Topological properties lie at the heart of many fascinating phenomena in solid state systems such as quantum Hall systems or Chern insulators. The topology can be captured by the distribution of Berry curvature, which describes the geometry of the eigenstates across the Brillouin zone. Employing fermionic ultracold atoms in a hexagonal optical lattice, we generate topological bands using resonant driving and show a full momentumresolved measurement of the ensuing Berry curvature. Our results pave the way to explore intriguing phases of matter with interactions in topological band structures.Topology is a fundamental concept for our understanding of many fascinating systems that have recently attracted a lot of interest, such as topological superconductors or topological insulators, which conduct only at their edges [1]. The topology of the bulk band is quantified by the Berry curvature [2] and the integral over the full Brillouin zone is a topological invariant, called the Chern number. According to the bulk boundary correspondence principle, the Chern number determines the number of chiral conducting edge states [1]. While in a variety of lattice systems ranging from solid state systems to photonic waveguides and even coupled mechanical pendula, edge states have been directly observed [3][4][5][6][7], the underlying Berry curvature as the central measure of topology is not easily accessible. In recent years, ultracold atoms in optical lattices have emerged as a platform to study topological band structures [8,9] and these systems have seen considerable experimental and theoretical progress. Whereas in condensed matter systems, topological properties arise due to external magnetic fields or intrinsic spin-orbit coupling of the material, they can in cold atom systems be engineered by periodic driving analogous to illuminated graphene [10]. Interestingly, the resulting Floquet system can have totally new topological properties [11]. The driving can, for example, be realized by lattice shaking [12][13][14][15][16] or Raman coupling [17][18][19] with high precision control in a large parameter space. In particular, the driving can break time-reversal symmetry [13,14,16] and thus allows for engineering non-trivial topology [16,18]. In quantum gas experiments, topolog- ical properties have been probed via the Hall drift of accelerated wave packets [16,18], via an interferometer in momentum space [20,21]

Recently, the identification of non-equilibrium signatures of topology in the dynamics of such systems has attracted particular attention [3][4][5][6] . Here, we experimentally study the dynamical evolution of the wavefunction using time-and momentum-resolved full state tomography for spin-polarized fermionic atoms in driven optical lattices 7 . We observe the appearance, movement and annihilation of dynamical vortices in momentum space after sudden quenches close to the topological phase transition. These dynamical vortices can be interpreted as dynamical Fisher zeros of the Loschmidt amplitude 8 , which signal a so-called dynamical phase transition 9,10 . Our results pave the way to a deeper understanding of the connection between topological phases and non-equilibrium dynamics.The discovery of topological matter has revolutionized our understanding of band theory: not only are the dispersions of the energy bands important, but so is the geometry of the corresponding eigenstates 1 . The non-local nature of the topological invariants characterizing such phases goes beyond the Landau paradigm of local order parameters and leads to topological protection, for example, against disorder. Ultracold quantum gases in optical lattices allow for controlled studies of archetypal topological models [11][12][13][14] . In addition, compared with, for example condensed-matter systems, they also allow for detailed studies of the relation between dynamics and topology as the timescales are experimentally easier to access. Dynamical studies of driven systems have recently attracted attention in terms of their high T c superconductivity 15 . A particular challenge is to identify non-equilibrium signatures of topology in the dynamics of highly excited states 3,4,16 . Here, we observe the time evolution of the wavefunction after a sudden quench in a Haldanelike model and find dynamical vortices as a signature of the topological nature of the underlying ground state.In the experiments described here, the state tomography method allows mapping of the full quantum-mechanical wavefunction of non-interacting ultracold fermionic quantum gases in an optical lattice for any time after a sudden quench of the system close to or into a Chern insulating phase. As a key result, we identify in an intense series of measurements the appearance, movement and annihilation In the initial system, tunnelling J AB between the A and B sites is suppressed by a large energy offset. In the final Floquet system, tunnelling is re-established by means of near-resonant driving. b, At each momentum, the Hamiltonian describes the coupling between the states of the A and B sublattices, and can be visualized on a Bloch sphere. In the initial system, the Hamiltonian for all momenta points to the north pole, whereas in the Floquet system, the Hamiltonian covers a large surface of the Bloch sphere. c, Phase diagram for the Floquet Hamiltonian as a function of shaking amplitude and detuning with respect to the sublattice offset for the case of circular lattice shaking...

Integer-valued topological indices, characterizing nonlocal properties of quantum states of matter, are known to directly predict robust physical properties of equilibrium systems. The Chern number, e.g., determines the quantized Hall conductivity of an insulator. Using non-interacting fermionic atoms in a periodically driven optical lattice, here we demonstrate experimentally that the Chern number determines also the far-from-equilibrium dynamics of a quantum system. Extending a respective proposal to Floquet systems, we measure the linking number that characterizes the trajectories of momentum-space vortices emerging after a strong quench. We observe that it directly corresponds to the ground-state Chern number. This one-to-one relation between a dynamical and a static topological index allows us to experimentally map out the phase diagram of our system. Furthermore, we measure the instantaneous Chern number and show that it remains zero under the unitary dynamics.

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...

Topological defects in Bloch bands, such as Dirac points in graphene, and their resulting Berry phases play an important role for the electronic dynamics in solid state crystals. Such defects can arise in systems with a two-atomic basis due to the momentum-dependent coupling of the two sublattice states, which gives rise to a pseudo-spin texture. The topological defects appear as vortices in the azimuthal phase of this pseudo-spin texture. Here, we demonstrate a complete measurement of the azimuthal phase in a hexagonal optical lattice employing a versatile method based on time-of-flight imaging after off-resonant lattice modulation. Furthermore we map out the merging transition of the two Dirac points induced by beam imbalance. Our work paves the way to accessing geometric properties in optical lattices also with spin-orbit coupling and interactions.

Spectroscopic tools are fundamental for the understanding of complex quantum systems. Here we demonstrate high-precision multi-band spectroscopy in a graphene-like lattice using ultracold fermionic atoms. From the measured band structure, we characterize the underlying lattice potential with a relative error of 1.2 · 10 −3 . Such a precise characterization of complex lattice potentials is an important step towards precision measurements of quantum many-body systems. Furthermore, we explain the excitation strengths into the different bands with a model and experimentally study their dependency on the symmetry of the perturbation operator. This insight suggests the excitation strengths as an suitable observable for interaction effects on the eigenstates. arXiv:1801.05614v2 [cond-mat.quant-gas]

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