Ramsey Fringe & Estimation Convergence: SQL vs Heisenberg Limit

Probe EnsembleSQL

Number of probes N 6

True phase φ (unknown) 0.30π

Measurement rate 12 shots/s

Live Estimate

Shots taken

Est. φ

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Estimation error |φ_est − φ_true|

—

Theoretical uncertainty Δφ

—

Precision gain over SQL

1×

How it works

This is the 2D counterpart to the 3D quantum-metrology sim. Instead of animating a ring of atoms in a 3D scene, it draws the two objects a metrologist actually looks at: the Ramsey fringe — the click probability as a function of the unknown phase φ — and a live convergence chart tracking how fast the running phase estimate locks onto the truth as shots accumulate.

Standard Quantum Limit (SQL) — N atoms prepared and measured independently. Each atom clicks 0/1 with probability p = (1+cosφ)/2. Averaging m shots over N atoms:

p̂ = (total clicks) / (N·m)
φ̂ = arccos(2p̂ − 1)
Δφ_SQL = 1 / √(N·m)

Heisenberg Limit — the N atoms are entangled into a GHZ state (|00…0⟩ + |11…1⟩)/√2. A collective parity measurement oscillates N times faster in φ, so on the fringe plot the GHZ curve completes a full period across the same φ range the SQL curve barely dents:

p = (1 + cos(N·φ)) / 2
φ̂ = arccos(2p̂ − 1) / N
Δφ_Heis = 1 / (N·√m)

The per-atom Fisher information of this measurement is exactly 1 regardless of φ (the derivative and the binomial variance both vanish as φ→0 at matching rates), so the 1/√(Nm) and 1/(N√m) bounds hold uniformly across the fringe, not just at the most sensitive point — verified numerically alongside this sim (see report). Entanglement narrows the effective fringe by exactly N, which is why the convergence chart shows the GHZ estimate's error collapsing onto the truth dramatically faster than SQL for the same number of atoms and shots (Giovannetti, Lloyd & Maccone, Nat. Photonics 2011).

Fringe plot (top) — the two theory curves in normalized units f = φ·N/π ∈ [0,1]; dots are real binomial Monte Carlo outcomes landing near the active curve at the true phase; the dashed marker is φ_true, the solid marker is the live estimate φ̂ converging toward it, with a shaded band showing ±Δφ.
Convergence chart (bottom) — precision in decades, −log₁₀(error), rising as the estimate locks on; the dashed curve is the theoretical −log₁₀(Δφ) envelope the empirical trace should track statistically.
N — number of probe atoms/photons in the ensemble.
Rate — measurement shots simulated per second.

SQL fringe / estimate GHZ fringe / estimate Outcome "0" sample Outcome "1" sample