Counting statistics in multistable systems

Using a microscopic model for stochastic transport through a single quantum dot that is modified by the Coulomb interaction of environmental (weakly coupled) quantum dots, we derive generic properties of the full counting statistics for multistable Markovian transport systems. We study the temporal crossover from multimodal to broad unimodal distributions depending on the initial mixture, the long-term asymptotics and the divergence of the cumulants in the limit of a large number of transport branches. Our findings demonstrate that the counting statistics of a single resonant level may be used to probe background charge configurations.

Figure 1

(Color online) (a) Scheme of the model. (b) Region of multistability: current-voltage (transport) characteristics for $N=7$ control dots; thin red curves: $N+1$ individual currents $I_k$ $\left(k=0,1,\dots ,7\right)$, thick blue line: average current $⟨I⟩$ which is actually observed; borders of colored regions: standard deviation $\pm c\sqrt{⟨⟨I^2⟩⟩}/2$ of the current which becomes divergent for $\eta \to 0$ in the multistability region. Inset: broad unimodal distribution function $\text{log}\left(P\right)/t$ vs current for large times and $V^{\ast }=7$ (thick blue curve); distributions for individual currents $I_k$ (thin red curves). Parameters: $N=7$, ${ϵ}_d=3$, $\beta =3$, $U=0.15$, and $\eta /\Gamma =0.001$.

Figure 2

(Color online) Current-voltage characteristics for a nontrivial nonthermal stationary state [Eq. (10)]. (a) For various temperatures $\beta \Delta$ and $N=10$; at small temperatures $\left(\beta \Delta ⪢1\right)$ the number of control dots $N$ can be probed in a current measurement, whereas at $\beta \Delta \lesssim 1$ one has to make use of counting statistics at $t\lesssim {\eta }^{-1}$ (see Fig. 3). (b) For various $N$ and $\beta \Delta =40$; for $N\to \infty$, $U\to 0$, $\left|V\right|<\Delta$, and $\beta \Delta ⪢1$ the current becomes linear $I=\frac{e\Gamma }{2\Delta }V$ since transport takes place through a continuum with finite band-width $\Delta \equiv NU$ (as sketched in the inset).

Figure 3

(Color online) Probability distribution $P_n\left(t\right)$ for the number of tunneled particles $n$ at different times $t$ (orange region). For increasing times $\left[\left(a\right)\to \left(d\right)\right]$ the average of the distribution moves linearly toward larger $n$. It proceeds a crossover from multimodal to unimodal with a transition time of ${\eta }^{-1}$ and becomes (nearly) a broad Gaussian for $t⪢{\eta }^{-1}$ (d). For $\eta =0$ even the long-term distribution depends on the initial mixture (symbols) here chosen as analytic continuation of $\overline{\rho }$ to $\eta =0$, which reflects in the different peak weights. For comparison: distributions $P_n^{\left(k\right)}\left(t\right)$ for individual currents $I_k$ (blue, bold green, bold red, and black curves in the background). The saddle-point approximation (dashed lines) only captures the long-term behavior $t⪢{\eta }^{-1}$. Parameters: $N=3$, ${ϵ}_d=1$, $\Gamma =0.1$, $\eta =0.0001$, $U=1$, $\beta =1$, $V=V^{\ast }=5$, and $\overline{V}\to \infty$.