Analytic model for electrostatic fields in surface-electrode ion traps

The design of surface-electrode ion traps is studied analytically. The classical motion of a single ion in such a trap is described by coupled Mathieu equations. An analytic boundary-value solution for the Laplace equation in three dimensions with a piecewise-constant Dirichlet boundary condition is described. This solution is used to model the electrostatic potential field generated by a series of electrodes lying in a single plane. The model is applied to the problem of designing surface-electrode ion traps, including calculating important trap design parameters such as the center position of the trapped ion, trap depth, and stability parameters. The model is used to determine optimized dimensions for trap electrodes in various cases.

Figure 1

Configuration of surface electrodes in the $xz$ plane. The two segments marked “rf” are the rf electrodes; they are separated by a central electrode that is rf grounded. Outside of the rf electrodes are many independent control electrodes that are segmented in the $z$ dimension in order to provide control over the ion’s position in the $z$ dimension. The parameters $a$, $b$, $c$, and $w$ define the dimensions of the trap electrodes. The rf and central electrodes are assumed to extend infinitely in the $z$ dimension, while the control electrodes are semi-infinite in $x$. Static voltages are applied to each electrode as numbered (see text).

Figure 2

Illustration of the pseudopotential well created by the electric fields of the surface electrodes, with the electrodes diagrammed at bottom. Dark shades correspond to low pseudopotential energy. The ion is to be trapped in the potential well near the center of the figure. The lowest energy path for the ion to escape the trap is through the saddle point in the pseudopotential directly above; the difference in pseudopotential between the center of the well and this escape point represents the maximum energy the ion can have and remain trapped. The top surface of the electrodes represent the $y=0$ plane. The electrodes labeled “rf” have the oscillating voltage applied to them; the others are rf grounded.

Figure 3

Potential curvature in the $z$ dimension at the trap center as a function of $z∕w$ for a single electrode segment with a $1\mathrm{V}$ applied voltage. $z=0$ is aligned with the center of the segment. The five wire configuration was used with parameters $a=1$, $b=1.2$. For small $w$, the curvature is relatively small. For large $w$, the curvature is reduced near the center of the segment, and a zero in the curvature approaches $z∕w=\pm 0.5$ as $w$ becomes larger. The neighboring segments, centered at $z=\pm w$, will therefore have a zero near the same point which means that the two nearest electrodes cannot contribute much to the curvature experienced by the ion at that point without applying large voltages to them.

Figure 4

Voltages applied to control electrodes in order to move an ion along the axial dimension of the trap while maintaining a constant $A_{zz}$. The trap parameters used are described in the text. Voltages are show for two control electrode widths: (a) $w=400\mu \mathrm{m}$, (b) $w=1000\mu \mathrm{m}$. The $z=0$ position corresponds to the center of electrode 3, and the $z$ scale ranges from one edge of electrode 3 to the next. If the ion is moved past the edge of electrode 3 the roles of the electrodes can be switched so that the adjacent electrode is now electrode 3. Voltages 6–9, not shown for clarity, are small and nearly constant in both cases except near the edges of electrode 3. In the $w=1000\mu \mathrm{m}$ case much larger voltages are required to achieve the same axial trap curvature $A_{zz}$, especially near the edges of the control electrodes.