We present a direct comparison between the stroboscopic and nonstroboscopic effective approaches for ultracold atoms in shaken honeycomb lattices, focusing specifically on the optimal driving introduced by A. Verdeny and F. Mintert []. In the fast-driving regime, we compare the effective nonstroboscopic Hamiltonian derived through a perturbative expansion with a nonperturbative calculation of the stroboscopic Floquet Hamiltonian, obtained through a direct numerical approach. We find that the leading off-diagonal tunneling coefficients, which define the undriven model, retain their isotropic nature under the effect of the driving, and their values are independent of the computational approach used. This consistency ensures the robustness of certain topological properties, such as the position of Dirac points, as expected. Conversely, the next-to-leading diagonal tunneling coefficients, generated dynamically, can vary across different approaches without affecting the system topology. This suggests that the topological phase diagram is most conveniently represented in terms of the physical parameters characterizing the driving and directly accessible in experiments, rather than in terms of the tunneling parameters. Published by the American Physical Society 2024
We present a direct comparison between the stroboscopic and nonstroboscopic effective approaches for ultracold atoms in shaken honeycomb lattices, focusing specifically on the optimal driving introduced by A. Verdeny and F. Mintert []. In the fast-driving regime, we compare the effective nonstroboscopic Hamiltonian derived through a perturbative expansion with a nonperturbative calculation of the stroboscopic Floquet Hamiltonian, obtained through a direct numerical approach. We find that the leading off-diagonal tunneling coefficients, which define the undriven model, retain their isotropic nature under the effect of the driving, and their values are independent of the computational approach used. This consistency ensures the robustness of certain topological properties, such as the position of Dirac points, as expected. Conversely, the next-to-leading diagonal tunneling coefficients, generated dynamically, can vary across different approaches without affecting the system topology. This suggests that the topological phase diagram is most conveniently represented in terms of the physical parameters characterizing the driving and directly accessible in experiments, rather than in terms of the tunneling parameters. Published by the American Physical Society 2024
This thesis reports on the realization of Floquet topological systems with ultracold atoms in an optical honeycomb lattice. Using periodically driven quantum systems, topological phases of matter can be simulated by an effective, static Hamiltonian related to the time-evolution at integer multiples of the driving period, which is known as Floquet engineering. However, periodic driving can also give rise to genuinely time-dependent settings without static counterparts. One example is the anomalous Floquet phase in two dimensions, in which all bulk bands have a Chern number of zero but nevertheless robust chiral edge modes appear, which would be precluded by the bulk-edge correspondence in a static system. In an optical honeycomb lattice, anomalous Floquet systems can be created by continuous, periodic modulation of the laser intensities. This driving scheme results in different topological regimes, three of which are investigated, including the anomalous Floquet phase and a Haldane-like phase. Periodically driven systems feature periodic quasienergies and can be characterized in terms of winding numbers, which count the number of chiral edge modes in each quasienergy gap. By interferometric measurements of the quasienergy gaps between the first two Floquet bands, the topological phase transitions, emerging as gap closings, are located. For periodically driven systems, a modified bulk-edge correspondence can be formulated. In particular, the change of the winding number at a phase transition is related to the sign change of the local Berry curvature. The Berry curvature at the gap closing points is probed by Hall deflection measurements to obtain the winding numbers in each of the topological regimes, revealing the existence of chiral edge modes also in a setting with smooth boundaries as used in the experiments. The measured quasienergy gaps and transverse deflections are quantitatively well described by a numerical calculation of the Floquet bandstructure that includes coupling to higher bands during the driving period. To derive the spectrum of the modulated lattice in a geometry with edges, a tight-binding description of the system is discussed. Circular phase modulation of the honeycomb lattice can also give rise to an anomalous Floquet system when inversion symmetry is broken. The topological phase diagrams and the experimental feasibility are compared for both modulation schemes. Due to the vanishing Chern numbers, the bulk states in the anomalous Floquet regime can be fully localized by a disorder potential. This could prevent heating in interacting, periodically driven systems, resulting in a many-body localized bulk coexisting with thermalizing edge states. The experimental realization of disorder and strong interactions, as well as independent probes of the edge states are investigated. Bloch band geometry can be extended to multiband systems, using Wilson lines. The possible symmetry protection of their eigenvalues is discussed, along with measurements in the optical honeycomb lattice. Moreover, the onset of heating in weakly-interacting, periodically driven systems, triggered by parametric instabilities, is studied experimentally in a one-dimensional lattice.
Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting (1+1)D Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a (1+1)D Wilson-Dirac theory with one quarter of the spatial sites.
Inspired by the recent creation of a honeycomb optical lattice and the realization of a Mott-insulating state in a square lattice by shaking, we study here the shaken honeycomb optical lattice. For a periodic shaking of the lattice, Floquet theory may be applied to derive a time-independent Hamiltonian. In this effective description, the hopping parameters are renormalized by a Bessel function, which depends on the shaking direction, amplitude, and frequency. Consequently, the hopping parameters can vanish and even change sign, in an anisotropic manner, thus yielding different band structures. Here, we study the merging and the alignment of Dirac points and dimensional crossovers from the two-dimensional system to one-dimensional chains and zero-dimensional dimers. We also consider next-nearest-neighbor hopping, which breaks the particle-hole symmetry and leads to a metallic phase when it becomes dominant over the nearest-neighbor hopping. Furthermore, we include weak repulsive on-site interactions and find the density profiles for different values of the hopping parameters and interactions, both in a homogeneous system and in the presence of a trapping potential. Our results may be experimentally observed by use of momentum-resolved Raman spectroscopy.
Tilted lattice potentials with periodic driving play a crucial role in the study of artificial gauge fields and topological phases with ultracold quantum gases. However, driving-induced heating and the growth of phonon modes restrict their use for probing interacting many-body states. Here, we experimentally investigate phonon modes and interaction-driven instabilities of superfluids in the lowest band of a shaken optical lattice. We identify stable and unstable parameter regions and provide a general resonance condition. In contrast to the high-frequency approximation of a Floquet description, we use the micromotion of the superfluids to analyze the growth of phonon modes from slow- to fast-driving frequencies. The model describes phonon excitations in both resonantly and nonresonantly driven systems, with or without a tilted potential. Our observations enable the prediction of stable parameter regimes for quantum-simulation experiments aimed at studying driven systems with strong interactions over extended time scales. Published by the American Physical Society 2024
Periodically driven quantum systems offer an exceptional platform for quantum simulations due to the possibility to approximate their dynamics in terms of time-independent effective Hamiltonians. This, together with the recent experimental advances, has situated driven systems at center stage of engineered quantum-mechanical devices. The aim of this thesis is to develop theoretical methods in order to design optimal quantum simulations with driven systems. By applying the derived tools to experimentally relevant models, the applicability and significance of the methods are furthermore demonstrated. First, we introduce a method to derive accurate effective Hamiltonians by merging two seemingly unrelated tools: Floquet theory and flow equations. With this, the required analytical identification of the effective Hamiltonian in terms of the system's parameters is achieved. Second, we identify structural properties that determine the accessible effective dynamics of a system of particles on shaken optical lattices, which is arguably one of the most remarkable systems for many-body quantum simulations. In particular, we identify fundamental symmetries of the underlying lattice geometry that determine the emergence of new tunneling processes. Third, we develop an optimal control scheme to design polychromatic driving protocols that optimally simulate specifically targeted dynamics. We apply this scheme to demonstrate an optimal realization of Raman transitions with a Lambda system, a building block in many quantum simulations. Then, we employ it to implement a topological Chern insulator through suitably engineering the geometry-dependent tunneling of particles on a shaken hexagonal lattice. Hereby, a realistic route to experimentally test strongly-correlated topological phases of matter is provided. By determining structural properties of driven systems and suitable driving protocols, the methods described in this thesis open substantial possibilities for the development of optimal quantum simulations and, ultimately, reliable quantum technologies.
Complex quantum many-body systems are ubiquitous in nature, yet their behaviour often remains very challenging to predict with analytical or numerical calculations.In this thesis a different approach is presented, using ultracold fermionic 40 K atoms in a lattice with a flexible geometry, formed by interfering laser beams.This system constitutes a precisely tunable, yet very accessible complex system, which can be used to probe -or "quantum simulate" -phenomena predicted for condensed-matter systems.We demonstrate, theoretically and experimentally, how a periodically modulated system can be described by an effective Hamiltonian on longer time-scales.This approach based on Floquet theory applies even when driving the system far from equilibrium.Using oscillating magnetic field gradients, we create an effectively spin-dependent lattice, where the tunnelling rate for different internal atomic states can be freely tuned.The flexible geometry of the lattice is used to create a honeycomb lattice and study the linear band crossings -known as Dirac cones -characteristic of its band structure.We distort the lattice and use momentum-resolved interband transitions probe the deformation and movement of the Dirac cones and their eventual disappearance when moving towards a square or dimerised geometry.We show that circular modulation of a honeycomb lattice induces imaginary next-nearest neighbour tunnelling, which breaks time-reversal symmetry.In addition, a tunable site-offset is used to break inversion symmetry.We thereby realize a model proposed by F.D.M. Haldane in 1988 and study the predicted topological transition from a trivial to a Chern-insulating regime, characterised by a closing band gap.Furthermore, effects of the Berry curvature are observed in both regimes, using transverse drifts of a driven atomic cloud.Using a mixture of two internal states with tunable scattering between them, we study the Hubbard model on a variety of geometries and with a range of observables.The crossover between metal, Mott-insulator and band insulator in a two-and threedimensional Honeycomb lattice is probed by observing the double occupancy, compressibility and excitation spectrum.The addition of a site-offset leads to the appearance of a competing type of density order -an explicit charge-density wave -which is directly probed and leads to new types of excitations.At lower temperatures, magnetic correlations appear.By freezing out the dynamics of the system and adiabatically merging neighbouring sites,the probability to find singlet or triplet states on different patterns of bonds is directly probed.Using this method we study the dependence of magnetic correlations on geometry in seven different lattice structures, and crossovers between them.We find that lower connectivity quantum effects enhance short-range correlations and compare our results to sophisticated numerical approximations.Ramping the lattice from one geometry to another allows for probing the quantum dynamics of re-distributing magnetic correlations, from the sudden to the adiabatic regime.v