Spin Density Vector Research Articles

Water-Wave Vortices and Skyrmions.

Topological wave structures-phase vortices, skyrmions, merons, etc.-are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i)water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii)skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii)meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv)spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.

Evolution and particle trapping dynamics of circular Pearcey-Airy Gaussian vortex beams in tightly focused systems.

This study investigates the propagation and evolution of self-focusing circular Pearcey-Airy Gaussian vortex beams (CPAGVB) through high numerical aperture objective lenses. CPAGVB demonstrates a unique light field distribution compared to the circular Pearcey vortex beam and circular Airy Gaussian vortex beam. By adjusting optical distribution factors, main radii, and off-axis vortex pair positions, a variety of light field structures can be generated, including asymmetric micro-optical bottles, quasi-flat-top beam micro-optical bottles, and dual optical bottles. The particle trapping performance of CPAGVB is examined, revealing a gradient force eight orders of magnitude larger than its scattering force, up to twice the peak gradient force, and 2.5 times the scattering force of CAGVB. Further analysis of lateral power flow density, spin density vector, and total angular momentum distribution at the focal plane unveils the dynamics of particle motion toward the center. The Gouy phase difference under varying main radii reveals two types of normalized spin density vectors, characterized by helical and oscillating distributions. Additionally, the study examines the two-dimensional polarization ellipse distribution at the focal plane, elucidating the formation of central polarization singularities with axial vortices and the impact of peripheral polarization rearrangement on phase singularities. This research advances the comprehension of CPAGVB's distinctive properties and potential applications in micro-optical systems and particle manipulation.

Tight-focusing evolution of linearly polarized auto-focusing circular Bessel-Gaussian vortex beam

In this paper, we analyzed the influence of on-axis and off-axis vortex on the intensity distribution of the linearly polarized auto-focusing Bessel-Gaussian vortex beam (LABGVB). By comparing its tight-focusing properties with that of the linearly polarized circular Airy beam, it is found that when transmitted in the tight-focusing system, the LABGVB not only generates multiple focusing, but also allows the focal points to be located away from the geometric focus of the high numerical aperture (HNA) lens. In addition, changing the focal length of the HNA lens not only causes the splitting and merging of the focal points, but also results in different control effects on the depth of focus and the distance between focal points due to different focusing times. In particular, the LABGVB can not only form multiple optical bottles for particle capture in the focusing region, but also exhibit two different distributions of the spin density vectors along the optical axis after choosing the appropriate scale factor and main ring radius, one being a circular line and the other being a helical line.

Three-dimensional polarization state and spin structure of a tightly focused radially polarized Gaussian Schell-model beam

In this work, we revisit the tight focusing of a radially polarized partially coherent Gaussian Schell-model beam and examine the effect of spatial coherence of the incident beam on the focal-plane three-dimensional (3D) polarization characters, including the 3D degree of polarization, polarimetric dimensionality, and polarimetric structure. The polarimetric structure and the associated structure for the spin density vector of the tightly focused field are obtained via the 3D characteristic decomposition of the $3\ifmmode\times\else\texttimes\fi{}3$ polarization matrix. We demonstrate that the generated focused field shows the genuine 3D polarization state and a sophisticated spin structure due to the reduced spatial coherence in the incident beam. We find the contributions to whole 3D polarization state of the focused field from the pure state, middle-term state, and 3D unpolarized state in the characteristic decomposition, as well as the contributions to the total spin density vector from the spins of pure state and middle-term state can be controlled by the transverse spatial coherence width of the incident Gaussian Schell-model beam. In addition, the genuine 3D polarization and the spin structure are analyzed in the connection of the recently introduced concept of polarimetric nonregularity.

Topological flowers and spider webs in 3D vector fields.

Topological structures currently are of special interest in the integration of singular optics and topological photonics. In this article, the topological flowers and spider webs, which got less attention comparing to the lemon-, star- structures of the same category, are investigated in 3D vector fields. We show that by strongly focusing higher-order singular beams, both the spin density (SD) vectors and the polarization states of the transverse fields on the focal plane exhibit flowers and spider webs structures in topology with 2|m - 1| folds/sectors (m is the beam order), and the topological structures of the SD vectors are demonstrated to have a 90°/|m - 1| rotation. On the other hand, the topological theory also needs to be developed according to the rapid growth of topological photonics. Here, by defining a 'relative topological charge', we have observed and analyzed the topological reactions of the loops (composed of SD singularities) rather than the 'point-type' singularities in conventional reactions. More specially, the 'radial index' and 'azimuthal index' are proposed to characterize the topological features of the flowers and spider webs, and have been verified that the 'radial index' is peculiar to 3D vector fields and proportional to 1/|m - 1| in general. Our work provides a way to describe the topological behaviors of groups of singularities and supplies new parameters for measuring the topological patterns in 3D vector fields, which will rich the topological theory and may have applications in topological photonics.

Constructing Spin Density Vector Twists With Spin Density Singularity

In this article, we propose a non-orbital angular momentum (OAM) method for constructing the spiral twists of spin density (SD) vectors in three-dimensional (3D) optical fields. This is realized by generating strings of SD singularities in a longitudinal plane of a strongly focused HG10 mode beam with circular polarization. We demonstrate that in this plane, the SD singularities manifest themselves as the Gouy phase difference. Through observing the Gouy phase difference, the strings of the SD singularities and their topological reactions are examined. It is found that the spiral twists of SD vectors are constructed in the two-dimensional (2D) space between two strings and their twisting behaviors can be adjusted by the topological reactions of the strings. This finding shows that the spiral twists can be formed in a 2D space rather than the 1D space in previous studies, which provides a more flexible way to utilize this rotational degree of freedom. Our result also clarifies that although the spiral twists of SD vectors can be a result of spin-orbital angular momentum interaction, the OAM is not a necessary condition for the generation of these twists. Our theoretical proposal for structuring SD vectors may have applications in 3D optical manipulation.

Influence of Rashba spin–orbit and Rabi couplings on the spin-mixing and ground state phases of binary Bose–Einstein condensates

We study the miscibility properties and ground state phases of two-component spin–orbit (SO) coupled Bose–Einstein condensates (BECs) in a harmonic trap with strong axial confinement. By numerically solving the coupled Gross–Pitaevskii equations in the two-dimensional setting, we analyze the SO-coupled BECs for two possible permutations of the intra- and interspecies interactions, namely (i) weak intra- and weak interspecies interactions (W–W) and (ii) weak intra- and strong interspecies interactions (W–S). Considering the density overlap integral as a miscibility order parameter, we investigate the miscible–immiscible transition by varying the coupling parameters. We obtain various ground state phases, including plane wave, half quantum vortex, elongated plane wave, and different stripe wave patterns for W–W interactions. For finite Rabi coupling, an increase in SO coupling strength leads to the transition from the fully miscible to the partially miscible state. We also characterize different ground states in the coupling parameter space using the root mean square sizes of the condensate. The spin density vector for the ground state phases exhibits density, quadrupole and dipole like spin polarizations. For the W–S interaction, in addition to that observed in the W–W case, we witness semi vortex, mixed mode, and shell-like immiscible phases. We notice a wide variety of spin polarizations, such as density, dipole, quadrupole, symbiotic, necklace, and stripe-like patterns for the W–S case. A detailed investigation in the coupling parameter space indicates immiscible to miscible state phase transition upon varying the Rabi coupling for a fixed Rashba SO coupling. The critical Rabi coupling for the immiscible–miscible phase transition decreases upon increasing the SO coupling strength.

Chiral Bloch states in single-layer graphene with Rashba spin-orbit coupling: Equilibrium spin current

We study the Bloch spectrum and spin physics of 2D massless Dirac electrons in single layer graphene subject to a one dimensional periodic Kronig-Penney potential and Rashba spin-orbit coupling. The Klein paradox exposes novel features in the band dispersion and in graphene spintronics. In particular it is shown that: (1) The Bloch energy dispersion $\veps(p)$ has unusual structure: There are {\it two Dirac points} at Bloch momenta $\pm p \ne 0$ and a narrow band emerges between the wide valence and conduction bands. (2) The charge current and the spin density vector vanish. (3) Yet, all the non-diagonal elements of the spin current density tensor are finite and their magnitude increases linearly with the spin-orbit strength. In particular, there is a spin density current whose polarization is perpendicular to the graphene plane. (4) The spin density currents are space-dependent, hence their continuity equation includes a finite spin torque density.

Matter-wave stripe solitons induced by helicoidal spin–orbit coupling

We present the exact analytical solution of stripe solitons on different backgrounds in Bose–Einstein condensates with helicoidal spin–orbit coupling (SOC). They are divided into two categories, one is stationary stripe soliton on zero background or period-wave background, the other is moving stripe soliton on zero background or plane-wave background. The period of stripe is determined by both strength and frequency of helicoidal SOC, and the oscillation amplitude of stripe is adjustable in the case of fixed total density. The corresponding spin polarization is expressed on Bloch sphere. For stationary soliton, the spin density vector take values in three directions, which is different from the Rashba–Dresslhaus coupling scheme. The spin density vectors display more complicated spatial distribution for moving stripe solitons. Meanwhile, the formation mechanism of stripe soliton and the interaction properties of moving stripe solitons are discussed. Our work enhances the understanding of stripe solitons, and is helpful for the observation of stripe phase.

Evolution of spin density vectors in a strongly focused composite field

The evolution of the spin density vectors (SDVs) is studied in a strongly focused composite field. It is found that the SDVs can be spiral along the propagation axis, and they are perpendicular to the ys direction on the ys axis. This behavior is governed by the Gouy phase difference between the field polarization components. The 60° rotation of the spatial distribution of the transverse SDVs is also generated, which is found to be controlled by the Gouy phase difference between the field orbital angular momentum modes. Additionally, the spin density singularities are observed in the evolution of the SDVs.

Tightly focusing evolution of the auto-focusing linear polarized circular Pearcey Gaussian vortex beams

Abstract We report tightly focused characteristics of the auto-focusing linear polarized circular Pearcey Gaussian vortex beams (CPeGVBs) with on-axis and off-axis vortex pairs through high numerical aperture in this paper. Compared with the linear polarized Airy beam, we find that the autofocusing linear polarized CPeGVBs are rotating during the tightly focusing progress. We analyse the relationship between the autofocusing and the tightly focusing of CPeGVBs. At the same time, some cases of the flat-top beam and the quasi optical comb mode generated by the linear polarized CPeGVBs are analyzed in detail, and the changing of the focus depth with beam parameters is also discussed. To show the superiority of linear polarization CPeGVBs, we explore the tightly focusing intensity contrasts of the beams, and we characterize the resulting optical polarization singularity using zero-, one- and two-order topological structures. We present the spin density vector and the polarization ellipse distributions of the CPeGVBs in the x − y plane. Furthermore, the CPeGVBs present two kinds of the spin density vector on the optical axis that one is helix shaped and the other one is oscillating in the one axial direction of CPeGVBs by choosing the appropriate factors. Our results have the potential applications in reverse shaping of focal fields and optical trapping control.

Transfer of spin angular momentum to a dielectric particle

We show here that in the sharp focus of a linearly polarized laser beam the spin vector flux has only transverse components (the effect of photonic wheels or photonic helicopter). For a linearly polarized optical vortex, the orbit-spin conversion leads to the appearance of both longitudinal and transverse components of the spin density vector in the focus. Spin-orbit conversion is experimentally demonstrated for a circularly polarized Gaussian beam when a transverse energy flux (orbital angular momentum) arises in the focus, which is transmitted to a microparticle and makes it rotate. Switching the handedness of circular polarization (from left to right) switches the microparticle rotation direction. It is also shown here that an azimuthally polarized vortex beam with an arbitrary integer topological charge generates in the focus a spin density vector that only has an axial component (pure magnetization), while the transverse spin flux is absent.

Generation of Spiral Spin Density Vectors With A Circularly Polarized, Vortex Beam

Spiral spin density vectors refer to the spiral trajectory of the spin density vectors in three dimensional (3D) space, and similar to the polarization Mobius strip, polarization knots and photonic wheel, are also special polarization structures in 3D optical field and only exist in three dimensions. In this article, we propose a new way to generate the spiral spin density vectors with a long spiral range. This is realized by strongly focusing a circularly polarized beam with one off-axis vortex. In this field, the spiral interval of spin density vectors theoretically always exists on the propagation axis, and its position can be controlled by the beam size parameter $f/w_0$ . The polarization variation and the spiral spin density vectors are discussed analytically along the propagation direction. It is found that the spiral shape and spiral ‘speed’ can be adjusted by the off-axis distance of the vortex and the semi-aperture angle of the focusing system respectively. A special spiral behavior of the spin density vector at the geometrical focus is also observed in this focused field. The spiral spin density vectors supply a new rotational degree of freedom in 3D optical field and our finding may have applications in optical micro-manipulations.

Tight-focusing properties of linearly polarized circular Airy Gaussian vortex beam

We show the tight-focusing properties of a linearly polarized circular Airy Gaussian vortex beam (CAiGVB) with a high-numerical-aperture objective lens; the light intensity distributions exhibit diversity with different positions of the vortex pairs (on-axis or off-axis vortex pairs). By choosing different optical distribution factors, the CAiGVB turns into a circular Airy vortex beam or Gaussian vortex beam, and the depth of focus can also be controlled. It is known that the vortex beam possesses both orbital and spin angular momentum. The spin density vector changes its direction in three-dimensional space during beam propagation, as long as it is not purely transverse or longitudinal, which would cause 3D polarization ellipse topologies. In contrast, the polarization topologies degenerate into 2D when the spin density vector is purely transverse or longitudinal. Furthermore, the direction of the spin density vector is closely related to the Gouy phase difference between longitudinal and transverse electric field components of the vectorial beam.

The evolutions of spin density and energy flux of strongly focused standard full Poincaré beams

Nonzero transverse spin density, which describes phenomenon in which the electromagnetic fields of localized light spin in a plane containing its wavenumber vector, has gained enormous interest recently because of its useful applications like spin-direction coupling and routing. In this Letter, using the Richards–Wolf vectorial method for standard full Poincaré beams, we present an analytical model for the high-numerical-aperture focusing system to calculate all components of the electric and magnetic field strength vectors as well as spin density and Poynting vector. The role and contribution of the optical degrees of freedom including ellipticity, handedness, and orientation when the transverse spin density is present, are revealed based on this analytical model. Ellipticity affects the localization and magnitude of the transverse spin density for both transverse and longitudinal components. In contrast, handedness only affects the longitudinal component whereas orientation only affects the transverse component. Furthermore, the energy flux in the focal plane are also studied in detail for the standard full Poincaré beams. These findings may be help in spin-controlled directive coupling and optical tweezers.

Spinning spin density vectors along the propagation direction.

It has been known that light possesses both spin and orbital angular momenta (AM) arising from the twisting behaviors of the electric (and magnetic) field vector and the wavefront of the field, respectively. The spin (AM) density is also a vector in the field of three dimensions (3D), since its orientation can be in any direction. In this Letter, we show that through focusing a Gaussian beam with both on-axis and off-axis vortices in a high-numerical-aperture system, the spin (AM) density vector in the focal region exhibits nontrivial behaviors: rotating around the central axis along the propagation direction. We demonstrate that this helical behavior of the spin (AM) density vector is mainly caused by the different Gouy phases of the three field components. By changing the position of the off-axis vortex and the semiaperture angle, the helical shape and the helical length can be adjusted. This is a new type of optical twist, to the best of our knowledge, and it may supply another rotational degree of freedom in optical tweezers.

Electron-Electron Cross-Relaxation and Spectral Diffusion during Dynamic Nuclear Polarization Experiments on Solids.

Recently it has been shown that experimental electron-electron double resonance (ELDOR) spectra of amorphous glasses containing free radicals with inhomogeneously broadened electron paramagnetic resonance (EPR) spectra can be analyzed using a set of coupled rate equations for the electron polarizations of frequency bins composing these spectra, named the eSD (electron spectral diffusion) model. The rate matrix defining these equations has elements depending on the microwave, the spin-lattice relaxation rates and on eSD rate constants responsible for polarization exchange. In this study, we show that in addition to the static dipolar flip-flop terms in the Hamiltonian a zero-quantum electron cross-relaxation mechanism can be responsible for the polarization exchange process in our samples. This conclusion was reached by calculating the EPR lineshapes of a system of 11 coupled electrons exposed to microwave irradiation using an eigenstate population rate equation derived from the spin density vector rate equation in Liouville space. These equations involve all terms of the Hamiltonian and in addition rate constants representing longitudinal relaxation and cross-relaxation mechanisms as well as MW irradiation. The results of these calculations are compared with the results obtained from the eSD model.

Analysis of a coupled spin drift–diffusion Maxwell–Landau–Lifshitz system

The existence of global weak solutions to a coupled spin drift–diffusion and Maxwell–Landau–Lifshitz system is proved. The equations are considered in a two-dimensional magnetic layer structure and are supplemented with Dirichlet–Neumann boundary conditions. The spin drift–diffusion model for the charge density and spin density vector is the diffusion limit of a spinorial Boltzmann equation for a vanishing spin polarization constant. The Maxwell–Landau–Lifshitz system consists of the time-dependent Maxwell equations for the electric and magnetic fields and of the Landau–Lifshitz–Gilbert equation for the local magnetization, involving the interaction between magnetization and spin density vector. The existence proof is based on a regularization procedure, L2-type estimates, and Moser-type iterations which yield the boundedness of the charge and spin densities. Furthermore, the free energy is shown to be nonincreasing in time if the magnetization–spin interaction constant in the Landau–Lifshitz equation is sufficiently small.

Analysis of a drift–diffusion model with velocity saturation for spin-polarized transport in semiconductors

A system of drift–diffusion equations with electric field under Dirichlet boundary conditions is analyzed. The system of strongly coupled parabolic equations for particle density and spin density vector describes the spin-polarized semi-classical electron transport in ferromagnetic semiconductors. The presence of a nonconstant and nonsmooth magnetization vector, solution of the Landau–Lifshitz equation, causes the diffusion matrix to be dependent on space and time and to have in general poor regularity properties, thus making the analysis challenging. To partially overcome the analytical difficulties the velocity saturation hypothesis is made, which results in a bounded drift velocity. The global-in-time existence and uniqueness of weak solutions is shown by means of a semi-discretization in time, which yields an elliptic semilinear problem, and a quadratic entropy inequality, which allow for the limit of vanishing time step size. The convergence of the weak solutions to the steady state, under some restrictions on the parameters and data, is shown. Finally, the higher regularity of solutions for a smooth magnetization in two space dimensions is shown through a diagonalization argument, which allows to get rid of the cross diffusion terms in the fluid equations, and the iterative application of Gagliardo–Nirenberg inequalities and a generalized version of Aubin lemma.

Spin turbulence in spinor Bose-Einstein condensates

We summarize the recent theoretical and numerical works on spin turbulence (ST) in spin-1 spinor Bose-Einstein condensates. When the system is excited from the ground state, it goes through hydrodynamic instability to ST in which the spin density vector has various disordered direction. The properties of ST depend on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic. ST has some characteristics different from other kinds of turbulence in quantum fluids. Firstly, the spectrum of the spin-dependent interaction energy exhibits the characteristic power law different from the usual Kolmogorov -5/3 law. Secondly, ST can show the spin-glass-like behavior; the spin density vectors are spatially random but temporally frozen.