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...

Superconductivity and superfluidity of fermionic and bosonic systems are remarkable many--body quantum phenomena. In liquid helium and dilute gases, Bose and Fermi superfluidity has been observed separately, but producing a mixture in which both the fermionic and the bosonic components are superfluid is challenging. Here we report on the observation of such a mixture with dilute gases of two Lithium isotopes, 6 Li and 7 Li. We probe the collective dynamics of this system by exciting center--of--mass oscillations that exhibit extremely low damping below a certain critical velocity. Using high precision spectroscopy of these modes we observe coherent energy exchange and measure the coupling between the two superfluids. Our observations can be captured theoretically using a sum--rule approach that we interpret in terms of two coupled oscillators.In recent years, ultracold atoms have emerged as a unique tool to engineer and study quantum many--body systems. Examples include weakly interacting Bose--Einstein condensates [1,2], two--dimensional gases [3], and the superfluid--Mott insulator transition [4] in the case of bosonic atoms, and the crossover between Bose--Einstein condensation (BEC) and fermionic superfluidity described by Bardeen, Cooper and Schrieffer's theory (BCS) for fermionic atoms Bose gas is sympathetically driven to quantum degeneracy.The two clouds reach the superfluid regime after a 4--second evaporation ramp [10]. As the 7 Li Bose gas is weakly interacting, the onset of BEC is detected by the growth of a narrow peak in the density profile of the cloud. From previous studies on atomic Bose--Einstein condensates, we conclude that the 7 Li BEC is in a superfluid phase. Superfluidity in a unitaryFermi gas is notoriously more difficult to detect because of the absence of any qualitative modification of the density profile at the phase transition. To demonstrate the superfluidity of the fermionic component of the cloud, we slightly imbalance the two spin populations. In an imbalanced gas, the cloud is organized in concentric layers, with a fully paired superfluid region at its center where Cooper pairing maintains equal spin populations. This 6 Li superfluid core can be detected by the presence of a plateau in the doubly integrated density difference [12].Examples of density profiles of the bosonic and fermionic superfluids are shown in Fig is the geometric mean trapping frequency for 7 Li ( 6 Li). Combined with the observation of the 6 Li plateau, this implies that the Fermi cloud is also superfluid with a temperature below 0.8 !,! .Here !,! is the critical temperature for superfluidity of a spin--balanced, harmonically trapped Fermi gas at unitarity,The superfluid mixture is very stable with a lifetime exceeding 7s for our coldest samples.As seen in Fig. 1, the Bose--Fermi interaction is too weak to alter significantly the density profiles of the two species [14]. To probe the interaction between the two superfluids we study the dynamics of the mass centers of the two isotopes (dipole modes), a schem...