Dispersive interaction of atoms with nearly resonant light leads to effective potentials restricting the atomic motion to e.g. periodic structures due to standing waves of counter-propagating beams. Interaction between atoms may be changed at will leading to fully controlled models. Those models, in particular allow to propose (and realize) topological states of matter -- interesting for possible future applications in quantum computing due to their stability induced by geometric topological protection.

Figure 1. Left: Quantum simulator at work. The optical lattice potential keeps green atoms. Their tunneling is affected by other red particles sitting in another lattice. All interaction terms denoted by $U_i$ may be controlled at will. Depending on the density the system reveals a devil's staircase of superluids or Peierls topological insulators. Right: atoms in optical lattice are placed in cavity. Interaction with cavity mode induces long-range coupling between atoms. |

Figure 2. Quantum simulator at work. Semi-classical sketch of the confining dynamics of Bosonic Schwinger Model. A well separated pair of particle and antiparticle connected by an electric-flux tube is our initial state. Initially the pair spreads as if the constituents were free, however their trajectories bend due to the energetic cost of creating larger electric-flux tubes. New dynamical charges are also created during the evolution and partially screen the electric field. Still the electric field oscillates coherently and can form an anti-string, creating a central core of strongly correlated bosons. The density of bosons in the core can get depleted through the radiation of lighter mesons that freely propagate. |

Over the last few years several interesting topologically schemes were proposed by us ranging from subwavelength comb potential [Phys. Rev. A 100, 033610] to atoms in double optical lattices (arXiv:2011.09228) or in the situation when an additional cavity induces long-range interaction between atoms (arXiv:2011.01687). In the case depicted above in the left panel the effective tunneling of green atoms between potential wells does not depend only on the depth of the potential but also on the state of different (red) atoms that sit in the second lattice and their state moderates the tunneling via the so called density dependent tunneling. By contrast, the standard typical kinetic tunneling is suppressed due to the lattice height. In the right panel the cavity mode serves a very similar purpose. The light scattered into the cavity mode leads to long-range interactions between atoms. For an appropriate choice of frequencies of the laser producing the standing wave and that of the resonant cavity mode the long-range interactions affect strongly the tunnelings via density dependent tunneling terms (which are long-ranged in this model) that, for mean density of 0.5 ( on the average one atom per two sites) leads to topological insulating state

Similar systems may be applied to answer questions coming from the lattice gauge theories. Here atomic simulations are performed at entirely different energy scale than typically envisioned for high energy physics, yet spectacular situations may be realized. We have studied, in particular, the bosonic Schwinger model, coupling particles and fields in a combined system which reveals quantum confinement, production of mesons and other aspects of relativistic quantum mechanics. The confinement is due to global conservation laws, notably Gauss law for the fictitious electric field. For Klein-Gordon model, by comparison no confinement appears and ballistic spread is observed.

Figure 3. The time dynamics of the entanglement entropy on different bonds. Confinenement in entropy production is apparent for bosonic Schwinger model (BSM) for large effective mass, while no confinement appears in Klein-Gordon model where no electric field conservation law is imposed. |

For a number of years it has been assumed that interaction between atoms results in thermalizaiton of the system leading to ergodic behavior. Recent years brought several counterexamples to this statement. Notably, strong disorder may lead to many-bosy localization; the disorder leads to a strong memory of the initial state that may lead to the creation of robust quantum memories. We are involved in studies of disordered models looking for limitiations of nonergodic behavior. On the other side recent developments suggest that nonergodicity may appear in simple potentials due to local electric field localization referred to as a Stark many body localization. We observed and theoretically explained the phase separation between localized and delocalized regions in space for the harmonic potential $V(x)=Ax^2/2$ with $A$ being the curvature. The system is localized when the local Stark field $F=Ax_0$ at point $x_0$ exceeds the border for Stark many-body localization (see arXiv:2004.00954)

Figure 4. Left: Time dependence of initial state of spinless fermions with every second site occupied and every second empty for different cirvatures of the harmonic trap potential. Right: Time dependence of the entanglement entropy on bonds between sites. Entropy rapidly grows in the delocalized middle regime staying low in the Stark many body localized parts. |

Interestingly, the very existence in the thermodynamic limit is questioned recently. Here top of the art numerics is used pushing the borders to larger sizes and new interesting models (see e.g. Phys. Rev. Lett. 124 186601 (2020) -- arXiv:1911.06221 or Phys. Rev. Lett. 125 156601 (2020) -- arXiv:2005.09534 or Phys. Rev. Research 2 032045(R) (2020)-- arXiv:2006.02860). It seems that a very important role in many body localization is played by possible conserved symmetries. These interesting issues are under intensive current research.

Figure 5. 1D space crystal with periodic boundary conditions. If a disordered potential is present, Anderson localization can take place. Then, when we travel along the ring we see an exponential localization of a particle around a certain space point. Right panel: when we switch from space to time crystals, we have to exchange the roles of space and time. That is, we fix position in space and ask how the probability density for the detection of a particle at this space point changes in time. If there is Anderson localization in the time domain, then this probability is exponentially localized around certain moment of time and such behavior is repeated periodically due to the periodic boundary conditions in time. |

Figure 6. Comparison of the Mott insulator phase in an ordinary 1D crystalline structure in space and in the time dimension. Panel (a) illustrates well defined numbers of bosons that occupy each well of a 1D spatially periodic potential on a ring. When we want to switch from space crystals (a) to time crystals (b) we have to exchange the role of space and time. That is, we fix position in space and ask how probability for the detection of particles at this fixed position changes in time. In the Mott insulating phase we observe that well defined numbers of bosons are arriving at the chosen space point like on a conveyor belt or in a machine gun. |

Our research is carried out thanks to support coming from National Science Centre (Poland) under grants:

- Many-body Localization - Cold atoms approach 2 NCN OPUS18 2019/35/B/ST2/00034
- Quantera H2020 QTFLAG: Quantum Technology For LAttice Gauge theories NCN Unisono 2017/25/Z/ST2/03029
- Periodically Driven Many-body systems NCN OPUS11 2016/21/B/ST2/01086
- Attoseconds in Biology, Chemistry and Physics - new experimental and theoretical approaches NCN SYMFONIA4 2016/20/W/ST4/00314
- Ultracold atoms in many-level systems, NCN OPUS18, 2019/35/B/ST2/00838

We acknowledge fantastic long time collaboration with teams of Dominique Delande (LKB, Paris) and Maciej Lewenstein (ICFO Barcelona) as well as no less fantastic recent collaboration with teams of Marcello Dalmonte and Antonello Schardicchio (ICTP, Trieste) or last but not least Giovanna Morigi (Univ. Saarland).