  Fast-rotating pulsars and magnetars have been suggested as the central
engines of super-luminous supernovae (SLSNe) and fast radio bursts, and this
scenario naturally predicts non-thermal synchrotron emission from their nascent
pulsar wind nebulae (PWNe). We report results of high-frequency radio
observations with ALMA and NOEMA for three SLSNe (SN 2015bn, SN 2016ard, and SN
2017egm), and present a detailed theoretical model to calculate non-thermal
emission from PWNe with an age of about 1-3 yr. We find that the ALMA data
disfavors a PWN model motivated by the Crab nebula for SN 2015bn and SN
2017egm, and argue that this tension can be resolved if the nebular
magnetization is very high or very low. Such models can be tested by future
MeV-GeV gamma-ray telescopes such as AMEGO.
