# Detection of On-Chip Generated Weak Microwave Radiation Using Superconducting Normal-Metal SET
## 4 Discussion  
Two important characteristics of the NISIN detector will be addressed below in more detail with the help of a theoretical analysis of tunneling rates: the amplification factor and the device resolution in respect of the lowest absorption rate of the microwave photons. In order to evaluate the amplification factor \(G\), we calculated the photon-assisted quasiparticle tunneling rate \(\Gamma_{\text{ph}}=\Gamma_{\text{ph}}(E_{\text{ph}})\), where \(E_{\text{ph}}=2 eV_{\text{JJ}}=hf_{\text{fj}}\) is a photon energy corresponding to the Josephson frequency, and compared it with the value \(I_{\text{NISIN}}/e\) measured in experiment. The NISIN transistor was assumed to be symmetrical, and the calculation was carried out for the gate setting \(n_{\text{g}}\approx 0\) corresponding to the highest device sensitivity. At low experimental temperature, \(T\sim\)15 mK, and small energy \(k_{\text{B}}T\sim 1\)\(\text{{\rm{\,ueV}}}\ll eV_{\text{b}}\sim 200\)\(\mu\)eV, the main direction of tunneling is basically defined by the bias voltage \(V_{\text{b}}\) and only two photon-assisted tunneling processes (one in each of the junctions, L and R, see Figure 1) dominate in creation of quasiparticles in the superconducting island, so that \(\Gamma_{\text{ph}}\approx\Gamma_{\text{ph}}^{L}+\Gamma_{\text{ph}}^{R}\). In order to calculate the tunneling rates \(\Gamma_{\text{ph}}^{R,L}\), we follow the argument of [38] based on the \(P\)-function theory [39]. In particular, we assign the role of the noise source as in [38] to the Josephson oscillations in our case and make use of the low impedance of the current leads attached to the NISIN double junction. Thus, the function \(P(E)\) peaks around \(E=0\) and for substantially negative energy values, \(E<-k_{\text{B}}T\), the value of \(P(E)\) is determined by its radiation-induced component [38]  
\[P(E)=\frac{\pi S_{\text{V}}(|E|/\hbar)}{R_{\text{k}}E^{2}}, \tag{2}\]  
with \(R_{\text{k}}=h/e^{2}\approx 26\)\(\text{k}\Omega\) being the resistance quantum and \(S_{\text{V}}(\omega)\) the spectral density of the voltage oscillations appearing across the NIS junction under study due to the Josephson radiation. The physical meaning of the function \(P(E)\) is the probability for the tunneling electron to release energy \(E\) into or absorb it from the electromagnetic environment [39]. Furthermore, we take into account a narrow linewidth of Josephson oscillations in our source (see Table 2) and approximate the Lorentzian shape of the spectral function \(S_{\text{V}}(\omega)\) by the delta function, thus obtaining a simplified expression for \(P(E)\)  
\[P(E)\approx\frac{\pi V_{\text{A}}^{2}}{4R_{\text{k}}}\times\frac{1}{E^{2}} \times\delta\left(\frac{|E|-E_{\text{ph}}}{\hbar}\right). \tag{3}\]  
Here \(V_{\text{A}}=aV_{1}\) is an amplitude of voltage oscillations across the NIS junction generated by the [PERSON] source and delivered by the transmission line. The factor \(a=2C_{\text{eff}}E_{\text{C}}/e^{2}=C_{\text{eff}}/C_{\Sigma}\approx 0.028\) is the coupling coefficient of the transmission line to the transistor island. The effective coupling capacitance \(C_{\text{eff}}\) of the superconducting island to the transmission line was obtained from the DC modulation period of the NISIN transistor. For simplicity, we also neglect the losses in the transmission line (we consider the case \(B=0\)), which should however look plausible once we have found the output signals measured for both types of the coupling methods, _i.e._, the transmission line and the capacitive coupling, to be comparable.