Ramsey Interferometry

Ramsey interferometry, devised by Norman Ramsey in 1950 for molecular beam spectroscopy, is the single most important phase-estimation protocol in quantum sensing. Atomic clocks, magnetometers, and most qubit-calibration routines are Ramsey sequences at heart. It is the concrete realisation of the abstract "encode a phase, then read it out" pipeline.

The three-step sequence

A Ramsey sequence on one qubit (or one two-level atom) is:

Open the interferometer. A Hadamard (a $\pi/2$ pulse) takes $|0\rangle$ to $|+\rangle = (|0\rangle + |1\rangle)/\sqrt{2}$ , splitting the amplitude into two coherent paths.
Accumulate phase. During a free-evolution time $\tau$ , the two levels acquire a relative phase $\theta = \omega\tau$ , where $\omega$ is the energy splitting we want to measure. This is modelled by $P(\theta)$ .
Close the interferometer. A second Hadamard recombines the paths, converting the relative phase into a population difference that a computational-basis measurement can read.

Tracking the state

Starting from $|0\rangle$ :

|0\rangle \xrightarrow{H} \frac{|0\rangle + |1\rangle}{\sqrt 2} \xrightarrow{P(\theta)} \frac{|0\rangle + e^{i\theta}|1\rangle}{\sqrt 2} \xrightarrow{H} \frac{1+e^{i\theta}}{2}|0\rangle + \frac{1-e^{i\theta}}{2}|1\rangle .

The outcome probabilities are the Ramsey fringes

p_0(\theta) = \cos^2\frac{\theta}{2} = \frac{1+\cos\theta}{2}, \qquad p_1(\theta) = \sin^2\frac{\theta}{2} = \frac{1-\cos\theta}{2},

oscillating as a function of the accumulated phase. Counting the fraction of $1$ s and inverting the fringe gives an estimate of $\theta$ , hence of $\omega$ and $\tau$ .

The sweet spot

The fringe's steepest slope — and therefore the most sensitive working point — is at $\theta = \pi/2$ , where $p_0 = p_1 = \tfrac12$ but $|\partial_\theta p_1| = \tfrac12$ is maximal. At this point the pre-measurement state is

\frac{1+i}{2}|0\rangle + \frac{1-i}{2}|1\rangle,

with both amplitudes of magnitude $1/\sqrt2$ . Operators bias real clocks to sit exactly here so a small drift in $\omega$ produces the largest measurable change in counts.

Try it

Build the full Ramsey sequence at the sweet spot $\theta = \pi/2$ : open with a Hadamard, imprint $P(\theta)$ , and close with a second Hadamard — no measurement, since the grader compares the full pre-measurement state vector.

Run your code to see the quantum state.

After running, the Probabilities tab should show both outcomes at exactly $0.5$ — the balanced fringe of the most sensitive Ramsey working point.