Influence of wave-packet dynamics on the medium gain of an atomic system

A sequence of two femtosecond pulses—a strong driving $\pi$-polarized pulse and a weak propagating $\sigma$-polarized pulse—excites resonantly the $S_{1∕2}\to P_{1∕2}$ transition of an atomic system. Strong interference effects take place in the system between absorption and emission paths leading to a substantial amplification of the $\sigma$ pulse. We study the influence of the fine structure on the medium gain when the contribution of the off-resonant $P_{3∕2}$ level is taken into account. A drastic reduction of the medium gain is obtained. This effect is explained within the bright-state–dark-state formalism where the strong driving pulse creates a wave packet that can be trapped in a state—the bright state—leading to a significant reduction of the gain for the $\sigma$ pulse. Finally, we also show that periodical gain dependence with the driving pulse energy exhibits a significant change in its period value (compared with expected Rabi oscillations).

Figure 1

Energy levels and optical transitions involved in our system. (a) Bare state representation. (b) Bright-state–dark-state representation. See text for notations.

Figure 2

Output energy for the $\sigma$ pulse (normalized to the initial energy) as a function of the driving pulse area ${\theta }_1$ for the set of parameters $e_{\mathit{disp}}=0.11$, ${\theta }_2=0.1\pi$, $\Delta =45\mathrm{THz}$, and ${\tau }_0=90\mathrm{fs}$. Solid cure: without $P_{3∕2}$ contribution, dashed curve: with $P_{3∕2}$ contribution.

Figure 3

Population in the bright state $\mid B⟩$ (dashed line), dark state $\mid D⟩$ (solid line), and the total (dotted line) for ${\theta }_1=\pi$, ${\theta }_2=0.1\pi$, $\Delta =45\mathrm{THz}$, and ${\tau }_0=90\mathrm{fs}$. The envelope in dash-dotted line represents the Gaussian envelope of the driving pulse. Initially, the atoms are in state ∣1⟩.

Figure 4

Population in the dark state $\mid D⟩$ for (a) ${\theta }_1=\pi$, (b) ${\theta }_1=3.2\pi$, (c) ${\theta }_1=7.9\pi$, corresponding respectively to the first, second, and fourth maxima of the transmitted energy (curve in dashed line, Fig. 2). The envelope in dotted line represents the Gaussian envelope of the driving pulse. The other parameters are ${\theta }_2=0.1\pi$, $\Delta =45\mathrm{THz}$, and ${\tau }_0=90\mathrm{fs}$. Initially, the atoms are in state ∣1⟩. The population transfer to the dark state is reduced during the action of the driving pulse.