Relaxation vs decoherence: Spin and current dynamics in the anisotropic Kondo model at finite bias and magnetic field

Using a nonequilibrium renormalization group method we study the real-time evolution of spin and current in the anisotropic Kondo model (both antiferromagnetic and ferromagnetic) at finite magnetic field $h_0$ and bias voltage $V$. We derive analytic expressions for all times in the weak-coupling regime $\max \{V,h_0,1\mathrm{/}t\}\gg T_c$ ($T_c=$ strong coupling scale). We find that all observables decay both with the spin relaxation and decoherence rates ${\mathrm{\Gamma }}_{1\mathrm{/}2}$. Various $V$-dependent logarithmic, oscillatory, and power-law contributions are predicted. The low-energy cutoff of logarithmic terms is generically identified by the difference of transport decay rates. For small times $t\ll \max \{V,h_0{\}}^{-1}$, we obtain universal dynamics for spin and current.