Coherent accumulation of excitation in the electromagnetically induced transparency of an ultrashort pulse train

We investigate the excitation of an atomic degenerate-$\Lambda$ system by a train of ultrashort pulses. An iterative analytic solution to the optical Bloch equations is found describing the temporal evolution of the density matrix elements. We show that electromagnetically induced transparency of the excitation pulses develops through the coherent accumulation of excitation (population and coherence) between the degenerate lower states. The influence of the pulse train parameters (area, repetition period, and phase between successive pulses) on establishing the transparency is investigated.

Figure 1

The model atomic system: a degenerate-$\Lambda$ system interacting with a train of ultrashort pulses of arbitrary shape and repetition period $T$. The atomic population is initially equally distributed between the lower $b$ and $c$ states, but there is no initial coherence between these states. The driving pulses couple both lower states equally to the excited state, which has a lifetime of $T_1$.

Figure 2

(Color online) (a) The lower states’ coherence term ${\rho }_{bc}$ evaluated with the iterative solution (symbols) and the numerical solution (solid lines) of the optical Bloch equations for a small number of excitation pulses. The plot of the numerical solution actually corresponds to the evolution of ${\rho }_{bc}$ as a function of time in units of the pulse repetition period $T$. (b) The coherence term ${\rho }_{bc}$ as a function of the number of excitation pulses evaluated using the iterative solution. For clarity, the results are plotted for only one out of each ten excitation pulses. In both (a) and (b), $\theta =\pi ∕50$.

Figure 3

(Color online) (a) The coherence between the lower atomic states as a function of the number of excitation pulses evaluated using the iterative solution for different pulse areas. For clarity, we show only the results for one out of each ten driving pulses; the dashed lines are a visual guide. (b) The atomic coherence ${\rho }_{bc}$ and population ${\rho }_{aa}$ in the excited state for a smaller number of pulses and $\theta =\pi ∕15$. In both figures, $T=0.10T_1$.

Figure 4

(Color online) The lower states’ coherence ${\rho }_{bc}$ as a function of the number of excitation pulses for various pulse phase difference $\Delta \phi$. Results are plotted for only one out of each ten pulses, and the dashed lines are a visual guide. We took $T=0.10T_1$ and $\theta =\pi ∕15$.

Figure 5

(Color online) Population in the bright $\left({\rho }_{BB}\right)$ and dark $\left({\rho }_{DD}\right)$ states plotted as a function of the number of driving pulses. For clarity, we plot the results for only one out of each ten driving pulses, and the dashed lines are a visual guide. Here, $\Delta \phi =0$, $T=0.10T_1$, and $\theta =\pi ∕15$.