Controlled Zeno-Induced Localization of Free Fermions in a Quasiperiodic Chain

Abstract

We investigate measurement-induced localization in a continuously monitored one-dimensional Aubry--Andr\'e--Harper model, focusing on the quantum Zeno regime in which the measurements dominate coherent dynamics. The presence of a quasiperiodic potential renders the problem analytically tractable and enables a controlled study of the interplay between monitoring and disorder. We develop an analytical description based on an instantaneous Schr\"odinger equation with a measurement-induced effective potential constructed self-consistently from individual quantum trajectories, without relying on postselection. In the quantum Zeno regime, an emergent dominant energy scale reduces the problem to a transfer-matrix formulation of an effective non-Hermitian Hamiltonian, which allows direct computation of the Lyapunov exponent. Complementarily, we extract the localization length numerically from long-time steady-state quantum state diffusion trajectories by reconstructing the intrinsic localized single-particle wave functions and analyzing their spatial decay. These numerical results show quantitative agreement with the effective theory predictions, with controlled corrections of order $J^2/[\lambda^2+(\gamma/2)^2]$ (where $J$ is the hopping amplitude, $\gamma$ the measurement strength, and $\lambda$ the quasiperiodic potential). Our results underscore the connection between the effective non-Hermitian description and the stochastic monitored dynamics, showing the interplay between Zeno-like localization, coherent hopping, and quasiperiodic-disorder-induced localization, while also laying the groundwork for understanding and exploiting measurement-induced localization as a tool for quantum control and state preparation.