Ions are atoms that have lost or gained electrons and thus carry charge. Dehmelt developed the Penning trap (static electric field plus strong magnetic field) while Paul invented the Paul trap (oscillating radio-frequency electric field). Both methods can confine single ions for milliseconds to many hours. This enabled precise g-factor measurements of the electron and single-ion laser cooling, marking a leap in quantum-level control of matter. Ion traps now underpin optical frequency standards, quantum cryptography, and qubit implementations for quantum computers.

In a Penning trap a cylindrically symmetric static quadrupole electric potential combines with a multi-tesla axial magnetic field, localizing ions via cyclotron motion and a magnetic bottle effect. A Paul trap employs an RF quadrupole field; ions are dynamically stabilized within the Mathieu-equation stability region, experiencing a pseudo-harmonic potential. Dehmelt laser-cooled single Be^+ ions to below 10 mK and measured the electron g-2 to 10^{-11}. Paul traps evolved into linear traps and multi-ion chains, forming the backbone of multi-qubit quantum gates. Fluorescence detection and resolved-sideband cooling within traps are now indispensable for quantum control and high-precision spectroscopy.

In Dehmelt’s Penning trap, simultaneous measurements of ω_c and ω_z in a calibrated magnetic field determined e/m and the electron g-2 to 10^{-11} with a single ion, avoiding Mølmer–Sørensen couplings. Paul RF quadrupoles with r_0 ~ mm and drive Ω ≈ 10 MHz yield pseudo-potential depths of order e^2V^2/(4mΩ^2r_0^4), allowing laser cooling to the mK regime. Linear RF traps leverage collective motional modes for Mølmer–Sørensen and Cirac–Zoller gates, with reported error rates below 10^{-3}. Trap-based optical frequency standards with Al^+ and Yb^+ have achieved fractional uncertainties in the low 10^{-18} range, enabling geopotential tests and searches for physics beyond the Standard Model.