Analytical design of axial magnetic coils with systematically improved uniformity for miniature quantum devices

We describe an analytical design method of shielding-coupled uniform magnetic coils for miniature quantum devices. The theoretical and simulation results point out that the 99% range along the symmetrical axis and the 50% range along the radius of the proposed m = 3 coils are uniform, and more important is that both the uniformity and the uniform region for these kinds of coils can be systematically improved only by adding more loops at specific places obtained from our analytical formula. A relevant experiment demonstrates the feasibility of this method and realizes the m = 3 coils with the inhomogeneity below 2.6 × 10^-3 along nearly the whole symmetrical axis. In addition, a practical technology to remove the influence of the shielding's nonideal gaps and openings is proposed and realized. All of these results are crucial for the miniaturization and high performance of quantum devices.