Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micromotion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micromotion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time evolution of the system.
Quantum systems can show qualitatively new forms of behavior when they are driven by fast time-periodic modulations. In the limit of large driving frequency, the long-time dynamics of such systems can often be described by a time-independent effective Hamiltonian, which is generally identified through a perturbative treatment. Here, we present a general formalism that describes time-modulated physical systems, in which the driving frequency is large, but resonant with respect to energy spacings inherent to the system at rest. Such a situation is currently exploited in optical-lattice setups, where superlattice (or Wannier-Stark-ladder) potentials are resonantly modulated so as to control the tunneling matrix elements between lattice sites, offering a powerful method to generate artificial fluxes for cold-atom systems. The formalism developed in this work identifies the basic ingredients needed to generate interesting flux patterns and band structures using resonant modulations. Additionally, our approach allows for a simple description of the micromotion underlying the dynamics; we illustrate its characteristics based on diverse dynamic-lattice configurations. It is shown that the impact of the micromotion on physical observables strongly depends on the implemented scheme, suggesting that a theoretical description in terms of the effective Hamiltonian alone is generally not sufficient to capture the full time evolution of the system.
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.
Shaking optical lattices in a resonant manner offers an efficient and versatile method to devise artificial gauge fields and topological band structures for ultracold atomic gases. This was recently demonstrated through the experimental realization of the Harper–Hofstadter model, which combined optical superlattices and resonant time-modulations. Adding inter-particle interactions to these engineered band systems is expected to lead to strongly-correlated states with topological features, such as fractional Chern insulators. However, the interplay between interactions and external time-periodic drives typically triggers violent instabilities and uncontrollable heating, hence potentially ruling out the possibility of accessing such intriguing states of matter in experiments. In this work, we study the early-stage parametric instabilities that occur in systems of resonantly-driven Bose–Einstein condensates in optical lattices. We apply and extend an approach based on Bogoliubov theory (Lellouch et al 2017 Phys. Rev. X 7 021015) to a variety of resonantly-driven band models, from a simple shaken Wannier–Stark ladder to the more intriguing driven-induced Harper–Hofstadter model. In particular, we provide ab initio numerical and analytical predictions for the stability properties of these topical models. This work sheds light on general features that could guide current experiments to stable regimes of operation.
Periodic Wannier–Stark ladder structures of the energy resonances associated with Bloch oscillations can be readily modified into quasi-periodic ones that exhibit peculiar self-similar effects. A compact theoretical description of the dynamics of driven Bloch oscillations is developed here within the quasi-momentum representation. We identify a rather viable scheme based on ultracold atomic wavepackets subject to gravity in a driven optical lattice potential where a self-similar scaling could be observed. Its feasibility in terms of realistic experimental parameters is also discussed.
We study the dynamics and timescales of a periodically driven Fermi-Hubbard model in a three-dimensional hexagonal lattice. The evolution of the Floquet many-body state is analyzed by comparing it to an equivalent implementation in undriven systems. The dynamics of double occupancies for the near- and off-resonant driving regime indicate that the effective Hamiltonian picture is valid for several orders of magnitude in modulation time. Furthermore, we show that driving a hexagonal lattice compared to a simple cubic lattice allows us to modulate the system up to 1 s, corresponding to hundreds of tunneling times, with only minor atom loss. Here, driving at a frequency close to the interaction energy does not introduce resonant features to the atom loss.
Controlling and manipulating complex many-body quantum systems will be a key ingredient \nfor the development of next-generation technologies. While the realisation of a \nuniversal quantum machine is still out of reach, in recent years experimental systems of \nultracold atoms have already evolved into a vivid field of research for quantum simulation. \nCrucially, such systems even allow for the successful quantum engineering of targeted \nmany-body systems by means of coherent periodic driving. The essential properties of \nthese Floquet systems encompass two main aspects: fast driving facilitates the simulation \nof effective static systems, and interactions lead to unique heating effects as energy is only \nconserved modulo the driving frequency. Within this thesis we theoretically study both \nof these aspects in respective model systems. \nIn part I of this thesis, we investigate the dynamics of excitations of a bosonic Mott \ninsulator in a designed one-dimensional Floquet system. Here, periodic driving in combination \nwith breaking all mirror symmetries of the system can induce directed motion of \nparticles. In the limit of small excitation densities, the effectively non-interacting quantum \nratchet determines the motion of holes and doublons in the Mott insulator and can in fact \nbe used to manipulate the dynamics of such. This little quantum machine can also be \nused to drive particles against an external force, where transport is possible but requires \nthe fulfilment of a commensurability condition for long times. \nIn part II, we discuss the role of interactions for periodically driven systems by means \nof a Floquet version of the Boltzmann equation. Starting from the Keldysh approach, \nwe develop this semiclassical formalism based on a clear separation of time scales. The \nresult is a description of the dynamics and the scattering of Floquet quasiparticles in \nsuch systems. Here, the property of discrete energy violation is naturally encoded in our \nformalism predicting the heating of interacting Floquet systems to infinite temperatures in \nthe long-time limit. As a first application of this approach, we investigate a cold atom setup \nrealising the Haldane model by means of periodic shaking. While homogeneous systems \nheat up globally, a confining potential evokes thermoelectric transport effects resulting \nfrom spatially dependent heating characteristics. Moreover, we show that the interplay \nof intrinsic heating, macroscopic diffusion and non-trivial topological properties of the \nHaldane model lead to an anomalous Floquet-Nernst effect, which describes anomalous \nparticle transport as the result of developing temperature gradients. \nIn part III, we elaborate on the quantum simulator aspect of ultracold atoms by providing \na theoretical framework for a possible simulation of a topological edge state in a \none-dimensional optical lattice. In this case, the one-dimensional Dirac equation with \nspatially varying mass is important, which captures the topological properties of a corresponding \nsystem of the BDI symmetry class. We analytically discuss such system and \ninvestigate the role of mean-field interaction effects. We also identify the emergence of \ndynamical instabilities in a realisation with bosonic atoms.
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.