  Mechanical loss of dielectric mirror coatings sets fundamental limits for
both gravitational wave detectors and cavity-stabilized optical local
oscillators for atomic clocks. Two approaches are used to determine the
mechanical loss: ringdown measurements of the coating quality factor and direct
measurement of the coating thermal noise. Here we report a systematic study of
the mirror thermal noise from room temperature to 4 K by operating reference
cavities at these temperatures. The directly measured thermal noise is used to
extract the corresponding mechanical loss for SiO$_2$/Ta$_2$O$_5$ coatings,
which are compared with previously reported values.
