A novel property of anti-Helmholz coils for in-coil syntheses of antihydrogen atoms: formation of a focused spin-polarized beam

We demonstrate here that cold antihydrogen beams formed and extracted from a cusp magnet (anti-Helmholtz coils) are well focused and spin-polarized. A new discovery was the fact that the antihydrogen beam follows the well-known lens formula of optical lenses with its focal length properly scaled with the initial kinetic energy, the magnetic field strength and the magnetic moment. Furthermore, the simulation revealed that for a certain kinetic energy region of antihydrogen atoms, the optimum production position is upstream of the center of the cusp magnet, where a well-known nested potential configuration can be applied.


Introduction
The CPT (C: charge conjugation, P: parity operation, T: time reversal) symmetry is assumed to be the most fundamental symmetry in physics, and guarantees that atomic properties of antihydrogen (H ) and hydrogen are exactly the same. This symmetry might be violated if, for example, the gravitational interaction is taken into account, which causes the space-time to be curved [1]. The standard model extension developed by Kostelecky et al discusses physical We report here novel properties of the cusp magnet in transporting H beams; namely that the H atoms in LFS produced in the cusp trap are extracted along the magnet axis following a well-known lens formula for a broad range of experimental conditions.

Trajectory calculations of H beams
The force acting on an H atom in a magnetic field B is given by ( ) B   μ · , where μ is the magnetic moment of the H atom. In the magnetic field, the energy level of the H atom in the ground state splits into four sub-states. For a strong magnetic field, μ μ = is approximately constant for all the four hyperfine states, and the forces are given by B   μ ± depending on whether μ is parallel (HFS) or antiparallel (LFS) to B, respectively. In the following discussions, this strong magnetic field condition is assumed for the sake of simplicity.
The above consideration is valid as far as the magnetic moment μ of the H atom adiabatically follows B. Such an adiabatic condition might not be satisfied near the center of the cusp magnet where B 0 = , and μ of the H atom makes a transition, which is called the Majorana spin-flip [15]. The criterion for the Majorana spin-flip can be obtained carefully as follows. The axial and radial magnetic fields of the cusp magnet near the center are given by , where v z is the axial velocity of the H atom.

Because the Larmor frequency is
μ¯≫ , μ is expected to follow the direction of B adiabatically. For example, when v 1000 z = [m/s] (∼50 K), r 0 is estimated to be 20μ ∼ m for the ground state adopting the minimum B B r = . Considering that the maximum radial position of the H atoms at z = 0 reaching the cavity is about 5 mm (see the next paragraph), the fraction to make the Majorana spin-flip is about 10 4 − or less, which is negligibly small. 3.0 m = T, for LFS and HFS, respectively (K 0 is given as a temperature unit, i.e., the kinetic energy divided by Boltzmannʼs constant). It is seen that the number of H atoms in LFS traveling along the magnet axis is much larger than that in HFS; i.e., one can obtain an intensified and spin-polarized H beam automatically. The 10 cm disk 1.5 m downstream from the cusp magnet center actually corresponds to the size and the position of the entrance of the microwave cavity in the ASACUSA setup (see figure 1). In other words, the best position to synthesize H atoms is located upstream of the cusp magnet center, although synthesis at the center was anticipated in the original cusp trap scheme [13]. This is particularly true because the densities of antiproton and positron plasmas are higher at the higher magnetic field, resulting in a higher H yield. Actually, the first H production in the cusp trap was realized at the maximum magnetic field region taking this optimal property into account [9,10]. In the case of K  where a well-known nested potential configuration [14] can be applied, still optimizing the H beam intensity. For H atoms in HFS in figure 4(c), the behavior is just the opposite: the lower the kinetic energy, the lower the transport efficiency; i.e., the cusp magnetic field acts in itself as the spin filter as well as the beam intensifier of H atoms in LFS. Figure 4(d) shows the polarization of H beams calculated using the results of figures 4(b) and (c) assuming that an equal number of LFS and HFS are produced, where the polarization is defined by The polarization decreases gradually as K 0 increases.

Lens formula and the scaling law
Having observed that the cusp magnet has the ability to enhance the beam intensity of H atoms in LFS, the next step is to quantitatively study the transport property. In doing so, the image point, z i , is evaluated calculating trajectories with different ejection angles, o θ , from the axis at a It is observed that for different ejection angles, l is distributed in a narrow region for a fixed z o ; i.e., the concept of the focal length is well satisfied. In the following, the central value of l for H atoms arriving at the disk is re-defined as the focal length l. Figure 5 ). Because we know the lens position, z l , as well as l, z o and z i , the transport fraction of H atoms to the 10 cm disk is easily calculated, and is shown by the lines in figure 4(b). These lines are in good agreement with results obtained by numerical simulations (symbols in figure 4(b)).

Qualitative analysis of the property of the cusp magnet
To understand why the lens formula is applicable to the cusp magnet, which is quadrupolar in its nature but not sextupolar, the behavior of the magnetic field of the cusp magnet is studied in detail. The focusing property is primarily determined by the radial force acting on the beam; i.e. B F r r μ = − ∂ ∂ for H atoms in LFS. Figure 6(a) shows B r ∂ ∂ , and figure 6(b) is a closeup for 0 m z 0.2 < < m. When z 0 ≠ , B r ∂ ∂ is 0 at r = 0 and increases monotonically with r for r 0.05 ≲ m (actually the inner radius of the MRE in the ASACUSA setup is 0.04 m). Although B r ∂ ∂ is almost constant with respect to r at z = 0, this is just a singular point. Taking e.g.
B r ∂ ∂ at z = 0.004 m (the curve closest to z = 0 in figure 6(b)), it increases more or less linearly for r 0.02 m ≲ . As a whole, B r ∂ ∂ can be practically approximated as ( ) is a coefficient of the field gradient as a function of z. Accordingly, the radial force acting on the H atom is given by ; i.e., the cusp magnet here behaves like a sextupole magnet (F r r ∝ ). The change of the deflection angle, d θ , due to the radial momentum transfer from the magnetic field to the H atom is given by above, one can immediately obtain the focal length l as Furthermore, the small correction appearing in equation (1) can be understood as the result of the acceleration of H atoms in LFS by the magnetic field, which amounts to B m μ ∼ . By replacing  (1); i.e., the behavior of the cusp magnet is well understood. As shown in equation (3), l LFS also has the scaling property with respect to μ.
In the case of H atoms in HFS, the direction of the force is just the opposite; i.e., H atoms are axially decelerated and radially defocused. Accordingly, by replacing K 0 by K B m 0 μ − and F r by F r − , and employing more correct coefficients obtained in equation (1) (5), which is shown by the lines in figure 4(c). It is seen that these lines reproduce the results of numerical simulations quite well. The polarization using l LFS and l HFS is also shown by the lines in figure 4(d), which again reproduces the numerical simulations quite well; i.e., the cusp magnet works as a magnetic lens for LFS and also for HFS H beams. It is also noted that the cusp magnet can be used as a good spin-selector for molecular beam experiments.

Conclusion
Here, we study the transport properties of cold H atoms produced in the cusp magnet. Although the cusp magnet (anti-Helmholtz coils) has in principle a quadrupolar field distribution, it is found that the H beams prepared upstream of the center of the cusp magnet are well characterized and follow the lens formula of optical lenses. The focal length satisfies a scaling law on the initial kinetic energy (K 0 ), the maximum magnetic field strength(B m ) and the magnetic moment (μ) of H atoms. Because the cusp magnet focuses H atoms in LFS along the magnet axis and defocuses those in HFS, one gets a spin-polarized intensified H beam automatically. Furthermore, it is found that the best position to produce H atoms for a stronger H beam with higher polarization is upstream of the center of the cusp magnet for a broad range of H kinetic energy. This allows the use of the so-called nested trap configuration, which greatly simplifies the H production procedures.