Past exam of the atomic and laser physics course of the University of Oxford / 2021 / 2 / b / Solution

Assume an initially unexcited chemical atom, two nonoverlapping square pulses starting at $0,T$, and a common coherent field phase. In the weak-excitation limit $|\Omega |{\tau }_p\ll 1$, substitute $c_1=1+O({\Omega }^2)$ into the second probability amplitude equation. After the second pulse,where the sinc function means $\mathrm{sinc}u=\sin u/u$, with value one at zero. Squaring givesThe printed prefactor is a factor of four too small for the supplied equations and pulse durations. A direct counterexample is $\delta =0$: the exact resonant Rabi oscillation gives $P_2={\sin }^2(\Omega {\tau }_p)$ for two pulses of duration ${\tau }_p$, because the intervening resonant rotating-frame evolution is trivial. Its small-area limit is $(\Omega {\tau }_p{)}^2$, rather than $(\Omega {\tau }_p/2{)}^2$. The latter is the single-pulse result before coherent addition.

The condition $\omega {\tau }_p\ll 1$ printed in the source also does not justify the derivation from a physical carrier field. A consistent rotating-wave approximation pulse regime isIf the supplied equations are treated as the mathematical model regardless, their weak-excitation integral remains valid for any ${\tau }_p$; subcycle pulses of the original real field require the discarded counter-rotating term and depend on carrier phase. $T$ here is the separation of pulse starts, as specified, rather than the dark interval $T-{\tau }_p$.