## 3 Design and Construction
### B-Field Collimation
The four customized annular magnets were evaluated for geometric individuality. A total of 0.7% variance in thickness was the most differing characteristic noted. Figure 8a shows the typical measured transverse (x-axis) B\({}_{\text{z}}\) profile for a single annular magnet. The center face \"surface\" field is \(\sim\)400 G, and the center ring surface field  
Figure 6: Simple Biot-Savart axial (z-axis) simulation plots: (**a**) B\({}_{\text{z}}\)(z, Gap)—attracting pair (red), repelling pair (blue), sum (green), system position slice and sum curve (shaded plane and dotted line); (**b**) B\({}_{\text{z}}\)(z, Gap = 0.125\({}^{\prime\prime}\)), 0.125\({}^{\prime\prime}\) Gap size due to coils modeled at the center point of 1/16\({}^{\prime\prime}\) thick magnet, i.e., actual gap (1/16\({}^{\prime\prime}\)) + 2 \({}^{\star}\) coil offset (1/32\({}^{\prime\prime}\))—attracting pair (red), repelling pair (blue), sum (green); (**c**) single magnet face center B\({}_{\text{z}}\)(Ri/Ro, thickness)—colored Ri/Ro iso-bands highlight ratio behavior, maximum values (black dots), system slice and curve (shaded plane and green broken curve); (**d**) single magnet face center B\({}_{\text{z}}\)(for Ri/Ro = 0.67) vs. thickness L—system position at red line. Figure 7: Thick-magnet axial field behavior (N = coil turns per length, I = current). (Bring) is \(\sim\)1600 G. K&J Magnetics (manufacturer) [11] details the original 1/8'' ID annular Bring to be \(\sim\)1600 G. However, more precise face center measures exhibited \(\sim\)10% departures from theoretical/simulated values by K&J Magnetics for the 1/4'' ID, i.e., 441 G versus 487 G. Our precise measurements took account of a Lakeshore 450 Gauss meter 30 mils (1 mil = 0.001'') standoff of the active area from the contacted surface. Some hypotheses to explain the reduced field include localized ID grain boundary damage due to grinding or overall magnetization reduction during grinding despite cooling precautions. As a result, all in-house simulation work is referenced to the gap center measured B-field. Colloidal graphite was used to stabilize the exposed surfaces and minimize stray pump laser reflections. An ultra-high vacuum (UHV) chamber compatible mount was manufactured retaining the 1/16'' gap, illustrated in Figure 9. A total of 80 and 180 mil separation between the top and bottom brackets versions allowed vacuum experiment and bench probing, respectively. The transverse Gauss meter want is \(\sim\)160 mil wide. Profile results are shown in Figure 8, revealing a 0.1'' to 0.15'' collimated B-field diameter in (b). \(1/4''\) ID was selected based on minimum atom travel distance, d, dictated by the number of atomic beam cycles, n, to allow 1/100 depletion of the 23S1:mj = 0 level. The pump/applied laser will be broad bandwidth covering all relevant 2S-2P transitions. A simplistic estimate of the signal channel population, N, is achieved from the single excitation remaining ratio (portion of atoms still in 23S1:mj = 0 state), \(r\), excitation lifetime, \(\tau\), number of excitations/cycles, \(n\), and atom velocity, \(v_{rms}\), in the following relations:  
\[N=N_{0}r^{n}\Rightarrow n=\frac{log\left(\frac{N}{N_{0}}\right)}{logr} \tag{7}\]  
Figure 8: Magnet transverse B-field profiles (3/8′′ \(\times\) 1/4′′ \(\times\) 1/16′′): (**a**) single; (**b**) assembly. Figure 9: UHV optical pumping magnet mount. \[d=v_{rms}(n\tau)=\frac{v_{rms}\tau log\Big{(}\frac{N}{N_{0}}\Big{)}}{logr} \tag{8}\]  
A simplified three-level transition scheme [12] is shown in Figure 10. Atoms are coherently excited from Level 0 (representing our \(2^{3}\)S\({}_{1}\):m\({}_{1}\) = 0) to Level 2 (all excited states). Relaxation de-coherently branches equally between Level 0 and Level 1 (destination spin state). Therefore, r is 0.5, and \(\tau\) is treated as twice the natural lifetime due to Level 0-Level 2 cycling. Figure 7.0  
Figure 7.1  
Figure 7.2  
Figure 7.3  
Figure 7.4  
This minimum of 0.08 inch He* beam travel distance requirement is an infinite power result for an average velocity.