Voltage Quench Dynamics of a Kondo System

We examine the dynamics of a correlated quantum dot in the mixed valence regime. We perform numerically exact calculations of the current after a quantum quench from equilibrium by rapidly applying a bias voltage in a wide range of initial temperatures. The current exhibits short equilibration times and saturates upon the decrease of temperature at all times, indicating Kondo behavior both in the transient regime and in the steady state. The time-dependent current saturation temperature connects the equilibrium Kondo temperature to a substantially increased value at voltages outside of the linear response. These signatures are directly observable by experiments in the time domain.

Figure 1

(a) Interacting quantum dot embedded into a tunnel junction. The dot is described by a level spacing ${ϵ}_d$ and interaction $U$. The leads are described by the half-bandwidth $D$ and the chemical potential and ${ϵ}_d=0$. (b) At time $t=0$ a voltage $V$ is instantaneously applied to the system, changing the bath Fermi level to $\pm V/2$.

Figure 2

Current $I_T(t)$ as a function of time $t$ after a voltage quench, for temperatures $T$ indicated. Parameters $U=3$, ${ϵ}_d=0$, $\mathrm{\Gamma }=0.3$, $D=5$, and $V=2$. (a) Monte Carlo (solid symbols, full lines), NCA (dotted lines), and OCA (dash dotted lines) results. (b) The temperature dependence of $I_T(t)$ at times $t=0.2–0.9$. (c) Behavior of the NCA and OCA currents up to $t=5$.

Figure 3

Normalized transient current $I_T(t)/I_{T=0}(t)$ and static magnetic susceptibility ${\chi }_T/{\chi }_{T=0}$ as a function of temperature $T$ for a set of times at $V=2$, ${ϵ}_d=0$, $D=5$ and (a) $\mathrm{\Gamma }=0.3$, $U=3$; (b) $\mathrm{\Gamma }=0.2$, $U=5$. Dashed line: value of $1/2$. Vertical arrows: crossing points $T_K$ (susceptibility) and $T_t$ (current). (c) $I_T(t)/I_{T=0}(t)$ and ${\chi }_T/{\chi }_{T=0}$ [all points from panels (a) and (b)] rescaled as a function of $T/T_t$ and $T/T_K$ correspondingly.

Figure 4

(a) Dot occupancy and (b) current at $U=3$, ${ϵ}_d=0.0$$\mathrm{\Gamma }=0.3$, $D=5.0$ and temperature $T=0.1$ for voltages $V=0.8$, 2, 4, 6 (equal to $2U$), 8, and 10 (equal to $2D$).

Figure 5

(a) Normalized current $I_T(t)/I_{T=0}(t)$ as a function of $T$ at $t=0.2$, 0.4, 0.6, 0.8, 1.0 and $U=3$, ${ϵ}_d=0.0$, $\mathrm{\Gamma }=0.3$, $D=5.0$ and $V=6.0$. Dashed line: value of $1/2$. (b) Normalized current $I_T(t)/I_{T=0}(t)$ as a function of $T$ at $t=0.2$ and (c) $t=0.8$ for different $V$.

Figure 6

(a) Current saturation temperature $T_t$ ($T$ at half of zero-temperature current value) as a function of $t$ at different $V$ at $U=3$, ${ϵ}_d=0$, $\mathrm{\Gamma }=0.3$, $D=5$. Solid lines: estimates of the steady state $T_t$. (b) $T_t$ as a function of $V$ at a set of $t$. Dashed line: linear fit at $V>2$.