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arXiv:2210.13444 · Physical review. B./Physical review. B

Universality classes of thermalization for mesoscopic Floquet systems

Alan Morningstar, David A. Huse, Vedika Khemani

Published Nov 9, 2023• 12 citations
arXivHTMLPDF

Abstract

We identify several distinct phases of thermalization that describe regimes of behavior in isolated, periodically driven (Floquet), mesoscopic quantum chaotic systems. In doing so, we also identify a Floquet thermal ensemble, the ``ladder ensemble,'' that is qualitatively distinct from the ``featureless infinite-temperature'' state that has long been assumed to be the appropriate maximum-entropy equilibrium ensemble for driven systems. The phases we find can be coarsely classified by (i) whether or not the system irreversibly exchanges energy of order $\ensuremath{\omega}$ with the drive, i.e., Floquet thermalizes, and (ii) the Floquet thermal ensemble describing the final equilibrium in systems that do Floquet thermalize. These phases are representative of regimes of behavior in mesoscopic systems, but they are sharply defined in a particular large-system limit where the drive frequency $\ensuremath{\omega}$ scales up with system size $N$ as the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit is taken: we examine frequency scalings ranging from a weakly $N$-dependent $\ensuremath{\omega}(N)\ensuremath{\sim}logN$, to stronger scalings ranging from $\ensuremath{\omega}(N)\ensuremath{\sim}\sqrt{N}$ to $\ensuremath{\omega}(N)\ensuremath{\sim}N$. We show that the transition where Floquet thermalization breaks down happens at an extensive drive frequency and, beyond that, systems that do not Floquet thermalize are distinguished based on the presence or absence of rare resonances across Floquet zones. We produce a thermalization phase diagram that is relevant for numerical studies of Floquet systems and experimental studies on small-scale quantum simulators, both of which lack a clean separation of scales between $N$ and $\ensuremath{\omega}$. A striking prediction of our work is that, under the assumption of perfect isolation, certain realistic quench protocols from simple pure initial states can show Floquet thermalization to a type of Schrodinger-cat state that is a global superposition of states at distinct temperatures. Our work extends and organizes the theory of Floquet thermalization, heating, and equilibrium into the setting of mesoscopic quantum systems.

DOI
10.1103/physrevb.108.174303

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We identify several distinct phases of thermalization that describe regimes of behavior in isolated, periodically driven (Floquet), mesoscopic quantum chaotic systems. In doing so, we also identify a Floquet thermal ensemble, the ladder ensemble,'' that is qualitatively distinct from the featureless infinite-temperature'' state that has long been assumed to be the appropriate maximum-entropy equilibrium ensemble for driven systems. The phases we find can be coarsely classified by (i) whether or not the system irreversibly exchanges energy of order \ensuremathω\ensuremath{\omega}\ensuremathω with the drive, i.e., Floquet thermalizes, and (ii) the Floquet thermal ensemble describing the final equilibrium in systems that do Floquet thermalize. These phases are representative of regimes of behavior in mesoscopic systems, but they are sharply defined in a particular large-system limit where the drive frequency \ensuremathω\ensuremath{\omega}\ensuremathω scales up with system size NNN as the N\ensuremath→\ensuremath∞N\ensuremath{\rightarrow}\ensuremath{\infty}N\ensuremath→\ensuremath∞ limit is taken: we examine frequency scalings ranging from a weakly NNN-dependent \ensuremathω(N)\ensuremath∼logN\ensuremath{\omega}(N)\ensuremath{\sim}logN\ensuremathω(N)\ensuremath∼logN, to stronger scalings ranging from \ensuremathω(N)\ensuremath∼N\ensuremath{\omega}(N)\ensuremath{\sim}\sqrt{N}\ensuremathω(N)\ensuremath∼N​ to \ensuremathω(N)\ensuremath∼N\ensuremath{\omega}(N)\ensuremath{\sim}N\ensuremathω(N)\ensuremath∼N. We show that the transition where Floquet thermalization breaks down happens at an extensive drive frequency and, beyond that, systems that do not Floquet thermalize are distinguished based on the presence or absence of rare resonances across Floquet zones. We produce a thermalization phase diagram that is relevant for numerical studies of Floquet systems and experimental studies on small-scale quantum simulators, both of which lack a clean separation of scales between NNN and \ensuremathω\ensuremath{\omega}\ensuremathω. A striking prediction of our work is that, under the assumption of perfect isolation, certain realistic quench protocols from simple pure initial states can show Floquet thermalization to a type of Schrodinger-cat state that is a global superposition of states at distinct temperatures. Our work extends and organizes the theory of Floquet thermalization, heating, and equilibrium into the setting of mesoscopic quantum systems.
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.
Conventional wisdom suggests that the long time behavior of isolated interacting periodically driven (Floquet) systems is a featureless maximal entropy state characterized by an infinite temperature. Efforts to thwart this uninteresting fixed point include adding sufficient disorder to realize a Floquet many-body localized phase or working in a narrow region of drive frequencies to achieve glassy non-thermal behavior at long time. Here we show that in clean systems the Floquet eigenstates can exhibit non-thermal behavior due to finite system size. We consider a one-dimensional system of spinless fermions with nearest-neighbor interactions where the interaction term is driven. Interestingly, even with no static component of the interaction, the quasienergy spectrum contains gaps and a significant fraction of the Floquet eigenstates, at all quasienergies, have non-thermal average doublon densities. We show that this non-thermal behavior arises due to emergent integrability at large interaction strength and discuss how the integrability breaks down with power-law dependence on system size.
Periodically driven many-body systems generally heat towards a featureless 'infinite-temperature' state. As an alternative to uniform heating in a clean system, here we establish a Floquet superheating regime, where fast heating nucleates at ''hot spots" generated by rare fluctuations in the local energy with respect to an appropriate effective Hamiltonian. Striking macroscopic consequences include exceptionally long-lived prethermalization and non-ergodic bimodal distributions of macroscopic observables. Superheating is predicated on a heating rate depending strongly on the local fluctuation; in our example, this is supplied by a sharp state-selective spin-echo, where the energy absorption is strongly suppressed for low-energy states, while thermal fluctuations open up excessive heating channels. A simple phenomenological theory is developed to show the existence of a critical droplet size, which incorporates heating by the driving field as well as the heat current out of the droplet. Our results shine light on a new heating mechanism and suggest new routes towards stabilizing non-equilibrium phases of matter in driven systems.
The experimental control and observation of quantum many-body systems has become a reality with the advent of ultracold quantum matter. The high level of isolation of these experiments, together with the development of novel measurement techniques, has revived a fundamental debate concerning the thermal equilibration of isolated quantum systems, commonly named "quantum thermalization". In this thesis we make use of a quantum-gas microscope to explore the thermalizing dynamics of highly isolated systems of ultracold bosonic atoms in optical lattices. The ability to prepare and control quantum systems made up of hundreds of atoms makes it possible to explore regimes that represent a challenge for classical numeric simulations. A major part of this dissertation deals with Bose-Hubbard systems in the presence of quenched disorder. We begin by studying the microscopic properties of its phases near equilibrium, where by tuning the strength of the disorder, observe features consistent with the emergence of a so-called Bose-glass phase. We then continue by preparing states far from equilibrium and exploring their quantum many-body dynamics. In particular, we observe signatures of the phenomenon of "many-body localization", which implies a breakdown of quantum thermalization. In addition, we study whether localized systems can be thermalized via the coupling to a bath of few degrees of freedom, i.e. a quantum bath. We do so by preparing a mixture of two interacting atomic species, where one acts as the bath and the other as the localized system. We do observe delocalizing dynamics for large enough baths, though in regimes of weak coupling localization can survive for extremely long times. The second main topic of this thesis is the thermalization of periodically driven many-body systems, so-called Floquet thermalization. In these systems, the absence of energy conservation eventually brings any initial state into a featureless infinite-temperature one. However, for sufficiently high frequencies this thermalization process can take arbitrarily long times, which can enable the engineering of exotic long-lived prethermal states. We use the high isolation of our system, together with the high sensitivity of quantum-gas microscopy, to measure the heating rates for a range of driving frequencies and interaction regimes. Our results show a dramatic suppression of the heating as the frequency of the drive is increased, which is consistent with theoretical expectations.
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.
Clean, interacting, periodically driven systems are believed to thermalize to featureless, infinite-temperature states that preclude the formation of stable ordered states. Here, we show stable quantum order, distinct from prethermal order, and its phase transitions in the ground state of the average-energy operator. Decomposing the time evolution into geometric and average-energy dynamics, we apply Geometric Floquet theory to identify the Floquet ground state. We derive a general variational principle to compute the Floquet ground state in large interacting periodically driven systems, and use it to develop a matrix product state algorithm. In driven nonintegrable chains of more than 100 interacting spins, our numerical simulations exhibit a gapped, area-law entangled Floquet ground state in extended parameter regimes; the ground-state manifold can support both dressed equilibrium order (e.g., Floquet antiferromagnets) and inherently nonequilibrium order (e.g., ground-state time crystals). This quantum order is protected by a finite average-energy gap, which closes at phase transition points upon changing model parameters. We decompose Floquet phases of matter into their equilibrium and nonequilibrium elements: whereas the Floquet ground state at any fixed time is a representative of an equilibrium phase of matter, the nonequilibrium content is entirely produced by the drive holonomy and is geometric. Finally, we identify quasilocal counterdiabatic driving as a mechanism for the stability against heating of gapped Floquet ground states in nonintegrable quantum systems, and argue why Floquet ground states evade the Eigenstate Thermalization Hypothesis. Our work enables the study of interacting nonequilibrium matter deep into the many-body regime, and lays the foundations for the engineering of strongly correlated ordered quantum states.