Neutron Orbital Angular Momentum States

• Neutron orbital angular momentum states are quantum eigenstates with helical phase structures that create doughnut-shaped intensity profiles for advanced wave manipulation.
• They are generated using transmission holograms and microfabricated phase gratings, combining cubic (Airy) and helical phases to produce Airy-vortex beams.
• Applications include enhanced neutron scattering, phase-contrast imaging, and probing of fundamental quantum and topological effects with improved selectivity.

Neutron orbital angular momentum (OAM) states are quantum eigenstates in which free neutrons acquire a well-defined projection $\ell \hbar$ of angular momentum along their propagation direction, analogous to OAM states in optics and electron vortex beams. Recent experimental advances have enabled the generation, detection, and application of neutron beams with engineered OAM and related structured matter-wave modes—most notably Airy beams and their OAM-carrying hybrids. These developments extend the toolkit of neutron optics, furnishing new degrees of freedom for neutron scattering, spectroscopic selectivity, and studies of fundamental quantum behavior in neutron matter waves.

1. Theoretical Foundation: Neutron OAM and Airy States

Neutron OAM states arise from wavefunctions with an azimuthal phase dependence $\exp(i \ell \phi)$ about the beam axis, where $\ell \in \mathbb{Z}$ is the OAM (topological charge). The paraxial wave equation for neutrons in free space admits solutions with such phase windings, leading to doughnut-shaped intensity profiles with a phase singularity at the axis. Neutron Airy beams, by contrast, are solutions to the free-space Schrödinger equation exhibiting transverse self-acceleration, non-diffraction, and self-healing, with their intensity given by the square of Airy functions: $$\psi_0(x, y, 0) = N e^{- (x^2 + y^2) / (2 \sigma_t^2)} \, \text{Ai}(x / x_0) \, \text{Ai}(y / x_0)$$ Under propagation, these packets preserve their shape and accelerate along parabolic trajectories, as shown by Berry and Balazs, with leading lobe coordinate $x_a(z) = (1 / (4 k^2 x_0^3)) z^2$.

The combination of OAM and Airy features—achieved by superposing a cubic (Airy) phase with a helical (OAM) phase—produces "Airy-vortex" neutron beams that carry both quantized OAM and exhibit transverse acceleration (Sarenac et al., 28 Jul 2024, Lailey et al., 10 Nov 2025). This expands both the mathematical and physical landscape of structured neutron beams for experimental and theoretical investigations.

2. Generation of Structured Neutron OAM States

Traditional neutron-optical elements such as refractive lenses are highly inefficient due to the weak neutron-matter interaction (refractive index $n \approx 1 - 10^{-5}$). Instead, OAM and Airy-like states are typically generated by imposing spatially-varying phase patterns via transmission holograms or microfabricated phase gratings.

Phase Mask Design

For Airy beams, a binary phase mask with a cubic spatial profile is used. Each cell imparts a local phase shift determined by: $$\Phi(x, y) = \frac{1}{2} \left[ \text{sgn} \left\{ \cos \left[ 2\pi x / p + c_x x^3 - c_y y^3 \right] \right\} + 1 \right]$$ where $p$ is the grating period and $c_{x,y}$ are cubic coefficients controlling the Airy trajectory (Sarenac et al., 28 Jul 2024, Lailey et al., 10 Nov 2025).

To generate OAM modes or hybrid Airy-OAM beams, a helical phase factor $\ell \varphi$ is superposed: $$\psi(x, y) \sim \text{Ai}(x/x_0) \, \text{Ai}(y/x_0) \, e^{i \ell \varphi}$$ Such complex phase structures are fabricated by electron-beam lithography and reactive-ion etching into silicon wafers, with typical element sizes $\sim1\,\mu$m, etch depths $\sim300$ nm, and periods $p = 120$ nm.

3. Propagation and Measurement: SANS Geometry and Diagnostics

Experimental realization requires maximizing the neutron coherence and intensity transmitted through the phase mask, given cold neutron fluxes $10^5$–$10^7$ n/cm$^2$/s and coherence lengths $\sigma_\perp \sim 1$–$10\,\mu$m. Strategies include:

• Using large-area ($0.5$ cm $\times$ $0.5$ cm) phase masks integrating millions of coherently radiating regions in parallel.
• Employing small-angle neutron scattering (SANS) setups with optimized source (20 mm) and sample (4 mm) apertures to enhance transverse coherence ($\sigma_\perp \approx 3\,\mu$m).
• Monte Carlo simulations accounting for wavelength spread ($\Delta\lambda/\lambda \sim 0.13$), gravity, finite mask size, detector resolution, and vibrational blurring.
• Post-processing techniques such as low-pass filtering and Gaussian convolution to correct for Poisson noise and detector imperfections (Sarenac et al., 28 Jul 2024).

Detection of the resulting beams is accomplished with large-area position-sensitive detectors ($1$ m$^2$ active area, pixel size $\sim5.5$ mm $\times$ $4.3$ mm), allowing direct imaging of both OAM doughnut modes and Airy lobe structures at distances up to $19.4$ m downstream.

4. Experimental Results: Neutron OAM/Airy Beams and Their Properties

Neutron Airy beams and OAM states have now been robustly generated and characterized in the far field (Sarenac et al., 28 Jul 2024, Lailey et al., 10 Nov 2025). Key observations include:

• Non-diffracting propagation and transverse acceleration of Airy beams, with centroid shifts $x_a(z)$ following quadratic dependence on propagation distance.
• Self-healing, as demonstrated by occluding the main lobe and observing its downstream reformation at later $z$.
• Doughnut-shaped OAM modes (modes with helical phase) differing from Airy beams in lacking transverse acceleration and autofocusing.
• Comparable generation efficiencies for Airy and OAM beams, with first-order Airy lobes capturing a few percent of the incident flux (Lailey et al., 10 Nov 2025).

Hybrid multimode beams—combining multiple OAM modes and Airy components—have also been achieved by mask multiplexing, enabling simultaneous delivery of several quantized angular momentum projections and non-spreading, curved trajectories. This approach mirrors OAM-multiplexing in optical domains and permits efficient interrogation of scattering processes with a discretized OAM spectrum (Lailey et al., 10 Nov 2025).

5. Applications in Neutron Scattering and Fundamental Quantum Studies

Structured neutron OAM and Airy beams provide a spectrum of novel capabilities in both applied and fundamental neutron science:

• Enhanced selectivity and information retrieval in elastic and inelastic scattering by leveraging OAM and Airy-induced selection rules and momentum transfer patterns.
• Investigation of chiral and topological magnetic textures (e.g., skyrmions) using Airy-vortex beams which combine quantized OAM with self-accelerating trajectories (Sarenac et al., 28 Jul 2024).
• Depth-selective SANS and phase-contrast imaging using auto-focusing properties of counter-propagating Airy beams, circumventing the lack of neutron lenses.
• Studies of geometric (Berry or Aharonov–Bohm) phases accumulated during propagation in non-inertial frames or in the presence of external fields; neutrons are particularly sensitive due to their intrinsic magnetic moment and weak interaction with matter.
• Comparative analyses of scattering processes with plane-wave, Airy, and OAM neutron beams to expose novel selection rules and enhance probing capabilities in materials science (Lailey et al., 10 Nov 2025).

6. Implications, Future Directions, and Outlook

The ability to manipulate neutron orbital angular momentum and complex transverse modes opens a new paradigm in neutron optics, enabling:

• Airy-beam multiplexers and OAM beam splitters for parallelized measurements and advanced instrument design.
• Cascaded or hybrid mask strategies to engineer increasingly exotic structured beams, such as pure Airy-vortex or higher-order OAM-Airy hybrids.
• Probing geometric and topological effects in quantum matter, with neutron wave-packet structures sensitive to subtle electromagnetic and gravitational phases (Sarenac et al., 28 Jul 2024).
• Autofocusing and self-healing phenomena, with potential for imaging through scattering or absorptive media, and enhanced sensitivity to scattering signatures in complex environments.

A plausible implication is that as mask framework fabrication advances and higher neutron fluxes become available, the landscape of neutron OAM/Airy beam applications will extend further into both quantum foundational tests and high-resolution material investigations. The limitations posed by low fluence and coherence have been meaningfully mitigated by holographically engineered masking and SANS geometry, setting the stage for broader adoption in experimental neutron science.

This body of work establishes neutron OAM and Airy states—alongside their multimode, hybrid variants—as robust, reproducible platforms for structured wave-matter interaction studies. The intersection of macroscopic quantum state engineering and neutron optics thus continues to advance in both methodological sophistication and experimental reach (Sarenac et al., 28 Jul 2024, Lailey et al., 10 Nov 2025, Ferrante et al., 27 Oct 2025).