content_image/435160.jpg content_image/435162.jpg content_image/435163.jpg content_image/435164.jpg content_image/435165.jpg content_image/435166.jpg content_image/435167.jpg content_image/435168.jpg content_image/435169.jpg content_image/435159.jpg content_image/435161.jpg
<img src='content_image/435160.jpg'>

By indirect incoherent pumps with rates r$_{1}$ and r$_{2}$ the population of the ground level | 1 〉 can be pumped to the exited levels | 2 〉 and | 3 〉 via some unspecified auxiliary states [29, 30]. The spontaneous decay rates from states | 2 〉 and | 3 〉 to the ground level are denoted by γ$_{21}$ and γ$_{31}$ , respectively. The density matrix equations of motion under the rotating wave approximation and in a rotating frame are: 

where Γ$_{ij}$ are the coherence decay rates given by 

Δ$_{c}$ = ω$_{c}$ − ω$_{21}$ and Δ$_{p}$ = ω$_{p}$ − ω$_{31}$ are the detunings of the coupling field and the probe field, respectively. Note that the interference due to the different spontaneous emission channels has been ignored. The response of the atomic system to the applied fields is determined by the suscep- tibility χ , which is defined as [31] 

where N is the atom number density in the medium. The real and imaginary parts of χ correspond to the dis- 

<img src='content_image/435162.jpg'>

<img src='content_image/435163.jpg'>

<img src='content_image/435164.jpg'>

<img src='content_image/435165.jpg'>

<img src='content_image/435166.jpg'>

<img src='content_image/435167.jpg'>

<img src='content_image/435168.jpg'>

<img src='content_image/435169.jpg'>

<img src='content_image/435159.jpg'>

persion and the absorption, respectively. The so-called group index, n$_{g}$ = c/v$_{g}$ , is also introduced where c is speed of light in the vacuum and 

is the group velocity of the probe field [5, 31]. Here χ ′ is the real part of χ . Eq.(4) shows that, when χ ′ is negligible the slope of dispersion has the major role in determina- tion of the group velocity and the group index. In our notation the positive (negative) slope of dispersion cor- responds to the increase (reduction) of the group index and the positive (negative) value of χ ′′ = Im( χ ) shows the attenuation (amplification) of the probe field. It is apparent that a negative group index means a negative group velocity. The steady state solutions of Eqs.(1) for the weak probe field approximation i.e. | Ω$_{p}$ | ≪ γ$_{ij}$ in the 

<img src='content_image/435161.jpg'>

