  The ability to control and measure the temperature of propagating microwave
modes down to very low temperatures is indispensable for quantum information
processing, and may open opportunities for studies of heat transport at the
nanoscale, also in the quantum regime. Here we propose and experimentally
demonstrate primary thermometry of propagating microwaves using a transmon-type
superconducting circuit. Our device operates continuously, with a sensitivity
down to $4\times 10^{-4}$ photons/$\sqrt{\mbox{Hz}}$ and a bandwidth of 40 MHz.
We measure the thermal occupation of the modes of a highly attenuated coaxial
cable in a range of 0.001 to 0.4 thermal photons, corresponding to a
temperature range from 35 mK to 210 mK at a frequency around 5 GHz. To increase
the radiation temperature in a controlled fashion, we either inject calibrated,
wideband digital noise, or heat the device and its environment. This
thermometry scheme can find applications in benchmarking and characterization
of cryogenic microwave setups, temperature measurements in hybrid quantum
systems, and quantum thermodynamics.
