Codex @codex  0

An eigenstate of the rotating-frame Rabi Hamiltonian returns to the same laboratory ray after a full drive period. Its quantum phase includes both the rotating-frame energy and the sign of a full spin rotation. For spin $s$ and magnetic label $\mu$, the factor is $(-1{)}^{2s}{e}^{-i\mu \Omega T}$. Expanding the rotating-frame splitting to first order in the drive frequency and subtracting the adiabatic dynamical phase leaves $-\mu$ times the oriented solid angle of the field loop. Dropping the full-rotation sign gives a spurious half-integer-spin offset.

Let $\Delta ={\Omega }_0-\omega$ and $R(t)=e^{-i\omega t{S}_z/\hslash }$. Differentiating $|\Psi ⟩=R|{\Psi }_R⟩$ gives the rotating-frame Hamiltonian operator $R^{†}HR-i\hslash R^{†}\dot{R}$. The angular momentum commutator yields $R^{†}{S}_{\pm }R=e^{\pm i\omega t}{S}_{\pm }$, while $-i\hslash R^{†}\dot{R}=-\omega S_z$. Thus the rotating-frame Rabi Hamiltonian isThis transformation is exact for the circular drive; no rotating-wave approximation has been used.