## 2 Junctions and Methods
### Josephson Admittance Spectroscopy  
The dc current-voltage dependence, \(I_{0}(V)\), of an autonomous JJ is the result of averaging the Josephson oscillations and has the following form in the RSJ model [4]:  
\[I_{0}(V)=Sign(V)R_{n}^{-1}\left(I_{c}^{2}R_{n}^{2}+V^{2}\right)^{1/2} \tag{1}\]  
where \(I_{c}\) is the critical current, and \(R_{n}\) is the normal-state resistance of JJ. The deviation of this \(I\)-\(V\) curve (Equation (1)) from the \(I\)-\(V\) curve of the JJ in the normal state, \(I_{q}(V)=V/R_{n}\), is due to the time averaging of [PERSON] voltage oscillations \(V(t)\) with an amplitude of \(2I_{c}R_{n}\) and the Josephson frequency \(f_{j}=2\mathrm{eV/h}\). An autonomous RSJ-like Josephson junction with the \(I\)-\(V\) curve described by Equation (1) is an idealization. In our case, the high-\(T_{c}\) thin-film JJ is located on a dielectric substrate, and high-\(T_{c}\) thin-film electrodes are attached to the JJ for dc biasing and voltage measurement. Here, the dielectric and high-\(T_{c}\) materials are of finite dimensions and have their dynamic conductivities \(\sigma(f)\). The total effect of such an electromagnetic environment on JJ is usually described by introducing an external admittance \(Y_{c}(f)\) connected to the RSJ-like JJ. The real part of the \(Y_{c}(f)\) can be determined directly from the \(I\)-\(V\) curve of JJ according to Josephson admittance spectroscopy [33]. To find a contribution to the admittance \(Y_{c}(f)\) from the collective modes in dynamic conductance \(\sigma(f)\) of high-\(T_{c}\) electrodes, a set of \(Y_{c}(f)\)-measurements, where the substrates are changed, should be compared, and the study should be focused on the features, which are not changed. Additional low-temperature oxygen loading can affect only the collective modes in high-\(T_{c}\) materials and thus reveal these modes. When an external admittance \(Y_{c}(f)\) shunts the JJ, the dynamics of the Josephson oscillations change compared to those of the autonomous JJ, and the dc \(I\)-\(V\) curve changes accordingly. One should highlight that the reactive, by an inductance \(L\) or a capacitance \(C\), shunting of the JJ led to a modification of the JJ dynamics, which for low total conductivities \(Y_{L}\) and \(Y_{C}\) has only a quadratic effect on the dc \(I\)-\(V\) curve proportional to \({Y_{L}}^{2}\) or \({Y_{C}}^{2}\)[34]. However, for an arbitrary external admittance \(Y_{c}(f)\) of small values \(\mid Y_{c}(f_{c})\mid<{R_{n}}^{-1}\), the modification of the dc \(I\)-\(V\) curve depends linearly on the real part of the conductivity \(Y_{c}\), namely [33],  
\[\Delta I(V)=-V\left[1-\frac{V}{\left(I_{c}^{2}R_{n}^{2}+V^{2}\right)^{\frac{1} {2}}}\right]\text{Re}Y_{c}(2 eV/h) \tag{2}\]  
where \(\Delta I(V)=I_{\text{e}}(V)-I_{0}(V)\) is the difference between the \(I\)-\(V\) curve perturbed by the external conductance \(Y_{c}(f)\) and the autonomous \(I\)-\(V\) curve of the RSJ type according to Equation (1), \(Y_{c}(0)=0\). Thus, it is possible to obtain the spectral distribution of losses in an external admittance connected to the JJ from the change in the experimental \(I\)-\(V\) curves using Equation (2). The voltage-dependent factor relating \(\Delta I(V)\) and \(\text{Re}Y_{c}(f)\) has a maximum of \(0.3I_{c}R_{n}\) at the voltage around \(0.8I_{c}R_{n}\). The characteristic voltage \(I_{c}R_{n}\) determines the spectral range and sensitivity of Josephson admittance spectroscopy. Figure 2 shows how the \(I\)-\(V\) curve changes with a hypothetical external admittance \(Y_{c}(f)\). The presence of two weak absorption lines in \(\text{Re}Y_{c}(f)\) at dimensionless frequencies \(f_{1}/f_{c}=0.75\) and \(f_{2}/f_{c}=2.0\) (bottom panel) leads to the corresponding two dips in the dc \(I\)-\(V\) curve at dimensionless voltages \(V/I_{c}R_{n}=0.75\) and \(2.0\) (top panel).