= Ramsey interferometry
{c}
{title2=$P(0)=\cos^2(\phi/2)$}
{wiki}
<Ramsey interferometry> uses two coherent pulses with a free interval between them: the first creates a <quantum superposition>, the interval accumulates relative <quantum phase>, and the second converts that <quantum phase> into measurement <probabilities>. Its attraction is that an ordinary population readout can reveal the <quantum coherence> hidden from an <energy measurement> during the free interval.
Here is an ideal <qubit> model with an explicit pulse convention. Let $R_y(\theta)=\exp(-i\theta\sigma_y/2)$, prepare $|0\rangle$, apply $R_y(\pi/2)$, evolve freely with $U_\phi=\exp(-i\phi\sigma_z/2)$, and apply the inverse pulse $R_y(-\pi/2)$ before a <computational-basis measurement>. The first pulse gives $|+\rangle$. Direct <matrix multiplication> gives
$$
R_y(\theta)=\begin{pmatrix}\cos(\theta/2)&-\sin(\theta/2)\\\sin(\theta/2)&\cos(\theta/2)\end{pmatrix},\qquad
R_y(-\pi/2)U_\phi R_y(\pi/2)|0\rangle
=\cos(\phi/2)|0\rangle+i\sin(\phi/2)|1\rangle.
$$
Thus the <Born rule> gives $P(0)=\cos^2(\phi/2)$ and $P(1)=\sin^2(\phi/2)$. Before the second pulse, the same <computational-basis measurement> would give $1/2$ for both outcomes for every $\phi$. At $\phi=2\pi/3$, the second pulse changes these <probabilities> to $1/4$ and $3/4$. If the two pulses have the same sign instead, the output labels interchange: the pulse convention is part of the prediction.
\Image[/ramsey-phase-population-readout.png]
{title=An ideal Ramsey sequence turns relative phase into population}
{description=Original pulse-sequence schematic and exact probabilities for $R_y(-\pi/2)U_\phi R_y(\pi/2)|0\rangle$. Dashed lines show a separate readout with the second pulse omitted. Instantaneous perfect pulses and noiseless free evolution are idealizations, not experimental measurements.}
In a pulse experiment the phase is measured relative to the oscillator supplying the coherent pulses. In a <rotating frame>, write the free <Hamiltonian operator> as $H_{\mathrm{rot}}=\hbar\delta\sigma_z/2$ with signed <angular frequency> difference $\delta$ in the chosen labeling convention. A free interval $T$ then accumulates $\phi=\delta T$; changing the second pulse's phase changes the analysis direction. This comparison makes <Ramsey interferometry> useful for measuring <frequency> differences and diagnosing loss of <quantum coherence>.
For example, suppose the first pulse has prepared a <density matrix> with equal populations and transverse <Bloch vector> length $0\leq V\leq1$:
$$
\rho_\phi=\frac12\begin{pmatrix}1&Ve^{-i\phi}\\Ve^{i\phi}&1\end{pmatrix}.
$$
The same inverse analysis pulse yields $P(0)=(1+V\cos\phi)/2$. The <pure state> case $V=1$ gives the full ideal fringe; the <mixed state> $I/2$ gives the flat result $1/2$. This is a controlled mathematical comparison, not a fit to an experiment. Experimental contrast alone does not identify <quantum decoherence>: imperfect preparation, pulse errors and readout errors can also reduce it.
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