## 2 TES Physics and Models
### TES Electro-Thermal Equations  
The standard electro-thermal linear model for a dc biased TES calorimeter was originally developed by [PERSON] et al. [49] to describe the noise and the response to photons. No assumption were made on the TES physics, and the resistance, for each bias point in the transition, was linearly approximated as in Equation (1). This model has been successfully used to understand the behaviour of TES-based detectors in many works that followed. In this section, we present the Langevin electro-thermal equations for a TES calorimeter, generalized for the dc and ac biased case, assuming the detector physics being well described by the Josephson effects, formalized in the RSJ model. A schematic diagram of the detector thermal model and the electrical ac and dc bias circuit is shown in Figure 1.  
As suggested and observed in many experiments [50; 51; 52; 53; 54], a single-body thermal model for a TES calorimeter is rarely sufficient to explain the detector response. This is due to the presence of dangling heat capacitance and parasitic thermal conductance in the TES-absorber structures or to a not sufficiently high thermal conductance between TES and absorber. It can be shown that a two-body model [49; 50; 53], as drawn in Figure 1b, is sufficiently general to account for parasitic thermal effects in TES-based detectors. A more detailed analysis for even more complex thermal structures can be found in [51; 52; 53; 54]. However, one has to keep in mind that the model is unconstrained when too many thermal bodies are added into the system of equations. As a consequence, the physical interpretation of the results becomes impossible. In Figure 1c, the electrical circuit for a MHz biased TES is shown. The TES is placed in series with a high-\(Q\) superconducting \(LC\) filter [55; 56] and the input coil of the SQUID current amplifier [57]. An optional superconducting transformer, shown in the picture in dashed lines, could be used to optimize the impedance matching between detector and amplifier. The ac voltage bias is provided via a capacitive divider in parallel with an ac shunt resistor. From a simple Thevenin equivalent circuit analysis, the bias network is equivalent to a shunt resistance \(r_{sh}=r_{ac}C_{b}/(C_{F}+C_{b})\), in series with the \(LC\) resonator. \(r_{s}\) accounts for the intrinsic loss in the \(LC\)-filters [58]. The dc bias circuit is shown in Figure 1d. It is equivalent to the ac bias one with \(C_{F}=0\), and \(C_{b}\rightarrow\infty\) and after replacing the ac voltage source with a dc one. In this case, \(L_{f}\) is the Nyquist inductor added to limit the read-out bandwidth, \(r_{sh}\) is a low shunt resistors needed to provide a stiff dc voltage bias to the TES, and \(r_{s}\) indicates any parasitic resistance in the bias circuit. Both for the ac and dc read-out the following holds: \(r_{s}\ll r_{sh}\ll R_{N}\). More details on the ac and dc read-out will be given in Section 5. Within the RSJ model, the TES is electrically treated as a Josephson weak-link with a resistor \(R_{N}\) in parallel with a non-linear Josephson inductance. It obeys the standard Josephson equations relating the TES voltage \(V_{TES}\) and the Josephson current \(I_{I}\) to the gauge-invariant phase difference \(\varphi\) of the superconducting order parameter across the leads  
\[V_{TES}=\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t},\ \ I_{I}(t)=I_{c}(T)\sin \varphi(t), \tag{14}\]  
assuming that a sinusoidal current-phase relation holds. The total phase difference across the weak-link depends on the perpendicular magnetic flux coupled into the TES, i.e.,\(\varphi_{tot}=\varphi+2\pi\frac{\Phi(t)}{\Phi_{0}}\), where \(\Phi(t)=A_{eff}(B_{\perp,DC}+B_{\perp,AC}(t))\) and \(A_{eff}\) is the effective weak-link area crossed by the dc (\(B_{\perp,DC}\)) and ac (\(B_{\perp,AC}\)) perpendicular magnetic field, respectively. The \(B_{\perp}\) field can be generated both from an external source or self generated in the TES by the current flowing in the leads.