## 2 Two Loops  
### Capacitively-Shunted Model for an rf-SQUID  
We first consider the capacitively-shunted model for an radio frequency superconducting quantum interference device(rf-SQUID) loop, which has a Josephson junction with the critical current \(l_{c}\) and a capacitor with capacitance \(C\) in parallel. The current flowing through the rf-SQUID loop is given by [19]  
\[I=-l_{c}\sin\phi+C\dot{V}, \tag{1}\]  
where \(\phi\) is the superconducting phase difference across the Josephson junction. Here, the Cooper pair current \(I\) is represented by \(I=-(n_{c}Aq_{c}/m_{c})\hbar k\) with the Cooper pair density \(n_{c}\), the cross section \(A\) of the loop, \(q_{c}=2e\), and \(m_{c}=2m_{c}\). Using the Josephson voltage-phase relation \(V=-(\Phi_{0}/2\pi)\dot{\phi}\) with the superconducting unit flux quantum \(\Phi_{0}=h/2e\), the current relation of Equation (1) becomes  
\[-\frac{\Phi_{0}^{2}}{2\pi L_{K}}\frac{l}{2\pi}k=-E_{J}\sin\phi-C\left(\frac{ \Phi_{0}}{2\pi}\right)^{2}\dot{\phi}, \tag{2}\]  
where we introduce the kinetic inductance, \(L_{K}=m_{c}l/An_{c}q_{c}^{2}\)[5; 20; 21], and the Josephson coupling energy, \(E_{J}=l_{c}\Phi_{0}/2\pi\).  
From the Lagrange equation \((d/dt)\partial\mathcal{L}/\partial\dot{\phi}-\partial\mathcal{L}/\partial\dot {\phi}=0\) with the Lagrangian  
\[\mathcal{L}=\frac{C}{2}\left(\frac{\Phi_{0}}{2\pi}\right)^{2}\dot{\phi}^{2}-U _{\text{eff}}(\{\phi\}) \tag{3}\]  
and the effective potential of the system \(U_{\text{eff}}(\phi)\), we can obtain the equation of motion of the phase variable of a Josephson junction, \(C(\Phi_{0}/2\pi)^{2}\dot{\phi}=-\partial U_{\text{eff}}/\partial\phi\), and thus the relation,  
\[\frac{\Phi_{0}^{2}}{2\pi L_{K}}\frac{l}{2\pi}k-E_{J}\sin\phi=-\frac{\partial U _{\text{eff}}}{\partial\dot{\phi}}. \tag{4}\]  
The fluxoid quantization condition for a thin superconducting loop with the total threading magnetic flux \(\Phi_{t}\) is given by \(-\Phi_{t}+(m_{c}/q_{c})\oint\vec{v}_{c}\cdot d\vec{l}=n\Phi_{0}\)[19; 22], with the average velocity of Cooper pairs \(\vec{v}_{c}\), which can be represented as a periodic boundary condition, \(k_{n}l=2\pi n+2\pi f_{t}\), where \(k_{n}\) is the wave vector of the Cooper pairs, \(l\) the circumference of the loop, and \(f_{t}\equiv\Phi_{t}/\Phi_{0}\). Here, \(k_{n}l\) is the phase accumulated along the circumference of the loop. For an rf-SQUID loop, the boundary condition with the phase difference \(\phi\) across the Josephson junction becomes \(k_{n}l=2\pi n+2\pi f_{t}-\phi\). The total flux \(\Phi_{t}=\Phi_{\text{ext}}+\Phi_{\text{ind}}\) with the external flux \(\Phi_{\text{ext}}\) and induced flux \(\Phi_{\text{ind}}=L_{S}l\) can be written by \(f_{t}=f+f_{\text{ind}}\) with \(f\equiv\Phi_{\text{ext}}/\Phi_{0},f_{\text{ind}}\equiv\Phi_{\text{ind}}/\Phi_ {0}=L_{S}l/\Phi_{0}\) and the geometric self-inductance \(L_{S}\) of the rf-SQUID loop, resulting in the relation  
\[\left(1+\frac{L_{S}}{L_{K}}\right)k_{n}l=2\pi n+2\pi f-\phi. \tag{5}\]  
From the relations of Equations (4) and (5), we obtain  
\[-\frac{\partial U_{\text{eff}}}{\partial\dot{\phi}}=\frac{\Phi_{0}^{2}}{2\pi (L_{K}+L_{S})}\left(n+f-\frac{\phi}{2\pi}\right)-E_{J}\sin\phi. \tag{6}\]  
Hence, the relation in Equation (2) describes the dynamics of a particle with kinetic energy \(E_{\text{C}}=Q^{2}/2C\) with \(Q=C(\Phi_{0}/2\pi)\dot{\phi}\) in an effective potential, [23]  
\[U_{\text{eff}}(\phi)=\frac{\Phi_{0}^{2}}{2(L_{K}+L_{S})}\left(n+f-\frac{\phi}{ 2\pi}\right)^{2}-E_{J}\cos\phi. \tag{7}\]