Optical properties of an inhomogeneously broadened multilevel V-system in the weak and strong probe regimes

Abstract

We present a theoretical model, using density matrix approach, to study the effect of weak as well as strong probe field on the optical properties of an inhomogeneously broadened multilevel V-system of the $^{87}$Rb D2 line. We consider the case of stationary as well as moving atoms and perform thermal averaging at room temperature. The presence of multiple excited states results in asymmetric absorption and dispersion profiles. In the weak probe regime, we observe the partial transparency window due to the constructive interference that occurs between transition pathways at the line center. We present our results after carrying out Doppler averaging at room temperature atomic vapor and observe that the line width of transparency window is enhanced, whereas the positive slope of corresponding dispersion curve become less steep. In the presence of strong probe field, the transparency window (with normal dispersion) at line center switches to enhanced absorption (with anomalous dispersion). Here, we also present the dependence of electromagnetically induced transparency on the polarization of applied fields. In the end, we present transient behavior of our system which agrees with corresponding absorption and dispersion profiles. This study may help to understand optical switching and controllability of group velocity.

Appendices

Appendix 1: Density matrix equations for a five-level system

In this appendix, we discuss density matrix equations for a five-level system. These density matrix equations can be obtained from Eq. (3) by removing the contribution of the transition $\left| e_{5} \right\rangle \leftrightarrow \left| g \right\rangle$, in Figure 1(a). This will affect the population of the ground state $\rho _{gg}$ that will have to modify due to the different decay channels in the resulted five-level system Figure 1(b), so it can be written as

$$\begin{aligned} \dot{\rho }_{gg}&=-\,\frac{\rho _{gg}}{\tau _{d}}+\frac{1}{\tau _{d}}+ \frac{b_{e_{1}}}{3}\varGamma _{e_{1}}\rho _{e_{1}e_{1}}+ \frac{b_{e_{2}}}{3}\varGamma _{e_{2}}\rho _{e_{2}e_{2}}+ \frac{b_{e_{3}}}{3}\varGamma _{e_{3}}\rho _{e_{3}e_{3}}+ {b_{e_{4}}}\varGamma _{e_{4}}\rho _{e_{4}e_{4}} \nonumber \\&\quad+\,\frac{i}{2}\displaystyle \sum _{j=1}^3(\varOmega ^{*}_{cge_{j}}\rho _{ge_{j}} - \varOmega _{cge_{j}}\rho _{e_{j}g})+\frac{i}{2}\displaystyle \sum _{k=4}^5(\varOmega ^{*}_{pge_{k}}\rho _{ge_{k}}- \varOmega _{pge_{k}}\rho _{e_{k}g}). \end{aligned}$$

(9)

Appendix 2: Dressed-state analysis

In this appendix, we present semiclassical dressed-state picture for a three-level V-system. Our calculations show that for a case $\varOmega _{cge_{1}} > \varGamma _{e_{4}}$, we observe the partial transparency window at the zero probe detuning. This can be explained by considering the dressed-state analysis of a three-level V-system. The control field couples the states $\left| g \right\rangle$ and $\left| e_{1} \right\rangle$ that create two dressed states $\left| + \right\rangle$ and $\left| - \right\rangle$;

$$\left| + \right\rangle=\frac{1}{\sqrt{2}}(\left| e_{1} \right\rangle +\left| g \right\rangle)$$

(10)

$$\left| - \right\rangle=\frac{1}{\sqrt{2}}(\left| e_{1} \right\rangle -\left| g \right\rangle).$$

(11)

The probe absorption is from one of these dressed states to an excited state $\left| e_{4} \right\rangle$, i.e. there are two transition pathways for probe absorption $\left| e_{4} \right\rangle \rightarrow \left| + \right\rangle$ and $\left| e_{4} \right\rangle \rightarrow \left| - \right\rangle$. Due to two pathways, there is interfere between them. The transition amplitude at the (undressed) resonant frequency $\omega _{e_{4}g}=(E_{e_{4}}-E_{g})/\hbar$, from the excited state $\left| e_{4} \right\rangle$ to the dressed states will be the sum of the contributions to states $\left| + \right\rangle$ and $\left| - \right\rangle$, is given by

$$P\propto \left| \frac{\langle { e_{4} | {\mathbf{d.E }} | + \rangle }}{\varOmega _{cge_{1}}}+\frac{\langle { e_{4} | {\mathbf{d.E }} | - \rangle }}{-\varOmega _{cge_{1}}}\right| ^{2}=\frac{{\varOmega ^{2}_{pge_{4}}}}{\varOmega _{cge_{1}}}.$$

(12)

The transition amplitude is not zero. It means for V-type EIT system, there is a constructive interference between two transition pathways [48] and the probe absorption is enhanced at the zero probe detuning with transition amplitude given by Eq. (12). Due to this reason, we observe only partial transparency window at zero probe detuning for $\varOmega _{cge_{1}} > \varGamma _{e_{4}}$ case.

Appendix 3: Effect of Doppler shift on susceptibility

In this appendix, we discuss the numerical solution of the imaginary part of susceptibility ($\rho _{ge_{4}}$+$\rho _{ge_{5}}$) for an atom with the detuning $\varDelta _{p}$ of the probe field for six-level system with various values of Doppler shift ($\varDelta _{D}$).

Our results show that for $\varDelta _{D}=0$ MHz, there are two transparency windows located at $\varDelta _{p}=0$ and $-157$ MHz, as shown in figure 6(a). One of the transparency windows, which is located at $\varDelta _{p}=-157$ MHz, is due to the level $e_{5}$. This level $e_{5}$ is the nearest Zeeman sublevel which is $-157$ MHz far away from the probe field transition $\left| g \right\rangle \rightarrow \left| e_{4} \right\rangle$ . When we further increase the Doppler shift, our calculations shows that the shift is negative (red shift) for atoms moving in the same direction as the probe and control fields, and positive (blue shift) for atoms moving opposite to the probe and control fields. It can be seen in figure 6, one of the absorption peaks completely disappear as the Doppler shift increases.

Fig. 6

Probe absorption coefficient Im($\rho _{ge_{4}}+\rho _{ge_{5}}$) for atoms as a function of the probe detuning ($\varDelta _{p}$) in the weak probe regime with different velocities (a) $\varDelta _{D} = 0$ MHz, (b) $\varDelta _{D} = \pm 5$ MHz and (c) $\varDelta _{D} = \pm 15$ MHz. Solid and dashed curves correspond to the velocity classes with positive Doppler shifts ($\varDelta _{D}>0$) and negative Doppler shifts ($\varDelta _{D}<0$), respectively. In the present calculations, we take $\varDelta _{c}=0$ and $\varOmega _{cge_{1}}=24$ MHz