Time-dependent Bloch Equations Research Articles

Harnessing Electron Spin Hyperpolarization in Chromophore-Radical Spin Probes for Subcellular Resolution in Electron Paramagnetic Resonance Imaging: Concept and Feasibility.

Obtaining a subcellular resolution for biological samples doped with stable radicals at room temperature (RT) is a long-sought goal in electron paramagnetic resonance imaging (EPRI). The spatial resolution in current EPRI methods is constrained either because of low electron spin polarization at RT or the experimental limitations associated with the field gradients and the radical linewidth. Inspired by the recent demonstration of a large electron spin hyperpolarization in chromophore-nitroxyl spin probe molecules, the present work proposes a novel optically hyperpolarized EPR imaging (OH-EPRI) method, which combines the optical method of two-photon confocal microscopy for hyperpolarization generation and the rapid scan (RS) EPR method for signal detection. An important aspect of OH-EPRI is that it is not limited by the abovementioned restrictions of conventional EPRI since the large hyperpolarization in the spin probes overcomes the poor thermal spin polarization at RT, and the use of two-photon optical excitation of the chromophore naturally generates the required spatial resolution, without the need for any magnetic field gradient. Simulations based on time-dependent Bloch equations, which took into account both the RS field modulation and the hyperpolarization generation by optical means, were performed to examine the feasibility of OH-EPRI. The simulation results revealed that a spatial resolution of up to 2 fL can be achieved in OH-EPRI at RT under in vitro conditions. Notably, the majority of the requirements for an OH-EPRI experiment can be fulfilled by the currently available technologies, thereby paving the way for its easy implementation. Thus, the proposed method could potentially bridge the sensitivity gap between the optical and magnetic imaging techniques.

Distance Dependence of Electron Spin Polarization during Photophysical Quenching of Excited Naphthalene by TEMPO Radical.

Quenching of excited states by a free radical is generally studied in systems where these two are separate entities freely moving in a liquid solution. Random diffusive encounters bring them together to cause the quenching and leave the spins of the radical polarized. In the dynamics of the radical-triplet pair mechanism of the generation of electron spin polarization (ESP), the distance-dependent exchange interaction plays a crucial role. To investigate how the distance between the excited molecule and the radical influences the ESP, we have covalently linked a naphthalene moiety to a TEMPO free radical through a spacer group of three different lengths. We compared the ESP process of these linked compounds with that of the usual "unlinked system" of naphthalene and TEMPO through time-resolved EPR experiments at low temperature in n-hexane solution. The time evolution of both the linked and the "unlinked system" was treated on a similar footing. The time-dependent EPR signal was analyzed by combining photophysical kinetics and time-dependent Bloch equations incorporating spin dynamics. Sequential quenching of the singlet state and the triplet state of naphthalene was seen in all the systems, as revealed through the spin-polarized TREPR spectra of opposite phase. The magnitudes of the ESP in the linked molecules were higher than those of the "unlinked system," showing that when the two moieties are held together greater mixing of quartet-doublet states takes place. The magnitudes of ESP steadily decrease with increasing the length of the spacer group. The polarization magnitudes due to triplet quenching and singlet quenching are very similar, differing by a factor of only ∼2. These characteristics show that for all the linked molecules the quenching takes place in the "weak exchange" regime and at almost the same distance of separation between the two moieties. Our results also showed that observation of small absorptive TREPR signals does not necessarily imply that its magnitude of polarization is small.

Ultrafast optical switching in quantum dot-metallic nanoparticle hybrid systems.

We study ultrafast excitonic population inversion resulting from the interaction of a semiconductor quantum dot (SQD) with localized surface plasmons. The plasmonic enhanced fields are generated when a metallic nanoparticle (MNP) is subject to a nonlinear chirped few-cycle pulse train. By numerically solving the time-dependent Bloch equations beyond the rotating-wave approximation, we show that the complete population inversion can be achieved for small interparticle distance and the dynamic in population inversion exhibits a steplike transition between absorption and amplifying. This phenomenon can be exploited as an all-optical ultrafast switching device. Moreover, the final state of population inversion is shown to be modified significantly with the interparticle distances, which is not only robust against the variation of probe pulse parameters but also suggests a straightforward method for measuring the interparticle distances via probing the final populations.

A theoretical and numerical consideration of the longitudinal and transverse relaxations in the rotating frame

We previously derived a simple equation for solving time-dependent Bloch equations by a matrix operation. The purpose of this study was to present a theoretical and numerical consideration of the longitudinal (R1ρ=1/T1ρ) and transverse relaxation rates in the rotating frame (R2ρ=1/T2ρ), based on this method. First, we derived an equation describing the time evolution of the magnetization vector (M(t)) by expanding the matrix exponential into the eigenvalues and the corresponding eigenvectors using diagonalization. Second, we obtained the longitudinal magnetization vector in the rotating frame (M1ρ(t)) by taking the inner product of M(t) and the eigenvector with the smallest eigenvalue in modulus, and then we obtained the transverse magnetization vector in the rotating frame (M2ρ(t)) by subtracting M1ρ(t) from M(t). For comparison, we also computed the spin-locked magnetization vector. We derived the exact solutions for R1ρ and R2ρ from the eigenvalues, and compared them with those obtained numerically from M1ρ(t) and M2ρ(t), respectively. There was excellent agreement between them. From the exact solutions for R1ρ and R2ρ, R2ρ was found to be given by R2ρ=(2R2+R1)/2−R1ρ/2, where R1 and R2 denote the conventional longitudinal and transverse relaxation rates, respectively. We also derived M1ρ(t) and M2ρ(t) for bulk water protons, in which the effect of chemical exchange was taken into account using a 2-pool chemical exchange model, and we compared the R1ρ and R2ρ values obtained from the eigenvalues and those obtained numerically from M1ρ(t) and M2ρ(t). There was also excellent agreement between them. In conclusion, this study will be useful for better understanding of the longitudinal and transverse relaxations in the rotating frame and for analyzing the contrast mechanisms in T1ρ- and T2ρ-weighted MRI.

Behavior of the magnetization in spin-locking magnetic resonance imaging using numerical solutions to the time-dependent Bloch equations

We present a simple method for calculating the magnetization in spin-locking (SL) magnetic resonance imaging (MRI), in which a simple matrix equation was derived for solving the time-dependent Bloch equations in the 2-pool chemical exchange model. We also present a method for visualizing the trajectory of a magnetization vector in a three-dimensional (3D) space. The longitudinal relaxation time in the rotating frame (T1ρ) was calculated by fitting the z component of magnetization for a duration of SL (tSL) (Mz(tSL)) to Mz(tSL) = (M0 − Mzss)exp ( − tSL/T1ρ) + Mzss, where M0 and Mzss denote the thermal equilibrium and steady-state z component of magnetization, respectively, and was compared with that calculated from the solution given by Trott and Palmer. Our 3D plots clearly visualized the effect of SL. When the population of the two pools was highly asymmetric, there was good agreement between the T1ρ values obtained by our method and Trott and Palmer's solutions. The difference between them increased with decreasing asymmetry in the population of the two pools. Our method will be useful for better understanding and optimization of SL MRI, because it allows us to calculate the magnetization vector and to visualize its trajectory simply and quickly.

Generalized equation for describing the magnetization in spoiled gradient-echo imaging

The purpose of this study was to demonstrate a generalized equation for describing the magnetization in spoiled gradient-echo (SPGR) imaging in which the in-pulse relaxation and magnetization transfer (MT) effects are taken into account. First, the time-dependent Bloch equations for the two-pool exchange model with MT effect were reduced to an inhomogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. Second, the equations describing the magnetization before and after the radiofrequency (RF) pulse were derived based on the above solution for the RF-pulse excitation and evolution phases. Finally, a generalized equation describing the steady-state magnetization was derived. The validity of this equation was investigated by comparing with the transverse magnetization obtained by the regular Ernst equation and analytical solution in which the in-pulse transverse relaxation is considered. When the same assumption was made in our method, there were good agreements between them, indicating the validity of our method. The in-pulse transverse and longitudinal relaxations decreased the transverse magnetization compared to the case in which these effects were neglected, whereas MT increased it. In conclusion, we derived a generalized equation for describing the magnetization in SPGR imaging. This equation will provide a suitable basis for understanding the signal intensity in SPGR imaging and/or T 1 measurement using an SPGR sequence in cases in which the effect of in-pulse relaxation and/or MT cannot be neglected.

An all-optical scalar and vector spin-exchange relaxation-free magnetometer employing on–off pump modulation

It is demonstrated that a spin-exchange relaxation-free (SERF) atomic magnetometer can be used for scalar measurements with no additional hardware. Because of relaxation processes, an ensemble of alkali atoms needs a constant supply of polarized photons by a pump beam to maintain a polarized state. If the pump beam is shuttered off, the system decays to its equilibrium state. For a low enough relaxation rate and with a magnetic field present, the system will exhibit oscillations at its natural frequencies. In a SERF magnetometer, it happens at the Zeeman resonance frequency of the atoms (Larmor frequency). Thus, shuttering off the pump beam reveals oscillations at the Larmor frequency. From this frequency, one can deduce the scalar value of the applied magnetic field. As a result, all-optical scalar measurements can be performed. At the same time, either one or two vector components of the applied field can be measured by using one or two orthogonal probe beams, respectively. In a low-polarization SERF regime, the ground state can be well described by the Bloch equations for the electron spin polarization. By solving the time-dependent Bloch equations [neglecting the diffusion term and assuming that the nuclear slowing-down factor q(P) is constant], the oscillation frequency of the system is obtained. From this frequency, the scalar value of the applied magnetic field is derived. It is shown that applied fields down to 1 nT can be measured with a 0.1% relative uncertainty. Fields down to 50 pT can be measured with a 10% relative uncertainty. The time dependence acquired in the “off” periods is strongly correlated with the Zeeman sublevels population of the atomic ground state and reveals its spin dynamics.

Numerical solutions to the time-dependent Bloch equations revisited

The purpose of this study was to demonstrate a simple and fast method for solving the time-dependent Bloch equations. First, the time-dependent Bloch equations were reduced to a homogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. The validity of this method was investigated by comparing with the analytical solutions in the case of constant radiofrequency irradiation. There was a good agreement between them, indicating the validity of this method. As a further example, this method was applied to the time-dependent Bloch equations in the two-pool exchange model for chemical exchange saturation transfer (CEST) or amide proton transfer (APT) magnetic resonance imaging (MRI), and the Z-spectra and asymmetry spectra were calculated from their solutions. They were also calculated using the fourth/fifth-order Runge-Kutta-Fehlberg (RKF) method for comparison. There was also a good agreement between them, and this method was much faster than the RKF method. In conclusion, this method will be useful for analyzing the complex CEST or APT contrast mechanism and/or investigating the optimal conditions for CEST or APT MRI.

Understanding pulsed magnetization transfer.

Using a two-pool exchange model of magnetization transfer (MT), numeric simulations were developed to predict the time dependence of longitudinal magnetization in both semisolid and liquid pools for arbitrary pulsed radiofrequency (RF) irradiation. Whereas RF excitation of the liquid pool was modeled using the time-dependent Bloch equations, RF saturation of the semisolid pool was described by a time-dependent rate proportional to both the absorption lineshape of the semisolid pool and the square of the RF pulse amplitude. Simulations show good agreement with experimental results for a 4% agar gel aqueous system in which the two-pool kinetics have been well studied previously. These simulations provide a method for interpreting pulsed MT effects, are easily extended to biologic tissues, and provide a basis for optimizing clinical imaging applications that exploit MT contrast.

Correlations between successively emitted photons in resonance fluorescence

We examine the relationship between the probability of detection of successively emitted photons and the second-order coherence correlation function for resonance fluorescence. A simple method for computing both in terms of dressed atom probability amplitudes and steady state populations instead of time-dependent Bloch equations is developed.