# Fully Superconducting Josephson Bolometers for Gigahertz Astronomy
## 2 Sensors Structure and Operating Principle  
The sensors presented in this review, i.e., the nano-TES [33] and the JES [34], can be described as a 1D fully superconducting Josephson junction (1D-JJ). A 1D-JJ is realized in the form of a superconducting nanowire (\(A\)), with both lateral dimensions (thickness (\(t\)) and width (\(w\))) smaller than its coherence length (\(\xi_{A}\)) and the London penetration depth of the magnetic field (\(\lambda_{L,A}\)), separating two superconducting lateral banks (\(B\)), as schematically shown in Figure 2a. The 1D nature of the JJ provides a two-fold advantage. On the one hand, \(\lambda_{L,A}\gg w,t\) ensures homogeneous supercurrent density in the JJ and uniform penetration of \(A\) by an out-of-plane magnetic field. On the other hand, \(\xi_{A}>w,t\) leads to uniform superconducting properties and constant superconducting wave function along the nanowire cross section. The electronic transport properties of a 1D-JJ can be described through the over-damped resistively shunted junction (RSJ)) model [38], where the dependence of the stochastic phase difference (\(\varphi(t)\)) over the 1D-JJ) on the bias current (\(I\)) is given by  
\[\frac{2e}{h}\frac{\dot{\varphi}(t)}{R_{N}}+I_{C}\sin\varphi(t)=I+\delta I_{th }(t), \tag{1}\]  
where \(e\) is the electron charge, \(R_{N}\) is the normal-state resistance, \(h\) is the reduced Planck constant and \(I_{C}\) is the critical current. The thermal noise generated by the shunt resistor  
Figure 2: Properties of a 1D fully superconducting Josephson junction (JJ). **(a)** Schematic representation of a 1D fully superconducting JJ, where a superconducting nanowire \(A\) is contacted by two superconducting leads \(B\) obeying to \(\Delta_{A}\ll\Delta_{B}\) (with \(\Delta_{A}\) and \(\Delta_{B}\) the superconducting energy gap of the nanowire and the leads, respectively). **(b)** Energy of the washboard potential normalized with respect to the zero-temperature Josephson energy as a function of the phase difference across a JJ for different values of bias current. The energy barrier (\(\delta U\)) for the escape of the phase particle from the WP is indicated. **(c)** Energy barrier normalized with respect to the zero-temperature Josephson energy (\(\delta U/E_{J,0}\)) calculated as a function of critical current (top) and bias current (bottom). **(d)** Resistance (R) versus temperature (T) characteristics calculated for different values of \(I\). Inset: Temperature derivative of \(R\) calculated for different values of \(I\). **(e)** Calculated temperature dependence of the JJ resistance, where the escape temperature (\(T_{e}\)) and the critical temperature (\(T_{C}\)) are indicated. **(f)** Escape temperature (\(T_{e}\)) and critical temperature (\(T_{C}\)) calculated as a function of the bias current (\(I\)). (\(R_{N}\)) is given by \(\langle\delta I_{th}(t)\delta I_{th}(t^{\prime})\rangle=\frac{K_{B}T}{R_{N}}\delta(t-t ^{\prime})\), where \(k_{B}\) is the Boltzmann constant and \(T\) is the temperature. Within the RSJ model, the transition of the junction to the resistive state is attributed to \(2\pi\) phase-slip events [38; 39]. The phase slips can be viewed as the movements of a phase particle in a tilted washboard potential model (WP) under the action of friction forces. In particular, the WP takes the form  
\[U(\varphi)=-\frac{\hbar I}{2e}\varphi-\delta U\cos\varphi. \tag{2}\]  
For a 1D-JJ composed of a superconducting nanowire, the escape energy barrier can be written as [40]  
\[\delta U(I,E_{J})\sim 2E_{J}(1-I/I_{C})^{5/4}=\frac{\Phi_{0}I_{C}}{\pi}(1-I/I_{C} )^{5/4}. \tag{3}\]  
Indeed, the effective WP profile strongly depends on both bias current (\(I\)) through the junction and Josephson energy \(E_{J}=\Phi_{0}I_{C}/2\pi\)[40], where \(\Phi_{0}\simeq 2.067\times 10^{-15}\) Wb is the flux quantum. In particular, by rising \(I\) the phase-slip energy barrier (\(\delta U\)) lowers, similarly to the result of decreasing \(E_{J}\), and the WP tilts proportionally to the value of the bias current, as shown in Figure 2b.