Relaxation, prethermalization, and diffusion in a noisy quantum Ising chain

We study the dynamics of thermalization resulting from a time-dependent noise in a quantum Ising chain subject to a sudden quench of the transverse magnetic field. For weak noise the dynamics shows a prethermalized state at intermediate time scales, eventually drifting towards an asymptotic infinite temperature steady state characterized by diffusive behavior. By computing analytically the density of kinks, as well as the transverse and longitudinal magnetic field correlators, we characterize these two regimes, their observability, and their signatures in the various physical quantities.

Figure 1

The density of kinks vs time for a quench with $\Gamma =0.01$, $\Delta =3$, ${\Delta }_0=0.1$ (blue full line). While the red dotted curve is the value attained by $n_{\mathrm{kink}}$ without perturbation (the red horizontal straight line sets the asymptotic value predicted by GGE), the full blue line shows the time evolution with noise that first appears to saturate at the GGE value, but later runs away towards the infinite temperature state (the blue oblique straight line is $n_{\text{drift}}$). Inset: $\Delta n_{\text{kink}}$ vs time. The dephasing of the quantum oscillations is accelerated in the presence of the noise (blue full line) in comparison with the case without perturbation (red dotted line).

Figure 2

The spreading of quantum and thermal correlations in the noisy quantum Ising model: The transverse field correlator has a first crossover when ballistic quasiparticles, carrying quantum correlations, propagate at the distance $r$. Thermal correlations propagate at a second stage, leading to a crossover to a diffusive form, consistent with thermal Glauber dynamics.