# Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Amplifier
## 3 Increase in Dynamic Range Using bi-SQUID Cells
For this purpose, one can substitute the dc SQUID cells for bi-SQUID cells as suggested recently in [21] and shown in Figure 3. In the cell, the required improvement may be achieved through redirection of the currents flowing inside the cell with the current via the added third Josephson junction. The mechanism can be explained in the following way. In the absence of both the signal and idler waves (i.e., at \(I_{b}=0\)) when only the dc magnetic flux \(\Phi_{dc}\) is applied to an assigned operation point, Josephson phases \(\varphi_{1}\) and \(\varphi_{2}\) of the junctions J1 and J2 in symmetric cells of both types, dc SQUID and bi-SQUID, are equal in magnitude and opposite in their signs in regard to the direction of transport current, i.e., \(\varphi_{2}=-\varphi_{1}\), due to the induced counterclockwise circular screening supercurrent. Figure 4a schematically shows the phases and the currents flowing through the junctions J1 and J2 when only the dc magnetic flux is applied to the cells. At a negligibly small geometric inductance of the device loops, as assumed in [16], \(\varphi_{1}=-\varphi_{2}=\varphi_{dc}/2=\pi\Phi_{dc}/\Phi_{0}\). Figure 3: After [21]. The inductively coupled elementary cells of the pump and signal transmission lines of JTWPA based on using bi-SQUIDs. \(C\) and \(C_{1}\) are ground capacitances of the two lines: \(L_{1}C_{1}\) line for the pump wave and \(LC\) line for the signal and idler waves, where \(L\) is inductance of the bi-SQUID cell. When the waves propagate in an artificial line and produce a relatively slow-varying transport current \(I_{b}\) flowing via a symmetric dc SQUID and a symmetric bi-SQUID both in the superconducting state, this current (when the wave frequencies are much less than both the Josephson characteristic and plasma frequencies \(\omega_{c}=2\pi I_{c}R_{n}/\Phi_{0}\), \(\omega_{0}=\sqrt{2\pi I_{c}/\left(\Phi_{0}C_{j}\right)}\); \(C_{j}\) and \(R_{n}\) are the junction capacitance and normal resistance) is carried practically by only the superconducting currents through Josephson junctions J1 and J2; therefore, the Josephson phases \(\varphi_{1}\) and \(\varphi_{2}\) of the junctions obey the current relation  
\[i_{b}\equiv I_{b}/I_{c}=i_{2}+i_{1}=\sin\!\varphi_{2}+\sin\varphi_{1} \tag{10}\]  
and the phase equation which is  
\[\varphi_{1}-\varphi_{2}=\varphi_{e}-l(\sin\varphi_{1}-\sin\varphi_{2})/2 \tag{11}\]  
for dc SQUID (in detail, see the macroscopic quantum interference effect in [20] and  
\[\varphi_{1}-\varphi_{2}=\varphi_{e}-l(\sin\varphi_{1}-\sin\varphi_{2})/2 \tag{12}\]  
for bi-SQUID in accordance with the schematic in Figure 4, where the junctions J1 and J2 are on the left and right sides, respectively, and J3 is above them. In both equations, \(l\) is the normalized loop inductance (8), \(\varphi_{e}=2\pi\Phi_{e}/\Phi_{0}\) is the normalized value of the applied magnetic flux \(\Phi_{e}=\Phi_{dc}+\Phi_{rf}\), where \(\Phi_{dc}\) is the applied dc magnetic flux setting operation point, and \(\Phi_{rf}\) is the _rf_ magnetic flux which is applied by the pump wave to modulate the cell dipole inductance; correspondingly, \(\varphi_{e}=\varphi_{dc}+\varphi_{rf}\). In dc SQUID, transport current \(i_{b}\) causes displacement of both the points 1 and 2 shown in Figure 4a in the sinusoid in the direction corresponding to either the increase in the junction phases at \(i_{b}>0\) or the decrease in the ones at \(i_{b}<0\), and some drop in the Josephson phase \(\Delta\varphi\) appears on the cell: \(\Delta\varphi>0\) or \(\Delta\varphi<0\), respectively. The relation between the phase drop and the transport current is evidently nonlinear. As seen from the sketch (Figure 4a), the nonlinearity results from either the first junction at the positive transport current (corresponding to red color of the shown arrows) or the second junction (green arrows) at the negative transport current. In bi-SQUID, the nonlinearity problem is considerably mitigated through the current flowing via the added Josephson junction J3.