In order to demonstrate how the optimal bandwidth can be obtained from the random telegraph signal, we have chosen for the binning time \(t_{\textrm{bin}}\) 13 logarithmically distributed values between \(\hbox {2}\,\upmu \hbox {s}\) and \(\hbox {100}\,\upmu \hbox {s}\). FigureÂ 3a,c,e show the histograms of the binned photon counts *k* for \(t_{\textrm{bin}}\) values of \(\hbox {100}\,\upmu \hbox {s}\), \(\hbox {10}\,\upmu \hbox {s}\) and \(\hbox {2}\,\upmu \hbox {s}\). They all exhibit Poisson distributions for both, the on and the off state. The corresponding expectation values are \(\mu _{\textrm{on}}\approx\) 175, 17, 3.6 (red dotted lines Fig.Â 3a,c,e) and \(\mu _{\textrm{off}}\ll 1\) photons per \(t_{\textrm{bin}}\). We have normalized the histograms in each case with the value at \(\mu _{\textrm{on}}\). There is also a constant background in between. This can be seen for example for \(t_{\textrm{bin}} = 100\,\upmu \hbox {s}\) (see Fig.Â 3a), where the background is \(\approx 0.1\) between about 1 and 120 photons per \(t_{\textrm{bin}}\). It reflects those time intervals in which a tunnel event takes place. Since the binning intervals are not correlated with the tunnel events, their numbers of photons are equally distributed between \(\mu _{\textrm{on}}\) and \(\mu _{\textrm{off}}\)^{26}.

The choice of the threshold photon number \(\tilde{k}\) (red dashed line in Fig.Â 3a,c,e) is not critical for high values of \(t_{\textrm{bin}}\), because the distributions of the on and off states are well separated (see Fig.Â 3a). Additionally, for \(t_{\textrm{bin}}=100\,\upmu \hbox {s}\), the \(\tau _{\textrm{on}}\) and \(\tau _{\textrm{off}}\) values exhibit very low noise, see Fig.Â 3b. From both these facts we conclude that the evaluation of the data in Fig.Â 3a,b is accurate and reliable. The fitted rates are \(\gamma _{\textrm{in}}\approx (1.531\pm 0.014)\,\textrm{ms}^{-1}\) and \(\gamma _{\textrm{out}}\approx (1.477\pm 0.010)\,\textrm{ms}^{-1}\). These are very similar, so both processes equally contribute to the electron transport between the dot and the reservoir. This is usually quantified by the asymmetry of the tunnel coupling \(A = (\gamma _{\textrm{out}}-\gamma _{\textrm{in}})/(\gamma _{\textrm{out}}+\gamma _{\textrm{in}})\)^{9,11}. For \(t_{\textrm{bin}}=100\,\upmu \hbox {s}\), we find \(A\approx -0.018\pm 0.006\).

When we evaluate the same experimental data with decreasing \(t_{\textrm{bin}}\) in order to improve the time resolution, it becomes increasingly difficult to separate the distributions of the on and off states (see Fig.Â 3c,e). Thus, the evaluation becomes less and less reliable.

For \(t_{\textrm{bin}}=10\,\upmu \hbox {s}\), \(\tau _{\textrm{on}}\) and \(\tau _{\textrm{off}}\) start to visibly differ (see Fig.Â 3d), resulting in an asymmetry \(A\approx -0.074\pm 0.004\) (\(\gamma _{\textrm{in}}\approx (1.826\pm 0.007)\,\textrm{ms}^{-1}\) and \(\gamma _{\textrm{out}}\approx (1.574\pm 0.009)\,\textrm{ms}^{-1}\)). Finally, for \(\hbox {2}\,\upmu \hbox {s}\) binning time, Fig.Â 3f, they differ so strongly, that the derived asymmetry \(A\approx -0.7194\pm 0.0016\) is close to the maximum possible value of \(\left| A\right| =1\). This is due to the increasing overlap of the on and off state distributions (left column of Fig.Â 3), as the overlap causes data points in the binned time trace to be on the wrong side of the threshold and, thus, be assigned to the wrong state. Such a false assignment will break up an otherwise continuous \(\tau _{\textrm{on}}\) interval into two shorter ones and thus skew the statistical distribution towards higher rates \(\gamma _{\textrm{on}}\). The same holds for the \(\tau _{\textrm{off}}\) statistics.

The evaluation of the WTD for 100Â \(\upmu \hbox {s}\) in Fig.Â 3b is not influenced by wrongly assigned data points and represents the correct tunneling rates in the system. Therefore we use the rates determined there and especially the asymmetry \(A\approx 0\) as a criterion for the optimal choice of the threshold photon number \(\tilde{k}\) for shorter binning times. Based on this determination of \(\tilde{k}\), we can now proceed to assess how well the transport statistics can be determined for a given \(t_{\textrm{bin}}\) using full counting statistics.

We characterize our probability distributions \(P_{\Delta t}(N)\) by the Fano factor \(f=c_2/c_1\), where \(c_i\) is the *i*-th cumulant of \(P_{\Delta t}(N)\), more specifically \(c_1\) is the mean and \(c_2\) is the variance. The Fano factor reflects whether the transport is limited by a single tunnel process (\(f=1\); Poisson-distributed \(P_{\Delta t}(N)\)) or by two tunnel processes (\(f=0.5\); sub-Poissonian distributed \(P_{\Delta t}(N)\))^{9}. The more complex evaluation of the higher order cumulants, as well as the factorial cumulants is possible^{17,26}, however, this evaluation is not relevant here and we will concentrate on the first and second cumulant and the Fano factor.

For the measurement presented in Fig.Â 3, both tunneling processes contribute equally to the transport. Accordingly, we expect a Fano factor of 0.5. In Fig.Â 4a (blue data points), we show the evaluated Fano factors for the already mentioned 13 logarithmically distributed binning rates. All blue data points were evaluated from the same single-photon stream recorded at a bias voltage of \(V_{\textrm{g}} = 0.3743\)Â V. For low binning rates, the Fano factor is very close to the expected value. However, above about \(1/(2t_{\textrm{bin}})=70\,\textrm{kHz}\) (dotted blue line), the Fano factor diverges. To show that this divergence is reproducible and not only applies to the described asymmetry of 0, we used the bias voltage \(V_{\textrm{g}}\) to change the tunneling rates. The transition energy of the neutral exciton shifts due to the quantum-confined Stark effect^{27}. To compensate for this, we applied a magnetic field of 2 T in the growth direction to lock the transition to the incident laser by â€™draggingâ€™, due to coupling to the nuclear spin bath of the quantum dot^{28,32,33,31}. For \(V_{\textrm{g}}=0.401\)Â V, the tunneling rate into the dot is significantly faster than before (\(\gamma _{\textrm{in}}\approx 2.9\,\textrm{ms}^{-1}\)) but the tunneling rate out of it is strongly suppressed (\(\gamma _{\textrm{out}}\approx 42.5\,\textrm{s}^{-1}\)) and, thus, an asymmetry \(A\approx -1\) is obtained (see Fig.Â 4b). In this situation, where the transport is limited by only one process, we expect a Fano factor which equals unity. The corresponding data (orange data points in Fig.Â 4a) shows qualitatively the same diverging behavior as for a Fano factor of 0.5. It also shows that we can resolve the single-electron transport statistics with a bandwidth of about 175Â kHz (dotted orange line).

The question to be clarified now is how to maximize the binning rate in order to increase the optimum bandwidth. Therefore, we will discuss in the following why the increase of the Fano factor occurs, when it occurs and which parameters influence it. As already discussed in connection to the WTD in Fig.Â 3, high binning rates lead to an incorrect state assignment for some data points. This manifests itself not only in shortened time intervals \(\tau _{\textrm{on}}\) and \(\tau _{\textrm{off}}\) but also in falsely detected, additional tunnel events. These have no physical meaning, so we refer to them as binning-induced tunnel events in the following. When these binning-induced events outnumber the correct tunnel events, the Fano factor starts to diverge and an evaluation of the real transport statistics is no longer possible.

Two main approaches are now promising to increase the time resolution: (1) Somewhat counter-intuitively, this can be achieved in systems with faster dynamicsÂ (here: higher tunneling rates). As mentioned above, the divergence of the Fano factor occurs when too many binning-induced (false) events are recorded compared to the number of correctly observed tunneling events. When the frequency of correct events is increased, the divergence occurs at higher binning rates, which increases the time resolution. For the present experiment, an increase in the tunneling rate can be realized in sample structures, which have a more transparent tunnel barrier. (2) Another approach is to reduce the overlap between the on and off state distributions. In our optical method, the off state is already close to the technical limit, where the observed photon events are exclusively given by the dark counts of the APD. The mean value of the on state distribution \(\mu _{\textrm{on}}\), however, can be shifted to higher values, e.g. by increasing the laser intensity or optimizing the detection efficiency. To demonstrate this, we have repeated the measurement for asymmetry \(A=0\) with a total of four different laser intensities (see Fig.Â 4c). We denote the different data sets according to the average rate of detected photons of the on state \(\Gamma _{\textrm{on}}\). The previously presented data set (blue data points) corresponds to \(\Gamma _{\textrm{on}}\approx 1.72\,\textrm{MCounts}/\textrm{s}\).

At low binning rates (\(1/(2t_{\textrm{bin}})\le 15\)Â kHz) in Fig.Â 4c, the determined Fano factor for all laser intensities is almost identical and close to 0.5, indicating that the different excitation of the QD does not affect the system dynamics. This is remarkable since in electrical measurements with quantum point contacts^{32} and radio-frequency single electron transistors^{33} as detectors, a backaction on the investigated system is observed. FigureÂ 4c also shows that the critical binning rate \(1/t_{\textrm{bin}}^{\textrm{crit}}\), at which the Fano factor starts to diverge, increases monotonically with the laser intensity. This can be seen, for example, by comparing the measurement with \(\Gamma _{\textrm{on}}\approx 0.23\,\textrm{MCounts}/\textrm{s}\) (red data points in Fig.Â 4c), which diverges already at about 17Â kHz, with the data set for \(\Gamma _{\textrm{on}}\approx 1.72\,\textrm{MCounts}/\textrm{s}\) (blue data points in Fig.Â 4c), which does not diverge until 70Â kHz. This is in agreement with our discussion about the overlap of the on and off state distributions.

In order to also quantitatively test our reasoning and enable statements about the possible scalability of the method, we consider a simple model for the divergence of the Fano factor for high \(1/t_{\textrm{bin}}\). It starts from the premise that an evaluation of the probability distribution \(P_{\Delta t}(N)\) will definitely no longer be meaningful when the mean number of correct \(N_{\textrm{corr}}\) and binning-induced events \(N_{\textrm{false}}\) are equal. \(N_{\textrm{corr}}\) is given by the effective tunnel rate

$$\begin{aligned} 1/\gamma _{\textrm{tun}}=1/\gamma _{\textrm{in}}+1/\gamma _{\textrm{out}} \end{aligned}$$

and the FCS time interval \(\Delta t\). The number of binning-induced events \(N_{\textrm{false}}\) contains the misassigned on and off state data points

$$\begin{aligned} N_{\textrm{false}}=N_{\textrm{false}}^{\textrm{on}}+N_{\textrm{false}}^{\textrm{off}}. \end{aligned}$$

The number \(N_{\textrm{false}}^{\textrm{on}}\) is the product of the number of data points that should be assigned to the on state \(N^{\textrm{on}}\) and the probability that one of these data points is wrongly assigned \(P^{\textrm{on}}_{\textrm{false}}\):

$$\begin{aligned} N_{\textrm{false}}^{\textrm{on}}=P^{\textrm{on}}_{\textrm{false}}N^{\textrm{on}}. \end{aligned}$$

An analogous relation applies to the off state.

For an asymmetry \(A=0\), the numbers \(N^{\textrm{on}}\) and \(N^{\textrm{off}}\) are equal (\(N^{\textrm{on}} = N^{\textrm{off}} = \Delta t/(2t_{\textrm{bin}}^{\textrm{crit}})\)). To obtain the probability \(P^{\textrm{on}}_{\textrm{false}}\), we start from a Poisson distribution \(P_{\mu _{\textrm{on}}}(k)\) for the number of photons *k* per \(t_{\textrm{bin}}\). The expectation value of the distribution is given by \(\mu _{\textrm{on}}=\Gamma _{\textrm{on}}\cdot t_{\textrm{bin}}^{\textrm{crit}}\). Then, \(P^{\textrm{on}}_{\textrm{false}}\) is given by the fraction of \(P_{\mu _{\textrm{on}}}(k)\) that falls below the threshold photon count \(\tilde{k}\) and is therefore falsely assigned to the off state:

$$\begin{aligned} P_{\textrm{false}}^{\textrm{on}}=\sum _{k=0}^{\tilde{k}}P_{\mu _{\textrm{on}}}(k). \end{aligned}$$

An analog derivation holds for the off state. Combining the above equations, we find

$$\begin{aligned} \gamma _{\textrm{tun}}\Delta t =\sum _{k=0}^{\tilde{k}}P_{\mu _{\textrm{on}}}(k)\frac{\Delta t}{2t_{\textrm{bin}}^{\textrm{crit}}} +\sum _{k=\tilde{k}}^{\infty }P_{\mu _{\textrm{off}}}(k)\frac{\Delta t}{2t_{\textrm{bin}}^{\textrm{crit}}}. \end{aligned}$$

(1)

This relation does not have a simple analytical solution for \(t_{\textrm{bin}}^{\textrm{crit}}\), therefore we have solved it numerically for an effective tunnel rate \(\gamma _{\textrm{tun}} \approx 0.75\,\textrm{ms}^{-1}\) and various combinations of \(\tilde{k}\), \(\Gamma _{\textrm{on}}\), and \(\Gamma _{\textrm{off}}\). For \(\Gamma _{\textrm{off}}<10^3\)Â counts/s, both \(\Gamma _{\textrm{off}}\) and \(\tilde{k}\) have little effect on the critical binning rate calculated from our model (see Fig.Â 4e). In the experiments, a small value of \(\Gamma _{\textrm{off}}\) corresponds to a low detector dark count rate and well suppressed scattered laser light intensity.

The dependence of \(t_{\textrm{bin}}^{\textrm{crit}}\) on the average photon rate of the on state is found to be \(t_{\textrm{bin}}^{\textrm{crit}}\propto \Gamma _{\textrm{on}}^{-0.8}\) (see Fig.Â 4e inset). Rescaling the laser-power-dependent data in Fig.Â 4c using this power law, shows that the curves for 0.23Â Mcounts/s, 0.82Â Mcounts/s and 1.72Â Mcounts/s all coincide (see Fig.Â 4d). Only the measurement with 3.8Â Mcounts/s (green line in Fig.Â 4d) is an exception as here the increase of the Fano factor is shifted towards smaller rescaled binning rates. This shift can be attributed to the fact that the condition above, \(\Gamma _{\textrm{off}}<10^3\)Â counts/s, is no longer fulfilled due to laser light scattered to the detector. Indeed, \(\Gamma _{\textrm{off}}\) for this data set is about 1.3Â kcounts/s, very close to the value, where an abrupt increase of \(t_{\textrm{bin}}^{\textrm{crit}}\) is observed in Fig.Â 4e. Similarly, the data set for \(\Gamma _{\textrm{on}}=1.72\)Â Mcounts/s (blue data points in Fig.Â 4d) has a spurious off state count rate of 1.4Â kcounts/s, which may explain why one of the data points (\(1/(2t_{\textrm{bin}})\approx 100\)Â kHz) does not fall onto the scaled curves. For the data sets with 0.23 and 0.82Â Mcounts/s, \(\Gamma _{\textrm{off}}\) is very low with 0.12 and 0.19Â kcounts/s, respectively, and thus in a range where \(\Gamma _{\textrm{off}}\) has no influence on the critical binning rate. FigureÂ 4e shows that the value of \(\Gamma _{\textrm{off}}\), where the critical binning time starts to diverge, is not fixed. For an experimental situation, where a high \(\Gamma _{\textrm{off}}\) is unavoidable, the divergence point can be shifted to much higher values of \(\Gamma _{\textrm{off}}\) by a proper choice of \(\tilde{k}\), while only moderately increasing the critical binning time \(t_{\textrm{bin}}^{\textrm{crit}}\).

Our optical measurements, their analysis and interpretation show that in contrast to an electrical measurement of a random telegraph signal, the single photon detection offers an optimum bandwidth that can be determined in post-processing, i.e. after the actual experiment in the lab.

We have shown that a critical binning rate and thereby optimal bandwidth exists, above which the evaluation of the random telegraph signal is not valid anymore. The simple model we have considered shows that the reason for this is the overlap of the photon distributions of the on and off states. This occurs when the binning rate is too large and leads to a misassignment of the states. In our experiment, this overlap is enhanced by technical imperfections, such as dark counts of the photodetector, reflected laser photons and an imperfect photon collection rate. However, even for an ideal measurement, the increasing overlap with decreasing binning time would impose a limit on the maximum time resolution and thus represents an absolute physical limit for this method.