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

Abstract

We investigate measurement-induced localization in a continuously monitored one-dimensional Aubry--André--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 Schrödinger 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 $J^2/[λ^2+(γ/2)^2]$ (where $J$ is the hopping amplitude, $γ$ the measurement strength, and $λ$ 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.

Figures (4)

• Figure 1: (a) Schematic of the Aubry--André--Harper lattice under continuous homodyne detection. (b) Illustration of quantum Zeno localization: increasing the measurement strength $\gamma$ suppresses transport and localizes the wave function.
• Figure 2: Steady-state energy statistics under continuous monitoring ($\gamma = 8$, $J = 1$). (Top) Stationary distributions $P_{\mathrm{ss}}(E)$ of the instantaneous energy for system sizes $L = 50, 100, 200, 300$. Shaded regions show normalized steady-state histograms, and solid lines denote kernel density estimates. Panels correspond to the quasiperiodic potential strength of (a) $\lambda = 0$ (clean system), (b) $\lambda = 0.5$, (c) $\lambda = 2.0$, and (d) $\lambda = 5.0$. (Bottom)(e) Finite-size scaling of the energy fluctuations $\sigma_E$ with particle number $N$. Symbols indicate numerical results for different $\lambda$, while the dashed line shows the theoretical scaling $\sigma_E \propto N^{-1/2}$ [Eq. \ref{['Fluc']}]. For $\lambda = 0$, fluctuations vanish and are size independent, whereas for $\lambda > 0$ the instantaneous energy is self-averaging in the thermodynamic limit. Error bars are smaller than the symbol size.
• Figure 3: Effective-theory Lyapunov exponent $\kappa$ in the $\lambda$--$\gamma$ parameter space for $J=1$. (a) Density plot of $\kappa(\lambda,\gamma)$ showing approximate regimes separated by wiggly black lines indicating gradual crossovers rather than sharp transitions. Regimes I, II, III ($\gamma \gtrsim 4$): Zeno (measurement-dominated) regime with varying quasiperiodic contributions. Regime I ($\lambda \lesssim 1$): weak quasiperiodic effect, dynamics primarily governed by measurements. Regime II ($1 \lesssim \lambda \lesssim 3$): intermediate crossover where measurement and quasiperiodic effects compete. Regime III ($\lambda \gtrsim 3$): strong quasiperiodic coupling dominates even with strong measurements. Regime IV ($\gamma \lesssim 4$): weak measurement regime with dominant quasiperiodic localization. The wavy boundaries emphasize that transitions between regimes are continuous and lack strict phase boundaries. The gold circle at $\lambda=2$, $\gamma = 0$ marks the AAH critical point, and the line at $\gamma=0$ corresponds to the unmonitored AAH model. (b) $\kappa$ versus $\lambda$ for fixed $\gamma$ values.
• Figure 4: Localization length $\xi$ as a function of the measurement rate $\gamma$ and strength of the quasiperiodic potential $\lambda$, with $J=1$ in all cases. (a) $\gamma = 0$, the unmonitored AAH model. (b) $\lambda = 0$, corresponding to purely measurement-induced (Zeno) localization. (c) $\lambda = 0.5$, representing the measurement-dominated regime with a weak quasiperiodic potential, where localization is primarily governed by measurement backaction. (d) $\lambda = 2.0$, a crossover regime with intermediate potential strength. (e) $\lambda = 5.0$, corresponding to the strong-potential regime, where intrinsic localization due to the quasiperiodic potential dominates the dynamics. In all panels, symbols denote numerical results obtained from left--right averaged orbitals after QSD evolution, while lines indicate the corresponding theoretical predictions from the effective theory. In panel (d), the effective theory is calculated numerically using the transfer-matrix approach (green dashed line), whereas the other panels use closed-form expressions found in the text (red solid lines). Plotted error bars (smaller than the markers) indicate the standard error across trajectories.