A transmission phenomenon for a quantal particle scattered through a multi-well potential in one dimension is observed by means of an amplitude-phase method. The potential model consists of $N$ identical potential cells, each containing a symmetric well. Typical transmission bands contain $N-1$ possible energies of total transmission. It is found that certain band types contain $n$ energies of total transmission. A fusion phenomenon of this type of band with a typical neigboring band is also found. As the transmission gap between them collapse and disappear, a resulting fused single band is seen to contain $2N-1$ energy peaks of total transmission.
Two connected equiperiodic one-dimensional multi-well potentials of different well depths are studied. Floquet/Bloch energy bands for respective multi-well potential are found to be relevant for understanding level structures. Althoug energies are classically allowed in both multi-well potentials, a band gap of one multi-well potential makes this potential quantum-mechanically 'forbidden'. All energy levels are located in the union of the band regions.
This work provides a convenient and powerful means towards the engineering of Floquet bands via Bloch oscillations, by adding a tilted linear potential to periodically driven lattice systems. The added linear field not only restricts the spreading of a time-evolving wavepacket but also, depending on the ratio between the Bloch oscillation frequency and the modulation frequency of the periodic driving, dramatically modifies the band profile and topology. Specifically, we consider a driven Aubry-André-Harper model as a working example, in the presence of a linear field. Almost flat Floquet bands…
We propose a general framework to solve tight binding models in D dimensional lattices driven by ac electric fields. Our method is valid for arbitrary driving regimes and allows to obtain effective Hamiltonians for different external fields configurations. We establish an equivalence with time independent lattices in D+1 dimensions, and analyze their topological properties. Further, we demonstrate that non-adiabaticity drives a transition from topological invariants defined in D+1 to D dimensions. Our approach provides a theoretical framework to analyze ac driven systems, with potential applic…
Electrons in a lattice exhibit time-periodic motion, known as Bloch oscillation, when subject to an additional static electric field. Here we show that a corresponding dynamics can occur upon replacing the spatially periodic potential by a time-periodic driving: Floquet oscillations of charge carriers in a spatially homogeneous system. The time lattice of the driving gives rise to Floquet bands that take on the role of the usual Bloch bands. For two different drivings (harmonic driving and periodic kicking through pulses) of systems with linear dispersion we demonstrate the existence of such o…
Recent works have demonstrated that the Floquet-Bloch bands of periodically-driven systems feature a richer topological structure than their non-driven counterparts. The additional structure in the driven case arises from the periodicity of quasienergy, the energy-like quantity that defines the spectrum of a periodically-driven system. Here we develop a new paradigm for the topological classification of Floquet-Bloch bands, based on the time-dependent spectrum of the driven system's evolution operator throughout one driving period. Specifically, we show that this spectrum may host topologicall…
Floquet theory has spawned many exciting possibilities for electronic structure control with light with enormous potential for future applications. The experimental realization in solids, however, largely remains pending. In particular, the influence of scattering on the formation of Floquet-Bloch states remains poorly understood. Here we combine time- and angle-resolved photoemission spectroscopy with time-dependent density functional theory and a two-level model with relaxation to investigate the survival of Floquet-Bloch states in the presence of scattering. We find that Floquet-Bloch state…
We study the open system dynamics and steady states of two dimensional Floquet topological insulators: systems in which a topological Floquet-Bloch spectrum is induced by an external periodic drive. We solve for the bulk and edge state carrier distributions, taking into account energy and momentum relaxation through radiative recombination and electron-phonon interactions, as well as coupling to an external lead. We show that the resulting steady state resembles a topological insulator in the Floquet basis. The particle distribution in the Floquet edge modes exhibits a sharp feature akin to th…
The unique electronic properties of the surface electrons in a topological insulator are protected by time-reversal symmetry. Circularly polarized light naturally breaks time-reversal symmetry, which may lead to an exotic surface quantum Hall state. Using time- and angle-resolved photoemission spectroscopy, we show that an intense ultrashort mid-infrared pulse with energy below the bulk band gap hybridizes with the surface Dirac fermions of a topological insulator to form Floquet-Bloch bands. These photon dressed surface bands exhibit polarization-dependent band gaps at avoided crossings. Circ…
The Lieb Lattice exhibits intriguing properties that are of general interest in both the fundamental physics and practical applications. Here, we investigate the topological Landau-Zener Bloch oscillation in a photonic Floquet Lieb lattice, where the dimerized helical waveguides is constructed to realize the synthetic spin-orbital interaction through the Floquet mechanism, rendering us to study the impacts of topological transition from trivial gaps to non-trivial ones. The compact localized states of flat bands supported by the local symmetry of Lieb lattice will be associated with other band…
We study the DC current generation in a periodically driven Bloch system connected to a heat bath. Under a relaxation time approximation, the density matrix for such a system is obtained, which is related to two equilibria: a Floquet quasi-equilibrium where the density matrix is diagonal under the Floquet-Bloch eigenbasis and an instantaneous Bloch thermal equilibrium. Then, the current responses and their power features, i.e. the power input behavior, are discussed in a unified manner, which reveals that there exist an intrinsic current and an extrinsic correction. Remarkably, the intrinsic p…
We introduce an efficient algorithm for the computation of the $W_3$ invariant of general unitary maps, which converges rapidly even on coarse discretization grids. The algorithm does not require extensive manipulation of the unitary maps, identification of the precise positions of degeneracy points, or fixing the gauge of eigenvectors. After construction of the general algorithm, we explain its application to the $2+1$ dimensional maps that arise in the Floquet-Bloch theory of periodically driven two-dimensional quantum systems. We demonstrate this application by computing the $W_3$ invariant…
We introduce $\mathbb Z_2$-valued bulk invariants for symmetry-protected topological phases in $2+1$ dimensional driven quantum systems. These invariants adapt the $W_3$-invariant, expressed as a sum over degeneracy points of the propagator, to the respective symmetry class of the Floquet-Bloch Hamiltonian. The bulk-boundary correspondence that holds for each invariant relates a non-zero value of the bulk invariant to the existence of symmetry-protected topological boundary states. To demonstrate this correspondence we apply our invariants to a chiral Harper, time-reversal Kane-Mele, and parti…
The coherent optical manipulation of solids is emerging as a promising way to engineer novel quantum states of matter. The strong time periodic potential of intense laser light can be used to generate hybrid photon-electron states. Interaction of light with Bloch states leads to Floquet-Bloch states which are essential in realizing new photo-induced quantum phases. Similarly, dressing of free electron states near the surface of a solid generates Volkov states which are used to study non-linear optics in atoms and semiconductors. The interaction of these two dynamic states with each other remai…
Floquet engineering offers a path to optically-controlled materials, but experimental implementations have frequently relied on femtosecond pulses to achieve the high peak fields required to maximise interactions and obtain a measurable response. The Floquet formalism, however, is based on a continuous, periodic drive. Here we use femtosecond laser pulses to drive the optical Stark effect, a simple realization of Floquet engineering, in monolayer WS2. By monitoring the coherent evolution of the free-induction decay, we show that the system evolves adiabatically, and that the finite duration of…
External driving is emerging as a promising tool for exploring new phases in quantum systems. The intrinsically non-equilibrium states that result, however, are challenging to describe and control. We study the steady states of a periodically driven one-dimensional electronic system, including the effects of radiative recombination, electron-phonon interactions, and the coupling to an external fermionic reservoir. Using a kinetic equation for the populations of the Floquet eigenstates, we show that the steady-state distribution can be controlled using the momentum and energy relaxation pathway…
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, specific lattice geometries, strong spin-orbit coupling, synthetic dimensions, 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-…
Recent experiments showed that the surface of a three dimensional topological insulator develops gaps in the Floquet-Bloch band spectrum when illuminated with a circularly polarized laser. These Floquet-Bloch bands are characterized by non-trivial Chern numbers which only depend on the helicity of the polarization of the radiation field. Here we propose a setup consisting of a pair of counter-rotating lasers, and show that one-dimensional chiral states emerge at the interface between the two lasers. These interface states turn out to be spin-polarized and may trigger interesting applications i…
Quantum walks are processes that model dynamics in coherent systems. Their experimental implementations proved key to unveil novel phenomena in Floquet topological insulators. Here we realize a photonic quantum walk in the presence of a synthetic gauge field, which mimics the action of an electric field on a charged particle. By tuning the energy gaps between the two quasi-energy bands, we investigate intriguing system dynamics characterized by the interplay between Bloch oscillations and Landau-Zener transitions. When both gaps at quasi-energy values 0 and $π$ are vanishingly small, the Floqu…
We investigate the relation between the symmetries of a Schrödinger operator and the related topological quantum numbers. We show that, under suitable assumptions on the symmetry algebra, a generalization of the Bloch-Floquet transform induces a direct integral decomposition of the algebra of observables. More relevantly, we prove that the generalized transform selects uniquely the set of "continuous sections" in the direct integral decomposition, thus yielding a Hilbert bundle. The proof is constructive and provides an explicit description of the fibers. The emerging geometric structure is a …
Quantum particle transmission through locally periodic potentials surrounded by symmetric exterior potentials is analyzed. Closed-form conditions for locating energy peaks of total transmission are derived. Floquet/Bloch energy band types are defined and found to affect the number of peaks in transmission bands. Modifications of band types and a band fusion phenomenon are discussed. The theoretical approaches suggested consist of several ways to express a Schrödinger wave as an amplitude function multiplying a harmonic function of a phase function. In particular, an approach containing a Floqu…
Tracking real-space quantum state breathing through Floquet-projector geometry
Institute for Theoretical Physics and Astrophysics, University of Würzburg（维尔茨堡大学理论物理与天体物理研究所）；Würzburg-Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群ctd.qmat）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）
Engineering quantum states through light-matter interaction has created a new paradigm in condensed matter physics. A representative example is the Floquet-Bloch state, which is generated by time-periodically driving the Bloch wavefunctions in crystals. Previous attempts to realise such states in condensed matter systems have been limited by the transient nature of the Floquet states produced by optical pulses, which masks the universal properties of non-equilibrium physics. Here, we report the generation of steady Floquet Andreev (F-A) states in graphene Josephson junctions by continuous micr…
Topological materials are fertile ground for investigating topological phases of matter and topological phase transitions. In particular, the quest for novel topological phases in 2D materials is attracting fast growing attention. Here, using Floquet-Bloch theory, we propose to realize chiral topological phases in 2D hexagonal FeX2 (X=Cl, Br, I) monolayers under irradiation of circularly polarized light. Such 2D FeX2 monolayers are predicted to be dynamical stable, and exhibit both ferromagnetic and semiconducting properties. To capture the full topological physics of the magnetic semiconducto…
Topological band structures can be designed by subjecting lattice systems to time-periodic modulations, as was recently demonstrated in cold atoms and photonic crystals. 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 measurements. In this work, we analyse how such loading sequences can be optimized in view…
Université Grenoble-Alpes, CNRS, Grenoble INP, Institut NEEL（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、NEEL研究所）；Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院量子材料与技术研究所）；Department of Electrical Engineering, The Pennsylvania State University（宾夕法尼亚州立大学电气工程系）
Graphene and phosphorene are monolayer of graphite and phosphorous, respectively. Graphene is completely relativistic (Dirac) fermionic system, but phosphorene is pseudorelativistic fermionic system. In phosphorene, electronic spectrum of phosphorene has a Dirac like (linear) band in one direction and Schrodinger like (parabolic) band in other direction. Conventional Rabi oscillations are studied by using rotating wave approximation in resonance case. The Floquet theory is an alternative way of study Rabi oscillations in off-resonance case and dominating in case of low energy physics. In this …
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 dress…
Energy or quasienergy (QE) band spectra depending on two parameters may have a nontrivial topological characterization by Chern integers. Band spectra of 1D systems that are spanned by just one parameter, a Bloch phase, are topologically trivial. Recently, an ensemble of 1D Floquet systems, double kicked rotors (DKRs) depending on an external parameter, has been studied. It was shown that a QE band spanned by both the Bloch phase and the external parameter is characterized by a Chern integer, which determines the change in the mean angular momentum of a state in a band when the external parame…
In this work we use Floquet-Bloch theory to study the influence of circularly and linearly polarized light on two-dimensional band structures with semi-Dirac band touching points, taking the anisotropic nearest neighbor hopping model on the honeycomb lattice as an example. We find circularly polarized light opens a gap and induces a band inversion to create a finite Chern number in the two-band model. By contrast, linearly polarized light can either open up a gap (polarized in the quadratically dispersing direction) or split the semi-Dirac band touching point into two Dirac points (polarized i…
In this work we theoretically study, using Floquet-Bloch theory, the influence of circularly and linearly polarized light on two-dimensional band structures with Dirac and quadratic band touching points, and flat bands, taking the nearest neighbor hopping model on the kagome lattice as an example. We find circularly polarized light can invert the ordering of this three band model, while leaving the flat-band dispersionless. We find a small gap is also opened at the quadratic band touching point by 2-photon and higher order processes. By contrast, linearly polarized light splits the quadratic b…
Rabi flopping between Bloch bands induced by a weak ac resonant field is a coherent effect involving interband transitions. Here we consider the fundamental processes of emission/absorption of quanta and Rabi oscillations in non-Hermitian two-band lattices exhibiting unbalanced non-Hermitian skin effect, and unveil an unprecedented scenario of Rabi flopping. The effective dipole moment of the transition - usually considered a bulk property - is however strongly dependent on boundary conditions, being greatly enhanced with increased Rabi frequency only when open boundaries are present. As the f…
We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons in gapped crystalline solids, as e.g. insulators and semiconductors. We shortly review the Bloch-Floquet decomposition of periodic operators, and the related concepts of Bloch frames and composite Wannier functions. We show that the latter are almost-exponentially localized if and only if there exists a smooth periodic Bloch frame, and that the obstruction to the latter condition is the triviality of a Hermitian vector bundle, call…
