Floquet Hamiltonian Research Articles

Dynamic tunable nonreciprocal single-photon scattering mediated by a giant atom assisted with a time-modulated cavity

Nonreciprocal single-photon scattering in a one-dimensional waveguide coupled to a giant two-level atom assisted with a time-modulated single-mode cavity is investigated. The analytic expressions of the single-photon scattering amplitudes are derived by using an effective Floquet Hamiltonian in real space. The scattering characteristics are discussed detail in both the Markovian and the non-Markovian regimes, and the corresponding conditions for achieving perfect nonreciprocal single-photon transmission are obtained. In the Markovian regime, a frequency-tunable single-photon diode with an ideal transmission contrast ratio can be realized by adjusting the frequency of the cavity mode, the local coupling phase difference, and the accumulated phase between the two coupling points. Furthermore, the influence of the intrinsic energy dissipations on the photon transport is discussed in detail. It is found that the dissipations of the cavity and the giant atom affect discriminatively the nonreciprocal single-photon scattering process. In the non-Markovian regime, the influence of the non-Markovian retarded effect induced by the time delay on the nonreciprocal single-photon scattering is discussed in detail. The results reveal that, although the retarded effect leads to a complex nonreciprocal scattering spectrum, dynamic tunable perfect nonreciprocal transmission with more abundant physical phenomena suitable for photons with different frequencies within a larger range can also be achieved. Such a nonreciprocal single-photon device can be used as an elementary unit for various quantum information processing and may have potential applications in quantum network engineering.

Floquet-Engineered Photodissociation Simulated Using Coupled Potential Energy and Dipole Matrices.

We simulate the nonadiabatic molecular dynamics of ammonia photodissociation in the presence of an external laser field by using an approximate Floquet Hamiltonian. The dipole-field interaction gives rise to seams of light-induced conical intersection (LICI), which can significantly change the topography of the coupled potential energy surfaces. We perform quasiclassical trajectories based on recently reported diabatic potential energy matrices (DPEM) and dipole matrices. It is shown that the branching ratio of ground and excited state NH2 is drastically altered by laser-dipole interaction, which is a signature of nonadiabatic effects induced by light.

Counterdiabatic Driving for Periodically Driven Systems.

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

Exactly solvable floquet dynamics for conformal field theories in dimensions greater than two

We find classes of driven conformal field theories (CFT) in d + 1 dimensions with d > 1, whose quench and floquet dynamics can be computed exactly. The setup is suitable for studying periodic drives, consisting of square pulse protocols for which Hamiltonian evolution takes place with different deformations of the original CFT Hamiltonian in successive time intervals. These deformations are realized by specific combinations of conformal generators with a deformation parameter β; the β < 1 (β > 1) Hamiltonians can be unitarily related to the standard (Lüscher-Mack) CFT Hamiltonians. The resulting time evolution can be then calculated by performing appropriate conformal transformations. For d ≤ 3 we show that the transformations can be easily obtained in a quaternion formalism. Evolution with such a single Hamiltonian yields qualitatively different time dependences of observables depending on the value of β, with exponential decays characteristic of heating for β > 1, oscillations for β < 1 and power law decays for β = 1. This manifests itself in the behavior of the fidelity, unequal-time correlator, and the energy density at the end of a single cycle of a square pulse protocol with different hamiltonians in successive time intervals. When the Hamiltonians in a cycle involve generators of a single SU(1, 1) subalgebra we calculate the Floquet Hamiltonian. We show that one can get dynamical phase transitions for any β by varying the time period of a cycle, where the system can go from a non-heating phase which is oscillatory as a function of the time period to a heating phase with an exponentially damped behavior. Our methods can be generalized to other discrete and continuous protocols. We also point out that our results are expected to hold for a broader class of QFTs that possesses an SL(2, C) symmetry with fields that transform as quasi-primaries under this. As an example, we briefly comment on celestial CFTs in this context.

Dynamical phase-field model of cavity electromagnonic systems

Cavity electromagnonic system, which simultaneously consists of cavities for photons, magnons (quanta of spin waves), and acoustic phonons, provides an exciting platform to achieve coherent energy transduction among different physical systems down to single quantum level. Here we report a dynamical phase-field model that allows simulating the coupled dynamics of the electromagnetic waves, magnetization, and strain in 3D multiphase systems. As examples of application, we computationally demonstrate the excitation of hybrid magnon-photon modes (magnon polaritons), Floquet-induced magnonic Aulter-Townes splitting, dynamical energy exchange (Rabi oscillation) and relative phase control (Ramsey interference) between the two magnon polariton modes. The simulation results are consistent with analytical calculations based on Floquet Hamiltonian theory. Simulations are also performed to design a cavity electro-magno-mechanical system that enables the triple phonon-magnon-photon resonance, where the resonant excitation of a chiral, fundamental (n = 1) transverse acoustic phonon mode by magnon polaritons is demonstrated. With the capability to predict coupling strength, dissipation rates, and temporal evolution of photon/magnon/phonon mode profiles using fundamental materials parameters as the inputs, the present dynamical phase-field model represents a valuable computational tool to guide the fabrication of the cavity electromagnonic system and the design of operating conditions for applications in quantum sensing, transduction, and communication.

Floquet analysis perspective of driven light–matter interaction models

In this paper, we analyze the harmonically driven Jaynes–Cummings and Lipkin–Meshkov–Glick models using both numerical integration of time-dependent Hamiltonians and Floquet theory. For a separation of time scales between the drive and intrinsic Rabi oscillations in the former model, the driving results in an effective periodic reversal of time. The corresponding Floquet Hamiltonian is a Wannier–Stark model, which can be analytically solved. Despite the chaotic nature of the driven Lipkin–Meshkov–Glick model, moderate system sizes can display qualitatively different behaviors under varying system parameters. Ergodicity arises in systems that are neither adiabatic nor diabatic, owing to repeated multi-level Landau–Zener transitions. Chaotic behavior, observed in slow driving, manifests as random jumps in the magnetization, suggesting potential utility as a random number generator. Furthermore, we discuss both models in terms of a Floquet Fock state lattice.

Robust Effective Ground State in a Nonintegrable Floquet Quantum Circuit.

An external periodic (Floquet) drive is believed to bring any initial state to the featureless infinite temperature state in generic nonintegrable isolated quantum many-body systems in the thermodynamic limit, irrespective of the driving frequency Ω. However, numerical or analytical evidence either proving or disproving this hypothesis is very limited and the issue has remained unsettled. Here, we study the initial state dependence of Floquet heating in a nonintegrable kicked Ising chain of length up to L=30 with an efficient quantum circuit simulator, showing a possible counterexample: the ground state of the effective Floquet Hamiltonian is exceptionally robust against heating, and could stay at finite energy density even after infinitely many Floquet cycles, if the driving period is shorter than a threshold value. This sharp energy localization transition or crossover does not happen for generic excited states. The exceptional robustness of the ground state is interpreted by (i)its isolation in the energy spectrum and (ii)the fact that those states with L-independent ℏΩ energy above the ground state energy of any generic local Hamiltonian, like the approximate Floquet Hamiltonian, are atypical and viewed as a collection of noninteracting quasiparticles. Our finding paves the way for engineering Floquet protocols with finite driving periods realizing long-lived, or possibly even perpetual, Floquet phases by initial state design.

Topological phase diagram of optimally shaken honeycomb lattices: A dual perspective from stroboscopic and nonstroboscopic Floquet Hamiltonians

We present a direct comparison between the stroboscopic and nonstroboscopic effective approaches for ultracold atoms in shaken honeycomb lattices, focusing specifically on the optimal driving introduced by A. Verdeny and F. Mintert []. In the fast-driving regime, we compare the effective nonstroboscopic Hamiltonian derived through a perturbative expansion with a nonperturbative calculation of the stroboscopic Floquet Hamiltonian, obtained through a direct numerical approach. We find that the leading off-diagonal tunneling coefficients, which define the undriven model, retain their isotropic nature under the effect of the driving, and their values are independent of the computational approach used. This consistency ensures the robustness of certain topological properties, such as the position of Dirac points, as expected. Conversely, the next-to-leading diagonal tunneling coefficients, generated dynamically, can vary across different approaches without affecting the system topology. This suggests that the topological phase diagram is most conveniently represented in terms of the physical parameters characterizing the driving and directly accessible in experiments, rather than in terms of the tunneling parameters. Published by the American Physical Society 2024

Frame change technique for phase transient cancellation

The precise control of complex quantum mechanical systems can unlock applications ranging from quantum simulation to quantum computation. Controlling strongly interacting many-body systems often relies on Floquet Hamiltonian engineering that is achieved by fast switching between Hamiltonian primitives via external control. For example, in our solid-state NMR system, we perform quantum simulation by modulating the natural Hamiltonian with control pulses. As the Floquet heating errors scale with the interpulse delay, δt, it is favorable to keep δt as short as possible, forcing our control pulses to be short duration and high power. Additionally, high-power pulses help to minimize undesirable evolution from occurring during the duration of the pulse. However, such pulses introduce an appreciable phase-transient control error, a form of unitary error. In this work, we detail our ability to diagnose the error, calibrate its magnitude, and correct it for π/2-pulses of arbitrary phase. We demonstrate the improvements gained by correcting for the phase transient error, using a method which we call the “frame-change technique”, in a variety of experimental settings of interest. Given that the correction mechanism adds no real control overhead, we recommend that any resonance probe be checked for these phase transient control errors, and correct them using the frame-change technique.

Floquet Insulators and Lattice Fermions.

Floquet insulators are periodically driven quantum systems that can host novel topological phases as a function of the drive parameters. These new phases exhibit features reminiscent of fermion doubling in discrete-time lattice fermion theories. We make this suggestion concrete by mapping the spectrum of a noninteracting (1+1)D Floquet insulator for certain drive parameters onto that of a discrete-time lattice fermion theory with a time-independent Hamiltonian. The resulting Hamiltonian is distinct from the Floquet Hamiltonian that generates stroboscopic dynamics. It can take the form of a discrete-time Su-Schrieffer-Heeger model with half the number of spatial sites of the original model, or of a (1+1)D Wilson-Dirac theory with one quarter of the spatial sites.

Time-dependent Gutzwiller simulation of Floquet topological superconductivity

Periodically driven systems provide a novel route to control the topology of quantum materials. In particular, Floquet theory allows an effective band description of periodically-driven systems through the Floquet Hamiltonian. Here, we study the time evolution of d-wave superconductors irradiated with intense circularly-polarized laser light. We consider the Floquet t–J model with time-periodic interactions, and investigate its mean-field dynamics by formulating the time-dependent Gutzwiller approximation. We observe the development of the idxy-wave pairing amplitude along with the original dx2−y2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${d}_{{x}^{2}-{y}^{2}}$$\\end{document}-wave order upon gradual increasing of the field amplitude. We further numerically construct the Floquet Hamiltonian for the steady state, with which we identify the system as the fully-gapped d + id superconducting phase with a nonzero Chern number. We explore the low-frequency regime where the perturbative approaches in the previous studies break down, and find that the topological gap of an experimentally-accessible size can be achieved at much lower laser intensities.

Controllable single-photon transport mediated by a time-modulated Jaynes–Cummings model

Controllable single-photon scattering in a one-dimensional waveguide coupled to a Jaynes–Cummings structure containing a time-modulated two-level atom interacting with a single-mode cavity is investigated. The photon transmission and reflection amplitudes are calculated by using an effective Floquet Hamiltonian in real space. The results show that the coupling between the atom and the cavity mode can dynamically be tuned via periodically modulating the atomic transition frequency. As a consequence, the scattering behaviors of the waveguide photons can be actively manipulated, and a controllable single-photon switch with high on-off ratio could be realized. More interestingly, the switch works well within a wide frequency region, i.e., the transmission of both resonant and off-resonant waveguide photons can be effectively switched on or off with appropriate system parameters. Furthermore, the proposed dynamically tunable switching scheme is robust against atomic dissipation associated with the help of atom-cavity coupling mismatch. Such single-photon device can be used as an elementary unit for various quantum information processing.

Long term behavior of the stirred vacuum on a Dirac chain: geometry blur and the random Slater ensemble

We characterize the long-term state of the 1D Dirac vacuum stirred by an impenetrable object, modeled as the ground state of a finite free-fermionic chain dynamically perturbed by a moving classical obstacle which suppresses the local hopping amplitudes. We find two different regimes, depending on the velocity of the obstacle. For a slow motion, the effective Floquet Hamiltonian presents features which are typical of the Gaussian orthogonal ensemble, and the occupation of the Floquet modes becomes roughly homogeneous. Moreover, the long term entanglement entropy of a contiguous block follows a Gaussian analogue of Page’s law, i.e. a volumetric behavior. Indeed, the statistical properties of the reduced density matrices correspond to those of a random Slater determinant, which can be described using the Jacobi ensemble from random matrix theory. On the other hand, if the obstacle moves fast enough, the effective Floquet Hamiltonian presents a Poissonian behavior. The nature of the transition is clarified by the entanglement links, which determine the effective geometry underlying the entanglement structure, showing that the one-dimensionality of the physical Hamiltonian dissolves into a random adjacency matrix as we slow down the obstacle motion.

Sensitivity of radio-frequency electric field sensor based on Rydberg Stark effect

Rydberg atoms hold special attraction in electric applications due to their large transition electric dipole moments and huge polarization, which leads to a strong response of atom to electric fields. In radio-frequency (RF) fields, the Rydberg levels are AC Stark shift and splitting, which can realize the study of high-sensitivity electric field sensor of Rydberg atoms. In this work, we use the simpler Shirley’s time-independent Floquet Hamiltonian model to calculate the AC Stark energy spectrum of Cs Rydberg atoms. This model can reduce the basic Hamiltonian into such a Hamiltonian that includes only those Rydberg states that have direct dipole-allowed transitions with the target state, thereby significantly improving the speed of computation. The accuracy of the calculation is proved by fitting with the calculated frequency shift of DC Stark energy levels in the weak fields, and the polarizability of 60D&lt;sub&gt;5/2&lt;/sub&gt; and 70D&lt;sub&gt;5/2&lt;/sub&gt; Rydberg atomic states are obtained by fitting with the measured ion spectra of DC Stark Cs ultra-cold Rydberg atoms in magneto-optical trap. In addition, we calculate the AC Stark shift of Cs Rydberg atom &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \left| {60{{\text{D}}_{5/2}},{m_j} = 1/2} \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240162_M2.jpg"/&gt;&lt;graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240162_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; state in electric fields with different frequencies with &lt;i&gt;ε&lt;/i&gt; = 100 mV/m. Rydberg atoms provide a structured spectrum of sensitivity to electric fields due to strong resonant interaction and off-resonant interaction with many dipole-allowed transitions to nearby Rydberg states. This kind of the frequency response structure is of significance to a broadband sensor. And we calculate the sensitivity and the scaling of the signal-to-noise ratio (SNR), &lt;i&gt;β&lt;/i&gt;, varying with detuning from the &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \left| {60{{\text{D}}_{5/2}}} \right\rangle \to \left| {61{{\text{P}}_{3/2}}} \right\rangle $\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic specific-use="online" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240162_M3.jpg"/&gt;&lt;graphic specific-use="print" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="9-20240162_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; transition. The value of &lt;i&gt;β&lt;/i&gt; allows one to use the result for any Rydberg state sensor to determine the SNR for any &lt;i&gt;Ε&lt;/i&gt; in a 1 s measurement. Therefrom, Rydberg sensor can preferentially detect many RF frequencies spreading across its carrier spectral range without modification while effectively rejecting large portions where the atom response is significantly weaker, and the signal depends primarily on the detuning of the RF field to the nearest resonance which does not convey the RF frequency directly.

The conditions for the analog of QED photons in semi-classical periodically driven systems

The Floquet and quantum electrodynamics (QED) Hamiltonians are widely used in various contexts for light-matter interactions. While they exhibit structural similarity, the QED Hamiltonian has a bounded spectrum while the Floquet Hamiltonian does not. Thus, it remains uncertain if they share the same or similar spectra, even at high energy with a substantial average photon count. Using the Gershgorin circle theorem, we bound analytically the difference between the spectra of the QED and Floquet Hamiltonians. We establish a common spectrum by imposing the constraints of high photon numbers and narrow photon statistics. Following the analytic proof, we numerically demonstrate this bound’s implications on a model Xe atom previously used in high harmonic generation, showing correspondence between Floquet and QED photons.

Absolutely Stable Time Crystals at Finite Temperature.

We show that locally interacting, periodically driven (Floquet) Hamiltonian dynamics coupled to a Langevin bath support finite-temperature discrete time crystals (DTCs) with an infinite autocorrelation time. By contrast to both prethermal and many-body localized DTCs, the time crystalline order we uncover is stable to arbitrary perturbations, including those that break the time translation symmetry of the underlying drive. Our approach utilizes a general mapping from probabilistic cellular automata to open classical Floquet systems undergoing continuous-time Langevin dynamics. Applying this mapping to a variant of the Toom cellular automaton, which we dub the "π-Toom time crystal," leads to a 2D Floquet Hamiltonian with a finite-temperature DTC phase transition. We provide numerical evidence for the existence of this transition, and analyze the statistics of the finite temperature fluctuations. Finally, we discuss how general results from the field of probabilistic cellular automata imply the existence of discrete time crystals (with an infinite autocorrelation time) in all dimensions, d≥1.

Chiral anomaly in a (1+1)-dimensional Floquet system under high-frequency electric fields

We investigate the chiral anomaly in a Floquet system under a time-periodic electric field in (1+1) dimensions. Using the van Vleck high-frequency expansion, we analytically calculate the chiral current and the pseudo-scalar condensate for massless/massive fermions and how they are balanced with the topological charge. In the high-frequency limit, we find that finite-mass effects are suppressed and the topological charge is dominated by the chirality production. Our calculations show that the information about the chiral anomaly is stored not in the static Floquet Hamiltonian but in the periodic kick operator. The computational steps are useful as the theoretical foundation for higher-dimensional generalization.

Electric circuit simulation of Floquet topological insulators in Fourier space

We present a method for simulating any non-interacting and time-periodic tight-binding Hamiltonian in Fourier space using electric circuits made of inductors and capacitors. We first map the time-periodic Hamiltonian to a Floquet Hamiltonian, which converts the time dimension into a Floquet dimension. In electric circuits, this Floquet dimension is simulated as an extraspatial dimension without any time dependency in the electrical elements. The number of replicas needed in the Floquet Hamiltonian depends on the frequency and strength of the drive. We also demonstrate that we can detect the topological edge states (including the anomalous edge states in the dynamical gap) in an electric circuit by measuring the two-point impedance between the nodes. Our method paves a simple and promising way to explore and control Floquet topological phases in electric circuits.

Energy transfer in random-matrix ensembles of Floquet Hamiltonians

Energy transfer in random-matrix ensembles of Floquet Hamiltonians

Exceptional Points and Exponential Sensitivity for Periodically Driven Lindblad Equations

In this contribution to the memorial issue of Göran Lindblad, we investigate the periodically driven Lindblad equation for a two-level system. We analyze the system using both adiabatic diagonalization and numerical simulations of the time-evolution, as well as Floquet theory. Adiabatic diagonalization reveals the presence of exceptional points in the system, which depend on the system parameters. We show how the presence of these exceptional points affects the system evolution, leading to a rapid dephasing at these points and a staircase-like loss of coherence. This phenomenon can be experimentally observed by measuring, for example, the population inversion. We also observe that the presence of exceptional points seems to be related to which underlying Lie algebra the system supports. In the Floquet analysis, we map the time-dependent Liouvillian to a non-Hermitian Floquet Hamiltonian and analyze its spectrum. For weak decay rates, we find a Wannier-Stark ladder spectrum accompanied by corresponding Stark-localized eigenstates. For larger decay rates, the ladders begin to dissolve, and new, less localized states emerge. Additionally, their eigenvalues are exponentially sensitive to perturbations, similar to the skin effect found in certain non-Hermitian Hamiltonians.