# Ultracold Bosons in Optical Superlattices

## Experiments with ultracold bosons in optical superlattices

In our experiments with ultracold bosonic rubidium atoms in optical superlattices, we are studying quantum phases and phase transitions in specific condensed matter models as well as questions of quantum many-body dynamics (more). Every experimental run starts out with the creation of a Bose-Einstein condensate of Rb-78 using a magneto-optical trap (MOT) and forced evaporation in a magnetic trap and, later, in an optical dipole trap.Beyond this initial step, our main tool is an optical superlattice formed by overlapping two standing-wave laser fields with a commensurate wavelength ratio of 2 (more). Two perpendicular optical lattices complete the setup of a regular three-dimensional lattice with a unit cell of two sites. Full dynamical control over the individual laser-beam intensities as well as over the relative phase between the two standing waves forming the superlattice allows us to set all relevant parameters of the atomic many-body system. We are currently extending this system by replacing one of the transverse lattice by another superlattice, thus obtaining a unit cell of four sites – a plaquette.

## The past

Some examples of experiments performed with the single superlattice potential – still in Mainz – include the direct observation of atom co-tunneling of repulsively interacting atoms, the observation and control of superexchange interactions between neighboring atoms and the controlled creation and detection of spin-correlations in the optical lattice. In terms of dynamics, we have investigated the relaxation of strongly correlated bosons in one-dimensional chains following a quantum quench. Another experiment dealt with many-body Landau-Zener sweeps with one-dimensional quantum gases and a drastic breakdown of adiabaticity far from equilibrium.

Even before the implementation of our first superlattice, the machine went through a long history of experiments with single-wavelength optical lattices. Prominent examples from the list of results are the observation of the quantum phase transition from a superfluid to a bosonic Mott insulator, the measurement of the collapse-and-revival dynamics of a matter-wave field and the realization of a Tonks-Girardeau gas. More recently, we could also demonstrate the storage of light in a Mott insulator by means of electrically induced transparency and determine a quantitative phase diagram of the Bose-Hubbard model at finite temperatures.

With the extension to the two-axes superlattices, we aimed at, for example, realizing minimal versions of topologically ordered quantum phases of bosons. Such phases with topological order cannot be classified by an order parameter and represent a new class of many-body systems without any local order. Only by carrying globals measurements on the system, one can reveal the hidden order in the system.

Furthermore, we are able to create, control and detect spin-correlated states in two dimensions in analogy to 2D valence bond solid states, and to follow their dynamics. Such states are of general interest in the context of quantum magnetism and high-temperature superconductivity.

## Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms

The plateaux of the Hall conductivity observed in the quantum Hall effect can be attributed to a topological invariant characterizing Bloch bands: the Chern number. Until now, topological transport associated with non-zero Chern numbers has only been observed in electronic systems. In the context of artificial gauge fields for ultracold atoms, however, the implementation of experimental probes revealing the non-trivial topology of energy bands is one of the most challenging goals. In this work, the Chern number of artificially generated Hofstadter bands in an optical lattice was measured by looking at the transverse deflection of an atomic cloud in response to an external gradient. When applying a force, the atoms acquire an anomalous transverse velocity which is caused by the non-trivial topology of the band structure and for a uniformly populated band it can be be related to its Chern number.

The Chern number could thus be determined by measuring the center-of-mass position of the atom cloud in situ. In addition to this, a novel band-mapping technique for determining the populations of the Hofstadter bands as well as a new all-optical scheme for generating uniform artificial gauge fields in optical superlattices were developed. This experiment constitutes the first measurement of a Chern number in a non-electronic systems and the applied method can easily be generalized to a wide range of physical systems. For the experimental parameters, the lowest band of the Hofstadter model is very flat and thus a good candidate for the realization of novel topological states of matter like fractional Chern insulators.

## A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice

The concept of a topological charge pump was first introduced by David Thouless more than 30 years ago. It would enable the robust transport of charge through an adiabatic cyclic evolution of the underlying Hamiltonian. In contrast to classical transport, the charge that is transported per cycle is quantized and purely determined by the topology of the pump cycle, making it robust to perturbations. On a fundamental level, the quantized charge transport can be connected to a topological invariant, the Chern number, and a Thouless quantum pump may therefore be regarded as a 'dynamical' version of the integer quantum Hall effect. In this work, such a pump was realized using ultracold bosonic atoms forming a Mott insulator in a dynamically controlled optical superlattice. By taking in situ images of the cloud, a quantized deflection per pump cycle was observed for the first time and its robustness against external perturbations could be shown by comparing the deflection for different pump parameters. The pump's genuine quantum nature was revealed by showing that, in contrast to groundstate particles, a counterintuitive reversed deflection occurs for particles in the first excited band.

Furthermore it was directly demonstrated that this system undergoes a controlled topological transition in higher bands when tuning the superlattice parameters. These results open a route to the implementation of more complex pumping schemes. By adding a spin degree of freedom, a Z2 spin pump could be implemented in a spin-dependent superlattice. Moreover, extending the Thouless pump to 2D systems would enable the realization of an analogon of the 4D integer quantum Hall effect.

## Spin pumping and measurement of spin currents in optical superlattices

Exposing materials to strong magnetic fields has led to remarkable discoveries, most prominently the observation of the integer and fractional quantum Hall effect. These quantum phenomena surprise due to their robustness and independence of material properties, arising from their topological nature. More recently, a fundamentally different quantum state was observed, the topological insulator, which preserves time-reversal symmetry. Such time-reversal symmetric systems in 2D that also conserve spin, exhibit the quantum spin Hall effect. It is characterized by a quantized spin but vanishing charge conductance. Analogous to topological charge pumps proposed by Thouless 1983, a dynamical version of a topological insulator can be designed: a quantum spin pump. We report on an experimental implementation of a spin pump with ultracold bosonic atoms in an optical superlattice.

Starting from an antiferromagnetically ordered spin chain, we periodically vary the underlying spin-dependent Hamiltonian by changing a global magnetic gradient. We show, that in the limit of isolated double wells the system is a dynamical version of the quantum spin Hall effect and observe the associated response: a spin current without charge current. The response was detected by both a direct verification of spin transport through in-situ measurements of the spins' center of mass displacement and the measurement of local spin currents. To observe these spin currents in optical lattices, we demonstrate a novel detection method using the amplitude of superexchange oscillations emerging after a projection onto static double wells.

## Exploring 4D quantum Hall physics with a 2D topological charge pump

The quantum Hall effect - a prominent example for topological states of matter in 2D - can be generalized to 4D systems. In 4D, a novel quantized Hall response appears, which is nonlinear and described by a 4D topological invariant - the second Chern number. We realize a dynamical version of the 4D quantum Hall effect by implementing a 2D topological charge pump for ultracold bosonic atoms in an angled optical superlattice. We observe a bulk response with intrinsic 4D topology and demonstrate its quantization by measuring the associated second Chern number.