Minimizing rf-induced excess micromotion of a trapped ion with the help of ultracold atoms

Abstract

We report on the compensation of excess micromotion due to parasitic rf-electric fields in a Paul trap. The parasitic rf-electric fields stem from the Paul trap drive but cause excess micromotion, e.g., due to imperfections in the setup of the Paul trap. We compensate these fields by applying rf-voltages of the same frequency, but adequate phases and amplitudes to Paul trap electrodes. The magnitude of micromotion is probed by studying elastic collision rates of the trapped ion with a gas of ultracold neutral atoms. Furthermore, we demonstrate that also reactive collisions can be used to quantify micromotion. We achieve compensation efficiencies of about 1 $\,\text {Vm}^{-1}$, which is comparable to other conventional methods.

Appendix: Some general considerations on compensating excess micromotion

Here, we discuss in a very general way excess micromotion in a Paul trap⁴. The motion of a confined ion is determined by the exposure to constant and rf-electrical fields⁵. Concretely, we consider the three electrical fields $\boldsymbol{\mathcal{E}}_c, \boldsymbol{\mathcal{E}}_{\cos }, \boldsymbol{\mathcal{E}}_{\sin }$, characteristic for a Paul trap. $\boldsymbol{\mathcal{E}}_c (\mathbf {r})$ is time-independent, while $\boldsymbol{\mathcal{E}}_{\cos }(\mathbf {r}) \propto \cos (\varOmega t)$ and $\boldsymbol{\mathcal{E}}_{\sin }(\mathbf {r}) \propto \sin (\varOmega t)$ are quadrature components of the rf-field. The electrical fields $\boldsymbol{\mathcal{E}}_c,$ $\boldsymbol{\mathcal{E}}_{\mathrm{cos} }, \boldsymbol{\mathcal{E}}_{\mathrm{sin} }$ can each be Taylor expanded around the trap center position $\mathbf {r}_0$. For each expansion the first term is a homogeneous offset field and the second term is a quadrupole field, followed by higher multipole terms such as the octupole field. In the Paul trap it is the quadrupole field which is used for trapping. For excess micromotion to vanish at $\mathbf {r}_0$, we need $\boldsymbol{\mathcal{E}}_c (\mathbf {r}_0) = \boldsymbol{\mathcal{E}}_{\mathrm{cos} }(\mathbf {r}_0) = \boldsymbol{\mathcal{E}}_{\mathrm{sin} } (\mathbf {r}_0) = 0$, which is equivalent to the vanishing of the respective offset field terms of the Taylor expansions. The remaining quadrupole fields at location $\mathbf {r}$ in the direct vicinity of $\mathbf {r}_0$ will generally dominate over the higher multipole fields, as long as $\mathbf {r}- \mathbf {r}_0$ is much smaller than the distance to any of the trap electrodes. The total quadrupole field $\boldsymbol{\mathcal{E}}_\text {qp}(\mathbf {r})$ (i.e., the sum of the three quadrupole fields) is fully determined by the second derivatives of the corresponding electrostatic potential $\phi$, i.e., $\boldsymbol{\mathcal{E}}_\text {qp}(\mathbf {r}) = H(\phi ) (\mathbf {r}-\mathbf {r}_0)$. Here, $H(\phi )$ is the Hessian matrix of $\phi$, i.e., $H_{i,j}(\phi )={\partial ^2 \phi \over \partial x_i \partial x_j}$, where $x_i, x_j \in \{ x, y, z \}$. Since $H(\phi )$ is symmetric it can be diagonalized. In the corresponding coordinate system $\{x', y', z'\}$ the electrical quadrupole field can be written as $\boldsymbol{\mathcal{E}}_\text {qp} = a x' {\hat{x}}' + b y' {\hat{y}}' + c z' {\hat{z}}'$, similarly as in Eq. (1). Here, a, b, c are time-dependent coefficients. Therefore, the motions of the ion along directions ${\hat{x}}', {\hat{y}}', {\hat{z}}'$ are decoupled and can be described by Mathieu equations. Thus, the main requirements for a Paul trap are fulfilled.

To cancel the offset field components of each of the $\boldsymbol{\mathcal{E}}_c, \boldsymbol{\mathcal{E}}_{\mathrm{cos} }, \boldsymbol{\mathcal{E}}_{\mathrm{sin} }$ fields at location $\mathbf {r}_0$ we can use three compensation electrodes to which we apply suitable dc- and rf-voltages. These electrodes produce electrical fields at $\mathbf {r}_0$ which are preferentially (but not necessarily) perpendicular to each other.