= Finite-pulse Ramsey transition probability
{title2=$P_{g\to e}(\delta;\tau,T)$}
Use the <Driven two-level Rabi model> in the ordered <basis> $(|e\rangle,|g\rangle)$, with <Pauli matrices> satisfying $\sigma_z=\operatorname{diag}(1,-1)$. Let the <Rabi detuning> be $\delta=\omega_0-\omega=2\pi(\nu_0-\nu)$, in <angular frequency>, while $\nu_0$ and $\nu$ are ordinary <frequencies> in <hertz>. Assume two identical square pulses, equal drive <oscillation phases>, constant positive <Rabi angular frequency> $\Omega$, constant $\delta$ during pulses and flight, and no <quantum decoherence> or loss. The <rotating-wave approximation> and isolation of the chosen <two-level system> are required, just as for the <Driven two-level Rabi model>. Free flight means no transverse drive; the reference <oscillator> continues running.
The pulse and free-flight <Hamiltonian operators> in the <rotating frame> are
$$
H_p=\frac{\hbar}{2}(\delta\sigma_z+\Omega\sigma_x),\qquad
H_f=\frac{\hbar\delta}{2}\sigma_z,\qquad
q=\sqrt{\Omega^2+\delta^2},\quad c_p=\cos(q\tau/2),\quad s_p=\sin(q\tau/2).
$$
The <Pauli matrix multiplication law> gives $(\delta\sigma_z+\Omega\sigma_x)^2=q^2I$. Thus the exact <special unitary group> $\mathrm{SU}(2)$ pulse <matrix> for this effective <Hamiltonian operator>, and the flight <matrix>, are
$$
U_p=\begin{pmatrix}
c_p-i(\delta/q)s_p&-i(\Omega/q)s_p\\
-i(\Omega/q)s_p&c_p+i(\delta/q)s_p
\end{pmatrix},\qquad
U_f=\begin{pmatrix}e^{-i\delta T/2}&0\\0&e^{i\delta T/2}\end{pmatrix}.
$$
The sequence is $U_pU_fU_p$. Starting in $|g\rangle$, <matrix multiplication> and the <Born rule> give
$$
\boxed{P_{g\to e}=4\frac{\Omega^2}{q^2}s_p^2
\left[c_p\cos\frac{\delta T}{2}-\frac{\delta}{q}s_p\sin\frac{\delta T}{2}\right]^2.}
$$
This retains the tilted rotation axis and <quantum phase> accumulated during each finite pulse. At resonance it gives $P_{g\to e}(0)=\sin^2(\Omega\tau)$; with no flight it reduces to the single square-pulse response for duration $2\tau$. Both checks would fail if arbitrary ideal $\pi/2$ rotations were silently substituted at nonzero <Rabi detuning>.
For resonant pulse area $\Omega\tau=\pi/2$, $|\delta|/\Omega\ll1$ and $\tau\ll T$, the central fringe approaches
$$
P_{g\to e}\simeq\cos^2(\delta T/2)=\cos^2[\pi(\nu-\nu_0)T].
$$
Its half-height points are $|\nu-\nu_0|=1/(4T)$, so the central peak's <full width at half maximum> is $1/(2T)$ in <hertz>, or $\pi/T$ in <angular frequency>. This width assumes an effectively flat pulse envelope across that central peak, one flight time and unit contrast; it is neither a universal finite-pulse width nor a <frequency> uncertainty. Longer flight lets a smaller <frequency> difference accumulate the same observable <quantum phase>.
The two pulses here have the same sign and the plotted outcome is transfer to $|e\rangle$. The earlier <Ramsey interferometry> example uses an inverse analysis pulse and reports return to its initial label. Specifying the measured outcome and pulse <oscillation phases> reconciles the two conventions. In a beam the lengths and <speed> determine $\tau$ and $T$; in the <Quantronium Ramsey interference experiment> electronics set the pulse times for a stationary circuit.
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