We present theoretical methods for studying quantum mechanical systems subjected to fast periodic driving and apply them to model systems with long-range interaction. We provide a comparison between the methods and insight facilitated by these methods. The methods recently occurred in scientific papers, which supports the need for a scrutinized exposition of the theory. One of the main objects of the theory is a so-called Floquet Hamiltonian-an artificial stationary Hamiltonian describing important features of a quantum system. The methods construct Floquet Hamiltonians in the form of series in the powers of the time period. We present the spectra of Floquet Hamiltonians-the so-called quasienergy spectra-computed by the methods and computed numerically (with higher precision). The quasienergy spectra were computed using various approximations of Floquet Hamiltonians and compared. We discuss an interesting topic of the classical limit of an artificial stationary system. We also mention the kicked rotor system and its connection with the kicked top system-one of our model systems. In summary, the method characterized by simultaneous construction of a Floquet Hamiltonian and a so-called kick operator (operator capturing fast changes of the system) was found universal and accurate. The thesis presents an...
We present theoretical methods for studying quantum mechanical systems subjected to fast periodic driving and apply them to model systems with long-range interaction. We provide a comparison between the methods and insight facilitated by these methods. The methods recently occurred in scientific papers, which supports the need for a scrutinized exposition of the theory. One of the main objects of the theory is a so-called Floquet Hamiltonian-an artificial stationary Hamiltonian describing important features of a quantum system. The methods construct Floquet Hamiltonians in the form of series in the powers of the time period. We present the spectra of Floquet Hamiltonians-the so-called quasienergy spectra-computed by the methods and computed numerically (with higher precision). The quasienergy spectra were computed using various approximations of Floquet Hamiltonians and compared. We discuss an interesting topic of the classical limit of an artificial stationary system. We also mention the kicked rotor system and its connection with the kicked top system-one of our model systems. In summary, the method characterized by simultaneous construction of a Floquet Hamiltonian and a so-called kick operator (operator capturing fast changes of the system) was found universal and accurate. The thesis presents an...
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 demonstrate the strength of the method of Peres lattices in periodically driven quantum systems. The method, which has previously been used mostly in stationary systems, enables us to efficiently detect resonances in the driven system, to monitor the onset of chaos, and to recognize critical properties of the Floquet modes. It also allows quick comparisons of the spectra of Floquet modes for various driving Hamiltonians and transparent tests of the iterative approximation techniques based on effective stationary Hamiltonians.
Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating.
We present an approach for efficiently simulating strongly damped quantum systems subjected to periodic driving, employing a periodic matrix product operator representation of the influence functional. This representation enables the construction of a numerically exact Floquet propagator that captures the non-Markovian open system dynamics, thus providing a dissipative analogue to the Floquet Hamiltonian of driven isolated quantum systems. We apply this method to study the asymptotic heating of a reservoir in spin-boson models, characterizing the deviation from equilibrium conditions. Moreover, we show how a local driving of two qubits can be utilized to stabilize a transient entanglement buildup of the qubits originating from the interaction with a common environment. Our results make it possible to directly study both stationary and transient dynamics of strongly damped and driven quantum systems within a transparent theoretical and numerical framework.
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.
Periodically driven quantum systems exhibit many fascinating phenomena absent in equilibrium systems, but their simulation is more challenging than that of static systems. Consequently, quantum simulation of these systems offers greater opportunity for achieving quantum advantage. To build the foundation for simulating time-periodic Hamiltonians, we utilize the Floquet-Hilbert space formalism to transform the Hamiltonian into a time-independent form and provide its Pauli decomposition. Within this formalism, we develop a hybrid quantum-classical algorithm, Floquet-ADAPT-VQE, that produces three key quantities that govern periodically driven quantum dynamics: the quasienergy, the associated Floquet eigenstates, and expectation values of observables with respect to these eigenstates. The latter is achieved by applying a fixed-depth quantum circuit to an initial Floquet eigenstate—an efficient construction made possible by the Floquet-Hilbert space formalism. We also identify a suitable auxiliary initial state whose squared Floquet energy is independent of the number of auxiliary qubits as well as the driving frequency, which leads to better convergence with fewer ADAPT iterations. We demonstrate our algorithm by performing numerical simulations on a periodically driven XYZ model with a magnetic field. We also explore potential applications of our algorithm for studying various non-equilibrium phenomena in periodically driven systems.