Measurements of higher-order noise correlations in a quantum dot with a finite bandwidth detector

We present measurements of the fourth and fifth cumulants of the distribution of transmitted charge in a tunable quantum dot. We investigate how the measured statistics is influenced by the finite bandwidth of the detector and by the finite measurement time. By including the detector when modeling the system, we use the theory of full counting statistics to calculate the noise levels for the combined system. The predictions of the finite bandwidth model are in good agreement with measured data.

Figure 1

(Color online) (a)–(d) Normalized cumulants $C_n∕C_1$ versus dot asymmetry, $a=({\Gamma }_{\mathrm{in}}-{\Gamma }_{\mathrm{out}})∕({\Gamma }_{\mathrm{in}}+{\Gamma }_{\mathrm{out}})$. The solid lines are theoretical predictions assuming a perfect detector, $C_2∕C_1=(1+a^2)∕2$, $C_3∕C_1=(1+3a^4)∕4$, $C_4∕C_1=(1+a^2-9a^4+15a^6)∕8$, and $C_5∕C_1=(1+30a^4-120a^6+105a^8)∕16$. The dashed lines show the cumulants calculated from the model defined by Eq. (1) in the text. The inset in (b) shows the quantum dot with integrated charge readout used in the experiment. The inset in (c) shows the variation of the total tunneling rate ${\Gamma }_{\mathrm{tot}}={\Gamma }_{\mathrm{in}}+{\Gamma }_{\mathrm{out}}$ for the different measurement points.

Figure 2

(Color online) (a) Probability density of time needed for an electron to tunnel into the dot. Note the sharp decrease in counts for $t<100\mu \mathrm{s}$ due to the finite bandwidth of the detector. The black curve is a long-time exponential fit with $\Gamma =1.39\mathrm{kHz}$. (b) Model for the dot-detector system. A state $(n,m)$ corresponds to $n$ electrons on the dot while the detector at the same time is measuring $m$ electrons. (c) Higher cumulants versus relative detection bandwidth ${\Gamma }_{\det }∕({\Gamma }_{\mathrm{in}}+{\Gamma }_{\mathrm{out}})$, calculated from the model in (b). The cumulants are normalized to the results from the infinite bandwidth case. The influence of the finite bandwidth is maximal when the asymmetry $a=({\Gamma }_{\mathrm{in}}-{\Gamma }_{\mathrm{out}})∕({\Gamma }_{\mathrm{in}}+{\Gamma }_{\mathrm{out}})$ is zero.

Figure 3

(Color online) Normalized cumulants evaluated for different lengths of the time interval $t_0$. The symbols show the experimental data, extracted from a time trace of length $T=10\min$, containing 350 595 events, with $a=0.053$ and ${\Gamma }_{\mathrm{tot}}=3062\mathrm{Hz}$. The solid lines are calculations from the FCS given by Eq. (4) in the text, while the dashed lines are the asymptotes for $t_0\to \infty$. The inset shows a magnification of the vertical axis (horizontal axis unchanged) for $C_4∕C_1$ and $C_5∕C_1$ for $⟨N⟩>0.6$.