Pulse splitting in light propagation through $N$-type atomic media due to an interplay of Kerr nonlinearity and group-velocity dispersion

We investigate the spatiotemporal evolution of a Gaussian probe pulse propagating through a four-level $N$-type atomic medium. At two-photon resonance of probe and control fields, weaker probe pulses may propagate through the medium with low absorption and pulse shape distortion. In contrast, we find that increasing the probe pulse intensity leads to a splitting of the initially Gaussian pulse into a sequence of subpulses in the time domain. The number of subpulses arising throughout the propagation can be controlled via a suitable choice of the probe and control field parameters. Employing a simple theoretical model for the nonlinear pulse propagation, we conclude that the splitting occurs due to an interplay of Kerr nonlinearity and group-velocity dispersion.

Figure 1

(a) A block diagram of the system of interest. We assume that the control field intensity is constant along the propagation direction. (b) Schematic representation of four-level atoms in $N$-type configuration. We consider two $J=1/2$ Zeeman manifolds with excited states $|1\rangle =|m_e=-1/2\rangle$ and $|2\rangle =|m_e=1/2\rangle$ and ground states $|3\rangle =|m_g=-1/2\rangle$ and $|4\rangle =|m_g=1/2\rangle$. The ${\sigma }^{-}$-polarized control field ${\mathit{E}}_c$ couples to the transition $|1\rangle \leftrightarrow |4\rangle$, and the $\pi$-polarized probe field ${\mathit{E}}_p$ interacts with the $|1\rangle \to |3\rangle$ and $|2\rangle \to |4\rangle$ transitions.

Figure 2

Intensity $|{\mathrm{\Omega }}_p{/\gamma |}^2$ of a weak probe pulse as a function of position and time for different detuning conditions. The parameters in (a) and (b) are ${\mathrm{\Omega }}_c=\gamma ,{\mathrm{\Omega }}_p=\sqrt{0.015}\gamma ,{\sigma }_p=30/\gamma$, and $\mathrm{\Gamma }=0$.

Figure 3

Normalized probe pulse intensities $I/I_{\text{max}}$ at the medium exit at $\eta \zeta /\gamma =200$. Parameters are as in Fig. 2. To facilitate the analysis of the pulse widths, all pulses are shown normalized to their respective peak values.

Figure 4

(a) Intensity $|{\mathrm{\Omega }}_p{/\gamma |}^2$ of a stronger probe pulse as function of the propagation distance. Parameters are $\mathrm{\Delta }={\mathrm{\Delta }}_{P^{\prime }}=4\gamma$ and ${\mathrm{\Omega }}_p=\sqrt{0.15}\gamma$; all other parameters are chosen as in Fig. 2. (b) shows the corresponding probe pulse power spectra obtained by Fourier transformation as function of the propagation distance.

Figure 5

Propagation dynamics of a probe pulse with ${\mathrm{\Omega }}_p=\sqrt{0.5}\gamma$ and $\mathrm{\Delta }={\mathrm{\Delta }}_{P^{\prime }}=4\gamma$. All other parameters are chosen as in Fig. 2.

Figure 6

(a) Real and imaginary parts of the linear susceptibility $\chi$ in Eq. (A12) as a function of the Fourier component $\omega$ of the probe pulse envelope. We have ${\beta }_0=\chi \left(0\right)=0$, and ${\beta }_n$ is the $n\mathrm{th}$ derivative of $\chi$ at $\omega =0$, see Appendix pp1. The real and imaginary parts of ${\beta }_2 \left[{\beta }_3\right]$ are shown in (b) [(c)]. (d) Real and imaginary parts of $R_p$ as a function of the Kerr detuning ${\mathrm{\Delta }}_{p^{{}^{\prime }}}$ for $\mathrm{\Delta }=4\gamma$. In (a)–(d), blue solid (red dashed) lines show real (imaginary) parts and ${\mathrm{\Omega }}_c=\gamma$.

Figure 7

Comparison between the full numerical solution to Eqs. (6) and (8) (green solid line) and the simple model in Eq. (14) (blue dot-dashed line). The parameters in (a) are ${\mathrm{\Omega }}_p=0.025\gamma ,{\sigma }_p=50/\gamma ,\mathrm{\Delta }=\gamma$ and ${\mathrm{\Delta }}_{P^{\prime }}=3\gamma$. Results in (b) correspond to ${\mathrm{\Omega }}_p=\sqrt{0.15}\gamma ,{\sigma }_p=30/\gamma$, and $\mathrm{\Delta }={\mathrm{\Delta }}_{P^{\prime }}=4\gamma$.