Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 6 / c / Solution

Subtract the common sector energy and call the remaining eigenvalue $\hslash \lambda$. Its characteristic equation isThe dressed-state energies are consequentlyOn resonance their splitting is $2\hslash |g|\sqrt{n+1}$; far off resonance the eigenstates approach the uncoupled quantum states with small dispersive energy shifts. The eigenvalue splitting is an angular-frequency difference ${\Omega }_n$, so an ordinary frequency would be ${\Omega }_n/(2\pi )$.