Temporal dissipative solitons in a three-level atomic medium confined in a photonic-band-gap fiber

We obtained a propagation equation for an optical pulse at an electromagnetically induced transparency window guided on a gas-filled hollow-core photonic crystal fiber. This equation admits dissipative solitons whose analytical expression was also obtained. Depending on the parameter region, they may be stable or unstable. We simulated a typical experimental arrangement and found some cases for which the equation parameters are such that it admits stable solitons.

Figure 1

Energy-level diagram and excitation scheme of the three-level $\Lambda$ system considered.

Figure 2

Curve of ${\sigma }_s$ defined by Eq. (19) when $Ds=-1$ (blue solid line) and $Ds=+1$ (red dashed line). The existence of solutions requires that $\sigma$ is above ${\sigma }_s$ when $\alpha +\frac{{\xi }^2}{4\nu }<0$ or below ${\sigma }_s$ if $\alpha +\frac{{\xi }^2}{4\nu }>0$.

Figure 3

The real part of the propagation constant for both fibers and two peak values of the control Rabi frequency (in ${\mathrm{s}}^{-1}$). The population decay and dephasing rates are (a) all equal to $6.28\times {10}^7$ ${\mathrm{s}}^{-1}$ (case 1) and (b) ${\gamma }_{13}={\gamma }_{23}=18.1\times {10}^6$ ${\mathrm{s}}^{-1},{\gamma }_{33}=36.1\times {10}^6$ ${\mathrm{s}}^{-1}$ (case 2). Circles correspond to values determined numerically, and the lines correspond to fourth-degree polynomial fittings.

Figure 4

The imaginary part of the propagation constant for fiber 2 and two peak values of the control Rabi frequency (in ${\mathrm{s}}^{-1}$). The population decay and dephasing rates are (a) all equal to $6.28\times {10}^7$ ${\mathrm{s}}^{-1}$ (case 1) and (b) ${\gamma }_{13}={\gamma }_{23}=18.1\times {10}^6$ ${\mathrm{s}}^{-1},{\gamma }_{33}=36.1\times {10}^6$ ${\mathrm{s}}^{-1}$ (case 2). Circles correspond to values determined numerically, and the lines correspond to fourth-degree polynomial fittings.

Figure 5

Transverse distribution of the fundamental mode for the probe field at fiber 2, with $\Delta =0$ and ${\Omega }_c^p=7\times {10}^{11}$ ${\mathrm{s}}^{-1}$.

Figure 6

Dependence of $\sigma$ with $\Delta$ as calculated using $\sigma =-\mathrm{Im}\left[G\left({\Omega }_c^{\mathrm{eff}}\right)\right]/\left|\mathrm{Re}\left[G\left({\Omega }_c^{\mathrm{eff}}\right)\right]\right|$ for (a) case 1 using ${\Omega }_c^{\mathrm{eff}}=7\times {10}^{11}{s}^{-1}$ and (b) case 2 using ${\Omega }_c^{\mathrm{eff}}=7\times {10}^{10}{s}^{-1}$.

Figure 7

Soliton evolution as simulated using Eq. (12) with $D=1,s=1,\alpha =0.0246,\xi =-0.0011,\nu =3.5\times {10}^{-4}$, and $\sigma =-91.5$ up to $Z=40$.