Structured Multimode Neutron Beams

• Multimode structured neutron beams are engineered wave packets that combine controllable superpositions of spatial, spin, energy, and orbital angular momentum modes using magnetic, RF, and phase-grating techniques.
• They enhance neutron interferometry and scattering by enabling high-dimensional multiplexing, entanglement, and contextuality tests, broadening quantum probing capabilities.
• Recent experiments demonstrate generation of OAM and Airy modes with measurable entanglement witnesses, opening new avenues in imaging, spectroscopy, and material characterization.

Multimode structured neutron beams are neutron wave packets engineered to propagate with controllable superpositions of spatial, spin, energy, and orbital angular momentum (OAM) modes. Such beams, produced by combinations of magnetic, radio-frequency, and phase-grating optical elements, expand the capabilities of neutron interferometry and scattering, allowing simultaneous access to quantum degrees of freedom beyond conventional plane-wave or spin-polarized neutron probes. These advances underpin new methodologies in neutron quantum optics, scattering spectroscopy, and entanglement/contextuality research, and enable high-dimensional multiplexing analogous to techniques in photonics and electron optics.

1. Quantum-Mechanical Formalism for Neutron Multimode States

The formalism underlying multimode neutron beams treats each accessible neutron degree of freedom—spin, path, energy, orbital angular momentum, etc.—as a two-dimensional quantum subsystem. In the finite tensor-product approximation, the total Hilbert space is represented as $H = \bigotimes_{l=0}^{n-1} H_l$, where each $H_l \simeq \mathbb{C}^2$ (Lu et al., 2019). States are vectors in this space, and operators take tensor-product forms.

For instance, using Pauli operators for each mode,

$$\sigma_x^l = |0\rangle\langle 1| + |1\rangle\langle 0|, \quad \sigma_y^l = -i|0\rangle\langle 1| + i|1\rangle\langle 0|,$$

any rotated measurement is represented as

$$\sigma_{u(\phi_l)}^l = \cos\phi_l \,\sigma_x^l + \sin\phi_l \,\sigma_y^l.$$

Eigenstates such as $$|\pm,\phi_l\rangle_l = (|0\rangle + e^{i\phi_l}|1\rangle)/\sqrt{2}$$ allow parametric control of mode phases.

In OAM optics, neutron OAM eigenstates are represented by Laguerre–Gaussian (LG) modes,

$$\psi_{p,\ell}(r,\phi,z) = \sqrt{\frac{2p!}{\pi(p+|\ell|)!}}\frac{1}{w(z)}\left(\frac{\sqrt{2}r}{w(z)}\right)^{|\ell|}L_p^{|\ell|}\left(\frac{2r^2}{w^2(z)}\right) e^{-r^2/w^2(z)} e^{i\ell\phi + ik r^2/(2R(z)) - i\zeta_{p,\ell}(z)},$$

where each neutron then carries OAM $\ell\hbar$ along the propagation axis (Lailey et al., 10 Nov 2025).

2. Experimental Realization: Magnetic, Radio-Frequency, and Phase-Grating Techniques

Magnetic Wollaston Prism (MWP) and Radio-Frequency Spin Flipper (RFNSF)

A Magnetic Wollaston Prism consists of two adjacent triangular regions of antiparallel magnetic fields, bounded by superconductors. The neutron’s spin interacts with the field via Zeeman energy $V = -\mu \cdot B$, creating spin-dependent refraction. The one-dimensional single-particle Hamiltonian for the boundary-normal direction is $H = p^2/2m + V_\pm \theta(x)$, with $V_\pm = \pm |\mu B|$ for spin up/down (Lu et al., 2019). Matching solutions at the interface yields spin-dependent spatial separation; operators such as

$$U_\text{ent}^{(MWP)} = (e^{i\phi} P_\uparrow + e^{-i\phi} P_\downarrow) \otimes U_{BS}$$

act as spin–path entanglers.

Inclined RF spin flippers produce entanglement among spin, path, and energy by coupling RF photons and static field gradients. Two RF flippers can produce three-mode Greenberger–Horne–Zeilinger (GHZ) states, e.g.,

$$U_{\text{ent}}^{(\text{RF},3)} |\uparrow\rangle \otimes |0\rangle \otimes |E_0\rangle = (|{\uparrow},1,E_- \rangle + |\downarrow,2,E_+ \rangle) / \sqrt{2}.$$

Phase-Grating Arrays

Nanofabricated silicon phase gratings impart prescribed phase profiles onto neutron wave functions. Fork-dislocation gratings of topological charge $q$ have transmission functions

$$T_\text{fork}(x,y) = \tfrac12 \left[1 + \mathrm{sgn}\left(\sin\left(\tfrac{2\pi}{p} x + q \arctan(y/x)\right)\right)\right],$$

generating OAM modes $\ell = m q$ in the $m$th diffraction order. Cubic (Airy) gratings,

$$T_\text{cubic}(x,y) = \tfrac12 \left[1 + \mathrm{sgn}\left(\sin\left(\tfrac{2\pi}{p} x + c_x x^3 - c_y y^3\right)\right)\right],$$

produce Airy beam profiles (Lailey et al., 10 Nov 2025). Stacking phase gratings along the beam axis enables superpositions of multiple OAM and Airy modes within the same pulse.

3. Multimode Superposition, Interferometry, and Spin–OAM Entanglement

When multiple subsystems (spin, path, energy, OAM) are entangled via MWPs, RFNSFs, and phase gratings, one can engineer quantum states of the form

$$|\Psi\rangle = \sum_i c_i |s_i\rangle \otimes |p_i\rangle \otimes |e_i\rangle \otimes |\ell_i\rangle \otimes |\mathrm{Ai}_i\rangle,$$

realizing simultaneous control of high-dimensional Hilbert-space components.

In the interferometric context, the measurement sequence follows the “entanglement–phase–recombine” architecture:

1. Prepare an initial product state,
2. Apply entangler (MWP, RF or grating stack),
3. Insert commuting phase shifters in each mode,
4. Recombine using an inverse entangler,
5. Rotate and projectively analyze spin, then count neutrons.

Adjustment of phase settings across modes (e.g., spin coil $U_s(\alpha)$, path shifter $U_p(\chi)$, energy precession $U_e(\gamma)$) modulates the combined modes: $$N(\{\phi_l\}) \propto 1 + \cos\left(\sum_l \phi_l\right)$$ for an ideal $n$-mode GHZ state (Lu et al., 2019).

The extension to path-integral methods and magnetic Snell’s law is necessary for predicting OAM singularities and spin textures induced by MWPs (Thien et al., 2022). The combined action of orthogonally oriented MWP pairs generates transverse phase patterns $e^{i\ell\phi}$, imparting spin-dependent OAM lattice sites.

4. Experimental Evidence and Performance

Experiments conducted on the GP-SANS beamline at the HFIR reactor (ORNL) and ISIS neutron facility have demonstrated:

• Simultaneous generation of multi-OAM mode beams by serially stacking fork phase gratings (e.g. $q=3, q=7$), yielding two concentric doughnut rings in far-field intensity profiles (Lailey et al., 10 Nov 2025). Diffraction efficiencies for first-order OAM modes range from ~1%–2% for $\ell=3$ and ~0.5% for $\ell=7$.
• Hybrid OAM–Airy beams, combining doughnut and lobe structures in a single spatial profile.
• Experimental realization of spin–path–energy GHZ states in single-neutron interferometry, for which measured contextuality witness values (CHSH S ≈ 2.16 and Mermin M ≈ 3.05) exceed classical bounds (Lu et al., 2019).

Performance is fundamentally limited by neutron–silicon phase shift depths, intrinsic beam coherence, field nonuniformity, and analyzer efficiency. The multimode stacking strategy ameliorates low individual mode efficiencies by producing several useful beams in parallel.

5. Measurement of Entanglement and Contextuality: CHSH, Mermin, and n-Mode Witnesses

Entanglement and contextuality tests within multimode neutron experiments use operator-based construction of witness inequalities. For two modes (spin and path), the Clauser–Horne–Shimony–Holt (CHSH) witness

$$S = E(\alpha_1, \chi_1) + E(\alpha_1, \chi_2) + E(\alpha_2, \chi_1) - E(\alpha_2, \chi_2)$$

is built from expectation values $$E(\alpha, \chi) = \langle\psi|\sigma_{u(\alpha)}^s \otimes \sigma_{v(\chi)}^p|\psi\rangle$$, reconstructed via four phase scan counts at settings shifted by $\pi$.

For three modes (spin–path–energy),

$M = E(XXX) - E(XYY) - E(YXY) - E(YYX),$

with classical and quantum bounds at 2 and 4, respectively (Lu et al., 2019).

Generalization to $n$ modes yields Mermin-like witnesses

$$K_n = \sum_{\text{even }Y} E(\ldots) - \sum_{\text{odd }Y} E(\ldots)$$

over all phase contexts. Measurement of each witness involves systematic scans over the accessible phase space, projecting the beam onto the corresponding eigenstates.

6. Applications: Neutron Scattering, Spectroscopy, and Quantum Probing

Multimode structured neutron beams have significant implications in both fundamental and applied physics:

• Selection Rule Exploration: OAM-structured neutrons enable mode-resolved scattering cross-section measurements, providing sensitivity to orbital angular momentum conservation effects in nuclear and magnetic processes (Lailey et al., 10 Nov 2025). This allows direct probing of selection rules and topological phenomena in materials such as skyrmion and meron lattices.
• Auto-Focusing and Self-Healing Beams: Airy beams’ auto-focusing and self-healing properties enhance contrast and interrogation depth in SANS and tomography, allowing probing of extended or inhomogeneous samples.
• 2D Multiplexing: By combining OAM multiplexing with time-of-flight energy resolution, experiments gain parallel channels for simultaneously measuring spatial, angular-momentum, and spectral correlations.
• Imaging and Tomography: Spin–OAM entanglement provides new contrast mechanisms for imaging complex quantum materials with spatially structured polarization patterns (Thien et al., 2022).

7. Future Directions and Theoretical Extensions

The operator formalism and multimodal engineering enable several future directions:

• Greater Mode Complexity: Incorporation of OAM entanglers (e.g. neutron-optical spin–orbit scattering) could yield four-mode (or higher) entangled states.
• Extended Linear Response Theory: A generalization of van Hove’s linear response, incorporating neutron and matter entanglement prior to scattering, is suggested as a pathway to probing many-body quantum correlations in strongly correlated systems such as quantum magnets and superconductors (Lu et al., 2019).
• Multiplexed Neutron Spectroscopy: The analogy to optical OAM multiplexing indicates that multimode neutron beams may dramatically enhance the throughput and dimensionality of neutron scattering experiments.
• Experimental Tests of Decoherence: Systematic mapping of decoherence and phase contrast as modal overlap is tuned is proposed as a test of the tensor product approximation and an avenue for engineering self-testing and fidelity witnesses for single-particle entanglement.

A plausible implication is that routine neutron-structured beam generation, especially OAM–Airy multiplexing, will become central to future neutron science, paralleling advances in photonics and electron optics. The ability to simultaneously address multiple quantum channels per neutron opens prospects in time-resolved, high-dimensional, and non-classical probe technologies.