Optimal Surface-Electrode Trap Lattices for Quantum Simulation with Trapped Ions

Trapped ions offer long internal state (spin) coherence times and strong interparticle interactions mediated by the Coulomb force. This makes them interesting candidates for quantum simulation of coupled lattices. To this end, it is desirable to be able to trap ions in arbitrary conformations with precisely controlled local potentials. We provide a general method for optimizing periodic planar radio-frequency electrodes for generating ion trapping potentials with specified trap locations and curvatures above the electrode plane. A linear-programming algorithm guarantees globally optimal electrode shapes that require only a single radio-frequency voltage source for operation. The optimization method produces final electrode shapes that are smooth and exhibit low fragmentation. Such characteristics are desirable for practical fabrication of surface-electrode trap lattices.

Figure 1

Optimized dimensionless curvatures $\kappa$ (black, left axis) and trap depths $\eta$ (cyan, right axis) as functions of the ratio of microtrap height $z$ to inter-ion spacing $d$, for several lattice geometries. Curvature tensors are cylindrically symmetric out-of-plane quadrupoles as for the optimized ring trap ($\widehat{\kappa }=0.298$ and $\widehat{\eta }=0.0196$; see text). The decreases in dimensionless curvatures and trap depths depend strongly on the typical electrode structure size $\Delta =2\pi /Q$ (see text). Spurious microtraps may be present for some parameters.

Figure 2

Top (bottom) row: optimized electrodes for square (triangular) lattices of microtraps at varying height $z/d=0.2$, 0.3, 0.4, 0.5 (0.25, 0.5, 0.75, 1) in units of the ion spacing $d$. Ground and rf electrodes are shown in white and blue (with arbitrary assignment). Microtrap locations are marked with triangles. Curvature tensors are cylindrically symmetric out-of-plane quadrupoles as for the optimized ring trap (see text). For $z/d=0.2$, 0.3 (0.75, 1) the electrodes maximizing the dimensionless curvature $\kappa$ (outlines shown) give rise to spurious trapping sites which have been eliminated by imposing additional constraints (see text) reducing $\kappa$ by 0.07% and 0.04% (4% and 18%), respectively.

Figure 3

Optimized electrodes for a bilayer honeycomb lattice. The desired microtraps, shown as equipotential ellipsoids, are at heights $z_{\pm }/b=0.4\pm \frac{1}{2\sqrt{6}}$, with $b$ the honeycomb lattice periodicity and $d=b/\sqrt{2}$ the inter-ion distance. Vibrational frequencies of the microtraps are $({\omega }_x,{\omega }_y,{\omega }_z)={\omega }_0({\varphi }^{-1},1,\varphi )$ in the three mutually orthogonal nearest-neighbor directions (marked with solid lines and projections; lighter shades indicate higher frequencies), where $\varphi =(1+\sqrt{5})/2$ is the golden ratio. Dimensionless curvatures are ${\kappa }_{-}/\widehat{\kappa }=0.0022$ and ${\kappa }_{+}/\widehat{\kappa }=0.020$ (differing because of the different heights $z_{\pm }$). The locations of spurious microtraps are marked with small spheres. Thin vertical lines are added for clarity of perspective.