Energy-current relation
The photon count rate and dark count rate measurements as a function of bias current and photon energy are shown in Fig. 3.6a. Each colour represents one measurement run with a specific incident photon wavelength. Each solid line traces the error function fit for the respective data curve, as explained later in this section. The dashed red line indicates the fraction η of the saturated detection efficiency η = 50% used to extract the energy-current relation. The integration time was 10 seconds at each bias current and energy point. In order to compare the PCR of various wavelengths, we normalise the data to a count rate value situated just below the critical current, i.e. in the plateau region where the efficiency is saturated.
The energy-current relation is shown in Fig. 3.7. For each wavelength we plot the amount of bias current Ibη required to achieve a certain fraction η of the saturated detection efficiency. The setup allows to measure from 0.6 eV to more than 1.6 eV in the single-photon absorption regime. The relation is plotted for two different efficiency threshold, η = 50%andη= 1%, at 0.8 K and 1.5 K. The relation between bias current and photon energy is nonlinear throughout the entire energy range, for both temperatures, and for both η values.
3.3. Results
Figure 3.6: (a) Relative photon count rate as a function of the bias current Ib at 0.75 K.
Each colour represents one measurement run with a specific incident photon wavelength.
Each solid line traces the error function fit for the respective data curve, as explained in the manuscript. The dashed red line indicates the fraction η of the saturated detection efficiency η= 50%. The leftmost and rightmost curves correspond to 750 and 2050 nm, respectively. (b) Dark count rate of the detector as a function of the bias current. Below 4µA, the DCR is due to the amplifier noise. The first plateau region between 5 µA and 9 µA correspond to short wavelength black body radiation. The critical current of the detector occurs at 14.7 µA.
Transition width and Fano fluctuations
The long plateau and the broad response of our detector allows us to carefully characterise the full shape of the PCR curves. The curves have a transition region where the detection efficiency increases, followed by a plateau region. One theory attributes the shape of the transition region to Fano fluctuations, which are the result of the statistical nature of the quasiparticle creation process [39, 40]. Since only a finite fraction of the incoming photon energy ends up in the quasiparticle bath, the number of quasiparticles generated by a photon of energyE fluctuates as∆N =p
F E/, whereF is the Fano factor andis the energy of a single quasiparticle. These fluctuations have recently been analysed in the context of a model of quasiparticle recombination [41, 38]. In this model, the transition region occurs because for some currents, the photon only occasionally produces enough quasiparticles to trigger a detection. This results in a predicted sigmoidal shape with a width that is set by the microscopic details of the down-conversion process.
To check whether the Fano fluctuation theory agrees with the measurements, the PCR experimental data is fitted with an error function R(Ib) = erfh
(Ib−Ib50%)/σ√ 2i
, where σ quantifies the width of the transition. The fits are visible in Fig. 3.6a, and the transition width, defined as∆Ib=Ib80%−Ib20%, is plotted in Fig. 3.8. At low photon energies the fit agrees very well with the data. However, at high energies, the shape of the curves starts
0.6 0.8 1.0 1.2 1.4 1.6
Figure 3.7: Energy-current relation. The threshold current Ib1% (red squares) andIb50%
(blue points) are plotted as a function of the photon energy and corresponding wavelength, at 0.8 K. Inset: Energy-current relation at 1.5 K.
to deviate from the R(Ib) fits. The inset in Fig. 3.8 shows the highest and lowest energy scans, which are overlapped to facilitate comparison. This discrepancy is statistically significant: the difference in the reduced χ2, which quantifies the quality of the fit, is over two orders of magnitude between the lowest and highest photon energies. From Fig. 3.6a, it is clear that the transition becomes narrower as the photon energy is increased. While this effect is observed in previous studies [42, 43], a complete and quantitative description are presented in this section. The interpretation of this effect is still an open problem and could originate both from Fano fluctuations and position-dependent effects [41, 38, 44].
Following the Fano fluctuations formalism, the results presented in Fig. 3.8 might seem contradictory: one would expect to have a larger ∆Ib as the photon energy increases.
However, the non-linearity of the energy-current relation actually explains this phe-nomenon. To prove this point, the following approach was taken: to each wavelength corresponds a current fluctuation ∆Ib that is translated to energy fluctuations through the energy-current relation. By doing so, the energy fluctuation clearly increases as a function of the photon energy, see Fig. 3.9. As explained above, for all wavelengths, the current fluctuation ∆Ib is extracted from Fig. 3.8 and translated to the energy scale through the energy-current relation of Fig. 3.7. The energy fluctuations gets bigger as the photon energy increases, qualitatively following a square root dependence, similarly to ∆N =p
F E/. The exact relation between∆N and ∆E has yet to be determined.
3.3. Results
Normalized bias current (Ib/Ib50%)
0.0
Figure 3.8: Transition width defined as ∆Ib = Ib80% −Ib20% obtained from the energy-current relation. Inset: photon count rate curve for 750 nm and 2050 nm as a function of the relative bias current. The solid lines represent the error function fit and the red arrows indicate the two inflection points where the error function fit significantly differs from the data.
Figure 3.9: Energy fluctuations as a function of the photon energy. The energy fluctuation is obtained by combining the current fluctuations of the Fig. 3.8 with the energy-current relation of the Fig. 3.7, see main text. The energy fluctuations gets bigger as the photon energy increases, qualitatively following a similar behaviour as ∆N =p
F E/.