Nonequilibrium dynamics of a noisy quantum Ising chain: Statistics of work and prethermalization after a sudden quench of the transverse field

We discuss the nonequilibrium dynamics of a quantum Ising chain following a quantum quench of the transverse field and in the presence of a Gaussian time-dependent noise. We discuss the probability distribution of the work done on the system both for static and dynamic noise. While the effect of static noise is to smooth the low energy threshold of the statistic of the work, appearing for sudden quenches, a dynamical noise protocol affects also the spectral weight of such features. We also provide a detailed derivation of the kinetic equation for the Green's functions on the Keldysh contour and the time evolution of observables of physical interest, extending previously reported results [Marino and Silva, Phys. Rev. B 86, 060408 (2012)], and discussing the extension of the concept of prethermalization which can be used to study noisy quantum many-body Hamiltonians driven out of equilibrium.

Figure 1

Out-of-equilibirum protocol studied in this paper for the QIC: the system is prepared in the ground state of the Ising chain with $g_0>1$ and is evolved according to the Ising Hamiltonian with a different value of the transverse field $g>1$, plus a Gaussian $\delta$-correlated noise on top of it. For simplicity, both $g_0$ and $g$ are chosen within the paramagnetic phase.

Figure 2

Diagrammatic representation of the Dyson series (37). Crossed diagrams are neglected according to the self-consistent Born approximation.

Figure 3

Populations $n_k=\langle {\gamma }_k^{†}{\gamma }_k\rangle$ vs wave vector $k$ at different times: from bottom to top, $\Gamma t=0.1$, 1, 10, ${10}^2$, ${10}^3$, and ${10}^4$ ($g_0=2$ and $g=4$).

Figure 4

Density of kinks vs time for a quench with $\Gamma =0.01$, $g_0=1.1$, and $g=4$. While the red line shows the value attained by $n_{\mathrm{kink}}$ without perturbation and predicted by the GGE, the full time evolution (blue line) shows first a saturation towards the GGE value and later a runaway towards the infinite temperature state.

Figure 5

Red line: populations contribution, $n_{\text{drift}}$, for the case of a quantum quench ($\Gamma =0$). Blue line: populations contribution in the case of a quench with noise ($\Gamma =0.01$). $g_0=1.1$ and $g=4$.

Figure 6

Red line: coherences contribution, $\Delta n_{\text{kink}}$, for the case of a quantum quench ($\Gamma =0$). Blue line: coherences contribution in the case of a quench with noise ($\Gamma =0.01$). $g_0=1.1$ and $g=4$.

Figure 7

Spreading of quantum and thermal correlations in the noisy quantum Ising model ($J=1$): 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 dynamics.