## 2 TES Physics and Models
### TES Electro-Thermal Equations
The total current in the TES \(I(t)\) is considered to be the sum of two components, the Josephson current \(I_{I}(t)\) and a quasi-particle current \(I_{qp}=V(t)/R_{N}\). In the general case, \(V(t)=V_{dc}+V_{pk}\cos\omega_{0}t\), where \(V_{dc}\) is a constant voltage across the TES and the second term account for any potential ac excitation of peak amplitude \(V_{pk}\) injected into the bias circuit. In the pure ac voltage bias case, \(V_{dc}=0\), and \(V_{pk}\cos\omega_{0}t\) is the applied voltage bias with \(\omega_{bias}=\omega_{0}\). The ac voltage across the TES forces the gauge invariant superconducting phase \(\varphi\) to oscillate at the bias frequency \(\omega_{bias}\), \(\pi/2\) out-of-phase with respect to the voltage. The \(\varphi\) peak value depends on \(V_{pk}/\omega_{0}\). A rigorous description of the electro-thermal equations for a TES, following the RSJ model, will be given below. However, the main features observable in the TES current can already be understood by a simple voltage-source model in the small signal limit, with \(\varphi=2e/\hbar\int V(t)dt\) from the ac Josephson relation in Equation (14). This is valid both for a dc and an ac biased TES. In the general case, the Josephson current becomes  
\[I_{J}(t)=I_{c}(T)\sin\left[\frac{2 eV_{dc}}{\hbar}t+\frac{2 eV_{pk}}{\hbar\omega }\sin\omega_{0}t\right], \tag{15}\]  
which, using standard trigonometric identities and the Bessel function relations, can be written in the form  
\[I_{J}(t)=I_{c}(T)\sum_{n=-\infty}^{\infty}(-1)^{n}J_{n}\left(\frac{2 eV_{pk}}{ \hbar\omega_{0}}\right)\sin(\omega_{J}-n\omega_{0})t. \tag{16}\]  
In the equation above, \(J_{n}\) is the ordinary Bessel function of the first kind and \(\omega_{I}=2 eV_{dc}/\hbar\) is the Josephson oscillation frequency. Equation (16) says that, when a dc biased TES is excited by a small ac signal with frequency \(\omega_{0}\), spikes in the TES current can be observed at the TES voltage \(V_{dc}=n\hbar\omega_{0}/2e\), with \(\omega_{J}=n\omega_{0}\). For the ac bias case, thanks to the high-\(Q\)\(LC\) filters in the bias circuits, \(V_{dc}\)=0, \(\omega_{J}=0\) and only the \(n=1\) frequency component survives in the sum, with \(\omega_{0}=\omega_{bias}=1/\sqrt{L_{F}C_{F}}\). At a fixed bias point and with \(V_{dc}=0\) and for small value of \(V_{pk}\), the non-linear Josephson inductance in parallel to the TES resistance is simply defined by  
\[L_{J}=\frac{V(t)}{dI/dt}=\frac{\hbar}{2 el_{c}(T)\cos\varphi}=\frac{\hbar}{2 el_ {c}J_{0}\left(\frac{2 eV_{pk}}{\hbar\omega_{0}}\right)}. \tag{17}\]  
In the ac bias read-out, \(L_{J}\) can be derived from the quadrature component of the current measured in the \(I-V\) characteristics. The generalized system of coupled thermal and electrical differential equations for a TES and its bias circuit, which includes the Josephson relations, becomes  
\[\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t} =I_{c}(T)\sin\varphi(t)+I(t)R_{N}, \tag{18a}\] \[L_{F}\frac{\partial I}{\partial t} =V_{dc}+V_{ac}(t)-\left(\frac{Q(t)}{C_{F}}+r_{sh}I(t)+\frac{ \hbar}{2e}\frac{\partial\varphi}{\partial t}\right)+e_{int}+e_{ext},\] (18b) \[\frac{\partial Q}{\partial t} =I(t),\] \[C\frac{\partial T}{\partial t} =\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t}I(t)-G_{12}T+ G_{12}T_{2}-P_{bath}+P_{X}-I(t)e_{int}+p_{bath}+p_{12}+p_{ext,1},\] (18c) \[C_{2}\frac{\partial T_{2}}{\partial t} =-G_{2,b}T_{2}+G_{12}T-G_{12}T_{2}+p_{12}+p_{2,b}+p_{ext,2},\]where Equation (18a) is the superconducting phase relation according to the Jospebson and the RSJ model, Equations (18b) and (18c) are, respectively, the electrical bias circuit equations and the thermal equations for the two-body model shown in \[V_{TES}=\frac{\hbar}{2e}\frac{\partial\varphi}{\partial t},\ \ I_{I}(t)=I_{c}(T)\sin \varphi(t), \tag{14}\]0 In Equation (18b), \(Q\) is the capacitor charge, and \(I\) and \(T\) are the TES current and temperature, respectively. For the ac-bias case, \(V_{dc}=0\).