Nonadiabatic electron pumping through interacting quantum dots

We study nonadiabatic charge pumping through single-level quantum dots taking into account Coulomb interactions. We show how a truncated set of equations of motion can be propagated in time by means of an auxiliary-mode expansion. This formalism is capable of treating time-dependent electronic transport for arbitrary driving parameters. We verify that the proposed method describes very precisely the well-known limit of adiabatic pumping through quantum dots without Coulomb interactions. As an example we discuss pumping driven by short voltage pulses for various interaction strengths. Such finite pulses are particularly suited to investigation of transient nonadiabatic effects, which may also be important for periodic drivings, where they are much more difficult to reveal.

Figure 1

(a) Time dependence of the resonance energy according to Eq. (29). (b, c) Time-resolved current $J_{\mathrm{L}}$ for different pulse lengths $t_{\mathrm{p}}$ and $U=0$ and $U/{\Gamma }_0=10$. Parameters used: ${\varepsilon }_0={\Gamma }_0$, ${\varepsilon }_1=-2{\Gamma }_0$, ${\mu }_{\alpha }=0$, $k_{\mathrm{B}}T=0.1{\Gamma }_0$, and $N_{\mathrm{F}}=160$ (number of auxiliary modes). Circles denote the adiabatic limit.[28]

Figure 2

Time dependence of the resonance energy ${\varepsilon }_{\mathrm{d}}$ (upper row) and the decay widths ${\Gamma }_{\mathrm{L}}$ [lower row; solid (blue) line] and ${\Gamma }_{\mathrm{R}}$ [lower row; dashed (red) line] for three cases: (a) ${\delta }_{\mathrm{L}}=-1$, (b) ${\delta }_{\mathrm{L}}=0$, and (c) ${\delta }_{\mathrm{L}}=1$. Dotted lines indicate the chemical potential in the reservoirs and the shaded area shows the times when the resonance energy is below the chemical potential.

Figure 3

Charge per pulse $Q_{\mathrm{p}}$ vs pulse shift ${\delta }_{\mathrm{L}}$ in the long-pulse limit (upper panel) and at $t_{\mathrm{p}}=1.7{\Gamma }_0^{-1}$ (lower panel). Dashed lines indicate half of the no-interacting result.

Figure 4

Charge pumped per pulse $Q_{\mathrm{p}}$ versus pulse length $t_{\mathrm{p}}$ for the pulse scheme given by Eq. (31). The solid (blue) line represents the noninteracting case, while the dashed (red) line represents $U/{\Gamma }_0=10$. We consider threecases: (a) ${\delta }_{\mathrm{L}}=-1$, (b) ${\delta }_{\mathrm{L}}=0$, and (c) ${\delta }_{\mathrm{L}}=1$.