We study the dynamics under continuous measurements for free fermions in a quasiperiodic potential by using the Aubry-André-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 $V_c/J\sim 2.3$. 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.

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• Open quantum systems
• Quantum entanglement
• Quantum measurements
• Quantum phase transitions
• Quasicrystalline structure
• Nonequilibrium systems
• Ultracold gases
• Tight-binding model