Fractional Angular Momentum Research Articles

Fractional angular momentum borne on rotating vortex solitons

We predict the existence of vortex solitons with one and two embedded off-centered phase singularities in competing media trapped in a rotating harmonic potential. The Coriolis force induced by the rotation of external potentials shifts the singularity of the single-charge vortex soliton from the origin to the periphery. In this process, the angular momentum carried by per photon varies continuously from 1 to 0. For vortex solitons with two separated opposite-charge singularities, the counterclockwise rotation leads to a variation of angular momentum per photon from 0 to −1, and vice versa. Particularly, for vortex solitons nesting two singularities of the same charge, two branches of vortices with symmetrical singularities and asymmetric singularities coexist when the rotation frequency exceeds a critical value. The angular momentum per photon still varies continuously with the rotation frequency. Linear stability analysis collaborated by direct propagation simulation demonstrates that rotating asymmetric vortex states carrying fractional angular momentum are extremely robust, provided that their power is high enough.

Fractional Optical Angular Momentum and Multi-Defect-Mediated Mode Renormalization and Orientation Control in Photonic Crystal Microring Resonators.

Whispering gallery modes (WGMs) in circularly symmetric optical microresonators exhibit integer quantized angular momentum numbers due to the boundary condition imposed by the geometry. Here, we show that incorporating a photonic crystal pattern in an integrated microring can result in WGMs with fractional optical angular momentum. By choosing the photonic crystal periodicity to open a photonic band gap with a band-edge momentum lying between that of two WGMs of the unperturbed ring, we observe hybridized WGMs with half-integer quantized angular momentum numbers (m∈Z+1/2). Moreover, we show that these modes with fractional angular momenta exhibit high optical quality factors with good cavity-waveguide coupling and an order of magnitude reduced group velocity. Additionally, by introducing multiple artificial defects, multiple modes can be localized to small volumes within the ring, while the relative orientation of the delocalized band-edge states can be well controlled. Our Letter unveils the renormalization of WGMs by the photonic crystal, demonstrating novel fractional angular momentum states and nontrivial multimode orientation control arising from continuous rotational symmetry breaking. The findings are expected to be useful for sensing and metrology, nonlinear optics, and cavity quantum electrodynamics.

Effects of the septic nonlinearity and the initial value of the radius of orbital angular momentum beams on data transmission in optical fibers using the cubic-quintic-septic complex Ginzburg-Landau equation in presence of higher-order dispersions

In a three-dimensional (3D) dissipative medium described by the higher-order (3+1)-dimensional cubic-quintic-septic complex Ginzburg-Landau [(3+1)D CQS-CGL] equation with the viscous (spectral-filtering) term, self steepening, Raman effect, dispersion terms up to six, diffraction and cubic-quintic-septic nonlinearities, we demonstrate that necklace ring beams with initial spherical shape carrying integer and even fractional angular momentum and whose intensities are azimuthally periodically modulated, can evolve into caterpillar, hexagonal, rectangular, scorpion, diamond and pillow dissipative optical bullets. The outcome of the evolution is controlled by the radius and the value of the septic nonlinearity in the initial necklace ring. We reveal numerically that spatiotemporal necklace-ring solitons carrying integer, and even fractional angular momentum can be self-trapped over a huge propagation distance even in the presence of random perturbations.

Necklace beams carrying fractional angular momentum in fractional systems with a saturable nonlinearity

We report the propagation dynamics of necklace beams in the framework of nonlinear fractional Schrödinger equation. The fractional diffraction can be partially compensated by a saturable nonlinearity and their combination leads to the quasi-stable propagation of necklace beams with integer or fractional angular momentum. The expansion of necklace can be slowed down remarkably by the decrease of Lévy index. There is an energy exchange between the neighboring pearls of necklace with fractional angular momentum, in sharp contrast to the necklace with integer or half-integer angular momentum. We, thus, put forward the first example of necklace-ring beams carrying fractional angular momentum in a fractional configuration.

Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry

The authors develop a topological field theory of crystalline gauge fields, which they use to classify the fractional charge, angular momentum, and linear momentum of anyons and lattice defects in (2+1)D topologically ordered phases of matter.

Fractional Angular Momentum and Anyon Statistics of Impurities in Laughlin Liquids.

The elementary excitations of a fractional quantum Hall liquid are quasiparticles or quasiholes that are neither bosons nor fermions, but are so-called anyons. Here we study impurity particles immersed in a quantum Hall liquid that bind to the quasiholes via repulsive interactions with the liquid. We show that the angular momentum of an impurity is given by the multiple of a fractional "quantum" of angular momentum, and can directly be observed from the impurity density. In a system with several impurities bound to quasiholes, their total angular momentum interpolates between the values for free fermions and for free bosons. This interpolation is characterized by the fractional statistical parameter of the anyons, which is typically defined via their braiding behavior.

Chern–Simons quantum mechanics and fractional angular momentum in atom system

The model of a planar atom which possesses a nonvanishing electric dipole moment interacting with magnetic fields in a specific setting is studied. Energy spectra of this model and its reduced model, which is the limit of cooling down the atom to the negligible kinetic energy, are solved exactly. We show that energy spectra of the reduced model cannot be obtained directly from the full ones by taking the same limit. In order to get the energy spectra of the reduced model from the full model, we must regularize energy spectra of the full model properly when the limit of the negligible kinetic energy is taken. It is one of the characteristics of the Chern–Simons quantum mechanics. Besides this, the canonical angular momentum of the reduced model will take fractional values although the full model can only take integers. It means that it is possible to realize the Chern–Simons quantum mechanics and fractional angular momentum simultaneously by this model.

On Generalized Fractional Spin, Fractional Angular Momentum, Fractional Momentum Operators in Quantum Mechanics

In this study, we have extended the idea of fractional spin introduced recently in literature based on two orders fractional derivative operator. Generalizations of the fractional spin, the fractional angular momentum and the fractional momentum operators were obtained. The theory is characterized by a noncommutativity between the generalized fractional angular momentum and the fractional Hamiltonian. We have derived the corresponding fractional Schrodinger equation and we have discussed its implications on the problems of a free particle and a particle moving in an infinite well potential. Enhancements of their corresponding energies levels and ground energies were observed which are in agreement with phenomenological theories such as noncommutative quantum mechanics.

Fractional Angular Momentum of an Atom on a Noncommutative Plane**Supported by National Natural Science Foundation of China under Grant No. 11465006 and partially supported by 20190234-SIP-IPN and the CONACyT under Grant No. 288856-CB-2016

The mechanism of obtaining the fractional angular momentum by employing a trapped atom which possesses a permanent magnetic dipole moment in the background of two electric fields is reconsidered by using an alternative method. Then, we generalize this model to a noncommutative plane. We show that there are two different mechanisms, which include cooling down the atom to the negligibly small kinetic energy and modulating the density of electric charges to the critical value to get the fractional angular momentum theoretically.

Correlations, dynamics, and interferometry of anyons in the lowest Landau level

In this article, we present a systematic study of quantum statistics and dynamics of a pair of anyons in the lowerst Landau level (LLL), of direct relevance to quasiparticle excitations in the quantum Hall bulk. We develop the formalism for such a two-dimensional setting of two charged particles subject to a transverse field, including fractional angular momentum states and the related algebra stemming from the anyonic boundary condition, coherent state descriptions of localized anyons, and bunching features associated with such anyons. We analyze the dynamic motion of the anyons in a harmonic trap, emphasizing phase factors emerging from exchange statistics. We then describe non-equilibrium dynamics upon the application of a saddle potential, elaborating on its role as a squeezing operator acting on LLL coherent states, and its action as a beam splitter for anyons. Employing these potential landscapes as building blocks, we analyze anyon dynamics in a quantum Hall bulk interferometer. We discuss parallels between the presented LLL setting and other realms, extensively in the context of quantum optics, whose formalism we heavily borrow from, and briefly in that of black hole phenomena.

The fractional angular momentum realized by a neutral cold atom

Inspired by the electromagnetic duality, we propose an approach to realize the fractional angular momentum by using a cold atom which possesses a permanent magnetic dipole momentum. This atom interacts with two electric fields and is trapped by a harmonic potential which enable the motion of the atom to be planar and rotationally symmetric. We show that eigenvalues of the canonical angular momentum of the cold atom can take fractional values when the atom is cooled down to its lowest kinetic energy level. The fractional part of canonical angular momentum is dual to that of the fractional angular momenta realized by using a charged particle. Another approach of getting the fractional angular momentum is also presented. The differences between these two approaches are investigated.

Fractional Angular Momentum at Topological Insulator Interfaces.

Recently, two fundamental topological properties of a magnetic vortex at the interface of a superconductor (SC) and a strong topological insulator (TI) have been established: The vortex carries both a Majorana zero mode relevant for topological quantum computation and, for a time-reversal invariant TI, acharge of e/4. This fractional charge is caused by the axion term in the electromagnetic Lagrangian of the TI. Here we determine the angular momentum J of the vortices, which in turn determines their mutual statistics. Solving the axion-London electrodynamic equations including screening in both a SC and a TI, we find that the elementary quantum of angular momentum of the vortex is -n^{2}ℏ/8, where n is the flux quantum of the vortex line. Exchanging two elementary fluxes thus changes the phase of the wave function by -π/4.

Half theory fractional angular momentum and the application of fractional derivatives to quantum mechanics

In this paper, we review the fractional derivative and apply it to various problems in quantum mechanics. Among other things, we find fractional angular momentum, with “fractional spherical harmonics” as solutions to the squared quantum mechanical angular momentum operator, but with non-integer eigenvalues. These fractional functions might be interpreted as intermediary states visited by an orbiting electron as it jumps between the more stable, integer angular momentum values in an atom. Alternatively, these fractional states may be of interest in the mechanisms of chemical bonding. The justification for these new states, which are normalizable solutions to Hermitian operators, depends of course on experiment. We remember Feynmann’s statement about quantum mechanics, “Everything that is not expressly forbidden, is mandatory.”

Particle on a torus knot: Constrained dynamics and semi-classical quantization in a magnetic field

Kinematics and dynamics of a particle moving on a torus knot poses an interesting problem as a constrained system. In the first part of the paper we have derived the modified symplectic structure or Dirac brackets of the above model in Dirac’s Hamiltonian framework, both in toroidal and Cartesian coordinate systems. This algebra has been used to study the dynamics, in particular small fluctuations in motion around a specific torus. The spatial symmetries of the system have also been studied.In the second part of the paper we have considered the quantum theory of a charge moving in a torus knot in the presence of a uniform magnetic field along the axis of the torus in a semiclassical quantization framework. We exploit the Einstein–Brillouin–Keller (EBK) scheme of quantization that is appropriate for multidimensional systems. Embedding of the knot on a specific torus is inherently two dimensional that gives rise to two quantization conditions. This shows that although the system, after imposing the knot condition reduces to a one dimensional system, even then it has manifested non-planar features which shows up again in the study of fractional angular momentum. Finally we compare the results obtained from EBK (multi-dimensional) and Bohr–Sommerfeld (single dimensional) schemes. The energy levels and fractional spin depend on the torus knot parameters that specifies its non-planar features. Interestingly, we show that there can be non-planar corrections to the planar anyon-like fractional spin.

Fractional angular momentum in noncommutative generalized Chern-Simons quantum mechanics

The noncommutative generalized Chern-Simons quantum mechanics, i.e., the Chern-Simons quantum mechanics on the noncommutative plane in the presence of Aharonov-Bohm magnetic vector potentials, is studied in this paper. We focus our attention on the canonical orbital angular momentum and show that there are two different approaches to produce the fractional angular momentum in the noncommutative generalized Chern-Simons quantum mechanics.

Quantum states emerging from charged transformation-wave in a uniform magnetic field

It is the purpose of this paper to investigate the behavior of a charged transformation-wave in a uniform magnetic field. We first derive the equation which a transformation-wave needs to satisfy, and then, as the main application, the magnetic duality of the Hooke-Newton transmutation is performed by a conformal mapping which generates the novel states with the following characteristics: (a) bound or scattering states are classified by the signatures of the quantum number of the angular momentum and the charge of the particle; (b) the angular momentum satisfies the half-integer quantization rule; and (c) the mass parameter to reach the states is isotropic. Finally, as an alternative application, we report the construction of the fractional angular momentum and fractional Landau levels with the transformation design method.

Tailoring optical complex field with spiral blade plasmonic vortex lens.

Optical complex fields have attracted increasing interests because of the novel effects and phenomena arising from the spatially inhomogeneous state of polarizations and optical singularities of the light beam. In this work, we propose a spiral blade plasmonic vortex lens (SBPVL) that offers unique opportunities to manipulate these novel fields. The strong interaction between the SBPVL and the optical complex fields enable the synthesis of highly tunable plasmonic vortex. Through theoretical derivations and numerical simulations we demonstrated that the characteristics of the plasmonic vortex are determined by the angular momentum (AM) of the light, and the geometrical topological charge of the SBPVL, which is govern by the nonlinear superposition of the pitch and the number of blade element. In addition, it is also shown that by adjusting the geometric parameters, SBPVL can be utilized to focus and manipulate optical complex field with fractional AM. This miniature plasmonic device may find potential applications in optical trapping, optical data storage and many other related fields.

Fractional angular momentum in cold-atom systems.

The quantum statistics of bosons or fermions are manifest through the even or odd relative angular momentum of a pair. We show theoretically that, under certain conditions, a pair of certain test particles immersed in a fractional quantum Hall state possesses, effectively, a fractional relative angular momentum, which can be interpreted in terms of fractional braid statistics. We propose that the fractionalization of the angular momentum can be detected directly through the measurement of the pair correlation function in rotating ultracold atomic systems in the fractional quantum Hall regime. Such a measurement will also provide direct evidence for the effective magnetic field resulting from Berry phases arising from attached vortices, and of excitations with a fractional particle number, analogous to the fractional charge of the electron fractional quantum Hall effect.

Nonlinear Surface-Plasmon Whispering-Gallery Modes in Metallic Nanowire Cavities

We demonstrate that the surface second-harmonic generation can lead to the formation of nonlinear plasmonic whispering-gallery modes (WGMs) in microcavities made of metallic nanowires. Since these WGMs are excited by induced surface nonlinear dipoles, they can be generated even when they are not coupled to the radiation continuum. Consequently, the quality factor of these nonlinear modes can be as large as the theoretical limit imposed by the optical losses in the metal. Remarkably, our theoretical analysis shows that nonlinear plasmonic WGMs are characterized by fractional azimuthal modal numbers. This suggests that the plasmonic cavities investigated here can be used to generate multicolor optical fields with fractional angular momentum. Applications to plasmonic sensors are also discussed.

Generation of a vectorial elliptic hollow beam by an elliptic hollow fiber

We propose a handy scheme to generate a vectorial elliptic hollow beam by using a short dual-mode elliptic hollow fiber, study its propagation characteristics in free space, and calculate its total angular momentum at different propagating distances. Our study shows that the resulting elliptic hollow beam has a self-focusing effect in near field, an arbitrary polarization, and a position-dependent fractional angular momentum and that the directions of the major and minor axes of the elliptical intensity profile will be interchanged after the self-focusing plane.