Effects of electrode surface roughness on motional heating of trapped ions

Electric-field noise is a major source of motional heating in trapped-ion quantum computation. While the influence of trap-electrode geometries on electric-field noise has been studied in patch potential and surface adsorbate models, only smooth surfaces are accounted for by current theory. The effects of roughness, a ubiquitous feature of surface electrodes, are poorly understood. We investigate its impact on electric-field noise by deriving a rough-surface Green's function and evaluating its effects on adsorbate-surface binding energies. At cryogenic temperatures, heating-rate contributions from adsorbates are predicted to exhibit an exponential sensitivity to local surface curvature, leading to either a large net enhancement or suppression over smooth surfaces. For typical experimental parameters, orders-of-magnitude variations in total heating rates can occur depending on the spatial distribution of adsorbates. Through careful engineering of electrode surface profiles, our results suggests that heating rates can be tuned over orders of magnitudes.

Figure 1

Cartesian coordinates defining positions on smooth and rough surfaces. The $xyz$ coordinates describe the macroscopic smooth planar geometry and is the reference against which the rough surface is defined through the height function $z=h(x,y)$. The $\tilde{x}\tilde{y}\tilde{z}$ axes describe a local coordinate system tangent to the rough surface at position $(x,y,h(x,y))$.

Figure 2

Qualitative plot of interaction potential $U\left(z\right)$ from Eq. (44) for surfaces that are planar, $H=0$, have positive curvature, $H>0$, and negative curvature, $H<0$. The distance scale depicted is typical for adsorbates, in this case a hydrogen adatom on a gold surface. Note in particular the direction of the shift of the binding energy, defined as the minimum $\mathbf{U}=U\left(z_0\right)$, and the transition frequency, defined through the second derivative of $U\left(z\right)$ at $z=z_0$. In the case of H-Ag interactions, $z_0\approx 1.5$ Å.

Figure 3

Distribution of adatoms (dots) on a rough surface (line) in two limiting regimes. (a) The thermal regime where atoms equilibrate at positions of positive curvature where the binding energy is enhanced. (b) The uniform regime where atoms are uniformly distributed, such as at nonequilibrium, or if binding sites are full. The roughness (vertical axis) is exaggerated.

Figure 4

Ratio of heating rate for rough surfaces $S_{\text{rough}}$ to heating rate for planar surfaces $S_{\text{planar}}$ with respect to root-mean-squared surface curvature $H$ to first order in $H$. Normalized for the same number of adsorbates, a uniform distribution (thick) of adatoms sees a strong exponential enhancement. When the distribution relaxes to thermal equilibrium (dashed), heating rates are gradually suppressed depending on the filling fraction $\theta$ of binding sites. The shaded area depicts the estimated contribution of higher order $H^2$ terms for typical physical surfaces. Parameter values are $h{\nu }_p/k_B=200$ K, $z_0=3$ Å, $T=4$ K.

Figure 5

$\epsilon$ value for the van der Waals' potential (red) and the repulsion (blue) potential. For typical electrode surfaces, the error on the potential curve from approximating the whole surface as a parabolic surface is less than $5\%$, and therefore the parabolic approximation is valid.