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We investigate the dynamical evolution of a parity-time ($\mathcal{PT}$)symmetric extension of the Aubry-Andr\'{e} (AA) model, which exhibits thecoincidence of a localization-delocalization transition point with a$\mathcal{PT}$ symmetry breaking point. One can apply the evolution of theprofile of the wave packet and the long-time survival probability todistinguish 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 isballistic, which is different from that in the $\mathcal{PT}$ symmetry brokenregime. Furthermore, we discuss the distinctive features of the Loschmidt echowith the post-quench parameter being localized in different $\mathcal{PT}$symmetric regimes.

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