  We have embedded a small Josephson junction in a microwave resonator that
allows simultaneous dc biasing and dispersive readout. Thermal fluctuations
drive the junction into phase diffusion and induce a temperature-dependent
shift in the resonance frequency. By sensing the thermal noise of a remote
resistor in this manner, we demonstrate primary thermometry in the range from
300 mK to below 100 mK, and high-bandwidth (7.5 MHz) operation with a
noise-equivalent temperature of better than 10 $\mathrm{\mu K/\sqrt{Hz}}$. At a
finite bias voltage close to a Fiske resonance, amplification of the microwave
probe signal is observed. We develop an accurate theoretical model of our
device based on the theory of dynamical Coulomb blockade.

Mesoscale and Nanoscale Physics