Lagrangian Dynamics of the Musakhail Aether Dynamical ‎Lagrangian

Utilizing the spin degree of freedom breaks new ground for designing photonic devices. Seeking out a suitable platform for flexible steering of photonic spin states is a critical task. In this work, we demonstrate a versatile Liquid crystal (LC) based platform for manipulating photonic spin and orbital states. Owing to the photoalignment technique, the local and fine tuning of the LC medium is effectively implemented to form various anisotropic microstructures. The light-matter interaction in the corresponding medium is tailored to control the evolution of photonic spin states. The physical mechanism of such a system is investigated, and the corresponding dynamical equation is obtained. The high flexibility endows the LC-based photonic system with great value to be used for Hamiltonian engineering. As an illustration, the optical analogue of intrinsic spin Hall effect (SHE) in electronic systems is presented. The pseudospins of photons are driven to split by the anisotropic effective magnetic field arising from the inhomogeneous spin-orbit interaction (SOI) in the twisting microstructures. In virtue of the designability of the LC-based platform, the form of the interaction Hamiltonian is regulated to present diverse PSHE phenomena, which is hard to be realized in the solid electronic systems. Some representative samples are prepared for experimental observation, and the results are in good agreement with theoretical predictions. We believe the tunable LC system may shed new light on future photonic quantum applications.

This Review covers recent advances in the implementation of spin–photon interfaces in semiconductor quantum dots, nitrogen–vacancy centres in diamond and emerging systems such as colour centres in other wide-bandgap materials. Realization of a quantum interface between stationary and flying qubits is a requirement for long-distance quantum communication and distributed quantum computation. The prospects for integrating many qubits on a single chip render solid-state spins promising candidates for stationary qubits. Certain solid-state systems, including quantum dots and nitrogen–vacancy centres in diamond, exhibit spin-state-dependent optical transitions, allowing for fast initialization, manipulation and measurement of the spins using laser excitation. Recent progress has brought spin photonics research in these materials into the quantum realm, allowing the demonstration of spin–photon entanglement, which in turn has enabled distant spin entanglement as well as quantum teleportation. Advances in the fabrication of photonic nanostructures hosting spin qubits suggest that chips incorporating a high-efficiency spin–photon interface in a quantum photonic network are within reach.

Active manipulation of photonic spin state and optical chirality leads to some key applications, such as in multichannel communication, polarization‐sensitive imaging, chiral spectroscopy, and chiral sensing. Magneto‐optical materials have unique advantages in the intrinsic transmission and magnetic control of photonic chiral spin states. Here, a scheme for dynamic terahertz (THz) anisotropy and chirality manipulations in the transversely magnetized InSb and its hybrid magneto‐optical metasurface structure is presented. A special transverse photonic spin state in the InSb and a transverse−longitudinal spin coupling effect in the hybrid magneto‐optical metasurface are revealed by the eigenmode analysis and numerical simulations. The strong magnetic birefringence effect induced by this spin mode is demonstrated in the experiment. Moreover, the symmetry‐breaking mechanism in this magneto‐optical structure leads to strong intrinsic chirality and polarization conversion. The experimental results confirm the magnetically active manipulation of spin states and their asymmetric transmission in this hybrid magneto‐optical metasurface, which achieve a polarization conversion rate of near 100% and an induced intrinsic chirality of over 15 dB. This work opens a new development for active THz polarization control and chiral manipulation in the magneto‐optical microstructure.

Trajectory Design in Irregular Gravitational Fields Based on Center Manifold Theory

Dielectric metasurfaces with spatially varying birefringence and high transmission efficiency can exhibit exceptional abilities for controlling the photonic spin states. We present here some of our works on spin photonics and spin-photonic devices with metasurfaces. We develop a hybrid-order Poincare sphere to describe the evolution of spin states of wave propagation in the metasurface. Both the Berry curvature and the Pancharatnam-Berry phase on the hybrid-order Poincare sphere are demonstrated to be proportional to the variation of total angular momentum. Based on the spin-dependent property of Pancharatnam-Berry phase, we find that the photonic spin Hall effect can be observed when breaking the rotational symmetry of metasurfaces. Moreover, we show that the dielectric metasurfaces can provide great flexibility in the design of novel spin-photonic devices such as spin filter and spin-dependent beam splitter.

Some invariant relations in the problem of the motion of a body on a smooth horizontal plane

We theoretically investigate tunable photonic spin Hall effect and in-plane spin splitting of light employing a parity time (PT)-symmetric 1D multilayer structure consisting of dielectric layers and liquid crystal as a defect layer. With generalized beam propagation method, the spatial separations between opposite spin states (LCP and RCP) in vertical and horizontal directions are examined with varying loss/gain and molecular orientations of liquid crystal. The exceptional points and non-Hermitian optical properties of the proposed 1D structure are studied using scattering and transfer matrix methods. In addition, we report enhancement, suppression and directional switching of vertical as well as horizontal spatial separations with the variation of loss/gain and the molecule orientation for both forward and backward incidences. The molecular orientation of the liquid crystal and the reflection coefficients reveal spatial separation of LCP and RCP components of the incoming light. The active manipulation of photonic spin states in vertical and horizontal directions unfolds potential applications in optical metrology, quantum information, spintronics, and non-reciprocal devices etc.

The codes of matter and their applications

Dynamic control of photonic spin state and chirality plays a vital role in various applications, such as polarization control, polarization-sensitive imaging, and biosensing. Here, we present a scheme for the flexible and dynamic manipulation of terahertz spin state conversion and optical chirality by combining two achiral structures: an asymmetric metasurface and a layer of anisotropic liquid crystal. The proposed asymmetric metasurface can realize the polarization conversion effect. For the circularly polarized incidence, it exhibits the asymmetric transmission of the spin-flipped states but no spin-locked optical chirality since its geometry is mirror symmetric along with the wave propagation. The introduction of the liquid crystal makes the composite metasurface not only exhibit the spin state conversion but also spin-locked chirality and spin-flipped chirality on account of breaking mirror symmetry, which realizes an electrically active terahertz chiral device. The experimental results show that the asymmetric transmission of the terahertz spin states can be dynamically manipulated, resulting in a large controllable range 83.8% to \ensuremath{-}30.7% of spin-locked circular dichroism at 0.76 THz and \ensuremath{-}98.2% to 44.7% of spin-flipped circular dichroism at 0.73 THz. This work paves the way for the development of terahertz meta devices capable of enabling active photonic spin state and chirality manipulation.

The well defined spin and orbital angular momentum states of photons offers a practical realization of quantum digits and a means of secure single-photon optical communication. The orbital angular momentum is associated with the spatial distribution of the wavefunction and the number of orbital angular momentum eigenstates is unlimited, giving the possibility of arbitrary base-N digits. In this paper, free-space optical communications using angular momentum states of single photons is investigated and in particular the effect of atmospheric turbulence on the angular momentum of the photons is modelled. The refractive index fluctuations in the atmosphere perturb the complex amplitude of a propagating beam so that the photons that were launched in an eigenstate of orbital angular momentum are no longer guaranteed to be in the original eigenstate after propagation. By considering the resulting wave as a superposition of angular momentum states, the probability of obtaining correct or incorrect measurements of the transmitted digit is calculated. The effect on a free-space optical communication using orbital angular momentum and the use of adaptive optics is discussed. The information capacity per photon is quantified and compared to that using polarization states for binary digits.

Spin photonics revolutionizes photonic technology by enabling precise manipulation of photon spin states, with spin-decoupled metasurfaces emerging as pivotal in complex optical field manipulation. Here, we propose a folded-path metasurface concept that enables independent dispersion and phase control of two opposite spin states, effectively overcoming the limitations of spin photonics in achieving broadband decoupling and higher integration levels. This advanced dispersion engineering is achieved by modifying the equivalent length of a folded path, generated by a virtual reflective surface, in contrast to previous methods that depended on effective refractive index control by altering structural geometries. Our approach unlocks previously unattainable capabilities, such as achieving achromatic focusing and achromatic spin Hall effect using the rotational degree of freedom, and generating spatiotemporal vector optical fields with only a single metasurface. This advancement substantially broadens the potential of metasurface-based spin photonics, extending its applications from the spatial domain to the spatiotemporal domain.

The lack of low‐loss and high‐efficiency nonreciprocal isolators has become one of the limitations in the development of terahertz (THz) application systems. This work demonstrates that the longitudinally magnetized InSb can achieve one‐way transmission for one photonic spin state but not for linear polarization (LP) state due to the chiral mirror‐symmetry of the two spin states. To solve this issue, a silicon microstructure is fabricated on the InSb substrate to form a magnetoplasmon/dielectric metasurface, where both the time‐reversal and mirror‐reversal symmetric transmission of the two spin states can be broken. In this device, the forward LP state is efficiently transformed into one of the spin states and output with low loss, but the backward wave is forbidden, which achieves the one‐way transmission for the LP incidence with over 30 dB isolation and only 1.7 dB insertion loss. When a THz polarizer is added behind the device to obtain the LP output, the isolation can reach 40 dB, significantly better than the previous reports. This study is significant to understand the nonreciprocal transmission and manipulation mechanism of THz spin states in the magnetized semiconductor and promotes the development of high‐performance THz isolators under the weak magnetic field.

We give a review on Poor Man's Majorana (PMM) modes, which are theoretically established in the minimal Kitaev chain implementation consisting of two grounded, spinless quantum dots (QDs) operating at thesweet spotcondition, where electron cotunneling and crossed Andreev reflection amplitudes achieve precise balance. Particularly, we systematically review, within the Green's functions theoretical framework, the PMM hybridization dynamics under spin-exchange perturbations proposed by some of us in Sanches et al 2025 J. Phys.: Condens. Matter 37 205601, which demonstrates a characteristic spatial delocalization when subjected to an exchange couplingJmediated by a quantum spinS. This spin-exchange induced PMM spillover effect provides a spectroscopic protocol for determining the quantum statistics ofSthrough the emergent multi-level structure in the proximal QD's density of states. Our principal theoretical result establishes that the exchange interaction generates2S+2(2S+1) satellite states symmetrically distributed about the zero-bias anomaly, serving as a definitive signature of bosonic (fermionic) spin statistics. As novelty, we demonstrate that multi-terminal environmental coupling induces significant suppression of the spin-exchange spillover mechanism. Under constrained variations ofJ, this effectively localizes the perturbed PMM within its host QD, preventing spatial hybridization with adjacent site. The absence of topological protection in this minimal Kitaev realization is strategically leveraged to: (i) develop a novel spectroscopic technique for quantum spin characterization through PMM hybridization signatures; (ii) propose the 'environmentally induced protection', an engineered dissipative spectral stabilization for PMMs against exchange fluctuations in multi-terminal architectures.

Structural dynamic models are used to simulate the performance of structures in a dynamic environment. The predictive accuracy of a model has been defined as the accuracy of predicted structural response under conditions for which the structure has not yet been tested. This paper addresses the quantification of modeling uncertainty, not including the additional uncertainties that may be introduced by the dynamic environment itself. Uncertainty is quantified relative to measured quantities (i.e., experimentally derived modal frequencies and displacements). Although these “measurements” are known to be uncertain themselves, they are taken as the “truth” reference, so modeling uncertainty by definition includes experimental uncertainty as well as parametric uncertainty and the uncertainty of model form (the equations of motion). This paper shows how this modeling uncertainty can be derived by comparing analysis and test modes of generically similar structures and thereafter be used to evaluate the accuracy of numerical simulations based on prior analysis and test experience. Practical examples are given.

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.