Dynamical evolution in a one-dimensional incommensurate lattice with $\mathcal{PT}$ symmetry

We investigate the dynamical evolution of a parity-time ($\mathcal{PT}$) symmetric extension of the Aubry-Andr\'{e} (AA) model, which exhibits the coincidence of a localization-delocalization transition point with a $\mathcal{PT}$ symmetry breaking point. One can apply the evolution of the profile of the wave packet and the long-time survival probability to distinguish the localization regimes in the $\mathcal{PT}$ symmetric AA model. The results of the mean displacement show that when the system is in the $\mathcal{PT}$ symmetry unbroken regime, the wave-packet spreading is ballistic, which is different from that in the $\mathcal{PT}$ symmetry broken regime. Furthermore, we discuss the distinctive features of the Loschmidt echo with the post-quench parameter being localized in different $\mathcal{PT}$ symmetric regimes.