of the ring.

Effective thermal response time
-------------------------------

In the results that follow we report direct measurements of thermal fluxes in multi-stage interferometers and ring resonators patterned into the low-dimensional legs of SiN$_{\rm x}$ TESs. Furthermore, as an additional indicator of reduction in thermal differential conductance, we also measure the time constants of phononic devices.

![\[fig:Circuits\] (a): Thévenin equivalent representation of the TES bias circuit, where the TES is shown as a variable resistor $R(T,I)$. $R_L$ is the internal resistance of the voltage source, $V$. $R_L$ is the sum of a 1.45m$\Omega$ bias resistor and a $\approx 1$m$\Omega$ stray resistance. $L$ corresponds to the input inductance of the SQUID and any stray wiring inductance. $V_{TES}$ is the voltage across the TES. (b): Thermal circuit showing the TES with heat capacity $C$ coupled to the heat bath via thermal conductance $G$. $T$ and $T_B$ are the temperatures of the TES and the heat bath respectively . $P_J$ represents the Joule power dissipated in the TES, and $P_B$ the thermal power flowing to the heat bath.](Figure2.pdf){width="3.37in"}

The effective thermal time constant $\tau_{eff}$ of a TES may be determined from its response to a small step in bias voltage. The induced current response 