Weak measurement of cotunneling time

Quantum mechanics allows the existence of “virtual states” that have no classical analog. Such virtual states defy direct observation through strong measurement, which would destroy the volatile virtual state. Here, we show how a virtual state of an interacting many-body system can be detected employing a weak measurement protocol with post-selection. We employ this protocol for the measurement of the time it takes an electron to tunnel through a virtual state of a quantum dot (cotunneling). Contrary to classical intuition, this cotunneling time is independent of the strength of the dot-lead coupling and may deviate from that predicted by time-energy uncertainty relation. Our approach, amenable to experimental verification, may elucidate an important facet of quantum mechanics which hitherto was not accessible by direct measurements.

Figure 1

A measurement scheme of the cotunneling time. (a) Sketch of a quantum dot coupled to a QPC detector. The transmission through the QPC is affected by the presence of an extra electron in the QD. The top gates $V_L$ and $V_R$ control the tunneling rate to the dot, $V_{\text{QPC}}$ the unperturbed transmission through the QPC, and $V_g$ the charging energy in the dot. $V_g$ allows us to tune the system from the sequential tunneling to the cotunneling regime. The transport through the QPC and the QD is controlled by the voltage bias $\Delta \mu$ and $eV$, respectively. The current-current correlation $S_{IJ}$ is sensitive to the excess number of electrons $N$ in the dot. (b) Typical detector signal and current through the dot in the sequential tunneling regime, performing strong measurement with the QPC. The time an electron spends in the dot can be classified according to the occurrence of a subsequent positive current pulse in the drain $\left({\tau }_{+,i}\right)$, or the absence thereof: back-reflection $\left({\tau }_{-,i}\right)$. The average sequential tunneling time can be directly obtain by averaging over the durations $\left\{t_{+,i}^{\left(1\right)}\right\}$ of the relevant QPC signals. (c) Typical detector signal and current through the dot in the weak measurement regime. Straightforward classification of events as in (b) is not possible. The signal of the QPC preceding a pulse of current through the quantum dot has to be weighted by an appropriate weight function. The latter should account for the QPC current signal, which precedes and is directly related to the electron detected at the QD's drain terminal. In the cotunneling regime, the interval between successive tunneling events is longer than the cotunneling time, hence, one may discard the weighting function.

Figure 2

Feynman diagrams for $\langle I\rangle$ and $S_{IJ}$. Elastic and inelastic contributions to the cotunneling current $\langle I\rangle$ [panel (a)], and to the current-current correlator $S_{IJ}$ [panel (b)], in the zero-temperature particle-dominated regime. The propagators and vertices constituting the diagrams are defined in panel (c). Inset: An example of a diagram neglected in the zero-temperature particle-dominated limit. Here, we indicate explicitly the time and charging energy labelings along the time contour, as dictated by the Feynman rules (cf. Appendix pp2).

Figure 3

Feynman diagrams contributing to the cotunneling current. Each contribution to the probability consists of a coherent superposition of electronlike $\left(e\right)$ and holelike $\left(h\right)$ amplitudes. Upon squaring these amplitudes one obtains contributions of the type $e-e$, $e-h$, etc., which are represented by the various diagrams. The diagrams are grouped into elastic (A) and inelastic (B) contributions. Each group consists of “forward” and “backward” diagrams, depending on whether the associated charge transfer is from source to drain or vice versa, respectively. The contribution of the latter set of diagrams is vanishing at $T=0$. The relative weight of the electron and hole contributions is controlled by the gate voltage.

Figure 4

Example of Feynman diagrams for the correlation function $\langle I_{+}\left(t\right)[N_{-}(t-s)-\langle N\rangle ]\rangle$ obtained from the corresponding diagrams for the cotunneling current. The presented diagrams are all those contributing the correlation function in the limit of zero-temperature and particle-dominated processes in the elastic (a) and inelastic (b) cotunneling regimes. For each diagram contributing the cotunneling current there are four diagrams contributing the correlation function.

Figure 5

(a) Time contour for the Keldysh formalism. (b) Rules for drawing vertices in the diagrams for $\langle I\rangle$ and $\langle I\left(t\right)N(t-s)\rangle$, required by the rule R2. (c) Explicit expressions of propagators stipulated by the rule R8.