Stationary and quasistationary light pulses in three-level cold atomic systems

We have studied stationary and quasistationary signal light pulses in cold Λ-type atomic media driven by counterpropagating control laser fields at the condition of electromagnetically induced transparency. By deriving a dispersion relation we present spectral and temporal properties of the signal light pulse and a significant influence of atomic decoherence on the coupled stationary light pulses for spatial splitting. Finally we discuss quasistationary light pulse evolution characterized by frozen spatial spreading for a robust coherent control of slow light pulses.

Figure 1

A schematic of atomic levels for resonant interactions with weak probe fields $E_p^{\pm }$ and two control fields ${\Omega }_c^{\pm }$; ${\gamma }_2$ and ${\gamma }_3$ are the decay constants of the atomic coherences ${\widehat{P}}_{12}^j\left(t,z_j\right)$ and ${\widehat{P}}_{13}^j\left(t,z_j\right)$; $\Delta$ is a spectral detuning from the optical transition.

Figure 2

Dispersion: $(\mathrm{c}/{\gamma }_3)\mathrm{Re}\left[K\left(\omega \right)\right]$ (red solid line) and absorption $(\mathrm{c}/{\gamma }_3)\mathrm{Im}\left[K\left(\omega \right)\right]$ (green dashed line). The curves are presented in ${\gamma }_3$: ${\Omega }_{+}={\Omega }_{-}={\gamma }_3$, ${\gamma }_2\to 0$, $\Delta \to 0$.

Figure 3

EIT dispersion $(\mathrm{c}/{\gamma }_3)\mathrm{Re}\left[K\left(\omega \right)\right]$ for traveling (red solid line, ${\Omega }_{+}\ne 0,{\Omega }_{-}=0$) and standing (blue dashed line, ${\Omega }_{+}={\Omega }_{-}$) control light fields. Other parameters are the same as in Fig. 2: ${\Omega }_{+}={\Omega }_{-}={\gamma }_3$, ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$.

Figure 4

EIT absorption $(\mathrm{c}/{\gamma }_3)\mathrm{Im}\left[K\left(\omega \right)\right]$ for standing (red solid line, ${\Omega }_{+}={\Omega }_{-}$) and traveling (blue dashed line, ${\Omega }_{+}\ne 0,{\Omega }_{-}=0$) control light fields; other parameters are the same as in Fig. 2: ${\Omega }_{+}={\Omega }_{-}={\gamma }_3$, ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$.

Figure 5

Spectral properties of group velocity for standing control laser field ${\Omega }_{+}={\Omega }_{-}$, other parameters are the same as in Figs. 2–4: ${\Omega }_{\pm }={\gamma }_3$, ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$.

Figure 6

Group velocities for the standing (${\Omega }_{-}={\Omega }_{+}$, red solid line) and traveling (${\Omega }_{-}=0$, ${\Omega }_{+}\ne 0$, blue dashed line) control laser fields in cold atomic systems. Other parameters are the same as in Figs. 2–4: ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$.

Figure 7

Group velocities for different amplitudes of standing control laser field (${\Omega }_{\pm }=0.6{\gamma }_3$, green solid line), (${\Omega }_{\pm }={\gamma }_3$, red dashed dotted line), (${\Omega }_{\pm }=1.4{\gamma }_3$, blue dashed line), ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$. Other parameters are the same as in Figs. 2–4.

Figure 8

Spectral properties of the absorption for different amplitudes of standing control laser field. (${\Omega }_{\pm }=0.6{\gamma }_3$, green solid line), (${\Omega }_{\pm }={\gamma }_3$, red dashed dotted line), (${\Omega }_{\pm }=1.4{\gamma }_3$, blue dashed line), ${\gamma }_2\to 0$, $\Delta \to 0$, ${\gamma }_3=1$. Other parameters are the same as in Figs. 2–4.

Figure 9

Spatial profile of SLP mode $A_{+}\left(t,z\right)$ for different moments of time $t$ = 0 (red solid line), $t$ = 0.2 (green dashed line), $t$ = 0.4 (blue dashed dotted line). Time is denoted by $1/{\gamma }_3$; spectral width $\sigma ={\gamma }_3$.

Figure 10

Spatial profile of SLP mode $A_{+}\left(t,z\right)$ for $t$ = 0 and $t$ = 0.4 for three spectral widths ($\sigma =1/\sqrt{2}$, red dashed line), ($\sigma =1$, blue dashed dotted line), ($\sigma =1/2$, green solid line) in units of ${\gamma }_3$.

Figure 11

Spectral properties of group velocity $v_{gr}\left(\omega \right)$ for SLP mode $A_{+}\left(t,z\right)$ at different relaxation rates (${\gamma }_2=0$, red solid line), (${\gamma }_2=0.02$, green dashed line), (${\gamma }_2=0.04$, blue dashed dotted line) in units of ${\gamma }_3$; large atomic relaxation excludes any formation of SLP even in narrow spectral range around $\omega =0$; other parameters are the same as in Figs. 2–4: ${\Omega }_{\pm }={\gamma }_3$, $\Delta \to 0$.

Figure 12

Spectral properties of group velocity in particular cases: SLP, ${\Omega }_{-}={\Omega }_{+}$ (red solid line); usual slow light ${\Omega }_{-}=0$, ${\Omega }_{+}\ne 0$ (blue dashed dotted line); quasistationary light ${\Omega }_{-}=0.66{\Omega }_{+}$ (green dashed line).

Figure 13

Spectral properties of absorption $(\mathrm{c}/{\gamma }_3)\mathrm{Im}\left[K\left(\omega \right)\right]$ for SLP field ${\Omega }_{-}={\Omega }_{+}$ (red dashed dotted line), usual EIT slow light ${\Omega }_{-}=0$, ${\Omega }_{+}\ne 0$ (blue solid line) and quasistationary light ${\Omega }_{-}/{\Omega }_{+}=0.66$ (green dashed line).

Figure 14

Relation between the two control Rabi frequencies ${\Omega }_{-}$ and ${\Omega }_{+}$ providing the same group velocity of quasistationary light pulse.