Paul trap micromotion ( $u_{i} ≃ X_{i} [1 - (q_{i} /2) cos Ω t]$ )

In the small-parameter regime, the slow coordinate

X_{i}

describing Paul trap secular motion approximately obeys a harmonic oscillator equation at angular frequency

ω_{i}

. Its superposed fast motion is Paul trap micromotion. Substituting the leading fast displacement from the RF-averaged trapping potential gives the displayed expression in our phase convention. A small positional ripple can have substantial velocity, because differentiating it brings down the large angular frequency

Ω

. Precision spectroscopy must therefore address both Paul trap secular motion and Paul trap micromotion; merely obtaining a bounded trajectory does not eliminate Doppler shifts.

Table of contents
- Paul trap secular-motion example Paul trap micromotion

Paul trap secular-motion example ( $a_{x} = 0, q_{x} = 0.10, q_{z} = - 0.20$ )

 0  0

Choose these dimensionless parameters for an original example, with no collisions or damping. They are teaching inputs, not measurements from either lecture. With

s = Ω t

and slow initial amplitude

X_{0}

, the radial equation and its leading approximation are

\frac{d ^{2} x}{d s ^{2}} = \frac{q _{x}}{2} cos s x, X = X_{0} cos (\frac{q _{x} s}{2 2}), x_{app} = X (1 - \frac{q _{x}}{2} cos s) .

(1)

Set

x (0) = 0.95 X_{0}

and

d x / d s (0) = 0

to match the approximate initial conditions. The leading secular angular frequency is

ω_{x} /Ω = 0.0353553

. The velocity Verlet method integrates the full RF equation over 80 drive periods; a separate fourth-order Runge-Kutta method checks the computed trajectory. Halving the timestep tests numerical error independently of the averaging error.

Figure 1.
Full RF motion and the leading secular approximation
. Original chosen small-parameter calculation in units of $X_{0}$ . The lower panel resolves six drive cycles and includes the first micromotion correction. Agreement over this finite interval does not establish a stability diagram or an exact secular frequency. **Details**.

Using 400, 800, 1600 and 3200 steps per RF period gives successive maximum position differences of approximately

1.40 \times 1 0^{- 4}

3.50 \times 1 0^{- 5}

and

8.74 \times 1 0^{- 6}

in units of

X_{0}

, consistent with second-order convergence. The finest trajectory differs from the independent classical fourth-order Runge-Kutta method result by at most

2.92 \times 1 0^{- 6} X_{0}

, whereas

x_{app}

differs by as much as

0.0378 X_{0}

. Over a sufficiently long time, the omitted correction to the secular angular frequency produces a growing phase discrepancy even if the numerical trajectory has converged. For the chosen point only, integration of a fundamental solution matrix over one RF period checks that both the radial and axial Floquet multipliers lie on the unit circle, away from its endpoints. This tests simultaneous ideal confinement without extrapolating to larger