Counting Statistics of Non-Markovian Quantum Stochastic Processes

We derive a general expression for the cumulant generating function (CGF) of non-Markovian quantum stochastic transport processes. The long-time limit of the CGF is determined by a single dominating pole of the resolvent of the memory kernel from which we extract the zero-frequency cumulants of the current using a recursive scheme. The finite-frequency noise is expressed not only in terms of the resolvent, but also initial system-environment correlations. As an illustrative example we consider electron transport through a dissipative double quantum dot for which we study the effects of dissipation on the zero-frequency cumulants of high orders and the finite-frequency noise.

Figure 1

Zero-frequency cumulants of the current as functions of the level detuning $\epsilon$ with different dissipation strengths $\alpha$. Parameters are $\Gamma ={\Gamma }_L={\Gamma }_R=0.5$, $T_c=0.1$, $k_{B}T=0$, ${\omega }_c=500$, and $2\pi \alpha =0$ (full line), 0.2 (full line with dots), 0.5 (dashed line), 1 (dashed line with dots). A large bias is applied across the system.

Figure 2

Finite-frequency current noise spectrum for different temperatures (left figure) and dissipation strengths (right figure). Left figure: $\alpha =0.005$, $k_{B}T=0$ (full line), 1 (full line with dots), 2 (dashed line), 5 (dashed line with dots). Right figure: $k_{B}T=0$, and $\alpha =0.01$ (full line), 0.02 (full line with dots), 0.03 (dashed line), 0.05 (dashed line with dots). Other parameters are ${\Gamma }_L={\Gamma }_R=0.01$, $T_c=3$, $\epsilon =10$, $\Delta =\sqrt{{\epsilon }^2+(2T_c{)}^2}$, and ${\omega }_c=500$. A large bias is applied across the system.