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arXiv:0812.3380 · Physical Review A

Electric field noise above surfaces: A model for heating-rate scaling law in ion traps

Romain Dubessy, T. Coudreau, Luca Guidoni

Published Sep 15, 2009• 59 citations
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Abstract

We present a model for the scaling laws of the electric field noise spectral density as a function of the distance $d$ above a conducting surface. Our analytical approach models the patch potentials by introducing a correlation length $\ensuremath{\zeta}$ of the electric potential on the surface. The predicted scaling laws are in excellent agreement with two different classes of experiments (cold trapped ions and cantilevers), for which the $d$ parameter differs by four orders of magnitude. According to this model, heating rate in miniature ion traps could be greatly reduced by proper material engineering.

DOI
10.1103/physreva.80.031402

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We present a model for the scaling laws of the electric field noise spectral density as a function of the distance ddd above a conducting surface. Our analytical approach models the patch potentials by introducing a correlation length \ensuremathζ\ensuremath{\zeta}\ensuremathζ of the electric potential on the surface. The predicted scaling laws are in excellent agreement with two different classes of experiments (cold trapped ions and cantilevers), for which the ddd parameter differs by four orders of magnitude. According to this model, heating rate in miniature ion traps could be greatly reduced by proper material engineering.
In this work we present a model where we take explicitly into account the spatial dependence of the electric field noise density above a surface, introducing a characteristic length not yet considered in previous models: the correlation length, zeta of the noisy potential on the surface. In that picture, the noise arise from finite size sources distributed randomly on the surface. We show that this simple model accounts for the behavior of electric field noise scaling on the whole range covered by both ion heating measurements and cantilever-based measurements. We propose that this correlation length could be related to the characteristic size of the patches, opening a way to the control of noise intensity by engineering the material properties.
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, ddd, 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−4d^{-4}d−4 dependence currently found by most experiments and models, but also predicts geometry-driven deviations from this trend.
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 ddd 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 ${d}^{\ensuremath{-}4}$ dependence currently found by most experiments and models, but also predicts geometry-driven deviations from this trend.
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.
We probe electric-field noise in a surface ion trap for ion-surface distances ddd between 50 and 300 \textmu{}m in the normal and planar directions. We find the noise distance dependence to scale as ${d}^{\ensuremath{-}2.6}$ in our trap and a frequency dependence which is consistent with 1/f1/f1/f noise. Simulations of the electric-field noise specific to our trap geometry provide evidence that we are not limited by technical noise sources. Our distance scaling data is consistent with a noise correlation length of about 100 \textmu{}m at the trap surface, and we discuss how patch potentials of this size would be modified by the electrode geometry.
Motional heating of ions in microfabricated traps is one of the open challenges hindering experimental realizations of large-scale quantum processing devices. Recently, a series of measurements of the heating rates in surface-electrode ion traps characterized their frequency, distance, and temperature dependencies, but our understanding of the microscopic origin of this noise remains incomplete. In this work we develop a theoretical model for the electric field noise which is associated with a random distribution of adsorbed atoms on the trap electrode surface. By using first-principles calculations of the fluctuating dipole moments of the adsorbed atoms we evaluate the distance, frequency, and temperature dependence of the resulting electric field fluctuation spectrum. Our theory reproduces correctly the ${d}^{\ensuremath{-}4}$ dependence with distance of the ion from the electrode surface and calculates the noise spectrum beyond the standard scenario of two-level fluctuators by incorporating all the relevant vibrational states. Our model predicts a regime of 1/f1/f1/f noise which commences at roughly the frequency of the fundamental phonon transition rate and a thermally activated noise spectrum which for higher temperatures exhibits a crossover as a function of frequency.