Measurement Waiting Time Distributions as Probes of Dynamical Many-Body Quantum Phases Transitions

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This thesis investigates waiting time distributions as post-selection-free probes of measurement-induced phase transitions in free fermion systems. We study the transverse-field XY chain under local particle number monitoring in both projective and decay unravelings, leveraging Gaussian state techniques and Monte Carlo wave function methods to efficiently simulate quantum trajectories. Through numerical simulations for chains up to L = 128 sites, we confirm the measurement-induced phase transition from a critical phase with S ∼ ln(l) entanglement scaling to an area-law phase at finite monitoring rate η.

We employ spectral statistics of the entanglement Hamiltonian, including r-statistics from random matrix theory, revealing signatures consistent with the transition between ergodic and localized regimes. For waiting time distributions, we derive exact analytic expressions for the no-click survival probability from the maximally mixed steady state and compute first-jump time distributions. We find that waiting time distributions exhibit universal Poissonian behaviour at short times across all phases, with transition signatures emerging only at long times when survival probabilities decay to negligible values, effectively hiding the measurement-induced phase transition in rare-event tails.

While derivatives of survival probability show non-analyticities at η agreeing with effective Hamiltonian spectral transitions, the dominance of short-time exponential decay poses experimental challenges. Our results demonstrate that although waiting time distributions remain post-selection-free and experimentally accessible, extracting clear measurement-induced phase transition signatures requires accessing rare long-time events, highlighting fundamental limitations and suggesting future directions toward refined measurement protocols, larger-scale simulations, and extensions to alternative unravelings such as quantum diffusion.