Measuring cotunneling in its wake

We introduce a rate formalism to treat classically forbidden electron transport through a quantum dot (cotunneling) in the presence of a coupled measurement device. We demonstrate this formalism for a toy model case of cotunneling through a single-level dot while being coupled to a strongly pinched-off quantum point contact (QPC). We find that the detector generates three types of back-action: the measurement collapses the coherent transport through the virtual state, but at the same time allows for QPC-assisted incoherent transport, and widens the dot level. Last, we obtain the measured cotunneling time from the cross correlation between dot and QPC currents.

Figure 1

A sketch of the setup: a single-level dot is tunnel-coupled to two leads, $S$ and $D$ with amplitudes $t_S$ and $t_D$, respectively. The respective dot-leads chemical potentials are ${\mu }_S$ and ${\mu }_D$, corresponding to a voltage bias $e{V}_{SD}={\mu }_S-{\mu }_D$. A quantum point contact (QPC) is capacitively coupled to the dot, i.e., transport between the leads $L$ and $R$ is governed by a tunneling amplitude $\Omega$ when the dot is empty, and $\Omega -\delta \Omega$ when the dot is occupied. The respective QPC-leads chemical potentials are ${\mu }_L$ and ${\mu }_R$, corresponding to a voltage bias $e{V}_{LR}={\mu }_L-{\mu }_R$.

Figure 2

Sketch of the rate $W_{SD}^0$ of cotunneling through the dot accompanied by no tunneling through the QPC [cf. Eq. (8)]. Similar to Fig. 1, the upper part (brown) depicts the dot system and the lower (blue) the QPC.

Figure 3

Sketches of the relevant cotunneling rates in our toy model that involve an electron tunneling through the QPC [cf. Eq. (9)]. Similar to Fig. 1, the upper part (brown) depicts the dot system and the lower (blue) the QPC. (a) The rate $W_{SD}^1$. (b) The rate $W_{DS}^1$. (c) The rate $W_{SS}^1$. (d) The rate $W_{DD}^1$.

Figure 4

Plots of (a) $W_{SD}^0$ and (b) $W_{SD}^1$ as a function of $\hslash \mathcal{D}/({\mu }_D-{\epsilon }_d)$ with bias voltages $e{V}_{SD}=4\hslash \Gamma ,e{V}_{LR}=3.9\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. The different curves are for different dot gating ${\epsilon }_d$ such that ${\mu }_D-{\epsilon }_d=4\hslash \Gamma$ (dark blue); ${\mu }_D-{\epsilon }_d=5\hslash \Gamma$ (medium-dark blue); ${\mu }_D-{\epsilon }_d=6\hslash \Gamma$ (bright blue). We highlight the area in which our toy model is relevant, namely, where $W_{SD}^0$ slightly decreases, $W_{SD}^1$ is far from peaking, and $W_{SD}^n$ for $n>1$ are negligible. Two effects are seen here: (i) The higher the dot energy is, the higher the cotunneling rates are; (ii) as $\hslash \mathcal{D}/({\mu }_D-{\epsilon }_d)$ increases it is more probable for electrons to pass through the QPC during the time window in which the dot is empty. This is seen in the decrease in $W_{SD}^0$ vs the increase in $W_{SD}^1$. At the point in which $W_{SD}^1$ peaks, the probability for two electrons to tunnel through the QPC during a cotunneling event becomes relevant.

Figure 5

Plots of $I_{\mathrm{dot}}$ [cf. Eq. (10)] as a function of $e{V}_{LR}/({\mu }_D-{\epsilon }_d)$ and $\rho \Omega$ with $e{V}_{SD}=4\hslash \Gamma ,{\mu }_D-{\epsilon }_d=5\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. (a) A density plot of $I_{\mathrm{dot}}$. The $I_{\mathrm{dot}}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. The dashed lines represent equal-$d\equiv \hslash \mathcal{D}/({\mu }_D-{\epsilon }_d)$ lines. We can see that the current increases with the coupling $\mathcal{D}$ of the QPC. However, the effect does not depend exclusively on $\mathcal{D}$, but it also directly depends on the QPC parameters: the tunnel-coupling $\Omega$ and the voltage-bias $V_{LR}$.

Figure 6

Plots of $I_{\mathrm{QPC}}$ [cf. Eq. (11)] as a function of $e{V}_{LR}/({\mu }_D-{\epsilon }_d)$ and $e{V}_{SD}/\left(\hslash \Gamma \right)$ with $\rho \Omega =0.15,{\mu }_D-{\epsilon }_d=5\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. (a) A density plot of $I_{\mathrm{QPC}}$. The $I_{\mathrm{QPC}}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. Naturally, the current increases with the the QPC voltage bias $V_{LR}$. As the voltage on the dot $V_{SD}$ is increased the following occurs: (i) The cotunneling to the drain and back $W_{DD}^1$ remains constant; (ii) the cotunneling to the source and back $W_{SS}^1$ becomes less probable; (iii) the cotunneling from drain to source $W_{DS}^1$ decreases, and disappears once $V_{SD}>V_{LR}$; (iv) additional cotunneling processes from source to drain in $W_{SD}^1$ are allowed. The latter features dominate the behavior of $I_{\mathrm{QPC}}$. We see initially a slight decrease as $W_{DS}^1$ vanishes with an overall increase due to $W_{SD}^1$.

Figure 7

Plots of $S$ [cf. Eqs. (12) and (13)] as a function of $e{V}_{SD}/\left(\hslash \Gamma \right)$ and $({\mu }_D-{\epsilon }_d)/\left(\hslash \Gamma \right)$ with $\rho \Omega =0.15,e{V}_{LR}=1.5\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. (a) A density plot of $S$. The $S$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. As the voltage on the dot $V_{SD}$ is increased the following occurs: (i) The cotunneling from drain to source $W_{DS}^1$ decreases, and disappears once $V_{SD}>V_{LR}$; and (ii) additional cotunneling processes from source to drain in $W_{SD}^1$ are allowed. Hence $S$ initially increases with $V_{SD}$ slowly, and only afterward increases as $W_{SD}^1$. The deeper the dot level is, the shorter the virtual time in which the dot is empty. As a result, (i) cotunneling processes through the dot become less probable, and (ii) it becomes less probable for a QPC electron to manage to tunnel during this time window, leading to the descent of $S$.

Figure 8

Plots of ${\overline{\tau }}_{\mathrm{SD}}$ [cf. Eq. (14)] as a function of $e{V}_{SD}/\left(\hslash \Gamma \right)$ and $({\mu }_D-{\epsilon }_d)/\left(\hslash \Gamma \right)$ with $\rho \Omega =0.15,e{V}_{LR}=1.5\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. (a) A density plot of ${\overline{\tau }}_{\mathrm{SD}}$. The ${\overline{\tau }}_{\mathrm{SD}}$ along the vertical (green) and horizontal (blue) mesh lines is plotted in (b) and (c), respectively. Interestingly, as $V_{SD}$ increases, ${\overline{\tau }}_{\mathrm{SD}}$ increases as well, i.e., it appears that the increased phase space adds processes with a slower time into the average. As expected, the deeper the dot level is, the shorter ${\overline{\tau }}_{\mathrm{SD}}$ becomes.

Figure 9

LogLog plots of ${\overline{\tau }}_{\mathrm{SD}}$ and ${\overline{\tau }}_h$ [cf. Eqs. (14) and (15)] as a function of $({\mu }_D-{\epsilon }_d)/\left(\hslash \Gamma \right)$ with $\rho \Omega =0.15,e{V}_{LR}=1.5\hslash \Gamma$, and ${\Gamma }_S={\Gamma }_D$. The plots of ${\overline{\tau }}_{\mathrm{SD}}$ have an overall shorter time (blue) than those of ${\overline{\tau }}_h$ (red). For both plotted times the curves are for $V_{SD}=2.5\hslash \Gamma$ (light), $V_{SD}=5.5\hslash \Gamma$ (medium-light), and $V_{SD}=8.5\hslash \Gamma$ (dark). In the inset, we plot the approximate slopes of the LogLog curves, defined as $\left(\partial \text{ln}\overline{\tau }\right)/\left[\partial \text{ln}({\mu }_D-{\epsilon }_d)\right]$. The overall magnitude difference between the two time models can be attributed to an unknown prefactor in the time taken from the energy-time uncertainty principle. The other differences can be attributed to an inherent discrepancy between the two models, as the former incorporates phase-space contributions of slower QPC-assisted processes.