Anti-relaxation coating-induced velocity-dependent population re-distribution in electromagnetically induced transparency

Abstract

In an uncoated vapor cell, transmission spectra obtained for electromagnetically induced transparency (EIT) for D$_2$ line of $^{87}$Rb show asymmetry and (or) absorption in the presence of a magnetic field. In this study, complete conversion from asymmetry/absorption to transmission is found using octadecyltrichlorosilane (OTS) as an anti-relaxation coating for a $\Lambda$ system. The experimental results were interpreted in terms of velocity-dependent population re-distribution in the ground states induced by the coating, eventually resulting in the conversion from absorption to transmission. A simple theoretical model based on density matrix formalism is presented for qualitative interpretation of the results.

Appendix

A four-level system interacting with two lasers was taken as a model. Energy levels 1 and 2 are dipole forbidden (E$_{1}$ E$_{2}$). A locked weak probe excites the transitions 1$\rightarrow$3 and 1$\rightarrow$4 simultaneously, while a strong pump scans the transitions 2$\rightarrow$3 and 2$\rightarrow$4. The excited states 3 and 4 have a frequency separation of $\Delta$. The density matrix equations for the system were taken to be of the following form:

$$\begin{aligned} \dot{\rho _{11}}= & {} \frac{1}{2} i \Omega _ p (c_{13} \rho _{31}+c_{14} \rho _{41}-c_{31}\rho _{13}-c_{41} \rho _{14})\\{} & {} + \gamma _p \rho _{22} - (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+ \rho _{44}) \\ \dot{\rho _{12}}= & {} - \frac{1}{2} i [\Omega _ c (c_{32} \rho _{13}+c_{42} \rho _{14})-\Omega _p(c_{13}\rho _{32}\\{} & {} +c_{14} \rho _{42})- 2\delta \rho _{12}] - \frac{1}{2} (\gamma _p + \gamma _d) \rho _{12}\\ \dot{\rho _{13}}= & {} - \frac{1}{2} i [\Omega _ p (c_{13} \rho _{11} - c_{13} \rho _{33}- c_{14} \rho _{43})+\Omega _c c_{23}\rho _{12}\\{} & {} - 2(\delta _p- \Delta -k \text {v}) \rho _{13}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{14}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{11} - c_{14} \rho _{44}- c_{13} \rho _{34})+\Omega _c c_{24}\rho _{12}\\{} & {} - 2(\delta _p-k \text {v}) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{22}}= & {} \frac{1}{2} i \Omega _ c (c_{23} \rho _{32}+c_{24} \rho _{42}-c_{32}\rho _{23}-c_{42}\rho _{24})\\{} & {} - \gamma _p \rho _{22} + (\gamma _a+\gamma _p) \rho _{11} +\Gamma (\rho _{33}+\rho _{44})\\ \dot{\rho _{23}}= & {} - \frac{1}{2} i [\Omega _ c (c_{23} \rho _{22} - c_{23} \rho _{33}- c_{24} \rho _{43})+\Omega _p c_{13}\rho _{21}\\ {}{} & {} - 2(\delta _p-\Delta - k \text {v}-\delta ) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{24}}= & {} - \frac{1}{2} i [\Omega _ c (c_{24} \rho _{22} - c_{24} \rho _{44}- c_{23} \rho _{34})+\Omega _p c_{14}\rho _{21}\\ {}{} & {} - 2(\delta _p- k \text {v}-\delta ) \rho _{14}] - \frac{1}{2} (2\Gamma +\gamma _p + \gamma _d) \rho _{13}\\ \dot{\rho _{33}}= & {} - \frac{1}{2} i [\Omega _ p (c_{13} \rho _{31} - c_{31} \rho _{13})+\Omega _c (c_{23}\rho _{32}\\ {}{} & {} -c_{32}\rho _{23})] - 2\Gamma \rho _{33}\\ \dot{\rho _{34}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{31} - c_{31} \rho _{14})+\Omega _c (c_{24}\rho _{32}\\ {}{} & {} -c_{32}\rho _{24}+(2 (\delta _p+k \text {v})-\Delta )\rho _{34})]\\ \dot{\rho _{44}}= & {} - \frac{1}{2} i [\Omega _ p (c_{14} \rho _{41} - c_{41} \rho _{14})+\Omega _c (c_{24}\rho _{42}\\ {}{} & {} -c_{42}\rho _{24})] - 2\Gamma \rho _{44}, \end{aligned}$$

with $\rho _{ij}=(\rho _{ji})^*$. In the equations, $\delta _p$=$\omega _{44}-\omega _p$, $\delta$=$\delta _p-\delta _c$, k v is the Doppler shift, and $\gamma _d$ is the dephasing rate due to collisions (taken to be of the order of few Hz). It should be noted that the non-zero detuning in the probe beam with respect to the excited state where the excited states are within the Doppler width of the system is important for the re-distribution discussed in the article. Usually in a three-level system, two-photon absorption processes are resolved when the conditions $\delta _p$ $>>$ $\Gamma$ and $\Omega _c>>\Omega _p$ are satisfied. However, in four-level system, the condition $\delta _p$ $>>$ $\Gamma$ is not a strict one due to the presence of the nearby excited state. It can be argued that if the Doppler shift of light makes some atoms near resonant with respect to the excited state (say 4), the same class of atoms may be far resonant to the other excited state (3). Hence every class of velocity except the ones which satisfy the condition $\Vert \delta _p-k v\Vert \ll \Gamma$ can undergo two-photon absorption. If the frequency separation between the excited states is large as compared to the Doppler width, no such re-distribution can occur (for e.g., in D$_1$ line of $^{87}$Rb).