# Fully Superconducting Josephson Bolometers for Gigahertz Astronomy
## 3 Theory of Nano-TES and JES Bolometers
### Nano-TES Bolometer Performance  
The noise equivalent power (NEP) represents the minimum power that can be detected above the noise level. Thus, the NEP is the most important figure of merit for a bolometer based on the nano-TES sensing element. Taking into account the equivalent circuit shown in Figure 3, the total \(NEP\) of the nano-TES is given by the [47,48]  
\[NEP_{tot}=\sqrt{NEP_{TFN,nano-TES}^{2}+NEP_{Jo}^{2}+NEP_{R_{S}}^{2}}, \tag{8}\]  
where \(NEP_{TFN,nano-TES}\) is associated with the thermal fluctuations, \(NEP_{Jo}\) is due to the Johnson noise, while \(NEP_{R_{S}}\) arises from the shunt resistor. For a nano-TES operating at \(T_{A}\simeq T_{C}\), the thermal fluctuations limited \(NEP\) takes the form [49]  
\[NEP_{TFN,nano-TES}=\sqrt{4 YG_{th,nano-TES}k_{B}{T_{C}}^{2}}, \tag{9}\]  
where \(\mathrm{Y}=n/(2n+1)\) accounts for the temperature gradient between the quasiparticles in \(A\) and all the other thermal sinks (with \(n=5\) for pure and \(n=4,6\) for dirty metals), and \(G_{th,nano-TES}\) is the thermal conductance of all heat losses. Since \(T_{C}\ll T_{C,B}\), the main thermalization channel for \(A\) is the electron-phonon coupling (\(P_{e-ph}\)). Furthermore, the 1D-JJ is always kept in the partially dissipative state (at \(R_{N}/2\)) for the nano-TES operation. Therefore, the electron-phonon coupling of a normal-metal diffusive thin film (\(P_{e-ph,n}\), see Equation (6)) is employed [44]. The resulting thermal conductance for a nano-TES (\(G_{th,nano-TES}\)) can be calculated through the temperature derivative of the electron-phonon energy relaxation [17]  
\[G_{th,nano-TES}=\frac{\mathrm{d}P_{e-ph,n}}{\mathrm{d}T_{A}}=5\Sigma_{A} \mathcal{V}_{A}T_{A}^{4}. \tag{10}\]  
The Johnson noise is originated by the electronic transport through \(A\) when the nano-TES is in the normal-state and takes the form [49]  
\[NEP_{Jo}=\sqrt{2k_{B}R_{N}T_{C}}\frac{G_{th,nano-TES}T_{C}}{V\alpha}\sqrt{1+4 \pi^{2}f^{2}\tau_{eff}^{2}}, \tag{11}\]  
where \(V\) is the voltage drop, \(f\) is the signal bandwidth, \(\alpha=\frac{\mathrm{d}R}{\mathrm{d}T}\frac{T}{R}\) is the electrothermal parameter accounting for the sharpness of the phase transition from the superconducting to the normal-state [17] and \(\tau_{eff}\) is the effective circuit time constant (see Equation (15)). Finally, the charge fluctuations through \(R_{S}\) provide a contribution [49]  
\[NEP_{R_{S}}=\sqrt{4k_{B}R_{S}T_{bath}}\frac{G_{th,nano-TES}T_{C}}{V\alpha}\sqrt{(1 -L_{0})^{2}+4\pi^{2}f^{2}\tau_{eff}^{2}}, \tag{12}\]  
where \(L_{0}=\alpha/n\) is the loop gain. Since the shunting resistor needs to satisfy \(R_{sh}\ll R_{A}\), the contribution of \(NEP_{sh}\) is usually negligible compared to Johnson noise. Indeed, in superconducting bolometers, the thermal fluctuations contribution dominates over \(NEP_{lo}\) and \(NEP_{R_{S}}\). Thus, we can consider \(NEP_{tot}=NEP_{TFN,nano-TES}\). The time response of the detector can be calculated by solving the time dependent energy balance equation that takes into account all the exchange mechanisms after radiation absorption [44]. In particular, the re-thermalization (to \(T_{bath}\)) of the quasiparticles in \(A\) has an exponential dependence on time. The associated time constant (\(\tau_{nano-TES}\)) can be calculated as the ratio between the thermal capacitance and the thermal conductance of \(A\)  
\[\tau_{nano-TES}=\frac{C_{e,nano-TES}}{G_{th,nano-TES}}. \tag{13}\]  
where \(C_{e,nano-TES}\) is the electron heat capacitance. The latter is written  
\[C_{e,nano-TES}=\gamma_{A}\mathcal{V}_{A}T_{C}, \tag{14}\]  
where \(\gamma_{A}\) is the Sommerfeld coefficient of the active region. We note that \(\tau_{nano-TES}\) is the intrinsic recovery time of \(A\). This term does not take into account the Joule heating due to the current flowing through the sensor in the dissipative state.