Interference and interaction effects in adiabatic pumping through quantum dots

In order to investigate the effects of interference and interaction in adiabatic pumping, we consider an Aharonov-Bohm (AB) interferometer with a quantum dot embedded either in one or in both arms. We employ a real-time formalism and we perform an expansion both in the tunnel-coupling strengths between dot and leads and in the pumping frequency, taking into account the Coulomb interaction nonperturbatively. We find that pumping in a single-dot AB interferometer has a peristaltic but phase-coherent character. In a double-dot AB interferometer, we find a pumping mechanism that relies purely on quantum-mechanical interference and has no classical counterpart.

Figure 1

Setup of (a) single-dot and (b) double-dot Aharonov-Bohm interferometer.

Figure 2

(Color online) The dependence of pumped charge in zeroth order in $\Gamma$ and up to first oder in $\left|t^{\text{ref}}\right|$ on the average dot level $\overline{\epsilon }$ is given by the derivative of the average dot occupation $\frac{d}{d\overline{\epsilon }}{⟨\overline{n}⟩}^{\left(\text{i},0,0\right)}$, which is plotted here for various Coulomb-interaction strengths $U$. The temperature is $k_{\text{B}}T=2\overline{\Gamma }$.

Figure 3

(Color online) Charge transferred by rectification in lowest nonvanishing order, $Q_{\text{rec},\epsilon }^{\left(\text{i},1,0\right)}$, in units of $Q_0=e^2\frac{{\Gamma }_{\text{R}}{\Gamma }_{\text{L}}}{{\Gamma }^3}{\eta }_{\text{rec},\epsilon }$ as a function of the average dot level $\overline{\epsilon }$ for various Coulomb-interaction strengths $U$. The time-varying parameter is $\epsilon$ and the temperature is $k_{\text{B}}T=2\overline{\Gamma }$.

Figure 4

(Color online) First flux-dependent correction to the charge transferred by rectification, $Q_{\text{rec},\epsilon }^{\left(\text{i},1,1\right)}$, in units of $Q_0=e^2\frac{\sqrt{{\Gamma }_{\text{L}}{\Gamma }_{\text{R}}}}{{\Gamma }^2}\left|t^{\text{ref}}\right|{\eta }_{\text{rec},\epsilon }\text{cos}\varphi$ as a function of the average dot level $\overline{\epsilon }$ for various Coulomb-interaction strengths $U$. The time-varying parameter is $\epsilon$ and the temperature is $k_{\text{B}}T=2\overline{\Gamma }$.

Figure 5

(Color online) Pumped charge $Q_{\Delta \epsilon ,\epsilon }^{\left(\text{a},0\right)}$ in units of $Q_0=e\eta \left(\epsilon ,\delta \epsilon \right)/{\Gamma }^2$ for ${\Gamma }_{\text{L}}=0.8\Gamma$, ${\Gamma }_{\text{R}}=0.2\Gamma$, $\varphi =\pi /2$, and $k_{\text{B}}T=2\Gamma$. (a) and (b) show a density plot where $Q_{\Delta \epsilon ,\epsilon }^{\left(\text{a},0\right)}$ is a function of $\overline{\epsilon }$ and $\overline{\Delta \epsilon }$ for (a) vanishing and (b) infinite Coulomb interaction. Cuts through (b) are shown in (c) and (d). In (c) $Q_{\Delta \epsilon ,\epsilon }^{\left(\text{a},0\right)}$ is plotted as a function of $\overline{\Delta \epsilon }$ for different $\overline{\epsilon }$. In (d) $Q_{\Delta \epsilon ,\epsilon }^{\left(\text{a},0\right)}$ is plotted as a function of $\overline{\epsilon }$ for different $\overline{\Delta \epsilon }$.

Figure 6

(Color online) Pumped charge $Q_{\Delta \epsilon ,\epsilon }^{\left(\text{a},0\right)}$ in units of $Q_0=e\eta \left(\epsilon ,\delta \epsilon \right)/{\Gamma }^2$ and rectified charge $Q_{\text{rec},\epsilon }^{\left(\text{i},1\right)}$ in units of $Q_0=e^2{\eta }_{\text{rec},\epsilon }/\Gamma$ as a function of $\varphi$ for $U=\infty$, ${\Gamma }_{\text{L}}=0.8\Gamma$, ${\Gamma }_{\text{R}}=0.2\Gamma$, $\overline{\epsilon }=0$, $\overline{\Delta \epsilon }=0.5\Gamma$, and $k_{\text{B}}T=2\Gamma$.