# Detection of On-Chip Generated Weak Microwave Radiation Using Superconducting Normal-Metal SET
## 4 Discussion
Taking advantage of the low temperature \(T\approx 15\) mK, we can make use of the zero-temperature approximation and replace the Fermi factors by corresponding step functions, thus obtaining a simplified standard golden rule formula [40]  
\[\Gamma_{\text{ph}}(E)\approx\frac{1}{e^{2}R_{T}}\int_{-\infty}^{E}dE_{\text{n} }\int_{\Delta}^{\infty}dE_{\text{s}}n_{\text{s}}(E_{\text{s}})P(E_{\text{n}}- E_{\text{s}}), \tag{4}\]  
where \(n_{\text{s}}(E)=E/\sqrt[]{E^{2}-\Delta^{2}}\) is a BCS density of states in the superconducting island. The integral can be taken analytically resulting in\[\Gamma_{\rm ph}^{L,R}(E_{\rm ph})|_{E_{\rm ph}=2e\
u_{\rm jj}}\approx\frac{\Delta} {8R_{T}}\left[\frac{V_{A}(V_{\rm jj})}{E_{\rm ph}}\right]^{2}\sqrt{\left(\frac{ E_{\rm ph}-E_{t}^{L,R}}{\Delta}+1\right)^{2}-1}, \tag{5}\]  
where \(E_{t}^{L,R}\) is the activation threshold for tunneling of an electron in a left (right) junction of the NISIN transistor, respectively. In Figure 10, we plotted a set of \(\Gamma_{\rm ph}\)_vs._\(V_{\rm jj}\) dependencies calculated for an NISIN detector tuned to slightly different gate voltages around \(n_{\rm g}=0\). The curves exhibit discontinuities in the derivative at the tunneling threshold in the individual junctions, which would, however, be smeared in a more realistic approximation considering a finite linewidth of Josephson oscillations, a finite impedance of the environment or non-zero \(T\). As expected, the activation threshold achieves its highest value in the point, \(n_{\rm g}=0\), of the maximum Coulomb blockade. The decay at higher voltages is partially related to the decay of the Josephson oscillation amplitude \(V_{1}(V_{\rm jj})\), see Equation (1). Two important conclusions can be derived from the calculated curves. First, for the Josephson voltages noticeably exceeding the activation thresholds \(V_{\rm jj,t}\), the rate \(\Gamma_{\rm ph}\) becomes independent of the gate voltage. As a result, the response should be insensitive to a small variation of the offset (background) charges. Second, the comparison of the experimental response current \(I_{\rm NISIN}\approx 200\) fA (see Figure 6a, the data for transmission line) with the calculated rate \(\Gamma_{\rm ph}(200\ \rm\mu V)\approx 700\) Hz (see Figure 10) gives rise to the amplification factor \(G\approx 200\ \rm fA/(e\times 700\ Hz)\approx 1.8\times 10^{3}\). The lowest measurable photon-assisted tunneling rate is then as low as about \(\delta\Gamma_{\rm ph}\ \sim\delta I_{\rm NISIN}/eG\sim 35\) photons per second, corresponding to the noise-limited current resolution \(\delta I_{\rm NISIN}\ \sim 10\) fA achieved in our experiment by averaging the transistor output signal over the time of 1 s. Even a higher resolution can be expected in the case of using lower noise electronics for femtoampere current measurement. A relatively high value of \(G\sim 10^{3}\) obtained in our study noticeably exceeds \(G\sim 40\) reported in [20]. We attribute this difference presumably to a high quality of superconducting Al film evaporated in our dedicated ultrahigh vacuum system which ensures a low density of impurity domains in the island resulting in a low rate of quasiparticle relaxation. Figure 10: The rate \(\Gamma_{\rm ph}\) of photon-assisted generation of quasiparticles in the superconducting island calculated as a function of mean voltage across the Josephson source, for three slightly different gate settings around the even state \(n_{\rm g}=0\). For the parameters of the circuit used in the calculation, see Tables 1 and 2.