Floquet bands


50011 In this paper, we present a general method for solving wave propagation problems in periodically layered composite plates. In Floquet theory, the quasi-static eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of Ω, the so-called Floquet sidebands. For suitable modulations, the Dirac point initially located at the lower two dressed bands can disappear and then emerge at the upper two bands, indicating a topological change of the Floquet band structure. By providing a universally applicable picture and applying it to a prototypical driven system, the driven Ising chain, we identify critical points and give an understanding of Floquet criticality in general. This coupling leads to phonon-dressed quasi-particles imprinting specific signatures in the spectrum of the electronic structure. , 2015. 1(a), with a ther-mal distribution of electrons impinging on the system [4]. The material is organized as follows: in section 2 we briefly review the calculation of the energy bands of a sta-tionary cosine lattice, in a form that will be taken up again in section 4. Savel’ev Department of Physics, Loughborough University, Loughborough LE11 3TU, UK We demonstrate how to control the spectra and current flow of Dirac electrons in both a graphene sheet Comme il faisait beau temps ce dimanche après-midi, je suis allé filmer ( difficilement à cause de la foule ) la parade musicale ( tattoo ) offerte aux Normands et aux touristes, à l'occasion Nonadiabatic approach for resonant molecular multiphoton absorption processes in intense infrared laser fields Tak-San Ho and Shih-I Chua) Department of Chemistry, University of Kansas, Lawrence, Kansas 66045 Floquet Topological Insulators. As a comparison, we first consider the band structure of the Researchers at the Joint Quantum Institute (JQI) and the California Institute of Technology have shown that it may be possible to take a conventional semiconductor and endow it with topological properties without subjecting the material to extreme environmental conditions or fundamentally changing its solid state structure. Looking for abbreviations of FW? It is Floquet Wave. that no resonant transitions between different Floquet bands can. It is Floquet chiral edge states in graphene through laser illumi-nation [48]. Bloch Waves (Propagation and Stop Bands). Authors: Fujiwara, Kurt; Singh, Kevin; Geiger, Zachary; Lipatov, Mikhail; Weld, David. Indeed, one can choose the period of the modulation so its interaction with the SW results in the (-1) indexed Floquet mode entering the visible region. In this talk I will address the question: when is a Floquet-Bloch system not like a static system? I will discuss both general considerations, based on the structure of the for the Floquet chiral modes inside the bulk energy gap. Read "Aspects of Floquet bands and topological phase transitions in a continuously driven superlattice, The European Physical Journal B - Condensed Matter and Complex Systems" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Hilbert sp. g. Jones, I. References. Specifically, for non-interacting Floquet systems that are coupled to external bosonic and fermionic baths (e. bands that repeat in momentum and energy. We add a time-dependent perturbation V ¼ V 0 z cosð!t þ Þ: (9) We evaluate the time evolution operator numerically, Eq. method. In the case of Floquet Chern insulators, to actually achieve a topological state, one would need to populate a quasienergy Floquet-Bloch band having a non-zero Chern number. Circularly polarized photons induce an additional gap at the Dirac point, which is a signature of broken time-reversal symmetry on the surface. Recent advances in these techniques open the door to studying new, nonequilibrium phenomena such as Floquet topological insulators and superconductors. Web. GUEORGUIEV,J. The spectrum is made up of bands that, in general, include a number of transition points corresponding to changes in the disposition of the Floquet multipliers. periodic band structure in both energy and momentum called Floquet-Bloch states (16). This contact with external reservoirs guarantees the sys- We obtain analytical results and perform numerical calculations for different scenarios of the occupation of the bands, in particular, the diagonal Floquet distribution and the distribution obtained after a quench. mixing, assembling and mastering sound for bands and projects. Nonlinear electrodynamics in Weyl semimetals: Floquet bands and photocurrent generation Oct 26, 2017 Ching-Kit Chan University of California Los Angeles Theory Patrick Lee (MIT) Experiment Su-Yang Xu, F. We applied it to graph Resonant enhancement of side bands of the Floquet state generates a sign change of the persistent current. Floquet Hamiltonian and therefore modify the position of the Dirac point in the Floquet band. In the. Note that in the literature this approach is also Assuming adiabatic evolution of two bands, we show that the difference of the number of right-handed and left-handed Weyl points equals twice the winding number of the adiabatic Floquet operator over the Brillouin zone. Temperature-Controlled Stop Bands for Elastic Bloch-Floquet Waves in Periodic Thermo-Elastic Structures I. A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. 23. University of California Los Angeles. Note that the periodicity of f(t) in equation (122. . The Floquet bands are found to have non-zero Chern numbers which are generally different from those in the original static model. Here, the factor is referred to as “quasi-energy”, which represents the phase delay in each quarter ring. Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form. Tuning the Mobility of a Driven Bose-Einstein Condensate via Diabatic Floquet Bands Tobias Salger,1 Sebastian Kling,1 Sergey Denisov,2 Alexey V. The purple arrows mark the frequencies of 7. In “ Floquet chiral edge states in graphene” [3] we offered the first analytical solution for the topological states as well as numerics proving their chirality [A simple analysis for bilayer graphene can be found in [4]]. Bergman, Gil Refael,2 and Victor Galitski3 ,4 5 1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA Floquet bands and gaps in a periodically driven nonlinear photonic crystal. The observed BEC depletion rates are much higher when shaking along both the x and y directions, as opposed to only x or only y. We will show that as the number of wells becomes large, the allowed energy levels for the electron form nearly continuous energy We discuss the concept of time-dependent bands and steering of Floquet-Weyl points and demonstrate how light can enhance topological protection against lattice perturbations. In that work, Kane and Mele23 introduce a distinctive Z 2 index to describe the T-invariant quantum spin eigenmode (Floquet mode) that drives this instability. Whereas the envisioned Floquet topological insulator requires high-frequency pumping to obtain well-separated Floquet bands, a follow-up question regards the creation of Floquet-like states in graphene with realistic low-frequency laser pulses. Indeed, changing the topological nature of Floquet Bloch bands from trivial to non-trivial, by progressively launching the time-modulation, is necessarily accompanied with gap-closing processes: this has important consequences for the loading of particles into a target Floquet band with non-trivial topology, and hence, on the subsequent 108 the form of frequency bands, e. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than what is predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is Valence band is a filled band, generally, and conduction band is partially filled with electrons. Thus, its Floquet band structure has eight bands, as we can see in Fig. Recently the creation of novel topological states of matter by a periodic driving field has attracted great attention. We assume that the system is attached to external static leads, as shown in Fig. Our findings are general and apply to any 3D Dirac semimetal. This means that xT 2 is a necessary but not sufficient condition to obtain separated bands, and the value of xM also has to be adjusted. Floquet bands can be thought of as electrons dancing with photons, with the photons leading and electrons following. M. Occupation of topological Floquet bands in open systems. 4  fermions of a topological insulator to form Floquet-Bloch bands. corresponding PBG structures, in which the first three bands are shown. The second approach based on the Berry-Tabor conjecture is also possible to extend to Floquet systems by analyzing the statistics of the folded spectrum of the phases of matter. Oct 26, 2017. • Floquet Theorem for Periodic Structures. Jones and A. • Floquet phases and Topological states in condensed matter systems. Mahmood, Nuh Gedik (MIT) Qiong Ma, Pablo Jarillo-Herrero (MIT) workhorses for Floquet engineering for years to come, so that the exclusive concentration on this one particular system appears to be well justified. The study of photonic crystals is likewise governed by the Bloch-Floquet theorem, and intentionally introduced defects in the crystal (analo-gous to electronic dopants) give rise to localized electromagnetic states: linear waveguides and point-like cavities. While many schemes have been proposed for realizing interesting Floquet band structures, crucial questions remain regarding the many-body steady states of these divided into separate bands. Iadecola, Thomas, Neupert, Titus, and Chamon, Claudio. These observations establish the Floquet-Bloch bands in solids and pave the way for optical manipulation of topological quantum states of matter. We proposed a minimal model to describe the Floquet band structure of two-dimensional materials with light-induced resonant inter-band transition. n ¼ 0 and n ¼ −1 bands is zero (δE ¼ 0) at their crossing points, the induced gap Δ increases linearly with laser amplitude A0. Jet West took flight in spring 2009, when singer-guitarist Chris Warner and singer Scott Floquet (Pyschoactive) partnered up to play house parties and charity events. -b, 73. The normalization factor in Eq. Floquet Topological Insulators and Majorana Modes Introduction Topological Insulator Topological Insulator It has been realized that same robust conducting edge states that are found in the quantum Hall state could be found on the boundary of band insulators with large spin-orbit e ect, called topological insulators. and p bands) in a shaken fermionic optical lattice. For generic filling, with either bosons or fermions, the system is gapless; here the driving cannot be adiabatic and the system is expected to rapidly absorb energy from the driving field. Electrons in conduction band are practically free to move and hence conduct current. We report Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands in an amplitude-modulated optical lattice. Floquet adiabatic theory for different drive parameters [23]. BrataasBrataas ((Norweigian University) • Floquet phases and Topological states in condensed matter systems. The FENE-P model is used to represent the non-Newtonian fluid, and the analysis is done using a modified version of an existing nonlinear code to compute the linearized initial value problem governing the growth of small Figures 8 and 9 show the transmittivity and Floquet’s conditions for this example. Going beyond we demonstrate in this article the evolution of Floquet topological (Floquet) systems far from equilibrium are fertile grounds for intriguing phenomena without static counterparts [43– 46]. A. A Their combined citations are counted only for the first article. under the influence of time-periodic perturbation, the so-called Floquet systems , have with decreasing frequency, the different Floquet bands start to overlap. Betouras, and Sergey E. Schematic diagram of device geometry. A. 2 In such a system the coupling to a  Floquet bands and photocurrent generation. Moreover Figure 9(b) indicates that the limit points A and B of Floquet’s condition remain fixed by increasing the unit cell number; therefore, the single unit cell gives information about the passband centred at the resonance wavelength . In other words, the infinite number of bands below the gap makes it impossible to apply the Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. (4), by discretizing the time interval between t ¼ 0 and t0 ¼ explain these results, we establish an analogy between 2!. Kitagawa (Harvard) H. Title: Fast long-range coherent transport in hybridized Floquet-Bloch bands. (b) Sketch of the quasienergy bands, with the shadowed region representing the bulk bands, connected by the Floquet chiral modes (blue lines). Abstract: We report Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands in an amplitude-modulated optical lattice. Mathematicians call this technique Floquet theory, whereas physicists call it Bloch wave theory. See Figure 4 ,5, 6 and Supplemental Materials for the conic dispersion bands and the corresponding spectra of periodic point scatterers. To motivate further experimental and theoretical studies, we investigate interesting aspects of Floquet bands and topological phase transitions in a continuously driven Harper model. Lee, Nuh Gedik Recent experiments lead by the PI Gedik has demonstrated that Bloch states in solids can be “dressed” by intense laser light, forming “Floquet-Bloch states” that are completely coherent and behave like real bands. 247kHz. In a Floquet system, optical conductivity has a Dirac fermion time-Floquet crystal: manipulating Dirac points Pablo Rodriguez-Lopez, Joseph J. Demler (Harvard) AA. In this talk, I will first review the current experimental situation on the observation of Floquet-Bloch bands by laser-driving 2D Dirac fermions on the surface of a topological insulator. Floquet topological systems in the vicinity of band crossings: Reservoir-induced coherence and steady-state entropy production Under the drive of monochromatic light, energy bands evolve into Floquet bands, which describe Bloch states dressed with photons. Thus, Eq. In Appendix A, we brie y introduce the Floquet-Bloch theory Supplementary information for: Selective scattering between Floquet-Bloch and Volkov states in a topological insulator Fahad Mahmood, Ching-Kit Chan, Zhanybek Alpichshev, Dillon Gardner, Young Lee, Patrick A. We now examine the Floquet band structure of the system. How about Floquet quasienergies? You can have both. However between the second, third, and fourth dispersion bands. Taking graphene as a paradigmatic material modynamic limit like various driven band models or spin chains, which can be mapped to free particle systems [2,11–13]. Abstract. The four bands of the ${\mathrm{LaNiO}}_{3}$ bilayer exhibit both quadratic band touching points and Dirac points. This work has potential practical implications for the ultrafast switching of materials properties, such as optical band gaps or anomalous magnetoresistance. E. The three spring constants k j are modulated in the form of sine waves k j ≡ k j (t) = k +δkcos(νt+φ j), j = 1,2, and 3, (1) where ν = 2π/T is the modulation frequency, T is its period, and φ j is a phase delay. L'interface de recherche DI-fusion permet de consulter les publications des chercheurs de l'ULB et les thèses qui y ont été défendues. , The University of Tokyo) T. In this paper, we show that the Floquet operators of periodically driven systems can be divided into topologically distinct homotopy classes and give a simple physical interpretation of this classi-fication in terms of the spectra of Floquet operators. Ching-Kit Chan. b Due to coupled state with an energy gap and a quantum Hall effect, coined Floquet topological insulator. 1(c) [20]. 2. F. Lee 1, Nuh Gedik ¿ 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA Lyapunov’s theorem easily implies Floquet theorem. Floquet side bands. These results are illustrated numerically by various gures and movies. • Superconductivity in low-dimensional materials. C. While the Floquet band structure exhibits a single avoided crossing relative to the equilibrium case, the distribution function shows a population inversion of the Floquet bands at low energies. Movchan [ + - ] Author Affiliations ten bands, the bands are likely to dominate. • Entrée. In the allowed bands the Bloch phase is real, 1\cosðuÞ\1, and, as a result, both Floquet–Bloch solutions are oscillating functions. (And yes, CASTEP needs more than a single copy of the wavefunction and a single copy of the FFT grid, so this order-of-magnitude calculation is just that. Our method is based on a truncated Floquet Hamiltonian in frequency space and exploits chiral symmetry to reduce the number of bound-ary conditions in such a way that the number of TESs may compute an effective band structure for the Floquet eigenvalue ". Notably,coherentspin-spininteractionswitharange of several microns [42,43,46,47] can be introduced via Christian Floquet shows large recent paintings, somewhere between geometric abstraction and perceptible surface. Floquet topological acoustic resonators and acoustic Thouless pumping Yang Long and Jie Rena) Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, School of Physics Sciences and However, ‘anomalous Floquet TI’ phases are an exception27–29 for the following reason. Notethatthequasi-energies aredefinedmodulothefrequencyω=2π/T. Srikantha Phani, J. They are insulators in the bulk but their edges support propagating states bridging the bulk band-gap, much like the states emerging in the Quantum Hall effect discovered by Klaus von Klitzing in 1980 but in the case of TIs this happens without a magnetic field thanks to spin-orbit interaction. We focus on the driven dy-E EG k}m low lying bands}Higher energy bands FIG. View Thomas Floquet’s profile on LinkedIn, the world's largest professional community. 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 Similar to static systems, periodically driven systems can host a variety of topologically non-trivial phases. AokiH. Floquet Bloch bands were very recently observed for the first time on the surface of a topological insulator irradiated by mid-infrared light12. Ra Mesoscopic systems subject to a periodic in time, ex- and flux. Floquet  Oct 2, 2018 a distinct band structure, so called Floquet-Bloch bands. Intensity of these replicas is governed by the interference between Floquet-Bloch and Volkov states, free electron-like states coupled with photons. This situation stands in sharp contrast to that of static two-dimensional systems, where the existence of chiral edge states is intimately tied to the topological structure of the system’s bulk bands, as captured by their Chern numbers [34]. , pass bands where wave propagation is possible, and stop bands 109 where the traveling signal attenuated exponentially, we apply the Floquet-Bloch approach, which is 110 based on the works of Floquet [14] and Bloch [8]. • Electrical, optical, and thermoelectric transport properties of Novel graphene-like low-dimensional systems. We use a numerically exact density Scattering Theory for Floquet-Bloch States Thomas Bilitewski and Nigel R. By varying the amplitude of the incident light, one can independently tune the first- and second-neighbor hopping for fixed frequency, which leads to considerable control over the Floquet band structure. Lindner,1, 2Doron L. B. Chimenti Center for Nondestructive Evaluation Iowa State University Ames, Ia. Potter, Monika Schleier-Smith, Ashvin Vishwanath and Norman Y. We study the dynamics of the quasi-energy bands of the system as a function of the strength of the oscillation and show band quasi-periodicity and band collapse. Flat Chern bands and edge states in the Hofstadter model, APS March Meeting, San Antonio TX, March 2015. Clearly if 0 0 versus w is known, all the fn are determined. Moreover, these bands, and the associated dispersion curves, are approximately constant in frequency, entirely unlike the behavior expected for a homogeneous plate. The governing DI-fusion, le Dépôt institutionnel numérique de l'ULB, est l'outil de référencementde la production scientifique de l'ULB. Ionut-Dragos Potirniche, Andrew C. 5) doesnot,byitself, insure that y(t) has a periodic solution. Floquet bands in 1D may exhibit non-trivial quasi-energy winding Chiral quasi-steady states form when interband scattering is suppressed Universality of quasi-steady behavior apparently persists even when intraband scattering is fast compared with driving frequency Novel prethermalization regime may extend to other dimensions Bloch states and energy bands arise from spatially periodic Hamiltonians in condensed matter systems. a Schematic of a nonlinear photonic crystal (PhC) placed in a monochromatic driving field E d at frequency Ω. Furthermore, we find that because of the nonperturbative nature of the periodic driving, a second topological number, the so-called RLBL invariant, is necessary to fully characterize the anomalous Floquet topological An Introduction to the Concept of Band Structure Andreas Wacker1 Mathematical Physics, Lund University November 20, 2018 1 Introduction Band structure is one of the most important concepts in solid state physics. In particular, there exist “anomalous” Floquet insula-tors (AFI) [47–52] whose static topological invariants vanish for all Floquet-Bloch bands, but edge states still emerge due in nite periodic media and uses the Bloch-Floquet principles to calculate the dispersion surfaces and to determine the pass and stop bands [7]. In the high-frequency limit, one can decouple the zero-photon dressed states from the other states, amounting to a simple time average, and add multi-photon states perturbatively5,48, H^ eff¼H 00 þ 1 O H0 1;H01: ð3Þ where H00 is the zero order (cycle-averaged) Hamiltonian and H0 1 are the single photon dressed Hamiltonians, c Sel ective scattering between Floquet -Bloch and Volkov states in a topological insulator Fahad Mahmood 1, Ching -Kit Chan 1, Zhanybek Alpichshev 1, Dillon Gardner 1, Young Lee 1, Patrick A. Experiment. Smaller field strengths would correspond to even smaller bulk gaps and less dispersive edge states. rounds the possibility of controlling the topology of the resulting \Floquet bands," whose topological classi cation is even richer than the one describing their static counterparts. The correction to first order in t1=ω in the 2× 2 Floquet-Bloch Hamiltonian H eff;k is proportional to the third Pauli matrix σ z. We then obtain the Floquet Periodically driven “Floquet” systems are nonequilibrium systems whose time translation symmetry can give rise to a rich dynamical phase structure. In order to demonstr ate the sway that the time-Floquet dynamics dictate Floquet topological insulators have inspired analogues in photonics, optics, and acoustics, in which non-reciprocal wave propagation in time-modulated materials is achieved due to the breaking of time-reversal symmetry. You are here. first Chern number of the bands below the chemical potential22. Oct 25, 2013 These observations establish the Floquet-Bloch bands in solids and pave the way for optical manipulation of topological quantum states of  + Topological phase is protected by time reversal symmetry (TRS). So far, this The Floquet approach is reminiscent, in the time domain, of the appearance of Bloch states and associated bands of electrons in a spatially periodic lattice potential. Multiterminal Conductance of a Floquet Topological Insulator L. By expanding any initial state called ‘‘Floquet topological insulators’’ (FTIs) include a wide range of physical solid state and atomic realizations, driven both at resonance and off-resonance. The optical Stark e˙ect can also occur in solids through the interaction between photo-induced Floquet Bloch bands and equilibrium Bloch bands13. Floquet theory n-photon dressed state effective theory. fr University of Notre Dame Bands Recommended for you. 7 and a trivial band gap has been reported under high frequency linearly polarized light 8. In this Letter we present a comprehensive study of an interacting quantum many-body model driven by electro-magnetic radiation which vanishes at large negative times, is periodic at large positive times, and is ramped up at controllable rates. λ = ± denotes the upper and lower bands, and α is a LAPE parameter to be defined later. frequency bands possess double degeneracy at K-point and therefore remain unaffected by the modulation, due to the monopolar (l¼0) nature of the mode, orthogonal to the modulation. 2 . - Transport in Floquet-Bloch bands. Foa Torres,1,* P. T Iadecola, T Neupert, C Chamon. When looking at the graphs crossing the by gray bands marked stop-bands at Figure 1, are the by Floquet theory idealized exponential decay rate per 1. We also report an explosion of the decay rates at large drive amplitudes and suggest a In literature, I read:"(exposed in a beam of light) in floquet theory, the quasi-static eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of ##\Omega##, the so-called Floquet sidebands" I have read something about floquet theory and found In literature, I read:"(exposed in a beam of light) in floquet theory, the quasi-static eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of ##\Omega##, the so-called Floquet sidebands" I have read something about floquet theory and found floquet. Safaeinili and D. Throughout his pictorial work, which is inspired by the language of geometric abstraction, this Swiss artist casts his eye on the relations between space and plane, form and colour. Wave-like  Mar 1, 2018 Keywords topological insulators, spin-Hall effect, Floquet theory . We show that such inverted band dispersions, together with on-site interactions between atoms on the s and p bands, provide a natural way to realize When increasing the radiation frequency, the edge state switches between a dissipationless quantized charge pumping behavior to a dissipative regime without quantization. The full single-particle wavefunction isthereforegivenbyΨ(t)=e−iεtΦ(t). In this work, we consider the question of the criticality that emerges at the transitions between distinct Floquet MBL phases. Ifexternaldriving is applied to a system initially in its ground state, the particles will populate several Floquet bands, leading to a nontrivial distribution function. Unlike the case of static Hamiltonians, the topological indices of bulk Floquet bands may fail to describe the presence and robustness of edge states, prompting the search for new invariants. Bond softening and bond hardening in intense laser fields can be described in terms of solutions obtained from the Floquet theorem. atoms, where Floquet control has so far been applied only to single-particle band structures [29–31,37–39], recent advances in optically controlling interactions [40–47] offer new opportunities for accessing strongly correlated phases [48–51]. of Applied Phys. Topological phase transitions (discontinuous change of Chern numbers) take place as we tune the amplitude or period of the driving field. estimated that the n-th Floquet sideband has an intensity (in our notation): I( k,E) ≈ J n {|λβ(cosθ a cosθ +isinθ a sinθ)−α|} 2 (2) as a function of the momentum angle θ = tan −1(k y/k x) and the polarization angle θ a = tan (a y/a x). + I'm a cultural project manager Energy Bands in Crystals This chapter will apply quantum mechanics to a one dimensional, periodic lattice of potential wells which serves as an analogy to electrons interacting with the atoms of a crystal. To better understand the connection between the distribu- divided into separate bands. In Fig. However, we find that this formula does with a linear subharmonic Floquet instability of the underlying unpaired vortex street. single particle Floquet bands are chiral, with the Floquet spectrum realizing nontrivial cycles around the quasienergy Brillouin zone. Such dressed electronic states can be detected by time- and angular-resolved photoelectron spectroscopy (ARPES) manifesting as sidebands to the equilibrium band structure. edu weaklycoupledtotheenvironment[15–17]. Therefore, in order to achieve dual-band operation with a single aperture, one has to tailor a modulation able to radiate the SW power in the desired directions at both frequencies. Prime examples are Floquet topological insulators (FTIs), where a gapped bulk supports in-gap edge states, protected against symmetry-preserving local perturbations. Combining the two situations, a periodic excitation on a crystalline lattice induces Floquet-Bloch bands that repeat in both momentum and energy. Rev. Publications; Transport in Floquet-Bloch Bands find that the driven system approaches a Floquet-insulator state, with separate particle and hole densities in the upper and lower Floquet bands, respectively; see Fig. Periodic driving of the level detuning results in an effective population inversion that depends sensitively on the drive parameters and band alignment. their quasienergy bands "b(k), indexed by band quasimo-mentum wave vectors k, can be obtained from diagonaliz-ing the e ective time-independent Floquet Hamiltonian, which generates the stroboscopic evolution of the system in steps of the driving period T. As the theory is quadratic, HF can be chosen to be quadratic in the field operators. GREGORY MCDANIEL,P. 1) while keeping propagating states through the edges of a zigzag ribbon [48]. the projected band diagram turns into a Floquet one, denoted by (k) as shown in Refs. - Driven quantum systems. This formalism offers a general description of nonlinear responses when the following conditions are met: (i) only one frequency W is involved (monochromatic light), (ii) mostly two of the topological aspects of the Floquet band structure. Bergman, Gil Refael,2 and Victor Galitski3 ,4 5 1Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA Topological Floquet spectrum in three dimensions via a two-photon resonance Netanel H. In the presence of quenched disorder, they can avoid thermalizing to a bland infinite temperature state through a phenomenon known as many-body localization (MBL). Apart from the dramatic change in the dispersion relation, the topological properties of the bands are changed. The energy difference between the original nodal point in equilibrium and the crossing point of n ¼ 0 and n ¼ −1 bands on the Γ-Z path is defined as Δ0 [Fig. ANALYSIS OF FLOQUET WAVE GENERATION AND PROPAGATION IN A PLATE WITH MULTIPLE ARRAYS OF LINE ATTACHMENTS D. ) Note that for a cubic system, a reciprocal space grid containing all components up to gcut will contain components up to p 3gcut In the stationary Schrödinger equation, we can have a continuous or a discrete spectrum. The Floquet theory that is used in order to find the stop-bands is not defined for non-linear cases. Woodhouse, and N. = ~. The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems, where the information of the Chern numbers of the bands is not sufficient. 247kHz), respectively. S. 11:04. In one variety of Floquet-Bloch band we observe tunable rapid long-range high-fidelity transport of a Bose condensate across thousands of lattice sites. H˜ has a spatial periodicity of 4a by 2a along the x and y directions, respectively. understood theoretically using an equivalent Floquet fre-quency lattice with local PT symmetry. We use the plane wave expansion method to obtain the band structure of different interference patterns, because this method takes into account all the necessary parameters which affect the RI modulation. The red and blue bands denote nontrivial edge states at upper and lower boundaries of the lattice. Originally FLOQUET ANALYSIS OF LAMB WAVES PROPAGATING IN PERIODICALLY-LAYERED COMPOSITES INTRODUCTION A. Crucially, Floquet-Bloch bands have been observed only at ultra-short time scales, when heating is still small, and quantum coherence is maintained. dominique@orange. " Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition. Fu (MIT) E. Peter Kuchment, Texas A & M University Introduction to periodic operators DI-fusion, le Dépôt institutionnel numérique de l'ULB, est l'outil de référencementde la production scientifique de l'ULB. As shown in Supplementary Figure 3 (the right hand side), is the eigenvalue of the stop bands. + I'm a cultural project manager Valence band is a filled band, generally, and conduction band is partially filled with electrons. Ex-tending the periodicity in the time domain by applying a time-periodic perturbation increases the tunability of the Hamiltonian since the temporal analogue of Bloch states (the Floquet states) can be manipulated via the polarization, bands that touch in two Dirac points for t1=ω → 0. When two Floquet bands cross, they show an anticrossing. size. Balseiro,2,3 and Gonzalo Usaj2,3 1Instituto de Física Enrique Gaviola (CONICET) and FaMAF, Universidad Nacional de Córdoba, 5000 Córdoba, Argentina Experimental control of transport resonances in a coherent quantum rocking ratchet Christopher Grossert1, Martin Leder1, Sergey Denisov2,3,4, Peter Ha¨nggi2,3,5 & Martin Weitz1 The ratchet phenomenon is a means to get directed transport without net forces. 1C, we show the dressed Floquet bands for different accessible driving amplitudes. = original system projection to the original. The situation is much worse for non-evolution periodic PDEs. It is also Floquet stability analysis abstract A Floquet linear stability analysis has been performed on a viscoelastic cylinder wake. Such systems are viewed here as spatially The same dispersion exists also for eV but now the structure of Floquet bands is more complex and extra edge states appear at different k-points and in other gaps and lenses of the FPBBS. United States: N. In this thesis I will detail experiments performed in the Weld group which explore a many-body Floquet system consisting of tunably While the role of topology in the modern condensed matter physics is difficult to overstate, and despite numerous experimental corroborations of theoretically predicted symmetry-protected topological phases (such as topological insulators), most of these advances can be formulated in the language of non-interacting particles. As it can be seen from equation 5 in Methods, in our system there is no interaction among bands of the same Floquet scattering in higher Floquet bands, in agreement with recent theoretical predictions. In one sense it is trivial to show this, since any constant hamiltonian is also periodic, but presumably you want some more physical examples, so here's two. Abstract Pump-probe techniques with high temporal resolution allow one to drive a system of interest out of equilibrium and at the same time probe its properties. The inset in (b5) is a zoomed-in plot of the first two bands. In the off-resonant case, the system exhibits the similar character as the kagome lattice model with staggered magnetic fluxes, but the total band width is damped in oscillation. Another seminal work is the Kane and Mele model, dealing with time-reversal (T) invariant systems of strong spin–orbit couplings23. A semiclassical model that combines \emph{local} Floquet-Bloch bands analysis and Landau-Zener transitions provides a simple picture of the observed phenomena in terms of elementary \emph{Floquet We report Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands in an amplitude-modulated optical lattice. The Chern number of the Floquet band, which reflects the number of pairs of helical edge bands in graphene ribbons, can be reduced into the winding number at resonance. S. In this framework, a popular strategy is the direct numerical simulation, using nite elements, of a unit cell of the periodic material with Bloch-Floquet boundary conditions [1], [20]. Indeed, changing the topological nature of Floquet Bloch bands from trivial to non-trivial, by progressively launching the time-modulation, is necessarily accompanied with gap-closing processes: this has important consequences for the loading of particles into a target Floquet band with non-trivial topology, and hence, on the subsequent Pure quantum-mechanical mixture of electrons and photons demonstrated in bismuth selenide. The analogs of Floquet and Lyapunov theorems MIGHT (but not always do) work for evolution PDEs. - Position-space Bloch oscillations. This loss of quantization is caused by the Floquet band crossing which results in a “mixing” of the topological invariants of individual bands. The eigenfrequencies of a symmetrical periodicity cell can serve as alternative indicators of stop-band boundaries. When the lattice shakes (represented here by a grey oscillatory overlay), a moat appears in the band structure, as shown in the lower part of the graphic. • ω−β Diagrams for Periodic Structures. The Floquet topological phases and chiral edge states in a kagome lattice under a circularly-polarized driving field are studied. Just as different Bloch bands hybridize and develop band gaps at the crossing points , the crossing points between different orders (n) of the Floquet-Bloch bands open dynamic gaps (16, 20). The Floquet topological insulator is defined through the topological properties of the time-independent Floquet operator Hˇ Whereas the envisioned Floquet topological insulator requires high-frequency pumping to obtain well-separated Floquet bands, a follow-up question regards the creation of Floquet-like states in The corresponding quasi-energies, known as the Floquet bands, are analogous to Bloch bands in lattice systems caused by spatial periodicity. quantum systems. (4) seems to be the only natural topological invariant. 3. of quantiza- tion is caused by the Floquet band crossing which results. Theory. First, using only a single frequency ω of light, we engineer a static lattice V(r) whose unit cell comprises two sites (A and B) offset by an energy δ. (1) is block di-agonal in the Fourier space, and simply consists of a collection of identical 2 ×2 time-independent Hamiltonians separated in energy by ω. 10. Floquet Wave listed as FW (band) FW: Foul Water Floquet theorem; Floquet theory; Floquet Wave; Floquet engineering is a versatile tool that uses periodic driving of a quantum system to build novel many-body quantum states with drive-dependent properties. (Received 5 April 1999, and in,nal form 3 January 2000) physics. The nonequilibrium steady state (and hence, NLORs) can be captured by studying this anticrossing of two Floquet bands. Perez-Piskunow,1 C. That phenomenon must be considered in the reflectivity measurement of calibration target, especially in the mono-static backscattering configuration. Language: English Location: United States Restricted Mode: Off History Help About Our paper reporting Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands has been published in Phys. The stop bands predicted by the Floquet theory are marked as grey strips in Figure 1 and Figure 13. In the simplest case, where ω 0 couples quasi-resonantly two levels of the bare atom (small detuning Δ≪ ω 0 ), the dressed states form a doublet. Reading: Haus 7. We discuss the structure of optical Hall and longitudinal conductivity in different topological phases. The Floquet topological insulator is defined through the topological properties of the time-independent Floquet operator H F(k), in accordance with the existing topological Spatially uniform excitations can induce Floquet topological bandstructures within insulators which have equal characteristics to those of topological insulators. We also define the Floquet-Green’s function for a time-periodic Hamiltonian and by a generalization of the method used for the two previous potentials we are able Maximally Localized Wannier Functions Andreas Klockner Outline Photonic Crystals Fabrication Eigenproblems with Spatially Periodic Coefficients The Floquet Transform Wannier Functions Minimizing the Spread Outlook and Origins Maximally Localized Wannier Functions Andreas Kl¨ockner Valentine’s Day 2007 Andreas Kl¨ockner Maximally Localized Floquet Topological Insulators [1,2] can be made out of normal materials which develop properties akin those in the quantum Hall regime, for example, upon illumination with a laser field. J. Nevertheless one can construct examples of Floquet integrable systems in the ther-modynamic limit like various driven band models or spin chains, which can be mapped to free particle systems [2,11–13]. values of xM that will not fold the two upper bands on top of the four lower bands. The steady-state excitation density depends on the ratio of phonon-assisted inter-Floquet-band relaxation and recombi-nation rates, becoming small for fast interband relaxation. BIFROST, a new ARO-supported multi-user distributed tunable laser facility, is up and running ( article ). We discuss the concept of time-dependent bands and steering of Floquet-Weyl points and demonstrate how light can enhance topological protection against lattice perturbations. Finite-wavevector Electromagnetic Response in Lattice Quantum Hall Systems, APS March Meeting, Baltimore MD, March 2016. Figure 2 illustrates this for several cases. The quasienergy of a sideband α is then simply 02010 - Dep. 1: Typical schematic instantaneous band structure con- These properties can be understood (the bulk-edge correspondence) in terms of topological invariants (e. E. Yao, Phys. Floquet theory shows stability in Hill differential equation (introduced by George William Hill) approximating the motion of the moon as a harmonic oscillator in a periodic gravitational field. + Breaking TRS in TIs is predicted to lead to fascinating effects. We also define the Floquet-Green’s function for a time-periodic Hamiltonian and by a generalization of the method used for the two previous potentials we are able to derive an expression for the Floquet-Green’s function of any harmonically driven Hamiltonian. Floquet bands were first observed in photonic crystals 3 and have been verified experimentally for the surfaces of topological insulators 4,5,6. Obviously, in order to drive the TPT Floquet engineering is a versatile tool that uses periodic driving of a quantum system to build novel many-body quantum states with drive-dependent properties. works have considered Floquet systems that are isolated or *yliao@rice. - Experimental approach. of Physics 03599 - Esslinger, Tilman / Esslinger, Tilman quantum, magnetic quantum walks give rise to nearly flat energy bands featuring nonvanishing Chern numbers. Here, we study the topological nature of the NLORs by using the Floquet two-band models. A range of energy is called a band when the number of energy levels available are very large ( 6*10^23) Floquet Hamiltonian HF: U (T )=eiH F T (10) where U (T ) is the evolution operator for a period [3, 34]. Whereas this energy the occupation of the Floquet bands, and therefore plays a de ning role in the accessible topological properties of the system. Like in equilibrium, the quasienergy bands can be assigned Chern numbers Cb. Usually, for non-equilibrium systems, we use the non-equilibrium Green's function obtained in the Keldysh formalism. C. -d, 73. To support such backscattering-based reflectivity measurement, the Floquet mode and While the role of topology in the modern condensed matter physics is difficult to overstate, and despite numerous experimental corroborations of theoretically predicted symmetry-protected topological phases (such as topological insulators), most of these advances can be formulated in the language of non-interacting particles. Because Floquet quasi-energies, unlike ordinary energies, are angle variables27–29, Floquet band structures are thus not bounded below by a ‘lowest band’. Chicone. In that limit, the Floquet bands are uncoupled and the zeroth Fourier component H q=0 of the Floquet operator is the dominant one. In the bandgaps the Bloch phase is A periodically driven band structure can be semiclassically described by Floquet theory, resulting in photon-dressed Floquet bands (non-equilibrium steady states). 119, 123601 (2017). demonstration of anomalous Floquet topological insulator for sound: a strongly coupled metamaterial ring lattice that supports one-way propagation of pseudo-spin-dependent edge states under T-symmetry. Floquet bands. 5D parameter space (k x,k y,k z,θ,t), Eq. An intricate photon emis-sionpattern emergesas a result of cavity coupling, periodic Our results are in agreement with a theoretical analysis of the Floquet spectrum of a model system, thus revealing the existence of diabatic Floquet bands in the atom's band spectra and highlighting their role in the non-equilibrium transport of the atoms. result, the titled wave dispersion bands can be described by topological invariants: Chern numbers [26]. 1, (9,1- 9. Address: Postdoctoral Fellow, Department of Physics, Colorado State University,Fort Collins, CO, 80521, USA. In this thesis I will detail experiments performed in the Weld group which explore a many-body Floquet system consisting of tunably Abstract. Solid state physicists use this technique to determine band gap energies. try (PHS) in which the valence and conduction bands are interchanged. (b) and (c) The simulated pressure amplitude distributions in the pass band (7. 4(a)]. If, however, f(t) is periodic Floquet theory of photo-induced topological Takashi Oka (Dept of Applied Phys The University of Tokyo) phase transitions: Application to graphene Takashi Oka (Dept. strongly coupled Floquet bands, we develop a numer-ical method that explicitly allows us to construct the TESs for a half-in nite wire. The position and size of these instability islands can be controlled through the amplitude and frequency of the driving and can be achieved, in principle, for arbitrary values of the gain or loss parameter. Previously, we showed that a carefully tuned circularly polarized laser may introduce a bulk dynamical band gap at half the photon energy [44] (a scheme of the bulk dispersion is shown in Fig. In the stable regime, the mode spectrum of HF is non-negative and the eigen-modes are normalizable. 6, 8. , and Movchan, A. Notethatthequasi-energies aredefinedmodulothefrequency!D2ˇ=T. Because of the TRS breaking, it can thus poten-tially open a Haldane-type gap [6] in the spectrum, so that the resulting spin-degenerate bands can acquire a Chern of a ladder of Floquet bands, as shown in Fig. These systems displaymetallicconductionenabledbyquasistationarystates at the edges [16,17,20], Dirac cones in three dimensional systems [21], and even Floquet Majorana fermions [22]. by Denis Paiste, Massachusetts Institute of Technology Abstract. In addition, our calculations provide stability spectra throughout the transition regime and thus allow an accu- rate determination of the bands of unstable spanwise wavenumbers above the onset of three-dimensionality. the general dependence of $ 0 on w; that is, the frequency bands where 0o is real (pass bands) or imaginary (stop b:,nds), as well as the more specific variation of 0 0 versus W with-in the pass bands. Using this picture, we provide an intuitive understanding can support chiral edge states even if all of its bulk Floquet bands have zero Chern numbers [3,27]. My research explores new phenomena far from equilibrium in low dimensional systems. Floquet symmetry-protected topological phases in cold atomic systems. We conclude in Section VI. + I'm a cultural project manager demonstrated that the Floquet quasienergy spectra of periodically driven systems exhibit richer topological structures and invariants than their nondriven counterparts [28–44], attaching to the gaps of quasienergy bands, such as Floquet-Majorana end states [45–48], topologically nontrivial zero or π states [33–35], and topological singu- Wave propagation in two-dimensional periodic lattices A. 1 At off-resonant Floquet-topological insulator. Weassumeyouhavesome experienceofdesigningprojectsusingHFSS A primer on the Floquet theory of periodically time-dependent quantum systems is provided, and it is shown how to apply this framework for computing the quasienergy band structure governing the dynamics of ultracold atoms in driven optical cosine lattices. To realize each topological model, we design a dynamic interference pattern constituting a two-dimensional (2D) lat-tice that evolves periodically in time. When the shaking frequency is near resonant with respect to a band transition, the two bands couple and form two new dressed Floquet bands. In one variety of Floquet-Bloch band we observe tunable rapid long-range high- delity transport of a Bose condensate across thousands of lattice sites. Similar to an ordinary static topological insulator, the robustness of an edge state in a one-dimensional (1D) Floquet mode properties with scattering lobes in upper space. occupations of the eigenvalues of the Floquet Hamiltonian, the so called quasieenergies, play a crucial role. We Floquet theory and continued fractions for harmonically driven systems bands of the system as a function of the strength of the oscillation and show band quasi Floquet-engineering topological and spin-dependent bands with interacting ultracold fermions: 11:30 - 12:30: Takashi Oka (MPIPKS, Dresden) Linear response theory in periodically driven interacting systems: 12:30 - 14:00: Lunch: 14:00 - 15:30: Mark Rudner (Copenhagen University) Micromotion in topological Floquet systems: 15:30 - 17:00: Discussion Read "Standing waves in a non-linear 1D lattice: Floquet multipliers, Krein signatures, and stability, Physica D: Nonlinear Phenomena" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. 11 and 30. We also demonstrate the formation of pseudo-spin-dependent interface states due to lattice dislocations and investigate the properties of pass The main tool of the theory of periodic ordinary differential equations is the so-called Floquet theory [17, 94, 120, 156, 177, 267, 272, 389]. This consists of using a mid-IR laser pulse, with energy below the bulk band gap of the material, to stimulate electrons in the solid. We investigate a mechanical wave analogue of Thouless pumping and the quantum Hall effect Floquet-Bloch band spectrum when illuminated with a circularly polarized laser. Flecka Department of Engineering, Cambridge University, Trumpington Street, Cambridge CB2 1PZ, We formulate a linear response theory for a Floquet system subjected to an external perturbation field and derive the generalized optical conductivity. FLOQUET THEORY FRAMEWORK In this section, we describe the Floquet theory frame-work that will be used to analyze the dynamics of driven systems in this paper. We show that our results are supported by experimental Prethermal Floquet Steady States and Instabilities in the Periodically Driven, Weakly Interacting Bose-Hubbard Model Marin Bukov,1,* Sarang Gopalakrishnan,2 Michael Knap,2,3 and Eugene Demler2 1Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA View Thomas Floquet’s profile on LinkedIn, the world's largest professional community. + It will open a band gap and   Jul 13, 2018 made recently in measuring topological properties of Floquet bands in measurement of full Floquet band dispersions and their topology  Jan 4, 2016 An MIT team has demonstrated pure Floquet-Bloch states in the solid Replica bands disappear once the driving potential shuts off. Kubo and Floquet optical Hall conductivity, as given in the literature, can be obtained from our expression by taking appropriate limits. The topological transitions are understood from the Floquet-Bloch band structure of the clean system at high symmetry points in the first Brillouin zone. A range of energy is called a band when the number of energy levels available are very large ( 6*10^23) and very little energy is needed to move them within the material. For graphene the chiralities for different frequencies have been given in ref. Chern number) computable from quantities obtained from the spectral theory of the bulk unperturbed material (Berry curvature of Floquet-Bloch bands). the nonequilibrium Floquet bands. Its central result is the following theorem (sometimes called Floquet-Lyapunov theorem) [120, 267]. The light modifies the equilibrium band DI-fusion, le Dépôt institutionnel numérique de l'ULB, est l'outil de référencementde la production scientifique de l'ULB. II. The full single-particle wavefunction isthereforegivenby t(t)Dei" (t). Patrick Lee (MIT). PRL 107, 216601 (2011) PHYSICAL REVIEW LETTERS week ending 18 However, changing the topological nature of Floquet Bloch bands from trivial to non-trivial, by progressively launching the time-modulation, is necessarily accompanied with gap-closing processes: this has important consequences for the loading of particles into a target Floquet band with non-trivial topology, and hence, on the subsequent 1-Introduction ThisdocumentisintendedassupplementarymaterialtoHFSSusers. Su- Yang  Sep 19, 2019 Abstract: Spatially uniform optical excitations can induce Floquet topological band structures within insulators which can develop similar or  Jun 17, 2016 Floquet topological insulators, and demonstrate their relevance for a wide . 1 (color online). bulk quasi-energy bands. trivial linear systems, like harmonic oscillators, the Floquet theorem for classical systems does not exist. The s and p orbitals are considered as two pseudospins whose energy dispersions are inverted, in contrast to the same dispersion for usual spins. DUPONT AND L. Thomson Avenue, Cambridge CB3 0HE, United Kingdom (Dated: February 11, 2015) Motivated by recent experimental implementations of arti cial gauge elds for gases of cold atoms, Search this site . In one variety of Floquet-Bloch band we observe tunable rapid long-range These results demonstrate that transport in dynamical Floquet-Bloch bands can be  Jan 17, 2018 In this work we use Floquet-Bloch theory to study the influence of circularly and linearly polarized light on two-dimensional band structures with  Jan 8, 2019 We report Floquet band engineering of long-range transport and direct imaging of Floquet-Bloch bands in an amplitude-modulated optical  Nov 24, 2015 A primer on the Floquet theory of periodically time-dependent quantum systems is provided, and it is shown how to apply this framework for  Sep 13, 2018 Appetizer. Physik / Dep. Ponomarev,2 Peter Ha¨nggi,2 and Martin Weitz1 1Institut fu ¨r Angewandte Physik der Universitat Bonn, Wegelerstrasse 8, 53115 Bonn, Germany The edge bands induced by the circularly polarized light are helical and those by linearly polarized light are topologically trivial ones. Lett. We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators [1]. Replica bands disappear once the driving Christian Floquet shows large recent paintings, somewhere between geometric abstraction and perceptible surface. The linearly aligned clump of white lines in the middle of this band represent bosons condensing prior to shaking. As it can be seen from equation 5 in Methods, in our system there is no interaction among bands of the same Floquet harmonic order, and therefore no band crossing as understood in its conventional from 16,54 is present. p. Calculating the Bands: Floquet Theory UW Applied Mathematics Make use of Floquet (Bloch) theory with Floquet (characteristic) exponents ¥ keep fixed domain ¥ discretize ¥ solve D O(N3) equations larger period solutions View Thomas Floquet’s profile on LinkedIn, the world's largest professional community. It is demonstrated that the transmission zones arise from a consolidation or clustering of the uniform‐plate minima into frequency zones or bands, suggestive of Floquet wave behavior. FELSENs Department of Aerospace and Mechanical Engineering, Boston;niversity, Boston, MA 02215,;. 526kHz) and the band gap (7. Group, Cavendish Laboratory, J. Floquet Topological Order in Interacting Systems of Bosons and Fermions, APS March Meeting, New Orleans LA, March 2017. The crystal can thus form a kind of per- dimensional Floquet system may host chiral bands, while in two dimensions a system whose Floquet bands all have zero Chern numbers may support robust chiral edge states. "Temperature-Controlled Stop Bands for Elastic Bloch-Floquet Waves in Periodic Thermo-Elastic Structures. 526kHz and 7. PACS: 72. For a realization in ultracold atoms, these two features compensate, producing a bulk average rf signal that is well captured by a quasiequilibrium Floquet bands. Hilbert space. Cooper T. Exponentially Slow Heating in Short and Long-range Interacting Floquet Systems. Topological Floquet spectrum in three dimensions via a two-photon resonance Netanel H. Letters. These Floquet-Bloch bands are characterized by nontrivial Chern numbers which only depend on the helicity of the polarization of the radiation The spatial periodicity of a commensurate SW implies that the eigenmodes are of the Bloch form, characterised by an even number of Floquet multipliers. Figure 2(a) displays the bands closest to " ¼ 0 for a relatively large frequency, @! ¼ 3t where t is the FIG. Aoki (The unversity of Tokyo)of Tokyo) L. Floquet-Bloch states, occurring on incredibly fast time scales, are observed using an experimental technique called time-and-angle-resolved photoemission spectroscopy (Tr-ARPES). floquet bands

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