Floquet-Bloch Bands Research Articles

Non-reciprocal band structures in an exciton–polariton Floquet optical lattice

Periodic temporal modulation of Hamiltonians can induce geometrical and topological phenomena in the dynamics of quantum states. Using the interference between two lasers, we demonstrate an off-resonant optical lattice for a polariton condensate with controllable potential depths and nearest-neighbour coupling strength. Temporal modulation is introduced via a gigahertz frequency detuning between pump lasers, creating a polariton ‘conveyor belt’. The breaking of time-reversal symmetry causes band structures to become non-reciprocal and acquire a universal tilt given by Planck’s constant and the frequency of modulation (hΔf). The non-reciprocal tilting is connected to the non-trivial topology of the Floquet–Bloch bands, which have a finite Chern number. Detailed characterization of the lattice potential depth and its dynamics highlights the role of high-energy carriers in the formation of optical potential landscapes for polaritons, demonstrating the possibility of modulation faster than the polariton lifetime and opening a pathway towards Floquet engineering in polariton condensates.

Floquet Fermi Liquid.

We demonstrate the existence of a nonequilibrium "Floquet Fermi liquid" state arising in partially filled Floquet Bloch bands weakly coupled to ideal fermionic baths, which possess a collection of "Floquet Fermi surfaces" enclosed inside each other, resembling matryoshka dolls. We elucidate several properties of these states, including their quantum oscillations under magnetic fields which feature slow beating patterns of their amplitude reflecting the different areas of the Floquet Fermi surfaces, consistent with those observed in microwave induced resistance oscillation experiments. We also investigate their specific heat and thermodynamic density of states and demonstrate how by controlling properties of the drive, such as its frequency, one can tune some of the Floquet Fermi surfaces toward nonequilibrium Van Hove singularities without changing the electron density.

Tracking electron motion within and outside of Floquet bands from attosecond pulse trains in time-resolved ARPES

Floquet engineering has recently emerged as a technique for controlling material properties with light. Floquet phases can be probed with time- and angle-resolved photoelectron spectroscopy (Tr-ARPES), providing direct access to the laser-dressed electronic bands. Applications of Tr-ARPES to date focused on observing the Floquet-Bloch bands themselves, and their build-up and dephasing on sub-laser-cycle timescales. However, momentum and energy resolved sub-laser-cycle dynamics between Floquet bands have not been analyzed. Given that Floquet theory strictly applies in time-periodic conditions, the notion of resolving sub-laser-cycle dynamics between Floquet states seems contradictory—it requires probe pulse durations below a laser cycle that inherently cannot discern the time-periodic nature of the light-matter system. Here we propose to employ attosecond pulse train probes with the same temporal periodicity as the Floquet-dressing pump pulse, allowing both attosecond sub-laser-cycle resolution and a proper projection of Tr-ARPES spectra on the Floquet–Bloch bands. We formulate and employ this approach in ab-initio calculations in light-driven graphene. Our calculations predict significant sub-laser-cycle dynamics occurring within the Floquet phase with the majority of electrons moving within and in-between Floquet bands, and a small portion residing and moving outside of them in what we denote as ‘non-Floquet’ bands. We establish that non-Floquet bands arise from the pump laser envelope that induces non-adiabatic electronic excitations during the pulse turn-on and turn-off. By performing calculations in systems with poly-chromatic pumps we also show that Floquet states are not formed on a sub-laser-cycle level. This work indicates that the Floquet-Bloch states are generally not a complete basis set for sub-laser-cycle dynamics in steady-state phases of matter.

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 nonequilibrium distribution function. We 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 paper 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. Published by the American Physical Society 2024

Mixing-spacetime symmetry in the Floquet-Bloch band theory

Mixing-spacetime symmetry in the Floquet-Bloch band theory

Build-up and dephasing of Floquet-Bloch bands on subcycle timescales.

Strong light fields have created opportunities to tailor novel functionalities of solids1-5. Floquet-Bloch states can form under periodic driving of electrons and enable exotic quantum phases6-15. On subcycle timescales, lightwaves can simultaneously drive intraband currents16-29 and interband transitions18,19,30,31, which enable high-harmonic generation16,18,19,21,22,25,28-30 and pave the way towards ultrafast electronics. Yet, the interplay of intraband and interband excitations and their relation to Floquet physics have been key open questions as dynamical aspects of Floquet states have remained elusive. Here we provide this link by visualizing the ultrafast build-up of Floquet-Bloch bands with time-resolved and angle-resolved photoemission spectroscopy. We drive surface states on a topological insulator32,33 with mid-infrared fields-strong enough for high-harmonic generation-and directly monitor the transient band structure with subcycle time resolution. Starting with strong intraband currents, we observe how Floquet sidebands emerge within a single optical cycle; intraband acceleration simultaneously proceeds in multiple sidebands until high-energy electrons scatter into bulk states and dissipation destroys the Floquet bands. Quantum non-equilibrium calculations explain the simultaneous occurrence of Floquet states with intraband and interband dynamics. Our joint experiment and theory study provides a direct time-domain view of Floquet physics and explores the fundamental frontiers of ultrafast band-structure engineering.

Semiclassical dynamics of electrons in a space-time crystal: Magnetization, polarization, and current response

A space-time crystal is defined as a quantum-mechanical system with both spatial and temporal periodicity. Such a system can be described by the Floquet-Bloch (FB) theory. We first formulate a semiclassical theory by constructing a wave packet through the superposition of the FB wave functions, and we derive the equations of motion of FB electrons subjected to slowly varying external fields (not to be confused with the fast-changing Floquet drive), revealing behaviors similar to ordinary Bloch electrons but with quantities modified in the Floquet context. Specifically, we study the local magnetic moment due to the self-rotation of the wave packet, a contribution to total magnetization from the Berry curvature in $\mathbf{k}$-space, and the polarization of a fully occupied FB band. Based on the semiclassical theory, we can also show the fingerprint of the energy flow in such an energy-nonconserved system. We then discuss the density matrix of a FB system attached to a thermal bath, which allows us to investigate quantities involving many electrons in the noninteracting limit. As an application, we calculate the intrinsic current response in an oblique space-time metal showing the nonequilibrium nature of the FB system. The current response can also be related to the acoustoelectric effect. Overall, we develop a systematic approach for studying space-time crystals, and we provide a powerful tool to explore the electronic properties of this exotic system with coupled space and time.

Topological Pumping in a Floquet-Bloch Band.

Constructing new topological materials is of vital interest for the development of robust quantum applications. However, engineering such materials often causes technological overhead, such as large magnetic fields, spin-orbit coupling, or dynamical superlattice potentials. Simplifying the experimental requirements has been addressed on a conceptual level-by proposing to combine simple lattice structures with Floquet engineering-but there has been no experimental implementation. Here, we demonstrate topological pumping in a Floquet-Bloch band using a plain sinusoidal lattice potential and two-tone driving with frequencies ω and 2ω. We adiabatically prepare a near-insulating Floquet band of ultracold fermions via a frequency chirp, which avoids gap closings en route from trivial to topological bands. Subsequently, we induce topological pumping by slowly cycling the amplitude and the phase of the 2ω drive. Our system is well described by an effective Shockley model, establishing a novel paradigm to engineer topological matter from simple underlying lattice geometries. This approach could enable the application of quantized pumping in metrology, following recent experimental advances on two-frequency driving in real materials.

Floquet engineering of individual band gaps in an optical lattice using a two-tone drive

The dynamic engineering of band structures for ultracold atoms in optical lattices represents an innovative approach to understand and explore the fundamental principles of topological matter. In particular, the folded Floquet spectrum determines the associated band topology via band inversion. We experimentally and theoretically study two-frequency phase modulation to asymmetrically hybridize the lowest two bands of a one-dimensional lattice. Using quasi-degenerate perturbation theory in the extended Floquet space we derive an effective two-band model that quantitatively describes our setting. The energy gaps are experimentally probed via Landau-Zener transitions between Floquet-Bloch bands using an accelerated Bose-Einstein condensate. Separate and simultaneous control over the closing and reopening of these band gaps is demonstrated. We find good agreement between experiment and theory, establishing an analytic description for resonant Floquet-Bloch engineering that includes single- and multi-photon couplings, as well as interference effects between several commensurate drives.

Detecting light-induced Floquet band gaps of graphene via trARPES

We propose a realistic regime to detect the light-induced topological band gap in graphene via time-resolved angle-resolved photoelectron spectroscopy (trARPES), that can be achieved with current technology. The direct observation of Floquet-Bloch bands in graphene is limited by low-mobility, Fourier-broadening, laser-assisted photoemission (LAPE), probe-pulse energy-resolution bounds, space-charge effects and more. We characterize a regime of low driving frequency and high amplitude of the circularly polarized light that induces an effective band gap at the Dirac point that exceeds the Floquet zone. This circumvents limitations due to energy resolutions and band broadening. The electron distribution across the Floquet replica in this limit allow for distinguishing LAPE replica from Floquet replica. We derive our results from a dissipative master equation approach that gives access to two-point correlation functions and the electron distribution relevant for trARPES measurements.

Observing light-induced Floquet band gaps in the longitudinal conductivity of graphene

Floquet engineering presents a versatile method of dynamically controlling material properties. The light-induced Floquet-Bloch bands of graphene feature band gaps, which have not yet been observed directly. We propose optical longitudinal conductivity as a realistic observable to detect light-induced Floquet band gaps in graphene. These gaps manifest as resonant features in the conductivity, when resolved with respect to the probing frequency and the driving field strength. The electron distribution follows the light-induced Floquet-Bloch bands, resulting in a natural interpretation as occupations of these bands. Furthermore, we show that there are population inversions of the Floquet-Bloch bands at the band gaps for sufficiently strong driving field strengths. This strongly reduces the conductivity at the corresponding frequencies. Therefore our proposal puts forth not only an unambiguous demonstration of light-induced Floquet-Bloch bands, which advances the field of Floquet engineering in solids, but also points out the control of transport properties via light, that derives from the electron distribution on these bands.

Single-shot measurement of the photonic band structure in a fiber-based Floquet-Bloch lattice

Floquet-Bloch lattices are systems in which wave packets are subjet to periodic modulations both in time and space, showing rich dynamics. While this type of lattice is difficult to implement in solid-state physics, optical systems have provided excellent platforms to probe their physics: among other effects, they have revealed genuine phenomena such as the anomalous Floquet topological insulator and the funnelling of light into localised interface modes. Despite the crucial importance of the band dispersion in the photon dynamics and the topological properties of the lattice, the direct experimental measurement of the Floquet-Bloch bands has remained elusive. Here we report the direct measurement of the Floquet-Bloch bands of a photonic lattice with a single shot method. We use a system of two coupled fibre rings that implements a time-multiplexed Floquet-Bloch lattice. By Fourier transforming the impulse response of the lattice we obtain the band structure together with an accurate characterization of the lattice eigenmodes, i. e. the amplitudes and the phases of the Floquet-Bloch eigenvectors over the entire Brillouin zone. Our results open promising perspectives for the observation of topological effects in the linear and nonlinear regime in Floquet systems.

Floquet–Dirac fermions in monolayer graphene by Wannier functions

Wannier functions have been widely applied in the study of topological properties and Floquet–Bloch bands of materials. Usually, the real-space Wannier functions are linked to the k-space Hamiltonian by two types of Fourier transform (FT), namely lattice-gauge FT (LGFT) and atomic-gauge FT (AGFT), but the differences between these two FTs on Floquet–Bloch bands have rarely been addressed. Taking monolayer graphene as an example, we demonstrate that LGFT gives different topological descriptions on the Floquet–Bloch bands for the structurally equivalent directions which are obviously unphysical, while AGFT is immune to this dilemma. We introduce the atomic-laser periodic effect to explain the different Floquet–Bloch bands between the LGFT and AGFT. Using AGFT, we showed that linearly polarized laser could effectively manipulate the properties of the Dirac fermions in graphene, such as the location, generation and annihilation of Dirac points. This proposal offers not only deeper understanding on the role of Wannier functions in solving the Floquet systems, but also a promising platform to study the interaction between the time-periodic laser field and materials.

Electromagnetic dressing of the electron energy spectrum of Au(111) at high momenta

Light-engineering of quantum materials via electromagnetic dressing is considered an on-demand approach for tailoring electronic band dispersions and even inducing topological phase transitions. For probing such dressed bands, photoemission spectroscopy is an ideal tool, and we employ here a novel experiment based on ultrafast photoemission momentum microscopy. Using this setup, we measure the in-plane momentum-dependent intensity fingerprints of the electromagnetically-dressed sidebands from a Au(111) surface for s- and p-polarized infrared driving. We find that at metal surfaces, due to screening of the driving laser, the contribution from Floquet-Bloch bands is negligible, and the dressed bands are dominated by the laser-assisted photoelectric effect. Also, we find that in contrast to general expectations, s-polarized light can dress free-electron states at large photoelectron momenta. Our results show that the dielectric response of the material must carefully be taken into account when using photoemission for the identification of light-engineered electronic band structures.

Modulation-induced long-range magnon bound states in one-dimensional optical lattices

Magnon excitations play an important role in understanding quantum magnetism and magnon bound states observed with ultracold atoms in optical lattices. Here, we investigate how gradient magnetic field and periodically modulated spin-exchange strength affect the two-magnon excitations. In the Stark resonance where the driving frequency matches and smooths the potential bias, the system gains translational invariance in both space and time in the rotating frame, and thus we can develop a Floquet–Bloch band theory for two magnons. We find a new kind of bound states with relative distance no less than two sites, apart from the conventional bound states with relative distance at one site, which indicates the modulation-induced long-range interaction. We analytically derive an effective Hamiltonian via the many-body perturbation theory for a deeper understanding of such novel bound states and explore the interplay between these two types of bound states. Moreover, we propose to probe modulation-induced bound states via quantum walks. Our study not only provides a scheme to form long-range magnon bound states, but also lays a cornerstone for engineering exotic quantum states in multi-particle Floquet systems.

Exponentially long lifetime of universal quasi-steady states in topological Floquet pumps

We investigate a mechanism to transiently stabilize topological phenomena in long-lived quasi-steady states of isolated quantum many-body systems driven at low frequencies. We obtain an analytical bound for the lifetime of the quasi-steady states which is exponentially large in the inverse driving frequency. Within this lifetime, the quasi-steady state is characterized by maximum entropy subject to the constraint of fixed number of particles in the system's Floquet-Bloch bands. In such a state, all the non-universal properties of these bands are washed out, hence only the topological properties persist.

The magnetic, electronic, and light-induced topological properties in two-dimensional hexagonal FeX2 (X = Cl, Br, I) monolayers

Using Floquet–Bloch theory, we propose to realize chiral topological phases in two-dimensional (2D) hexagonal FeX2 (X = Cl, Br, I) monolayers under irradiation of circularly polarized light. Such 2D FeX2 monolayers are predicted to be dynamically stable and exhibit both ferromagnetic and semiconducting properties. To capture the full topological physics of the magnetic semiconductor under periodic driving, we adopt ab initio Wannier-based tight-binding methods for the Floquet–Bloch bands, with the light-induced bandgap closings and openings being obtained as the light field strength increases. The calculations of slabs with open boundaries show the existence of chiral edge states. Interestingly, the topological transitions with branches of chiral edge states changing from zero to one and from one to two by tuning the light amplitude are obtained, showing that the topological Floquet phase of high Chern number can be induced in the present Floquet–Bloch systems.

Probing Nonexponential Decay in Floquet–Bloch Bands

Abstract Exponential decay laws describe systems ranging from unstable nuclei to fluorescent molecules, in which the probability of jumping to a lower-energy state in any given time interval is static and history-independent. These decays, involving only a metastable state and fluctuations of the quantum vacuum, are the most fundamental nonequilibrium process and provide a microscopic model for the origins of irreversibility. Despite the fact that the apparently universal exponential decay law has been precisely tested in a variety of physical systems, it is a surprising truth that quantum mechanics requires that spontaneous decay processes have nonexponential time dependence at both very short and very long times. Cold-atom experiments have proven to be powerful probes of fundamental decay processes; in this article, we propose the use of Bose condensates in Floquet–Bloch bands as a probe of long-time nonexponential decay in single isolated emitters. We identify a range of parameters that should enable observation of long-time deviations and experimentally demonstrate a key element of the scheme: tunable decay between quasi-energy bands in a driven optical lattice.

Contribution of Floquet-Bloch states to high-order harmonic generation in solids

We identify the contribution of Floquet-Bloch states to the high-order harmonic generation (HHG) in solids by numerically solving the time-dependent Schr\odinger equation for both a one-dimensional and a two-dimensional model. Results from the single $k$ point and the full Brillouin zone are compared to each other and the symmetry- breaking effect is discussed. We show that the observed phenomena can be explained under the framework of the Floquet-Bloch theory and the strong-field approximation, respectively. Our results indicate that the total yield of the harmonic radiation increases nonmonotonically with the intensity of the driving pulse. After a rough consideration of the focusing volume effects, we find that the yield of the harmonics shows a steplike structure, which is similar to several recent experimental observations. The present work can contribute to a better understanding of the channel-closing effect in the HHG of solids and may provide a way to detect the Floquet-Bloch bands in a laser field.

Microscopic theory for the light-induced anomalous Hall effect in graphene

We employ a quantum Liouville equation with relaxation to model the recently observed anomalous Hall effect in graphene irradiated by an ultrafast pulse of circularly polarized light. In the weak-field regime, we demonstrate that the Hall effect originates from an asymmetric population of photocarriers in the Dirac bands. By contrast, in the strong-field regime, the system is driven into a non-equilibrium steady state that is well-described by topologically non-trivial Floquet-Bloch bands. Here, the anomalous Hall current originates from the combination of a population imbalance in these dressed bands together with a smaller anomalous velocity contribution arising from their Berry curvature. This robust and general finding enables the simulation of electrical transport from light-induced Floquet-Bloch bands in an experimentally relevant parameter regime and creates a pathway to designing ultrafast quantum devices with Floquet-engineered transport properties.