Temporal optical memory based on coherent population and two-photon coherence oscillations

We consider the $F_g=1\to F_e=1$ transition between the ground and excited hyperfine levels in alkali-metal vapor interacting with $\sigma$ linearly polarized control and probe fields whose polarizations can be either parallel or perpendicular to each other. We develop a matrix formulation that allows a solution of the Bloch equations to all orders in the pump and probe Rabi frequencies. Using this formalism, we calculate the steady-state probe absorption spectrum, the coherent population oscillations (CPOs), and two-photon coherence, in the absence (degenerate case) and presence (nondegenerate case) of a longitudinal magnetic field. We then calculate the probe storage when the pump is switched off and on again. We are particularly interested in whether the probe regains its original temporal shape when the pump is switched on again for the case of identical pump and probe frequencies. We show that, in the nondegenerate case, the restored probe does not regain its original shape whereas, in the degenerate case, the original shape is restored. This can be explained by considering the relative magnitudes of the CPOs which do not remember the temporal shape of the probe and the two-photon coherence oscillations which store the probe shape when the pump is switched off, as in electromagnetically induced transparency memories. In the nondegenerate case, the CPOs are much stronger than the two-photon coherence oscillations whereas, in the degenerate case, they are of similar magnitudes. Thus it is the two-photon coherence oscillations that are responsible for the restoration of the temporal shape of the probe in the degenerate case.

Figure 1

Energy-level scheme for $F_g=1\to F_e=1$ transition in the presence of longitudinal magnetic field interacting with $\sigma$ polarized control (red) and probe (green) fields. The control frequency ${\omega }_c$ is resonant with the $F_g=1,m_g=0\to F_e=1,m_e=0$ transition and the probe can have a polarization which is parallel or perpendicular to that of the control.

Figure 2

Probe absorption spectrum as a function of pump-probe detuning for $F_g=1\to F_e=1$ transition in the presence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The quantity plotted is $\mathrm{Im}({\rho }_{31}^{\prime \left(1\right)}/V_p+{\rho }_{32}^{\prime \left(1\right)}/r{V}_p)$, which is proportional to the probe absorption [26]. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=1, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }_t=0.001, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.001$ (closed system, solid red line) and ${\gamma }_{3t}=0.2$ (open system, dotted blue line). The plot for the open system is magnified fivefold. All the parameters are normalized by the population decay rate $\mathrm{\Gamma }={\mathrm{\Gamma }}_{31}={\mathrm{\Gamma }}_{32}$ and are thus dimensionless.

Figure 3

Probe absorption spectrum as a function of pump-probe detuning for $F_g=1\to F_e=1$ transition in the absence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The quantity plotted is $\mathrm{Im}({\rho }_{31}^{\prime \left(1\right)}/V_p+{\rho }_{32}^{\prime \left(1\right)}/r{V}_p)$, which is proportional to the probe absorption [26]. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=0, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }_t=0.001, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.001$ (closed system, solid red line) and ${\gamma }_{3t}=0.2$ (open system, dotted blue line). All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 4

Absolute values of the coherent population oscillations ${\rho }_{11}^{\left(1\right)}$ (full red line) and ${\rho }_{22}^{\left(1\right)}$ (dashed green line) as a function of pump-probe detuning for an open $F_g=1\to F_e=1$ transition in the presence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=1, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.2$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 5

Absolute values of the coherent population oscillations ${\rho }_{11}^{\left(1\right)}$ (full red line) and ${\rho }_{22}^{\left(1\right)}$ (dashed green line) as a function of pump-probe detuning for an open $F_g=1\to F_e=1$ transition in the absence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=0, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.2$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 6

Two-photon coherences ${\rho }_{21}^{\left(1\right)}$ (full red line) and ${\rho }_{21}^{(-1)}$ (dashed green line) as a function of pump-probe detuning for $F_g=1\to F_e=1$ transition in the presence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=1, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.2$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 7

Two-photon coherences ${\rho }_{21}^{\left(1\right)}$ (full red line) and ${\rho }_{21}^{(-1)}$ (dashed green line) as a function of pump-probe detuning for $F_g=1\to F_e=1$ transition in the absence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The parameters used are $|V_c|=0.3, |V_p|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=0, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }^{*}=0$, and ${\gamma }_{3t}=0.2$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 8

Probe amplitude as a function of time for $z=3\times {10}^{-4}$ at line center ($\delta =0$) for $F_g=1\to F_e=1$ transition in the presence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations (the inset contains a magnified version of the restored probe). The parameters used are $|V_c(t=0,z=0)|=0.3, |V_p(t=0,z=0)|=0.001, {\mathrm{\Delta }}_{31}=-{\mathrm{\Delta }}_{32}=1, {\gamma }_{3t}=0.2, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\gamma }^{*}=0, t_p=2.5, {\tau }_p=1.5, {\tau }_1=4, {\tau }_2=11$, and $\alpha ={10}^4$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.

Figure 9

Probe amplitude as a function of time for various propagation distances at line center ($\delta =0$) for $F_g=1\to F_e=1$ transition in the absence of longitudinal magnetic field for lin $\parallel$ lin and lin $\perp$ lin polarizations. The parameters used are $|V_c(t=0,z=0)|=0.3, |V_p(0,z=0)|=0.001, {\gamma }_t=0.001, {\gamma }_{21}={\gamma }_{12}={\gamma }_t, {\mathrm{\Delta }}_{31}={\mathrm{\Delta }}_{32}=0, {\gamma }_{3t}=0.2, {\gamma }^{*}=0, t_p=2.5, {\tau }_p=1.5, {\tau }_1=4, {\tau }_2=11$, and $\alpha ={10}^4$. All the parameters are normalized by $\mathrm{\Gamma }$ and are thus dimensionless.