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 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.
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.
Floquet engineering of isolated systems is often based on the concept of the effective time-independent Floquet Hamiltonian, which describes the stroboscopic evolution of a periodically driven quantum system in steps of the driving period and which is routinely obtained analytically using high-frequency expansions. The generalization of these concepts to open quantum systems described by a Markovian master equation of Lindblad type turns out to be nontrivial: On the one hand, already for a two-level system two different phases can be distinguished, where the effective time-independent Floquet generator (describing the stroboscopic evolution) is either again Markovian and of Lindblad type or not. On the other hand, even though in the high-frequency regime a Lindbladian Floquet generator (Floquet Lindbladian) is numerically found to exist, this behavior is, curiously, not correctly reproduced within analytical high-frequency expansions. Here, we demonstrate that a proper Floquet Lindbladian can still be obtained from a high-frequency expansion, when treating the problem in a suitably chosen rotating frame. Within this approach, we can then also describe the transition to a phase at lower driving frequencies, where no Floquet Lindbladian exists, and show that the emerging non-Markovianity of the Floquet generator can entirely be attributed to the micromotion of the open driven system.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.
We study the stroboscopic and nonstroboscopic dynamics in the Floquet realization of the Harper-Hofstadter Hamiltonian. We show that the former produces the evolution expected in the high-frequency limit only for observables, which commute with the operator to which the driving protocol couples. On the contrary, nonstroboscopic dynamics is capable of capturing the evolution governed by the Floquet Hamiltonian of any observable associated with the effective high-frequency model. We provide exact numerical simulations for the dynamics of the number operator following a quantum cyclotron orbit on a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, as well as the chiral current operator flowing along the legs of a $2\ifmmode\times\else\texttimes\fi{}20$ ladder. The exact evolution is compared with its stroboscopic and nonstroboscopic counterparts, including finite-frequency corrections.
We generalize the Schrieffer-Wolff transformation to periodically driven systems using Floquet theory. The method is applied to the periodically driven, strongly interacting Fermi-Hubbard model, for which we identify two regimes resulting in different effective low-energy Hamiltonians. In the nonresonant regime, we realize an interacting spin model coupled to a static gauge field with a nonzero flux per plaquette. In the resonant regime, where the Hubbard interaction is a multiple of the driving frequency, we derive an effective Hamiltonian featuring doublon association and dissociation processes. The ground state of this Hamiltonian undergoes a phase transition between an ordered phase and a gapless Luttinger liquid phase. One can tune the system between different phases by changing the amplitude of the periodic drive.
We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold-atom experiments.