  Thermal voltage noise measurements are performed in order to determine the
electron temperature in nanopatterned channels of a GaAs/AlGaAs heterostructure
at bath temperatures of 4.2 and 1.4 K. Two narrow two-dimensional (2D) heating
channels, close to the transition to the one-dimensional (1D) regime, are
connected by a quasi-1D quantum interferometer. Under dc current heating of the
electrons in one heating channel, we perform cross-correlated noise
measurements locally in the directly heated channel and nonlocally in the other
channel, which is indirectly heated by hot electron diffusion across the
quasi-1D connection. We observe the same functional dependence of the thermal
noise on the heating current. The temperature dependence of the electron
energy-loss rate is reduced compared to wider 2D systems. In the quantum
interferometer, we show the decoherence due to the diffusion of hot electrons
from the heating channel into the quasi-1D system, which causes a thermal
gradient.
