Floquet Formalism Research Articles

Floquet formalism for tunable optically induced orbital magnetization in twisted bilayers

Floquet formalism for tunable optically induced orbital magnetization in twisted bilayers

Analysis of Photo-Induced Chirality and Magnetic Toroidal Moment Based on Floquet Formalism

Analysis of Photo-Induced Chirality and Magnetic Toroidal Moment Based on Floquet Formalism

Qocttools: A program for quantum optimal control calculations

The qocttools code performs optimization calculations on quantum systems using optimal control theory. The problem that it solves is the following: given a system governed by a Hamiltonian whose precise form can be tuned through the variation of some control parameters, and given a user-defined merit function of the system trajectory, what are the values of those parameters that lead to the time evolution that maximizes the merit function? The code permits to work on generic systems, and uses either Schrödinger's or Lindblad's equation in order to deal with closed or open systems, respectively. It may also use the tools of Floquet formalism when dealing with periodic perturbations. It is written in Python, and the user can interface with it by preparing driver Python scripts that define the model, and load the appropriate qocttools modules. It is open and free software, and also relies on open and free widely used libraries and packages, such as QuTiP (to handle the quantum model definition and manipulation), and NLopt (to perform the optimizations). Program summaryProgram Title:qocttoolsCPC Library link to program files:https://doi.org/10.17632/49z3vydmwk.1Developer's repository link:https://gitlab.com/acbarrigon/qocttoolsCode Ocean capsule:https://codeocean.com/capsule/7560215Licensing provisions: GPLv3Programming language: PythonNature of problem: When a quantum system can be driven by external perturbations, one may be interested in finding the form of those perturbations that lead to a certain behavior. For example, driving the system from its ground to a given excited state, maximizing the value of some observable, etc. This can be interpreted as an inverse spectroscopy: the goal is not to predict the reaction of a piece of matter to an external probe or perturbation (e.g. an electromagnetic field), but instead to predict the precise shape of the perturbation that leads to a given predefined reaction. This problem may be posed for closed systems evolving coherently, or in the context of open quantum systems, in the presence of an environment. Also, when considering periodic perturbations, the problem may be formulated in the language of quantum Floquet theory.Solution method: This code implements the fundamental equations of quantum optimal control theory [1], the mathematical framework that addresses the problem described above. It (1) handles both closed or open quantum systems; (2) allows the user to specify any form for the target function (the one that is to be optimized, and encodes the desired system behavior), as it may be specified through any user-defined Python function; (3) allows the user to specify general parameterized forms for the control functions, and to set bounds and constraints on those parameters; (4) implements formulas, based on the adjoint method in most cases, for the computation of the gradient of the target functions with respect to the control parameters, as these gradients are the key tools for an efficient optimization, and (5) interfaces to a generic optimization library (NLopt), that permits to perform the optimization with numerous different algorithms, allowing for the use of bounds and linear or nonlinear constraints.

Precision bound and optimal control in periodically modulated continuous quantum thermal machines.

We use Floquet formalism to study fluctuations in periodically modulated continuous quantum thermal machines. We present a generic theory for such machines, followed by specific examples of sinusoidal, optimal, and circular modulations, respectively. The thermodynamic uncertainty relations (TUR) hold for all modulations considered. Interestingly, in the case of sinusoidal modulation, the TUR ratio assumes a minimum at the heat engine to refrigerator transition point, while the chopped random basis optimization protocol allows us to keep the ratio small for a wide range of modulation frequencies. Furthermore, our numerical analysis suggests that TUR can show signatures of heat engine to refrigerator transition, for more generic modulation schemes. We also study bounds in fluctuations in the efficiencies of such machines; our results indicate that fluctuations in efficiencies are bounded from above for a refrigerator and from below for an engine. Overall, this study emphasizes the crucial role played by different modulation schemes in designing practical quantum thermal machines.

An on-demand source of energy-entangled electrons using levitons

We propose a source of purely electronic energy-entangled states implemented in a solid-state system with potential applications in quantum information protocols based on electrons. The proposed device relies on the standard tools of electron quantum optics and exploits entanglement of the Cooper pairs of a BCS superconductor. The latter is coupled via an adjustable quantum point contact to two opposite spin-polarized electron wave-guides, which are driven by trains of Lorentzian pulses. This specific choice for the drive is crucial to inject purely electronic entangled states devoid of spurious electron–hole pairs. In the Andreev regime, a perturbative calculation in the tunnel coupling confirms that entangled electrons states are generated at the output of the normal side. For arbitrary tunnel coupling and for a periodic drive, direct current and noise (auto and cross correlations) are computed numerically using a Keldysh–Nambu–Floquet formalism. Importantly, for a periodic drive, the production of these states can be controlled in time, thus implementing an on-demand source of entangled states. We exploit realistic experimental parameters for our device to identify its optimal functioning point.

Floquet formulation of the dynamical Berry-phase approach to nonlinear optics in extended systems

We present a Floquet scheme for the ab-initio calculation of nonlinear optical properties in extended systems. This entails a reformulation of the real-time approach based on the dynamical Berry-phase polarisation (Attaccalite and Grüning 2013 Phys. Rev. B 88 1–9) and retains the advantage of being non-perturbative in the electric field. The proposed method applies to periodically-driven Hamiltonians and makes use of this symmetry to turn a time-dependent problem into a self-consistent time-independent eigenvalue problem. We implemented this Floquet scheme at the independent particle level and compared it with the real-time approach. Our reformulation reproduces real-time-calculated 2nd and 3rd order susceptibilities for a number of bulk and two-dimensional materials, while reducing the associated computational cost by one or two orders of magnitude.

Cascaded dynamics of a periodically driven dissipative dipolar system

Recent experiments show that periodic drives on dipolar systems lead to long-lived prethermal states. These systems are weakly coupled to the environment and reach prethermal states in a timescale much shorter than the timescale for thermalization. Such nearly-closed systems have previously been analyzed using Floquet formalism, which shows the emergence of a prethermal plateau. We use a fluctuation-regulated quantum master equation (FRQME) to describe these systems. In addition to the system-environment coupling, FRQME successfully captures the dissipative effect from the various local interactions in the system. Our investigation reveals a cascaded journey of the system to a final steady state. The cascade involves a set of prethermal or arrested states characterized by a set of quasi-conserved quantities. We show that these prethermal states emerge in a timescale much shorter than the relaxation timescale. We also find and report the existence of a critical limit beyond which the prethermal plateau ceases to exist.

The Dynamic Proximity Effect in Superconductor –Ferromagnetic Insulator Hybrid Structures

This paper presents a theoretical study of the dynamics of the induced magnetization and spin cur-rent arising in a layer of an impure superconductor due to the proximity to a ferromagnetic dielectric with a uniform periodically precessing magnetization. The dynamics of the observed physical quantities is described within the semi-classical Usadel–Floquet formalism, which makes it possible to study the effect of a periodic perturbation on an inhomogeneous superconducting system. The spatial distributions and temporal evolu-tion of the induced magnetization and the superconducting spin current inside the superconductor layer are found from the numerical solutions of the system of Usadel–Floquet equations.

Magneto-transport signatures in periodically-driven Weyl and multi-Weyl semimetals

We investigate the influence of a time-periodic driving (for example, by shining circularly polarized light) on three-dimensional Weyl and multi-Weyl semimetals, in the planar Hall and planar thermal Hall set-ups. We incorporate the effects of the drive by using the Floquet formalism in the large frequency limit. We evaluate the longitudinal magneto-conductivity, planar Hall conductivity, longitudinal thermo-electric coefficient, and transverse thermo-electric coefficient, using the semi-classical Boltzmann transport equations. We demonstrate the explicit expressions of these transport coefficients in certain limits of the system parameters, where it is possible to perform the integrals analytically. We cross-check our analytical approximations by comparing the physical values with the numerical results, obtained directly from the numerical integration of the integrals. The answers obtained show that the topological charges of the corresponding semimetals have profound signatures in these transport properties, which can be observed in experiments.

Optical conductivity of ultrathin Floquet topological insulators

In this paper, we theoretically investigate the topological aspects of a 3D ultrathin topological insulator in the presence of an irradiated off-resonant circularly-polarized light field. Based on the Floquet formalism we treat the light field as an external perturbation which introduces an effective energy term in the system Hamiltonian and hence, gives mass to the massless Dirac fermions at the surfaces of the Floquet topological insulators (FTIs). By controlling the strength of the off-resonant light field we manipulate the interplay between the hybridization energies of the top and bottom surface states (SSs), and the effective energies induced by the irradiated light field to open and close band gaps of the SSs Dirac electrons. This gives rise to topologically nontrivial and trivial phases. We calculate the longitudinal and transverse conductivities of the FTI thin film and show that these strongly depend on the strength of the irradiated light field. The behavior of conductivities signatures TQPTs between distinct topological insulating phases.

Floquet topological superconductivity induced by chiral many-body interaction

Non-equilibrium engineering is becoming a seminal way for realising novel quantum phases that are unimaginable in equilibrium. In particular, Floquet theory applied to quantum mechanics revealed that we can even control the band topology in semimetallic/insulating systems, while the straightforward application to topological superconductivity fails for typical superconductors because the supercondicting gap function does not couple to the electromagnetic field in a direct manner. Here we show that we can overcome this difficulty by taking account of correlation effects. Namely, we study how a d-wave superconductivity is changed when illuminated by circularly-polarised light (CPL) in the repulsive Hubbard model in the strong-coupling regime. We adopt the Floquet formalism for the Gutzwiller-projected effective Hamiltonian with the time-periodic Schrieffer-Wolff transformation. We find that CPL induces a topological superconductivity with a d + id pairing, which arises from the chiral spin coupling and the three-site term generated by the CPL. The latter term remains significant even for low frequencies and low intensities of the CPL. This is clearly reflected in the obtained phase diagram against the laser intensity and temperature for various frequencies red-detuned from the Hubbard U, with the transient dynamics also examined. The phenomenon revealed here can open a novel, dynamical way to induce a topological superconductivity.

Bandgap merging with double-negative metabeam

We theoretically investigate the bandgap formation in an Euler–Bernoulli beam-based metastructure with periodically attached double negative 3-dof local resonators. The idiosyncratic physical phenomena found in mechanical metamaterials, such as negative stiffness and mass, are introduced in the attached local resonator to design the double negative metabeam. The transfer matrix method with the Bloch–Floquet formulation is implemented in a unit cell of the double negative metabeam. The complex band structure describes the existence of various locally resonant attenuation bandgaps on the frequency spectrum. Further system analysis is conducted to illustrate the merging of the negative stiffness and mass controlled locally resonant bandgaps. A significant increase of 164% is found in the attenuation bandwidth due to the merging of locally resonant bandgaps.

Radiation modulated spin coupling in a double-stranded DNA model

The spin activity in macromolecules such as DNA and oligopeptides, in the context of the chiral induced spin selectivity has been proposed to be due to the atomic spin–orbit coupling (SOC) and the associated chiral symmetry of the structures. This coupling, associated with carbon, nitrogen and oxygen atoms in biological molecules, albeit small (meV), can be enhanced by the geometry, and strong local polarization effects such as hydrogen bonding. A novel way to manipulate the spin degree of freedom is by modifying the spectrum using a coupling to the appropriate electromagnetic radiation field. Here we use the Floquet formalism in order to show how the half filled band Hamiltonian for DNA, can be modulated by the radiation to produce up to a tenfold increase of the effective SOC once the intrinsic coupling is present. On the other hand, the chiral model, once incorporating the orbital angular momentum of electron motion on the helix, opens a gap for different helicity states (helicity splitting) that chooses spin polarization according to transport direction and chirality, without breaking time reversal symmetry. The observed effects are feasible in physically reasonable parameter ranges for the radiation field amplitude and frequency.

Floquet graphene antidot lattices

We establish the theoretical foundation of the Floquet graphene antidot lattice, whereby massless Dirac fermions are driven periodically by a circularly polarized electromagnetic field, while having their motion excluded from an array of nanoholes. The properties of interest are encoded in the quasienergy spectra, which are computed non-perturbatively within the Floquet formalism. We find that a rich Floquet phase diagram emerges as the amplitude of the drive field is varied. Notably, the Dirac dispersion can be restored in real time relative to the gapped equilibrium state, which may enable the creation of an optoelectronic switch or a dynamically tunable electronic waveguide. As the amplitude is increased, the ability to shift the quasienergy gap between high-symmetry points can change which crystal momenta dominate in the scattering processes relevant to electronic transport and optical emission. Furthermore, the bands can be flattened near the $\Gamma$ point, which is indicative of selective dynamical localization. Lastly, quadratic and linear dispersions emerge in orthogonal directions at the $M$ point, signaling a Floquet semi-Dirac material. Importantly, all our predictions are valid for experimentally accessible near-IR radiation, which corresponds to the above bandwidth limit for the graphene antidot lattice. Cycling between engineered Floquet electronic phases may play a key role in the development of next-generation on-chip devices for optoelectronic applications.

Dynamic impurities in two-dimensional topological-insulator edge states

Helical edge states of two-dimensional topological insulators show a gap in the density of states (DOS) and suppressed conductance in the presence of ordered magnetic impurities. Here we will consider the dynamical effects on the DOS and transmission when the magnetic impurities are driven periodically. Using the Floquet formalism and Green's functions, the system properties are studied as a function of the driving frequency and the potential energy contribution of the impurities. We see that increasing the potential part closes the DOS gap for all driving regimes. The transmission gap is also closed, showing a pronounced asymmetry as a function of energy. These features indicate that the dynamical transport properties could yield valuable information about the magnetic impurities.

Floquet vacuum engineering: Laser-driven chiral soliton lattice in the QCD vacuum

What happens to the QCD vacuum when a time-periodic circularly polarized laser field with a sufficiently large intensity and frequency is applied? Based on the Floquet formalism for periodically driven systems and the systematic low-energy effective theory of QCD, we show that for a sufficiently large frequency and above a critical intensity, the QCD vacuum is unstable against the chiral soliton lattice of pions; a crystalline structure of topological solitons that spontaneously breaks parity and continuous translational symmetries. Our work would pave the way for novel "Floquet vacuum engineering."

Ultrafast quantum imaging in the dissociation of H2+ via the induced conical intersection of two lowest adiabatic states by strong field laser pulses

• A non-Born-Oppenheimer time-dependent Schrödinger equation is numerically solved to study the dissociative-ionization of H 2 + molecule subjected to strong field six-cycle laser pulses ( I = 4 × 10 14 W / cm 2 , λ = 800 nm ) . • Analysis of the time evolution of the H 2 + nuclear wave packet is performed using Floquet formalism in the adiabatic representation to gain newly insightful images of the nonlinear processes such as vibrational trapping, above-threshold dissociation, tunneling, and bond-softening of the laser-driven nuclear wave packet on two lowest field-dressed adiabatic potential energy curves. • Starting from an initial state-selected vibrational level of H 2 , the images enable ones to fully comprehend the dissociative-ionization dynamics of H 2 + , notably in the vicinity of a conical intersection up to eventually entering the two exit channels with the main one producing fragments H + H + via the H 2 + ( 2 ∑ u + ) exit channel. • The direct treatment of the electronic-nuclear wave packet of H 2 enables one to image the one-electron nuclear localization of H 2 + on the right or left of field-distorted double-well Coulomb potential to reveal the influence of the field on the charge-resonance enhanced ionization coupled with the other nonlinear processes. A non-Born–Oppenheimer time-dependent Schrödinger equation is numerically solved to study the dissociative-ionization of H 2 + subjected to strong field six-cycle laser pulses ( I = 4 × 10 14 W / cm 2 , λ = 800 nm ) . Analysis of the evolution of H 2 + nuclear wave packet is performed using Floquet formalism in the adiabatic representation to gain newly insightful images of the nonlinear processes in the vicinity of a conical intersection of two lowest field-dressed adiabatic states. The coherent superposition of the states leads to the right or left localization of H 2 + on the field-distorted double-well Coulomb potential.

Coherent dynamics of Floquet-Bloch states in monolayer WS2 reveals fast adiabatic switching

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 the pulses does not introduce any effects beyond Floquet theory. Furthermore, we demonstrate that the induced energy shift follows a linear dependence on instantaneous intensity, even for ultrafast driving fields of fewer than 15 optical cycles.

Quantum-classical nonadiabatic dynamics of Floquet driven systems.

We develop a trajectory-based approach for excited-state molecular dynamics simulations of systems subject to an external periodic drive. We combine the exact-factorization formalism, allowing us to treat electron-nuclear systems in nonadiabatic regimes, with the Floquet formalism for time-periodic processes. The theory is developed starting with the molecular time-dependent Schrödinger equation with the inclusion of an external periodic drive that couples to the system dipole moment. With the support of the Floquet formalism, quantum dynamics is approximated by combining classical-like, trajectory-based, nuclear evolution with electronic dynamics represented in the Floquet basis. The resulting algorithm, which is an extension of the coupled-trajectory mixed quantum-classical scheme for periodically driven systems, is applied to a model study, exactly solvable, with different field intensities.

Manipulating Intertwined Orders in Solids with Quantum Light.

Intertwined orders exist ubiquitously in strongly correlated electronic systems and lead to intriguing phenomena in quantum materials. In this Letter, we explore the unique opportunity of manipulating intertwined orders through entangling electronic states with quantum light. Using a quantum Floquet formalism to study the cavity-mediated interaction, we show the vacuum fluctuations effectively enhance the charge-density-wave correlation, giving rise to a phase with entangled electronic order and photon coherence, with putative superradiant behaviors in the thermodynamic limit. Furthermore, upon injecting even one single photon in the cavity, different orders, including s-wave and η-paired superconductivity, can be selectively enhanced. Our study suggests a new and generalizable pathway to control intertwined orders and create light-matter entanglement in quantum materials. The mechanism and methodology can be readily generalized to more complicated scenarios.