Applications of Floquet–Magnus expansion,average Hamiltonian theory and Fer expansion to study interactions in solid state NMR when irradiated with the magic-echo sequence

This work presents the possibility of applying the Floquet–Magnus expansion and the Fer expansion approaches to the most useful interactions known in solid-state nuclear magnetic resonance using the magic-echo scheme. The results of the effective Hamiltonians of these theories and average Hamiltonian theory are presented.     

Multiband Adiabatic Inversion Pulses

The conditions necessary to obtain multiple adiabatic population inversion at different frequencies in a two-level system are defined. It is shown how any pulse that produces adiabatic inversion in a single frequency band can be modified to become a multiband adiabatic inversion pulse. Using Floquet formalism, the interaction between the different inversion bands is described and shown to create effective nonlinear irradiation fields. By controlling the reference phase of the single-inversion-frequency-band pulses, these effective irradiation fields can be minimized or canceled. These pulses can be used for multiple selective excitation or selective population inversion in coherent spectroscopy. NMR experiments confirming the theoretical predications are shown. The experimental results agree very well with the theoretical predictions.     

On relativistic energy band corrections in the presence of periodic potentials

A previously developed formalism to compute relativistic corrections of bound state energies for spin-1/2 particles is applied to relativistic corrections of energy bands of one-dimensional, periodic Hamiltonians. We explicitly describe Floquet theory for periodic Dirac operators on the line. Extensions including impurity potentials and/or v2 dimensions are straightforward and sketched at the end.     

Analysis of multiple-pulse techniques under fast MAS conditions

The combination of magic-angle spinning and multiple-pulse sequences for line-narrowing in solids with homogeneous spin interactions is analyzed using Floquet theory. It is found that, for quasi-static conditions and for special synchronization conditions, line-narrowing is possible while for other conditions destructive interference of the two techniques takes place. However, even for optimum line-narrowing conditions, fundamental limitations with respect to the achievable linewidth are found, whereas the conditions of recoupling spin interactions are more easily realized. The implications of these results with respect to improving existing line-narrowing techniques or techniques for the design of specific Hamiltonians are discussed.     

Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering

We give a general overview of the high-frequency regime in periodically driven systems and identify three distinct classes of driving protocols in which the infinite-frequency Floquet Hamiltonian is not equal to the time-averaged Hamiltonian. These classes cover systems, such as the Kapitza pendulum, the Harper–Hofstadter model of neutral atoms in a magnetic field, the Haldane Floquet Chern insulator and others. In all setups considered, we discuss both the infinite-frequency limit and the leading finite-frequency corrections to the Floquet Hamiltonian. We provide a short overview of Floquet theory focusing on the gauge structure associated with the choice of stroboscopic frame and the differences between stroboscopic and non-stroboscopic dynamics. In the latter case, one has to work with dressed operators representing observables and a dressed density matrix. We also comment on the application of Floquet Theory to systems described by static Hamiltonians with well-separated energy scales and, in particular, discuss parallels between the inverse-frequency expansion and the Schrieffer–Wolff transformation extending the latter to driven systems.     

Quantum computation based on magic-angle-spinning solid state nuclear magnetic resonance spectroscopy

Magic-angle spinning (MAS) solid state nuclear magnetic resonance (NMR) spectroscopy is shown to be a promising technique for implementing quantum computing. The theory underlying the principles of quantum computing with nuclear spin systems undergoing MAS is formulated in the framework of formalized quantum Floquet theory. The procedures for realizing state labeling, state transformation and coherence selection in Floquet space are given. It suggests that by this method, the largest number of qubits can easily surpass that achievable with other techniques. Unlike other modalities proposed for quantum computing, this method enables one to adjust the dimension of the working state space, meaning the number of qubits can be readily varied. The universality of quantum computing in Floquet space with solid state NMR is discussed and a demonstrative experimental implementation of Grover's search is given. Received 19 April 2001     

Irreducible spin precession theory applied to some topics in atomic and nuclear radio-frequency spectroscopy

Several problems from radio-frequency spectroscopy of atoms and nuclei are treated with irreducible spin precession theory. In the first part, effective field techniques are used to derive analytically single and multiple quantum double resonance lineshapes for atoms with a hyperfine structure in a high magnetic field. In the second part (as an extension to previous work), nuclear resonance signals are calculated for oriented nuclei subject to an electric hexadecapole interaction. Lineshapes of acoustically driven hexadecapole transitions are derived in closed form and compared to experiment. Further, multiple quantum NMR transitions within a hexadecapole shifted nuclear Zeeman structure are calculated, and some distinct features of hexadecapole effects on NMR lineshapes are pointed out. This last case is of current interest due to recent progress in NMR-line narrowing techniques. — In the Appendix, we give lineshape equations for single and double quantum NMR transitions on oriented (I=1)-nuclei subject to an electric quadrupole interaction; these equations are also being used in the atomic rf-spectroscopy calculations. The equations are exact to all orders of the interaction with the external fields.     

Simulations of chemical exchange lineshapes in CP/MAS spectra using floquet theory and sparse matrix methods

This paper presents a general method for simulating the effect of chemical exchange on MAS NMR spectra of solid samples. The complication in MAS spectra is that the Hamiltonian itself is time-dependent, due to the spinning of the sample. The approach taken in this work is to use Floquet theory to convert the problem into a time-independent form, and then use established methods (used in liquid NMR simulations) to calculate the lineshape. Floquet theory has been admired for its elegance, but criticized for its computational inefficiencies. This is because it removes the time dependence of the system by expanding the problem in a Fourier-like series. This makes a relatively small, time-dependent calculation into a much larger time-independent one. Typically, we use twice as many Floquet blocks as there are spinning sidebands, so the increase in size is substantial. The problem that this creates stems from the fact that the usual Householder methods for diagonalizing a matrix scale as the cube of the size of the matrix. This would make a Floquet calculation prohibitively long. However, the Floquet matrix is inherently sparse, so sparse matrix methods can produce substantial computational savings. Also, fully diagonalizing a matrix is expensive, but converting the matrix to a tridiagonal form (using iterative Lanczos methods) is much cheaper. The use of the Lanczos methods makes the Floquet calculations feasible as a general method for systems of more than one spin. We show how to set up the full matrix describing chemical exchange in a spinning sample, but the details of how the Lanczos methods work are not included-they are described elsewhere. We then validate the theory by simulating the MAS spectra of dimethyl sulfone both with natural abundance (13)C and with methyl groups labeled with (13)C. The latter system has both dipolar and chemical shielding anisotropy terms contributing to the spectrum. Copyright 2000 Academic Press.     

A New Treatment for Some Periodic Schrdinger Operators I: The Eigenvalue

We study the problem of how the Floquet property manifests for periodic Schr¨odinger operators, which are known to have multiple of asymptotic spectral solutions. The main conclusions are made for elliptic potentials,we demonstrate that for each period of the elliptic function there is a relation about the Floquet exponent and the monodromy of wave function. Among them there are two relations not explained by the classical Floquet theory. These relations produce both old and new asymptotic solutions consistent with results already known.     

Effect of Floquet engineering on the p-wave superconductor with second-neighbor couplings

The influence of the Floquet engineering on a particular one-dimensional p-wave superconductor, Kitaev model, with second-neighbor couplings is investigated in this paper. The effective Hamiltonians in the rotated reference frames have been obtained, and the convergent regions of the approximated Hamiltonian as well as the topological phase diagrams have been analyzed and discussed. We show that by modulating the external driving field amplitude, frequency as well as the second-neighbor hopping amplitude, the rich phase diagrams and transitions between different topological phases can be obtained.     

Floquet wave homogenization of periodic anisotropic media

A dynamic homogenization method based on Floquet wave theory is developed. The theory is based on the equivalency within the homogenization domain of Floquet waves in a periodic anisotropic medium and plane waves in a dispersive homogeneous anisotropic medium. A simple procedure has been developed to estimate analytically critical angles and the upper frequency bound of this homogenization domain. Using this method, the frequency dependent effective elastic constants are obtained and examples for [0/90] and [0/45/90/-45] composites are given. By comparison with an exact theory, it is shown that the time domain signal propagation in a periodic laminate is well described by the Floquet wave homogenization theory in the homogenization domain. It is also shown that in the static limit the results are identical to those calculated by static homogenization theory (the generalized method of cells). The potential applications of the method are discussed.     

Construction of model hamiltonians in framework of Rayleigh-Schrödinger perturbation theory

A theory of the model Hamiltonians within the framework of Rayleigh-Schrödinger perturbation theory is elaborated. The approach of a model Hamiltonian is based on the assumption that if it is diagonalized in a chosen model space it will yield eigenvalues of the original Hamiltonian in the entire Hilbert space. The theory of the model Hamiltonians may be fruitful as a theoretical background for the study of effective Hamiltonians and as natural extension of the standard Rayleigh-Schrödinger perturbation theory.     

Hamilton系统Noether理论的新型逆问题

研究Hamilton系统Noether理论新型的逆问题,得到利用Noether理论从已知的第一积分构建Hamilton函数和对称性的一般解法和若干特殊解法,提出由Hamilton函数直接导出守恒量的两条推论.举例说明所得结果的应用.     

Periodically driven ergodic and many-body localized quantum systems

We study dynamics of isolated quantum many-body systems whose Hamiltonian is switched between two different operators periodically in time. The eigenvalue problem of the associated Floquet operator maps onto an effective hopping problem. Using the effective model, we establish conditions on the spectral properties of the two Hamiltonians for the system to localize in energy space. We find that ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local and global driving. In contrast, many-body localized systems with quenched disorder remain localized at finite energy. We support our conclusions by numerical simulations of disordered spin chains. We argue that our results hold for general driving protocols, and discuss their experimental implications.     

A four-level “Ξ-system”: Exact solvability and effective evolution operators via multiple scales

Properties of a four-level atomic system interacting with one and two modes of the electromagneticfield in a “Ξ”-configuration are investigated. By linearization of the Hamiltonians we show that the corresponding mathematical models are exactly solvable. To obtain simpler effective Hamiltonians the perturbative method of multiple scales is applied. The lowest-order corrections to the resulting effective evolution operators are also calculated.     

Amplitude-modulated decoupling in rotating solids: a bimodal Floquet approach

This paper centers on a theoretical study of amplitude-modulated heteronuclear decoupling in solid-state NMR under magic-angle spinning (MAS). A spin system with a single isolated rare spin coupled to a large number of abundant spins is used in the analysis. The phase-alternating decoupling scheme (XiX decoupling) is analyzed using bimodal Floquet theory and the operator-based perturbation method developed by van Vleck. An effective Hamiltonian correct to second order is calculated for the spin system under XiX decoupling. The results of these calculations indicate that under XiX decoupling the main contribution to the residual line width comes from a cross-term between the heteronuclear and the homonuclear dipolar couplings. This is in contrast to continuous-wave decoupling, where the residual line width is dominated by the cross-term between the heteronuclear dipolar coupling and the chemical-shielding tensor of the irradiated spin. For high-power decoupling the method results in very good decoupling provided that certain unfavorable recoupling conditions, imposed by specific ratios of the amplitude modulation frequency and the MAS frequency, are avoided. For low-power decoupling, the method leads to acceptable decoupling when the pulse length corresponds to an integer multiple of a 2pi rotation and the rf-field amplitude is less than a quarter of the MAS frequency. The performance of the XiX scheme is analyzed over a range of values of the rf power, and numerical results that agree well with the most recent experimental observations are presented.     

Effective Hamiltonian of the Jaynes–Cummings model beyond rotating-wave approximation

The Jaynes–Cummings model with or without rotating-wave approximation plays a major role to study the interaction between atom and light. We investigate the Jaynes–Cummings model beyond the rotating-wave approximation. Treating the counter-rotating terms as periodic drivings, we solve the model in the extended Floquet space. It is found that the full energy spectrum folded in the quasi-energy bands can be described by an effective Hamiltonian derived in the highfrequency regime. In contrast to the Z_2 symmetry of the original model, the effective Hamiltonian bears an enlarged U(1)symmetry with a unique photon-dependent atom-light detuning and coupling strength. We further analyze the energy spectrum, eigenstate fidelity and mean photon number of the resultant polaritons, which are shown to be in accordance with the numerical simulations in the extended Floquet space up to an ultra-strong coupling regime and are not altered significantly for a finite atom-light detuning. Our results suggest that the effective model provides a good starting point to investigate the rich physics brought by counter-rotating terms in the frame of Floquet theory.     

The continuous wave electron paramagnetic resonance experiment revisited

When the modulation frequency used in continuous wave electron paramagnetic resonance (cw EPR) spectroscopy exceeds the linewidth, modulation sidebands appear in the spectrum. It is shown theoretically and experimentally that these sidebands are actually multiple photon transitions, sigma(+)+kxpi, where one microwave (mw) sigma(+) photon is absorbed from the mw radiation field and an arbitrary number k of radio frequency (rf) pi photons are absorbed from or emitted to the modulation rf field. Furthermore, it is demonstrated that both the derivative shape of the lines in standard cw EPR spectra and the distortions due to overmodulation are caused by the unresolved sideband pattern of these lines. The single-photon transition does not even give a contribution to the first-harmonic cw EPR signal. Multiple photon transitions are described semiclassically in a toggling frame and their existence is proven using second quantization. With the toggling frame approach and perturbation theory an effective Hamiltonian for an arbitrary sideband transition is derived. Based on the effective Hamiltonians an expression for the steady-state density operator in the singly rotating frame is derived, completely describing all sidebands in all modulation frequency harmonics of the cw EPR signal. The relative intensities of the sidebands are found to depend in a very sensitive way on the actual rf amplitude and the saturation of single sidebands is shown to depend strongly on the effective field amplitude of the multiple photon transitions. By comparison with the analogous solutions for frequency-modulation EPR it is shown that the field-modulation and the frequency-modulation technique are not equivalent. The experimental data fully verify the theoretical predictions with respect to intensities and lineshapes.     

Effect of a phase factor on the minimum time of a quantum gate

A relationship between the phase factor of a quantum gate, the layout of energy levels of its effective Hamiltonian, and the implementation time of the gate is demonstrated. By an example of the direct and inverse quantum Fourier transforms (QFT) gates for a qutrit represented by a quadrupole nucleus with spin I = 1, as well as for a system of two qubits (I = 1/2), effective Hamiltonians and minimum implementation times corresponding to different global phases are obtained. Implementation schemes are proposed for these Hamiltonians by the nuclear magnetic resonance (NMR) technique with the use of sequences of radio-frequency (RF) pulses separated by intervals of free evolution. Analytic results for the minimum times of gates are in agreement with the results obtained by numerical optimization methods. The phase considered is divided into dynamic and geometric parts.