Twisting Neutral Particles with Electric Fields

We demonstrate that spin-orbit entangled states are generated in neutral spin 1/2 particles travelling through an electric field. The quantization axis of the orbital angular momentum is parallel to the electric field, hence both longitudinal and transverse orbital angular momentum can be created. Furthermore we show that the total angular momentum of the particle is conserved. Finally we propose a neutron optical experiment to measure the effect.

We demonstrate that spin-orbit entangled states are generated in neutral spin 1/2 particles travelling through an electric field. The quantization axis of the orbital angular momentum is parallel to the electric field, hence both longitudinal and transverse orbital angular momentum can be created. Furthermore we show that the total angular momentum of the particle is conserved. Finally we propose a neutron optical experiment to measure the effect.

Introduction.
Intrinsic orbital angular momentum (OAM) has been observed in free photons [1] and electrons [2]. Furthermore extrinsic OAM states have also been observed in neutrons, using spiral phase plates [3] and magnetic gradients [4]. In the latter case spin-orbit entangled states are generated [5]. It has also been demonstrated that magnetic quadrupoles can generate spin-orbit states in neutral spin 1/2 particles [6]. The aforementioned methods require a beam with exceptional collimation (0.01 • -0.1 • divergence) if instrinsic OAM is the goal. Furthermore the incident particles must be on the optical axis. These two requirements limit the available flux to an inpractical level. For this reason intrinsic OAM has not been observed in neutrons to date [7]. The additional quantum degree of freedom offered by OAM provides utility in the realm of quantum information [8][9][10]. Additionally in neutrons the additional degree of freedom may help improve existing tests of quantum contextuality [11,12]. Furthermore neutrons carrying net OAM may reveal additional information on atomic nuclei in scattering experiments [13] In this paper we propose a method by which instrinsic spin-orbit states can be generated in an arbitrarily collimated beam of neutral spin 1/2 particles. This removes flux limitations and allows for the construction of practical spin-orbit optical equipment for neutrons. We show that a static homogeneous electric field polarized along the direction of particle propagation induces longitudinal spin orbit states, while a transversely polarized electric field generates transverse spin orbit states. The latter has not yet been observed in massive free particles. Furthermore we demonstrate that the total angular momentum of a particle is conserved in static electric fields, due to the fact that in an electric field the particle spin couples to the cross product between the electric field strength and the particle momentum [14]. So any change in particle spin can be conserved by a change in the particle (angular) momentum. Theoretical Framework. An observer moving through an electric field, E, will experience a magnetic field B . In the low velocity limit when v << c the magnetic field * niels.geerits@tuwien.ac.at can be written as [15] Inversely in the lab frame a moving magnetic moment will appear to have a small electric dipole moment d = v× µ c 2 . Hence a spin 1/2 particle with magnetic moment µ experiences a Zeeman shift d · E = µ · B when moving through an electric field. Thus we can determine applicable the Schroedinger equation with γ the gyromagnetic ratio and σ the Pauli matrices. The wavefunction is described by a spinor ψ = ψ + (x, y, z) ψ − (x, y, z) , where the index ± refers to the spin state parallel or anti-parallel to the z-axis respectively. Transmission Geometry -Longitudinal OAM. First we will consider the longitudinal spin-orbit effect, hence we will assume that the extent of the electric field is semi-infinite and that it is parallel to the z-axis. Hence the Schroedinger equation can be written as with C = γEz c 2 . The incident wave can be described by a solution of the free space Schroedinger equation ψ I ± = b ± J m± (k ρ r)e im±φ e −ikz . By applying a Fourier transform in x and y the PDE 3 is simplified to a coupled second order ODE.
Here we have also transformed the equation to cylindrical coordinates with k 2 r = k 2 x + k 2 y and k x ± ik y = k r e ±iφ . It is noteworthy that in the specrtal domain the potential, C(k x σ y + k y σ x ) closely resembles that of the quadrupole described in [6]. This gives an intuitive reason as to why a static electric field mimics the action of a quadrupole in reciprocal space. Hence an electric field is more effective for large divergences (i.e. large k r ). We diagonalize 4, arXiv:2007.03345v1 [quant-ph] 7 Jul 2020 by applying a transformation of the formψ = Tψ and multiplying the by T −1 from the left.
for this particular diagonalization T is given by The general solution to 5 is simply a superposition of a forward and backward propagating planewaves for each spin state.
The backward propagating amplitudest 1 andt 3 represent reflections from +∞ and are therefore zero. The general solution forψ is simply found by applying the transformation Tψ .
To determine the values oft 2 ,t 4 and the reflection coefficientsr ± we apply the boundary conditionŝ Here the subscript z denotes the partial derivative to the z coordinate.f ± (k r , φ) = δ(kr−kρ) kr e im±φ denotes the 2D Fourier transform of the incident wavefunction. This boundary value problem can be formulated as the following matrix vector problem r + Ck r and k 2 = − k 2 r − Ck r . By inverting the above 4x4 matrix we find the transmission and reflection coefficientŝ which leads us to the solution for the transmitted waveŝ Looking at this expression we can see that the total angular momentum J = S + L of the wave is conserved in a static electric field. Furthermore it is obvious that the sign of the electric field strength is irrelevant to the direction of the OAM which is generated. Sincef ± (k r , φ) can be expanded such thatf ± (k r , φ) = lf l ± (k r )e ilφ , with f l ± (k r ) = 2π 0 f ± (k r , φ)e −ilφ dφ, the solution in real space can be obtained by applying the Bessel transform to 11.
It is instructive to look at the solution of 12 for an incoming Bessel beam carrying no OAM. In this case In this case the solution is trivial where ψ 0 ± and ψ 1 ± are the components with and without OAM respectively, such that ψ ± = ψ 0 ± + e ∓iθ ψ 1 ± . For a collimated beam geometry we may use k ρ = k z tan(α) ≈ k z α, where α is the beam divergence. Furthermore if Ck ρ is sufficiently small we may linearize the square root terms in equation 13 and obtain a much simpler expression for the wavefunction.
A longitudinal beam twister device may be constructed using a parallel plate capacitor, with the surfaces of the plates normal to the beam. The voltage required to fully twist the beam from the l = 0 state into the l = ±1 state is given by These equations are valid for single Bessel planewaves. In a realistic setup we must contend with a superposition of Bessel planewaves which interfere, resulting in damping or amplification of spin orbit production. This interference can be described by calculating 12 for an arbitrary divergence profile, which is related to f ± . Though we may also determine the probability of the particle being in the mth OAM state as a function of z without the inverse transform 12, by simply calculating the projection A m = |ψ m | 2 k r dk r (16) withψ m =< e imφ |ψ >. At neutron beamlinesf ± (k r ) can be shaped using appropriate collimators. Figure 1 shows the amplitude A 1 for various distributions of k r , assuming that the incident wave carries no OAM and is polarized along the z-axis. Equation 15 demonstrates that for neutrons with a divergence of 1 • propagating through a capacitor we require a voltage drop of 88.4GV to put a neutron into an OAM state with l = ±1. Obviously this is not feasible, so we should consider waves interacting with the interface at grazing incidence angles. Reflection Geometry -Quasi Transverse OAM. This results in a more pronounced coupling, since it increases the ratio between k z and k r . Furthermore the OAM carried by the transmitted and reflected waves in this case is quasi-transverse to the wavevector k. Since the quantization axis of OAM is normal to interface, the incident wave must be described by a superposition of all azimuthal modes. Nonetheless transmitted and reflected waves will have a mean OAM of ±1 with respect to the mean incident OAM. The reflection probability |r ± | 2 as a function of incident angle is shown in figure 2, for an electric field of 10 10 V /m (found in electric double layers [16,17]), a neutron wavelength of 2Å and an initial spin allgined along the −z direction. We can deduce that the optimal angle of reflection is around 0.001 • . Hence this method of OAM generation would suffer from similar flux limitations as the quadrupole method. Transmission Geometry -Transverse OAM The flux limitations can be overcome by considering transmission through a transversly polarized electric field which leads to the generation of transverse spin-orbit states. To demonstrate this we consider the time dependent Schroedinger equation for a neutral spin 1/2 particle in an electric field again we will assume that the electric field is polarized along the z-direction. However this time we will consider a field which extends infinitely in space. To reduce the problem to an ordinary differential equation we apply an unbounded Fourier transform to the spatial cooordinates. In cylindrical coordinates this leads to now denotes the kinetic energy paramter k 2 r + k 2 z . Once again we diagonalize this set of equations using the transformψ = Tψ Applying the initial conditionsψ ± (t = 0) = b ± (k r , φ, k z ) we can determine the particular solution of equation 18.
which appears almost equivalent to equation 14. If the wave propagates along the y-direction the value of k r , which may be approximated by k y is a factor 10 2 − 10 3 larger than in the previously described case (equation 14). Hence the required electric field integral to obtain maximal OAM is reduced to a more practical level. The incident wave in this case must be described by an infinite superposition of transverse OAM modes. Upon being transmitted through an ideal beam twister device the mean l value of this superposition will be raised or lowered by one. For example if we consider b ± to follow a Gaussian distribution with e − (Jacobi-Anger expansion) [18] e − 2kr k y sin(φ) where I(z) denotes the modified Bessel function of first kind. Hence the amplitude of the lth OAM mode is simply i l I l ( 2krk y σ 2 ). Upon passing through an appropriate electric beam twister the index is shifted by ±1. Such a wavefunction (real space) carrying a single unit of OAM is shown in figure 3. Proposed Methodology. Based on the previous theoretical analysis we propose a proof of concept experiment with neutrons to demonstrate that neutral spin 1/2 particles can obtain quanta of transverse OAM when traversing an electric field polarized perpendicular to the flight direction. The beam twister device will consist out of a one meter long evacuated flight tube loaded with two electrodes 5 mm apart. A voltage is applied to across the electrodes to generate the maximal permissible electric field in a high vacuum environment (10 7 − 10 8 V /m). Such a beam twister can generate an OAM amplitude between 2 and 20%. To measure the OAM we propose an experiment similar to [19], which was designed for photons. The experimental setup employs two supermirrors to spin polarize and analyze the beam, two beam twisters to generate and analyze spin-orbit entanglement and a set of three mirrors inbetween the two beam twisters as a means of rotating the image, thereby imprinting an OAM dependent phase on the wavefunction. By rotating the mirror set around the beam-axis the OAM dependent phase shift can be altered. This is the neutron optical equivalent of a Dove prism. Since the effects of all components described in this setup are wavelength independent, the experiment can exploit the high thermal flux available at a white neutron beam. The proposed setup is shown in figure 4. Conclusion. We have provided a theoretical framework which predicts that neutral spin 1/2 particles propagating through a static electric field acquire OAM parallel to the electric field axis. Furthermore we have illustrated a proof of concept experiment which could verify the generation of transverse OAM in neutrons.