Here we report on the experimental investigation of thermally induced
fluctuations of the relative phase between two Bose-Einstein condensates which
are coupled via tunneling. The experimental control over the coupling strength
and the temperature of the thermal background allows for the quantitative
analysis of the phase fluctuations. Furthermore, we demonstrate the application
of these measurements for thermometry in a regime where standard methods fail.
With this we confirm that the heat capacity of an ideal Bose gas deviates from
that of a classical gas as predicted by the third law of thermodynamics.Comment: 4 pages, 4 figure