Tunable phase control of rotary photon drag by superposition of three probes coherence in atomic medium

Abstract

We use the superposition of three probes fields to modify the rotatory photon drag by strength and phases of the applied fields. Subluminal propagation is tuned to superluminal propagation with strength, phases of control field and superposition states. We measure a group index of $\pm \,2000$ with phases of the control fields and superposition states. The group velocity is modified to $\pm \,1.5\times 10^5$ m/s in the medium. We investigate the rotary photon drag in the region positive and negative group velocity regions. Significant tunability of rotary photon drag is reported with the phases and strength of applied fields and superposition states. We note a maximum of $\pm \,5$ micro-radians rotary photon drag in the medium. The significant tunability of photon drag from positive to negative values in micro-radians shows potential application in the spacial modes imaging coding.

Appendices

Appendix I

$$\begin{aligned} \dot{{\widetilde{\rho }}}_{24}= & {} A_1{\widetilde{\rho }}_{24}+\frac{i}{2}\Omega _{1p}{\widetilde{\rho }}_{21}+\frac{i}{2}\Omega _{2p}({\widetilde{\rho }}_{22}-{{\widetilde{\rho }}}_{44}) +\frac{i}{2}\Omega _{3p}{{\widetilde{\rho }}}_{23}+\frac{i}{2}\Omega ^*_c{{\widetilde{\rho }}}_{25},\\ \dot{{\widetilde{\rho }}}_{25}= & {} A_2{\widetilde{\rho }}_{25}+\frac{i}{2}\Omega _{1c}{\widetilde{\rho }}_{21}+ \frac{i}{2}\Omega _{2c}{\widetilde{\rho }}_{23}+\frac{i}{2}\Omega _{c}{\widetilde{\rho }}_{24}-\frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{45},\\ \dot{{\widetilde{\rho }}}_{21}= & {} -A_3{\widetilde{\rho }}_{21}+\frac{i}{2}\Omega ^*_{1p}{\widetilde{\rho }}_{24}+ \frac{i}{2}\Omega ^*_{1c}{\widetilde{\rho }}_{25}-\frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{41},\\ \dot{{\widetilde{\rho }}}_{23}= & {} A_4{\widetilde{\rho }}_{23}+\frac{i}{2}\Omega ^*_{3p}{\widetilde{\rho }}_{24}+ \frac{i}{2}\Omega ^*_{2c}{\widetilde{\rho }}_{25}-\frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{43},\\ \dot{{\widetilde{\rho }}}_{14}= & {} B_1{\widetilde{\rho }}_{14}+\frac{i}{2}\Omega _{1p}({\widetilde{\rho }}_{11}-{\widetilde{\rho }}_{44})+ \frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{12}+\frac{i}{2}\Omega _{3p}{\widetilde{\rho }}_{13}-\frac{i}{2}\Omega _{1c}{\widetilde{\rho }}_{54}+ \frac{i}{2}\Omega ^*_c{\widetilde{\rho }}_{15},\\ \dot{{\widetilde{\rho }}}_{54}= & {} -B_2{\widetilde{\rho }}_{54}+\frac{i}{2}\Omega _{1p}{\widetilde{\rho }}_{51}+\frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{52}+ \frac{i}{2}\Omega _{3p}{\widetilde{\rho }}_{53}-\frac{i}{2}\Omega ^*_{1c}{\widetilde{\rho }}_{14}-\frac{i}{2}\Omega ^*_{2c}{\widetilde{\rho }}_{34}+ \frac{i}{2}\Omega ^*_c({\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{44}),\\ \dot{{\widetilde{\rho }}}_{15}= & {} B_3{\widetilde{\rho }}_{15}+\frac{i}{2}\Omega _{1c}({\widetilde{\rho }}_{11}-{\widetilde{\rho }}_{55})+ \frac{i}{2}\Omega _{2c}{\widetilde{\rho }}_{13}+\frac{i}{2}\Omega _c{\widetilde{\rho }}_{14}-\frac{i}{2}\Omega _{1p}{\widetilde{\rho }}_{45},\\ \dot{{\widetilde{\rho }}}_{13}= & {} B_4{\widetilde{\rho }}_{13}+\frac{i}{2}\Omega ^*_{3p}{\widetilde{\rho }}_{14}+\frac{i}{2}\Omega ^*_{2c}{\widetilde{\rho }}_{15}- \frac{i}{2}\Omega _{1p}{\widetilde{\rho }}_{43}-\frac{i}{2}\Omega _{1c}{\widetilde{\rho }}_{53},\\ \dot{{\widetilde{\rho }}}_{53}= & {} -B_5{\widetilde{\rho }}_{53}+\frac{i}{2}\Omega ^*_{3p}{\widetilde{\rho }}_{54}+ \frac{i}{2}\Omega ^*_{2c}({\widetilde{\rho }}_{55}-{\widetilde{\rho }}_{33})-\frac{i}{2}\Omega ^*_{1c}{\widetilde{\rho }}_{13}- \frac{i}{2}\Omega ^*_c{\widetilde{\rho }}_{43},\\ \dot{{\widetilde{\rho }}}_{34}= & {} B_6{\widetilde{\rho }}_{34}+\frac{i}{2}\Omega _{1p}{\widetilde{\rho }}_{31}+ \frac{i}{2}\Omega _{3p}({\widetilde{\rho }}_{33}-{\widetilde{\rho }}_{44})-\frac{i}{2}\Omega _{2c}{\widetilde{\rho }}_{54}+ \frac{i}{2}\Omega _{2p}{\widetilde{\rho }}_{32}+\frac{i}{2}\Omega ^*_c{\widetilde{\rho }}_{35}. \end{aligned}$$

Appendix II

$$\begin{aligned} A_1= & {} i\Delta _{2p}-\frac{1}{2}(\gamma _{42}+\gamma _{52}+\gamma _{54}),~ A_2=i(\Delta _{2p}+\Delta _c)-\frac{1}{2}(\gamma _{42}+\gamma _{52}),\\ A_3= & {} -(i(\Delta _{1p}-\Delta _{2p})+\frac{1}{2}(\gamma _{41}+\gamma _{42}+\gamma _{51}+\gamma _{52})),\\ A_4= & {} i(\Delta _{2p}+\Delta _c-\Delta _{2c})-\frac{1}{2}(\gamma _{42}+\gamma _{43}+\gamma _{52}+\gamma _{53}),\\ B_1= & {} i\Delta _{1p}-\frac{1}{2}(\gamma _{41}+\gamma _{51}+\gamma _{54}), ~ B_2=-(i\Delta _c+\frac{1}{2}\gamma _{54}),~ B_3=i\Delta _{1c}-\frac{1}{2}(\gamma _{41}+\gamma _{51}),\\ B_4= & {} i(\Delta _{1c}-\Delta _{2c})-\frac{1}{2}(\gamma _{41}+\gamma _{43}+\gamma _{51}+\gamma _{53}),~ B_5=-i\Delta _{2c}-\frac{1}{2}(\gamma _{43}+\gamma _{53})\\ B_6= & {} i\Delta _{3p}-\frac{1}{2}(\gamma _{43}+\gamma _{53}+\gamma _{54}),\\ Q_1= & {} \frac{1}{16}B_2B_4e^{-2i\varphi }\left( 4B_1B_3+|\Omega ^2|\right) \left( 4B_5B_6e^{2i\varphi }-e^{-2i\varphi _2}|\Omega _2^2|\right) ,\\ Q_2= & {} \frac{1}{16}(|\Omega _1^2|\\&\times \left( B_2B_6|\Omega ^2|+B_3(4B_4B_5B_6-B_4e^{-2i(\varphi -\varphi _2)}|\Omega ^2|+B_6|\Omega _1^2|))+B_1B_3(B_2B_6|\Omega _1^2|+B_4B_5|\Omega _2^2|)\right) ,\\ Q_3= & {} \frac{1}{16}e^{-2i\varphi }\\&\times \left( -B_1B_2e^{4i\varphi _2}|\Omega ^2|+B_5B_6e^{2i(\varphi +\varphi _2)}(4B_1B_2|\Omega _1^2|)+e^{2i\varphi }(B_4B_5|\Omega ^2|+B_1B_3|\Omega _1^2|)\right) |\Omega _2^2|,\\ Q_4= & {} \frac{1}{16}e^{-i(\varphi -2\varphi _2)}\left( B_1B_5e^{i\varphi }|\Omega _2^4|-i|\Omega ^2||\Omega _1^2||\Omega _2^2|\cos (\varphi -\frac{3\varphi _2}{2})\sin (\frac{\varphi _2}{2})\right) . \end{aligned}$$