High-Order Vortex States Overview

Updated 10 July 2026

High-Order Vortex States are characterized by quantized phase winding and singularities with topological charges (ℓ > 1) that define their unique intensity profiles.
They are described through modal frameworks such as Laguerre-Gaussian, Ince-Gaussian, and two-mode Fock states, enabling precise phase control and entanglement quantification.
These states drive advances in diverse applications including optical communications, quantum cryptography, superconductivity, and high-energy vortex detection.

High-order vortex states are non-plane-wave or topological states with a phase singularity and an azimuthal phase factor such as $e^{i\ell\varphi}$ , so that the phase winds by $2\pi \ell$ around the axis and the field vanishes on axis; in alternative conventions the winding is denoted by $n$ , while higher-order modal descriptions may require additional discrete and continuous labels such as $p$ , $m$ , and $\epsilon$ . In optics, matter waves, magnetic textures, fluid vortices, superconductors, and driven Dirac systems, these states are unified by quantized winding, but they differ in how the winding is generated, stabilized, detected, and coupled to other degrees of freedom such as spin, polarization, or Floquet photon sectors (Ivanov, 2022, Banerji et al., 2018).

A high-order optical vortex is a phase singularity of strength $\ell > 1$ ; more generally, vortex states are characterized by a topological charge or winding number, and by a ring-shaped or core-singular intensity or density profile. For scalar particles, the orbital angular momentum projection along the propagation axis is $L_z=\ell\hbar$ , while for spinful or vector particles one often works with a total angular momentum projection that combines spin and orbital parts. The orbital degree of freedom is distinct from spin and is topologically protected in the sense used for vortex states and twisted states (Barros et al., 2023, Ivanov, 2022).

In photonic mode theory, Ince-Gauss modes provide a higher-order generalization beyond the standard Laguerre-Gaussian and Hermite-Gaussian bases. They are complete solutions to the paraxial wave equation, or equivalently the $2$D quantum harmonic oscillator, in elliptical coordinates. Their defining labels are the order $p$ , the degree $2\pi \ell$ 0, and the ellipticity parameter $2\pi \ell$ 1, with the parity restriction $2\pi \ell$ 2. The helical Ince-Gauss states are formed as

$2\pi \ell$ 3

and the parameter $2\pi \ell$ 4 interpolates continuously between the Laguerre-Gaussian regime $2\pi \ell$ 5 and the Hermite-Gaussian regime $2\pi \ell$ 6 (Banerji et al., 2018).

Symbol	Meaning	Context
$2\pi \ell$ 7	topological charge, OAM quantum number	photons, electrons, optical vortices
$2\pi \ell$ 8	winding number	magnetic vortices, BEC vortices
$2\pi \ell$ 9	order, degree, ellipticity parameter	Ince-Gauss modes
$n$ 0	topological charge in lattice solitons	azimuthal waveguide arrays

This shared notation conceals substantial physical variation. In some settings, “high-order” means a multiply quantized singularity with $n$ 1 or $n$ 2; in others, it refers to higher-order spatial-mode families with additional structure in radial, angular, or elliptic coordinates. The distinction is operationally important because instability, splitting, and selection rules depend on whether the relevant object is a single multiply charged singularity, a superposition of several unit-charge vortices, or a higher-order mode family.

The quantum-optical formulation based on Ince-Gauss modes treats high-order vortex states as entangled two-mode photon states with a continuous ellipticity degree of freedom. In the two-mode Fock description, the order $n$ 3 is interpreted as the total number of photons $n$ 4, while the degree $n$ 5 controls the angular structure. Helical Ince-Gauss states admit an expansion in the Laguerre-Gaussian basis with $n$ 6, and the coefficients are determined by mode overlaps. The same framework quantifies entanglement through the von Neumann entropy

$n$ 7

and reports that entanglement increases with increasing photon number, saturates for high $n$ 8, is higher for odd photon numbers than for even ones, and depends sensitively on both $n$ 9 and $p$ 0. The Wigner function of these states displays negativity and nontrivial interference features, especially for matched $p$ 1, establishing nonclassicality in phase space (Banerji et al., 2018).

A distinct generation route starts from two-mode photon-number squeezed states produced by Spontaneous Parametric Down Conversion. In that scheme, a $p$ 2-converter performs a parametric rotation in quadrature space,

$p$ 3

and for $p$ 4 the rotation converts Hermite-Gauss structure into a superposition of Laguerre-Gauss modes. For an input state containing $p$ 5 photons, the resulting quadrature-space profile exhibits $p$ 6 vortices or singularities, so that photon-number fluctuations are mapped into interference effects in quadrature space. The proposal is explicitly framed as a route to improve measurement sensitivity beyond the Standard Quantum Limit $p$ 7 in quantum cryptography, quantum metrology, and sensing (Puentes et al., 2021).

This optical literature also assigns an information-theoretic role to higher-order modal parameters. In particular, the ellipticity parameter $p$ 8 in Ince-Gauss modes is treated as an additional continuous variable expected to play a major role in continuous variable quantum key distribution protocols, while the large modal Hilbert space of vortex and OAM states is used for high-dimensional encoding (Banerji et al., 2018).

3. Nonlinear conversion, interference, aberration, and dissipative optical vortices

Continuous-wave nonlinear conversion can transfer high-order vortex structure across spectral domains. In a two-crystal, singly resonant optical parametric oscillator, one crystal is pumped by a Gaussian beam and the other by optical vortices with orders $p$ 9 to $m$ 0. In this configuration the idler vortices of orders $m$ 1 to $m$ 2 are tunable across $m$ 3, with a maximum output power of $m$ 4 at $m$ 5 for a pump power of $m$ 6, corresponding to a near-IR vortex to mid-IR vortex conversion efficiency as high as $m$ 7. The key mechanism is coherent energy coupling between the resonant signals of the two crystals, which facilitates the transfer of pump vortex of any order to the idler wavelength without stringent operation threshold condition (Aadhi et al., 2018).

At much shorter wavelengths, a linearly polarized Laguerre-Gaussian laser pulse impinging on a solid foil generates intense high-order vortex harmonics in the extreme ultraviolet. In three-dimensional particle-in-cell simulation, both reflected and transmitted fields contain high-order harmonics of the Laguerre-Gaussian mode, and the generated harmonic mode follows the scaling

$m$ 8

so each harmonic photon carries $m$ 9 orbital angular momentum. The reported harmonics are close to the relativistic region in intensity and can have attosecond-scale duration; for the third harmonic, the conversion efficiency is approximately $\epsilon$ 0 in the cited simulation (Zhang et al., 2014).

Interference studies reveal that high-order perfect optical vortices behave differently from Laguerre-Gaussian beams. For two displaced perfect optical vortex beams, the critical normalized separation for splitting into two distinct individual perfect vortices is numerically found at $\epsilon$ 1, and, for fixed beam width and beam radius, this splitting point does not depend on the topological charge $\epsilon$ 2 or on the phase shift $\epsilon$ 3. The interference pattern remains sensitive to axial separation, phase shift, beam radius, and the constituent topological charges, but the comparison with high-order Laguerre-Gauss beams shows that splitting is more rapid in the Laguerre-Gauss case (Das et al., 2024).

Reflection adds another layer of structure through topological aberration. Upon reflection from an interface, a field containing a high-order optical vortex can split into a constellation of unit-charge vortices. The reflected constellation is described in terms of the elementary symmetric polynomials of the vortex coordinates, and the aberration acts as a linear transformation on these polynomials through Wirtinger derivatives of the reflection coefficient. Experimentally, constellations of up to three vortices reflected from a thin metallic film were verified, with first-, second-, and third-order aberration effects directly extracted, and barycenter shifts greater than $\epsilon$ 4 wavelengths observed near surface plasmon resonance (Barros et al., 2023).

Dissipative optical media can also stabilize high-charge vortices rather than destroy them. In azimuthally modulated waveguide arrays with localized gain and nonlinear loss, the accessible topological charge is set by the number of waveguide channels: for odd $\epsilon$ 5, $\epsilon$ 6; for even $\epsilon$ 7, $\epsilon$ 8. These vortex solitons can be excited with nearly vanishing power thresholds, and higher-charge vortices display enhanced propagation stability compared with lower-charge states. In large arrays, power curves for different charges separate clearly; in small arrays they become less distinguishable (Huang et al., 20 Nov 2025).

4. Matter-wave and cold-atom realizations

In ultracold quantum gases, macroscopic superpositions of vortex states have been created by transferring an optical vortex-state superposition to the center-of-mass rotational state of atoms through Raman coupling. The experiment realizes two-vortex and three-vortex superposition states in a nearly spherical $\epsilon$ 9 Bose-Einstein condensate, with the optical OAM superposition generated on a spatial light modulator and mapped onto the atomic rotational state. The winding numbers are verified by the number of interference fringes, $\ell > 1$ 0; the relative phase is obtained from

$\ell > 1$ 1

and the relative amplitude determines the visibility through

$\ell > 1$ 2

The reported lifetime of the vortex superposition state in quantum gases is as large as $\ell > 1$ 3, about two orders of magnitude longer than the storage time in atomic ensembles (Kong et al., 2024).

Spin-orbit-coupled spin-1 Bose-Einstein condensates support another route to high-order vortices, this time through ground-state phase transitions. In the presence of a gradient magnetic field and spin-orbit coupling, the linearized problem admits exact eigenstates when $\ell > 1$ 4, and the chemical potential of the $\ell > 1$ 5-th vortex state is

$\ell > 1$ 6

At critical values $\ell > 1$ 7, determined by $\ell > 1$ 8, higher-order vortex states become the ground state. The paper states that arbitrary winding numbers can be achieved as corresponding to stable ground states; for repulsive spin-spin interactions $\ell > 1$ 9, pure vortex ground states remain stable in the nonlinear regime, while attractive spin-spin interactions $L_z=\ell\hbar$ 0 produce mixed states near the transition points. Harmonic trapping is essential, since removing the trap leads to expansion and destabilization (Zhang et al., 10 Oct 2025).

The cold-atom literature therefore contains both coherent superposition protocols and equilibrium stabilization mechanisms. In the first, the vortex degree of freedom is treated as a controllable qubit or qutrit coordinate on Bloch or nested Bloch spheres; in the second, it is selected by a parameter-driven sequence of ground-state changes. The combination shows that high-order vortex states in matter waves are not restricted to metastable excitations.

5. Magnetic, fluid, and magnonic vortex states

In curved magnets, high-order vortex textures arise from exchange frustration and geometry. For single-wall VSe $L_z=\ell\hbar$ 1 nanotubes, the magnetism is described by a Heisenberg Hamiltonian with ferromagnetic nearest-neighbor coupling $L_z=\ell\hbar$ 2, antiferromagnetic longer-range couplings $L_z=\ell\hbar$ 3 and $L_z=\ell\hbar$ 4, and anisotropy terms $L_z=\ell\hbar$ 5 and $L_z=\ell\hbar$ 6. As the tube diameter increases, competition between $L_z=\ell\hbar$ 7 and $L_z=\ell\hbar$ 8 stabilizes non-collinear $L_z=\ell\hbar$ 9 vortex states with spin texture

$2$0

Magnon excitations in these backgrounds obey an orbital-angular-momentum selection rule

$2$1

For the fundamental vortex $2$2, $2$3 and no hybridization occurs; for the $2$4 state, $2$5 hybridizes with $2$6, producing an eight-petal magnon density pattern. The estimated Néel temperatures for the $2$7 states are $2$8 (Li et al., 10 Sep 2025).

Electron plasmas in a Penning-Malmberg trap realize a different class of high-order vortices through the isomorphism between the drift-Poisson equation and the Euler equation for an inviscid, incompressible $2$9D fluid. In that setting, a V-state is the generalization of the $p$ 0D Kirchhoff elliptical vortex to a generic Kelvin-Helmholtz mode in the nonlinear regime. Multipolar rotating electric fields selectively excite the azimuthal mode $p$ 1, and destructive imaging plus angular Fourier decomposition provide mode-resolved diagnostics of the deformation amplitude. High-order V-states reach the nonlinear regime within a few tens of rotation periods, exhibit saturation, filamentation, and frequency downshift, and at large amplitude the $p$ 2-th mode can collapse into the $p$ 3 daughter mode. Autoresonant excitation is also demonstrated, with the threshold scaling as $p$ 4 (Maero et al., 2024).

These two platforms emphasize different aspects of high-order vortex dynamics. In VSe $p$ 5 nanotubes, the principal novelty is an intrinsic angular-momentum hybridization mechanism that is forbidden for $p$ 6; in the electron-plasma fluid analogue, the focus is controlled access to nonlinear polygonal vortex states and their mode-coupling cascades. In both cases, higher winding enables behaviors not present in fundamental vortices.

6. Superconducting, Dirac, and solid-state topological states

Vortex structure in superconductors depends strongly on pairing symmetry. In spin-singlet chiral superconductors, $p$ 7-wave vortices support only finite-energy bound states, whereas $p$ 8-wave vortices bind zero-energy states that are dispersionless along the vortex line, forming a doubly degenerate Majorana flat band. Exact diagonalization and analytical solutions of tight-binding Bogoliubov-de Gennes Hamiltonians show that the local density of states at the vortex core differs sharply between these cases: the $p$ 9 state exhibits a zero-bias peak, while the $2\pi \ell$ 00 state does not. The tunneling conductance peak of the vortex is considerably broader than that of the antivortex, which is proposed as a direct signature of the chiral order-parameter symmetry (Lee et al., 2015).

High magnetic fields in a quasi-two-dimensional FeSe-based superconductor produce a broader vortex-matter landscape that extends beyond the classical paradigm of vortex solid, glass, and liquid. High-field transport measurements on $2\pi \ell$ 01 map an $2\pi \ell$ 02- $2\pi \ell$ 03 phase diagram to $2\pi \ell$ 04 and identify a current-sensitive fragile superconducting state at low temperatures and high fields, a phase-fluctuating vortex state with finite longitudinal resistance and vanishing Hall resistance, an anomalous non-Ohmic vortex liquid, and an Ohmic vortex liquid at higher temperature or field. The material is described as an extremely type II superconductor with significant thermal fluctuations, and the reported Ginzburg number satisfies $2\pi \ell$ 05 (Li et al., 26 Jan 2026).

Driven massive Dirac systems furnish a light-induced analogue of multiply quantized vortex matter. Under circularly polarized vortex light, a dynamical gap opens in which topological edge states coexist with photoinduced multiply quantized vortex states. The beam vector potential carries OAM $2\pi \ell$ 06, and the effective Floquet-BdG mapping shows that the optical vorticity imprints $2\pi \ell$ 07 on the electronic system. Exactly $2\pi \ell$ 08 vortex-state branches appear in the gap, with even-odd structure in the quasienergy spectrum; in real space and in the local density of states, the number of central bright rings tracks $2\pi \ell$ 09. Localized impurities and impurity clusters mix angular momentum channels and reshape the states, while polarization detuning gradually fills the gap with bulk-derived states; nevertheless, vortex and edge signatures remain observable in the presence of impurities and realistic polarization deviations, including deviations up to approximately $2\pi \ell$ 10 (Walsh et al., 15 Jun 2026).

7. High-energy vortex states, diagnostics, and collisions

In high-energy and particle physics, vortex states are treated as twisted states of photons, electrons, neutrons, and atoms with helicoidal wave fronts and nonzero OAM projection along the average propagation direction. Canonical examples include Bessel and Laguerre-Gaussian wave packets, and the field of a scalar vortex state contains the azimuthal factor $2\pi \ell$ 11. Low-energy realizations exist for photons, electrons, neutrons, and helium atoms, and the review literature emphasizes proposals for extending this degree of freedom to high energies through Compton backscattering, acceleration of vortex electrons, plasma-based techniques, and scattering-based state preparation. The same literature stresses that high-energy collisions of vortex states are theoretically distinctive because double-vortex collisions allow interference between two plane-wave momentum configurations leading to the same final state, giving sensitivity to the overall phase of the scattering amplitude, a quantity not measurable in plane-wave collisions (Ivanov, 2022).

A central diagnostic issue is that ring-shaped intensity alone is not sufficient to prove the existence of a true high-energy vortex state. The proposed superkick method addresses this by colliding a wide vortex wave packet with a compact Gaussian probe at controlled impact parameter $2\pi \ell$ 12. For a genuine vortex with $2\pi \ell$ 13, the average total final transverse momentum $2\pi \ell$ 14 is nonzero, orthogonal to $2\pi \ell$ 15, and can be as large as

$2\pi \ell$ 16

Control calculations show that removing the phase factor $2\pi \ell$ 17 removes the orthogonal superkick effect even if the intensity profile remains ring-shaped. The proposal is presented as an unambiguous detection method for high-energy vortex states and as a proof-of-principle route to the first observation of the superkick effect with vortex electrons using existing technology (Li et al., 2024).

The high-energy perspective also sharpens a recurring theme from lower-energy platforms: the physically decisive quantity is not merely the presence of a ring or a central node, but the phase winding and the coherent angular-momentum structure. In high-order vortex states, whether in optics, condensed matter, or particle beams, experimental access therefore depends on techniques that resolve phase, interference, OAM transfer, or symmetry-protected spectral signatures rather than intensity alone.