Influence of propagation on the coherent accumulation of excitation induced by an ultrashort pulse train

We study the effects of propagation through an extended sample of two- and three-level atoms on the coherent accumulation of excitation by an ultrashort pulse train. In the two-level case, previous pulses in the train may prepare the medium such that a later pulse may experience amplification and absorption in different positions inside the sample. For a large number of pulses, the atomic medium may be saturated by the train, and some pulses propagate without experiencing either absorption or amplification. In general, absorption of the resonant pulse frequency during propagation compromises the accumulation efficiency. In the three-level case, the coherent accumulation of excitation leads to electromagnetically induced transparency of the pulse train, and the pulses exit the medium without any distortion in their temporal profile.

Figure 1

A train of ultrashort pulses propagating through an extended collection of two-level atoms. $\mu z$ is the optical depth. At the input $\left(\mu z=0\right)$, the pulses are Gaussians and are separated in time by $T$. At the output, the pulses are distorted and may exhibit oscillations in time. Only one output pulse is shown, and the time scale is expanded by several orders of magnitude with respect to that at the input. The inset shows the atomic levels: $b$ is the ground state and $a$ is the excited state, which has a lifetime of $T_1$.

Figure 2

(Color online) Atomic population inversion as a function of the number of excitation pulses in the train at an optical depth $\mu z$ equal to (a) 0, (b) 58, (c) 114, and (d) $226{\left(\text{ns}\right)}^{-1}$.

Figure 3

(Color online) Spatial evolution of three pulses in the train: $N=1$, 15, and 300. The figure shows the pulses’ spectral amplitude (in arbitrary units) as a function of optical depth. The side panels show the population inversion along the medium. The pulse’s spectral frequencies are calculated relative to the atomic resonance.

Figure 4

The model three-level atomic system: a degenerate $\Lambda$ system interacting with a train of ultrashort pulses with 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 5

(Color online) Coherence between the lower atomic states as a function of the pulse number for three different positions along the medium: $\mu z=0$ (dotted black line), $\mu z=3.2{\left(\text{ps}\right)}^{-1}$ (dashed blue line), and $\mu z=6.4{\left(\text{ps}\right)}^{-1}$ (solid red line).

Figure 6

(Color online) Spatial evolution of two pulses in the train: $N=1$ (left column) and $N=2400$ (right column). The figure shows the pulses’ spectral and temporal amplitudes (in arbitrary units) as functions of optical depth. The pulse’s spectral frequencies are calculated relative to the atomic transition. The side panels, in the top plots, show the ground-state coherence ${\rho }_{bc}$ along the medium.

Figure 7

(Color online) Pulse temporal profile at $\mu z=0$ (dotted black line) and $\mu z=6.4{\left(\text{ps}\right)}^{-1}$ for pulses $N=1$ (dashed blue line) and $N=2400$ (solid red line).