Adiabatic pumping through a quantum dot with coulomb interactions: A perturbation expansion in the tunnel coupling

We present a diagrammatic real-time approach to adiabatic pumping of electrons through interacting quantum dots. Performing a systematic perturbation expansion in the tunnel-coupling strength, we compute the charge pumped through a single-level quantum dot per pumping cycle. The combination of coulomb interaction and quantum fluctuations, accounted for in contributions of higher order in the tunnel coupling, modifies the pumping characteristics via an interaction-dependent renormalization of the quantum-dot level. The latter is even responsible for the dominant contribution to the pumped charge when pumping via time-dependent tunnel-coupling strengths.

Figure 1

Example for the time evolution of the reduced density matrix. The upper and lower lines represent the forward and backward time propagation along the Keldysh contour. Tunneling lines connecting vertices represent tunneling events with the left (right) reservoir. Next to the propagators the respective dot states are indicated.

Figure 2

Examples of diagrams contributing to the irreducible diagram part $W_{00}$.

Figure 3

Example on how to add the external frequency lines. The line $-iz$ is needed for the evaluation of $\partial \mathbf{W}$; $-i{z}_1$, $-i{z}_2$, and $-i{z}_4$ are needed for the contributions to ${\mathbf{W}}^{\left(\mathrm{a}\right)}$ due to both $\frac{d\Gamma }{dt}$ and $\frac{dϵ}{dt}$; $-i{z}_3$ does not contribute and can be omitted; and $-i{z}_5$ is additionally needed for the evaluation of the contribution to ${\mathbf{W}}^{\left(\mathrm{a}\right)}$ due to $\frac{dϵ}{dt}$.

Figure 4

Pumped charge up to first order $\Gamma$ in units of $e{\eta }_1∕{\overline{\Gamma }}^2$ as a function of the time-average level position in units of $\overline{\Gamma }$ for different values of $U$. Pumping parameters are $ϵ$ and ${\Gamma }_{\mathrm{L}}$. The temperature is $k_{\mathrm{B}}T=2\overline{\Gamma }$.

Figure 5

Pumped charge up to first order $\Gamma$ in units of $e{\eta }_2∕{\overline{\Gamma }}^2$ as a function of the time-average level position in units of $\overline{\Gamma }$ for different values of $U$. Pumping parameters are ${\Gamma }_{\mathrm{L}}$ and ${\Gamma }_{\mathrm{R}}$. The temperature is $k_{\mathrm{B}}T=2\overline{\Gamma }$.

Figure 6

Diagrams contributing to the first-order-$\Gamma$ part of the adiabatic correction to the matrix element ${\left({\mathbf{W}}_t\right)}_{0,0}$. All appearing reservoir and spin indices $\alpha$, $\sigma$ are to be summed over and the energy $\omega$ is to be integrated over.

Figure 7

Diagrams contributing to the second-order-$\Gamma$ instantaneous part of the matrix element ${\left({\mathbf{W}}_t\right)}_{0,0}$. All appearing reservoir and spin indices $\alpha$, ${\alpha }^{\prime }$, $\sigma$, ${\sigma }^{\prime }$ are to be summed over and the energy $\omega$ is to be integrated over.