## 2 Magnet Requirements and Dimensions  
Our atomic experiment collects \(10^{4}\) counts in each counting interval. The Poisson noise threshold is therefore \(1/100\) using \(\varepsilon=\sigma/\sqrt{N}\), where \(\varepsilon\) is error, \(\sigma\) is standard deviation, and N is quantity of events [6]. This threshold is expected to continue into projected higher signal levels coupled with reduced count interval time length. Less than 0.1 radian deviation of the magnetic field from the z-direction is required to suppress competing depolarizing transitions to less than \(1/100\). Gaussmeter measurements around the neighboring electron  
Figure 1: Atomic beam apparatus: Helium Source is the area of interest.  
Figure 2: Atomic beam preparation laser polarization, \(\mathbf{k}\), and \(\mathbf{B_{magnet}}\) arrangement (Right or \(\sigma^{+}\) shown).  
gun magnets that use a return flux yoke revealed a 5 G perpendicular surrounding field suggesting a 50 G minimum preparation magnetic field strength.  
Any fringing field in the spectroscopic laser interaction region needs to be minimized. In addition, Helium energy levels change at ~1.4 MHz/G, so sub-kHz precision in helium spectroscopy makes field gradients of ~1 mG/mm desirable. The atomic beam pump laser is approximately 3.5\({}^{\prime\prime}\) from the interaction region, as seen in Figure 1. The far field is thus defined as greater than 3\({}^{\prime\prime}\) removed from the preparation magnet center. Shielding would impact other needed magnetic fields in the apparatus.  
For compactness and convenience, the assembly should fit in the existing space for linear-polarized atomic beam pumping. The critical dimensions are 3/8\({}^{\prime\prime}\) along the atomic beamline (outer diameter of the annular magnet), and enough atomic beam pumping transit distance to allow a minimum of 90% and preferred 99% or better pumping to m\({}_{\text{s}}\) = +1 or \(-\)1 (internal diameter). In Figure 3, the upper left corner shows a cross-sectional layout.  
Faraday isolators such as [PERSON]'s design [7] offer some inspiration toward the fringe field requirement. As shown in Figure 3, stacked pairs of magnets with opposed magnetization sum to a zero far field. The central field is collimated in a modified Helmholtz geometry as the inner surfaces of the inner magnet pair make the most contribution to the field in the pumping region (see the current loop model discussed below).  
Note that coincident points on axial (z-axis) and transverse (x-axis or \(\rho\) due to cylindrical symmetry) B-field profiles within the gap must have the same value, i.e., the physical center point. Axial (z) and transverse (x or \(\rho\)) B\({}_{\text{z}}\)-field plots are shown in the bottom and top-right of Figure 3, respectively. Greater than three inches from the origin (center of the gap) is the far field. The gap is labeled.  
In Section 3.1, we discuss a simple current loop model for each magnet to verify that the physical constraints can support the B\({}_{\text{z}}\) performance requirements. In Section 3.3, benchtop prototypes, experimental magnetic field measurements, and 3-D simulation models are used to detail and validate various arrangements against our requirements and to allow the selection of an optimum configuration. In Section 4, adequate far-field behavior is verified by Gauss meter measurements and compared to COMSOL modeling.  
Figure 3: Created magnet assembly: bottom, axial (z) B\({}_{\text{z}}\) field; right: mid-plane transverse (x or \(\rho\)) B\({}_{\text{z}}\) field.  
In Section 5, the central field performance, as well as the overall experimental optical pumping efficiency of atomic helium, is presented. Pumping results are also compared to the predictions of electronic transition rate equations.