We investigate a heating phenomenon in periodically driven integrable systems that can be mapped to free-fermion models. We find that heating to the high-temperature state, which is a typical scenario in nonintegrable systems, can also appear in integrable time-periodic systems; the amount of energy absorption rises drastically near a frequency threshold where the Floquet-Magnus expansion diverges. As the driving period increases, we also observe that the effective temperatures of the generalized Gibbs ensemble for conserved quantities go to infinity. By the use of the scaling analysis, we reveal that, in the limit of infinite system size and driving period, the steady state after a long time is equivalent to the infinite-temperature state. We obtain the asymptotic behavior L^{-1} and T^{-2} as to how the steady state approaches the infinite-temperature state as the system size L and the driving period T increase.
We investigate a heating phenomenon in periodically driven integrable systems that can be mapped to free-fermion models. We find that heating to the high-temperature state, which is a typical scenario in nonintegrable systems, can also appear in integrable time-periodic systems; the amount of energy absorption rises drastically near a frequency threshold where the Floquet-Magnus expansion diverges. As the driving period increases, we also observe that the effective temperatures of the generalized Gibbs ensemble for conserved quantities go to infinity. By the use of the scaling analysis, we reveal that, in the limit of infinite system size and driving period, the steady state after a long time is equivalent to the infinite-temperature state. We obtain the asymptotic behavior L^{-1} and T^{-2} as to how the steady state approaches the infinite-temperature state as the system size L and the driving period T increase.
We discuss the universal nature of relaxation in isolated many-body quantum systems subjected to global and strong periodic driving. Our rigorous Floquet analysis shows that the energy of the system remains almost constant up to an exponentially long time in frequency for arbitrary initial states and that an effective Hamiltonian obtained by a truncation of the Floquet-Magnus expansion is a quasiconserved quantity in a long time scale. These two general properties lead to an intriguing classification on the initial stage of relaxation, one of which is similar to the prethermalization phenomenon in nearly integrable systems.
Does a closed quantum many-body system that is continually driven with a time-dependent Hamiltonian finally reach a steady state? This question has only recently been answered for driving protocols that are periodic in time, where the long-time behavior of the local properties synchronizes with the drive and can be described by an appropriate periodic ensemble. Here, we explore the consequences of breaking the time-periodic structure of the drive with additional aperiodic noise in a class of integrable systems. We show that the resulting unitary dynamics leads to new emergent steady states in at least two cases. While any typical realization of random noise causes eventual heating to an infinite-temperature ensemble for all local properties in spite of the system being integrable, noise that is self-similar in time leads to an entirely different steady state (which we dub the ``geometric generalized Gibbs ensemble'') that emerges only after an astronomically large time scale. To understand the approach to the steady state, we study the temporal behavior of certain coarse-grained quantities in momentum space that fully determine the reduced density matrix for a subsystem with size much smaller than the total system. Such quantities provide a concise description for any drive protocol in integrable systems that are reducible to a free-fermion representation.
We investigate a class of periodically driven many-body systems that allows us to extend the phenomenon of prethermalization to the vicinity of isolated intermediate-to-low drive frequencies away from the high-frequency limit. We provide numerical evidence for the formation of a parametrically long-lived prethermal plateau, captured by an effective Floquet Hamiltonian computed using the replica inverse-frequency expansion, and demonstrate its stability with respect to random perturbations in the drive period. Considering exclusively nonintegrable Floquet Hamiltonians, we find that heating rates are nonuniversal: we observe Fermi's golden rule scaling, power-law scaling inconsistent with the golden rule, and non-power-law scaling, depending on the drive. Despite the asymptotic character of the inverse-frequency expansion, we show that it describes the thermostatic properties of the state all along the evolution up to infinite temperature, with higher-order terms improving the accuracy. Our results suggest a dynamical mechanism to gradually increase the temperature in isolated quantum simulators, such as ultracold atoms, and open up an alternative possibility to investigate thermal phase transitions and the interplay between thermal and quantum criticality using Floquet drives.
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.
We investigate a class of periodically driven many-body systems that allows us to extend the phenomenon of prethermalization to the vicinity of isolated intermediate-to-low drive frequencies away from the high-frequency limit. We provide numerical evidence for the formation of a parametrically long-lived prethermal plateau, captured by an effective Floquet Hamiltonian computed using the replica inverse-frequency expansion, and demonstrate its stability w.r.t.~random perturbations in the drive period. Considering exclusively nonintegrable Floquet Hamiltonians, we find that heating rates are non-universal: we observe Fermi's Golden Rule scaling, power-law scaling inconsistent with the Golden Rule, and non-power law scaling, depending on the drive. Despite the asymptotic character of the inverse-frequency expansion, we show that it describes the thermostatic properties of the state all along the evolution up to infinite temperature, with higher-order terms improving the accuracy. Our results suggest a dynamical mechanism to gradually increase temperature in isolated quantum simulators, such as ultracold atoms, and open up an alternative possibility to investigate thermal phase transitions and the interplay between thermal and quantum criticality using Floquet drives.
We investigate a long-time asymptotic state of periodically driven open quantum systems analytically. The model we consider in this paper is a free fermionic system coupled to an energy and particle reservoir. We clarify some generic properties of the system which are independent of the details of the reservoir in the high-frequency regime of the external driving. When the frequency of the external driving is much larger than the energy cutoff of the system-reservoir coupling, the low-energy properties of the system are equivalent to those of the Gibbs distribution of a Floquet effective Hamiltonian. Furthermore, we investigate the effect of finite dissipation on the system, and elucidate that we cannot suppress excitations by merely increasing the system-reservoir coupling because the excitations are created through the reservoir by the external driving when the frequency is smaller than the energy cutoff of the system-reservoir coupling.