Probing Single-Particle Spatial Extent With Helical Neutron Wavefronts

Technical Summary: Probing Single-Particle Spatial Extent With Helical Neutron Wavefronts

Problem Statement
A fundamental challenge in neutron physics is distinguishing between the transverse coherence length of a beam and the spatial extent of an individual neutron's wavepacket. In conventional experiments, both quantities manifest as spatial broadening of observed intensity profiles, making them difficult to separate. This ambiguity has led to conceptual misinterpretations in the literature, where the transverse coherence length is sometimes incorrectly identified with the spatial extent of a single neutron's wavefunction. Such conflation has resulted in theoretical predictions, such as orbital angular momentum (OAM) dependent cross sections, that contradict experimental observations. While the Van Cittert-Zernike (VCZ) theorem describes beam coherence as arising from the angular distribution of an ensemble of extended wavepackets, the intrinsic spatial size of a single neutron wavepacket remains experimentally elusive because standard measurements are primarily sensitive to ensemble coherence.

Methodology
The authors introduce a method utilizing helical neutron wavefronts carrying well-defined orbital angular momentum (OAM) to decouple these two physical quantities. The experimental setup was conducted on the SANS2D instrument at the ISIS Neutron and Muon Source. The geometry included a source aperture ($20 \times 20$ mm), a collimation length of 12 m, a sample aperture ($6 \times 6$ mm), and a sample-to-detector distance of 12 m, resulting in a geometric beam divergence of approximately 1.1 mrad.

The experiment employed four fork-dislocation phase gratings etched into a silicon wafer to generate neutron OAM states with topological charges $\ell = 0, 3, 5, 7$. The gratings featured micron-scale phase elements with periods of 120 nm (for $\ell = 0, 3, 7$) and 100 nm (for $\ell = 5$). The study focused on neutrons with wavelengths between 1.75 and 12.5 Å, with a spectral peak near 3 Å.

The core theoretical insight relies on the behavior of helical states. While conventional intensity profiles broaden due to both wavepacket extent and coherence effects, helical states produce annular intensity profiles. The radius of the intensity maximum ($r_{peak}$) in these profiles depends explicitly on the transverse wavepacket extent ($w$), whereas the transverse coherence length ($\sigma$) primarily contributes to the broadening of the profile without dictating the peak's location. The authors simulated the propagation of these states, accounting for the interaction between the wavepacket size and the grating dimensions, to establish a relationship between $r_{peak}$, $\ell$, and $w$.

Key Results
By comparing experimental azimuthally integrated intensity profiles with numerical simulations, the authors independently determined the beam divergence and the lower bound of the single-particle wavepacket extent:

• Beam Divergence and Coherence: The measured beam divergence was found to be $\theta \approx 1.1$ mrad. Using the relationship derived from the VCZ theorem, this corresponds to a transverse coherence length of $\sigma \approx 180$ nm (for $\lambda = 12$ Å).
• Wavepacket Extent: The analysis of the helical intensity profiles established a lower bound for the spatial extent of individual neutron wavepackets of $w \ge 2$ µm.
• Disparity: The spatial extent of the individual wavepackets is more than an order of magnitude larger than the transverse coherence length.

The data showed that as the assumed wavepacket width ($w$) in simulations increased, agreement with experimental data improved, with optimal agreement occurring for $w \gtrsim 2$ µm. Conversely, varying the divergence angle in simulations confirmed the experimental value of 1.1 mrad. The study notes that while coherence-induced broadening can shift the fitted intensity maximum, the annular structure remains resolvable, allowing the method to provide a robust lower bound on the wavepacket size.

Significance and Claims
The paper provides direct experimental evidence that transverse coherence length and single-particle wavepacket extent are distinct physical quantities. This resolves a longstanding source of confusion in neutron literature where these concepts were often conflated.

The authors emphasize that post-selection techniques, such as apertures and collimation, modify the coherence properties of the beam by selecting subsets of the ensemble with narrower angular distributions. This increases the measured coherence length but does not correspond to an increase in the spatial extent of the individual neutron wavepackets. The work clarifies that beam coherence arises from ensemble correlations, whereas the single-particle wavepacket extent is an intrinsic property that evolves via Schrödinger dynamics, potentially becoming comparable to beam dimensions during free propagation.

The study concludes that this distinction holds regardless of specific assumptions regarding Gaussian wavepackets or source profiles, as the separation between ensemble coherence and individual-particle spatial extent is a fundamental consequence of the VCZ theorem and wave mechanics. The authors note that extending this approach to investigate the relationship between longitudinal coherence length and longitudinal wavepacket extent remains a direction for future study.