Floquet Theory Research Articles

Floquet Schrieffer-Wolff transform based on Sylvester equations

We present a Floquet Schrieffer-Wolff transform (FSWT) to obtain effective Floquet Hamiltonians and micromotion operators of periodically driven many-body systems for any nonresonant driving frequency. The FSWT perturbatively eliminates the oscillatory components in the driven Hamiltonian by solving operator-valued Sylvester equations with systematic approximations. It goes beyond various high-frequency expansion methods commonly used in Floquet theory, as we demonstrate with the example of the driven Fermi-Hubbard model. In the limit of high driving frequencies, the FSWT Hamiltonian reduces to the widely used Floquet-Magnus result. We anticipate this method will be useful for designing Rydberg multiqubit gates, controlling correlated hopping in quantum simulations in optical lattices, and describing multiorbital and long-range interacting systems driven in-gap. Published by the American Physical Society 2024

Stability and bifurcation analysis of simply supported rectangular nanoplate embedded on visco-Pasternak foundation based on first-order shear deformation theory

Vibrations of simply supported nanoplates embedded on visco-Pasternak foundation and stressed by in-plane periodic forces are investigated. The governing size-dependent equations employ the first-order shear deformation theory, nonlinear von Kármán strains, and the nonlocal elasticity theory. Reducing the governing system is carried out by the Bubnov-Galerkin method, which is based on two-mode model, thus the resulting system contains two ordinary differential equations with size- and time-dependent coefficients. Analysing the linearized system, the stability of the nanoplate structure is studied depending on the foundation parameters, force parameters and small-scale parameter. Nonlinear dynamics approaches are employed to determine the nature of vibration after the loss of stability. Time history, Poincare section, Lyapunov exponent are analysed for chosen values of excitation parameters, visco-Pasternak foundation parameters to indicate small-scale effects and nonlinear effects. The novelty of the work consists of the combined application of the nonlocal theory, first-order shear deformation theory, Floquet theory, methods of nonlinear dynamics and obtaining new results by a numerical experiment.

Manipulating Floquet-driven topological insulator with off-resonant elliptically polarized light in presence of hexagonal fermi surface warping.

The irradiation of topological insulator surface with elliptically polarized light modifies the topological properties in a phase-dependent manner impacting the Floquet Chern number which is a crucial topological invariant associated with such driven systems. Employing Floquet theory in presence of hexagonal warping term in the Dirac fermion Hamiltonian under off-resonant conditions, we derive an effective Hamiltonian that highlights distinct features in the Floquet-Dirac surface states. Specifically, we identify a helicity and ellipticity-dependent mass term in the quasi-static Hamiltonian, breaking time reversal symmetry. This mass term changes sign with variations in the helicity and ellipticity of incident radiation, potentially driving the system into trivial or Floquet-Chern topological insulating phases. We calculate the Berry curvature associated with the bandstructure and the resulting anomalous Hall conductivity, finding a strong dependence on the ellipticity and amplitude of the radiation.

Stability of Thermal Convection in Two Superimposed Miscible Viscous Fluids Under Gravitational Modulation

Abstract The thermal stability of a system of two superimposed miscible viscous fluid layers, under gravitational modulation, is investigated using linear stability analysis. It is well known in the literature that in the absence of gravitational modulation, there exist two convection regimes depending on the buoyancy number: the single-cell regime, where convection is oscillating and occupies the entire volume of the two fluid layers, and the stratified regime where stationary convection occurs in each fluid layer. Using Chebyshev's spectral collocation method and Floquet's theory, the numerical results show that under gravitational modulation, single-cell convection oscillates at half the frequency of the external oscillation (subharmonic threshold), while the stratified regime oscillates at a frequency equal to that of the forcing (harmonic threshold). The critical buoyancy number corresponding to the transition between the two convection regimes depends, in addition to the viscosity ratio and the depth of the two layers, on certain dimensionless frequencies permeating this transition and also on the amplitude of oscillation via the Froude number. In the plane of the critical Rayleigh number versus dimensionless modulation frequency, the branch associated with stratified convection does not depend on the buoyancy number for equal viscosities whereas this is the case for the whole-convection regime. For different viscosities, both convection regimes depend on the buoyancy number. At last, in the stratified regime (harmonic threshold), the critical value of the viscosity contrast, corresponding to the passage from an active layer to a passive layer and vice versa, increases with the modulation frequency.

Dynamic Modeling and Stability Analysis for Spinning Circular Solar Sail Considering Periodically Time-Varying Characteristics

The spinning circular solar sail is a promising spacecraft for long-duration missions. This work reveals its structural dynamic and stability behavior under the periodically time-varying solar radiation pressure and gravitational force. The geometric stiffness generated by the centrifugal force due to spinning and the coupling effect between the deformation and solar radiation pressure are taken into account. The von Kármán plate theory is adopted by neglecting the high-frequency in-plane vibrations and considering the effect of the in-plane internal force on the transverse vibration. The partial differential equation of the spinning solar sail is derived and further spatially discretized into periodically time-varying equations of motion. Effects of Poisson ratio and radius ratio on natural frequencies and mode shapes are analyzed, and curve veering phenomena are then observed. Steady-state periodic responses of the solar sail under the solar radiation pressure with different orbit distances, incident angles, and spinning angular velocities are analyzed. The stability analysis is rigorously performed by the Floquet theory rather than the commonly used approach of conducting the eigenvalue analysis at a series of specific discrete time nodes. Moreover, the stability boundary associated with transverse vibrations is determined, which contributes to the parameter design of the spinning solar sail.

Floquet engineering of anomalous Hall effects in monolayer MoS2

Light-matter interactions have emerged as a new research focus recently offering promises of unveiling novel physics and leading to applications under nonequilibrium conditions. The quantized Hall conductivities predicted by Floquet theory assuming a Fermi-Dirac distribution however deviate from experimental observations. To resolve these puzzles, we consider the effect of nonequilibrium electron occupation to study the anomalous, valley, and spin Hall effects of a prototype monolayer transition metal dichalcogenide MoS2. We find that spin Hall conductivity can be effectively suppressed approaching zero value by linearly polarized light under near resonant excitations. In contrast, it is substantially enhanced by circularly polarized light, originating from optical selection rules and topological phase transitions. Besides, the quantized anomalous Hall conductivity is much reduced if nonequilibrium occupations of Floquet bands are considered. Our study provides a novel avenue for engineering various Hall effects in two-dimensional materials using light, holding great promises for future device applications.

Large Amplitude Dynamic Instability of a Graphene-Reinforced Pipe Conveying Pulsating Flow

In many industries’ applications, one deals with pipes conveying fluid. Besides, the implementation of advanced materials in manufacturing of new-brand structures is developing. Consequently, in this paper, one of the most important dynamics analyzes that an advanced pipe conveying fluid may meet it is addressed. For the first time, linear and nonlinear dynamics of a pipe reinforced with graphene nanoplatelets (GPL pipe) that conveys pulsating flow is presented. The Euler–Bernoulli beam model follows the plug flow model in order to formulate the problem. The method of multiple scales (MMS) applied to the discretized couple nonlinear gyroscopic governing equations with the aim of specifying the instability region and the steady-state response of the GPL pipe subjected to the principal parametric resonance of one of its modes, as a consequence of pulsating flow. Respectively, the Floquet theory and the Runge–Kutta fourth-order (RK4) method are applied to the linear governing equations and nonlinear governing equations for the sake of confirming the current MMS results. It is deduced that a small amount usage of GPL reinforcement phase improves substantially the resistance against static instability, and the critical fluid excitation amplitude fraction, meanwhile declines the instability region bandwidth, and the steady-state response of the pipe. In contrary, it is confirmed that a heavier fluid makes reverse the preceding statements with respect to a lighter fluid. The aforementioned findings shed light on the essential design keys that outstandingly affect the mechanics of the GPL pipe conveying pulsating fluid that lead and conduct forthcoming studies and open new horizons for designers in industry implementations.

Damping-destabilized parametric resonances of a rotating beam magnetically plucking a stationary beam

A beam subjected to rotary magnetic plucking plays an important role in energy harvesting from rotational motion. Rotary magnetic plucking involves impulse-like parametric excitation, the instability mechanisms of which are not well understood. This paper presents a second-order perturbation analysis using the method of multiple scales to study the parametric resonances of a rotating beam magnetically plucking a stationary beam. Stability criteria are derived to predict the principal parametric resonance and combination parametric resonance. The criteria are used to determine when strain-rate damping in the beams has the destabilizing effect. The stability criteria and the destabilizing effect of damping are demonstrated through numerical examples and validated using Floquet theory. The perturbation analysis reveals that when the beams have closely spaced modal frequencies, damping in the rotating beam destabilizes the principal parametric resonance of the stationary beam if damping in the stationary beam is below a certain threshold, and vice versa.

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Thin liquid film stability in the presence of bottom topography and surfactant

We consider gravity-driven fluid flow down a wavy inclined surface in the presence of surfactant. The periodicity of the bottom topography allows us to leverage Floquet theory to determine the form of the solution to the linearized governing partial differential equations. The result is that perturbations from steady state are wavelike, and a dispersion relation is identified which relates the wavenumber of an initial perturbation, κ, to its complex frequency, ω. The real part of ω provides a criterion for determining linear flow stability. We observe that the addition of surfactant generally has a stabilizing effect on the flow, but has a destabilizing effect for small wavenumbers. These results are compared and validated against numerical simulations of the nonlinear system. The linear and nonlinear analyses show good agreement, except at small wavenumbers, where the linear results could not be replicated.

Model of Faraday waves in a cylindrical container with force detuning

Recent experiments by Shao et al. [“Surface wave pattern formation in a cylindrical container,” J. Fluid Mech. 915, A19 (2021)] have revealed complex wave dynamics on the surface of a liquid bath in a vertically vibrated cylindrical container that are related to the presence of a meniscus on the container sidewall. We develop a corresponding theoretical model for this system by detuning the driving acceleration of the container, which results in an inhomogeneous Mathieu equation that governs the wave dynamics whose spatial structure is defined by the mode number pair (n,m), with n and m the radial and azimuthal mode numbers, respectively. Asymmetric m≠0 modes are unaffected by the detuning parameter, which is related to the meniscus shape and satisfy a homogeneous Mathieu equation with the shape of the instability tongues computed by the Floquet theory. The Poincaré–Lindstedt method is used to compute the instability tongues for the axisymmetric m=0 modes, which have a lower threshold acceleration and larger bandwidth that depend upon the detuning parameter. Our model results explicitly show how the shape of the meniscus and spatial structure of the wave determine the temporal response and are in good agreement with prior experimental observations for both pure modes and mixed modes.

Periodic orbits of solar sails in the Sun-Earth system

Abstract This paper studies the periodic orbits of solar sails using reflectivity control devices (RCDs) in the circular Sun-Earth system. Significantly, the dynamical equations about the solar sail are given separately in the rotational coordinate system and adjoint coordinate system. Analytical solutions about the periodic orbits of solar sails are given. So as to gain the periodic orbits of solar sails, we discuss the conditions of the attitude angles and reflectivity modulation ratio of solar sails. This paper simulates some periodic orbits of solar sails and analyzes the stability of a periodic orbit of a solar sail by using the Floquet theory. The outcomes show that it is unstable for the periodic orbit of a solar sail. Based on the linear quadratic regulator (LQR), this paper designs a control algorithm to achieve tracking in the target orbit of solar sails.

Electronic states bound by repulsive potentials in graphene irradiated by a circularly polarized electromagnetic field

In the framework of the Floquet theory of periodically driven quantum systems, it is demonstrated that irradiation of graphene by a circularly polarized electromagnetic field induces an attractive area in the core of repulsive potentials. Consequently, the quasi-stationary electron states bound by the repulsive potentials appear. The difference between such field-induced states in graphene and usual systems with the parabolic dispersion of electrons is discussed and possible manifestations of these states in electronic transport and optical spectra of graphene are considered.

Microwave-optical spectroscopy of Rydberg excitons in the ultrastrong driving regime

Abstract We study the ultrastrong driving of Rydberg excitons in Cu$_2$O by a microwave field. The effect of the field is studied using optical absorption spectroscopy, and through the observation of sidebands on the transmitted laser light. A model based on Floquet theory is constructed to study the system beyond the rotating wave approximation. We obtain near quantitative agreement between theory and experiment across a 16-fold range of microwave field strengths spanning from the perturbative to the deep strong driving regime. Compared to Rydberg atoms, the large non-radiative widths of Rydberg excitons leads to new behaviour such as the emergence of an absorption continuum without ionization.

Sensing force gradients with cavity optomechanics while evading backaction

We study force-gradient sensing with a coherently driven mechanical resonator and phase-sensitive detection of motion through the two-tone backaction evading measurement of cavity optomechanics. The response of the optomechanical system, solved by numerical integration of the classical equations of motion, shows an extended region which is monotonic to changes in force gradient. We use Floquet theory to model the fluctuations, which rise only slightly above that of the usual backaction evading measurement in the presence of the mechanical drive. The monotonic response and minimal backaction are advantageous for applications such as atomic force microscopy. Published by the American Physical Society 2024

Raman-phonon-polariton condensation in a transversely pumped cavity

Phonon polaritons are hybrid states of light and matter that are typically realised when optically active phonons couple strongly to photons. We suggest a new approach to realising phonon polaritons, by employing a transverse-pumping Raman scheme, as used in experiments on cold atoms in optical cavities. This approach allows hybridisation between an optical cavity mode and any Raman-active phonon mode. Moreover, this approach enables one to tune the effective phonon–photon coupling by changing the strength of the transverse pumping light. We show that such a system may realise a phonon-polariton condensate. To do this, we find the stationary states and use Floquet theory to determine their stability. We thus identify distinct superradiant and lasing states in which the polariton modes are macroscopically populated. We map out the phase diagram of these states as a function of pump frequencies and strengths. Using parameters for transition metal dichalcogenides, we show that realisation of these phases may be practicably obtainable. The ability to manipulate phonon mode frequencies and attain steady-state populations of selected phonon modes provides a new tool for engineering correlated states of electrons.

EFFECT OF EXTERNAL HARMONIC VIBRATION ON THE FORMATION OF SOLITARY WAVES IN FALLING TWO-LAYER LIQUID FILMS

Abstract We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Operational Modal Analysis of Self-excited Vibrations in Milling Considering Periodic Dynamics

Abstract A new method is presented to identify the dynamics of regenerative chatter from measured process vibrations in milling. This method combines the synchronous once-per-revolution sampling of process vibrations with Operational Modal Analysis to estimate the Floquet multipliers of the delayed linear time-periodic dynamics in milling, all from the natural process vibrations without external excitation. The identified multipliers quantify vibration stability, enabling chatter prediction before it occurs. In addition to this, they can be used to calibrate physics-based chatter models based on vibration measurements solely within the stable region. The method's accuracy in identifying Floquet multipliers is validated through extensive numerical simulations and two experimental case studies. The results show that chatter due to both Hopf and period-doubling bifurcations can be predicted from the process vibrations during stable cuts. Moreover, the experimental case studies demonstrate a vibration measurement system for implementing the presented method in standard milling operations and confirm its effectiveness in practice.

Self-resonance during preheating: The case of [formula omitted]-attractor models

In this paper, for the first time, we obtain a new class of solutions for the Hill-type differential equations, which emerge in the preheating self-resonance phase of the expanding Universe. We study, in particular, the class of symmetric and asymmetric scalar field potentials coming from the so-called α-attractor models of the early Universe cosmology. By making a series expansion of the potential and employing perturbative techniques we reformulate the Mukhanov–Sasaki equation, which captures the dynamics of the curvature perturbation in these models, into a Hill equation. This last includes higher-order terms that were never solved in the literature. Namely, those coming from the cubic and quartic contributions of the scalar field potential. Then, we derive the expressions for the Floquet exponents of the Mukhanov–Sasaki variable. Our analytical results are then compared with numerical computations, showing a good agreement and thus making this method valuable for obtaining theoretical predictions with new observational applications in the contexts of Primordial Black Holes and Scalar-Induced Gravitational Waves.