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Keep the first pulse's oscillation phase zero and give the second phase $\phi$ in the Driven two-level Rabi model. Its upper off-diagonal Hamiltonian operator entry is $\hslash \Omega e^{-i\phi }/2$. Multiplying the two pulse unitary operators and the free-flight unitary operator givesThe Born rule gives the displayed phase-dependent form using the $p,b,\theta$ of the Ramsey fringe envelope. Thus $\phi =0$ reproduces the existing expression, while $\phi =\pi$ turns a resonant transfer maximum into zero transfer. A single pulse's population is insensitive to its drive phase; the second coherent interaction makes their difference observable. An uncontrolled, uniformly random $\phi$ instead averages this probability to $2p(1-p)$, leaving the broad pulse response but removing the dark-flight fringes.

Finite pulses also contribute to the near-resonant frequency slope. For $\Omega \tau =\pi /2$, differentiating $\theta$ at $\delta =0$ gives ${\theta }^{\prime }(0)=1/\Omega$. Consequently the phase near resonance isAt $\phi =\pi /2$ the exact resonant slope is $\partial P/\partial \nu =-\pi T_{\mathrm{eff}}$. For the plotted parameters $T_{\mathrm{eff}}\simeq 5.637\mathrm{ms}$; the near-resonant estimate $1/(2T_{\mathrm{eff}})\simeq 88.71\mathrm{Hz}$ explains why the exact finite-pulse width below differs from $100\mathrm{Hz}$. The slope is exact at resonance under these assumptions, while this width estimate uses the local phase expansion. A velocity distribution requires averaging the full response, not substituting one effective flight time.

Use the Driven two-level Rabi model in the ordered basis $(|e⟩,|g⟩)$, with Pauli matrices satisfying ${\sigma }_z=\mathrm{diag}(1,-1)$. 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 areThe Pauli matrix multiplication law gives $(\delta {\sigma }_z+\Omega {\sigma }_x{)}^2=q^{2}I$. Thus the exact special unitary group $\mathrm{SU}(2)$ pulse matrix for this effective Hamiltonian operator, and the flight matrix, areThe sequence is $U_p{U}_f{U}_p$. Starting in $|g⟩$, matrix multiplication and the Born rule giveThis 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 $\Omega \tau =\pi /2$, $|\delta |/\Omega \ll 1$ and $\tau \ll T$, the central fringe approachesIts 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 $|e⟩$. 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.

Write ${\omega }_z=\gamma B_z$ and use the ordered spin basis $(|\uparrow ⟩,|\downarrow ⟩)$. The free Hamiltonian operator is $H_0=-\hslash {\omega }_z{\sigma }_z/2$, soResolving the cosine into its two complex exponentials gives the exact upper off-diagonal factorwith the complex conjugate in the lower entry. Choose the field orientation with ${\omega }_z>0$, as implicit in the printed near-resonance condition, and put $\delta =\omega -{\omega }_z$. The rotating-wave approximation discards the terms at $\omega +{\omega }_z$, retaining the slow terms at $\delta$. The resulting interaction picture equations areThe factor $1/4$ contains $1/2$ from $S_x=\hslash {\sigma }_x/2$ and $1/2$ from the cosine decomposition. This is a Driven two-level Rabi model, with signed on-resonance coupling $\Omega =\gamma B_x/2$ and generalized Rabi angular frequency ${\omega }_R=\sqrt{{\delta }^2+{\Omega }^2}$. Here the Rabi detuning uses the paper's convention, drive minus free frequency.

Small detuning alone is insufficient to derive the printed equations. One also requires $|\gamma B_x|,|\delta |\ll \omega +{\omega }_z$, and an evolution time over which neglected terms stay small. At resonance this is the weak transverse-field regime $|B_x|\ll |B_z|$. The supplied solution is thus an approximate solution for the linearly oscillating field, not an exact one at arbitrary drive strength. If $\gamma B_z<0$, the other exponential is resonant and the physical positive resonance frequency is $|\gamma B_z|$.

The Rabi angular frequency is the on-resonance population oscillation rate in the specified Driven two-level Rabi model. The rotating-frame off-diagonal Hamiltonian operator entry has magnitude $\hslash \Omega /2$. Stating this convention avoids a factor-of-two ambiguity between drive coefficients and oscillation rates. The generalized rate off resonance is ${\Omega }_{\mathrm{eff}}=\sqrt{{\Omega }^2+{\delta }^2}$.

For a constant-amplitude square pulse on resonance, a Rabi pi pulse lasts ${\tau }_{\pi }=\pi /\Omega$ and gives $|g⟩\mapsto -i{e}^{-i\varphi }|e⟩$ in the rotating frame. A half-duration pulse prepares $(|g⟩-i{e}^{-i\varphi }|e⟩)/\sqrt{2}$. The quantum pulse area $\Omega t$ determines these rotations only under the stated resonant, constant-axis assumptions; a detuned drive rotates about a different axis.

As an original numerical illustration, choose $\Omega =2\pi \times 1000\mathrm{rad}{\mathrm{s}}^{-1}$ and hence ${\tau }_{\pi }=0.500\mathrm{ms}$. These are chosen model parameters, not historical apparatus settings. Holding that duration fixed givesThus $P_{\pi }(0)=1$, $P_{\pi }(1)=\frac{1}{2}{\sin }^2(\pi /\sqrt{2})\simeq 0.3166$, and $P_{\pi }(\sqrt{3})=0$. The last zero comes from completing a full rotation about the tilted axis, rather than switching the drive off. At zero Rabi detuning, doubling the pulse duration restores the ground-state population; continuous driving does not keep transferring population irreversibly.

In the Driven two-level Rabi model, the transition probability is $P(t)={\Omega }^2{\sin }^2(\sqrt{{\Omega }^2+{\delta }^2}t/2)/({\Omega }^2+{\delta }^2)$. Maximizing over time at each detuning gives a Lorentzian line shape, with first maximizing time $\pi /\sqrt{{\Omega }^2+{\delta }^2}$ and full width at half maximum $2|\Omega |$ in angular frequency. The amplitude parameter $\Omega$ must use the specified Rabi angular frequency convention.

This envelope differs from a spectrum obtained with a fixed pulse duration: optimizing the time separately removes the sine-squared factor, while fixing time retains it. A finite linewidth here describes detuning-limited transfer under coherent driving, without requiring dissipative decay.