Doppler broadening and squeezing-induced sub- and super-luminal group velocity in a driven qubit model

Abstract

We investigate the absorption and dispersion spectra of the light pulse propagation through Doppler broadened driven qubit by exponentially weak decaying field in the presence of an off-resonant broadband squeezed vacuum (SV) reservoir. Combined effects of Doppler broadening and SV reservoir parameters induce enhanced sub- and super-luminal behavior of light through the qubit medium, as compared with the normal vacuum (NV) and thermal field (TF) reservoir cases. The SV phase, as a control parameter, induce enhanced sinusoidal (positive and negative) peaks of group index of the Doppler broadened medium.

Appendix A

The functions $C^*_{+,z}(s=iD)$ appearing in (4) are listed in the general case of SV reservoir as follows [48],

$$\begin{aligned} C^*_+(s=iD)&=\left\{ \dfrac{1}{2r}\left[ \dfrac{r+iq}{s+\Gamma -r+i\delta }+ \dfrac{r-iq}{s+\Gamma +r+i\delta }\right] \right. \nonumber \\&\quad \left. \Omega ^2\left[ \dfrac{-1}{4r(s+\Gamma -r+i\delta )} \left( \dfrac{r+iq}{\Gamma +r+k^*} \left[ \dfrac{r+iq+2\eta }{\eta (\eta +r)}+\dfrac{\gamma M}{k^*(r+k^*)} \right] \right. \right. \right. \nonumber \\&~\left. \left. \left. +\dfrac{\gamma M^*}{\Gamma +r+k}\left[ \dfrac{r+iq+2k}{k(r+k)}+\dfrac{\gamma M}{\eta (\eta +r)} \right] \right) \nonumber \right. \right. \nonumber \\&~\left. \left. +\dfrac{1}{4r( s+\Gamma +r+i\delta )}\left( \dfrac{r-iq}{\Gamma -r+k^*}\left[ \dfrac{r-iq-2\eta }{\eta (\eta -r)} +\dfrac{\gamma M}{k^*(r-k^*)}\right] \right. \right. \right. \nonumber \\&~\left. \left. \left. + \dfrac{\gamma M^*}{\Gamma -r+k}\left[ \dfrac{r-iq-2k}{k(r-k)}+\dfrac{\gamma M}{\eta (\eta -r)}\right] \right) \right. \right. \nonumber \\&~+\left. \left. \dfrac{1}{4r\eta (r-\eta )(s+\Gamma -r+2\eta +i\delta )}\left( \dfrac{(r+iq)(r+iq-2\eta )}{\Gamma +r-k}-\dfrac{\gamma ^2|M |^2}{\Gamma +r-k^*} \right) \right. \right. \nonumber \\&~+\left. \left. \dfrac{1}{4r\eta (r+\eta )(s+\Gamma +r+2\eta +i\delta )} \left( \dfrac{(r-iq)(r-iq+2\eta )}{\Gamma -r-k}+\dfrac{\gamma ^2|M |^2}{\Gamma -r-k^*} \right) \right. \right. \nonumber \\&\quad \left. \left. 2\left( \dfrac{k-\Gamma +iq}{(\Gamma -k)^2-r^2}+ \dfrac{\gamma M}{(\Gamma -k^*)^2-r^2} \right) \right. \right. \nonumber \\&~\times \left. \left. \left( \dfrac{\Gamma +k^*-iq}{\left[ (\Gamma +k^*)^2 -r^2\right] (s+2\Gamma +k^*+i\delta ) } - \dfrac{\gamma M^*}{\left[ (\Gamma +k)^2 -r^2\right] (s+2\Gamma +k+i\delta ) } \right) \right. \right. \nonumber \\&\qquad \left. \left. +\dfrac{\gamma M}{4rk^*}\left( \dfrac{r+iq-2k^*}{(\Gamma +r-k^*)(r-k^*)(s+\Gamma +2k^*-r+i\delta )} \right. \right. \right. \nonumber \\&\qquad \left. \left. -\left( \dfrac{r-iq-2k^*}{(\Gamma -r-k^*)(r+k^*)(s+\Gamma +2k^*+r+i\delta )} \right) \right. \right. \nonumber \\&\qquad \left. \left. +\dfrac{\gamma M^*}{4rk}\left( \dfrac{r+iq}{(\Gamma +r-k)(r-k)(s+\Gamma +2k-r+i\delta )} \right. \right. \right. \nonumber \\&\qquad \left. \left. -\left( \dfrac{r-iq}{(\Gamma -r-k)(r+k)(s+\Gamma +2k+r+i\delta )} \right) \right] \right\} _{s=iD} \end{aligned}$$

(A.1)

$$\begin{aligned} C^*_z(s=iD)&=-2i\Omega \left\{ \dfrac{1}{2r(s+\Gamma +r+i\delta )} \left( \dfrac{r-iq}{\Gamma +k^*-r}-\dfrac{\gamma M^*}{\Gamma +k-r}\right) \right. \nonumber \\&\qquad \left. +\dfrac{1}{2r(s+\Gamma -r+i\delta )}\left( \dfrac{r+iq}{\Gamma +k^*+r}+\dfrac{\gamma M^*}{\Gamma +k+r} \right) \right. \nonumber \\&\qquad \left. -\dfrac{\Gamma +k^*-iq}{[(\Gamma +k^*)^2-r^2](s+2\Gamma +k^*+i\delta )}\right. \nonumber \\&\qquad \left. +\dfrac{\gamma M^*}{[(\Gamma +k)^2-r^2](s+2\Gamma +k+i\delta )}\right\} _{s=iD} \end{aligned}$$

(A.2)

where, $k=\eta +\delta ,q=\delta +\Delta$, $r=\sqrt{\gamma |M |^2-q^2}$. Consideration of Doppler broadening means the replacement $q\equiv \delta +\Delta \rightarrow \delta +\Delta -kV,D\equiv \nu -\omega _L\rightarrow \nu -\omega _L+kV.$