Generation and detection of spin-orbit coupled neutron beams

TL;DR

This paper introduces a method to generate and detect neutron beams with coupled spin and orbital angular momentum, enabling advanced studies in topological materials and neutron optics.

Contribution

It presents a novel technique for creating and detecting spin-orbit coupled neutron beams using $^3$He spin-filters and triangular coils, a new tool in neutron optics.

Findings

Neutron beams with tailored spin-orbit correlations were successfully generated.

Spin-dependent intensity profiles demonstrated the presence of spin-orbit coupling.

Potential applications in studying topological materials were identified.

Abstract

Spin-orbit coupling of light has come to the fore in nano-optics and plasmonics, and is a key ingredient of topological photonics and chiral quantum optics. We demonstrate a basic tool for incorporating analogous effects into neutron optics: the generation and detection of neutron beams with coupled spin and orbital angular momentum. $^3$He neutron spin-filters are used in conjunction with specifically oriented triangular coils to prepare neutron beams with lattices of spin-orbit correlations, as demonstrated by their spin-dependant intensity profiles. These correlations can be tailored to particular applications, such as neutron studies of topological materials.

Equations

$$\sigma =\frac{\lambda L_1}{s}\sim 0.4\mu \text{m}$$

$$|{\Psi }_{\mathrm{in}}⟩=|{\uparrow }_z⟩.$$

$${\hat{U}}_y=e^{-i\frac{\pi }{a}y{\hat{\sigma }}_x}{\hat{U}}_x=e^{-i\frac{\pi }{a}x{\hat{\sigma }}_y}$$

$$a=\frac{2\pi v_z}{{\gamma }_{\mathrm{n}}|B|\tan (\theta )}$$

$$|{\Psi }_Q⟩\sim \left[\cos \left(\frac{\pi r}{a}\right)|{\uparrow }_z⟩+i{e}^{-i\varphi }\sin \left(\frac{\pi r}{a}\right)|{\downarrow }_z⟩\right],$$

$$|{\Psi }_{\mathrm{LOV}}^N⟩=({\hat{U}}_x{\hat{U}}_y{)}^N|{\Psi }_{\mathrm{in}}⟩.$$

Full text

††preprint: APS/123-QED

I Introduction

Studies of optical OAM have blossomed since the early 1990’s and are now encompassed in a larger framework of structured waves of light and matter Rubinsztein-Dunlop et al. (2016); Barnett et al. (2017). OAM has been induced in beams of light Allen et al. (1992), electrons Bliokh et al. (2007); Uchida and Tonomura (2010); McMorran et al. (2011), and neutrons Clark et al. (2015). Photonic OAM has demonstrated usefulness in edge-detection microscopy, quantum information processing protocols, encoding and multiplexing of communications, and optical manipulation of matter. Mair et al. (2001); Wang et al. (2012); Andersen et al. (2006); He et al. (1995); Friese et al. (1996); Brullot et al. (2016); Simpson et al. (1997); Zhang et al. (2019). Electron OAM beams have found applications in the characterization of nanoscale magnetic fields in materials  Grillo et al. (2017) and exploration of magnetic monopoles Béché et al. (2014). Neutron OAM has shown promise in the detection of buried interfaces Sarenac et al. (2016). Furthermore, it has been theorized that neutron OAM can modify Schwinger scattering of neutrons on nuclei Afanasev et al. (2019), and might also enable studies of neutron’s internal structure Larocque et al. (2018).

A related set of techniques have been developed for preparing and characterizing beams in which the spin and OAM are correlated. In the case of photons, these “spin-orbit” beams possess correlations between polarization and the OAM Maurer et al. (2007); Marrucci et al. (2006), whereas for electrons and neutrons the correlations are between the spin and the OAM Karimi et al. (2012); Nsofini et al. (2016). Photonic spin-orbit beams have been demonstrated and they have enriched the application range of OAM beams by increasing the number of accessible degrees of freedom Marrucci et al. (2011); Milione et al. (2015); Schmiegelow et al. (2016); Vallone et al. (2014). Although analogous preparation methods have been proposed for electrons Karimi et al. (2012, 2014) they have yet to be implemented in the laboratory.

In this paper we demonstrate not only the first preparation and characterization of spin-orbit beams using neutrons, but also neutron beams with a lattice of spin-orbit correlations. Our technique, which we previously demonstrated with light Sarenac et al. (2018a), involves the preparation of a spin-orbit textured “lattice of vortices” wavefront. A variety of textures can be generated by this method Sarenac et al. (2018b), including skyrmion-like geometries analogous to those recently observed in evanescent electromagnetic fields Tsesses et al. (2018). We expect the techniques shown here to pave the way for neutron OAM and spin-orbit applications in material characterization and fundamental physics.

II Methods

The experiment was carried out the on the Polarized 3He And Detector Experiment Station (PHADES) pha at the National Institute for Standards and Technology Center for Neutron Research (NCNR). A monochromatic beam of neutrons with wavelength $\lambda=0.41$ nm was directed into the setup, as shown in Fig. 1. The beam divergence was $\sim 1\degree$ in both $x$ and $y$ directions. The setup is composed of a slit, two 3He neutron spin filters, guide field coils, two pairs of specifically orientated triangular coils, a permalloy tube, and a neutron camera. The neutron camera has a 25 mm diameter active area and a spatial resolution of 100 $\mu$m Dietze et al. (1996).

A $1\times 1$ mm2 slit was placed at the start of the setup. The slit sets the lower limit on the transverse coherence length at the first triangular coil to:

[TABLE]

where $L_{1}=0.965$ m is the distance from the slit to the first triangular coil and $s$ is the slit width. Although it might be desirable for some specific applications to reduce the slit width to the point that the transverse coherence length extends over the beam diameter, it is not practical in this experiment as the neutron peak count rate was $\sim 15$ $s^{-1}$ at the camera.

Two 3He cells were used as the spin polarizer and the spin analyzer due to their spatially homogeneous neutron polarization Chen et al. (2014a). The cells were polarized in an off-line lab using spin-exchange optical pumping Chen et al. (2014b), and they were changed three times during the experiment. Their initial 3He polarization at the beamline was measured to be between $73~{}\%$ and $82~{}\%$, while their relaxation time was measured to be between $365$ h and $516$ h. The polarization of the neutron beam would reduce from $\sim~{}94~{}\%$ to $\sim~{}90~{}\%$ during a 2-3 day time period. The spin filter direction could be aligned with the $\pm z-$axis at the beamline using the adiabatic fast passage nuclear magnetic resonance method Abragam and Abragam (1961).

To prepare neutron beams with lattices of spin-orbit correlations we used specifically orientated triangular coils that acted as “Lattice of Optical Vortices” (LOV) prism pairs Sarenac et al. (2018a, b). They were arranged to induce magnetic phase gradients perpendicular to each other as well as to the incoming neutron beam. Therefore the coil arrangement differs from the Wollaston arrangement where two triangular coils with anti-parallel fields are placed with their inclined sides facing each other Li et al. (2017); Bouwman et al. (2008). The triangular coils have side lengths of 8.5 cm, 12 cm, and 14.7 cm with an overall height of 7.3 cm. At an applied 10 A current their inner magnetic field was $\sim 0.014$ T which provided a magnetic phase gradient of $\sim 1.5\pi$ rad/mm. The triangular coils were run between 2.5 A and 10 A throughout the experiment, and in every configuration the current in each coil was optimized to compensate for beam divergence.

As shown in Fig. 1 there is a spatially dependant path difference between the 2${}^{\text{nd}}$ and the 3${}^{\text{rd}}$ triangular coils due to their inclined sides. Therefore it is necessary to minimize the magnetic field in this region to avoid an unwanted phase gradient across the beam. In our setup this was accomplished via a permalloy tube. The tube was built from 15 layers of a $500$ $\mu$m thick nickel-iron soft ferromagnetic sheets. The sheets were wrapped around a thin-walled aluminium pipe with an inner diameter of 3.18 cm and whose ends were cut to match the angled prism faces. Guide coils were placed between other triangular coils to provide a uniform magnetic field along the spin quantization axis.

III Results and Discussion

The first 3He polarizer filters the neutrons with spin along the beam propagation axis, thereby setting the neutron wavefunction to

[TABLE]

The triangular coils induce perpendicular phase gradients along the directions that are also perpendicular to the direction of the incoming spin state. Pairs of triangular coils then effectively act as LOV prism pairs, as described in Ref. Sarenac et al. (2018b). In this particular case their individual operators are given by:

[TABLE]

where $\hat{\sigma}_{y}$ and $\hat{\sigma}_{x}$ are the Pauli spin operators, and $a$ is the spatial spin oscillation period. For the case of no beam divergence:

[TABLE]

where $|B|$ is the magnetic field inside the triangular coils, $v_{z}$ is the neutron velocity, $\gamma_{\mathrm{n}}$ is the neutron gyromagnetic ratio Mohr et al. (2016), and $\theta$ is the incline angle of the triangle coils. For example, for a field of $|B|\sim 0.005$ T inside the triangular coils the corresponding period of a non-diverging beam would be $a\sim 3.8$ mm.

A pair of specifically orientated triangular coils, or a LOV prism pair, approximates the action of a quadrupole magnetic field Sarenac et al. (2018a). The state induced by a quadrople acting on $\ket{\Psi_{\mathrm{in}}}=\ket{\uparrow_{z}}$ has the following form Nsofini et al. (2016):

[TABLE]

where $(r,\phi)$ are the cylindrical coordinates. It follows from Eq. 5 that two spin states possess a differing spatial amplitude profile and that there is an azimuthal phase difference between the two spin states which indicates the OAM difference between the two spin states of $\Delta\ell=\ell_{\uparrow}-\ell_{\downarrow}=1$.

In addition to approximating the quadrupole operator, LOV prism pairs possess a periodic structure which induces a 2D lattice structure in the output state Sarenac et al. (2018a). The state after N sets of LOV prism pairs is given by:

[TABLE]

After passing through one of the triangular coils the spin polarization of the beam oscillates along the direction of the coil incline. Therefore the intensity profile post-selected on one spin state exhibits linear fringes with period $a$, as shown in Fig. 2a. After passing through a pair of perpendicular triangular coils, or a LOV prism pair, an N=1 lattice of spin-orbit correlations is prepared. The intensity profile post-selected on $\ket{\downarrow_{z}}$ is shown in Fig. 2b, and it resembles a checkerboard pattern. The spin direction before the post-selection, is overlaid on the intensity profile via the red arrows and it elucidates why the N=1 lattice is is composed of a vortex anti-vortex structure.

Passing a polarized neutron beam through two pairs of LOV prisms pairs prepares a beam with a lattice of spin-orbit correlations as described by Eq. 5. The spin dependant intensity profile has the doughnut/ring structure as shown in Fig. 2c. This is a consequence of the cosine/sine amplitude terms in Eq. 5. The major features can be seen between the simulated and observed profiles in Fig. 2c. Note that the spin analyzer sets the spin filter direction, and the two profiles in Fig. 2c are from two separate setup configurations.

The slight differences between the simulated and observed profiles shown in Fig. 2a and Fig. 2b can be attributed to the interface region between the longitudinal field of the guide coils and the transverse field of the triangular coils. However, when triangular coils 2 and 3 are used to prepare the N=1 lattice the observed profile is significantly more distorted, indicating that the permalloy tube is not sufficiently removing the field between the triangular coils.

The phase difference between the two spin states of the N=2 lattice is shown in Fig. 1. This phase structure can be mapped via the spin-dependant intensity profile after mixing the two spin states. That is, we require to post-select the spin along a direction that is perpendicular to the spin quantization axis, which in our case would be the x and y directions.

It can be noted that translating one of the triangular coils along its incline direction induces an additional uniform phase shift. This provides a convenient method of obtaining the $\ket{\downarrow_{\text{x,y}}}$ dependant intensity profiles without changing the 3He polarization direction Sarenac et al. (2018b). Fig. 3 shows the simulated and observed spin dependant intensity profile as the first coil in the setup is translated along the y-direction. Fig. 3a and Fig. 3c correspond to the spin-up and spin down intensity profiles of a single cell of the N=2 lattice, as shown in Fig. 2. Fig. 3b corresponds to the intensity profile after mixing the two spin states. It can be seen that the intensity varies azimuthally, indicating the phase structure of the N=2 lattice as shown in Fig. 1.

IV Conclusion

Photon spin-orbit coupling arises naturally in nano-optics, photonics, plasmonics and optical metamaterials Bliokh et al. (2015); Stav et al. (2018) and is a core construct of chiral quantum optics Lodahl et al. (2017) and topological photonics Ozawa et al. (2019). In this work we explore spin-orbit coupling in the context of freely-propagating beams in which spin and orbital angular momentum (OAM) degrees of freedom are correlated. We have prepared and characterized neutron beams with lattices of spin-orbit correlations in which the OAM of one spin state is different from the OAM of the other spin state. This was achieved via sets of specifically oriented triangular coils which acted as LOV prism pairs. The beams were characterized via their spin dependant intensity profiles.

The triangular coils induced good quality magnetic phase gradients, as can be observed in Fig. 2a and Fig. 2b. However, in our experiment the permalloy tube did not sufficiently remove the magnetic field between the two sets of triangular coils. This resulted in distortions when all four coils were on simultaneously. For more pronounced results a better mechanism of removing the field is required.

We expect the described techniques to be the forerunners of neutron OAM applications in material characterization and fundamental physics. Superconducting triangular coils with higher fields may be employed to prepare lattices with smaller periods. The next set of experiments will focus on the preparation of spin-orbit correlations over the coherence length of neutron wave packets and the characterization of these spin-orbit states via the correlations between spin and projected linear momentum.

V Acknowledgements

This work was supported by the Canadian Excellence Research Chairs (CERC) program, the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery program, Collaborative Research and Training Experience (CREATE) program, the Canada First Research Excellence Fund (CFREF), and the National Institute of Standards and Technology (NIST) Quantum Information Program. The work utilized facilities supported in part by the National Science Foundation under Agreement No. DMR-1508249. The authors thank S. Watson and M. T. Hassan for their help in the experimental setup on PHADES.