Motion of Single Ion in a Linear Paul Trap

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.
A linear Paul trap features translational symmetry along one of its axes (, not shown), while the quadrupole extends along the transversal plane . The motion of the ion in this plane is shown for selected conditions , , and . This motion is described by what is normally a fast oscillation of small amplitude at the rf, which is called micromotion, on top of an averaged, slower oscillation with larger amplitude, called the secular motion (in green). The parameter characterizes the strength of the trap and is part of the definition of the micromotion amplitude and the secular frequency. The parameter controls the total length of the ion path in time. Micromotion can be minimized, which requires that the ion oscillates around the center of the quadrupole field or rf-null. You can select "excess micromotion" to offset the location of the ion away from the origin by and , or add an oscillating field in the direction with Kac,x=R α ϕac (for details see [1]). In both cases the ion will experience excess micromotion [1, equation (18)].


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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[-ϕ/ω]==u1(1+1/2q Cos[ϕ Ω/ω]),ui'[-ϕ/ω]⩵1/2q u1 Ω Sin[ϕ Ω/ω]},ui, t]
/. ω->Ω RealAbs[q]/(2 Sqrt[2])
[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.
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