Raman process under condition of radiation trapping in a disordered atomic medium

We consider the Raman process developing in a disordered medium of alkali-metal atoms when the scattered modes are trapped on a closed transition. Our theoretical analysis, based on numerical simulations of the Bethe-Salpeter equation for the light correlation function, which includes all Zeeman states and light polarization, lets us track the stimulated amplification as well as the losses associated with the inverse anti-Stokes scattering channel. We discuss possible conditions when this process could approach the instability point and enter the regime of random lasing.

Figure 1

Energy levels and the excitation diagram for the Raman process initiated in an ensemble of ${}^{85}$Rb atoms via $F_0=2\to F=2,3\to F_0=3$ transitions. The Raman emission is a result of the simultaneous action of microwave $\Omega$ and optical ${\omega }_c$ excitations, both are linearly polarized along the quantization direction. The emitted light is trapped on the closed $F_0=3\to F=4$ transition in the optically thick atomic ensemble. While propagating through the atomic sample this light stimulates additional Raman emission and may increase the output fluorescence emerging from the sample.

Figure 2

Excitation geometry: The linearly polarized control mode (shown as a pink arrow-tube) and microwave field (not shown) initiate the Raman emission for the process shown in Fig. 1. The observation channels are parametrized by the polar angle $\theta =0^{\circ },{45}^{\circ },{90}^{\circ }$ from the control mode polarization direction.

Figure 3

Spectral dependence of the imaginary part (upper panel) and real part (lower panel) of the ${\chi }_{\perp }\left(\omega \right)$ component of the susceptibility tensor in units of the dimensionless local density of atoms $n_0{\lambda -}^3$. The gray dashed curve reproduces the original undisturbed multiplet of the $D_2$ line of ${}^{85}$Rb probed from the $F_0=3$ ground state hyperfine sublevel (Fig. 1). The red dashed curve indicates how this multiplet is modified by the interaction with the optical control mode and the blue solid line shows the spectral profile affected by both the optical and microwave modes. The Rabi frequency for the optical transition has a value of $2\overline{V}=25\gamma$, for the microwave mode it was selected for the “clock” transition $F_0=2,M_0=0\to F_0=3,M_0=0$, and chosen as $2\overline{U}=0.05\gamma$. The details of spectral behavior near the $F_0=3\to F=4$ resonance are magnified in the insets, where the black dashed curve shows how the isotropic part (gray dashed) is reduced by depopulation of the $F_0=3$ state.

Figure 4

Same as in Fig. 3 but for the ${\chi }_{\parallel }\left(\omega \right)$ component of the susceptibility tensor. This component is insensitive to the external fields near the $F_0=3\to F=4$ resonance, where it only tracks the depopulation of the $F_0=3$ state. Other resonances, associated with $F_0=3\to F=3,2$ transitions, are split by the control mode, which partly resolves their Zeeman structure.

Figure 5

Upper panel: Extinction length (upper curves) vs scattering length (lower curves) in the presence of the control optical mode only (red dashed line) and in the presence of both the optical and microwave fields (blue solid line). The parameters are the same as in Fig. 3. Lower panel: Related lengths of losses, see definition (3.4). The filled area (below the line corresponding to the absence of pump or microwave excitation) indicates the region where the trapped light can be amplified by the stimulated Raman process due to microwave induced repopulation of atoms on level $F_0=2$.

Figure 6

Intensity distribution over the scattering orders for a pointlike isotropic light source located at the center of the atomic sample. The dashed gray curve in both panels indicates the reference (without driving fields) distribution for light trapping associated with elastic scattering on the closed $F_0=3\to F=4$ transition. The upper panel shows how this distribution is modified by the action of the control optical mode: The red solid and the red dashed curves respectively indicate the distribution for elastic and anti-Stokes inelastic scattering channels; the black solid curve gives the total outcome. The lower panel shows the similar distributions (blue solid for elastic, and blue dashed for inelastic) when both the optical and microwave modes are turned on. The output intensity for the overall outgoing light via both elastic and inelastic channels overcomes the input intensity. The inset shows the total amplification given by the sum over all the scattering orders as a function of the amplitude of the microwave mode.

Figure 7

Intensity of the light scattered at the angle $\theta =0^{\circ }$ (Fig. 2). The control mode frequency is scanned near the $F_0=2\to F=4$ forbidden transition such that ${\Delta }_c={\omega }_c-{\omega }_{42}-{\Delta }_L$, where ${\Delta }_L$ is the light shift of level $F_0=2$ induced by this mode. The scattered intensity (number of photons) is given by Eq. (2.2) calculated for the same driving field parameters as in Fig. 5. The curve thickness in the plotted graphs is associated with optical thickness changed from lower to higher values $b_0=1,5,10,15,20$. For this observation angle only elastic scattering contributes to the emitted intensity.

Figure 8

Same as Fig. 7 but for the observation angle $\theta ={45}^{\circ }$. The lower panel shows the contribution of the inverse anti-Stokes scattering, the middle panel is the contribution of elastic scattering, and the upper panel gives the sum for both channels. The curve thickness in the plotted graphs is associated with optical thickness changed from lower to higher values $b_0=1,5,10,15,20$.

Figure 9

Same as Fig. 8 but for the observation angle $\theta ={90}^{\circ }$.