Crossover from nonequilibrium to equilibrium behavior in the time-dependent Kondo model

We investigate the equilibration of a Kondo model that is initially prepared in a nonequilibrium state towards its equilibrium behavior. Such initial nonequilibrium states can, e.g., be realized in quantum-dot experiments with time-dependent gate voltages. We evaluate the nonequilibrium spin-spin correlation function at the Toulouse point of the Kondo model exactly and analyze the crossover between nonequilibrium and equilibrium behavior as the nonequilibrium initial state evolves as a function of the waiting time for the first spin measurement. Using the flow equation method, we extend these results to the experimentally relevant limit of small Kondo couplings.

Figure 1

Universal curves for the spin-spin correlation function $C(t_w,\omega )$ at the Toulouse point for various waiting times ($t_w=0,t_K∕4,t_K∕2,t_K,2t_K,\infty$ from top to bottom). The inset depicts the same curves on a linear scale, which shows more clearly the onset of the nonanalyticity $(C(t_w,\omega )-C(t_w,0)\propto \mid \omega \mid )$ for waiting times $t_w>0$ leading to the algebraic long-time decay.

Figure 2

Universal curves for the spin-spin correlation function $C(t_w,\omega )$ in the limit of small Kondo couplings (Kondo limit) for various waiting times ($t_w=0,t_K∕4,t_K∕2,t_K,2t_K,\infty$ from top to bottom as in Fig. 1). The inset shows the spin expectation value $P\left(t\right)$; the dashed line is an asymptotic fit $P_{\mathrm{asym}}\left(t\right)=0.11\exp (-1.51t∕t_K)$.