Time periodic forcing in the form of coherent radiation is a standard tool for the coherent manipulation of small quantum systems like single atoms. In the last years, periodic driving has more and more also been considered as a means for the coherent control of many-body systems. In particular, experiments with ultracold quantum gases in optical lattices subjected to periodic driving in the lower kilohertz regime have attracted a lot of attention. Milestones include the observation of dynamic localization, the dynamic control of the quantum phase transition between a bosonic superfluid and a Mott insulator, as well as the dynamic creation of strong artificial magnetic fields and topological band structures. This article reviews these recent experiments and their theoretical description. Moreover, fundamental properties of periodically driven many-body systems are discussed within the framework of Floquet theory, including heating, relaxation dynamics, anomalous topological edge states, and the response to slow parameter variations.

Magnetism plays a key role in modern technology and stimulates research in several branches of condensed matter physics. Although the theory of classical magnetism is well developed, the demonstration of a widely tunable experimental system has remained an elusive goal. Here, we present the realization of a large-scale simulator for classical magnetism on a triangular lattice by exploiting the particular properties of a quantum system. We use the motional degrees of freedom of atoms trapped in an optical lattice to simulate a large variety of magnetic phases: ferromagnetic, antiferromagnetic, and even frustrated spin configurations. A rich phase diagram is revealed with different types of phase transitions. Our results provide a route to study highly debated phases like spin-liquids as well as the dynamics of quantum phase transitions.

We demonstrate that the transition from a superfluid to a Mott insulator in the Bose-Hubbard model can be induced by an oscillating force through an effective renormalization of the tunneling matrix element. The mechanism involves adiabatic following of Floquet states, and can be tested experimentally with Bose-Einstein condensates in periodically driven optical lattices. Its extension from small to very large systems yields nontrivial information on the condensate dynamics.

We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. The necessary Peierls phase of the hopping parameters between neighboring lattice sites is generated by applying a suitable periodic inertial force such that the method does not rely on any internal structure of the particles. We experimentally demonstrate the realization of such artificial potentials, which generate ground state superfluids at arbitrary non-zero quasi-momentum. We furthermore investigate possible implementations of this scheme to create tuneable magnetic fluxes, going towards model systems for strong-field physics.First introduced in electromagnetism, gauge fields play a central role in the description of interactions in physics, from particle physics to condensed matter. Currently there is a large interest to introduce gauge fields into model systems in order to study fundamental aspects of physics [1]. Especially the emulation of synthetic electric and magnetic fields for of ultracold atomic systems is crucial in order to extend their proven quantum simulation abilities further, e.g. to quantum Hall physics or topological insulators. In this context, the analogy between inertial and Lorentz forces triggered the simulation of homogeneous artificial magnetic fields using rapidly rotating trapped ultracold gases [2]. Recently, several proposals ([3, 4] and references therein) and experimental realizations focused on the simulation of a gauge vector potential A either in a bulk system [5,6] or in optical lattices [7,8]. The realized schemes exploit the Berry phase which arises when the atomic ground state is split in several space-dependent sublevels, as in the presence of an electromagnetic field. Hence they rely on the coupling between internal and external degrees of freedom induced by laser fields. Here we demonstrate the generation of artificial gauge potentials for neutral atoms in an optical lattice without any requirements on the specific internal structure. As the realized scheme only relies on the trapability of the particle, it can be very widely applied to many atomic systems, in principle also to molecules and other complex particles. It is particularly interesting for fermionic systems, where in a many-body state governed by Pauli principle, the use of internal degrees of freedom often lead to conflicts with the creation of gauge potentials. As an important additional benefit, the internal degrees of freedom of the particles can be addressed independently, e.g. by real magnetic fields or microwave excitations. In general, the presence of a gauge vector potential modifies the kinetic part of the Hamiltonian describing the system. In a lattice, an artificial field can then be simulated by engineering a complex tunneling parameter J = |J| · e iθ , where θ is the Peierls phase. The central approach here is to control this phase via a suitable forcing of the lattice potential, acting at the single-particle level. We describe the general scheme for the...

We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach (Rahav et al 2003 Phys. Rev. A 68 013820), (Goldman and Dalibard 2014 Phys. Rev. X 4 031027) and can also be related to the Floquet-Magnus expansion (Casas et al 2001 J. Phys. A 34 3379). We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking. The high-frequency approach is illustrated using the example of a periodically driven Hubbard model. Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.