Polynomially scaling spin dynamics simulation algorithm based on adaptive state-space restriction

We report progress with an old problem in magnetic resonance -- that of the exponential scaling of simulation complexity with the number of spins. It is demonstrated below that a polynomially scaling algorithm can be obtained (and accurate simulations performed for over 200 coupled spins) if the dimension of the Liouville state space is reduced by excluding unimportant and unpopulated spin states. We found the class of such states to be surprisingly wide. It actually appears that a majority of states in large spin systems are not essential in magnetic resonance simulations and can safely be dropped from the state space. In restricted state spaces the spin dynamics simulations scale polynomially. In cases of favourable interaction topologies (sparse graphs, e.g. in protein NMR) the asymptotic scaling is linear, opening the way to direct fitting of molecular structures to experimental spectra.     

Challenges in numerical simulations of solid-state NMR experiments: spin exchange pulse sequences

While simulations are essential for interpretation of solid-state NMR experiments, large spin systems involved in e.g. spin-diffusion experiments and/or dynamic effects like chemical exchange pose great challenges for the numerical simulations, where we typically want to include effects of finite pulses and other external manipulations in the simulation. In this paper, these topics are reviewed and addressed by simple numerical simulations. The numerical simulations comprise a 2-spin simulation of (2)H two-site jumps and a 332-spin simulation of a CHHC experiment for a small protein.     

Quantum parallelism as a tool for ensemble spin dynamics calculations

Efficient simulations of quantum evolutions of spin-1/2 systems are relevant for ensemble quantum computation as well as in typical NMR experiments. We propose an efficient method to calculate the dynamics of an observable provided that the initial excitation is "local." It resorts to a single entangled pure initial state built as a superposition, with random phases, of the pure elements that compose the mixture. This ensures self-averaging of any observable, drastically reducing the calculation time. The procedure is tested for two representative systems: a spin star (cluster with random long range interactions) and a spin ladder.     

Concepts of spin quantum computing

Quantum computing is based on two-state quantum systems called qubits. Recent proposals for quantum computing included nuclear spin 1/2 as qubits and implementations of several quantum algorithms were demonstrated by applying liquid-state NMR. Here I want to lay out some of the concepts of spin quantum computing including nuclear and electron spins. This article is intended to serve a dual purpose. It should on one hand introduce the reader who is not familiar with magnetic resonance to the spin quantum computing terminology and the concepts of pulsed magnetic resonance. At the same time I want to introduce the NMR expert to the recent developments in quantum computing.     

Dynamic nuclear spin resonance in n-GaAs

The dynamics of optically detected nuclear magnetic resonance is studied in n-GaAs via time-resolved Kerr rotation using an on-chip microcoil for rf field generation. Both optically allowed and optically forbidden NMR are observed with a dynamics controlled by the interplay between dynamic nuclear polarization via hyperfine interaction with optically generated spin-polarized electrons and nuclear spin depolarization due to magnetic resonance absorption. Comparing the characteristic nuclear spin relaxation rate obtained in experiment with master equation simulations, the underlying nuclear spin depolarization mechanism for each resonance is extracted.     

Polynomially scaling spin dynamics II: further state-space compression using Krylov subspace techniques and zero track elimination

We extend the recently proposed state-space restriction (SSR) technique for quantum spin dynamics simulations [Kuprov et al., J. Magn. Reson. 189 (2007) 241-250] to include on-the-fly detection and elimination of unpopulated dimensions from the system density matrix. Further improvements in spin dynamics simulation speed, frequently by several orders of magnitude, are demonstrated. The proposed zero track elimination (ZTE) procedure is computationally inexpensive, reversible, numerically stable and easy to add to any existing simulation code. We demonstrate that it belongs to the same family of Krylov subspace techniques as the well-known Lanczos basis pruning procedure. The combined SSR+ZTE algorithm is recommended for simulations of NMR, EPR and Spin Chemistry experiments on systems containing between 10 and 10(4) coupled spins.     

Improvement of duty-cycle heating compensation in NMR spin relaxation experiments

To reliably measure NMR relaxation properties of macromolecules is a prerequisite for precise experiments that identify subtle variations in relaxation rates, as required for the determination of rotational diffusion anisotropy, CSA tensor determination, advanced motional modeling or entropy difference estimations. An underlying problem with current NMR relaxation measurement protocols is maintaining constant sample temperature throughout the execution of the relaxation series especially when rapid data acquisition is required. Here, it is proposed to use a combination of a heating compensation and a proton saturation sequence at the beginning of the NMR relaxation pulse scheme. This simple extension allows reproducible, robust and rapid acquisition of NMR spin relaxation data sets. The method is verified with (15)N spin relaxation measurements for human ubiquitin.     

A complete Hermitian operator basis set for any spin quantum number.

A new Hermitian operator basis set for spins of any quantum number is presented for use in simulations of NMR experiments. The advantage with a Hermitian operator basis is that the Liouville-von Neumann equation, including relaxation with dynamic frequency shifts, is real. Real algebra makes numerical calculations faster and simplifies physical interpretation of the equation system as compared to complex algebra. The unity operator is included in the Hermitian operator basis, which makes it easy to rewrite the inhomogeneous Liouville-von Neumann equation into a homogeneous form. The unity operator also simplifies physical interpretation of the equation system for coupled spin systems.     

Spin dynamics of polarization inversion spin exchange at the magic angle in multiple spin systems

Polarization inversion spin exchange at the magic angle (PISEMA) [J. Magn. Reson. A 109, 270 (1994)] is an important experiment in NMR structural characterization of membrane proteins in oriented lipid bilayers. This paper presents a theoretical and experimental study of the spin dynamics in PISEMA to investigate the line-narrowing mechanism. The study focuses on the effect of neighboring protons on the spin exchange of a strongly coupled spin pair. The spin exchange is solved analytically for simple spin systems and is numerically simulated for many-spin systems. The results show that the dipolar couplings from the neighboring protons of a strongly coupled spin pair perturb the spin exchange only in the second order, therefore it has little contribution to the linewidth of PISEMA spectra in comparison to the separated-local-field spectra. The effects from proton resonance offset and the mismatch of the Hartmann-Hahn condition are also discussed along with experimental results using model single-crystal samples.     

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids

Multiple-quantum NMR spin dynamics of inhomogeneous one-dimensional systems in solids is investigated by analytical and numerical methods. A fermion approach for MQ spin dynamics of one-dimensional inhomogeneous systems is developed in the approximation of the dipole–dipole interactions (DDI) of nearest neighbors. It is shown that only MQ coherences of the zeroth and plus/minus second orders appear in the approximation of the DDI of the nearest neighbors even in inhomogeneous one-dimensional systems. We also investigate MQ dynamics of inhomogeneous chains numerically. Intensities of MQ NMR coherences for a linear chain consisting of 3000 spins are calculated.     

Tapping spin glasses and ferromagnets on random graphs.

We consider a tapping dynamics, analogous to that in experiments on granular media, on spin glasses and ferromagnets on random thin graphs. Between taps, zero temperature single spin flip dynamics takes the system to a metastable state. Tapping corresponds to flipping simultaneously any spin with probability p. This dynamics leads to a stationary regime with a steady state energy E(p). We analytically solve this dynamics for the one-dimensional ferromagnet and +/-J spin glass. Numerical simulations for spin glasses and ferromagnets of higher connectivity are carried out; in particular, we find a novel first order transition for the ferromagnetic systems.     

Nonlinear identification of NMR spin systems by adaptive filtering

In this paper, we present two new methods for identifying NMR spin systems. These methods are based on nonlinear adaptive filtering. The spin system is assumed to be time-invariant with memory. The first method uses a truncated discrete Volterra series to describe the nonlinear relationship between excitation (input) and system response (output). First-, second-, and third-order kernels of this series are estimated employing the least mean square (LMS) algorithm. Three parallel filters can then model the NMR spin system so that its output is no more than simple sum of three convolution products between combinations of the input signal and filters coefficients. It is also shown that the contribution of the Volterra second-order term to the total system response is neglected compared with the contributions of the first- and the third-order terms. In the second identification method, the output signal is related to the input signal through a recursive nonlinear difference equation with constant coefficients. The LMS algorithm is used again to estimate the equation coefficients. The two methods are validated with a simulated NMR system model based on Bloch equations. The results and the performances of these methods are analyzed and compared. It is shown that our methods permit a simple identification of NMR spin systems. The field of applications of this study is promising in the optimization of NMR signal detection, especially in the cases of low signal-to-noise ratios where optimum signal filtering and analysis must be performed.     

Correlation inequalities for vector spin models

Correlation inequalities forn-vector spin models (n 2) are reviewed. A relatively simple and unified derivation of the inequalities is achieved, using duplicate variable methods, for spin dimensionalitiesn=2 (plane rotator model),n=3 (classical Heisenberg model), andn=4. Although correlation inequalities are lacking forn > 4, new proofs are presented for the comparison inequalities relating correlations for systems with arbitrary spin dimensionality to corresponding correlations for systems with low spin dimensionality (n = 1 or 2).Research supported by National Science Foundation under Grant DMR 76-23071.     

Field-induced negative differential spin lifetime in silicon

We show that the electric-field-induced thermal asymmetry between the electron and lattice systems in pure silicon substantially impacts the identity of the dominant spin relaxation mechanism. Comparison of empirical results from long-distance spin transport devices with detailed Monte?Carlo simulations confirms a strong spin depolarization beyond what is expected from the standard Elliott-Yafet theory even at low temperatures. The enhanced spin-flip mechanism is attributed to phonon emission processes during which electrons are scattered between conduction band valleys that reside on different crystal axes. This leads to anomalous behavior, where (beyond a critical field) reduction of the transit time between spin-injector and spin-detector is accompanied by a counterintuitive reduction in spin polarization and an apparent negative spin lifetime.     

Entanglement assisted metrology

We propose a new approach to the measurement of a single spin state, based on nuclear magnetic resonance (NMR) techniques and inspired by the coherent control over many-body systems envisaged by quantum information processing. A single target spin is coupled via the magnetic dipolar interaction to a large ensemble of spins. Applying radio frequency pulses, we can control the evolution so that the spin ensemble reaches one of two orthogonal states whose collective properties differ depending on the state of the target spin and are easily measured. We first describe this measurement process using quantum gates; then we show how equivalent schemes can be defined in terms of the Hamiltonian and thus implemented under conditions of real control, using well established NMR techniques. We demonstrate this method with a proof of principle experiment in ensemble liquid state NMR and simulations for small spin systems.     

核磁共振中的量子控制

近年来, 随着量子信息科学的发展, 对由量子力学原理描述的微观世界的主动调控已成为重要的前沿研究领域. 为构造实际的量子信息处理器, 一个关键的挑战是: 如何对处于噪声环境下的量子体系实现一系列高精度的任意操作, 以完成目标量子信息处理任务. 为此, 人们将经典系统控制论的思想方法延伸到量子体系的领域, 提出了大量的量子控制方法以及相关的数值技术(如量子优化控制、量子反馈控制等), 并取得了丰富的研究成果. 核磁共振自旋体系具备成熟的系统理论和操控技术, 为量子控制方法的实用性研究提供了优秀的实验测试平台. 因此, 基于核磁共振的量子控制成为量子控制领域的重要方向. 本文简要介绍了量子控制的基本概念和方法; 从系统控制论的角度对核磁共振自旋体系的基本原理和重要控制任务做了阐述; 介绍了近些年来在该领域发展的相关控制方法及其应用; 对基于核磁共振体系的量子控制的进一步的研究做了几点展望.     

A simulation algorithm based on Bloch equations and product operator matrix: application to dipolar and scalar couplings

A product operator matrix is proposed to describe scalar couplings in liquid NMR. Combination of the product operator matrix and non-linear Bloch equations is employed to describe effects of chemical shift, translational diffusion, dipolar field, radiation damping, and relaxation in multiple spin systems with both scalar and dipolar couplings. A new simulation algorithm based on this approach is used to simulate NMR signals from dipolar field effects in the presence of scalar couplings. Several typical coupled spin systems with both intra-molecular scalar couplings and inter-molecular dipolar couplings are simulated. Monte Carlo methods are incorporated into simulations as well to analyze diffusion process in these complicated spin systems. The simulated results of diffusion and relaxation parameters and 2D NMR spectra are coincident with the experimental measurements, and agree with theoretical predictions as well. The simulation algorithm presented herein therefore provides a convenient means for designing pulse sequences and quantifying experimental results in complex coupled spin systems.     

Long-lived nuclear spin states in the solution NMR of four-spin systems

The existence of long-lived nuclear spin states in four-spin systems is explored by solution-state NMR experiments. Long-lived states are proved to exist in three different natural product molecules, each containing either a AA'BB' or a AA'XX' proton spin system. The measured state lifetimes are between four and eight times the spin-lattice relaxation time constants.