Magnetic Field Configuration Research Articles

Self-similar Solutions of Oscillatory Reconnection: Parameter Study of Magnetic Field Strength and Background Temperature

Oscillatory reconnection is a specific type of time-dependent reconnection which involves periodic changes in the magnetic topology of a null point. The mechanism has been reported for a variety of magnetic field strengths and configurations, background temperatures, and densities. All these studies report an oscillation in the current density at the null point, but also report a variety of periods, amplitudes, and overall behaviors. We conduct a parametric study for equilibrium magnetic field strength and initial background temperature, solving two-dimensional resistive magnetohydrodynamic equations around a magnetic X-point. We introduce a parameter space for the ratio of internal to magnetic energy and find self-similar solutions for simulations where this ratio is below 0.1 (which represents a magnetically dominated environment or, equivalently, a low-beta plasma). Self-similarity can be seen in oscillations in the current density at the null (including amplitude and period), ohmic heating, and the temperature generated via reconnection jets. The parameter space of energy ratios also allows us to contextualize previous studies of the oscillatory reconnection mechanism and bring those different studies together into a single unified understanding.

Analysis and control of Hall effect thruster using optical emission spectroscopy and artificial neural network

This study presents a proof-of-principle for using optical emission spectroscopy and artificial neural networks for real-time monitoring and control of the operational parameters of a Hall effect thruster: the anode voltage, the anode xenon injection, the discharge current, and the coil current. In that regard, we build an optical database of 26 spectral lines across 6469 operating conditions to train and test the neural network. We then reduced the learning lines from 26 to 15 based on their statistical correlation with the target parameters. After tuning the hyperparameters of the network, the network predicted the thruster’s parameters with notable accuracies: 95% for the anode voltage, 84% for the coil current, and 99% for both the anode flow rate and the discharge current. The estimated uncertainty of predictions, at 3σ, is ±51V for voltage, ±1A for coil current, ±0.15A for discharge current, and ±0.15mgs−1 for anode flow rate. The prediction calculations were within milliseconds and enabled real-time monitoring of the thruster parameters. Therefore, a proportional-integrator-derivative controller (PID) controller was implemented to regulate the anode voltage and flow rate based on the optical emission of the plume. The PID showcased short settling times from 0.1 to 0.4 s and overshoot levels up to 3% of the target value for the voltage and 10% of the target value for the flow rate. These results were for a fixed coil current at 4A. The study showed that changing the coil current may necessitate more sophisticated prediction models and control strategies. Future work will expand the model’s generalizability to different thruster types, propellants, and magnetic field configurations.

Observation of super-Alfvénic slippage of reconnecting magnetic field lines on the Sun

AbstractSlipping motions of magnetic field lines are a distinct signature of three-dimensional magnetic reconnection, a fundamental process driving solar and stellar flares. While being a key prediction of numerical experiments, the rapid super-Alfvénic field line slippage driven by the ‘slip-running’ reconnection has remained elusive in previous observations. New frontiers into exploring transient flare phenomena were introduced by recently designed high cadence observing programs of the Interface Region Imaging Spectrograph (IRIS). By exploiting high temporal resolution imagery (~2 s) of IRIS, here we reveal slipping motions of flare kernels at speeds reaching thousands of kilometres per second. The fast kernel motions are direct evidence of slip-running reconnection in quasi-separatrix layers, regions where magnetic field strongly changes its connectivity. Our results provide observational proof of theoretical predictions unaddressed for nearly two decades and extend the range of magnetic field configurations where reconnection-related phenomena can occur.

First experimental confirmation of island SOL geometry effects in a high radiation regime on W7-X

Abstract This work characterizes the detachment behavior and radiation characteristics of the low
iota configuration in the Wendelstein 7-X (W7-X) stellarator. The island scrape-off layer (SOL) of the
low iota has a poloidal mode number of 6 islands surrounding the last closed flux surface (LCFS).
The island geometry of the low iota configuration is significantly different to that of the standard
magnetic field configuration, whose detachment characteristics have already been described in previous
work[2, 3, 4]. Experimental results show that the radiation pattern in the low iota configuration
is starkly different to that of the standard magnetic field configuration, with radiation concentrated
at the island SOL O-points, rather than the X-points. Additionally, this O-point localized radiation
pattern is associated with unstable detachment, with both radiation oscillations in experiments and the
lack of a self-consistent plasma solution at high radiated power fraction in EMC3-Eirene simulations.
EMC3-Eirene simulations are used to understand the radiation distribution. It was found that the O-
point localized radiation arises first from local impurity accumulation near the parallel flow stagnation,
which is located close to the geometrical center of the island (”O-point”). The local cooling in this
region leads to plasma condensation in the islands in closest magnetic connection to the divertor target
plates. The heat source to this region of the island, which is thermally isolated from the upstream heat
source in terms of parallel transport, must arise via perpendicular heat transport. This heat source is
expected to be large for the low iota configuration due to its very small internal island field line pitch.
This work highlights the importance (complementary to previous work, e. g. [5, 6]) of the internal
island field line pitch not only on the radiation pattern, but also the detachment performance of the
island divertor.

MHD simulations of astrophysical and laboratory jets under different magnetic field configurations

This paper presents the results of MHD simulations of astrophysical and laboratory supersonic jets under a superposition of poloidal (Br, Bz) and toroidal (Bϕ) magnetic fields. It is shown that the escaping matter is quickly collimated by the magnetic field. A shock wave of an elongated shape is formed, which moves from the target to the boundary of the chamber, leaving behind a stable flow. A periodic shock wave structure is observed inside the main conical expanding shock wave. It is shown that the toroidal component of the magnetic field remains in the region throughout the entire calculation and plays a role in the collimation of the flow. The poloidal magnetic field decreases in the region of the jet cone, but remains in the simulation region throughout the calculation and also participates in flow collimation. Thus, both components Bz and Bϕ take part in the collimation of the flow by the magnetic field. The width of the jet and the opening angle of the cone Ɵ depend on the magnitude of the magnetic field induction. As the field increases, the jet becomes narrower and the cone angle decreases. Initially, we do not specify the rotation of the jet Ω. However, due to the presence of the Bϕ field, the substance acquires angular velocity and twists along the z axis. The simulation results are in agreement with laboratory jets arising in the experiment at the Neodymium laser installation, and with the previously obtained results of MHD modeling of jet formation separately, in poloidal or toroidal magnetic field.

Energetics of Buchdahl stars and the magnetic Penrose process

Buchdahl star is the most compact object without an event horizon and is an excellent candidate for a black hole mimicker. Unlike black holes, rotating Buchdahl star can be over-extremal with respect to the black hole, sustaining a larger spin. We show that it can also develop an ergosphere above the threshold spin β>1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta > 1/\\sqrt{2}$$\\end{document}, which allows extraction of its rotational energy. Electromagnetic field around Buchdahl star is also expected to differ from that of black hole in both strength and topology. In this paper, we explore the energetics of Buchdahl star focusing on the magnetic Penrose process in the two magnetic field configurations, i.e., uniform and dipole. Below the threshold spin, Buchdahl star is expected to be quiet, while above the threshold it can be much more efficient than the black hole if a dipolar magnetic field is developed on its surface.

Simulation of Interaction Between Proton‐Electron Plasma Beam and Arched Magnetic Field

ABSTRACTThe present study investigates the interaction between a proton‐electron plasma beam and an arched magnetic trap using the particle‐in‐cell (PIC) simulation method. The results show that when the magnetic field is sufficiently weak, the beam disrupts the arched magnetic field configuration, causing the breaking and reconnection of magnetic field lines. The relationships between the coil current threshold (the coil current at which the disruption of the magnetic induction just happens) and initial plasma velocity, and between the critical beam velocity (the plasma is just enough to reach the arched magnetic field) and coil current, are quantified. Additionally, the interaction between the beam and the arched magnetic field is observed to induce microwave excitation. These findings lay the foundation for theoretically simulating the dynamic behavior of space plasma in a unique magnetic field environment within a laboratory setting.

Toroidal distribution of heat load on castellated plasma facing components for lower divertor target in EAST

In tokamak devices, the performance of plasma facing components (PFCs) under high heat loads is a key challenge for achieving long-pulse and high-performance plasma operations. The lower divertor of EAST adopts an ITER-like cassette modules (CMs) structure. However, regular distribution of damage along the toroidal direction have been observed on the divertor target plates, which not only affects the lifespan of PFCs but also limits the long-pulse and high-performance plasma operations. Therefore, investigating the toroidal distribution of the heat load on the divertor target in EAST is crucial for understanding the mechanism of damage on the target plate surface. The toroidal characteristics of heat loads on CMs of the lower divertor in EAST was analyzed using a three-dimensional magnetic field line tracing program (PFCFlux). For the inner vertical target (IVT) and outer vertical target (OVT) within a single CM, the heat load varies monotonically along the toroidal direction. In contrast, the heat load distribution on the outer horizontal target (OHT) is uniform, except at the chamfers, which is directly correlated with the target’s geometric structure and the magnetic field configuration. The surface heat load distribution exhibits toroidal non-uniformity, with the heat deposition pattern being generally consistent and displaying periodic variations. The toroidally uneven heat load distribution is attributed to the change in the magnetic field’s tilt angle. Additionally, the peak heat load is observed at the chamfer positions between cassettes, and an analysis is conducted on the relationship between peak heat load and the size of the chamfer. Smaller tilt angles result in lower surface heat load. Furthermore, a comparative analysis of peak heat loads at the chamfer under different plasma current conditions reveals that higher plasma currents generally result in increased peak heat loads. This investigation into the toroidal heat load distribution on the divertor surface contributes to understanding of the damage mechanism of W PFC as the target plate and provides valuable data for divertor design of future fusion devices.

Afterglow Linear Polarization Signatures from Shallow GRB Jets: Implications for Energetic GRBs

Gamma-ray bursts (GRBs) are powered by ultrarelativistic jets. The launching sites of these jets are surrounded by dense media, which the jets must cross before they can accelerate and release high-energy emission. Interaction with the medium leads to the formation of a mildly relativistic sheath around the jet, resulting in an angular structure to the jet’s asymptotic Lorentz factor and energy per solid angle, which modifies the afterglow emission. We build a semi-analytical tool to analyze the afterglow light curve and polarization signatures of jets observed from a wide range of viewing angles, and focus on ones with slowly declining energy profiles known as shallow jets. We find overall lower polarization compared to the classical top hat jet model. We provide an analytical expression for the peak polarization degree as a function of the energy profile power-law index, magnetic field configuration, and viewing angle, and show that it occurs near the light-curve break time for all viewers. When applying our tool to GRB 221009A, suspected to originate from a shallow jet, we find that the suggested jet structures for this event agree with the upper limits placed on the afterglow polarization in the optical and X-ray bands. We also find that at early times the polarization levels may be significantly higher, allowing for a potential distinction between different jet structure models and possibly constraining the magnetization in both forward and reverse shocks at that stage.

Unveiling complex magnetic field configurations in red giant stars

The recent measurement of magnetic field strength inside the radiative interior of red giant stars has opened the way toward full 3D characterization of the geometry of stable large-scale magnetic fields. However, current measurements, which are limited to dipolar (ℓ = 1) mixed modes, do not properly constrain the topology of magnetic fields due to degeneracies on the observed magnetic field signature on such ℓ = 1 mode frequencies. Efforts focused toward unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigated the detectability of complex magnetic field topologies (such as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focused on a field composed of a combination of a dipole and a quadrupole (quadrudipole) and on an offset field. We explored the potential of probing such magnetic field topologies from a combined measurement of magnetic signatures on ℓ = 1 and quadrupolar (ℓ = 2) mixed mode oscillation frequencies. We first derived the asymptotic theoretical formalism for computing the asymmetric signature in the frequency pattern for ℓ = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically performed a grid search over the parameter space to map the degeneracy of the signatures of given topologies. We demonstrate the crucial role played by ℓ = 2 mixed modes in accessing internal magnetic fields with a quadrupolar component. The degeneracy of the quadrudipole compared to pure dipolar fields is lifted when considering magnetic asymmetries in both ℓ = 1 and ℓ = 2 mode frequencies. In addition to the analytical derivation for the quadrudipole, we present the prospect for complex magnetic field inversions using magnetic sensitivity kernels from standard perturbation analysis for forward modeling. Using this method, we explored the detectability of offset magnetic fields from ℓ = 1 and ℓ = 2 frequencies and demonstrate that offset fields may be mistaken for weak and centered magnetic fields, resulting in underestimating the magnetic field strength in stellar cores. We emphasize the need to characterize ℓ = 2 mixed-mode frequencies, (along with the currently characterized ℓ = 1 mixed modes), to unveil the higher-order components of the geometry of buried magnetic fields and to better constrain angular momentum transport inside stars.

Survey of non-thermal electrons around supermassive black holes through polarization flips

Abstract Optically thick non-thermal synchrotron sources notably produce linear polarization vectors that are parallel to projected magnetic field lines on the observer’s screen, although they are perpendicular in well-known optically thin cases. To elucidate the complex relationship between the vectors and fields, and to investigate the energy and spatial distribution of non-thermal electrons through the images, we perform polarization radiative transfer calculations at submillimeter wavelengths. Here the calculations are based on semi-analytic force-free jet models with non-thermal electrons with a power-law energy distribution. In the calculated images, we find a $90{^\circ }$-flip of linear polarization vectors at the base of counter-side (receding) jet near a black hole, which occurs because of large optical depths for the synchrotron self-absorption effect. The $90{^\circ }$-flip of LP vectors is also seen on the photon ring at a high frequency, since the optical depth along the rays is large there due to the light bending effect. In addition, we see the flip of the sign of circular polarization components on the counter-jet and photon ring. Furthermore, we show that these polarization flips are synthesized with large values in the spectral index map, and also give rise to outstanding features in the Faraday Rotation Measure map. Since the conditions of flipping depend on the magnetic field strength and configuration and the energy distribution of electrons, we can expect that the polarization flips will provide us with observational evidence for the presence of non-thermal electrons around the black hole, and a clue to the magnetic driving mechanism of plasma jets.

Evaluating the Viability and Optimization of Plasma Pilot Plants

Nuclear fusion stands as a promising avenue for achieving a sustainable and abundant energy source. Central to this endeavor is the development of effective pilot plants capable of demonstrating the feasibility of fusion power on a commercial scale. This paper provides a comprehensive evaluation of three leading magnetic confinement fusion configurations: the Advanced Tokamak (AT), Spherical Tokamak (ST), and Compact Stellarator (CS).The Tokamak, characterized by its toroidal shape and strong magnetic fields, has been the most researched and developed fusion device. The analysis focuses on the challenges related to maintaining stability and minimizing disruptions. Furthermore, the optimization strategies involving advanced materials, superconducting magnets, and innovative plasma control techniques are discussed to enhance the Tokamak's viability. In contrast, the Spherical Tokamak, a variant with a more compact and spherical design, promises improved current advancements including the handling of higher heat loads and magnetic field configurations. The Stellarator, with its complex, twisted magnetic field structure, eliminates the need for continuous external current drive, addressing some intrinsic issues of the Tokamak. By comparing these configurations, the paper identifies the relative strengths and weaknesses of each approach in terms of confinement efficiency, operational stability, and engineering feasibility. The evaluation is supported by recent experimental data, simulation results, and technological advancements. Finally, the paper proposes a roadmap for the future development of fusion pilot plants, highlighting the need for an integrated approach that leverages the strengths of each configuration while addressing their individual challenges. The synthesis of this evaluation underscores the importance of continued research, cross-collaboration, and investment in advanced technologies to realize the goal of practical and economically viable fusion energy. This comparative analysis aims to provide a strategic framework for policymakers, researchers, and engineers in the fusion community, fostering informed decisions and prioritizing research efforts towards the most promising fusion energy configurations.

Stability optimization of energetic particle driven modes in nuclear fusion devices: the FAR3d gyro-fluid code

The development of reduced models provide efficient methods that can be used to perform short term experimental data analysis or narrow down the parametric range of more sophisticated numerical approaches. Reduced models are derived by simplifying the physics description with the goal of retaining only the essential ingredients required to reproduce the phenomena under study. This is the role of the gyro-fluid code FAR3d, dedicated to analyze the linear and nonlinear stability of Alfvén Eigenmodes (AE), Energetic Particle Modes (EPM) and magnetic-hydrodynamic modes as pressure gradient driven mode (PGDM) and current driven modes (CDM) in nuclear fusion devices. Such analysis is valuable for improving the plasma heating efficiency and confinement; this can enhance the overall device performance. The present review is dedicated to a description of the most important contributions of the FAR3d code in the field of energetic particles (EP) and AE/EPM stability. FAR3d is used to model and characterize the AE/EPM activity measured in fusion devices as LHD, JET, DIII-D, EAST, TJ-II and Heliotron J. In addition, the computational efficiency of FAR3d facilitates performing massive parametric studies leading to the identification of optimization trends with respect to the AE/EPM stability. This can aid in identifying operational regimes where AE/EPM activity is avoided or minimized. This technique is applied to the analysis of optimized configurations with respect to the thermal plasma parameters, magnetic field configuration, external actuators and the effect of multiple EP populations. In addition, the AE/EPM saturation phase is analyzed, taking into account both steady-state phases and bursting activity observed in LHD and DIII-D devices. The nonlinear calculations provide: the induced EP transport, the generation of zonal structures as well as the energy transfer towards the thermal plasma and between different toroidal/helical families. Finally, FAR3d is used to forecast the AE/EPM stability in operational scenarios of future devices as ITER, CFETR, JT60SA and CFQS as well as possible approaches to optimization with respect to variations in the most important plasma parameters.

Vector magnetometry employing a rotating RF field in a single-beam optically pumped magnetometer

Magnetic field vector information is crucial for many advanced applications, such as navigation and biomedical imaging. However, existing methods often lack high sensitivity or require complex setups. This study addresses these challenges by proposing a novel vector magnetometry method using a single-beam optically pumped magnetometer. A rotating radio-frequency field is innovatively utilized to excite atomic spin precession, enabling accurate measurement of the magnetic field direction based on scalar measurement. The method is tested through physical experiments with different magnetic field configurations to validate its performance. The experimental results demonstrate high accuracy, and achieve a magnetic field amplitude sensitivity of 800 fT/Hz1/2, an azimuth sensitivity of 100 μrad/Hz1/2, and a polar angle sensitivity of 13 μrad/Hz1/2. The proposed method facilitates sensor miniaturization and is suitable for applications in high magnetic field environments, such as geomagnetic field.

“Cosmic alternator” for the onset of large-scale magnetic fields

Magnetic field configurations extending over macroscopic scale distances are shown to be generated in rarefied collisionless plasmas when non-thermal and spatially inhomogeneous electron distributions in phase space emerge. The analyzed representative case is where, in the presence of a significant spatial gradient of the electron pressure and of a sheared magnetic field configuration, an alternating magnetic field reconnection process can develop associated with the anisotropy of the fluctuating electron temperature. The relevant process is intrinsically different from that known as the “Biermann Battery,” which does not involve magnetic reconnection and requires that the unperturbed spatial gradient of the (isotropic) electron temperature be misaligned relative to that of the density gradient.

Galaxy populations and redshift dependence of the correlation between infrared and radio luminosity

We argue that the difference in infrared-to-radio luminosity ratio between local and high-redshift star-forming galaxies reflects the alternative physical conditions –including magnetic field configurations– of the dominant population of star-forming galaxies in different redshift ranges. We define three galactic types, based on our reference model, with reference to ages of stellar populations. “Normal” late-type galaxies dominate the star formation in the nearby Universe; “starburst” galaxies take over at higher redshifts, up to z ∼ 1.5; while “protospheroidal” galaxies dominate at high redshift. A reanalysis of data from the COSMOS field combined with literature results shows that, for each population, the data are consistent with an almost redshift-independent mean value of the parameter qIR, which quantifies the infrared–radio correlation. However, we find a hint of an upturn of the mean qIR at z ≳ 3.5 consistent with the predicted dimming of synchrotron emission due to cooling of relativistic electrons by inverse Compton scattering off the cosmic microwave background. The typical stellar masses increase from normal, to starburst, and to protospheroidal galaxies, accounting for the reported dependence of the mean qIR on stellar mass. Higher values of qIR found for high-z strongly lensed dusty galaxies selected at 500 μm might be explained by differential magnification.

Performance and discharge characteristics of Applied–Field MPD thruster in accordance with ring magnet position

The magnetic field has a significant impact on the performance and discharge characteristics of magnetoplasmadynamic (MPD) thrusters. A low-power MPD thruster with an operating power of 10 kW was developed, and its performance characteristics were examined by altering the applied magnetic field. The thrust increased from 92 mN to 310 mN, and specific impulse from 479 s to 1264 s, respectively, within the input power range of 4–10 kW. Spectral analysis of the plume revealed that the increase in performance was significantly affected by the ion current density rather than by the ion energy. Furthermore, the shape and flux density of the magnetic fields inside the discharge channel were altered by adjusting the position of the permanent magnets from z = 0 to −4 cm at 1 cm intervals relative to the alumina insulator inside the discharge channel. Consequently, the thrust increased by approximately 14–40 % at the specific magnetic field configuration with a ratio of Br/Bz = 0.3, despite not being at the maximum magnetic field strength. The ratio of the radial to axial magnetic fields is strongly correlated with thruster performance and should be considered in optimizing design.

A fully implicit, asymptotic-preserving, semi-Lagrangian algorithm for the time dependent anisotropic heat transport equation

In this paper, we extend the operator-split asymptotic-preserving, semi-Lagrangian algorithm for time dependent anisotropic heat transport equation proposed in Chacón et al. (2014) [18] to use a fully implicit time integration with backward differentiation formulas. The proposed implicit method can deal with arbitrary heat-transport anisotropy ratios χ∥/χ⊥⋙1 (with χ∥, χ⊥ the parallel and perpendicular heat diffusivities, respectively) in complicated magnetic field topologies in an accurate and efficient manner. The implicit algorithm is second-order accurate temporally and demonstrates an accurate treatment at boundary layers (e.g., island separatrices), which was not ensured by the operator-split implementation. The condition number of the resulting algebraic system is independent of the anisotropy ratio, and is inverted with preconditioned GMRES. We propose a simple preconditioner that renders the finite-dimensional linear operator compact, resulting in mesh-independent convergence rates for topologically simple magnetic fields, and convergence rates scaling as ∼(NΔt)1/4 (with N the total mesh size and Δt the timestep) in topologically complex magnetic-field configurations. We demonstrate the accuracy and performance of the approach with test problems of varying complexity, including an analytically tractable boundary-layer problem in a straight magnetic field, and a topologically complex magnetic field featuring magnetic islands with extreme anisotropy ratios (χ∥/χ⊥=1010).

Origin of the Twice-90° Rotations of the Polarization Angle in GRB 170114A and GRB 160821A

The observed abrupt twice-90° rotations of the polarization angle (PA) in the prompt phase of gamma-ray bursts (GRBs) are difficult to understand within the current one-emitting-shell models. Here, we apply a model with multiple emitting shells to solve this new challenging problem. Two configurations of large-scale ordered magnetic fields in the shells are considered: toroidal and aligned. Together with the light curves and the spectral peak energy evolutions, the twice-90° PA rotations in GRB 170114A and GRB 160821A could be well interpreted with the multishell aligned magnetic field configuration (MFC). Our numerical calculations also show that the multiple shells with the toroidal MFC could not explain the observed twice-90° PA rotations. An aligned MFC in the GRB outflow usually indicates the preference of a magnetar central engine, while a toroidal field configuration is typically related to a central black hole. Therefore, the magnetar central engines for the two GRBs are favored.

Constructions of optimal-speed quantum evolutions: a comparative study

We present a comparative analysis of two different constructions of optimal-speed quantum Hamiltonian evolutions on the Bloch sphere. In the first approach (Mostafazadeh’s approach), the evolution is specified by a traceless stationary Hermitian Hamiltonian and occurs between two arbitrary qubit states by maximizing the energy uncertainty. In the second approach (Bender’s approach), instead, the evolution is characterized by a stationary Hermitian Hamiltonian which is not traceless and occurs between an initial qubit state on the north pole and an arbitrary final qubit state. In this second approach, the evolution occurs by minimizing the evolution time subject to the constraint that the difference between the largest and the smallest eigenvalues of the Hamiltonian is kept fixed. For both approaches we calculate explicitly the optimal Hamiltonian, the optimal unitary evolution operator and, finally, the optimal magnetic field configuration. Furthermore, we show in a clear way that Mostafazadeh’s and Bender’s approaches are equivalent when we extend Mostafazadeh’s approach to Hamiltonians with nonzero trace and, at the same time, focus on an initial quantum state placed on the north pole of the Bloch sphere. Finally, we demonstrate in both scenarios that the optimal unitary evolution operator is a rotation about an axis that is orthogonal to the unit Bloch vectors that correspond to the initial and final qubit states.