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The robustness of topological properties, such as quantized currents, generally depends on the existence of gaps surrounding the relevant energy levels or on symmetry-forbidden transitions. Here, we observe quantized currents that survive the addition of bounded local disorder beyond the closing of the relevant instantaneous energy gaps in a driven Aubry-Andr\'e-Harper chain, a prototypical model of quasiperiodic systems. We explain the robustness using a local picture in \textit{configuration-space} based on Landau-Zener transitions, which rests on the Anderson localisation of the eigenstates. Moreover, we propose a protocol, directly realizable in for instance cold atoms or photonic experiments, which leverages this stability to prepare topological many-body states with high Chern numbers and opens new experimental avenues for the study of both the integer and fractional quantum Hall effects., Comment: 12 pages, 8 figures. Slightly extended discussion on the effect of pumping rate. Added further connections to literature on mobility gaps. Additional minor edits

In the curved spacetime of a black hole, quantum physics gives rise to distinctive effects such as Hawking radiation. Here, we present a scheme for an analogue quantum simulation of (1 + 1)- dimensional black holes using ultracold atoms in a locally Floquet-driven 1D optical lattice. We show how the effective dynamics of the driven system can generate a position-dependent tunnelling profile that encodes the curved geometry of the black hole. Moreover, we provide a simple and robust scheme to determine the Hawking temperature of the simulated black hole based solely on on-site atom population measurements. Finally, we show how this scheme can be directly applied to simulate (2 + 1)D black holes by utilising 2D optical lattices. By incorporating the effect of atom-atom interactions, our simulator can probe the scrambling of quantum information which is a fundamental property of black holes.

Ultracold atoms in optical lattices have emerged as powerful quantum simulators of translationally invariant systems with many applications in e.g.\ strongly-correlated and topological systems. However, the ability to locally tune all Hamiltonian parameters remains an outstanding goal that would enable the simulation of a wider range of quantum phenomena. Motivated by recent advances in quantum gas microscopes and optical tweezers, we here show theoretically how local control over individual tunnelling links in an optical lattice can be achieved by incorporating local time-periodic potentials. We propose to periodically modulate the on-site energy of individual lattice sites and employ Floquet theory to demonstrate how this provides full individual control over the tunnelling amplitudes in one dimension. We provide various example configurations realising interesting topological models such as extended Su-Schrieffer-Heeger models that would be challenging to realise by other means. Extending to two dimensions, we demonstrate that local periodic driving in a Lieb lattice engineers a 2D network with fully controllable tunnelling magnitudes. In a three-site plaquette, we show full simultaneous control over the relative tunnelling amplitudes and the gauge-invariant flux piercing the plaquette, providing a clear stepping stone to building a fully programmable 2D tight-binding model. We also explicitly demonstrate how utilise our technique to generate a magnetic field gradient in 2D. This local modulation scheme is applicable to many different lattice geometries., Comment: 17 pages, 14 figures

The combined effect of disorder and interactions is central to the richness of condensed matter physics and can lead to novel quantum states such as the Bose glass phase in disordered bosonic systems. Here, we report on the experimental realisation of the two-dimensional Bose glass using ultra-cold atoms in an eight-fold symmetric quasicrystalline optical lattice. By probing the coherence properties of the system, we observe a Bose glass to superfluid transition and map out the phase diagram in the weakly interacting regime. Moreover, we reveal the non-ergodic nature of the Bose glass by probing the capability to restore coherence. Our observations are in good agreement with recent quantum Monte Carlo predictions and pave the way for experimentally testing the connection between the Bose glass, many-body localisation, and glassy dynamics more generally.

Quasicrystals are long-range ordered, yet not periodic, and thereby present a fascinating challenge for condensed matter physics, as one cannot resort to the usual toolbox based on Bloch's theorem. Here, we present a numerical method for constructing the Hubbard Hamiltonian of non-periodic potentials without making use of Bloch's theorem and apply it to the case of an eightfold rotationally symmetric 2D optical quasicrystal that was recently realized using cold atoms. We construct maximally localised Wannier functions and use them to extract on-site energies, tunneling amplitudes, and interaction energies. In addition, we introduce a configuration-space representation, where sites are ordered in terms of shape and local environment, that leads to a compact description of the infinite-size quasicrystal in which all Hamiltonian parameters can be expressed as smooth functions. This configuration-space picture allows one to construct arbitrarily large tight-binding graphs for numerical many-body calculations and enables new analytic arguments on the topological structure and many-body physics of these models, for instance the conclusion that this quasicrystal will host unit-filling Mott insulators in the thermodynamic limit., Comment: 9 pages, 12 figures, plus appendix; minor changes

We study the effect of spatially correlated classical noise on both Anderson and many-body localization of a disordered fermionic chain. By analyzing the evolution of the particle density imbalance following a quench from an initial charge density wave state, we find prominent signatures of localization also in the presence of the time-dependent noise, even though the system eventually relaxes to the infinite temperature state. In particular, for sufficiently strong static disorder we observe the onset of metastability, which becomes more prominent the stronger the spatial correlations of the noise. In this regime we find that the imbalance decays as a stretched-exponential - a behavior characteristic of glassy systems. We identify a simple scaling behavior of the relevant relaxation times in terms of the static disorder and of the noise correlation length. We discuss how our results could be exploited to extract information about the localization length in experimental setups., Comment: 6 pages, 3 figures; v2 close to published version

Discontinuous quantum phase transitions and the associated metastability play central roles in diverse areas of physics ranging from ferromagnetism to false vacuum decay in the early universe. Using strongly-interacting ultracold atoms in an optical lattice, we realize a driven many-body system whose quantum phase transition can be tuned from continuous to discontinuous. Resonant shaking of a one-dimensional optical lattice hybridizes the lowest two Bloch bands, driving a novel transition from a Mott insulator to a $\pi$-superfluid, i.e., a superfluid state with staggered phase order. For weak shaking amplitudes, this transition is discontinuous (first-order) and the system can remain frozen in a metastable state, whereas for strong shaking, it undergoes a continuous transition toward a $\pi$-superfluid. Our observations of this metastability and hysteresis are in good quantitative agreement with numerical simulations and pave the way for exploring the crucial role of quantum fluctuations in discontinuous transitions., Comment: 5 pages, 4 figures + Supplementary

Quasicrystals are long-range ordered but not periodic, representing an interesting middle ground between order and disorder. We experimentally and numerically study the ground state of non- and weakly-interacting bosons in an eightfold symmetric quasicrystalline optical lattice. We find extended states for weak lattices but observe a localisation transition at a lattice depth of $V_0=1.78(2)\,E_{\mathrm{rec}}$ for the non-interacting system. We identify this transition by measuring the timescale required for adiabatic loading into the lattice, which diverges at the critical lattice depth for localisation. Gross-Pitaevskii simulations show that in interacting systems the transition is shifted to deeper lattices, as expected from superfluid order counteracting localisation. Our experimental results are consistent with such a mean-field shift. Quasiperiodic potentials, lacking conventional rare regions, provide the ideal testing ground to realise many-body localisation in 2D.

We introduce a two-dimensional generalisation of the quasiperiodic Aubry-Andr\'e model. Even though this model exhibits the same duality relation as the one-dimensional version, its localisation properties are found to be substantially more complex. In particular, partially extended single-particle states appear for arbitrarily strong quasiperiodic modulation. They are concentrated on a network of low-disorder lattice lines, while the rest of the lattice hosts localised states. This spatial separation protects the localised states from delocalisation, so no mobility edge emerges in the spectrum. Instead, localised and partially extended states are interspersed, giving rise to an unusual type of mixed spectrum and enabling complex dynamics even in the absence of interactions. A striking example is ballistic transport across the low-disorder lines while the rest of the system remains localised. This behaviour is robust against disorder and other weak perturbations. Our model is thus directly amenable to experimental studies and promises fascinating many-body localisation properties., Comment: 10 pages, 8 figures, published version

Quasicrystals are long-range ordered and yet non-periodic. This interplay results in a wealth of intriguing physical phenomena, such as the inheritance of topological properties from higher dimensions, and the presence of non-trivial structure on all scales. Here we report on the first experimental demonstration of an eightfold rotationally symmetric optical lattice, realising a two-dimensional quasicrystalline potential for ultracold atoms. Using matter-wave diffraction we observe the self-similarity of this quasicrystalline structure, in close analogy to the very first discovery of quasicrystals using electron diffraction. The diffraction dynamics on short timescales constitutes a continuous-time quantum walk on a homogeneous four-dimensional tight-binding lattice. These measurements pave the way for quantum simulations in fractal structures and higher dimensions.

We have investigated scaling properties of the Aubry-Andr\'e model and related one-dimensional quasiperiodic Hamiltonians near their localisation transitions. We find numerically that the scaling of characteristic energies near the ground state, usually captured by a single dynamical exponent, does not obey a power law relation. Instead, the scaling behaviour depends strongly on the correlation length in a manner governed by the continued fraction expansion of the irrational number $\beta$ describing incommensurability in the system. This dependence is, however, found to be universal between a range of models sharing the same value of $\beta$. For the Aubry-Andr\'e model, we explain this behaviour in terms of a discrete renormalisation group protocol which predicts rich critical behaviour. This result is complemented by studies of the expansion dynamics of a wave packet under the Aubry-Andr\'e model at the critical point. Anomalous diffusion exponents are derived in terms of multifractal (R\'enyi) dimensions of the critical spectrum; non-power-law universality similar to that found in ground state dynamics is observed between a range of critical tight-binding Hamiltonians., Comment: 18 pages, 13 figures

Periodic driving of optical lattices has enabled the creation of novel bandstructures not realizable in static lattice systems, such as topological bands for neutral particles. However, especially driven systems of interacting bosonic particles often suffer from strong heating. We have systematically studied heating in an interacting Bose-Einstein condensate in a driven one-dimensional optical lattice. We find interaction-dependent heating rates that depend both on the scattering length and the driving strength and identify the underlying resonant intra- and interband scattering processes. By comparing experimental data and theory, we find that for driving frequencies well above the trap depth, the heating rate is dramatically reduced by the fact that resonantly scattered atoms leave the trap before dissipating their energy into the system. This mechanism of Floquet evaporative cooling offers a powerful strategy to minimize heating in Floquet engineered quantum gases., Comment: 5 pages + Appendix

In a many-body localized (MBL) quantum system, the ergodic hypothesis breaks down completely, giving rise to a fundamentally new many-body phase. Whether and under which conditions MBL can occur in higher dimensions remains an outstanding challenge both for experiments and theory. Here, we experimentally explore the relaxation dynamics of an interacting gas of fermionic potassium atoms loaded in a two-dimensional optical lattice with different quasi-periodic potentials along the two directions. We observe a dramatic slowing down of the relaxation for intermediate disorder strengths and attribute this partially to configurational rare-region effects. Beyond a critical disorder strength, we see negligible relaxation on experimentally accessible timescales, indicating a possible transition into a two-dimensional MBL phase. Our experiments reveal a distinct interplay of interactions, disorder, and dimensionality and provide insights into regimes where controlled theoretical approaches are scarce., Comment: 5 pages, 4 figures. Comments and suggestions welcome

In the presence of sufficiently strong disorder or quasiperiodic fields, an interacting many-body system can fail to thermalize and become many-body localized. The associated transition is of particular interest, since it occurs not only in the ground state but over an extended range of energy densities. So far, theoretical studies of the transition have focused mainly on the case of true-random disorder. In this work, we experimentally and numerically investigate the regime close to the many-body localization transition in quasiperiodic systems. We find slow relaxation of the density imbalance close to the transition, strikingly similar to the behavior near the transition in true-random systems. This dynamics is found to continuously slow down upon approaching the transition and allows for an estimate of the transition point. We discuss possible microscopic origins of these slow dynamics.

In the presence of disorder, an interacting closed quantum system can undergo many-body localization (MBL) and fail to thermalize. However, over long times even weak couplings to any thermal environment will necessarily thermalize the system and erase all signatures of MBL. This presents a challenge for experimental investigations of MBL, since no realistic system can ever be fully closed. In this work, we experimentally explore the thermalization dynamics of a localized system in the presence of controlled dissipation. Specifically, we find that photon scattering results in a stretched exponential decay of an initial density pattern with a rate that depends linearly on the scattering rate. We find that the resulting susceptibility increases significantly close to the phase transition point. In this regime, which is inaccessible to current numerical studies, we also find a strong dependence on interactions. Our work provides a basis for systematic studies of MBL in open systems and opens a route towards extrapolation of closed system properties from experiments., Comment: 5+7 pages, 9 figures

We experimentally study a periodically driven many-body localized system realized by interacting fermions in a one-dimensional quasi-disordered optical lattice. By preparing the system in a far-from-equilibrium state and monitoring the remains of an imprinted density pattern, we identify a localized phase at high drive frequencies and an ergodic phase at low ones. These two distinct phases are separated by a dynamical phase transition which depends on both the drive frequency and the drive strength. Our observations are quantitatively supported by numerical simulations and are directly connected to the change in the statistical properties of the effective Floquet Hamiltonian., Comment: 4 pages main text + 4 pages supplementary mat

Topology and geometry are essential to our understanding of modern physics, underlying many foundational concepts from high energy theories, quantum information, and condensed matter physics. In condensed matter systems, a wide range of phenomena stem from the geometry of the band eigenstates, which is encoded in the matrix-valued Wilson line for general multi-band systems. Using an ultracold gas of Rb atoms loaded in a honeycomb optical lattice, we realize strong-force dynamics in Bloch bands that are described by Wilson lines and observe an evolution in the band populations that directly reveals the band geometry. Our technique enables a full determination of band eigenstates, Berry curvature, and topological invariants, including single- and multi-band Chern and $Z_2$ numbers., Comment: 5+9 pages, final ublished version

We experimentally study the effects of coupling one-dimensional Many-Body Localized (MBL) systems with identical disorder. Using a gas of ultracold fermions in an optical lattice, we artifically prepare an initial charge density wave in an array of 1D tubes with quasi-random onsite disorder and monitor the subsequent dynamics over several thousand tunneling times. We find a strikingly different behavior between MBL and Anderson Localization. While the non-interacting Anderson case remains localized, in the interacting case any coupling between the tubes leads to a delocalization of the entire system., Comment: 4+3 pages, 5+4 figures final version using stretched exponential fits to improve fit quality

We experimentally observe many-body localization of interacting fermions in a one-dimensional quasi-random optical lattice. We identify the many-body localization transition through the relaxation dynamics of an initially-prepared charge density wave. For sufficiently weak disorder the time evolution appears ergodic and thermalizing, erasing all remnants of the initial order. In contrast, above a critical disorder strength a significant portion of the initial ordering persists, thereby serving as an effective order parameter for localization. The stationary density wave order and the critical disorder value show a distinctive dependence on the interaction strength, in agreement with numerical simulations. We connect this dependence to the ubiquitous logarithmic growth of entanglement entropy characterizing the generic many-body localized phase., Comment: 6 pages, 6 figures + supplementary information

The geometric structure of an energy band in a solid is fundamental for a wide range of many-body phenomena in condensed matter and is uniquely characterized by the distribution of Berry curvature over the Brillouin zone. In analogy to an Aharonov-Bohm interferometer that measures the magnetic flux penetrating a given area in real space, we realize an atomic interferometer to measure Berry flux in momentum space. We demonstrate the interferometer for a graphene-type hexagonal lattice, where it has allowed us to directly detect the singular $\pi$ Berry flux localized at each Dirac point. We show that the interferometer enables one to determine the distribution of Berry curvature with high momentum resolution. Our work forms the basis for a general framework to fully characterize topological band structures and can also facilitate holonomic quantum computing through controlled exploitation of the geometry of Hilbert space., Comment: 5+5 pages

In this comment we argue that negative absolute temperatures are a well-established concept for systems with bounded spectra. They are not only consistent with thermodynamics, but are even unavoidable for a consistent description of the thermal equilibrium of inverted populations., Comment: 3 pages

We experimentally and numerically investigate the expansion of initially localized ultracold bosons in homogeneous one- and two-dimensional optical lattices. We find that both dimensionality and interaction strength crucially influence these non-equilibrium dynamics. While the atoms expand ballistically in all integrable limits, deviations from these limits dramatically suppress the expansion and lead to the appearance of almost bimodal cloud shapes, indicating diffusive dynamics in the center surrounded by ballistic wings. For strongly interacting bosons, we observe a dimensional crossover of the dynamics from ballistic in the one-dimensional hard-core case to diffusive in two dimensions, as well as a similar crossover when higher occupancies are introduced into the system., Comment: 6 pages, 4 figures + supplementary material; V3: Published version

Absolute temperature, the fundamental temperature scale in thermodynamics, is usually bound to be positive. Under special conditions, however, negative temperatures - where high-energy states are more occupied than low-energy states - are also possible. So far, such states have been demonstrated in localized systems with finite, discrete spectra. Here, we were able to prepare a negative temperature state for motional degrees of freedom. By tailoring the Bose-Hubbard Hamiltonian we created an attractively interacting ensemble of ultracold bosons at negative temperature that is stable against collapse for arbitrary atom numbers. The quasi-momentum distribution develops sharp peaks at the upper band edge, revealing thermal equilibrium and bosonic coherence over several lattice sites. Negative temperatures imply negative pressures and open up new parameter regimes for cold atoms, enabling fundamentally new many-body states and counterintuitive effects such as Carnot engines above unity efficiency., Comment: 5 pages, 4 figures + supplementary material

We theoretically investigate the thermodynamics of an interacting inhomogeneous two-component Fermi gas in an optical lattice. Motivated by a recent experiment by L. Hackerm\"uller et al., Science, 327, 1621 (2010), we study the effect of the interplay between thermodynamics and strong correlations on the size of the fermionic cloud. We use dynamical mean-field theory to compute the cloud size, which in the experiment shows an anomalous expansion behavior upon increasing attractive interaction. We confirm this qualitative effect but, assuming adiabaticity, we find quantitative agreement only for weak interactions. For strong interactions we observe significant non-equilibrium effects which we attribute to a dynamical arrest of the particles due to increasing correlations., Comment: 4.5 pages, 5 figures (slightly different from published version)

We analyze the possibility to prepare a Heisenberg antiferromagnet with cold fermions in optical lattices, starting from a band insulator and adiabatically changing the lattice potential. The numerical simulation of the dynamics in 1D allows us to identify the conditions for success, and to study the influence that the presence of holes in the initial state may have on the protocol. We also extend our results to two-dimensional systems., Comment: 5 pages, 4 figures + Supplementary Material (5 pages, 6 figures), published version

We have experimentally studied few-body impurity systems consisting of a single fermionic atom and a small bosonic field on the sites of an optical lattice. Quantum phase revival spectroscopy has allowed us to accurately measure the absolute strength of Bose-Fermi interactions as a function of the interspecies scattering length. Furthermore, we observe the modification of Bose-Bose interactions that is induced by the interacting fermion. Because of an interference between Bose-Bose and Bose-Fermi phase dynamics, we can infer the mean fermionic filling of the mixture and quantify its increase (decrease) when the lattice is loaded with attractive (repulsive) interspecies interactions., Comment: 4+ pages, 5 figures, updated to published version

Transport properties are among the defining characteristics of many important phases in condensed matter physics. In the presence of strong correlations they are difficult to predict even for model systems like the Hubbard model. In real materials they are in general obscured by additional complications including impurities, lattice defects or multi-band effects. Ultracold atoms in contrast offer the possibility to study transport and out-of-equilibrium phenomena in a clean and well-controlled environment and can therefore act as a quantum simulator for condensed matter systems. Here we studied the expansion of an initially confined fermionic quantum gas in the lowest band of a homogeneous optical lattice. While we observe ballistic transport for non-interacting atoms, even small interactions render the expansion almost bimodal with a dramatically reduced expansion velocity. The dynamics is independent of the sign of the interaction, revealing a novel, dynamic symmetry of the Hubbard model., Comment: 5 pages, 4 figures +supplementary information, shortened, published version

The collapses and revivals of a coherent matter wave field of interacting particles can serve as a sensitive interferometric probe of the interactions and the number statistics of the underlying quantum field. Here we show how the ability to observe long time traces of collapse and revival dynamics of a Bose-Einstein condensate loaded into a three-dimensional (3D) optical lattice allowed us to directly reveal the atom number statistics and the presence of effective coherent multi-particle interactions in a lattice. The multi-particle interactions are generated via virtual transitions to higher lattice orbitals and can find use for simulations of effective field theories with ultracold atoms in optical lattices. We measured their absolute strengths up to the case of six-particle interactions and compare our findings with theory., Comment: 6 pages, 4 figures

Strong correlations can dramatically modify the thermodynamics of a quantum many-particle system. Especially intriguing behaviour can appear when the system adiabatically enters a strongly correlated regime, for the interplay between entropy and strong interactions can lead to counterintuitive effects. A well known example is the so-called Pomeranchuk effect, occurring when liquid 3He is adiabatically compressed towards its crystalline phase. Here, we report on a novel anomalous, isentropic effect in a spin mixture of attractively interacting fermionic atoms in an optical lattice. As we adiabatically increase the attraction between the atoms we observe that the gas, instead of contracting, anomalously expands. This expansion results from the combination of two effects induced by pair formation in a lattice potential: the suppression of quantum fluctuations as the attraction increases, which leads to a dominant role of entropy, and the progressive loss of the spin degree of freedom, which forces the gas to excite additional orbital degrees of freedom and expand to outer regions of the trap in order to maintain the entropy. The unexpected thermodynamics we observe reveal fundamentally distinctive features of pairing in the fermionic Hubbard model., Comment: 6 pages (plus appendix), 6 figures