Linear Paul Trap: Ion Chain (2D Cross-Section)

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

This is a 2D axial/radial cross-section of the same linear Paul trap as the 3D version: an oscillating RF quadrupole field on four rod electrodes confines ions radially (shown here as the vertical axis) while static DC endcaps confine them along the trap axis (horizontal). Each ion's motion obeys the Mathieu equation:

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

Radial (vertical, shown): q_r = q  (from RF),   a_r = −a_z / 2   (Laplace's eq.)
Axial   (horizontal):     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 radial secular frequency

β_r ≈ √(a_r + q_r²/2) = √(q²/2 − a_z/2)      ω_sec = β_r·Ω/2

Trapping is stable only inside the Mathieu stability region — for this trap that means |q| < 0.908 and β_r real. Ions repel each other by the Coulomb force; balanced against axial confinement they self-organise into a linear 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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