Abstract
We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-Andr\'{e}-Harper model with hopping rate J and potential strength V. On the basis of the quantum trajectory method, we obtain the phase diagram for the steady-state entanglement entropy and demonstrate that robust logarithmic system-size scaling emerges up to a critical potential strength V_c/J ∼ 2.3. Moreover, we find that the measurement induces entanglement phase transitions from the logarithmic-law phase to the area-law phase for the potential strength V< V_c, while any finite measurement stabilizes the area-law phase for V>V_c. This result is distinct from the entanglement scaling in the unitary limit, where volume-law and area-law phases undergo a transition at V/J=2. To further support the phase diagram, we analyze the connected correlation function and find that it shows algebraic decay in the logarithmic-law phase, while it decays quickly in the area-law phase. Our results can be tested in ultracold atoms by introducing quasiperiodic potentials and continuously monitoring the local occupation number with an off-resonant probe light.