We report upper limits to the radio and X-ray emission from the newly discovered ultracool dwarf binary WISE J104915.57$-$531906.1 (Luhman 16AB). As the nearest ultracool dwarf binary (2 pc), its proximity offers a hefty advantage to studying plasma processes in ultracool dwarfs which are more similar in gross properties (radius, mass, temperature) to the solar system giant planets than stars. The radio and X-ray emission upper limits from the Australia Telescope Compact Array (ATCA) and Chandra observations, each spanning multiple rotation periods, provide the deepest fractional radio and X-ray luminosities to date on an ultracool dwarf, with $log{(L_{rm r, u}/L_{rm bol}) [Hz^{-1}]} < -18.1$ (5.5 GHz), $log{(L_{rm r, u}/L_{rm bol}) [Hz^{-1}]} < -17.9$ (9 GHz), and $log{(L_{rm x}/L_{rm bol})} < -5.7$. While the radio upper limits alone do not allow for a constraint on the magnetic field strength, we limit the size of any coherently emitting region in our line of sight to less than 0.2% of the radius of one of the brown dwarfs. Any source of incoherent emission must span less than about 20% of the brown dwarf radius, assuming magnetic field strengths of a few tens to a few hundred Gauss. The fast rotation and large amplitude photometric variability exhibited by the T dwarf in the Luhman 16AB system are not accompanied by enhanced nonthermal radio emission, nor enhanced heating to coronal temperatures, as observed on some higher mass ultracool dwarfs, confirming the expected decoupling of matter and magnetic field in cool neutral atmospheres.
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