Quantum sensing via π/2 – free precession – π/2 protocol on the Bloch sphere
φ = 0.00 rad | δω_SQL = — | δω_HL = —
Ramsey protocol: a π/2 pulse tilts the Bloch vector to the equator; atoms precess freely for time T accumulating phase φ = ωT;
a second π/2 pulse converts phase to population difference. The fringe signal is P↑ = ½(1 + cos φ).
Shot-noise limit (SQL): δω = 1/(T√N).
Heisenberg limit: δω = 1/(TN) — achievable with entangled states.
Atomic clocks exploit the ultra-narrow resonance at large T and N.