A resistance bridge based on a cryogenic current comparator achieving sub-$10^{-9}$ measurement uncertainties

A new resistance bridge has been built at the Laboratoire national de m\'etrologie et d'essais (LNE) to improve the ohm realization in the $Syst\`eme International$ (SI) of units from the quantum Hall effect. We describe the instrument, the performance of which relies on two synchronized and noise-filtered current sources, an accurate and stable current divider and a cryogenic current comparator (CCC) having a low noise of $\mathrm{80~pA.t/Hz^{1/2}}$. The uncertainty budget for the measurement of the 100 $\Omega/(R_\mathrm{K}/2)$ ratio, where $R_\mathrm{K}$ is the von Klitzing constant, amounts to a few parts in $10^{10}$ only.