A Ramsey sequence is the workhorse of every atomic clock and quantum magnetometer. Starting from the ground state, a π/2 pulse tips the qubit into an equal superposition of |0⟩ and |1⟩ on the equator of the Bloch sphere. During the free-evolution time T that follows, the qubit's energy splitting ΔE (set by the field or frequency being sensed) makes the superposition accumulate a real relative phase φ = ΔE·T/ℏ = Δ·T, where Δ is the angular detuning. A second π/2 pulse converts that invisible phase back into a measurable population difference, giving the Ramsey fringe signal:
P(T) = cos²(φ/2) = ½(1 + cos(ΔT))
with decoherence: P(T) = ½(1 + e-T/T2·cos(ΔT))
Sweeping T (or, equivalently, the detuning) traces out oscillating fringes whose spacing directly reveals Δ — this is exactly how a Ramsey magnetometer or an atomic-clock local oscillator is locked to an unknown frequency. Environmental noise randomizes the phase over a coherence time T2, so the fringe contrast decays as e-T/T2: a real, unavoidable limitation of every real sensor, not a plotting artifact.
Each dot on the scan is not the true probability P — it is the outcome of N independent projective measurements (quantum projection noise), so the sampled population estimate scatters around the ideal curve with standard deviation √(P(1-P)/N). Averaging more repetitions N narrows that scatter as 1/√N: the standard quantum limit that sets how long a real sensor must integrate to resolve a given field.
- Detuning Δ/2π — the unknown quantity being sensed (a magnetic-field-induced or reference-frequency detuning); it sets the fringe spacing.
- T2 — the coherence time; shorter T2 washes out the fringes faster and caps how long T can usefully be extended.
- N — repetitions per data point; higher N shrinks the shot-noise scatter around the ideal curve, improving sensitivity as 1/√N.