Coupled Bloch Equations Research Articles

The application of the distributed-order time fractional Bloch model to magnetic resonance imaging

It is now well known that the magnetic resonance imaging (MRI) signal decay deviates from the classical mono-exponential relaxation. This deviation is referred to in the literature as anomalous relaxation. The modelling of this anomalous relaxation can provide a better understanding of MRI magnetization. The purpose of this work is to investigate the utility of the distributed-order time fractional Bloch equations to describe anomalous relaxation processes in human brain MRI data. Two choices of continuous distribution weight functions, which are parameterised by their mean μ and standard deviation σ, are studied to investigate their impact on the model solution behaviour. An implicit numerical method implemented on a graded mesh is proposed to solve the model and the stability and convergence analysis are presented. We also derive semi-analytical solutions of the fully coupled Bloch equations using the Laplace transform technique to assess the accuracy of the numerical scheme. Furthermore, three different voxel models of continuous distribution weight functions, namely a single continuous probability distribution (model 1), two distinct continuous probability distributions (model 2) and a mixture of two continuous probability distributions (model 3), are applied to the in vivo human brain MRI data, and a feasible and reliable parameter estimation method based on a modified hybrid Nelder-Mead simplex search and particle swarm optimization is presented to perform the voxel-level temporal fitting of the MRI data. The application of these distributed-order time fractional Bloch models highlights the validity of the proposed models, and based on the mean square error we conclude that models 2 and 3 might be more suitable than model 1 to characterize anomalous relaxation processes in human brain MRI data.

Response dynamics of an alkali-metal–noble-gas hybrid trispin system

With numerical calculation of coupled Bloch equations, we have simulated the spin dynamics of nuclear magnetic resonance gyroscope based on alkali metal-noble gas hybrid trispin system. From the perspective of damping harmonic oscillator, a thorough analysis of the response dynamics is demonstrated. The simulation results shows a linear increasing response of gyroscope signal while the noblge gas nuclear spin magnetization and alkali atomic spin lifetime parameters are at the over damping condition. An upper limit of response is imposed on the NMR gyroscope signal due to the inherent dynamics of the hybrid trispin system. The results agrees with present available experimental results and provide useful guidings for future experiments.

Magic density in a self-rephasing ensemble of trapped ultracold atoms

We investigate the collective spin dynamics of a self-rephasing bosonic ensemble of $^{87}$Rb trapped in a 1D vertical optical lattice. We show that the combination of the frequency shifts induced by atomic interactions and inhomogeneous dephasing, together with the spin self-rephasing mechanism leads to the existence of a `magic density': \textit{i.e} a singular operating point where the clock transition is first-order insensitive to density fluctuations. This feature is very appealing for improving the stability of quantum sensors based on trapped pseudo-spin-1/2 ensembles. Ramsey spectroscopy of the $|F=1,m_{F}=0\rangle\rightarrow|F=2,m_{F}=0\rangle$ hyperfine transition is in qualitative agreement with a numerical model based on coupled Bloch equations of motion for energy dependent spin vectors.

Towards an analytic solution for pulsed CEST.

Chemical exchange saturation transfer (CEST) is an imaging method based on magnetization exchange between solutes and water. This exchange generates changes in the measured signal after off-resonance radiofrequency irradiation. Although the analytic solution for CEST with continuous wave (CW) irradiation has been determined, most studies are performed using pulsed irradiation. In this work, we derive an analytic solution for the CEST signal after pulsed irradiation that includes both short-time rotation effects and long-time saturation effects in a two-pool system corresponding to water and a low-concentration exchanging solute pool. Several approximations are made to balance the accuracy and simplicity of the resulting analytic form, which is tested against numerical solutions of the coupled Bloch equations and is found to be largely accurate for amides at high fields, but less accurate at the higher exchange rates, lower offsets and typically higher irradiation powers of amines.

Low frequency magnetic field suppression in an atomic spin co-magnetometer with a large electron magnetic field

In a K-Rb-21Ne co-magnetometer, the Rb electron magnetic field which is experienced by the nuclear spin is about 100 times larger than that of the K in a K-3He co-magnetometer. The large electron magnetic field which is neglected in the K-3He co-magnetometer coupled Bloch equations model is considered here in the K-Rb-21Ne co-magnetometer to study the low frequency magnetic field suppression effect. Theoretical analysis and experimental results shows that in the K-Rb-21Ne spin co-magnetometer, not only the nuclear spin but also the large electron spin magnetic field compensate the external magnetic field noise. By comparison, only the 3He nuclear spins mainly compensate the external magnetic field noise in a K-3He co-magnetometer. With this study, in addition to just increasing the magnetic field of the nuclear spins, we can suppress the magnetic field noise by increasing the density of the electron spin. We also studied how the magnetic field suppression effect relates to the scale factor of the K-Rb-21Ne co-magnetometer and we compared the scale factor with that of the K-3He co-magnetometer. Lastly, we show the sensitivity of our co-magnetometer. The magnetic field noise, the air density fluctuation noise and pumping power optimization are studied to improve the sensitivity of the co-magnetometer.

Drift Error Analysis of Atomic Spin Gyroscope

With the rapid development of quantum manipulation technique, atomic spin gyroscope has been demonstrated in recent years. Understanding the sources and the influence of error on the performance of atomic spin gyroscope is important during the design of such devices. The basic principle of atomic spin gyroscope was reviewed and the analysis of error sources presented began with the coupled Bloch equations describing the dynamics of atomic spin gyroscope including magnetic field, light shift and pump-probe non-orthogonality. It was shown analytically that the first order effects of the transverse magnetic fields and imperfections can be eliminated by turning the longitudinal magnetic field on the magnetic field compensation point. The effects of longitudinal magnetic field and light shift imperfections with other terms zeroed contributed only to scale factor of the atomic spin gyroscope. Furthermore, the effects magneto-optical crystal and magnetic shielding due to temperature drift were described and according approaches to suppress these errors were also proposed.

Characterizing healthy and diseased white matter using quantitative magnetization transfer and multicomponent T2 relaxometry: A unified view via a four‐pool model

Nuclear magnetic resonance relaxation and magnetization transfer in cerebral white matter can be described using a four-pool model: two for water protons (in separate myelin and intra/extracellular compartments) and two for protons associated with the lipids and proteins of biologic membranes (of myelin and nonmyelin semisolids). This model was used to gain insight into the observations from multicomponent quantitative T(2) relaxometry and quantitative magnetization transfer imaging, both based on simplified white matter models and experimentally feasible in vivo. Using a set of coupled Bloch equations describing the behavior of the magnetization in a four-pool model of white matter, simulations of the quantitative T(2) relaxometry and quantitative magnetization transfer imaging techniques were performed. Pathology-inspired modifications were made to the four-pool model to gauge their impact on quantitative T(2) relaxometry and quantitative magnetization transfer imaging observations. Our results show that changes in the rate of water movement between microanatomic compartments may impact otherwise stable quantitative T(2) relaxometry observations; that the measure of the quantitative magnetization transfer imaging-based semisolid pool population is robust, despite the presence of two distinct semisolid components; and that quantitative magnetization transfer imaging compartment size estimates are not influenced by changes in the T(2) of the intra/extracellular water pool. The four-pool model, while impractical for in vivo characterization, yields important insight into the interpretation of changes observed with these quantitative MRI methods based on simplified models of white matter.

Microscopic modeling of the optical properties of semiconductor nanostructures

A brief overview of a consistent microscopic approach to model the optical and electronic properties of semiconductor nanostructures is presented. Coupled semiconductor Bloch and Maxwell equations are used to investigate the performance of semiconductor microcavity structures, photonic band gap systems, and lasers. The predictive potential of the microscopic theory is demonstrated for several examples of practical importance. Optical gain and output characteristics are computed for modern vertical external cavity surface emitting laser structures. It is shown how design flexibilities can be used to optimize the device performance. Nanostructures are proposed where semiconductor quantum wells are embedded in one-dimensional photonic crystals. For field modes spectrally below the photonic band edge it is shown that the optical gain and absorption can be enhanced by more than one order of magnitude over the value of the homogeneous medium. The increased gain can be used for laser action by placing quantum wells and a suitably designed photonic crystal structure inside a microcavity.

Analysis of transient interband light modulation by ultrashort intersubband resonant light pulses in semiconductor quantum wells

We explore the transient characteristics of an interband resonant light modulation process by ultrashort intersubband resonant light pulses in semiconductor quantum wells (QW's). The modulation characteristics in a three-level semiconductor QW system, including the effects due to in-plane momentum, have been investigated by a numerical analysis of the coupled Bloch equations using a density-matrix approach. We have studied the effect of the carrier density and the nature of the intersubband coupling light on the transient absorption features of the interband light. The modulation process has been compared in doped and undoped QW's. The switching behavior of the strong interband light field in presence of a train of intersubband light pulses has also been discussed.

Nonlinear coupling and radiation damping in oscillator-detected magnetic resonance of single spins

We consider the effects of close coupling between magnetic spin systems and oscillator detectors. The scaling of present experiments using both mechanical and electrical oscillator detectors for single-spin detection is examined. We show how nonlinear interactions ignored in previous descriptions may dominate the detector response in single-spin schemes using mechanical detectors. From a generalized form of the coupled Bloch equations, the difference between mechanical and electrical oscillator detection can be readily appreciated. Numerical solution of the coupled system using reasonable experimental parameters shows that complex dynamics may be observed in the single-spin detection scheme when the detector and the spin system are strongly coupled. The interpretation of spin system data obtained from the motion of a mechanical oscillator will require understanding of these detector effects. Chaotic regimes are predicted in extreme cases. These results help both in identifying realistic limits of sensitivity for mechanical oscillator-detected magnetic resonance and in designing experimental methods to realize those limits.

Pulsed magnetization transfer contrast in gradient echo imaging: a two-pool analytic description of signal response.

Magnetization transfer (MT) imaging with a rapid gradient-echo sequence and pulsed saturation provides an efficient means of acquiring high resolution three-dimensional data in vivo. This paper presents a derivation of the theoretical steady-state signal equation for this sequence based on the two-site coupled Bloch equations. Numerical simulations are used to validate the derived expression and experiments are performed on an agar gel model and normal brain. Experimental agar data indicate that direct saturation of the liquid component can be a major source of signal attenuation whereas MT normally dominates in brain tissue. The signal equation presented here establishes the necessary theory for sequence design and optimization and provides insight into model parameters and experimental results.

Oscillations, Stability, and Equilibrium in Magnetic Exchange Networks

Abstract The generalized N-site magnetic exchange network is described by coupled Bloch equations whose solution is in general stable and characterized by eigenvalues with negative real parts. The analysis of this system is greatly simplified when detailed balance is satisfied, being described by an exchange matrix with eigenvalues that are negative and purely real. Consequently, the multiple-site system in detailed balance approaches its equilibrium condition by purely exponential relaxation. In addition, satisfaction of detailed balance requires that the stationary solution of the Bloch equations describing the longitudinal magnetization of a spin be the equilibrium magnetization of that spin. Violation of detailed balance with cyclic exchange results in the relaxation of the system taking the form of a damped oscillatory function, with the amount of damping generally decreasing as the number of sites increases.

Transient Decay of Longitudinal Magnetization in Heterogeneous Spin Systems Under Selective Saturation. III. Solution by Projection Operators

A general method of solving equations governing the transient response of heterogeneous spin systems to continuous and pulsed RF saturation using the projection-operator technique is outlined. An effective rate equation is derived in which there is natural separation of relaxation and source terms. Our previously derived implicit solution to the coupled Bloch equations is reformulated using this more general approach, which avoids some of the restrictive assumptions made earlier. With this formulation, we provide an analytical and explicit solution to the full set of coupled Bloch equations with no additional assumptions or restrictive conditions attached limiting their general validity. The results obtained with this new method of solution are compared with the results previously obtained by the Laplace-transform technique for heat-denatured albumin.

Lineshape of magnetization transfer via cross relaxation

Magnetization transfer, a well-known technique in NMR ( 1) , has recently become of interest in magnetic resonance imaging. This is largely due to the discovery by Wolf and Balaban that a new form of tissue-specific contrast, which they refer to as magnetization-transfer contrast (MTC), can be generated with the technique (2). While the detailed mechanism for MTC is an area of active research, current consensus is that the contrast is generated through cross relaxation between protons in the mobile water and immobile protons in macromolecules (3). Magnetization transfer is usually observed by RF irradiation at an offset frequency which results in the partial saturation of immobile protons in macromolecules and minimal energy absorption by the mobile water. Subsequently, the cross relaxation between the mobile and immobile protons results in a reduction of the longitudinal magnetization 44, on the mobile water protons. Although For&n and Hofhnan (I ) and Mann (4) derived formulas relating the mobile water M, reduction to the crossrelaxation rate constant in the 1960s and 1970s respectively, their formulas do not describe the functional dependence of magnetization transfer on offset frequency and RF irradiation power. Experimental results indicate that the magnitude of the magnetization-transfer effect can be controlled by these factors (2). A theoretical understanding of the problem is of fundamental importance to applications of the technique and should be especially useful in MRI applications. Recently, Grad and Bryant derived an expression for the cross-relaxation spectroscopy lineshape, from which one can determine the effect of offset frequency and RF irradiation power on magnetization transfer (5). However, they employed a truncated set of the coupled Bloch equations, and their results are valid only if the transverse magnetization of mobile water spins is negligible. This assumption may not be met in MRI experiments where the offset frequency is relatively small or where the RF irradiation power is large. In these cases Grad and Bryant’s lineshape expression will overestimate the degree of magnetization transfer. In the present work, a complete set of coupled Bloch equations is used to obtain the cross-relaxation spectroscopy lineshape. The result is more general than Grad and Bryant’s and reduces to their expression when the transverse magnetization of mobile water spins is negligible. The coupled Bloch equations for two cross-relaxed spins (e.g., via the dipole-dipole coupling) are

Effective g-Factor for Two Strongly Exchange-Coupled Magnetization Vectors

A magnetic resonance without recourse to the linear response theory (as first formulated for the dynamics of nuclear induction by Bloch) can be characterized in terms of a limit cycle whose period is equal to the inverse of the Larmor frequency. This fact is used to investigate the effective g·factor for a system of two strongly coupled, macroscopic magnetizations (generally, with unequal g·factors) through an isotropic, spatially uniform (expressible as a mean field) exchange interaction. The result rigorously confirms Bloembergen and Rowlands' prediction, and is verified explicitly by a computer experiment. In a previous article (Hasegawa and Stew are») a resolution has been made of a question about the 'destination vector' in a set of coupled Bloch equations for two macroscopic magnetization vectors. Interacting with each other by an isotropic exchange coupling and also with external magnetic fields, the magnetization vectors, if denoted by MA,B, are assumed to obey (1)

Transformation Properties of Coupled Bloch Equations

The problem of the ‘destination vector’ in a pair of Bloch equations for two exchange-coupled magnetizations is reinvestigated. It is shown that a standard form (the one familiar in spin resonance in metals) with can be transformed equivalently into another pair with the same structure of damping but with the destination vectors such that each magnetization relaxes towards its equilibrium value determined only by the external field (i.e.,Ms′=χs′H and Md′=χd′H, where the prime indicates the transformed frame). A complete list of the relations between the coefficients of the standard and transformed pairs is given.

Transmission-electron spin resonance in dilute CuFe alloys

Alloys of Cu with nominally 5- and 10-at. ppm Fe were studied by transmission-electron spin resonance (TESR) in the temperature range 2-40 K. The samples were strained in order to remove background signals which hamper analysis of TESR in relatively pure metal films. These measurements represent the first reported observation of electron spin resonance in a dilute magnetic alloy below its Kondo temperature. The relaxation rate of these alloys roughly scales with Fe concentration and exhibits a clear minimum at approximately 25 K. The contribution of the Fe impurities to the relaxation rate of the alloys is (7.7 \ifmmode\pm\else\textpm\fi{} 0.9) \ifmmode\times\else\texttimes\fi{} ${10}^{8}$/sec. at. ppm at 25K. The $g$ value of the alloys did not deviate from the $g$ value of pure Cu within the experimental scatter. We analyze our results in terms of coupled Bloch equations and discuss possible mechanisms for the inferred local-moment-lattice and conduction-electron-lattice relaxation rates.

Rotational diffusion studied by passage saturation transfer electron paramagnetic resonance

A comprehensive description is given of instrumental and theoretical methods employed to make accurate measurements of rotational correlation times using passage saturation transfer electron paramagnetic resonance (ST–EPR). Saturation transfer methods extend by several orders of magnitude the sensitivity of EPR to very slow motion; for example, for nitroxide spin labels, correlation times as long as 10−3 sec become accessible to measurement. Two ST–EPR detection schemes are discussed in detail: dispersion, detected 90° out of phase with respect to the 100 kHz field modulation, and absorption, detected 90° out of phase with respect to the second harmonic of the 50 kHz field modulation. The sensitivities of these configurations are illustrated with experimental spectra obtained from a system obeying isotropic Brownian rotational diffusion; namely, maleimide spin labeled human oxyhemoglobin in aqueous glycerol solutions. Two theoretical approaches, one employing coupled Bloch equations and the other utilizing the stochastic Liouville equation for the density matrix with the orientation variables treated by transition rate matrix or orthogonal eigenfunction expansion methods, are in excellent agreement with each other and with model system spectra. Both experimental and theoretical spectra depend on a number of relaxation processes other than rotational diffusion; consequently, considerable care must be taken to ensure the accuracy of measured rotational correlation times. Although the absorption method is currently the more sensitive and convenient one to apply with most conventional (commercial) spectrometers, the dispersion ST–EPR method is potentially more powerful, providing strong motivation for future technological efforts to decrease noise levels in dispersion experiments.

Theory of spin resonance in dilute magnetic alloys. II

The previous derivation by two of the authors of coupled Bloch equations appropriate to conduction-electron spin resonance is extended to the case where the conduction electrons and localized spins have different $g$ values. As before, this derivation is carried out to second order in $J$, the exchange interaction between the two species. The derivation is also carried out in the presence of the hyperfine interaction between the nuclear spins and the electon spins of the localized impurities. Finally the form of the coupled equations when there exists "direct relaxation" is discussed and some rather unexpected features are found.

Electron spin resonance in metals with magnetic impurities

Starting with s-d interaction and an impurity spin-orbit interaction inducing electron-lattice relaxation, the coupled Bloch equations describing the motion of the magnetization density in dilute magnetic alloys are derived. Temperature Green function methods and Abrikosov's pseudofermion representation for spin are used. Vertex equations, similar to those of Holstein (1964) are obtained, but the treatment is to a higher degree of accuracy. The frequency dependence of self energies and vertex equations is investigated. This is shown to be necessary to obtain results consistently to first order in concentration and second order in s-d interaction (Kondo term). The instantaneous local effective fields the magnetizations relax to are expressed as integrals over certain vertex functions. They are found to be different from the effective fields driving the magnetizations, a result which was first derived by Langreth and Wilkins (1972) using a different method.