# Motion of Single Ion in a Linear Paul Trap

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A Paul trap uses electric fields oscillating at a radiofrequency (rf) to confine charged particles (such as ions) in space. The graphic shows the full path (in blue, only when "excess micromotion" is false) and a low-order approximation (in orange) of the motion of a single ion in a Paul trap. One of the simplest Paul traps is implemented with an oscillating quadrupole electric field. A static quadrupole vector field is shown for reference.

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Contributed by: Roland Hablützel (August 2022)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: excess micromotion due to ion displacement; the micromotion amplitude follows the local quadrupole

Snapshot 2: excess micromotion due to an additional oscillating term along the direction

Snapshot 3: comparison of secular, low-order and full solution of the ion motion

Snapshot 4: unstable orbit for

The factors in the and directions are equal and opposite.

A low-order approximation is not sufficient for an accurate description of the ion path for increasing values of the parameter.

The IonPath function is the solution of

DSolve[{ui''[t]+2q Cos[Ω t]Ω^2/4ui[t]==0,ui[-&straightpi;/ω]==u1(1+1/2q Cos[&straightpi; Ω/ω]),ui'[-&straightpi;/ω]⩵1/2q u1 Ω Sin[&straightpi; Ω/ω]},ui, t]

/. ω->Ω RealAbs[q]/(2 Sqrt[2])

Reference

[1] D. J. Berkeland, J. D. Miller, J. C. Bergquist, W. M. Itano and D. J. Wineland, "Minimization of Ion Micromotion in a Paul Trap," *Journal of Applied Physics*, 83(10), 1998 pp. 5025–5033. doi:10.1063/1.367318.

## Permanent Citation