Magnus and Fer expansions for matrix differential equations: the convergence problem

A critique on the suitability of Fer expansion in time-evolution studies in quantum mechanics

The present report examines the utility and exactness of time-propagators derived from Fer expansion (FE). While the mathematical intricacies of the FE scheme are well established, the operational aspects of the same in time-evolution studies remain less explored and authenticated in physical problems of relevance. Through suitable examples, the operational inconsistencies observed in time-evolution studies based on the FE scheme are identified and corroborated through rigorous comparisons with simulations emerging from exact numerical methods. The limitations outlined seriously undermine the advantages associated with the FE scheme over other existing analytic methods.

VarRCWA: An Adaptive High-Order Rigorous Coupled Wave Analysis Method

Semianalytical methods, such as rigorous coupled wave analysis, have been pivotal in the numerical analysis of photonic structures. In comparison to other numerical methods, they have a much lower computational cost, especially for structures with constant cross-sectional shapes (such as metasurface units). However, when the cross-sectional shape varies even mildly (such as a taper), existing semianalytical methods suffer from high computational costs. We show that the existing methods can be viewed as a zeroth-order approximation with respect to the structure's cross-sectional variation. We derive a high-order perturbative expansion with respect to the cross-sectional variation. Based on this expansion, we propose a new semianalytical method that is fast to compute even in the presence of large cross-sectional shape variation. Furthermore, we design an algorithm that automatically discretizes the structure in a way that achieves a user-specified accuracy level while at the same time reducing the computational cost.

Heating in Integrable Time-Periodic Systems

We investigate a heating phenomenon in periodically driven integrable systems that can be mapped to free-fermion models. We find that heating to the high-temperature state, which is a typical scenario in nonintegrable systems, can also appear in integrable time-periodic systems; the amount of energy absorption rises drastically near a frequency threshold where the Floquet-Magnus expansion diverges. As the driving period increases, we also observe that the effective temperatures of the generalized Gibbs ensemble for conserved quantities go to infinity. By the use of the scaling analysis, we reveal that, in the limit of infinite system size and driving period, the steady state after a long time is equivalent to the infinite-temperature state. We obtain the asymptotic behavior L^{-1} and T^{-2} as to how the steady state approaches the infinite-temperature state as the system size L and the driving period T increase.

Controlling the Dynamics of Quadrupolar Spin-1 Nuclei by Means of Average Hamiltonian Theory When Irradiated with Modified Composite Quadrupolar Echo Pulse Sequences

The importance of the average Hamiltonian theory and its antecedent the Magnus expansion are discussed. The investigation of its convergence in different situations is very important. In this paper, we introduced a well-established approach to minimize the zeroth-order average Hamiltonian for modified composite pulse sequence in quadrupolar spectroscopy of spin-1. We designed two modified composite pulse sequences constructed by modifying the timing sequence in the original composite pulse sequences.17 We tested various configurations of times associated with the free evolution of the spin system in the modified composite pulses. We found that by decreasing the time delays between the pulses associated with the free evolution of spin system, the line shapes become increasingly better until we obtained the new modified composite pulses showing improvement of the signal compared to the original composite pulse sequences.17 This promising work is expected to play an important role not only for recording high resolution spectra of amino acids, pharmaceutical samples, and peptides but also for probing structural and dynamic information in biomolecules. The generality of the present theoretical scheme points to potential applications in solid-state NMR, to problems in chemical physics, quantum mechanics and theoretical developments, chemistry, and physical chemistry, and in interdisciplinary research areas whenever they include spin dynamics approach.

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.

EXPONENTIAL PROPAGATORS (INTEGRATORS) FOR THE TIME-DEPENDENT SCHRÖDINGER EQUATION

The time-dependent Schrödinger Equation (TDSE) is a parabolic partial differential equation (PDE) comparable to a diffusion equation but with imaginary time. Due to its first order time derivative, exponential integrators or propagators are natural methods to describe evolution in time of the TDSE, both for time-independent and time-dependent potentials. Two splitting methods based on Fer and/or Magnus expansions allow for developing unitary factorizations of exponentials with different accuracies in the time step △t. The unitary factorization of exponentials to high order accuracy depends on commutators of kinetic energy operators with potentials. Fourth-order accuracy propagators can involve negative or complex time steps, or real time steps only but with gradients of potentials, i.e. forces. Extending the propagators of TDSE's to imaginary time allows to also apply these methods to classical many-body dynamics, and quantum statistical mechanics of molecular systems.

Investigation of the Effect of Finite Pulse Errors on BABA Pulse Sequence Using Floquet-Magnus Expansion Approach

This paper presents the study of finite pulse widths for the BABA pulse sequence using the Floquet-Magnus expansion (FME) approach. In the FME scheme, the first order F1 is identical to its counterparts in average Hamiltonian theory (AHT) and Floquet theory (FT). However, the timing part in the FME approach is introduced via the Λ1 (t) function not present in other schemes. This function provides an easy way for evaluating the spin evolution during "the time in between" through the Magnus expansion of the operator connected to the timing part of the evolution. The evaluation of Λ1 (t) is useful especially for the analysis of the non-stroboscopic evolution. Here, the importance of the boundary conditions, which provides a natural choice of Λ1 (0) is ignored. This work uses the Λ1 (t) function to compare the efficiency of the BABA pulse sequence with δ - pulses and the BABA pulse sequence with finite pulses. Calculations of Λ1 (t) and F1 are presented.