Suppression of potential roughness in atom-chip ac Zeeman traps

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We present several toy models for predicting the character and strength of the roughness of ac Zeeman trapping potentials generated by imperfections in wire traces on an atom chip. An ac Zeeman trapping potential is generated by targeting a microwave or rf magnetic field at hyperfine or Zeeman ground-state transitions, respectively, while a dc Zeeman trap uses a static magnetic field to manipulate an atom via its magnetic moment. We find that an ac Zeeman trap suppresses potential roughness by several orders of magnitude with respect to a comparable dc Zeeman trap. This suppression stems from the inherent differences between an atom's response to dc and ac (namely rf and microwave) magnetic fields, in concert with how the ac skin effect smooths out a current distribution disturbed by a conductivity variation. For chip wires fabricated by evaporation, we find that an ac Zeeman trap for $^{87}\mathrm{Rb}$ atoms, located $100\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{m}$ from the chip and operating on a 6.8 GHz hyperfine transition, is expected to suppress potential roughness by a factor of $4\ifmmode\times\else\texttimes\fi{}{10}^{5}$ compared with a similar dc Zeeman trap. For a comparable ac Zeeman trap operating on a 10 MHz Zeeman transition, the suppression is $2\ifmmode\times\else\texttimes\fi{}{10}^{4}$. Conversely, the ac skin effect also slightly exacerbates potential roughness, in ac Zeeman traps, from defects in the sidewall of a wire trace.