Zero-frequency noise in adiabatically driven interacting quantum systems

We investigate current-current correlations of adiabatic charge pumping through interacting quantum dots weakly coupled to reservoirs. To calculate the zero-frequency noise for a time-dependently driven system, possibly in the presence of an additional dc bias, we perform within a real-time diagrammatic approach a perturbative expansion in the tunnel coupling to the reservoirs in leading and next-to-leading orders. We apply this formalism to study the adiabatic correction to the zero-frequency noise, i.e., the pumping noise, in the case of a single-level quantum dot charge pump. If no stationary bias is applied, the adiabatic correction shows Coulomb-interaction-induced deviations from the fluctuation-dissipation theorem. Furthermore, we show that the adiabatic correction to the Fano factor carries information about the coupling asymmetry and is independent of the choice of the pumping parameters. When including a time-dependent finite bias, we find that there can be pumping noise even if there is zero adiabatically pumped charge. The pumping noise also indicates the respective direction of the bias-induced current and the pumping current.

Figure 1

Energy diagram of a single-level quantum dot attached to two leads via tunneling barriers. A symmetric voltage bias (with respect to the equilibrium chemical potential, see horizontal line) can be applied across the system and the energy level, the bias and the coupling strength may be chosen time dependent. Time-dependent charging and uncharging of the system goes along with a time-dependent difference between the currents flowing into the leads.

Figure 2

Diagrams for the cases of having (a) both current operators in the same irreducible block, and (b) for having them on different blocks. Diagrams (c) and (d) depict the last contribution due to the product of current expectation values, where for (c) $t_1<t^{\prime }$ and for (d) $t_1>t^{\prime }$.

Figure 3

Plot for (a) the instantaneous and (b) the adiabatic zero-frequency noise as a function of the mean energy level $\overline{\epsilon }$. The graphs are drawn for $U=0,20k_{\mathrm{B}}T$ (blue dashed line, black solid line). The left and right tunnel coupling rates are ${\Gamma }_{\mathrm{L}}=0.3\Gamma$ and ${\Gamma }_{\mathrm{R}}=0.7\Gamma$. The other parameters are $\Gamma =0.5k_{\mathrm{B}}T$, the pumping amplitudes $\delta \epsilon ,\delta {\Gamma }_{\mathrm{L}}=0.1\Gamma$, the phase difference between the pumping parameters $\phi =\pi /2$, and $\Omega =0.1\Gamma$.

Figure 4

The instantaneous noise (black solid line) and the sum of instantaneous and adiabatic (red dashed line) as a function of the mean energy level $\overline{\epsilon }$. The splitting of the resonances is visible for $\delta \epsilon >k_{\mathrm{B}}T$, here $\delta \epsilon =4k_{\mathrm{B}}T$ and $\delta \Gamma =0.3\Gamma$. The other parameters are $U=20k_{\mathrm{B}}T$, $\Omega =\Gamma /4$, and $\Gamma =0.5k_{\mathrm{B}}T$.

Figure 5

Plot of the adiabatic Fano factor $F^{\left(a\right)}$ as a function of the mean energy level $\overline{\epsilon }$. The two curves are for $({\overline{\Gamma }}_{\mathrm{L}}-{\Gamma }_{\mathrm{R}})/\overline{\Gamma }=-0.4,0.2$ (solid black line, dashed red line). The other parameters are $\Gamma =0.5k_{\mathrm{B}}T$, the pumping amplitudes $\delta \epsilon ,\delta {\Gamma }_{\mathrm{L}}=0.1\Gamma$, the phase difference between the pumping parameters $\phi =\pi /2$, and $\Omega =0.1\Gamma$.

Figure 6

The number of pumped charges $\tau {\overline{I}}_t^{(a,0)}$ (a) and the total noise (cross correlations) $S_{\mathrm{LR}}$ as a function of the average level energy $\overline{\epsilon }$. The time-dependent parameters are chosen as $\epsilon \left(t\right)=\overline{\epsilon }+\delta \epsilon \sin \left(\Omega t\right)$, ${\Gamma }_{\mathrm{L}}=\Gamma /2\left[1+\sin (\Omega t-\pi /2)\right]$, and ${\Gamma }_{\mathrm{R}}=\Gamma /2\left[1+\sin (\Omega t+\pi /2)\right]$. The amplitude of the level modulation is $\delta \epsilon =\left\{5\Gamma ,25\Gamma ,75\Gamma \right\}$ for the black (solid), blue (dotted), and red (dashed) lines, respectively. The other parameters are $U=100k_{\mathrm{B}}T$ and $k_{\mathrm{B}}T=2\Gamma$.

Figure 7

Density plot of the pumped charge ${\overline{I}}^{(a,0)}$ [(a) and (c)] and the adiabatic noise $S_{\mathrm{LR}}^{(a,0)}$ [(b) and (d)], as a function of the mean energy level $\overline{\epsilon }$ and the mean bias voltage $\overline{V}$, without magnetic field $\Delta =0$. Figures (a) and (b) depict the case for symmetric tunnel coupling, ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma /2$, and both (c) and (d) for asymmetric coupling, ${\Gamma }_{\mathrm{L}}=0.7\Gamma$ and ${\Gamma }_{\mathrm{R}}=0.3\Gamma$. The underlined dashed grid sketches the dot level resonance lines. The other parameters are $U=25k_{\mathrm{B}}T$, $\delta \epsilon =0.1\Gamma$, $\delta V=0.1\Gamma$, $\Omega =0.1\Gamma$, $\phi =-\pi /2$, and $\Gamma =0.5k_{\mathrm{B}}T$.

Figure 8

Density plot of the pumped charge ${\overline{I}}^{(a,0)}$ [(a) and (c)] and the adiabatic noise $S_{\mathrm{LR}}^{(a,0)}$ [(b) and (d)], as a function of the mean energy level $\overline{\epsilon }$ and the mean bias voltage $\overline{V}$, with finite magnetic field $\Delta =7.5k_{\mathrm{B}}T$. Figures (a) and (b) depict the case for symmetric tunnel coupling, ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma /2$, and both (c) and (d) for asymmetric coupling, ${\Gamma }_{\mathrm{L}}=0.7\Gamma$ and ${\Gamma }_{\mathrm{R}}=0.3\Gamma$. The underlined dashed grid sketches the dot level resonance lines. The other parameters are as in Fig. 7.

Figure 9

Upper graph: general scheme of a diagram element, where $t^{\prime }<t$. The index $x$ stands for the different kinds of diagrams $W$, $W_{I_{\alpha }}$, and $W_{I_{\alpha }{I}_{\beta }}$. The forward part of the Keldysh contour is the upper line going from left to right, the lower line represents the backward part of the contour. Lower graph: example for one specific diagram first order in $\Gamma$, of type $W_{I_{\alpha }{I}_{\beta }}$. The solid arrow line with index $\alpha$ is representing a tunneling line where an electron is tunneling in and out of reservoir $\alpha$. The solid dots represent external vertices due to the current operator. The dashed lines represent the extra imaginary energy lines that are used for either Laplace derivatives or adiabatic corrections of diagrams.

Figure 10

Diagrammatic representation of the current expectation value $\langle {\widehat{I}}_{\alpha }\left(t\right)\rangle$. The current operator originally placed at the turning point of the Keldysh contour can be equivalently put on the upper or the lower contour. By summing both possibilities, a factor $\frac{1}{2}$ is added.