= Driven two-level Rabi model
{c}
{title2=$H_{\mathrm{rot}}=\hbar[\delta\sigma_z+\Omega(\cos\phi\,\sigma_x+\sin\phi\,\sigma_y)]/2$}
Choose an isolated pair of <energy levels>, with <ground state> $|g\rangle=(0,1)^T$ and <excited state> $|e\rangle=(1,0)^T$. Use the <Pauli matrices> in the ordered <basis> $(|e\rangle,|g\rangle)$, so $\sigma_z=\operatorname{diag}(1,-1)$. After subtracting the mean <energy>, prescribe the laboratory <Hamiltonian operator>
$$
H_{\mathrm{lab}}(t)=\frac{\hbar\omega_0}{2}\sigma_z
+\hbar\Omega\cos(\omega t+\phi)\sigma_x,
\qquad \delta=\omega_0-\omega,\qquad \Omega\geq0.
$$
The unperturbed <energy> gap is $\hbar\omega_0$, the drive has <angular frequency> $\omega$, and $\phi$ fixes its <oscillation phase> at the chosen time origin. All of $\omega_0$, $\omega$, the <Rabi angular frequency> $\Omega$ and the <Rabi detuning> $\delta$ are in radians per second; ordinary <frequency> is angular frequency divided by $2\pi$. In particular, $\Omega$ here is the coefficient multiplying the cosine in $H_{\mathrm{lab}}/\hbar$, not twice that coefficient.
Write $|\psi_{\mathrm{lab}}\rangle=R(t)|\psi_{\mathrm{rot}}\rangle$ with $R(t)=\exp(-i\omega t\sigma_z/2)$. Transforming the <Schrödinger equation> gives $H_{\mathrm{rot}}=R^\dagger H_{\mathrm{lab}}R-i\hbar R^\dagger\dot R$. The <rotating-wave approximation> drops terms at angular frequency $2\omega$, leaving
$$
H_{\mathrm{rot}}=\frac{\hbar}{2}K,\qquad
K=\begin{pmatrix}\delta&\Omega e^{-i\phi}\\\Omega e^{i\phi}&-\delta\end{pmatrix},\qquad
\Omega_{\mathrm{eff}}=\sqrt{\Omega^2+\delta^2}.
$$
This approximation requires $\Omega,|\delta|\ll\omega_0,\omega$, and an interaction duration over which omitted corrections remain small. It also assumes that other <energy levels> can be neglected. Resonance alone does not justify either assumption. An exactly co-rotating drive has no discarded counter-rotating component; the linearly oscillating laboratory drive above does.
The <Pauli matrix multiplication law> gives $K^2=\Omega_{\mathrm{eff}}^2I$. Separating even and odd terms of the <matrix exponential> therefore proves
$$
U_{\mathrm{rot}}(t)=I\cos\frac{\Omega_{\mathrm{eff}}t}{2}
-i\frac{K}{\Omega_{\mathrm{eff}}}\sin\frac{\Omega_{\mathrm{eff}}t}{2},
$$
with the continuous limit $U=I$ when $\Omega=\delta=0$. Starting from $|g\rangle$, the upper <probability amplitude> is $-ie^{-i\phi}(\Omega/\Omega_{\mathrm{eff}})\sin(\Omega_{\mathrm{eff}}t/2)$. The <Born rule> gives
$$
P_{g\to e}(t)=\frac{\Omega^2}{\Omega^2+\delta^2}
\sin^2\!\left(\frac{\sqrt{\Omega^2+\delta^2}\,t}{2}\right).
$$
The rotating transformation only changes the <quantum phases> of these <basis> states, so their laboratory populations are the same. <Rabi detuning> tilts the rotation axis of the <Bloch vector>: it increases the oscillation rate to $\Omega_{\mathrm{eff}}$ but caps the transferred population at $\Omega^2/(\Omega^2+\delta^2)$. A fixed drive <oscillation phase> cancels from this single-pulse population when the initial state is $|g\rangle$. It still fixes the relative <quantum phase> of a resulting <quantum superposition> and matters when another pulse supplies a phase reference.
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