0 5 10 15 20 −1 0 1 2 3 4 5 E  n  WKB numeric analytic 15 FIG. 1: (Color online) Contour lines for u > 2. Sea levels are colored blue, Island levels are colored green, and the Separatrix is colored red (left panel). Energy spectrum for N = 20 and u = 10. WKB energies (red x) are compared with exact eigenvalues (blue +). Dashed lines indicate slopes ω$_{J}$ for low energies, ω$_{x}$ for near-separatrix energies, and ω + for high energies (right panel). FIG. 2: (Color online) An illustration of TwinFock ( n = 0) preparation (left), and of Pi (” π ”), Zero (”0”) and Edge (”e”) preparations (right) using Wigner plots on a sphere. The left panel is a 3D plot, while the right panel is a Mercator projection of the sphere using ( ϕ , n ) coordinates. −1.75 −1.25 −0.75 −0.25 0.25 −1 −0.5 0 0.5 1  /   n/j 0 e