Nonlinear charge and energy dynamics of an adiabatically driven interacting quantum dot

We formulate a general theory to study the time-dependent charge and energy transport of an adiabatically driven interacting quantum dot in contact with a reservoir for arbitrary amplitudes of the driving potential. We study within this framework the Anderson impurity model with a local ac gate voltage. We show that the exact adiabatic quantum dynamics of this system is fully determined by the behavior of the charge susceptibility of the frozen problem. At $T=0$, we evaluate the dynamic response functions with the numerical renormalization group (NRG). The time-resolved heat production exhibits a pronounced feature described by an instantaneous Joule law characterized by a universal Büttiker resistance quantum $R_0=h/\left(2e^2\right)$ for each spin channel. We show that this law holds in the noninteracting as well as in the interacting system and also when the system is spin polarized. In addition, in the presence of a static magnetic field, the interplay between many-body interactions and spin polarization leads to a nontrivial energy exchange between electrons with different spin components.

Figure 1

Sketch of the setup. A quantum dot described by a single electron level with Coulomb interaction $U$ and is driven by an ac gate voltage $V_g\left(t\right)=V_{0}sin\left(\mathrm{\Omega }t\right)$ and is connected to a normal lead. Top: representation of the setup in terms of a resistance connected in series with a capacitor.

Figure 2

Sketch of the circuit. Upper and lower branch corresponds to $\uparrow$ and $\downarrow$ spin channels.

Figure 3

Imaginary part of the dynamic susceptibility as a function of frequency for $\mathrm{\Omega }t=\pi /2, \mathrm{\Delta }=8x{10}^{-4}D, {\varepsilon }_0=\mu =0, U=0.05D, V_0=0.024D$, and $T=0$.

Figure 4

(a) Capacitance $C\left(t\right)$, (b) dissipation coefficient ${\mathrm{\Lambda }}^c\left(t\right)$, and (c) dissipated power $P_{\mathrm{diss}}\left(t\right)$ in the interacting nonlinear regime, as a function of time, calculated with two techniques. Other parameters as in Fig. 3.

Figure 5

Occupancy of the quantum dot as a function of time for different values of the Coulomb interaction (indicated in the figure). Other parameters as in Fig. 3.

Figure 6

Same as Fig. 4 calculated with BA for several values of $U$.

Figure 7

Frozen occupancies $n_{f,\uparrow }\left(t\right)$ and $n_{f,\downarrow }\left(t\right)$ for a Zeeman splitting ${\delta }_Z={10}^{-3}D$ and different values of $U$. Other parameters are the same as in the previous figures.

Figure 8

Analysis of the Korringa-Shiba laws of Eqs. (18) and (19). The functions ${\mathrm{\Lambda }}^{\uparrow }\left(t\right)$ and ${\mathrm{\Lambda }}^{\downarrow }\left(t\right)$ are compared with ${\left[{\chi }_t^{\uparrow \uparrow }\left(0\right)\right]}^2$ and ${\left[{\chi }_t^{\downarrow \downarrow }\left(0\right)\right]}^2$ for $U=0.05D$. Other parameters are the same as in Fig. 7.

Figure 9

Top panels: Functions ${\mathrm{\Lambda }}^{\sigma }\left(t\right)$. Middle panels: Power developed by the forces induced by electrons with spin $\sigma , P_{\sigma }\left(t\right)$. Lower panels: Joule power $P_{\mathrm{Joule},\sigma }\left(t\right)$ (see text). Solid and dashed lines correspond to $\sigma =\downarrow ,\uparrow$, respectively. Left (right) panels correspond to $U=0.01D$ ($U=0.05D$), respectively. Other parameters are the same as in Fig. 7.

Figure 10

Power developed by the forces induced by electrons with spin $\sigma$ averaged over the cycle as a function of $U$. The inset denotes the different components (see text) for small $U$. Other parameters are the same as in Fig. 7.