Cell-Dependent Criticality for Quantum Metrology

Exploiting enhanced sensitivity of a system in the vicinity of a phase transition boundary, critical quantum metrology to date still suffers from gap-closure related bottleneck effects, namely, critical slowing down of the sensing dynamics and a drastic shrinking of the parameter sensing window. To alleviate the said bottleneck inherent to any homogeneous lattice used for sensing, here we propose to leverage the intrinsic hopping inhomogeneity arising from bosonic ladder-operator matrix elements in Fock-space lattices (FSLs). Specifically, using a two-mode Jaynes--Cummings-type model, we show that the sensing parameter can be imprinted onto a topological zero-energy mode of the FSL. The key system parameters thus become cell dependent, effectively tracing out a curve in a topological phase diagram. Cell-dependent criticality emerges when this curve crosses or approaches a topological phase boundary, without globally tuning the lattice close to criticality. An external control parameter reshapes this curve, continuously tuning the scaling of the quantum Fisher information from the standard to the Heisenberg scaling while maintaining broad sensing coverage and a reduced gap cost. Furthermore, a local photon-number measurement on a single cavity saturates the quantum Fisher information. These results identify FSLs as a scalable and practical route to criticality-based quantum metrology.