Superluminal two-color light in a multiple Raman gain medium

We investigate theoretically the formation of two-component light with superluminal group velocity in a medium controlled by four Raman pump fields. In such an optical scheme only a particular combination of the probe fields is coupled to the matter and exhibits superluminal propagation; the orthogonal combination is uncoupled. The individual probe fields do not have a definite group velocity in the medium. Calculations demonstrate that this superluminal component experiences an envelope advancement in the medium with respect to the propagation in vacuum.

Figure 1

Raman amplification schemes with a single probe field: Raman singlet (a) and Raman doublet (b).

Figure 2

Raman amplification schemes with two probe fields: double Raman singlet (a) and double Raman doublet (b).

Figure 3

Propagation of the incoming Gaussian wave packet, described by Eq. (93) through an atomic cloud, calculated using Eqs. (86), (91), and (92). All the quantities shown are dimensionless. (a) Time evolution of the probe field. Dashed blue lines show the location of the atomic cloud. (b) Comparison of the transmitted wave packet (solid blue line) with the wave packet propagating in the vacuum (dashed red line) at the same time moment $t=90$. In order to make the amplitudes of the wave packets similar, the amplitude of the Gaussian packet propagating in the vacuum is increased by the factor $R$ given by Eq. (90). The parameters used in calculation are $\sigma =0.1$, $z_0=-75$, $\Delta =\sqrt{3}$, and $M=1$; the length of the atomic cloud is $L=10$. At these parameters the group velocity is $v_g=1.33$ and transmission coefficient $R=148.4$.

Figure 4

Propagation through the atomic cloud of the incoming Gaussian wave packet (93) in the first probe field ${\mathcal{E}}_1$ and the creation of the second probe field ${\mathcal{E}}_2$ for the scheme with two Raman gain doublets, calculated using Eqs. (106), (107), (108), (109), and (92). All the quantities shown are dimensionless. (a) Evolution of the probe fields with time. Solid black line shows the first probe field; dotted red line shows the second probe field. Dashed blue lines indicate the location of the atomic cloud. (b) Comparison of the wave packet of the first probe beam (solid blue line) and the second probe beam (dotted red line) with the incident wave packet of the first beam propagating in the vacuum (dashed green line) at the same time moment $t=90$. In order to make the amplitudes of the wave packets similar, the amplitude of the Gaussian packet propagating in the vacuum is increased by the factor $0.5R$, with $R$ given by Eq. (90). The parameters used in calculation are ${\Omega }_{1,1}/{\Omega }_1={\Omega }_{2,1}/{\Omega }_1=1/\sqrt{2}$; all other parameters are the same as in Fig. 3.