Quasienergy Spectrum Research Articles

New approach to measuring topological phase transitions utilizing Floquet technology

Abstract The Floquet technique provides a novel anomalous topological phase for non-equilibrium phase transitions. Based on the high symmetry of the quantum anomalous Hall model, the findings suggest a one-to-one correspondence between the average spin texture and the Floquet quasi-energy spectrum. A new approach is proposed to directly measure the quasi-energy spectrum, replacing previous measurements of the average spin texture. Finally, we proposed a reliable experimental scheme based on ion trap platforms. This scheme markedly reduces the measurement workload, improves the measurement fidelity, and is applicable to multiple platforms such as cold atoms and nuclear magnetic resonance.

Predicting edge-localized monovacancy defects in zigzag graphene nanoribbons from Floquet quasienergy spectrum

Predicting edge-localized monovacancy defects in zigzag graphene nanoribbons from Floquet quasienergy spectrum

Floquet insulators and lattice fermions beyond naive time discretization

Periodically driven quantum systems known as Floquet insulators can host topologically protected bound states known as “π modes” that exhibit response at half the frequency of the drive. Such states can also appear in undriven lattice field theories when time is discretized as a result of fermion doubling, raising the question of whether these two phenomena could be connected. Recently we demonstrated such a connection at the level of an explicit mapping between the spectra of a continuous-time Floquet model and a discrete-time undriven lattice fermion model. However, this mapping relied on a symmetry of the single-particle spectrum that is not present for generic drive parameters. Inspired by the example of the temporal Wilson term in lattice field theory, in this paper we extend this mapping to the full drive parameter space by allowing the parameters of the discrete-time model to be frequency-dependent. The spectra of the resulting lattice fermion models exactly match the quasienergy spectrum of the Floquet model in the thermodynamic limit. Our results demonstrate that spectral features characteristic of beyond-equilibrium physics in Floquet systems can be replicated in static systems with appropriate time discretization. Published by the American Physical Society 2024

Majorana bound states in a driven quantum dot

We study a periodically driven quantum dot in two different configurations. In the first setup, a quantum dot coupled to a topological superconductor and a normal metal lead. In the second setup, a T-shape quantum dot connected to two topological superconductors and side coupled to a normal metal lead. By a combination of non-equilibrium Green’s function techniques and Floquet’s formalism, we obtain the quasienergy spectra as a function of the amplitude, frequency, and superconducting phase difference. We show that the states develop unique electronic responses, such as the broken particle-hole symmetry that appears when considering the non-locality of Majorana bound states. Finally, we compute the time-average current and the differential conductance to reveal these spectra signatures through physically measurable magnitudes in the two proposed configurations.

Nonreciprocal topological mode conversion by encircling an exceptional point in dynamic waveguides.

The topology of exceptional points (EPs) has been revealed by taking stationary or dynamical encircling around them, which induces eigenstate exchange or chiral mode conversion. However, the conversions are usually reciprocal obeying restricted transmittances. Here we propose the concept of nonreciprocal encircling of EPs in a dynamic waveguide under complex modulation. The waveguide allows direction-dependent EPs in their quasienergy spectra due to different phase-matching conditions for opposite propagation direction. We design a closed loop that will encircle the EP in the backward direction but not in the forward direction. In this way, a nonreciprocal topological conversion is achieved as the forward transmittance from the even to odd mode significantly exceeds the backward transmittance from the odd to even mode. As a result, the forward propagation produces two modes with equal strength while the backward propagation leads to a specific mode regardless of the input. The structure is promising for making robust optical isolators.

Controllable sidebands of resonance fluorescence of a two-level system driven by bichromatic field

Strong polychromatic driving reshapes characteristics of the resonance fluorescence spectrum of a two-level system. Employing bichromatic driving feild with a low beat-frequency smaller than the emission rate of the system we demonstrate the exotic features of the fluorescence spectrum calculated by the numerical Floquet-Liouville approach and analytical method. It is found that fluorescence spectrum possesses two broadened sidebands in the place of the Rabi sidebands under certain conditions. Moreover, the heights and widths of the sidebands can be controlled by tuning the driving parameters. The properties of the spectrum is determined by the transitions between the Floquet states in a rotating frame. The broadened sidebands result from the quasi-continuous quasienergy spectrum which happens with steering the beat frequency lower. The present study provides insights into the Floquet engineering of the fluorescence spectral features with polychromatic excitation fields.

Probing topological signatures in an optically driven α−T3 lattice

The $\alpha$-$T_3$ lattice, an interpolation model between the honeycomb lattice of graphene($\alpha=0$) and the dice lattice($\alpha=1$), undergoes a topological phase transition across $\alpha=1/\sqrt{2}$ when exposed to a circularly polarized off-resonant light. We study Berry phase mediated bulk magnetic and anomalous thermoelectric responses in order to capture the topological signatures of a driven $\alpha$-$T_3$ lattice. It is revealed that both the Berry curvature and the orbital magnetic moment associated with the flat band change their respective signs across $\alpha=1/\sqrt{2}$. The off-resonant light distorts the flat band near the Dirac points when $0<\alpha<1$ which eventually introduces two distinct well separated forbidden gaps of equal width in the quasienergy spectrum. The orbital magnetization varies linearly with the chemical potential, in the forbidden gaps. The slopes of the linear regions in the orbital magnetization are closely related to the respective Chern numbers on either side of $\alpha=1/\sqrt{2}$. We find that the slope for $\alpha>1/\sqrt{2}$ is approximately two times of that for $\alpha<1/\sqrt{2}$ which essentially indicates a topological phase transition across $\alpha=1/\sqrt{2}$. However, the anomalous Nernst coefficient vanishes when the chemical potential is tuned in the forbidden gaps. The anomalous Hall conductivity in the forbidden gap(s) approaches different quantized values on either side of $\alpha=1/\sqrt{2}$. All these topological signatures can be observed experimentally.

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.

All-Optically Controlled Topological Current Switcher Based on Silicene-Like Nanoribbons

Low-dimensional topological insulator materials, such as silicene-like 2-D materials, have wide applications in electronic, spintronic, and optoelectronic nanodevice, by utilizing their topologically protected edge states. In theoretical studies, Floquet theorem is usually employed in the nanosystems under light irradiation with high-frequency approximation, which may not be accessible in experimental research. In this work, we study the quasienergy spectra and edge-state transport of silicene-like nanoribbons under low-frequency light irradiation (in visible wavelengths), by extracting the zero-photon component from the nonperturbative Floquet bands. These silicene-like nanoribbons undergo topological phase transition from quantum spin Hall (QSH) to quantum anomalous Hall (QAH) insulator under circularly polarized light, accompanied by helical edge modes to chiral ones. The propagation direction of edge-mode current in QAH nanoribbons can be switched by left-/right-circularly polarized (LCP/RCP) light, and such a current-switching effect is then confirmed in heterojunction composed of QSH and QAH nanoribbons. Based on the all-optically controlled current-switching effect, we design a three-terminal current switcher, with one input terminal, two output ones, and central device irradiated by LCP/RCP light. Conductance and real-space local current distribution of the proposed nanosystem reveal that the input current can indeed be switched into specific output terminal by LCP/RCP light. The proposed current switcher changes the electron movement very quickly, and we believe that it would have potential applications in high-speed photoelectric nanodevices.

Bulk-edge correspondence for Floquet topological phases in honeycomb nanoribbon

Possibility of having different topological phases by time-periodic external field encourages people who wish to manipulate the properties of the materials as their will for the various kinds of systems including a honeycomb nanoribbon. We study bulk-edge correspondence by looking at the change in the value of the first Chern number of a specific band, which reflects the bulk properties, together with the change of associated edge states in quasienergy spectrum when the honeycomb nanoribbon is exposed to circularly polarized laser field. Various topological phase transitions are observed as the intensity of the laser field is raised up and we classify them in two categories. In the first category, when the frequency is sufficiently high, the system undergoes simple transition from trivial phase to topological one, reflected by the sudden jump in the value of Chern number from 0 to 1 and the appearance of two crossing edge states in the spectrum. Whereas, in the second category where the frequency is low enough, anomalous transitions are drawn, having several nonzero values of Chern number, and band folding prevails, often disappearing depending on the laser intensities. The results confirm the bulk-edge correspondence in periodically driven systems and will contribute to future material designs.

Controllable discrete Talbot self-imaging effect in Hermitian and non-Hermitian Floquet superlattices.

We investigate the discrete Talbot self-imaging effect in Floquet superlattices based on a mesh of directional couplers with periodically varying separation between waveguides, both theoretically and numerically. The modulated discreteness of the lattices sets strong constraints to ensure the Talbot effect generation. We show that discrete Talbot effect occurs only if the incident periods are N = 1, 2, and 4 in dispersive regimes of the Hermitian superlattices. In both dynamic localized and rectification regimes, self-imaging effect can occur for arbitrary input period N. For the rectification case, Talbot distance equals the input period. In the regime of dynamical localization, the Talbot distance remains unchanged irrespective of the pattern period. For non-Hermitian Floquet superlattices, due to the non-zero imaginary part of quasi-energy spectrum arising at the center of the Brillouin zone, where the mode degeneracy occurs, Talbot revival is not preserved when the input period is an even number, and exists only as N = 1 in the dispersive regime. The theoretical calculations and numerical simulations verify each other completely.

Localization, multifractality, and many-body localization in periodically kicked quasiperiodic lattices

We study the combined effect of quasiperiodic disorder, driven and interaction in the periodically kicked Aubry-Andr\'{e} model. In the non-interacting limit, by analyzing the quasienergy spectrum statistics, we verify the existence of a dynamical localization transition in the high-frequency region, whereas the spectrum statistics becomes intricate in the low-frequency region due to the emergence of the extended/localized-to-multifractal edges in the quasienergy spectrum, which separate the multifractal states from the extended (localized) states. When the interaction is introduced, we find the periodically kicked incommensurate potential can lead to a transition from ergodic to many-body-localization phase in the high-frequency region. However, the many-body localization phase vanishes in the low-frequency region even for strong quasiperiodic disorder. Our studies demonstrate that the periodically kicked Aubry-Andr\'{e} model displays rich dynamical phenomena and the driving frequency plays an important role in the formation of many-body localization in addition to the disorder strength.

Transport properties of vertical heterostructures under light irradiation

Electronic and transport properties of bilayer heterostructure under light irradiation are of fundamental interest to improve functionality of optoelectronic devices. We theoretically study the modification of transport properties of bilayer graphene and bilayer heterostructures under a time-periodic external light field. The bulk electronic and transport properties are studied in a Landauer-type configuration by using the nonequilibrium Green's function formalism. To illustrate the behavior of the differential conductance of a bilayer contact under light illumination, we consider tight-binding models of bilayer graphene and graphene/hexagonal boron-nitride heterostructures. The non-adiabatic driving induces sidebands of the original band structure and opening of gaps in the quasienergy spectrum. In transport properties, the gap openings are manifested in a suppression of the differential conductance. In addition to suppression, an external light field induces an enhancement of the differential conductance if photoexcited electrons tunnel into or out of a Van~Hove singularity.

Hinge-mode dynamics of periodically driven higher-order Weyl semimetals

We study the stroboscopic dynamics of hinge modes of a second-order topological material modeled by a tight-binding free fermion Hamiltonian on a cubic lattice in the intermediate drive frequency regime for both discrete (square pulse) and continuous (cosine) periodic drive protocols. We analyze the Floquet phases of this system and show that its quasienergy spectrum becomes almost gapless in the large drive amplitude regime at special drive frequencies. Away from these frequencies, the gapped quasienergy spectrum supports weakly dispersing Floquet hinge modes. Near them, these hinge modes penetrate into the bulk and eventually become indistinguishable from the bulk modes. We provide an analytic, albeit perturbative, expression for the Floquet Hamiltonian using Floquet perturbation theory (FPT) which explains this phenomenon and leads to analytic expressions of these special frequencies. We also show that in the large drive amplitude regime, the zero energy hinge modes corresponding to the static tight-binding Hamiltonian display qualitatively different dynamics at these special frequencies. We discuss possible local density of state measurement using a scanning tunneling microscope which can test our theory.

Physics of manipulation of spin dynamics in a driven double well made transparent

We study the coherent manipulation of quantum spin dynamics via simple analytic solutions for a spin–orbit (SO) coupled ultracold atom held in a driven double well. In the high-frequency regime, we analytically obtain the quasienergies, the Floquet states, and the non-Floquet state of this system. When the system parameters are properly selected, the non-Floquet state reduces to special quantum states of simple forms which can provide a more intuitive understanding of some interesting quantum spin dynamical phenomena, such as the intrawell spin–flipping, the interwell spin-conserving tunneling, and the interwell spin–flipping tunneling. Specially, for the interwell spin–flipping tunneling, we surprisingly find that the selective coherent destruction of tunneling (SCDT) can occur at the anti-crossing (non-collapse) points of quasienergy spectrum, which breaks the conventional understanding that the SCDT is associated with the crossing of quasienergy levels. Further, we reveal the interwell spin–flipping tunneling strongly depends on the initial spin direction and the initial position of the SO-coupled atom. These novel results can be readily tested by the current experimental setups and may be useful for the design of spintronic devices.

Floquet analysis of extended Rabi models based on high-frequency expansion

The extended quantum Rabi models make a significant contribution to understanding of the quantum nature of the atom-light interaction. We transform two kinds of extended quantum Rabi model, the anisotropic Rabi model and the asymmetric Rabi model, into the rotating frame and regard them as periodically driven quantum systems. The analytical solutions of the quasienergy spectrum as well as the Floquet modes for both models are constructed by applying the Floquet theory and the high-frequency expansion, which is applied to the nonstroboscopic dynamics of physical observables such as atomic inversion, transverse magnetization, and atom-field correlation. For the anisotropic Rabi model, the quasienergy fits well with the numerical results even when the rotating-wave coupling is in the deep strong-coupling regime $g\ensuremath{\simeq}\ensuremath{\hbar}\ensuremath{\omega}$ if the counter-rotating term is small enough compared to the driving frequency. Avoided level crossing may occur for quasienergy with the same parity when the positive branch spectrum lines for the total excitation number $N$ cross the negative branch lines for $N+2$, while the high-frequency expansion fails to predict this due to the conservation of the total excitation number. Furthermore, we present analytical and numerical studies of the long-time evolution of population and conclude that the analytical method is credible for the population dynamics. For the asymmetric Rabi model, we find that the external bias field which breaks the parity symmetry of the total excitation number tends to cluster the upper and lower branches into two bundles, and the detuning induced gap in the first temporal Brillouin zone shows a quadratic dependence on the bias. Fourier analysis is applied to extract the frequency composition and two-frequency driving behavior is revealed. Varying the bias strength will change the time-averaged value of the oscillation, which shows how the bias competes with the detuning and atom-field coupling in the driving dynamics. Both models prove that treating the Hamiltonian in the rotating frame by Floquet theory gives an alternative tool in the study of interaction between atom and light.

Quasienergy operators and generalized squeezed states for systems of trapped ions

Collective many-body dynamics for time-dependent quantum Hamiltonians is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.

Proposal for realizing anomalous Floquet insulators via Chern band annihilation

Two-dimensional periodically driven systems can host an unconventional topological phase unattainable for equilibrium systems, termed the Anomalous Floquet-Anderson insulator (AFAI). The AFAI features a quasi-energy spectrum with chiral edge modes and a fully localized bulk, leading to non-adiabatic but quantized charge pumping. Here, we show how such a Floquet phase can be realized in a driven, disordered Quantum Anomalous Hall insulator, which is assumed to have two critical energies where the localization length diverges, carrying states with opposite Chern numbers. Driving the system at a frequency close to resonance between these two energies localizes the critical states and annihilates the Chern bands, giving rise to an AFAI phase. We exemplify this principle by studying a model for a driven, magnetically doped topological insulator film, where the annihilation of the Chern bands and the formation of the AFAI phase is demonstrated using the rotating wave approximation. This is complemented by a scaling analysis of the localization length for two copies of a quantum Hall network model with a tunable coupling between them. We find that by tuning the frequency of the driving close to resonance, the driving strength required to stabilize the AFAI phase can be made arbitrarily small.

Periodically driven perturbed CFTs: The sine-Gordon model

We analyze a version of the sine-Gordon model in which the strength of the cosine potential has a periodic dependence on time. This model can be considered as the continuum limit of the many-body generalization of the Kapitza pendulum. Based on the perturbed CFT point of view, we develop a truncated conformal space approach (TCSA) to investigate the Floquet quasienergy spectrum. We focus on the effective behaviour for large driving frequencies, which we also derive exactly. Depending on the driving protocol, we can recover the original sine-Gordon model or its two-frequency version. The rich structure of the two-frequency model implies that the time-periodic drive can break integrability, can lead to new states in the spectrum, or can result in a phase transition. Our method is applicable for any periodically driven perturbed conformal field theories.

Dynamical localization in non-Hermitian quasicrystals

We study the localization transition in periodically driven one-dimensional non-Hermitian lattices where the piece-wise two-step drive is constituted by uniform coherent tunneling and incommensurate onsite gain and loss. We find that the system can be in localized, delocalized, or mixed-phase depending on the driving frequency and the phase shift of complex potential. Two critical driving frequencies of the system are identified, the first one corresponds to the largest phase shift of the complex potential so that the quasi-energy spectrum is still real and all the states are extended, the second one corresponds to the disappear of full real spectrum, and very weak complex potential leads to the emergence of localized states when the driving frequency is lower than this critical frequency. In the high frequency limit, we find the critical phase shift that separates the two regions with respectively real and complex spectrum tends to a constant value that can be captured by an effective non-Hermitian Hamiltonian.