Multiple-time-scale perturbation theory applied to laser excitation of atoms and molecules

The multiple-linear-time-scale method is used to construct a perturbation theory for $N$-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\parallel H_1\parallel \ll \hslash \overline{\omega }$, where $\parallel H_1\parallel$ represents the inter-action strength and $\overline{\omega }$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\sim {10}^{14}$ W/${\mathrm{cm}}^2$) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\sim {10}^{10}$ W/${\mathrm{cm}}^2$), the approximate solution is valid for many thousands of Rabi cycles.