No abstract

More than 30 years ago, Thouless introduced the concept of a topological charge pump [1] that would enable the robust transport of charge through an adiabatic cyclic evolution of the underlying Hamiltonian. In contrast to classical transport, the transported charge was shown to be quantized and purely determined by the topology of the pump cycle, making it robust to perturbations [1,2]. On a fundamental level, the quantized charge transport can be connected to a topological invariant, the Chern number, first introduced in the context of the integer quantum Hall effect [3,4]. A Thouless quantum pump may therefore be regarded as a 'dynamical' version of the integer quantum Hall effect. Here, we report on the realization of such a topological charge pump using ultracold bosonic atoms that form a Mott insulator in a dynamically controlled optical superlattice potential. By taking in-situ images of the atom cloud, we observe a quantized deflection per pump cycle. We reveal the genuine quantum nature of the pump by showing that, in contrast to ground state particles, a counterintuitive reversed deflection occurs when particles are prepared in the first excited band. Furthermore, we were able to directly demonstrate that the system undergoes a controlled topological phase transition in higher bands when tuning the superlattice parameters.Charge pumping in solid-state systems has received much attention, mainly due to its potential for realizing novel current standards [5,6], but also for characterizing many-body systems [7,8]. Quantized transport of electrons without an external bias was observed in tunnel junctions with modulated gate voltages [9], in 1D channels using surface acoustic waves [10] and in quantum dots [11]. While the latter was used to realize an adiabatic quantum pump [12,13], topological charge pumping has so far remained out of reach in condensed matter experiments. In engineered bosonic systems, non-quantized topological pumping of edge states was observed in 1D quasicrystalline photonic waveguide arrays [14,15]. In cold atomic systems, topological Bloch bands have been realized, ranging from the Su-SchriefferHeeger (SSH) model [16], the Hofstadter model in real space [17,18] and synthetic dimensions [19,20] to the Haldane model [21]. Their geometric features have been probed using novel interferometric [16,22] and transport probes [21,23] that have e.g. enabled the direct measurement of the Chern number through bulk topological currents [23].Due to their versatility, ultracold atoms in optical superlattices constitute an ideal system for the implementation of quantized topological charge pumps [24][25][26]. A superlattice is formed by superimposing two lattices with periodicities d l and, where V s (V l ) denotes the depth of the short (long) lattice, respectively. The relative position of the lattices is determined by the variable phase ϕ. When applying Bloch's theorem, the resulting single-particle HamiltonianĤ(k x , ϕ) is periodic in both the quasi-momentum k x ∈ ]−π/d l , π/d l ] and the superl...

The discovery of topological states of matter has profoundly augmented our understanding of phase transitions in physical systems. Instead of local order parameters, topological phases are described by global topological invariants and are therefore robust against perturbations. A prominent example thereof is the two-dimensional integer quantum Hall effect [1]. It is characterized by the first Chern number which manifests in the quantized Hall response induced by an external electric field [2]. Generalizing the quantum Hall effect to four-dimensional systems leads to the appearance of a novel non-linear Hall response that is quantized as well, but described by a 4D topological invariant -the second Chern number [3,4]. Here, we report on the first observation of a bulk response with intrinsic 4D topology and the measurement of the associated second Chern number. By implementing a 2D topological charge pump with ultracold bosonic atoms in an angled optical superlattice, we realize a dynamical version of the 4D integer quantum Hall effect [5,6]. Using a small atom cloud as a local probe, we fully characterize the non-linear response of the system by in-situ imaging and site-resolved band mapping. Our findings pave the way to experimentally probe higher-dimensional quantum Hall systems, where new topological phases with exotic excitations are predicted [4].Topology, originally a branch of mathematics, has become an important concept in different fields of physics, ranging from particle physics [3,7] to solid state physics [8] and quantum computation [9]. In this context, a hallmark achievement was the discovery of the 2D integer quantum Hall (QH) effect [1]. It demonstrated that the Hall conductance, in response to an electric field E, is quantized for charged particles moving in 2D in the presence of a perpendicular magnetic field. In a cylindrical geometry, following Laughlin's gedankenexperiment, E can be generated by varying the magnetic flux φ x (t) along the axis of the cylinder [10] (Fig. 1a). Due to the interplay of the perpendicular magnetic field and the induced electric field E z , the particles start moving in the x-direction. This gives rise to the quantized Hall response through an integer number of particles that is transported from one edge to the other per flux quantum threaded through the cylinder. This quantized response is characterized by an integer topological invariant, the first Chern number [2].Dimensionality plays a crucial role for topological phases and many intriguing states have recently been discovered in 3D systems, e.g. Weyl semimetals [11,12] and 3D topological insulators [13]. Ascending further in dimensions, a generalization of the QH effect to 4D was proposed [4]. It has received much attention in theoretical studies [14][15][16], in particular since it might exhibit novel strongly correlated QH phases [4]. This interest was renewed recently due to the unprecedented control and flexibility offered by engineered systems like ultracold atoms and photonics. Indeed, such systems have...