Generalized Bloch Equations Research Articles

Tuning the size of skyrmion by strain at the Co/Pt3 interfaces

SummaryBased on density functional theory calculations, we elucidated the tunability of the atomic structures and magnetic interactions of Co/Pt3 interface (one layer of hcp(0001) Co and three layers of fcc(111) Pt) and thus the skyrmion sizes using strain. The dispersion relations of the spin spiral in the opposite directions, E(q) and E(-q), were evaluated based on generalized Bloch equations. Effective exchange coupling (EC) and Dzyaloshinsky-Moriya interaction (DMI) parameters between different neighbors Ji and di at different lattice constants were derived by fitting the resulting spin spiral dispersion E(q) to EC model with DMI and E(q)-E(-q) formula, respectively. We observed an increase in DMI and a significant decrease in EC with an increase in strain. Hence, the size of Néel-type skyrmions determined by the ratio of EC/DMI can be controlled by applying strain, leading to an effective approach to tailor the formation of skyrmion lattices by inducing slight structural modifications on the magnetic thin films.

Two-body neutral Coulomb system in a magnetic field at rest: From hydrogen atom to positronium

A simple locally accurate uniform approximation for the nodeless wave function is constructed for a neutral system of two Coulomb charges of different masses $(\ensuremath{-}q,{m}_{1})$ and $(q,{m}_{2})$ at rest in a constant uniform magnetic field for the states of positive and negative parity, $(1{s}_{0})$ and $(2{p}_{0})$, respectively. It is shown that by keeping the mass and charge of one of the bodies fixed, all systems with different second-body masses are related. This allows one to consider the second body as infinitely massive and to take such a system as basic. Three physical systems are considered in detail: the hydrogen atom with (in)finitely massive proton (deuteron, triton) and the positronium atom $(\ensuremath{-}e,e)$. We derive the Riccati-Bloch and generalized Bloch equations, which describe the domains of small and large distances, respectively. Based on the interpolation of the small- and large-distance behavior of the logarithm of the wave function, a compact ten-parametric function is proposed. Taken as a variational trial function, it provides accuracy of not less than six significant digits (SDs) ($\ensuremath{\lesssim}{10}^{\ensuremath{-}6}$ in relative deviation) for the total energy in the whole domain of considered magnetic fields $[0\phantom{\rule{0.16em}{0ex}},\phantom{\rule{0.16em}{0ex}}{10}^{4}]$ a.u. and not less than three SDs for the quadrupole moment ${Q}_{zz}$. To get reference points, the Lagrange mesh method with 16 K mesh points was used to get from ten to six SDs in energy from small to large magnetic fields. Based on the Riccati-Bloch equation, the first 100 perturbative coefficients for the energy, in the form of rational numbers, are calculated and, using the Pad\'e-Borel resummation procedure, the energy is found with not less than ten SDs at magnetic fields $\ensuremath{\le}1\phantom{\rule{0.16em}{0ex}}\mathrm{a}.\mathrm{u}.$

Tuning the Size of Skyrmion by Strain at the Co/Pt <sub>3</sub> Interfaces

Based on density functional theory calculations, we elucidated the tunability of the atomic and magnetic structures of Co/Pt3 interface and thus the skyrmion sizes using strain. The dispersion relations of spin spiral at opposite direction E(q) and E(-q) were evaluated based on generalized Bloch equations. Effective exchange coupling (EC) and Dzyaloshinsky-Moriya interaction (DMI) parameters between different neighbors Ji and di at different lattice constants were derived by fitting the resulting magnon dispersion E(q) to EC model with DMI and E(q)-E(-q) formula, respectively. We observed that DMI increases and EC decreases significantly with strain. Hence, the size of Neel-type skyrmions determined by the ratio of EC/DMI can be controlled by applying strain, leading to an effective approach to tailor the formation of skyrmion lattices by inducing slight structural modifications on the magnetic thin films.

Generalized open quantum system approach for the electron paramagnetic resonance of magnetic atoms

A recent experimental breakthrough has enabled probing the electronic parametric resonance of a single magnetic atom in a scanning tunneling microscopy setup. The results present intriguing features, such as an asymmetric line shape and an unusually large ratio of the decoherence and decay rates, which defy standard approaches using the conventional Bloch equations. To address these issues we employ generalized Bloch equations, together with proper microscopic modeling of the magnetic adatom, and show how all the experimental features can be naturally accounted for. The proposed approach may also be useful in treating any future similar experiments, as well as next generation hybrid quantum devices.

Spincaloritronic in Magnetic Nanostructures

Using one of the methods of quantum nonequilibrium statistical physics we have investigated the spin transport transverse to the normal metal/ferromagnetic insulator interface in hybrid nanostructures. An approximation of the effective parameters, when each of the interacting subsystems (electron spin, magnon, and phonon) is characterized by its own effective temperature have been considered. We have constructed the macroscopic equations describing the spin-wave current caused by both the resonantly exciting spin subsystem of conduction electrons and an inhomogeneous temperature field in the ferromagnetic insulator. In addition, we have derived the generalized Bloch equations describing the spin-wave current propagation in the insulator and considering the resonant-diffusion nature of the propagation of magnons and their relaxation processes. We have shown that the spin-wave current excitation under combined resonance conditions bears a resonant nature. The formation of the two: injected and thermally excited, different in energies magnon subsystems and the influence of its interaction with phonon drag effect under spin Seebeck effect conditions in the magnetic insulator part of the metal/ferromagnetic insulator/metal structure is studied.

Efficient continuous-wave noise spectroscopy beyond weak coupling

The optimization of quantum control for physical qubits relies on accurate noise characterization. Probing the spectral density $S(\omega)$ of semi-classical phase noise using a spin interacting with a continuous-wave (CW) resonant excitation field has recently gained attention. CW noise spectroscopy protocols have been based on the generalized Bloch equations (GBE) or the filter function formalism, assuming weak coupling to a Markovian bath. However, this standard protocol can substantially underestimate $S(\omega)$ at low frequencies when the CW pulse amplitude becomes comparable to $S(\omega)$. Here, we derive the coherence decay function more generally by extending it to higher orders in the noise strength and discarding the Markov approximation. Numerical simulations show that this provides a more accurate description of the spin dynamics compared to a simple exponential decay, especially on short timescales. Exploiting these results, we devise a protocol that uses an experiment at a single CW pulse amplitude to extend the spectral range over which $S(\omega)$ can be reliably determined to $\omega=0$.

Dzyaloshinsky-Moriya interaction in -(Zn, Cr)S(111) surface: First principle calculations

According to density functional theory calculations, we elucidate the atomic and electronic structure of -(Zn, Cr)S(111) surface. The magnetic interaction between Cr atoms is via S atoms close to the Cr layer. This interaction is shown by the analysis of spin charge contour plot and partial density of states (DOS) of each atom. The DOSs of other S atoms are non magnetic and have no magnetic exchange with the Cr layer. E(q) and E(-q) are the dispersions between energy E and wave vector q of spin spiral in the opposite directions. They are calculated with generalized Bloch equations and all the magnetic moments of Cr atoms are arranged in the plane perpendicular to the -(Zn, Cr)S(111) film. The differences between E(q) and E(-q) are caused by the interface of -(Zn, Cr)S(111), where the symmetry of space perpendicular to the film is broken. Effective Heisenberg exchange interaction (HBI) and Dzyaloshinsky-Moriya interaction (DMI) parameters between different neighbors (Ji and di) are derived by well fitting the ab initio spin spiral dispersion E(q) to HBI with DMI model and E(q)-E(-q) to DMI model, respectively. The J2 plays a major role with a large negative value of -9.04 meV. The J1 is about 2/5 of J2, and J3 is about 1/4 of J2 with positive value. The DMI d1 is -0.53 meV, and d2 is 0.07 meV. With these HBI parameters, E(0) is the largest one at which -(Zn, Cr)S(111) has no ferromagnetic interface. The E(q) has its lowest energy with the q at M=b1/2 in the first Brillouin zone. Hence, -(Zn, Cr)S(111) is an M-type antiferromagnetic (AFM) material. In this type of AFM configuration, magnetic moments of Cr atom in a line along b2 are parallel to each other, and antiparallel to the magnetic moments in adjacent lines. The E(q) at K=b1/2+ b2/2 is almost as large as that at point. The value of DMI parameter d1 is about 1/5 of that on Co/Pt3 interface and 1/2 of Co/graphene. However, it is a negative number, which shows the clockwise chirality. The -(Zn, Cr)S(111) interface has obvious DMI, and skyrmion may be formed at this transition-metal/semiconductor (TM/S) interface. It is a good option to search for DMI in different kinds of TM/S heterojunctions. The material that combines the advantage of heterojunction, and DMI may have new magnetic phenomenon, which is usefulfor the magnetic storage. This paper enriches the research on DMI.

First principle study of weak Dzyaloshinsky-Moriya interaction in Co/BN surface

Based on density functional theory calculations, we elucidate the atomic and electronic structures of Co atom of hexgonal BN (Co/h-BN). The interaction between magnetic moments of Co atoms is realized through Co-N_-B_ grid, which is indicated by the analysis of spin charge contour plot and partial density of states of each atom, where and denote the site of B or N atom close to and away from Co atom, respectively. Then the dispersion relations E(q) and E(-q) (q denotes the direction vector of spin spiral) between energy and wave vector of spin spiral in the opposite directions are calculated with generalized Bloch equations. In the incommensurate spin spiral calculations, all the magnetic moments of Co atom are arranged in the same plane that is perpendicular to the Co/h-BN film. The difference between E(q) and E(-q) is caused by the interface of Co/h-BN, where the symmetry of space perpendicular to the film is broken. Moreover, the effective Heisenberg exchange interaction (HBI) and Dzyaloshinsky-Moriya interaction (DMI) parameters between different neighbors (Ji and di) are derived by well fitting the ab initio magnon dispersion E(q) to HBI with DMI model and E(q)-E(-q) to DMI model, respectively. The J1 has a negative value and plays a major role, J3 is one order of magnitude smaller than J1, and other parameters are close to zero. Hence, Co/h-BN is triangular antiferromagnetic material with the q at k point in the first Brillouin zone. However, the spin spiral with the q at M point is only 2 meV larger than the basic state with the only negative J1 and smaller positive J2. The DMI is not shown in this interface with d1 and d2 close to zero. Based on the non DMI character and its stability in air, h-BN can be capped on other DMI interfaces. The reason that the DMI in Co/h-BN is much smaller than in Co/Gra is much larger height between Co and h-BN. It is 0.192 nm for h-BN but it is 0.156 nm for Co/Gra. We may reduce the height to enhance the DMI by other ways, such as adding electrical and magnetic fields in the future.

Theory of graphene saturable absorption

Saturable absorption is a non-perturbative nonlinear optical phenomenon that plays a pivotal role in the generation of ultrafast light pulses. Here we show that this effect emerges in graphene at unprecedentedly low light intensities, thus opening avenues to new nonlinear physics and applications in optical technology. Specifically, we theoretically investigate saturable absorption in extended graphene by developing a non-perturbative single-particle approach, describing conduction-electron dynamics in the atomically-thin material using the two-dimensional Dirac equation for massless Dirac fermions, which is recast in the form of generalized Bloch equations. By solving the electron dynamics non-perturbatively, we account for both interband and intraband contributions to the intensity-dependent saturated conductivity and conclude that the former dominates regardless of the intrinsic doping state of the material. The results are in excellent agreement with atomistic quantum-mechanical simulations including higher-band effects. Additionally, we find that the modulation depth of saturable absorption in graphene can be electrically manipulated through an externally applied gate voltage. Our results are relevant for the development of graphene-based optoelectronic devices, as well as for applications in mode-locking and random lasers.

Expansion formulas for quantum master equations including initial correlation

The quantum master equation is considered for an open system initially correlated with a thermal reservoir, where the initial correlation is created by an initial preparation of a relevant system. Perturbative expansion formulas for quantum master equations are systematically derived with respect to a system-reservoir interaction. The result is applied to the dispersive JC model and the spin-boson model. In the second order approximation, the generalized Bloch equations for a spin system are found. It is shown that the effect of the initial correlation between the spin and the thermal reservoir is equivalent to an effective external field applied to the spin system.

Tonks-Dattner resonances in a quantum plasma bubble.

We establish the spectrum of Tonks-Dattner mode resonances in a quantum plasma bubble and consider the spectral changes associated with plasma quenching and plasma expansion. The quantum corrections associated with the mode spectrum are specified, which can be used as a diagnostic tool to identify the quantum regime. The frequency shifts associated with time-varying plasma bubbles correspond to time refraction and can also be used as a plasma diagnostic. We also study the energy mode coupling, in the presence of a low-frequency perturbation. It will be shown that the mode coupling equations take the form of generalized Bloch equations, where a nonlinear Rabi frequency can also be identified.

Spin effects induced by thermal perturbation in a normal metal/magnetic insulator system

Using one of the methods of quantum nonequilibrium statistical physics, we have investigated the spin transport transverse to the normal metal/ferromagnetic insulator interface in hybrid nanostructures. An approximation of the effective parameters, when each of the interacting subsystems (electron spin, magnon, and phonon) is characterized by its own effective temperature, has been considered. The generalized Bloch equations which describe the spin-wave current propagation in the dielectric have been derived. Finally, two sides of the spin transport ``coin'' have been revealed: the diffusive nature of the magnon motion and magnon relaxation processes, responsible for the spin pumping, and the spin-torque effect.

Generalized spin precession equations

The Bloch equations, which describe spin precession and relaxation in external magnetic fields, can be generalized to include the evolution of polarization tensors of various ranks in arbitrary multipole fields. We show applications of the generalized spin precession equations using simple examples from atomic, nuclear and condensed matter physics, and compare the various approaches found in the literature. The derivation of the generalized Bloch equations can be considerably simplified using a particular bra–ket notation for irreducible tensors.

Mixed states in Rabi waves and quantum nanoantennas

The mixed states of Rabi waves in a one-dimensional chain of two-level quantum dots (QDs) with tunnel coupling between neighbor QDs are theoretically considered. The propagation of Rabi waves is described by a $2\ifmmode\times\else\texttimes\fi{}2$ statistical operator depending on two spatial and one temporal variables. For a statistical operator the generalized Bloch equations are derived. The eigenmodes of the statistical operator are considered, the general solution of the initial problem is obtained, and the frequency spectrum of the induced current is investigated. The high-frequency part of the spectrum corresponds to a Mollow triplet, but is distinct from the last one as it is the continuous spectral band located on both sides of the transition frequency. The low-frequency part is caused by tunnel current and includes a dc-component and continuous ac-band in the vicinity of the Rabi oscillation frequency. The application of Rabi waves to the electrically controlled quantum nanoantennas of terahertz range is considered.

Electronic structure and dynamics of low symmetry Cu 2+ complexes in kainite-type crystal KZnClSO 4·3H 2O: EPR and ESE studies

EPR measurements at X-band were performed in the temperature range 4.2–300 K with angular dependence measurements at 77 K for Cu 2+ in KZnClSO 4·3H 2O. Rigid lattice spin – Hamiltonian parameters are: g z = 2.4247, g y = 2.0331, g x = 2.1535, A z = −103 × 10 −4 cm −1, 63 × 10 −4 cm −1, and −31 × 10 −4 cm −1. The parameters were analyzed using MO-theory with the d x 2 - y 2 ground state containing admixture of the d z 2 -state in the rhombic symmetry D 2h. The analysis consistently explained unusual g-factor sequence and relatively small hyperfine splitting anisotropy as the consequence of the mixing and spin density delocalization via excited orbital states. We assigned that Cu 2+ ions substituting host Zn 2+ prefer one of the four structurally different zinc sites where they are coordinated by four water molecules and two SO 4 groups in an distorted octahedron elongated along SO 4–Cu–SO 4 direction. The distortion is due to the Jahn–Teller effect which is static at low temperatures but becomes dynamic above 20 K with jumps of the Cu 2+ complex between two lowest potential wells. The jumps produce continuous g-factor and hyperfine splitting averaging when temperature increases. This process is discussed in terms of two motional averaging theories: classical theory based on generalized Bloch equations and Silver–Getz model. Their limitations are discussed. Importance of the difference in the g-factors of the averaged line is explained and a new expression for calculation of jump frequency from the line shift is proposed. The jumps are described as phonon induced tunneling via excited vibrational level of energy 76 (±6) cm −1. This process is not effective enough at low temperatures and Boltzmann population of the two lowest energy potential wells is reached above 110 K. From electron spin–lattice relaxation measurements by electron spin echo methods the Debye temperature was determined as Θ D = 172 K. Fourier Transform of strongly modulated spin echo decay gives pseudo-ENDOR spectrum with peaks from 1H and 35Cl nuclei. From splitting of the peaks into doublets we determined the distance to the modulating nuclei and confirmed the position of the site where Cu 2+ ion is located.

Fast electromagnetic response of a thin film of resonant atoms with permanent dipole

We consider the propagation of extremely short pulses through a dielectric thin film containing resonant atoms (two-level atoms) with permanent dipole. Assuming that the film width is less than the field wavelength, we can solve the wave equation and reduce the problem to a system of generalized Bloch equations describing the resonant atoms. We compute the stationary solutions for a constant irradiation of the film. Superimposing a small amplitude linear wave we calculate the reflection and transmission coefficients. From these, one can then deduce the different parameters of the model. We believe that this technique could be used in experiments to obtain the medium atomic and relaxation parameters.

Generalized bloch equations for optical interactions in confined geometries

By combining the field-susceptibility technique with the optical Bloch equations, a general formalism is developed for the investigation of molecular photophysical phenomena triggered by nanometer scale optical fields in the presence of complex environments. This formalism illustrate the influence of the illumination regime on the fluorescence signal emitted by a single molecule in a complex environment. In the saturated case, this signal is proportional to the optical local density of states, while it is proportional to the near-field intensity in the non-saturated case.

Decoherence and relaxation of a quantum bit in the presence of Rabi oscillations

Dissipative dynamics of a quantum bit driven by a strong resonant field and interacting with a heat bath is investigated. We derive generalized Bloch equations and find modifications of the qubit's damping rates caused by Rabi oscillations. Nonequilibrium decoherence of a phase qubit inductively coupled to a LC-circuit is considered as an illustration of the general results. It is argued that recent experimental results give a clear evidence of effective suppression of decoherence in a strongly driven flux qubit.

Spectrum of qubit oscillations from generalized Bloch equations

We have developed a formalism suitable for calculation of the output spectrum of a detector continuously measuring quantum coherent oscillations in a solid-state qubit, starting from microscopic Bloch equations. The results coincide with that obtained using Bayesian and master equation approaches. The previous results are generalized to the cases of arbitrary detector response and finite detector temperature.

Interband transitions of quantum wells and device structures containing Ga(N, As) and (Ga, In)(N, As)

The unusual N-induced band formation and band structure of Ga(N, As) and (Ga, In)(N, As) alloys are also reflected in the electronic structure of quantum wells (QWS) and device structures containing these non-amalgamation-type alloys. This review is divided into three parts. The first part deals with band structure aspects of bulk Ga(N, As) and motivates the possibility of a k · p-like parameterization of the band structure in terms of the level repulsion model between the conduction band edge of the host and a localized N-level. The second part presents experimental studies of interband transitions in Ga(N, As)/GaAs and (Ga, In)(N, As)/GaAs QW structures addressing band offsets, electron effective mass changes and an intrinsic mechanism contributing to the blueshift of the (Ga, In)(N, As) band gap on annealing. The observed interband transitions can be well described using a ten-band k · p model based on the level repulsion scheme. The third part deals with (Ga, In)(N, As)-based laser devices. The electronic structure of the active region of vertical-cavity surface-emitting laser and edge-emitter laser structures is studied by modulation spectroscopy. The gain of such structures is measured by optical methods and analysed in terms of a model combining the ten-band k · p description of the band structure and generalized Bloch equations.