= Laboratory spin motion under a resonant linear drive
{title2=$\mathbf s=(\sin\Omega t\sin\omega_zt,\sin\Omega t\cos\omega_zt,\cos\Omega t)$}
For $H=-\gamma B_zS_z-\gamma B_xS_x\cos(\omega t)$, choose $\omega_z=\gamma B_z>0$ and nominal resonance $\omega=\omega_z$. Starting from up spin, the <rotating-wave approximation> gives the normalized <Bloch vector> in the title, with signed $\Omega=\gamma B_x/2$. It follows by transforming the spinor $(\cos(\Omega t/2),i\sin(\Omega t/2))^T$ from the <interaction picture> back to the <Schrödinger picture>. Thus weak transverse driving combines slow meridional <Rabi oscillations> with fast <Larmor precession>.
This result requires $|B_x|\ll|B_z|$ and cannot be extrapolated as a controlled strong-drive formula. The exact equation is $\dot{\mathbf s}=\gamma\mathbf s\times(B_x\cos\omega t,0,B_z)$. For $|\gamma B_z|t\ll1$, neglecting the bias gives the controlled leading solution
$$
\mathbf s\simeq(0,\sin\Theta,\cos\Theta),\qquad
\Theta=\frac{\gamma B_x}{\omega}\sin\omega t.
$$
The fixed-axis driving generators commute at all times, making this solution exact when the bias is zero at any prescribed drive frequency. At nominal resonance in a strong-drive experiment, the initial rotation scale is $\Theta\simeq\gamma B_xt$, twice the incorrectly extrapolated rotating-wave rate. Bias effects must be retained on longer times.
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