Inverse Doppler shift and control field as coherence generators for the stability in superluminal light

A gain-based four-level atomic medium for the stability in superluminal light propagation using control field and inverse Doppler shift as coherence generators is studied. In regimes of weak and strong control field, a broadband and multiple controllable transparency windows are, respectively, identified with significantly enhanced group indices. The observed Doppler effect for the class of high atomic velocity of the medium is counterintuitive in comparison to the effect of the class of low atomic velocity. The intensity of each of the two pump fields is kept less than the optimum limit reported in [M. D. Stenner and D. J. Gauthier, Phys. Rev. A 67, 063801 (2003)] for stability in the superluminal light pulse. Consequently, superluminal stable domains with the generated coherence are explored.

Figure 1

Schematics of the atomic system.

Figure 2

$\mathrm{Im}\left[\chi \right],\mathrm{Re}\left[\chi \right]$ vs $\frac{\Delta }{\Gamma }$ with $\lambda =586.9nm,{\omega }_{ac}={10}^3\Gamma ,{\delta }_3=0,{\delta }_2=50\Gamma ,{\delta }_1=30\Gamma$. The radiative decay rates are selected as $\Gamma =1\mathrm{MHz},{\Gamma }_{ad}={\Gamma }_{ac}={\Gamma }_{bd}={\Gamma }_{bc}=2.05\Gamma$ whereas ${\Gamma }_{cd}={\Gamma }_{ab}=0.5\Gamma$ represents the decay rates due to atomic collisions. Also, ${\Omega }_1=2.5\Gamma ,{\Omega }_2=2.55\Gamma ,{\Omega }_3=2\Gamma$. Further, in (a) and (b) [(c) and (d)] $V_D=0$ $\left(0\right)$ (red), $4\Gamma$ $\left(2\Gamma \right)$ (red dashed), $6\Gamma$ $\left(3\Gamma \right)$ (blue), $9\Gamma$ $\left(4\Gamma \right)$ (blue dashed), $12\Gamma$ $\left(5\Gamma \right)$ (black). Note that in this figure and in what follows $\frac{\Delta }{\Gamma }$ is shown as multiple of 10. The insets in (c) and (d) are, respectively, the gain and dispersion around $\frac{\Delta }{\Gamma }=50$.

Figure 3

$N_g$ vs $\frac{\Delta }{\Gamma }$ for average $v$ (a) $1.5\times {10}^3$ m/s (b) $5\times {10}^3$ m/s. The $V_D$ for (a) and (b) are chosen from Fig. 2, correspondingly. However, vs $V_D$ (c) $v$ for the classes of $5\times {10}^3$ m/s (solid), $10\times {10}^3$ m/s (dotted) and for the class of $15\times {10}^3$ m/s (dot-dashed).

Figure 4

In (a) $\mathrm{Im}\left[\chi \right]$, in (b) $\mathrm{Re}\left[\chi \right]$ and in (c) $N_g$ vs $\frac{\Delta }{\Gamma }$ with ${\Omega }_1=2.5\Gamma ,{\Omega }_2=4.3\Gamma$ and with ${\Omega }_3=6\Gamma$ (red), $10\Gamma$ (red dashed), $15\Gamma$ (blue dashed), $20.5\Gamma$ (blue). The other parameters are chosen from Fig. 2. In (d) $N_g$ vs $V_D$ with ${\Omega }_3=6\Gamma$ (red) for average $v$ of $5\times {10}^3$ m/s and $\Delta =40\Gamma ,30\Gamma$ (blue, blue dot-dashed), $15\times {10}^3$ m/s and $\Delta =40\Gamma ,30\Gamma$ (red, red dashed) and for average $v$ of $500\times {10}^3$ m/s and $\Delta =40\Gamma ,30\Gamma$ (black, black dotted).

Figure 5

The intensity of a Gaussian pulse at input and output versus the normalized time for (a) $\xi =0$, (b) $\xi =0.005\mathrm{MHz}$, (c) $\xi =0.01\mathrm{MHz},{\tau }_0=3.50\mu s,L=0.03m,{\omega }_0=970\Gamma ,V_D=6\Gamma ,{\Omega }_1=2.5\Gamma ,{\Omega }_2=4.3\Gamma ,{\Omega }_3=8\Gamma$. The values of the other parameters are the same as in Fig. 2