This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.
This dissertation presents a self-contained study of periodically-driven quantum systems. Following a brief introduction to Floquet theory, we introduce the inverse-frequency expansion, variants of which include the Floquet-Magnus, van Vleck, and Brillouin-Wigner expansions. We reveal that the convergence properties of these expansions depend strongly on the rotating frame chosen, and relate the former to the existence of Floquet resonances in the quasienergy spectrum. The theoretical design and experimental realisation (`engineering') of novel Floquet Hamiltonians is discussed introducing three universal high-frequency limits for systems comprising single-particle and many-body linear and nonlinear models. The celebrated Schrieffer-Wolff transformation for strongly-correlated quantum systems is generalised to periodically-driven systems, and a systematic approach to calculate higher-order corrections to the Rotating Wave Approximation is presented. Next, we develop Floquet adiabatic perturbation theory from first principles, and discuss extensively the adiabatic state preparation and the corresponding leading-order non-adiabatic corrections. Special emphasis is thereby put on geometrical and topological objects, such as the Floquet Berry curvature and the Floquet Chern number obtained within linear response in the presence of the drive. Last, pre-thermalisation and thermalisation in closed, clean periodically-driven quantum systems are studied in detail, with the focus put on the crucial role of Floquet many-body resonances for energy absorption.
We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.
We construct a systematic high-frequency expansion for periodically driven quantum systems based on the Brillouin-Wigner (BW) perturbation theory, which generates an effective Hamiltonian on the projected zero-photon subspace in the Floquet theory, reproducing the quasienergies and eigenstates of the original Floquet Hamiltonian up to desired order in $1/\ensuremath{\omega}$, with $\ensuremath{\omega}$ being the frequency of the drive. The advantage of the BW method is that it is not only efficient in deriving higher-order terms, but even enables us to write down the whole infinite series expansion, as compared to the van Vleck degenerate perturbation theory. The expansion is also free from a spurious dependence on the driving phase, which has been an obstacle in the Floquet-Magnus expansion. We apply the BW expansion to various models of noninteracting electrons driven by circularly polarized light. As the amplitude of the light is increased, the system undergoes a series of Floquet topological-to-topological phase transitions, whose phase boundary in the high-frequency regime is well explained by the BW expansion. As the frequency is lowered, the high-frequency expansion breaks down at some point due to band touching with nonzero-photon sectors, where we find numerically even more intricate and richer Floquet topological phases spring out. We have then analyzed, with the Floquet dynamical mean-field theory, the effects of electron-electron interaction and energy dissipation. We have specifically revealed that phase transitions from Floquet-topological to Mott insulators emerge, where the phase boundaries can again be captured with the high-frequency expansion.
We present a brief overview of some of the analytic perturbative techniques for the computation of the Floquet Hamiltonian for a periodically driven, or Floquet, quantum many-body system. The key technical points about each of the methods discussed are presented in a pedagogical manner. They are followed by a brief account of some chosen phenomena where these methods have provided useful insights. We provide an extensive discussion of the Floquet-Magnus (FM) expansion, the adiabatic-impulse approximation, and the Floquet perturbation theory. This is followed by a relatively short discourse on the rotating wave approximation, a FM resummation technique and the Hamiltonian flow method. We also provide a discussion of some open problems which may possibly be addressed using these methods.
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.
The manifestations of topology are ubiquitous in condensed matter physics. One of the most striking examples is the integer quantum Hall effect, which served as a theoretical basis for developing many classes of new materials in the last years such as topological insulators. It has been realized that topology can manifest itself also in time-dependent quantum systems. The question that we try to address in this thesis is how much of these topological structures can be harnessed in the realistic dynamics of a periodically driven closed quantum system. We investigate two kinds of topological periodically driven systems Thouless pumping and Floquet Chern insulators. Our analysis shows that the nonequilibrium occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasienergies, play a crucial role. In the case of Thouless pumping we analysed the pumped charge averaged over many cycles and discovered that the corrections to perfect quantization of the pumped charge are actually quadratic in the frequency of the driving for a suddenly switched-on driving. In the case of Floquet Chern insulators, we studied fermions on the honeycomb lattice subjected to a circularly polarised field. We found that while the quasienergy bulk states can be populated in a controlled way, edge states get unavoidable excitations due to exponentially small gaps preventing adiabaticity. These defects reflect in nonequilibrium currents flowing at the edges, that can be manipulated by tailoring the parameters of the periodic driving. Nevertheless, the nonequilibrium state undergoes a generalized form of topological transition, thanks to the controlled population of bulk states. We describe it with a generalized version of the Chern marker of Bianco and Resta
Periodically driven quantum systems have emerged as one of the most versatile tools for quantum simulation, enabling the engineering of effective Hamiltonians, topological phases, and inherently nonequilibrium phenomena such as discrete time crystals. Yet fundamental challenges persist: from a spectral ambiguity that precludes a unique Floquet ground state to an inability to manipulate Floquet eigenstates beyond the adiabatic regime. This thesis demonstrates that counterdiabatic driving—a technique from adiabatic quantum control—provides a unifying framework to address these challenges. We first show that the periodic lab-frame Hamiltonian generates counterdiabatic driving for the Floquet eigenstates, revealing Floquet theory as an inverse counterdiabatic driving problem. Exploiting this equivalence, we develop geometric Floquet theory: a reformulation based on the parallel-transport gauge, which uniquely decomposes Floquet dynamics into a geometric contribution, encoded in Berry phases, and a dynamical contribution, generated by the average-energy operator. The average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifies a unique Floquet ground state at all drive frequencies, and serves as a sensitive diagnostic for heating and symmetry-breaking transitions. Inherently nonequilibrium phenomena—the 𝜋-quasienergy splitting of discrete time crystals and the anomalous edge modes of Floquet topological insulators—are traced to geometric phases. On the practical side, we exploit the intrinsic robustness of prethermal discrete time crystals for a highly frequency-selective DTC-based AC magnetic field sensing scheme and demonstrate it experimentally on a nuclear spin quantum simulator. We then generalize counterdiabatic driving to periodically driven systems. In particular, we derive a non-perturbative variational principle for the Floquet adiabatic gauge potential; this yields local, experimentally feasible control protocols for transitionless manipulation of Floquet states. We demonstrate this formalism on Floquet state preparation in an ultracold atom quantum simulator and many-body state transfer across a quantum phase transition, with significant improvements over unassisted protocols in both cases.