Bidirectional single-electron counting and the fluctuation theorem

We investigate the direction-resolved full counting statistics of single-electron tunneling through a double quantum-dot system and compare with predictions of the fluctuation theorem (FT) for Markovian stochastic processes. Experimental data obtained for GaAs/GaAlAs heterostructures appear to violate the FT. After analyzing various potential sources for the discrepancy we conclude that the nonequilibrium shot noise of the quantum point contact electrometer, which is used to study the transport, induces strong dot-level fluctuations which significantly influence the tunneling statistics. Taking these modifications into account we find consistency with the FT.

Figure 1

(a) Setup of the system with two quantum dots (DQD) with single-level energies ${\epsilon }_L$ and ${\epsilon }_R$ coupled to a QPC. (b) The QPC current switches between three values corresponding to the three charge states of the DQD. (c) Test of FT, Eq. (3), at several times. Lines with symbols: logarithm of left-hand side (lhs) of Eq. (3) at several times; dashed line: $qe{V}_S/T^{\ast }$ with $T^{\ast }=1.37\text{K}$; dot-dashed line: $qe{V}_S/T$. Inset: the distribution $P_{\tau }\left(q\right)$ at $\tau =4\text{ms}$.

Figure 2

(a) Test of the integrated FT, Eq. (4), in the time interval $0<\tau <8\text{ms}$. The time scale is set by the frequency of electron tunneling $I/e\approx 370\text{Hz}$. Squares: lhs of Eq. (4) [denoted as $P_{\tau }\left(q\le 0\right)/P_{\tau }\left(q\ge 0\right)$]; dashed line: rhs of Eq. (4) [denoted as ${⟨\text{exp}\left(-qe{V}_S/T\right)⟩}_{q\ge 0}$]; solid line: rhs of Eq. (4) with $T$ replaced by $T^{\ast }=1.37\text{K}$. All three curves are multiplied by $\text{exp}\left(I\tau /e\right)$ for clarity. (b) Normalized second, ${\mathcal{C}}_2=⟨{\left(q-⟨q⟩\right)}^2⟩/⟨q⟩$, and third cumulants, ${\mathcal{C}}_3=⟨{\left(q-⟨q⟩\right)}^3⟩/⟨q⟩$, of the charge distribution $P_{\tau }\left(q\right)$. (c) The six transitions with rates ${\Gamma }_{ij}$ between three charge states.

Figure 3

(a) Effective temperature $T^{\ast }$ versus bias voltage $V_S$. Black squares and dotted line: $T^{\ast }$ needed to satisfy FT, Eq. (3); empty squares and solid line: $T^{\ast }$ obtained from Eq. (14); dashed line: temperature of the leads $T=130\text{mK}$. (b) Frequency of electron tunneling $I/e$ versus bias voltage.