Spectral and temporal modifications of a weak resonant ultrashort pulse propagating in a two-level system driven by a strong nonresonant field

An ultrashort strong nonresonant pulse induces transient phenomena in an optically thick medium consisting of an ensemble of nondegenerate two-level atoms (nonadiabatic transitions, transient light-shifts,…). A weak resonant pulse propagates in the system and we study theoretically and numerically the temporal and spectral modifications that are induced. Moreover, we show that the atomic media, modified by a strong pulse can act as a “pulse-shaper” device that can modify, in a controlled way, the shape of the weak pulse. It maps out the transient light shift induced by the strong pulse and reduces efficiently the long time dispersion tail of the weak pulse. We study and discuss the physical phenomena that allow such shaping effects and the necessary conditions to satisfy in order to observe them.

Figure 1

Incident (dotted line) and transmitted (solid line) intensity spectrum of the driving pulse. Each frequency is created at two times giving rise to interference effects. The parameters are ${\theta }_d=60$, $\Delta =10$, $e_{\mathit{disp}}=1$. The position of resonance is marked.

Figure 2

Adiabatic representation: each stationary state can be decomposed as a linear superposition of the two adiabatic states with time dependent energies. Three resonance frequencies appear at $W_d$ (solid arrows), $W_d+\Omega$ (dashed arrow), $W_d-\Omega$ (dotted arrow).

Figure 3

Intensity spectrum of the field emitted in the direction of the propagating pulse in (a) collinear and (b) noncollinear cases. Parameters are ${\theta }_d=60$, ${\theta }_p=0.01$, $\Delta =10$, ${\tau }_{\mathit{pd}}=2$, $e_{\mathit{disp}}=0.5$. In both cases dashed and solid lines correspond to $\beta =0$ and $\pi$, respectively whereas the incident spectrum is represented in dotted line. In (b), the curves for $\beta =0$ and $\pi$ are indistinguishable.

Figure 4

Time dependence (at the entrance of the medium) of the coherence amplitude $\mid {\rho }_p\mid$ (solid line) and the dominant path contribution $\mid K_{14}\mid$ (dashed line) corresponding to the transition $1\to 4$ of Fig. 2. Expression of $\mid K_{14}\mid$ is given by Eq. (22). Here, ${\theta }_d=60$, ${\theta }_p=0.01$, $\Delta =10$, $\beta =0$. The situations (a), (b), (c) correspond to the case ${\tau }_{\mathit{pd}}=2$, 1, and 0.5, respectively. When the propagating pulse is alone, the asymptotic value for $\mid {\rho }_p\mid$ is $5\times {10}^{-3}$. Finally, the difference between ${\rho }_p$ and $K_{14}$ becomes more important when the propagating pulse becomes narrower than the driving pulse. We represent in (d), the Gaussian pulse envelope of the propagating pulse in solid line (the same envelope than that of the driving pulse when ${\tau }_{\mathit{pd}}=1$), the driving pulse envelope when ${\tau }_{\mathit{pd}}=2$ (dashed line) and ${\tau }_{\mathit{pd}}=0.5$ (dotted line).

Figure 5

Intensity envelope of the propagating pulse at the entrance of the medium (dotted line), and at the exit, in the absence (dashed line) and the presence (solid line) of the driving pulse. The situations (a), (b), (c) correspond to the case ${\tau }_{\mathit{pd}}=2$, 1, and 0.5, respectively with $e_{\mathit{disp}}=1∕{\tau }_{\mathit{pd}}$. The others parameters are the same as in Fig. 4.

Figure 6

Amplitude spectrum of the transmitted propagating pulse for the cases ${\tau }_{\mathit{pd}}=2$ (solid line), ${\tau }_{\mathit{pd}}=1$ (dashed line), ${\tau }_{\mathit{pd}}=0.5$ (dotted line), and $e_{\mathit{disp}}=1∕{\tau }_{\mathit{pd}}$. The generated frequencies are more pronounced for ${\tau }_{\mathit{pd}}=2$ than the other cases. The frequency cut is the same for the three cases as shown in the inset. The others parameters are the same as in Fig. 4.

Figure 7

Intensity envelope of the transmitted propagating pulse for $e_{\mathit{disp}}=0.5$ (solid line), $e_{\mathit{disp}}=1$ (dashed line), $e_{\mathit{disp}}=20$ (dotted line). Here, ${\tau }_{\mathit{pd}}=2$ and the other parameters are the same as in Fig. 4.

Figure 8

Spatial configuration taking into account for the finite spatial extension of the beams.