Quantum Metrology: When Decoherence Breaks the Heisenberg Limit

Entangling N atoms into a GHZ (Greenberger-Horne-Zeilinger) "cat" state lets a Ramsey interferometer estimate a phase with precision Δφ = 1/N instead of the classical shot-noise floor Δφ = 1/√N reachable with N independent atoms — the Heisenberg limit of quantum metrology. But that Fisher-information gain is fragile: this simulator models individual dephasing acting on each of the N entangled particles during the Ramsey wait time, which decoheres the whole cat state at a rate that scales with N itself. The 3D cluster shows the entangled probe atoms losing their shared coherence over the wait time, while a live log-log curve plots the classical (1/√N), ideal Heisenberg (1/N) and noise-limited precision together so you can find the optimal particle number N* = 1/(Γt) — the exact point where adding one more entangled atom stops helping and starts hurting.