in off-resonantly driven Rydberg gases by separating the initial off-resonant excitation phase from the facilitation phase, in which successive facilitation events lead to excitation avalanches. We achieve this by creating a controlled number of initial seed excitations. Greater insight into the avalanche mechanism is obtained from an analysis of the full counting distributions. We also present simple mathematical models and numerical simulations of the excitation avalanches that agree well with our experimental results.

Imaging is central to gaining microscopic insight into physical systems, and new microscopy methods have always led to the discovery of new phenomena and a deeper understanding of them. Ultracold atoms in optical lattices provide a quantum simulation platform, featuring a variety of advanced detection tools including direct optical imaging while pinning the atoms in the lattice1,2. However, this approach suffers from the diffraction limit, high optical density and small depth of focus, limiting it to two-dimensional (2D) systems. Here we introduce an imaging approach where matter wave optics magnifies the density distribution before optical imaging, allowing 2D sub-lattice-spacing resolution in three-dimensional (3D) systems. By combining the site-resolved imaging with magnetic resonance techniques for local addressing of individual lattice sites, we demonstrate full accessibility to 2D local information and manipulation in 3D systems. We employ the high-resolution images for precision thermodynamics of Bose–Einstein condensates in optical lattices as well as studies of thermalization dynamics driven by thermal hopping. The sub-lattice resolution is demonstrated via quench dynamics within the lattice sites. The method opens the path for spatially resolved studies of new quantum many-body regimes, including exotic lattice geometries or sub-wavelength lattices3–6, and paves the way for single-atom-resolved imaging of atomic species, where efficient laser cooling or deep optical traps are not available, but which substantially enrich the toolbox of quantum simulation of many-body systems.