Atomic quantum gases in periodically driven optical lattices

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.Comment: Review, accepted for publication as Colloquium in Reviews of Modern Physic

High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective

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 [Phys.\ Rev.\ A {\bf 68}, 013820 (2003), Phys.\ Rev.\ X {\bf 4}, 031027 (2014)] and can also be related to the Floquet-Magnus expansion [J.\ Phys.\ A {\bf 34}, 3379 (2000)]. 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.Comment: 48 pages, 7 figure

Interband heating processes in a periodically driven optical lattice

We investigate multi-"photon" interband excitation processes in an optical lattice that is driven periodically in time by a modulation of the lattice depth. Assuming the system to be prepared in the lowest band, we compute the excitation spectrum numerically. Moreover, we estimate the effective coupling parameters for resonant interband excitation processes analytically, employing degenerate perturbation theory in Floquet space. We find that below a threshold driving strength, interband excitations are suppressed exponentially with respect to the inverse driving frequency. For sufficiently low frequencies, this leads to a rather sudden onset of interband heating, once the driving strength reaches the threshold. We argue that this behavior is rather generic and should also be found in lattice systems that are driven by other forms of periodic forcing. Our results are relevant for Floquet engineering, where a lattice system is driven periodically in time in order to endow it with novel properties like the emergence of a strong artificial magnetic field or a topological band structure. In this context, interband excitation processes correspond to detrimental heating.Comment: 11 pages, 4 figure

Orbital-driven melting of a bosonic Mott insulator in a shaken optical lattice

In order to study the interplay between localized and dispersive orbital states in a system of ultracold atoms in an optical lattice, we investigate the possibility to coherently couple the lowest two Bloch bands by means of resonant periodic forcing. Considering bosons in one dimension, it is shown that a strongly interacting Floquet system can be realized, where at every lattice site two (and only two) near-degenerate orbital states are relevant. By smoothly tuning both states into resonance we find that the system can undergo an orbital-driven Mott-insulator-to-superfluid transition. As an intriguing consequence of the kinetic frustration in the system, this transition can be either continuous or first-order, depending on parameters such as lattice depth and filling.Comment: 7 pages, 3 figure

Quantum crystal growing: Adiabatic preparation of a bosonic antiferromagnet in the presence of a parabolic inhomogeneity

We theoretically study the adiabatic preparation of an antiferromagnetic phase in a mixed Mott insulator of two bosonic atom species in a one-dimensional optical lattice. In such a system one can engineer a tunable parabolic inhomogeneity by controlling the difference of the trapping potentials felt by the two species. Using numerical simulations we predict that a finite parabolic potential can assist the adiabatic preparation of the antiferromagnet. The optimal strength of the parabolic inhomogeneity depends sensitively on the number imbalance between the two species. We also find that during the preparation finite size effects will play a crucial role for a system of realistic size. The experiment that we propose can be realized, for example, using atomic mixtures of Rubidium 87 with Potassium 41 or Ytterbium 168 with Ytterbium 174.Comment: 25 pages, 6 figure

Avoided level crossing spectroscopy with dressed matter waves

We devise a method for probing resonances of macroscopic matter waves in shaken optical lattices by monitoring their response to slow parameter changes, and show that such resonances can be disabled by particular choices of the driving amplitude. The theoretical analysis of this scheme reveals far-reaching analogies between dressed atoms and time-periodically forced matter waves.Comment: 4 pages, 3 figure

The optimal frequency window for Floquet engineering in optical lattices

The concept of Floquet engineering is to subject a quantum system to time-periodic driving in such a way that it acquires interesting novel properties. It has been employed, for instance, for the realization of artificial magnetic fluxes in optical lattices and, typically, it is based on two approximations. First, the driving frequency is assumed to be low enough to suppress resonant excitations to high-lying states above some energy gap separating a low energy subspace from excited states. Second, the driving frequency is still assumed to be large compared to the energy scales of the low-energy subspace, so that also resonant excitations within this space are negligible. Eventually, however, deviations from both approximations will lead to unwanted heating on a time scale $\tau$. Using the example of a one-dimensional system of repulsively interacting bosons in a shaken optical lattice, we investigate the optimal frequency (window) that maximizes $\tau$. As a main result, we find that, when increasing the lattice depth, $\tau$ increases faster than the experimentally relevant time scale given by the tunneling time $\hbar/J$, so that Floquet heating becomes suppressed.Comment: 11 pages, 8 figure

Bath-induced decay of Stark many-body localization

We investigate the relaxation dynamics of an interacting Stark-localized system coupled to a dephasing bath, and compare its behavior to the conventional disorder-induced many body localized system. Specifically, we study the dynamics of population imbalance between even and odd sites, and the growth of the von Neumann entropy. For a large potential gradient, the imbalance is found to decay on a time scale that grows quadratically with the Wannier-Stark tilt. For the non-interacting system, it shows an exponential decay, which becomes a stretched exponential decay in the presence of finite interactions. This is different from a system with disorder-induced localization, where the imbalance exhibits a stretched exponential decay also for vanishing interactions. As another clear qualitative difference, we do not find a logarithmically slow growth of the von-Neumann entropy as it is found for the disordered system. Our findings can immediately be tested experimentally with ultracold atoms in optical lattices

Tomography of band insulators from quench dynamics

We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing---via time-of-flight imaging---the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with $\pi$-flux and the Haldane model on a honeycomb lattice. Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise, and finite measurement resolution.Comment: v2: 5+5 pages, 3+5 figures; added analytical and numerical results for the presence of a harmonic confinement. v3: Minor changes; as accepted in PR

A unified theory for excited-state, fragmented, and equilibrium-like Bose condensation in pumped photonic many-body systems

We derive a theory for Bose condensation in nonequilibrium steady states of bosonic quantum gases that are coupled both to a thermal heat bath and to a pumped reservoir (or gain medium), while suffering from loss. Such a scenario describes photonic many-body systems such as exciton-polariton gases. Our analysis is based on a set of kinetic equations for a gas of noninteracting bosons. By identifying a dimensionless scaling parameter controlling the boson density, we derive a sharp criterion for which system states become selected to host a macroscopic occupation. We show that with increasing pump power, the system generically undergoes a sequence of nonequilibrum phase transitions. At each transition a state either becomes or ceases to be Bose selected (i.e. to host a condensate): The state which first acquires a condensate when the pumping exceeds a threshold is the one with the largest ratio of pumping to loss. This intuitive behavior resembles simple lasing. In the limit of strong pumping, the coupling to the heat bath becomes dominant so that eventually the ground state is selected, corresponding to equilibrium(-like) Bose condensation. For intermediate pumping strengths, several states become selected giving rise to fragmented nonequilibrium Bose condensation. We compare these predictions to experimental results obtained for excitons polaritons in a double-pillar structure [Phys. Rev. Lett. 108, 126403 (2012)] and find good agreement. Our theory, moreover, predicts that the reservoir occupation is clamped at a constant value whenever the system hosts an odd number of Bose condensates