Frequency-dependent counting statistics in interacting nanoscale conductors

We present a formalism to calculate finite-frequency current correlations in interacting nanoscopic conductors. We work within the $n$-resolved density matrix approach and obtain a multitime cumulant generating function that provides the fluctuation statistics solely from the spectral decomposition of the Liouvillian. We apply the method to the frequency-dependent third cumulant of the current through a single resonant level and through a double quantum dot. Our results, which show that deviations from Poissonian behavior strongly depend on frequency, demonstrate the importance of finite-frequency higher-order cumulants in fully characterizing transport.

Figure 1

(Color online) The third-order frequency-dependent Fano factor $F^{\left(3\right)}(\omega ,{\omega }^{\prime })$ for the single resonant level. (a) Contour plot of $F^{\left(3\right)}(\omega ,{\omega }^{\prime })$ as a function of its arguments for ${\Gamma }_R={\Gamma }_L$. (b) Sections $F^{\left(3\right)}(\omega ,0)$ and $F^{\left(3\right)}(\omega ,-\omega )$ show that the skewness is suppressed throughout frequency space both with respect to the Poissonian value of unity and to the noise $F^{\left(2\right)}\left(\omega \right)$. In contrast to the shotnoise, the skewness has a minimum at a finite frequency ${\omega }_m$, which exists in the coupling range $(3-\sqrt{5})∕2\le {\Gamma }_L∕{\Gamma }_R\le (3+\sqrt{5})∕2$. In direction $\omega =-{\omega }^{\prime }$, this minimum occurs, for ${\Gamma }_L={\Gamma }_R$, at ${\omega }_{m1}∕{\Gamma }_R=\sqrt{\sqrt{10}-2}∕\sqrt{3}$ whereas for ${\omega }^{\prime }=0$ the position of the minimum shifts slightly to ${\omega }_{m2}∕{\Gamma }_R=2∕\sqrt{3}$. (c) The maximum suppression occurs at ${\Gamma }_L={\Gamma }_R$.

Figure 2

(Color online) Frequency-dependent Fano skewness for the double quantum dot in Coulomb blockade. (a) Contour plot in the strong coupling regime, $T_c=3{\Gamma }_R$, with ${\Gamma }_L={\Gamma }_R$ and $ϵ=0$. (b) Sections $F^{\left(3\right)}(\omega ,0)$ and $F^{\left(3\right)}(\omega ,-\omega )$, and shot noise $F^{\left(2\right)}\left(\omega \right)$ show a series of abrupt increases with increasing $\omega$. Both noise and skewness exhibit both sub- and super-Poissonian behavior. (c) Varying the internal coupling $T_c$, the skewness shows rapid increases along the lines $\omega =\Delta$ and $\omega =\Delta ∕2$. For $\omega >\Delta$ the system is Poissonian (slightly super-Poissonian for $\omega \gtrsim \Delta$), while for $\omega <\Delta$ the transport is always sub-Poissonian. The skewness is strongly suppressed at low frequencies. (d) The derivative $d{F}^{\left(3\right)}(\omega ,-\omega )∕d\omega$ as a function of frequency and detuning $ϵ$ for $T_c=3{\Gamma }_L=3{\Gamma }_R$. Resonances occur at $\omega =\Delta$, $\Delta ∕2$, and $\sim {\Gamma }_R$.