Floquet Bands Research Articles

Counterdiabatic Driving for Periodically Driven Systems.

Periodically driven systems have emerged as a useful technique to engineer the properties of quantum systems, and are in the process of being developed into a standard toolbox for quantum simulation. An outstanding challenge that leaves this toolbox incomplete is the manipulation of the states dressed by strong periodic drives. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. To achieve fast control of nonequilibrium quantum matter, we generalize the notion of variational counterdiabatic driving away from equilibrium focusing on Floquet systems. We derive a nonperturbative variational principle to find local approximations to the adiabatic gauge potential for the effective Floquet Hamiltonian. It enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. We discuss applications to two-level, Floquet band, and interacting periodically driven models. The developed technique allows us to capture nonperturbative photon resonances and obtain high-fidelity protocols that respect experimental limitations like the locality of the accessible control terms.

Cavity Floquet engineering

Floquet engineering is a promising tool to manipulate quantum systems coherently. A well-known example is the optical Stark effect, which has been used for optical trapping of atoms and breaking time-reversal symmetry in solids. However, as a coherent nonlinear optical effect, Floquet engineering typically requires high field intensities obtained in ultrafast pulses, severely limiting its use. Here, we demonstrate using cavity engineering of the vacuum modes to achieve orders-of-magnitude enhancement of the effective Floquet field, enabling Floquet effects at an extremely low fluence of 450 photons/μm2. At higher fluences, the cavity-enhanced Floquet effects lead to 50 meV spin and valley splitting of WSe2 excitons, corresponding to an enormous time-reversal breaking, non-Maxwellian magnetic field of over 200 T. Utilizing such an optically controlled effective magnetic field, we demonstrate an ultrafast, picojoule chirality XOR gate. These results suggest that cavity-enhanced Floquet engineering may enable the creation of steady-state or quasi-equilibrium Floquet bands, strongly non-perturbative modifications of materials beyond the reach of other means, and application of Floquet engineering to a wide range of materials and applications.

Optically Tunable Many-Body Exciton-Phonon Quantum Interference.

This study introduces a novel paradigm for achieving widely tunable many-body Fano quantum interference in low-dimensional semiconducting nanostructures, beyond the conventional requirement of closely matched energy levels between discrete and continuum states observed in atomic Fano systems. Leveraging Floquet engineering, the remarkable tunability of Fano lineshapes is demonstrated, even when the original discrete and continuum states are separated by over 1eV. Specifically, by controlling the quantum pathways of discrete phonon Raman scattering using femtosecond laser pulses, the Raman intermediate states across the excitonic Floquet band are tuned. This manipulation yields continuous transitions of Fano lineshapes from antiresonance to dispersive and to symmetric Lorentzian profiles, accompanied by significant variations in Fano parameter q and Raman intensity spanning 2 orders of magnitude. A subtle shift in the excitonic Floquet resonance is further shown, achieved by controlling the intensity of the femtosecond laser, which profoundly modifies quantum interference strength from destructive to constructive interference. The study reveals the crucial roles of Floquet engineering in coherent light-matter interactions and opens up new avenues for coherent control of Fano quantum interference over a broad energy spectrum in low-dimensional semiconducting nanostructures.

Non-Bloch theory for spatiotemporal photonic crystals assisted by continuum effective medium

As one indispensable type of nonreciprocal mechanism, a system with temporal modulations is intrinsically open in the physical sense and inevitably non-Hermitian, but the space and time degrees of freedom are nonseparable in a large variety of circumstances, which restrains the application of the non-Bloch band theory. Here, we investigate the spatially photonic crystals (PhCs) composed of spatiotemporal modulation materials (STMs) and homogeneous media, dubbed as the STMPhC, wherein the spatial and temporal modulations are deliberately designed to be correlated. To bypass the difficulty of the spatiotemporal correlation, we first employ the effective medium theory to account for the dispersion of fundamental bands under the influence of Floquet sidebands. Based on the continuum generalized Brillouin zone condition, we then analytically give the criteria for the existence of the non-Hermitian skin effect in the STM. Assisted by developing a numerical method that embeds the plane wave expansion in the transfer matrix, we establish the non-Bloch band theory for the low-frequency Floquet bands in the STMPhCs, in which the identification of the generalized Brillouin zone is central. We finally delve into the topological properties, including non-Bloch Zak phases and non-Bloch bulk-boundary correspondence. Our work validates the idea that the effective medium assists the non-Bloch band theory applied to the STMPhCs, which delivers a prescription to broaden the horizons of non-Bloch theory. Published by the American Physical Society 2024

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.

Selection and enhancement of the frequency modes with Floquet exceptional points and chiral Zener tunneling

Mode selecting plays a vital role in the field of optoelectronics, such as optical communication, signal processing, on-chip light manipulation, mode conversion, and frequency synthesis. In this work, flexible selection and enhancement of the frequency modes in an unidirectional coupled Su–Schrieffer–Heeger (SSH) frequency lattice are obtained with Floquet exceptional points (EPs) and chiral Zener tunneling (ZT). The unidirectional coupled non-Hermitian SSH frequency lattices are synthesized by a double-ring system with complex dynamical modulations. Under an effective direct current (dc) force induced by the phase-mismatching of the modulations, the two Floquet bands of the non-Hermitian frequency lattices are degenerated and the Floquet EPs arise. Therefore, the unidirectional and irreversible frequency mode conversion takes place, which is the chiral ZT. Moreover, through perturbation analysis and numerical simulations, we prove that the frequency modes of the two-band system can be selected and enhanced by a multi-photon resonance dc force.

Floquet Weyl states at one-photon resonance: An origin of nonperturbative optical responses in three-dimensional materials

Electrons driven coherently by laser light can exhibit nonperturbative geometric effects. Drastic deformation and gap openings of the electrons' Floquet bands occur at one-photon resonances since the electron and hole bands hybridize through their replicas at the lowest-order photon exchange. We study the evolution of Floquet bands in three-dimensional (3D) materials driven by circularly polarized light (CPL) using the Dirac model. We find that the light-induced gap closes at a select few points in the momentum space where Floquet Weyl points are formed. The Weyl points are protected by their monopole charge and can merge, separate, or pair annihilate depending on the anisotropy of electrons and the ellipticity of the incident light. In isotropic 3D Dirac electrons driven by CPL, the Weyl points merge to form Floquet double-Weyl points with topological charge ±2. Our results reveal a universal aspect of light-matter interactions in 3D quantum materials and open a route towards controllable versatile electromagnetic responses associated with light-induced Floquet Weyl states. Published by the American Physical Society 2024

Investigation of Floquet engineered non-Abelian geometric phase for holonomic quantum computing

Holonomic quantum computing functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold Rb87 atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in SU(2), characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems. Published by the American Physical Society 2024

Mapping Light-Dressed Floquet Bands by Highly Nonlinear Optical Excitations and Valley Polarization.

Ultrafast nonlinear optical phenomena in solids have been attracting a great deal of interest as novel methodologies for the femtosecond spectroscopy of electron dynamics and control of the properties of materials. Here, we theoretically investigate strong-field nonlinear optical transitions in a prototypical two-dimensional material, hBN, and show that the k-resolved conduction band charge occupation patterns induced by an elliptically polarized laser can be understood in a multiphoton resonant picture, but, remarkably, only if using the Floquet light-dressed states instead of the undressed matter states. Our work demonstrates that Floquet dressing affects ultrafast charge dynamics and photoexcitation even from a single pump pulse and establishes a direct measurable signature for band dressing in nonlinear optical processes in solids, opening new paths for ultrafast spectroscopy and valley manipulation.

Rayleigh-Jeans prethermalization and wave condensation in a nonlinear disordered Floquet system

Periodically driven quantum systems make it possible to reach stationary states with new emerging properties. However, this process is notoriously difficult in the presence of interactions because continuous energy exchanges generally boil the system to an infinite temperature featureless state. Here, we describe how to reach nontrivial states in a periodically kicked nonlinear disordered system. One ingredient is crucial: both disorder and kick strengths should be weak enough to induce sufficiently narrow and well-separated Floquet bands. In this case, inter-band heating processes are strongly suppressed and the system can reach an exponentially long-lived prethermal plateau described by the Rayleigh-Jeans distribution. Saliently, the system can even undergo a wave condensation process when its initial state has a sufficiently low total quasi-energy.

Giant self-driven exciton-Floquet signatures in time-resolved photoemission spectroscopy of MoS2 from time-dependent GW approach

Time-resolved, angle-resolved photoemission spectroscopy (TR-ARPES) is a one-particle spectroscopic technique that can probe excitons (two-particle excitations) in momentum space. We present an ab initio, time-domain GW approach to TR-ARPES and apply it to monolayer MoS2. We show that photoexcited excitons may be measured and quantified as satellite bands and lead to the renormalization of the quasiparticle bands. These features are explained in terms of an exciton-Floquet phenomenon induced by an exciton time-dependent bosonic field, which are orders of magnitude stronger than those of laser field-induced Floquet bands in low-dimensional semiconductors. Our findings imply a way to engineer Floquet matter through the coherent oscillation of excitons and open the new door for mechanisms for band structure engineering.

Time Translation Symmetry Breaking in an Isolated Spin-Orbit-Coupled Fluid of Light.

We study the interplay between intrinsic spin-orbit coupling and nonlinear photon-photon interactions in a nonparaxial, elliptically polarized fluid of light propagating in a bulk Kerr medium. We find that in situations where the nonlinear interactions induce birefringence, i.e., a polarization-dependent nonlinear refractive index, their interplay with spin-orbit coupling results in an interference between the two polarization components of the fluid traveling at different wave vectors, which entails the breaking of translation symmetry along the propagation direction. This phenomenon leads to a Floquet band structure in the Bogoliubov spectrum of the fluid, and to characteristic oscillations of its intensity correlations. We characterize these oscillations in detail and point out their exponential growth at large propagation distances, revealing the presence of parametric resonances.

Fermi-Dirac staircase occupation of Floquet bands and current rectification inside the optical gap of metals: An exact approach

We consider a model of a Bloch band subjected to an oscillating electric field and coupled to a featureless fermionic heat bath, which can be solved exactly. We demonstrate rigorously that in the limit of vanishing coupling to this bath (so that it acts as an ideal thermodynamic bath) the occupation of the Floquet band is not a simple Fermi-Dirac distribution function of the Floquet energy, but instead it becomes a ``staircase'' version of this distribution. We show that this distribution generically leads to a finite rectified electric current within the optical gap of a metal even in the limit of vanishing carrier relaxation rates, providing a rigorous demonstration that such rectification is generically possible and clarifying previous statements in the optoelectronics literature. We show that this current remains non-zero even up to the leading perturbative second order in the amplitude of electric fields, and that it approaches the standard perturbative expression of the Jerk current obtained from a simpler Boltzmann description within a relaxation time approximation when the frequencies are small compared to the bandwidth.

Photonic Floquet Landau-Zener tunneling and temporal beam splitters.

Landau-Zener tunneling (LZT), i.e., the nonadiabatic transition under strong parameter driving in multilevel systems, is ubiquitous in physics, providing a powerful tool for coherent wave control both in quantum and classical systems. While previous works mainly focus on LZT between two energy bands in time-invariant crystals, here, we construct synthetic time-periodic temporal lattices from two coupled fiber loops and demonstrate dc- and ac-driven LZTs between periodic Floquet bands. We show that dc- and ac-driven LZTs display distinctive tunneling and interference characteristics, which can be harnessed to realize fully reconfigurable LZT beam splitter arrangements. As a potential application to signal processing, we realize a 4-bit temporal beam encoder for classical light pulses using a reconfigurable LZT beam splitter network. Our work introduces and experimentally demonstrates a new class of reconfigurable linear optics circuits harnessing Floquet LZT, which may find versatile applications in temporal beam control, signal processing, quantum simulations, and information processing.

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.

Floquet band engineering in action

Floquet band engineering in action

Controllable Floquet edge modes in a multifrequency driving system

A driven quantum system was recently studied in the context of nonequilibrium phase transitions and their responses. In particular, for a periodically driven system, its dynamics are described in terms of the multidimensional Floquet lattice with a lattice size depending on the number of driving frequencies and their rational or irrational ratio. So far, for a multifrequency driving system, the energy pumping between the sources of frequencies has been widely discussed as a signature of topologically nontrivial Floquet bands. However, the unique edge modes emerging in the Floquet lattice have not been explored yet. Here, we discuss how the edge modes in the Floquet lattice are controlled and result in the localization at particular frequencies, when multiple frequencies are present and their magnitudes are commensurate values. First, we construct the minimal model to exemplify our argument, focusing on a two-level system with two driving frequencies. For the strong frequency limit, one can describe the system as a quasi-one-dimensional Floquet lattice where the effective hopping between the neighboring sites depends on the relative magnitudes of the potential for two frequency modes. With multiple driving modes, nontrivial Floquet lattice boundaries always exist from controlling the frequencies, and this gives rise to states that are mostly localized at such Floquet lattice boundaries, i.e., particular frequencies. We suggest the time-dependent Creutz ladder model as a realization of our theoretical Hamiltonian and show the emergence of controllable Floquet edge modes.

Floquet engineering of two dimensional photonic waveguide arrays with [formula omitted] or [formula omitted] corner states

In this paper, we theoretically study the Floquet engineering of two dimensional photonic waveguide arrays in three types of lattices: honeycomb lattice with Kekulé distortion, breathing square lattice and breathing Kagome lattice. The Kekulé distortion factor or the breathing factor in the corresponding lattice is periodically changed along the axial direction of the photonic waveguide with frequency ω. Within certain ranges of ω, the Floquet corner states in the Floquet band gap of quasi-energy spectrum are found, which are localized at the corner of the finite two-dimensional arrays. Due to particle-hole symmetry in the model of honeycomb and square lattice, the quasi-energy level of the Floquet π corner states is ±ω/2. On the other hand, Kagome lattice does not have particle-hole symmetry, so that the quasi-energy level of the Floquet ±2π/3 corner states is near to ±1ω/3. The corner states are either protected by crystalline symmetry or reflection symmetry. The finding of Floquet fractional-π corner states could provide more options for engineering of on-chip photonic devices.

Pseudospin-selective Floquet band engineering in black phosphorus

Time-periodic light field has emerged as a control knob for manipulating quantum states in solid-state materials1-3, cold atoms4 and photonic systems5 through hybridization with photon-dressed Floquet states6 in the strong-coupling limit, dubbed Floquet engineering. Such interaction leads to tailored properties of quantum materials7-11, for example, modifications of the topological properties of Dirac materials12,13 and modulation of the optical response14-16. Despite extensive research interests over the past decade3,8,17-20, there is no experimental evidence of momentum-resolved Floquet band engineering of semiconductors, which is a crucial step to extend Floquet engineering to a wide range of solid-state materials. Here, on the basis of time and angle-resolved photoemission spectroscopy measurements, we report experimental signatures of Floquet band engineering in a model semiconductor, black phosphorus. On near-resonance pumping at a photon energy of 340-440 meV, a strong band renormalization is observed near the band edges. In particular, light-induced dynamical gap opening is resolved at the resonance points, which emerges simultaneously with the Floquet sidebands. Moreover, the band renormalization shows a strong selection rule favouring pump polarization along the armchair direction, suggesting pseudospin selectivity for the Floquetband engineering as enforced by the lattice symmetry. Our work demonstrates pseudospin-selective Floquet band engineering in black phosphorus and provides important guiding principles for Floquet engineering of semiconductors.

Tuning Anomalous Floquet Topological Bands with Ultracold Atoms.

The Floquet engineering opens the way to create new topological states without counterparts in static systems. Here, we report the experimental realization and characterization of new anomalous topological states with high-precision Floquet engineering for ultracold atoms trapped in a shaking optical Raman lattice. The Floquet band topology is manipulated by tuning the driving-induced band crossings referred to as band inversion surfaces (BISs), whose configurations fully characterize the topology of the underlying states. We uncover various exotic anomalous topological states by measuring the configurations of BISs that correspond to the bulk Floquet topology. In particular, we identify an unprecedented anomalous Floquet valley-Hall state that possesses anomalous helical-like edge modes protected by valleys and a chiral state with high Chern number.