The reliable detection of single quantum particles has revolutionized the field of quantum optics and quantum information processing. For several years, researchers have aspired to extend such detection possibilities to larger-scale, strongly correlated quantum systems in order to record in situ images of a quantum fluid in which each underlying quantum particle is detected. Here we report fluorescence imaging of strongly interacting bosonic Mott insulators in an optical lattice with single-atom and single-site resolution. From our images, we fully reconstruct the atom distribution on the lattice and identify individual excitations with high fidelity. A comparison of the radial density and variance distributions with theory provides a precise in situ temperature and entropy measurement from single images. We observe Mott-insulating plateaus with near-zero entropy and clearly resolve the high-entropy rings separating them, even though their width is of the order of just a single lattice site. Furthermore, we show how a Mott insulator melts with increasing temperature, owing to a proliferation of local defects. The ability to resolve individual lattice sites directly opens up new avenues for the manipulation, analysis and applications of strongly interacting quantum gases on a lattice. For example, one could introduce local perturbations or access regions of high entropy, a crucial requirement for the implementation of novel cooling schemes.

Ultracold atoms in optical lattices are a versatile tool to investigate fundamental properties of quantum many body systems. In particular, the high degree of control of experimental parameters has allowed the study of many interesting phenomena such as quantum phase transitions and quantum spin dynamics. Here we demonstrate how such control can be extended down to the most fundamental level of a single spin at a specific site of an optical lattice. Using a tightly focussed laser beam together with a microwave field, we were able to flip the spin of individual atoms in a Mott insulator with sub-diffraction-limited resolution, well below the lattice spacing. The Mott insulator provided us with a large two-dimensional array of perfectly arranged atoms, in which we created arbitrary spin patterns by sequentially addressing selected lattice sites after freezing out the atom distribution. We directly monitored the tunnelling quantum dynamics of single atoms in the lattice prepared along a single line and observed that our addressing scheme leaves the atoms in the motional ground state. Our results open the path to a wide range of novel applications from quantum dynamics of spin impurities, entropy transport, implementation of novel cooling schemes, and engineering of quantum many-body phases to quantum information processing.The ability to observe and control the position of single atoms on a surface of a solid via scanning tunnelling and atomic force microscopy has revolutionised the field of condensed matter physics [1,2]. In few-atom systems, coherent control of single particles in e.g. an ion chain has proven crucial for the implementation of high-fidelity quantum gates and the readout of individual qubits in quantum information processing [3]. Bringing such levels of control to the regime of large scale many-body systems has been a longstanding goal in quantum physics. In the context of ultracold atoms in optical lattices, a major challenge has been to combine degenerate atomic samples with single-site addressing resolution and singleatom sensitivity. This full control is essential for many applications in condensed matter physics, such as the study of spin impurities [4] and quantum spin dynamics [5,6] within quantum magnetism, entropy transport, the implementation of novel cooling schemes [7,8] or digital quantum simulations based on Rydberg atoms [9]. For scalable quantum information processing, a Mott insulator with unity filling provides a natural quantum register with several hundreds of qubits. In order to exploit the full potential of such a large scale system for quantum computation, coherent manipulation of individual spins is indispensable, both within a circuit-based [10] or a one-way quantum computer architecture [11,12].The quest to address atoms on single sites of an optical lattice has a long history [7,[13][14][15][16][17][18][19][20][21][22]. In one dimension, single-site addressing was accomplished optically † present address: Department of Physics and Astronomy, University of Aarhus, DK-8000 ...

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...

Quantum phases of matter are characterized by the underlying correlations of the many-body system. Although this is typically captured by a local order parameter, it has been shown that a broad class of many-body systems possesses a hidden non-local order. In the case of bosonic Mott insulators, the ground state properties are governed by quantum fluctuations in the form of correlated particle-hole pairs that lead to the emergence of a non-local string order in one dimension. Using high-resolution imaging of low-dimensional quantum gases in an optical lattice, we directly detect these pairs with single-site and single-particle sensitivity and observe string order in the one-dimensional case.The realization of strongly correlated quantum manybody systems using ultracold atoms has enabled the direct observation and control of fundamental quantum effects [1][2][3]. A prominent example is the transition from a superfluid (SF) to a Mott insulator (MI), occurring when interactions between bosonic particles on a lattice dominate over their kinetic energy [4][5][6][7][8]. At zero temperature, and in the limit where the ratio of kinetic energy over interaction energy vanishes, particle fluctuations are completely suppressed and the lattice sites are occupied by an integer number of particles. However, at a finite tunnel coupling, but still in the Mott insulating regime, quantum fluctuations create correlated particlehole pairs on top of this fixed-density background, which can be understood as virtual excitations. These particlehole pairs fundamentally determine the properties of the Mott insulator such as its residual phase coherence [9] and lie at the heart of superexchange-mediated spin interactions that form the basis of quantum magnetism in multi-component quantum gas mixtures [10][11][12].In a one-dimensional system, the appearance of correlated particle-hole pairs at the transition point from a superfluid to a Mott insulator is intimately connected to the emergence of a hidden string-order parameter O P [13,14]:Here δn j =n j −n denotes the deviation in occupation of the jth lattice site from the average background density, and k is an arbitrary position along the chain. In the simplest case of a Mott insulator with unity filling * Electronic address: manuel.endres@mpq.mpg.de (n = 1), relevant to our experiments, each factor in the product of operators in Eq. 1 yields −1 instead of +1, when a single-particle fluctuation from the unit background density is encountered. In the superfluid, particle and hole fluctuations occur independently and are uncorrelated, such that O P = 0. However, in the Mott insulating phase, density fluctuations always occur as correlated particle-hole pairs, resulting in O P = 0. For a homogeneous system, O P is expected to follow a scaling of Berezinskii-Kosterlitz-Thouless (BKT) type [15]. Non-local correlation functions, like the string-order parameter defined above, have been introduced in the context of low-dimensional quantum systems. They classify many-body quantum phases that are n...

Phase transitions are ubiquitous in our three-dimensional world. By contrast, most conventional transitions do not occur in infinite uniform low-dimensional systems because of the increased role of thermal fluctuations. The crossover between these situations constitutes an important issue, dramatically illustrated by Bose-Einstein condensation: a gas strongly confined along one direction of space may condense along this direction without exhibiting true long-range order in the perpendicular plane. Here we explore transverse condensation for an atomic gas confined in a novel trapping geometry, with a flat in-plane bottom, and we relate it to the onset of an extended (yet of finite-range) in-plane coherence. By quench crossing the transition, we observe topological defects with a mean number satisfying the universal scaling law predicted by Kibble-Zurek mechanism. The approach described can be extended to investigate the topological phase transitions that take place in planar quantum fluids.

Owing to thermal fluctuations, two-dimensional (2D) systems cannot undergo a conventional phase transition associated with the breaking of a continuous symmetry 1 . Nevertheless they may exhibit a phase transition to a state with quasi-longrange order via the Berezinskii-Kosterlitz-Thouless (BKT) mechanism 2 . A paradigm example is the 2D Bose fluid, such as a liquid helium film 3 , which cannot condense at non-zero temperature although it becomes superfluid above a critical phase space density. The quasi-long-range coherence and the microscopic nature of the BKT transition were recently explored with ultracold atomic gases 4-6 . However, a direct observation of superfluidity in terms of frictionless flow is still missing for these systems. Here we probe the superfluidity of a 2D trapped Bose gas using a moving obstacle formed by a micrometre-sized laser beam. We find a dramatic variation of the response of the fluid, depending on its degree of degeneracy at the obstacle location.'Flow without friction' is a hallmark of superfluidity 7 . It corresponds to a metastable state in which the fluid has a non-zero relative velocity v with respect to an external body such as the wall of the container or an impurity. This metastable state is separated from the equilibrium state of the system (v = 0) by a large energy barrier, so that the flow can persist for a macroscopic time. The height of the barrier decreases as v increases, and eventually passes below a threshold (proportional to the thermal energy) for a critical velocity v c . The microscopic mechanism limiting the barrier height depends on the nature of the defect and is associated with the creation of phonons and/or vortices 7 . Whereas the quantitative comparison between experiments and theory is complicated for liquid 4 He, cold atomic gases in the weakly interacting regime are well suited for precise tests of many-body physics. In particular, superfluidity was observed in 3D atomic gases by stirring a laser beam or an optical lattice through bosonic [8][9][10][11][12] or fermionic 13 fluids and by observing the resulting heating or excitations. Here we transpose this search for dissipation-less motion to a disc-shaped, non-homogeneous 2D Bose gas. We use a small obstacle to locally perturb the system. The obstacle moves at constant velocity on a circle centred on the cloud, allowing us to probe the gas at a fixed density. We repeat the experiment for various atom numbers, temperatures and stirring radii and identify a critical point for superfluid behaviour.Our experiments are performed with 2D Bose gases of N = 35,000-95,000 87 Rb atoms confined in the vertical direction by the harmonic potential W (z) and in the horizontal plane by the radially symmetric harmonic potential V (r) (see ref. perturbed by a focussed laser beam, which moves at constant velocity on a circle centred on the cloud. The stirring beam has a frequency greater than the 87 Rb resonance frequency ('blue detuning' of ≈2 nm) and thus creates a repulsive potential which causes a dip in the d...

We create supercurrents in annular two-dimensional Bose gases through a temperature quench of the normal-to-superfluid phase transition. We detect the magnitude and the direction of these supercurrents by measuring spiral patterns resulting from the interference of the cloud with a central reference disk. These measurements demonstrate the stochastic nature of the supercurrents. We further measure their distribution for different quench times and compare it with predictions based on the Kibble-Zurek mechanism.