Session N50: Dynamical Phenomena with Floquet or Quasi-Periodic Driving

Floquet Prethermalization with Lifetime Exceeding 90 s in a Bulk Hyperpolarized Solid

We report the observation of long-lived Floquet prethermal states in a bulk solid composed of dipolar-coupled ¹³C

nuclei in diamond at room temperature. For precessing nuclear spins prepared in an initial transverse

state, we demonstrate Floquet control that prevents their decay over multiple-minute long periods. We observe

Floquet prethermal lifetimes T₂'≈90.9s, extended >60,000-fold over the nuclear free induction decay times. The

spins themselves are continuously interrogated for ∼10min, corresponding to the application of ≈5.8M control

pulses. The ¹³C nuclei are optically hyperpolarized by lattice Nitrogen-Vacancy (NV) centers; the combination

of hyperpolarization and continuous spin readout yields significant signal-to-noise in the measurements. This

allows probing the Floquet thermalization dynamics with unprecedented clarity. We identify four characteristic

regimes of the thermalization process, discerning short-time transient processes leading to the prethermal

plateau, and long-time system heating towards infinite temperature. This work points to new opportunities

possible via Floquet control in networks of dilute, randomly distributed, low-sensitivity nuclei. In particular, the

combination of minutes-long prethermal lifetimes and continuous spin interrogation opens avenues for quantum

sensors constructed from hyperpolarized Floquet prethermal nuclei.   

Continuous Observation of a Long Lived Prethermal Discrete Time Crystal by Two Frequency Driving

We report the observation of long-lived Floquet prethermal discrete time crystals (PDTCs) in a three-dimensional

position disordered lattice of dipolar-coupled ¹³C nuclei in diamond at room temperature. We propose and

demonstrate a novel strategy of ``two frequency’’ driving, involving an interleaved application of slow and fast

drives of period τ and t respectively, that simultaneously prethermalize the spins along a Bloch sphere axis x,

while enabling observation of their dynamic evolution when periodically kicked away from x. The PDTC phase

appears as a robust period-doubling response of the spin magnetization interchanging between x and -x; our

experiments allow a unique means to study the formation and melting of this PDTC phase leveraging a

continuous interrogation of the spins during the fast drive. This permits the production of ``movies’’ of the time-

crystallization process with a clarity and experimental throughput orders of magnitude higher that previous

reports. With this, we report a highly long-lived PDTC lasting over 10s in time (corresponding to

>10⁵ pulses of the fast drive) and measurable in a single-shot of the experiment. Such rapid measurement enables a

detailed experimental characterization of the PDTC rigidity, informing on the role of prethermalization towards

stabilizing the DTC response. The two frequency drive approach represents the simplest generalization of DTCs to higher time

dimensions, and as such expands the toolkit for Floquet engineering. Overall, this work points to new opportunities available for

exciting and stabilizing far-from-equilibrium phases of matter in solids with hyperpolarizable, dilute, low-sensitivity nuclei.   

An absolutely stable open time crystal

Periodically driven (Floquet) systems can host a discrete time crystal (DTC) --- a phase of matter characterized by the spontaneous breaking of time translation symmetry. Time crystalline order with an infinite auto-correlation time relies upon the Floquet system's ability to avoid ergodicity; in particular, it must skirt the absorption of energy from the drive, which would otherwise lead to an infinite temperature final state. In quantum systems, this can be accomplished via strong disorder leading to many-body localization. In this talk, I will describe an entirely different setting where the DTC is stable: locally-interacting, disorder-free, Floquet Hamiltonian dynamics coupled to a finite temperature Langevin bath. Employing a mapping from probabilistic cellular automata to open classical dynamics, the resulting DTC is stable to arbitrary perturbations, including those that break the underlying time translation symmetry of the drive. Finally, I will discuss how general results in the context of probabilistic cellular automata imply the existence of discrete time crystals in all dimensions, D≥1.   

Emergence of Floquet time crystals in infinite-range spin systems with multi-body interactions

Time crystals are out-of-equilibrium phases of matter emerging as a consequence of time-translation symmetry breaking. In periodically driven systems, discrete time-translation symmetry breaking gives rise to Floquet time crystal (FTC) phases. A FTC phase is characterized by the emergence of ordering of the Floquet eigenstates, which manifests as a subharmonic system response. In this work we study infinite range spin systems with multi-body interactions, admitting an exact mean-field description, in presence of a periodic drive. Going beyond the usual case of two-body interactions, we show that the higher multi-body interactions give rise to FTC phases showing eigenstate ordering leading to system responses with periods as large as the degree of the interaction. Using the mean-field description of these models, and relying on the phenomenology of dynamical flows and area preserving maps, we construct a classical picture of FTC phases in driven Hamiltonian systems. Finally, in systems hosting more than one FTC phase, we explore control protocols to extract rigid system responses carrying more than one subharmonic frequency.   

Long-lived π edge modes of interacting and disorder-free Floquet spin chains

Floquet spin chains have been a venue for understanding topological states of matter that are qualitatively

different from their static counterparts by, for example, hosting  $\pi$ edge modes that show stable period-doubled dynamics.

However the stability of these edge modes to interactions has traditionally required the system to be many-body localized

in order to suppress heating. In contrast, here we show that even in the absence of disorder, and in the presence

of bulk heating, $\pi$ edge modes are long lived. Their lifetime is extracted from exact diagonalization and is found to

be non-perturbative in the interaction strength. A tunneling estimate for the lifetime is obtained by mapping the stroboscopic

time-evolution to dynamics of a single particle in Krylov subspace. In this subspace, the $\pi$ edge mode manifests as the

quasi-stable edge mode of an inhomogeneous Su-Schrieffer-Heeger model whose dimerization vanishes in the bulk of the Krylov chain.   

Universal transport in periodically driven systems without long-lived quasiparticles

An intriguing regime of universal charge transport at high entropy density has been proposed for periodically driven interacting one-dimensional systems with Bloch bands separated by a large single-particle band gap. For weak interactions, a simple picture based on well-defined Floquet quasiparticles suggests that the system should host a quasisteady state current that depends only on the populations of the system's Floquet-Bloch bands and their associated quasienergy winding numbers. Here we show that such topological transport persists into the strongly interacting regime where the single-particle lifetime becomes shorter than the drive period. Analytically, we show that the value of the current is insensitive to interaction-induced band renormalizations and lifetime broadening when certain conditions are met by the system's non-equilibrium distribution function, and show that these conditions correspond to a quasisteady state. We support these predictions through numerical simulation of a system of strongly interacting fermions in a periodically-modulated chain of Sachdev-Ye-Kitaev dots. Our work establishes universal transport at high entropy density as a robust far from equilibrium topological phenomenon, which can be readily realized with cold atoms in optical lattices.   

Floquet Proximity Effect

We introduce the concept of "Floquet proximity effect," in which the usual equilibrium proximity effect is modified by the presence of a strong periodic drive at the interface. We develop the general theory of Floquet proximity effect and obtain the Floquet mean field equations, which yield the frequency-dependent induced order parameter. To illustrate the effect, we consider a Dirac system such as the surface of a topological insulator in proximity to an s-wave superconductor. The Floquet proximity effect modifies and induces effective frequency-dependent pairing amplitudes in both systems. Thus, we provide an avenue to the theoretical and experimental study of dynamical superconducting pairing induced by Floquet proximity effect.   

Universal Nonadiabatic Energy Pumping in a Quasiperiodically Driven Extended System

The paradigm of Floquet engineering of topological states of matter can be generalized into the time-quasiperiodic scenario, where a lower dimensional time-dependent system maps into a higher dimensional one by combining the physical dimensions with additional synthetic dimensions generated by multiple incommensurate driving frequencies. Different than most previous works in which gapped topological phases were considered, we propose an experimentally realizable, one dimensional chain driven by two frequencies, which maps into a gapless Weyl semimetal in synthetic dimension. Based on analytical reasoning and numerical simulations, we found the nonadiabatic quantum dynamics of this system exhibit energy pumping behaviors characterized by universal functions. We also numerically found such behaviors are robust against a considerable amount of spatial disorder.   

Localization-protected energy pump

Localization can protect topological orders in interacting one-dimensional systems, even in the presence of quasiperiodic driving. We present an explicit model achieving such a non-trivial topological phase with protected edge modes. The system pumps energy between the drives at a quantized rate, even at infinite temperature.   

Interactions in the time-domain within the Floquet-Bloch formalism

The rich landscape of phenomena in condensed matter physics is strongly tied to quasi particle interactions. In the static domain and in equilibrium, treating interaction effects is conceptually established. Out of equilibrium, novel phenomena emerge. We report about an extension of the Hatsugai-Kohmoto model [1] into the time domain. This model, local in momentum and exactly solvable, captures the essential leading contributions of strongly correlated quantum systems. In the time-domain, it serves as a platform to investigate dynamic quantum phases of matter. We focus on the case of coherently driven matter which can be approximated in the Floquet-Bloch framework adopted recently [2].    

Non-equilibrium Floquet steady states of time-periodic driven Luttinger liquids

We study interplay of Floquet states and strong interactions using time-periodic fields in a Luttinger liquid with periodically changing interactions. By developing a time-periodic operator algebra, we are able to obtain the complete set of explicit steady state solutions of the time-dependent Schrödinger equation in terms of a Floquet-Bogoliubov ansatz and known analytic functions. Complex valued Floquet eigenenergies occur when integer multiples of the driving frequency approximately match twice the dispersion energy, which correspond to resonant states. Including damping effects we show that this resonant behavior leads to a large number of density excitations, which allow quantitative comparisons with experiments. This setup is one or the rare cases, where a complete Floquet solution can be obtained exactly for a time-periodically driven many-body system.   

Transport properties of Floquet Majorana modes and Floquet quasi-Majorana modes

We numerically investigate the properties of Floquet Majorana zero modes (FMZMs) and partially-separated Floquet quasi-Majorana modes (FQMMs) in a realistic Floquet topological superconductor. The model consists of a periodically driven nanowire proximitized to an s-wave superconductor, whose dissipative nature needs to be taken into account due to the periodic drive [1]. We study the spectrum of the driven nanowire in the two phases, topological and trivial, as well as the transport properties of the FMZMs and FQMMs when coupled to a quantum dot and external leads.

[1] Z. Yang, Q. Yang, J. Hu, and D. E. Liu, Phys. Rev. Lett. 126, 086801 (2021).   

Stability of Floquet Majorana box qubits

A topological superconductor in one dimension can host Majorana zero modes at its edge. By driving the system periodically, so-called π modes (also named Floquet-Majoranas) can arise. These are topologically protected modes with the quasi-energy π/T, where T is the period of the drive. We consider the role of π modes in the presence of long-range Coulomb interactions and therefore study a Cooper pair box made of two Josephson coupled superconducting topological quantum wires. Time-dependent gate voltages periodically drive the system and can induce π modes. The presence of four Majoranas and four π Majoranas in our setup allows us to define three topological qubits in a fixed fermion parity sector within one single box. We investigate how to obtain and control the π modes and study their stability in the presence of interactions. The stability of the Floquet-Majorana box qubit depends crucially on the initialization of the Floquet state. If the system is prepared by adiabatically increasing the amplitude of the oscillating gate voltage, the topological Floquet phase is always inherently unstable. The instability arises due to resonant quasi-particle creation mediated by interactions. However, a stable Floquet phase can be reached by using a two-step protocol. First, the amplitude of the oscillating gate voltage is adiabatically increased, while the frequency of the oscillation is small. Then, the oscillation frequency is increased slowly. With this frequency-sweep protocol, we can achieve a stable Floquet device despite interactions.   

An ideal rapid-cycle Thouless pump

Thouless pumping is a fundamental instance of quantized transport, which is topologically protected. Although its theoretical importance, the adiabaticity condition is an obstacle for further practical applications. Focusing on the Rice-Mele model, I will explain the way to create a family of finite-frequency examples that ensure both the absence of excitations and the perfect quantization of the pumped charge at the end of each cycle. This family, which contains an adiabatic protocol as a limiting case, is obtained through a mapping onto the zero curvature representation of the Euclidean sinh-Gordon equation.   

Anomalous Random Multipolar Driven Insulators

We show that in the absence of the time translation symmetry, non-equilibrium topological phases of mattercan exist for an exceptionally long time despite its eventual thermalization. As a prerequisite, we first demon-strate the existence of a long-lived prethermal Anderson localization in 2D with random multipolar driving. Later we show the localization is topologically non-trivial as its bulk orbital magnetization is precisely quantized even though there is no well-defined Floquet eigenergies and states. We further confirm the existence ofthe anomalous random multipolar driven insulators by detecting the quantized charge pumping at the boundries