Full counting statistics for the number of electrons in a quantum dot

Motivated by recent real-time electron counting experiments, we evaluate the full counting statistics for the probability distribution of the electron number inside a quantum dot which is weakly coupled to source and drain leads. A non-Gaussian exponential distribution appears when there is no dot state close to the lead chemical potentials. We propose the measurement of the joint probability distribution of current and electron number, which reveals correlations between the two observables. We also show that for increasing strength of tunneling, the quantum fluctuations qualitatively change the probability distribution of the electron number. In this paper, we derive the cumulant generating functions (CGFs) of the joint probability distribution for several cases. The Keldysh generating functional approach is adopted to obtain the CGFs for the resonant-level model and for the single-electron transistor in the intermediate conductance regime. The general form for the CGF of the joint probability distribution is provided within the Markov approximation in an extension of the master equation approach [D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. B 67, 085316 (2003)].

Figure 1

The time evolution of electron number for rare (upper panel) and frequent (lower panel) tunneling cases.

Figure 2

The closed time-path consisting of forward $C_{+}$, backward $C_{-}$, and imaginary time $C_{\tau }$ branches.

Figure 3

The logarithm of the probability distribution of electron number $P\left(N\right)$ for the noninteracting (a) and interacting (b) QDs in equilibrium for the symmetric case $({\Gamma }_L={\Gamma }_R)$. The vertical axes are normalized by the coupling strength $\Gamma$ and the horizontal axes are shifted by $-1∕2$. Solid lines are for various dot-levels. Dashed lines are for Gaussian approximations.

Figure 4

Contour plots of the joint probability distribution $\ln P(I,N)$ are given for the symmetric case in (a) and the asymmetric case in (b) $({\Gamma }_L=5{\Gamma }_R)$. The function $\ln P(I,N)$ has a maximum value $\ln P(I,N)=0$ at $I=⟨I⟩$ and $N=⟨N⟩=({\Gamma }_L-{\Gamma }_R)∕2\Gamma$. The counter interval is $t_0\Gamma ∕20$ and the shaded regions are for very small values. Vertical axes are shifted by $-1∕2$.

Figure 5

Diagrammatic expansion of $\mathcal{W}$. Solid, dashed, and wavy lines are for Dirac, Majorana, and particle-hole Green functions.

Figure 6

The nonequilibrium number distributions are given for the resonant-level model in (a) and the single-electron transistor in (b) at ${ϵ}_0={\Delta }_0=0$ and $T=0$. Solid lines are for various coupling strengths $\Gamma$ in (a) and for various dimensionless tunnel conductances ${\alpha }_0(={\sum }_{r=L,R}1∕2\pi e^2{R}_r)$ in (b). The dashed lines overlapping with curves for $\Gamma ∕eV=0.05$ (a) and for ${\alpha }_0={10}^{-4}$ (b) are obtained by the extension of the master equation approach. Horizontal axes are shifted by $-1∕2$.