Linear Paul Trap: Ion Coulomb Crystal

Trap Drive

RF amplitude — Mathieu q 0.30

Endcap DC — Mathieu a_z 0.0100

Ion chain length 5

Laser-cooling damping γ 0.30

Live State

Stability (q < 0.908)

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Secular freq. ω_sec/Ω

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Mean ion spacing

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Mean kinetic energy

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How it works

A linear Paul trap confines charged ions with a radio-frequency quadrupole field on four rod electrodes plus static DC on two endcaps. Each ion's motion along a trap axis u obeys the Mathieu equation:

d²u/dτ² + [a_u − 2 q_u·cos(2τ)]u = 0,   τ = Ωt/2

Radial (x,y):  q_r = q  (from RF),   a_r = −a_z / 2   (Laplace's eq.)
Axial   (z):   q_z = 0  (no RF),     a_z  set by the endcap slider

For small a,q the motion splits into slow secular oscillation plus fast micromotion at the drive frequency Ω, with secular frequency

β_u ≈ √(a_u + q_u²/2)      ω_sec = β_u·Ω/2

Trapping is stable only inside the Mathieu stability region — for this trap that means |q| < 0.908 and β real. Ions repel each other by the Coulomb force; balanced against axial confinement they self-organise into a linear (or, once radial confinement weakens relative to Coulomb repulsion, zig-zag) crystal — exactly the structure used as trapped-ion qubits in real devices (IonQ, Quantinuum).

RF amplitude (q) — stronger RF pins ions more tightly to the trap axis; push q above 0.908 and the crystal flies apart (unstable).
Endcap DC (a_z) — stronger axial confinement compresses the ion chain; too weak and Coulomb repulsion buckles it into a zig-zag.
Laser-cooling damping (γ) — a velocity-proportional drag standing in for Doppler/sympathetic laser cooling; raise it to watch a hot, jittering cloud settle into a still crystal.
All quantities are in scaled simulation units chosen so the crystal and its micromotion are visible in real time — the governing Mathieu/Coulomb physics is exact, only the numeric scale is compressed from real MHz-range RF drives.

Trapped ion RF rod electrode DC endcap

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