In quantum mechanics, adiabatic elimination is a standard tool that produces a low-lying reduced Hamiltonian for a relevant subspace of states, incorporating effects of its coupling to states with much higher energy. Suppose this powerful elimination approach is applied to quasi-energy states in periodically-driven systems, a critical question then arises that the violation of the adiabatic condition caused by driven forces challenges such a presence of spectral reduction in the non-equilibrium driven system. Here, both theoretically and experimentally, we newly reported two kinds of driven-induced eliminations universal in topologically-protected Floquet systems. We named them "quasi-adiabatic elimination" and "high-frequency-limited elimination", in terms of different driven frequencies that deny the underlying requirement for the adiabatic condition. Both two non-adiabatic eliminations are observed in our recently developed microwave Floquet simulator, a programmable test platform composed of periodically-bending ultrathin metallic coupled corrugated waveguides. Through the near-field imaging on our simulator, the mechanisms between the adiabatic and driven-induced eliminations are revealed, indicating the ubiquitous spectral decomposition for tailoring and manipulating Floquet states with quasi-energies. Finally, we hope our findings may open up profound and applicable possibilities for further developing Floquet engineering in periodically-driven systems, ranging from condensed matter physics to photonics.
In quantum mechanics, adiabatic elimination is a standard tool that produces a low-lying reduced Hamiltonian for a relevant subspace of states, incorporating effects of its coupling to states with much higher energy. Suppose this powerful elimination approach is applied to quasi-energy states in periodically-driven systems, a critical question then arises that the violation of the adiabatic condition caused by driven forces challenges such a presence of spectral reduction in the non-equilibrium driven system. Here, both theoretically and experimentally, we newly reported two kinds of driven-induced eliminations universal in topologically-protected Floquet systems. We named them "quasi-adiabatic elimination" and "high-frequency-limited elimination", in terms of different driven frequencies that deny the underlying requirement for the adiabatic condition. Both two non-adiabatic eliminations are observed in our recently developed microwave Floquet simulator, a programmable test platform composed of periodically-bending ultrathin metallic coupled corrugated waveguides. Through the near-field imaging on our simulator, the mechanisms between the adiabatic and driven-induced eliminations are revealed, indicating the ubiquitous spectral decomposition for tailoring and manipulating Floquet states with quasi-energies. Finally, we hope our findings may open up profound and applicable possibilities for further developing Floquet engineering in periodically-driven systems, ranging from condensed matter physics to photonics.
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.
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.
Periodic driving has emerged as a powerful experimental tool to engineer physical properties of isolated, synthetic quantum systems. However, due to the lack of energy conservation and heating effects, non-trivial (e.g., topological) many-body states in periodically driven (Floquet) systems are generally metastable. Therefore it is necessary to find strategies for preparing long-lived many-body states in Floquet systems. We develop a theoretical framework for describing the dynamical preparation of states in Floquet systems by a slow turn-on of the drive. We find that the dynamics of the system is well approximated by the initial state evolving under a slowly varying effective Hamiltonian Heff(s)(t), provided the ramp speed s≫t∗−1∼e−CJω, the inverse of the characteristic heating time-scale in the Floquet system. At such ramp speeds, the heating effects due to the drive are exponentially suppressed. We compute the slowly varying effective Hamiltonian Heff(s)(t), and show that at the end of the ramp it is identical to the effective Hamiltonian of the unramped Floquet system, up to small corrections of the order O(s). Therefore, the system effectively undergoes a slow quench from H0 to Heff. As an application, we consider the passage of the slow quench through a quantum critical point (QCP), and estimate the energy absorbed due to the non-adiabatic passage through the QCP via a Kibble-Zurek mechanism. By minimizing the energy absorbed due to both the drive and the ramp, we find an optimal ramp speed s∗∼t∗−z/(d+2z) for which both heating effects are exponentially suppressed. Our results bridge the gap between the numerous proposals to obtain interesting systems via Floquet engineering, and the actual preparation of such systems in their effective ground states.
Periodically-driven (Floquet) quantum systems are ubiquitous in science and technology. For example, when a laser illuminates a material or an AC voltage is applied to a device, the system is well-described by a time-periodic Hamiltonian. In recent years, periodic driving has been proposed, not just as a tool to excite and probe devices, but actually as a mechanism of engineering new phases of matter, some of which have no equilibrium analog. However, with this promise comes a serious problem. Intuitively, if energy is injected into and distributed throughout a system, it is no surprise that it tends to heat up indefinitely to infinite temperature. In this thesis, we study the mechanisms of heating, i.e. the process of thermalization, in Floquet systems and propose methods to control them. Specifically, for non-interacting Floquet systems that are coupled to external bosonic and fermionic baths (e.g. laser-driven electrons in a semiconductor that interact with phonons and an external lead), we classify the relevant scattering processes that contribute to cooling/heating in the Floquet bands and suggest methods to suppress heating via bandwidth-restrictions on the baths. We find that is possible, with appropriate dissipative engineering, to stabilize a controlled incompressible nonequilibrium steady-state resembling a ground state - a state we term the "Floquet insulator." We extend this analysis to include short-range interactions that contribute additional heating processes and show, under the same framework, that heating can be controlled with dissipation. In the process, we develop a simple effective model for the Floquet band densities that captures the essence of all the Floquet scattering processes and that is useful for ballparking experimentally-relevant estimates of heating. Next, we turn our attention to strongly-interacting closed Floquet systems and study how heating emerges through a proliferation of resonances. We find a novel integrable point governing the strong-interaction limit of the Floquet system and examine the breakdown of integrability via the proliferation of resonances. We observe two distinct scaling regimes, attributed to non-thermal and thermal behavior, and discover a power-law scaling of the crossover between them as a function of system size. The lingering ergodicity-breaking effects of the conserved quantities in the vicinity (in parameter space) of the integrable point at finite size is a phenomena we term "near-integrability." These results suggest that small quantum systems, which are accessible currently in many platforms (e.g. trapped ions, cold atoms, superconducting devices), intrinsically host non-thermal states that one may be able to utilize to avoid heating. Furthermore, our results suggest a "dual" interpretation, in the thermodynamic limit, that a periodically-driven system exhibits prethermalization as a power-law in interaction strength.
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 systems have emerged as a useful technique to engineer the properties of quantum systems, and are in the process of being developed into a standard toolbox for quantum simulation. An outstanding challenge that leaves this toolbox incomplete is the manipulation of the states dressed by strong periodic drives. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. To achieve fast control of nonequilibrium quantum matter, we generalize the notion of variational counterdiabatic driving away from equilibrium focusing on Floquet systems. We derive a nonperturbative variational principle to find local approximations to the adiabatic gauge potential for the effective Floquet Hamiltonian. It enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. We discuss applications to two-level, Floquet band, and interacting periodically driven models. The developed technique allows us to capture nonperturbative photon resonances and obtain high-fidelity protocols that respect experimental limitations like the locality of the accessible control terms.