= Solution
Insert the interaction-picture expansion into the Schrödinger equation. The <phases> multiplying the basis states already solve the unperturbed evolution, leaving
$$
i\hbar\dot c_r=\sum_s H'_{rs}\cos(\omega t)
\,e^{i(\omega_r-\omega_s)t}c_s.
$$
Hermiticity requires $H'_{21}=\hbar(\omega')^*$, not $\hbar\omega'$ unless the coupling is real. Since the diagonal <matrix elements> vanish and $\omega_0=\omega_2-\omega_1$, expanding the cosine gives
$$
\boxed{i\dot c_2=\frac{(\omega')^*}{2}c_1
[e^{i(\omega+\omega_0)t}+e^{-i(\omega-\omega_0)t}],}
$$
$$
\boxed{i\dot c_1=\frac{\omega'}{2}c_2
[e^{i(\omega-\omega_0)t}+e^{-i(\omega+\omega_0)t}].}
$$
The conjugate pair of couplings conserves $|c_1|^2+|c_2|^2$. Near resonance, the difference <frequency> is slow while the sum <frequency> is approximately $2\omega_0$. Rapid terms alternate sign before an appreciable slow transfer accumulates. The <rotating-wave approximation> retains the slow terms when $|\omega'|,|\omega-\omega_0|\ll\omega+\omega_0$; resonance alone is insufficient for a strong drive. Counter-rotating corrections include small shifts of the resonance and need not remain negligible at arbitrary strength or arbitrarily long times.
For the <spin> preparation, use $H=-\boldsymbol\mu\cdot\mathbf B=-\gamma\mathbf s\cdot\mathbf B$ and $\mathbf s=\hbar\boldsymbol\sigma/2$. Choose $B>0$ and $\gamma>0$ to fix helicity, with $|\uparrow\rangle=(1,0)^T$, $|\downarrow\rangle=(0,1)^T$ and $\omega_0=\gamma B$. The initial $|\uparrow\rangle$ is the lower state. For negative $\gamma$, reverse the RF rotation sense; <pulse areas> use $|\gamma|$, and signed <Larmor precession> determines the corresponding <phase> adjustment.
A <circularly rotating magnetic drive> uses two transverse fields in quadrature during $0<t<\tau$:
$$
\mathbf B(t)=(B_1\cos(\omega_0t+\varphi),\,-B_1\sin(\omega_0t+\varphi),\,B),
\qquad \Omega=\gamma B_1>0.
$$
With $U_0(t)=e^{+i\omega_0t\sigma_z/2}$, their interaction <Hamiltonian> is exactly constant,
$$
H_I=-\frac{\hbar\Omega}{2}(\cos\varphi\,\sigma_x-\sin\varphi\,\sigma_y).
$$
The minus sign in the second RF component selects co-rotation for the specified positive <gyromagnetic ratio>. Exponentiating this Pauli matrix gives
$$
|\psi_I(t)\rangle=\cos\frac{\Omega t}{2}|\uparrow\rangle
+i e^{-i\varphi}\sin\frac{\Omega t}{2}|\downarrow\rangle.
$$
The <laboratory phase of a resonant spin pulse> includes free precession. Returning to the laboratory frame and removing a common <phase>,
$$
\boxed{|\psi(t)\rangle\sim
\cos\frac{\Omega t}{2}|\uparrow\rangle+
i e^{-i(\varphi+\omega_0t)}\sin\frac{\Omega t}{2}|\downarrow\rangle.}
$$
Unlike a linearly polarized RF field, this exactly co-rotating field has no counter-rotating component. A single linearly polarized field $B_{x,\rm peak}\cos(\omega_0t+\varphi)$ gives the same slow rotation in the weak-drive <rotating-wave approximation> with \b[$\Omega=|\gamma|B_{x,\rm peak}/2$]. For the circular field above, \b[$\Omega=|\gamma|B_1$]. Keeping these amplitude conventions distinct avoids a factor-of-two pulse-duration error.
\Image[/past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2015/paper-1-spin-pulses.png]
{title=Resonant rotations in rotating axes and the maximum transfer permitted by a fixed detuned pulse}
For a single harmonic pulse detuned by $\Delta=\omega_0-\omega$, the effective rotating-frame field tilts out of the transverse plane. Its rotation rate becomes $\sqrt{\Omega^2+\Delta^2}$, and its maximum population transfer is $\Omega^2/(\Omega^2+\Delta^2)$. \b[Exact inversion to $-z$ is impossible with one fixed nonzero-detuning pulse]. Half transfer, needed for a transverse <eigenstate>, gives the <equatorial reachability of a detuned spin pulse> criterion \b[$|\Delta|\leq\Omega$]. With that condition, changing the duration and RF <phase> can still prepare $+x$ or $+y$ at the chosen laboratory stopping time. The mere statement $\omega\approx\omega_0$ does not ensure this condition, because the drive can be much weaker than either <frequency>. If $|\Delta|>\Omega$, neither transverse target is reached by that single pulse. These restrictions concern fixed-frequency, fixed-amplitude single-pulse control; retuning, chirping or phase-controlled composite pulses change the available controls and can remove the restriction. No dissipative <spin relaxation> is included in this coherent two-state model.
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