## 3 Measurement Results  
### DC Characteristics and Coupling Power Estimation  
Figure 5 shows the measured and simulated IVCs of the JJ mixers coupled with different antennas with and without the \(\sim\)0.21 THz radiation. The Mixer 1 coupled with Antenna 1 (bowtie loaded with a pair of lumped-element stubs) has a critical current (\(I_{\mathrm{C}}\)) of 550 \(\mu\)A and a normal-state resistance (\(R\)n) of 0.78 \(\Omega\). The Mixer 2 coupled with Antenna 2 (bowtie loaded with a pair of open-end stubs) has a \(I_{\mathrm{C}}\) of 900 \(\mu\)A and a \(R\)n of 1 \(\Omega\). When mixers are pumped by a \(\sim\)0.21 THz signal, \(I_{\mathrm{C}}\) are suppressed to \(\sim\)110 \(\mu\)A and \(\sim\)275 \(\mu\)A, respectively, but Shapiro steps [37] are located at the same voltages. Obviously, the voltages \(V_{m}\) where the Shapiro steps are located are related to the incident frequency \(f_{\mathrm{THz}}\), consistent with the Josephson voltage-frequency relationship  
\[V_{m}=m\Phi_{0}\,f_{\mathrm{THz}}, \tag{5}\]  
where \(m\) is an integer, and \(\Phi_{0}\) is the magnetic flux quantum [38],  
\[1/\Phi_{0}\equiv 2e/h=483.6\,\mathrm{GHz/mV}. \tag{6}\]  
As the first Shapiro steps are located at 0.429 mV, the \(f_{\mathrm{THz}}\) is estimated at 0.207 THz. According to the measured DC IVCs and resistively shunted junction (RSJ) model [39], the THz power PTHz, which has been coupled into the junctions, can be obtained through numerical simulation by MATLAB software. The circuit equation for this model is  
\[I_{\mathrm{DC}}+I_{\mathrm{RF}}cos\omega_{\mathrm{RF}}t=\frac{V(t)}{R_{n}}+l_{ c}sin\phi(t), \tag{7}\]  
where the phase difference \(\phi\) is related to the voltage \(V\) by the Josephson relation  
\[\frac{d\phi}{dt}=\frac{2\pi}{\Phi_{0}}V(t). \tag{8}\]  
Figure 4: (**a**) Mixer chip on a sample holder with packaged modules. (**b**) Three-dimensional view of the monopole antenna at LO input in a GM cryocooler. (**c**) Schematic diagram of the measurement setup for harmonic mixing.  
In our experiment, \(I_{\text{DC}}\) = 0 for JJ mixer is performed for zero-bias operation which means that there is no DC bias supply, and thus, there is no heating effect or shot noise in the junction due to DC bias current [40]. \(I_{\text{RF}}\) and \(\omega_{\text{RF}}\) are THz current in the junction and THz pump frequency here. Thus, the THz power \(P_{\text{THz}}\) coupled into the junction is  
\[P_{\text{THz}}=\frac{1}{2}I_{\text{RF}}^{2}R_{n}. \tag{9}\]  
The fourth-order Runge Kutta algorithm is used to numerically solve the first-order differential equation with a sinusoidal function. When the simulated DC IVCs matches the measured well, \(P_{\text{THz}}\) of mixers coupled with Antenna 1 and Antenna 2 are \(-23.6\) dBm and \(-36\) dBm, respectively. Minor errorbetween the simulated and the measured DC IVCs at \(\sim\)0.4 mV in Figure 5a may be caused by a capacitance in the junction.  
Figure 5: (**a**) Measured and simulated DC IVCs of the JJ Mixer 1 coupled with Antenna 1: bowtie loaded by a pair of lumped-element stubs under a 0.21 THz signal radiation or not at the bath temperature about 4.5 K. (**b**) Measured and simulated DC IVCs of the JJ Mixer 2 coupled with Antenna 2: bowtie loaded by a pair of open-end stubs under a 0.21 THz signal radiation or not at the bath temperature about 4.5 K.