Finite-geometry models of electric field noise from patch potentials in ion traps

We model electric field noise from fluctuating patch potentials on conducting surfaces by taking into account the finite geometry of the ion trap electrodes to gain insight into the origin of anomalous heating in ion traps. The scaling of anomalous heating rates with surface distance $d$ is obtained for several generic geometries of relevance to current ion trap designs, ranging from planar to spheroidal electrodes. The influence of patch size is studied both by solving Laplace's equation in terms of the appropriate Green's function as well as through an eigenfunction expansion. Scaling with surface distance is found to be highly dependent on the choice of geometry and the relative scale between the spatial extent of the electrode, the ion-electrode distance, and the patch size. Our model generally supports the $d^{-4}$ dependence currently found by most experiments and models, but also predicts geometry-driven deviations from this trend.

Figure 1

Examples of electrode geometries relevant to ion trapping. $d$ is the nearest ion-electrode surface distance. Electrodes are at either rf potential (shaded) or dc potential (unshaded). (a) Ion-in-a-sphere model used in Ref. [13]. (b) Cross sectional view of the hyperbolic Paul trap. (c) Needle electrode Paul trap. (d) Surface electrode Paul trap.

Figure 2

Graphic illustration of Eq. (3) for a given, arbitrary conducting surface $\sigma$, here represented by the shaded cylinder. The electrostatic potential $\Phi \left(\mathbf{r}\right)$ at point $\mathbf{r}$ is determined by adding the convolution of some charge density $\rho \left({\mathbf{r}}^{\prime }\right)$ with the Green's function $G(\mathbf{r},{\mathbf{r}}^{\prime })$ and the convolution of the surface potential $\phi \left({\mathbf{r}}^{\prime \prime }\right)$ with the surface Green's function $-{\partial }_{n^{\prime \prime }}G(\mathbf{r},{\mathbf{r}}^{\prime \prime })$.

Figure 3

Patch sizes and associated correlation functions, ranging from the infinite patch (IP) to the point patch (PP) from top to bottom.

Figure 4

Coordinate system used in the Green's function for the infinite plane in Eq. (20). The black dot represents the ion.

Figure 5

Ion (black dot) in a planar hole trap with radius $d$. rf is applied to the entire plane.

Figure 6

Coordinates system used for the hole trap. The ion (black dot) is suspended at point $\mathbf{r}$. The source is located at ${\mathbf{r}}^{\prime }$.

Figure 7

Approximations to the Deslauriers et al. needle trap [14]. Black dot: ion. Shaded electrode: rf. (a) Needle trap configuration. (b) Approximation to (a) assuming a single needle suffices to describe the scaling of the geometric factor with respect to $d$. (c) Approximation to (b) representing the needle by an effective sphere of radius $a$.

Figure 8

Coordinates system used for the sphere. The ion is suspended at point $\mathbf{r}$. The source is located at ${\mathbf{r}}^{\prime }$.

Figure 9

Scaling exponent $\alpha$ as a function of dimensionless distance $D=d/a$ above a sphere of radius $a$ for both normal ($r$) and transverse ($\theta$) modes. The labeled solid lines are for the IP and PP limits, and are based on Eq. (29). The shaded area between the IP and PP lines of each mode represents permissible values of $\alpha$. The dotted lines correspond to intermediate patches sizes ${\theta }_{\zeta }/a={10}^{-1},{10}^{-2},{10}^{-3},{10}^{-4}$ evaluated via Eq. (31) by truncating the expansion at $l_0=2/\left({\theta }_{\zeta }\right)$. The dashed line of $\alpha =4$ is plotted for reference.

Figure 10

(a) Prolate spheroidal coordinates defined in Eq. (32). (b) Oblate spheroidal coordinates defined in Eq. (33).

Figure 11

(a) Needle-shaped electrode in prolate spheroidal coordinates for a surface $\xi >1$. (b) Disk-shaped electrode in oblate spheroidal coordinates for a surface $\xi =0$. The ion is represented by the black dot.

Figure 12

Scaling exponent $\alpha$ as a function of dimensionless distance $D=d/r_{\mathrm{prolate}}$ above a needle of radius $r_{\mathrm{prolate}}=0.0100005\times a$ and half-length $100\times r_{\mathrm{prolate}}$ for both normal ($\xi$) and transverse ($\eta$) modes. The labeled solid lines are for the IP limits. The dotted lines correspond to intermediate patches sizes ${\theta }_{\zeta }=0.4,0.04$ evaluated via Eq. (37) by truncating the expression at $l+l^{\prime }=2l_0=4/\left({\theta }_{\zeta }\right)$. The dashed line of $\alpha =4$ is plotted for reference.

Figure 13

Scaling exponent $\alpha$ as a function of dimensionless distance $D=d/a$ above a disk of radius $a$ for both normal ($\xi$) and transverse ($\eta$) modes. The labeled solid lines are for the IP limits. The dotted lines correspond to intermediate patches sizes ${\theta }_{\zeta }=0.4,0.04$ evaluated via Eq. (38) by truncating the expression at $l+l^{\prime }=2l_0=4/\left({\theta }_{\zeta }\right)$. The dashed line of $\alpha =4$ is plotted for reference.