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Stability diagram (q_z, a_z): green = stable on both axes. Drag the dot to move the operating point.
Each button runs real simulations or evaluates the Mathieu equation for the current settings (a few seconds at most).
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How to use
The ion starts displaced from the centre. Press Guided tour for a narrated slow-motion walk-through, pick a preset, or change any slider. Drag the white dot in the stability diagram to move the operating point: green is stable, red dot means the ion will be lost. The ring electrode glows red or blue with the sign of its RF voltage, and the force arrows in the middle flip with it. Space pauses, R resets. For many ions the display magnifies positions (shown in the HUD); a scale bar in micrometres is drawn at the bottom. The spectrum has dashed lines at beta/2 and at 1 - beta/2, 1 + beta/2.
The science
The trap. A Paul trap has a hyperbolic ring electrode and two hyperbolic endcaps (here with r0^2 = 2 z0^2). The ring carries U + V cos(Omega t), the endcaps are grounded. Near the centre the potential is a saddle,
Phi(r, z, t) = (U + V cos Omega t) (r^2 - 2 z^2 + 2 z0^2) / (4 z0^2)
At one instant it confines radially and expels axially; half an RF period later the roles swap. A static field can never trap a charge (Earnshaw), but this oscillating one can.
Mathieu equation. With xi = Omega t / 2 the motion along each axis obeys u'' + (a_u - 2 q_u cos 2 xi) u = 0 with
a_z = -4 e U / (m z0^2 Omega^2), q_z = 2 e V / (m z0^2 Omega^2), a_r = -a_z / 2, q_r = -q_z / 2
This is a linear equation with a periodic coefficient, so Floquet theory applies: the solutions are e^(i beta xi) times a periodic function, and the motion is bounded only if |trace M| < 2 for the one-period transfer matrix M, with cos(pi beta) = trace M / 2. The stable region is the green area of the stability diagram. At a = 0 the first stability limit is q = 0.908.
Secular motion and micromotion. For small q the motion is a slow oscillation (the secular motion) at omega = beta Omega / 2 with beta ~ sqrt(a + q^2/2), plus a small fast jitter at Omega (micromotion). Averaged over the RF the ion sits in a pseudopotential bowl of depth about e q_z V / 8 (axial) or e q_z V / 16 (radial). The spectrum shows a peak at beta/2 and sidebands at 1 +/- beta/2 (in units of the RF frequency). The exact beta used here comes from the Floquet integration, not from the pseudopotential formula.
Many ions. Each ion also feels the Coulomb repulsion of the others, kappa = 4 e^2 / (4 pi eps0 m Omega^2 z0^3) in units of z0, so smaller traps and lighter ions give a stronger relative repulsion. With friction (laser or buffer-gas cooling) the ions settle into a Coulomb crystal along the weakest axis; random kicks heat them back into a cloud.
Honest simplifications. (1) The trap field is the ideal quadrupole: no electrode truncation, no stray fields, no patch potentials, no anharmonic terms. (2) Cooling is a linear friction and heating is white velocity noise, not a laser-atom model. (3) Ion positions of a crystal are tiny compared with the trap, so the picture magnifies them by the factor shown; the electrodes are not scaled. (4) The depth formulas are the a = 0 pseudopotential approximations. (5) RF cycles are shown in slow motion (a few per second instead of 10^6). (6) Ions are removed when they leave the region |z| < 0.98 z0, rho < 1.38 z0.
Checks run in the page. The first stability limit from the Floquet trace against 0.9080; beta from the Floquet integration against the continued-fraction solution of the Mathieu recursion; the secular frequency measured in a simulated trajectory against beta Omega / 2; the exact beta^2 against its second-order series q^2/2 + 3 q^4/16; the separation of a simulated two-ion crystal against 2 (kappa / 4 beta_r^2)^(1/3); and the decay of the amplitude under friction.