Time scales in the dynamics of an interacting quantum dot

We analyze the dynamics of a single-level quantum dot with Coulomb interaction, weakly tunnel coupled to an electronic reservoir, after it has been brought out of equilibrium, e.g., by a step-pulse potential. We investigate the exponential decay toward the equilibrium state, which is governed by three time scales. In addition to the charge and spin relaxation time there is a third time scale which is independent of the level position and the Coulomb interaction. This time scale emerges in the time evolution of physical quantities sensitive to two-particle processes.

Figure 1

Schematics of the models: (a) Single-level QD with Coulomb interaction, $U$, coupled to a normal lead with a tunneling strength $\Gamma$. Dot occupations can be measured via the current passing through a nearby quantum point contact (QPC) capacitively coupled to the dot. (b) QD attached to an additional superconducting contact. (c) QD coupled to a ferromagnetic lead.

Figure 2

Decay rates ${\gamma }_m$ (blue, dashed line), ${\gamma }_n$ (red, dash-dotted line), and ${\gamma }_s$ (green, solid line) in units of $\Gamma$ as a function of the dot level position $\epsilon$. The temperature is $k_{B}T=1.5\Gamma$ and the interaction energy is $U=10\Gamma$.

Figure 3

Equilibrium value of the quantity $\widehat{m}$ as a function of the dot level position $\epsilon$. The other parameters are $k_{B}T=1.5\Gamma$ and $U=10\Gamma$.

Figure 4

Equilibrium electron-hole occupation ${\langle {\widehat{n}}_{\mathrm{eh}}\rangle }^{\mathrm{eq}}$ (red, solid line) and the coefficient $S$ (blue, dashed line) as a function of the dot level position $\epsilon$. The other parameters are: $k_{B}T=1.5\Gamma$ and $U=10\Gamma$.

Figure 5

Field lines describing variations of $\epsilon$ and $U$ that lead to a response of the system with only the rate ${\gamma }_m$.

Figure 6

Log plot of the current in the QPC as a function of the time after a finite variation of $\epsilon$. The on-site Coulomb repulsion $U$ is constant and takes the value $U=5k_{B}T$. Dashed blue line: $\epsilon$ changes from ${\epsilon }_0=10k_{B}T$ to $\epsilon =-10k_{B}T$, its slope yields the relaxation rate ${\gamma }_m$. Red dot-dashed line: in this case ${\epsilon }_0=10k_{B}T$ to $\epsilon =2k_{B}T$, and the slope leads to ${\gamma }_n$. The black line is obtained if $\epsilon$ changes from ${\epsilon }_0=-13k_{B}T$ to $\epsilon =2k_{B}T$, in which both rates ${\gamma }_m$ and ${\gamma }_n$ are present. In all cases we have subtracted the corresponding value for the current in the long-time limit.