Fig. 3: Length profile of the thermal conductance and the upstream noise.

a False colors SEM image of the central part of Device B1. The mesa (gray) is divided into three arms by the etched regions (black). The three arms are connected by a floating metallic island (in red) with area \(15\times 2\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}^{2}\), serving as a heat source. When a current \(I\) from \({S}_{1}\) and \(\mbox{-}I\) from \({S}_{2}\) are sourced simultaneously, the floating island heats up to a temperature \({T}_{m}\). The resulting noise is measured simultaneously in the downstream and upstream amplifiers. The propagation length from the floating contact to the upstream amplifier can be varied using the metallic gates (yellow, as in Device A). Depicted is the case where the middle gate (darker yellow) redirects the path of the edge modes by the application of a gate voltage \({V}_{g}\), while the other gates are unbiased, and hence do not affect the propagation length. b Downstream noise (full shapes) and upstream noise (empty shapes) as a function of the current. Results are shown for \(\nu =\frac{2}{3}\) (triangles) and \(\nu =\frac{3}{5}\) (circles), and the propagation lengths \(15\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (red) and \(75\;{{{{{\rm{\mu}}}}}} {{{{{\rm{m}}}}}}\) (gray). The upstream-noise decays with length while the downstream noise does not. c Two-terminal thermal conductance \({\kappa }_{2T}\) (extracted from the downstream noise) (blue), and upstream-noise strength (red) as a function of length (See Methods). The thermal conductance is separately normalized for \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\) with respect to their respective means. For both \(\nu =\frac{2}{3}\) and \(\nu =\frac{3}{5}\), we observe that \({\kappa }_{2T}\) is length-independent, while the upstream-noise decays (similarly to Fig. 2b). This indicates an unequilibrated thermal regime. The empty (full) shapes mark the data measured in device B1 (B2), and error bars represent the 95% confidence bounds of the normalized thermal conductance and the upstream-noise strength.