Weak-coupling approximations in non-Markovian transport

We study the transport properties of the Fano-Anderson model with non-Markovian effects, which are introduced by making one tunneling-rate energy dependent. We show that the non-Markovian master equation may fail if these effects are strong. We evaluate the stationary current, the zero-frequency current noise, and the occupation dynamics of the resonant level by means of a quantum master-equation approach within different approximation schemes and compare the results to the exact solution obtained by the scattering theory and Green’s functions.

Figure 1

(Color online) Single resonant level coupled to two leads with one constant and one Lorentzian tunneling rate. In the infinite-bias limit, ${\mu }_L\to \infty$ and ${\mu }_R\to -\infty$.

Figure 2

(Color online) The left plot shows the right tunneling rate ${\Gamma }_R\left(\omega \right)$ as a function of the energy $\omega$. We further show how the current $I$ and Fano factor $F$ behave as a function of the detuning ${ϵ}_d-{\epsilon }_R$. The bandwidth ${\delta }_R$ takes the three values shown on the right. For all curves, we have ${\Gamma }_L={\Gamma }_{R,0}$. Results from the exact Green’s functions.

Figure 3

(Color online) Current $I$ as a function of the width ${\delta }_R$ on a logarithmic scale in absence (upper part) and presence (lower part) of detuning, exact (red and solid) and with the $\text{NMME}\stackrel{\wedge}{=}\text{MME}\stackrel{\wedge}{=}\text{DCG}$ (green and dashed) with the parameters ${\Gamma }_L={\Gamma }_{R,0}$. Inset: for the marked points at ${\delta }_R/{\Gamma }_{R,0}=0.1$, we plot as functions of $\omega$ the left tunneling rate ${\Gamma }_L$, the dot-spectral function $A\left(\omega \right)$, and the right tunneling rate ${\Gamma }_R\left(\omega \right)$, with a density plot of the transmission probability $T\left(\omega \right)$ in the background. The dashed green line marks the level of the dot state. The master equations depend only on ${\Gamma }_R\left({ϵ}_d\right)$ while the exact solution is sensitive to the detailed shape of $A\left(\omega \right)$ and $T\left(\omega \right)$.

Figure 4

(Color online) Fano factor $F$ as a function of the width ${\delta }_R$ on a logarithmic scale. Parameters are ${\Gamma }_L={\Gamma }_{R,0}$ and the detunings ${ϵ}_d-{\epsilon }_R$ indicated in each plot.

Figure 5

(Color online) Time-dependent occupation probability $n_d\left(t\right)$, exact solution, and solution in the three presented approximations parameters are ${ϵ}_d={\Gamma }_{R,0}$ and ${ϵ}_d-{ϵ}_R={\Gamma }_L=0.1{\Gamma }_{R,0}$. The width ${\delta }_R$ takes the values denoted in each plot.

Figure 6

(Color online) Frequency ${\omega }_0$ as a function of the detuning ${\omega }_d-{\omega }_R$ for the parameters ${\Gamma }_L=0.1{\Gamma }_{R,0}$ and ${ϵ}_d={\Gamma }_{R,0}$.