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We consider the emergence of edge states in a finite optical lattice and show that the boundaries of the lattice play a decisive role for their location in the corresponding energy spectrum. We introduce a simple parametrisation of the boundaries of the optical lattice and demonstrate the existence of an optimal choice of the values of the parameters which lead to an approximate restoration of chiral symmetry. A crucial property of this optimization is the suppression of tunneling between next-nearest neighboring wells of the lattice. This in turn allows the mapping of the optical lattice set-up to a finite SSH model. The topological character of the emerging edge states is discussed., Comment: 10 pages, 11 figures

Driving a many-body system out of equilibrium induces phenomena such as the emergence and decay of transient states, which can manifest itself as pattern and domain formation. The understanding of these phenomena expands the scope of established thermodynamics into the out-of-equilibrium domain. Here, we experimentally and theoretically study the out-of-equilibrium dynamics of a bosonic lattice model subjected to a strong DC field, realized as ultracold atoms in a strongly tilted triangular optical lattice. We observe the emergence of pronounced density wave patterns -- which spontaneously break the underlying lattice symmetry -- using a novel single-shot imaging technique with two-dimensional single-site resolution in three-dimensional systems, which also resolves the domain structure. Our study suggests that the short-time dynamics arises from resonant pair tunneling processes within an effective description of the tilted Hubbard model. More broadly, we establish the far out-of-equilibrium regime of lattice models subjected to a strong DC field, as an exemplary and paradigmatic scenario for transient pattern formation., Comment: 8 pages, 4 figures, 6 pages of Supplementary Information

Periodically driven systems play a prominent role in optical lattices. In these ultracold atomic systems, driving is used to create a variety of interesting behaviours, of which an important example is provided by topological states of matter. Such Floquet topological phases have a richer classification that their equilibrium counterparts. Although analogues of the equilibrium topological phases exist, which are characterised by a Chern number, the corresponding Hall conductivity, and protected edge states, there is an additional possibility. This is a phase that has vanishing Chern number and no Hall conductivity, but nevertheless hosts anomalous topological edge states. Due to experimental difficulties associated with the observation of such a phase, it has not been experimentally realised so far. In this paper, we show that optical lattices prove to be a good candidate for both its realisation and subsequent observation, because they can be driven in a controlled manner. Specifically, we present a simple shaking protocol that serves to realise this special Floquet phase, discuss the specific properties that it has, and propose a method to experimentally detect this fascinating topological phase that has no counterpart in equilibrium systems., Comment: 14 pages, 4 figures

Supersymmetric systems derive their properties from conserved supercharges which form a supersymmetric algebra. These systems naturally factorize into two subsystems, which, when considered as individual systems, have essentially the same eigenenergies, and their eigenstates can be mapped onto each other. We first propose a one-dimensional ultracold atom setup to realize such a pair of supersymmetric systems. We propose a Mach-Zehnder interference experiment which we demonstrate for this system, and which can be realized with current technology. In this interferometer, a single atom wave packet that evolves in a superposition of the subsystems, gives an interference contrast that is sharply peaked if the subsystems form a supersymmetric pair. Secondly, we propose a two-dimensional setup that implements supersymmetric dynamics in a synthetic gauge field., Comment: 8 pages, 9 figures

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 momentum-resolved measurement of the ensuing Berry curvature. Our results pave the way to explore intriguing phases of matter with interactions in topological band structures.

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., Comment: 9 pages, 7 figures