Analysis of the Trapping Potential of the Linear Radio
Frequency Ion Traps for Quantum Computing
The dynamics of an ion in a linear radio frequency ion trap is established by classical motion equation. For an radio frequency electric potential,
The x and y equations become Mathieu equations;
d2x / dℓ2 + (a – 2qcos (2ℓ)) x = 0 (2)
and, d2y/dℓ2 – (a – 2qcos (2ℓ)) x = 0 (3)
where, ℓ = Ωt/2, is time which is dimensionless, a = (4QV)/mr20Ω2 and q = (2QV)/mr20Ω2 are DC and RF dimensionless. Q and m are ion and mass.
In the assumptions regarding psuedopotential, q <<1, the motion of ion along the i axis is decomposed into secular motion of large and slow amplitude at the frequency, w and small and fast amplitude micromotion at