Fig. 4: Quantitative analysis of the thermal conductance and the upstream noise.

a Dissipated power \(P\) as a function of \({T}_{m}^{2}-{T}_{0}^{2}\), where \({T}_{m}\) and \({T}_{0}\) are the Ohmic contact and the base temperatures, respectively. The base temperature was separately calibrated (see Supplementary Note 2) and found to be \({T}_{0}\) = 11 mK at \(\nu =3/5\) and \({T}_{0}\) = 14 mK at \(\nu =2/3\). The colored markers (low temperature data - \({T}_{m} \, < \, 25{mK}\)) were linearly fitted to extract \({\kappa }_{2T}\) (fits marked by colored dashed and full lines for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\), respectively). The black markers are high temperature points, which were not fitted, since at these temperatures, the cooling of the central contact by lattice phonons becomes non-negligible. We plot the data for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\) for propagation lengths \(15\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (red) \(45\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (blue), and \(85\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (green). We find length-independent, thermal conductances \({\kappa }_{2T}/{\kappa }_{0}=1.00\pm 0.03\), \({\kappa }_{2T}/{\kappa }_{0}=1.45\pm 0.03\) for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\), respectively. The theoretically expected values for \(\nu =\frac{2}{3}\) (\(\nu =\frac{3}{5}\)) are plotted as a black dashed (full) line. We find excellent agreement with the data for \(\nu =\frac{2}{3}\), while the thermal conductance for \(\nu =\frac{3}{5}\) is somewhat smaller than predicted. b Excess upstream noise as a function of \({T}_{m}\) for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\), and for propagation lengths \(15\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (red), \(45\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (blue), and \(85\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (green). The slope of the linear fit, denoted as \(2{k}_{B}{G}_{2T}{f}_{T}\) in Eq. (3), increases with decreasing length (due to diminishing dissipation) and approaches a value of roughly \(0.5\) times that predicted by a microscopic calculation. The predicted values (see Supplementary Note 9) are depicted by the black, dashed, and solid line for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\), respectively.