Phase vortex lattices in neutron interferometry

Abstract

Neutron Orbital Angular Momentum (OAM) is an additional quantum mechanical degree of freedom, useful in quantum information, and may provide more complete information on the neutron scattering amplitude of nuclei. Various methods for producing OAM in neutrons have been discussed. In this work we generalize magnetic methods which employ coherent averaging and apply this to neutron interferometry. Two aluminium prisms are inserted into a nested loop interferometer to generate a phase vortex lattice with significant extrinsic OAM, 〈L_z〉 ≈ 0.35, on a length scale of ≈ 220 μm, transverse to the propagation direction. Our generalized method exploits the strong nuclear interaction, enabling a tighter lattice. Combined with recent advances in neutron compound optics and split crystal interferometry our method may be applied to generate intrinsic neutron OAM states. Finally, we assert that, in its current state, our setup is directly applicable to anisotropic ultra small angle neutron scattering.

Keywords: Matter waves and particle beams; Techniques and instrumentation.

Fig. 1. Setup schematic.

Sketch of the 4 plate interferometer (145 mm long), containing two (red) orthogonal prisms (blown up on the top portion) and two phase shifters (blue). The neutron beam, coming from the right, forms three loops, two small ones between the first and third and second and fourth plate respectively and a large loop between the first and last plate. The phase shifters can be rotated around the vertical to induce phase shifts between the paths in their respective loops. A position sensitive detector is shown in black. Additionally, in black the coordinate convention used in this paper is shown.

Fig. 2. OAM expectation value and bandwidth.

a Expectation value of the OAM for the test wavefunction (Eq. (4)) as given by the analytical expression in Eq. (8) for various transverse momentum shifts k_⊥ and phase shifts Δα. Around Δα = ± α/2 and k_⊥ = 4π the OAM attains a maximal/minimal value of ± 0.4 b The OAM bandwidth defined by Eq. (11) for ψ_t as a function of transverse momentum shift k_⊥ and phase shift Δα. Inserts (c) and (d) show the behavior of < L_z > and χ, respectively, for small k_⊥ in the vicinity of Δα = π. In all figures k_⊥ is in units of ζ. In the case of the described experiment the normalized k_⊥ ranges from 10⁻⁵ (vertical refraction) to 0.02 (horizontal refraction).

Fig. 3. Probability amplitudes of the first and zeroth order OAM modes.

The first order mode probabilities ℓ = 1 (blue), ℓ = − 1 (red dashed) and the zeroth order mode probability ℓ = 0 (black) are plotted against the phase shift Δα (centered on Δα = π) for various transverse momentum shifts, (a) equal to the experimental case k_⊥σ = 0.015, (b) ten times larger and (c) thirty times larger than in the experimental case. In (a) the ℓ = 0 amplitude is not plotted for improved visibility. It can be clearly discerned that ℓ = ± 1 probabilities widen for increasing refraction, k_⊥. In addition, the ℓ = 1 and ℓ = − 1 probabilities appear to be mirror images of one another (mirrored around Δα = π).

Fig. 4. Measurement results and fit.

a The processed, normalized and filtered image of the neutron vortex lattice, recorded using the position sensitive detector seen in Fig. 1. The contrast, according to the fit (b) based on Eq. (23), is 0.53. The lattice period is 1.83 mm.

Fig. 5. Reconstructed wavefunction and OAM expectation value.

a Image of the real part of the test wavefunction of a single vortex carrying extrinsic OAM. This test wavefunction is reconstructed using the fit parameters generated by the model shown in Fig. 4. A circle is drawn in the center of the image indicating the domain on which the spatially averaged AFTs are applied and the first order approximations used throughout the paper are valid. The axis around which the OAM is defined is centered on and normal to this ciruclar domain. b The average extrinsic OAM 〈L_z〉 over the image is shown. This is calculated using the spatially averaged AFT (Eq. (24)) and Eq. (25).

Fig. 6. Raw datasets.

Sum over the raw datasets used to generate the figures shown in this paper. a Image with the prisms inserted. b Image of the intensity distribution without prisms in the interferometer.

Fig. 7. Step by step illustration of the image processing technique.

The raw image (a) is binned (b) by a factor of 10 × 10 squared pixels. The first normalization is shown in (c), followed by the next normalization steps (d) by dividing by a quadratic polynomial and subsequently dividing by the mean intensity and subtracting one. Finally, the Fourier filter is applied resulting in the last image (e).