Pulse propagation in a medium of $\Lambda$-type atoms

The propagation of a weak-field pulse through a medium of three-level atoms is considered. Each atom has a $\Lambda$-type level scheme in which the two lower levels are stable. A strong control field drives one of the electronic transitions while the signal field drives the coupled transition under conditions where EIT (electromagnetically induced transparency) is usually operative. The input pulse is slowed and compressed as it enters the medium, adiabatically following the EIT solution. However, if the control field is changed suddenly when the pulse is in the medium, the signal field is transformed into two pulses, one of which propagates as a normal EIT pulse and the other with a different speed and an amplitude that oscillates in time. The temporal oscillations are transformed into both spatial and temporal oscillations as the pulse exits the medium. An analytic expression is derived for the pulse intensity which provides a good approximation to the exact result at all times. It is shown that the oscillating component of the exiting pulse can be spatially compressed in comparison with the input pulse.

Figure 1

A signal pulse is sent into a medium of three level atoms, each of whose level schemes is shown in the figure. A cw control field is also present. The quantity ${\chi }_0$ is one half the maximum Rabi frequency associated with the signal field $\chi (X,t)$, whose central frequency is ${\omega }_s={\omega }_{21}$. The quantity ${\chi }^{\prime }$ is one-half the Rabi frequency associated with the control field, whose frequency is ${\omega }_c={\omega }_{23}.$

Figure 2

Field intensity in the medium for ${\tilde{\chi }}^{\prime }=20$, $\tilde{\alpha }=40$, $\xi =-1$, and $\tau =6.75.$ Solution including dispersion (dashed blue line), solution neglecting dispersion (dotted red line), and the exact solution (solid black line).

Figure 3

Field intensity in the medium for ${\tilde{\chi }}^{\prime }=20$, $\tilde{\alpha }=40$, $\xi =-1$, and $\tau$ in the range 6.0–6.5$.$ The EIT component is present, as is the oscillating component.

Figure 4

Field intensity in the medium for ${\tilde{\chi }}^{\prime }=20$, $\tilde{\alpha }=40$, $\xi =-1$, and $\tau$ in the range 0–0.5. Interference between the two pulse components is seen.

Figure 5

Field intensity in the medium for ${\tilde{\chi }}^{\prime }=\tilde{\alpha }=20$, $\xi =-1$, and $\tau$ in the range 6.0–7.0$.$ The EIT component is no longer present.

Figure 6

Field intensity exiting the medium with ${\tilde{\chi }}^{\prime }=20$, $\tilde{\alpha }=20$, $\xi =-1$, $\tilde{L}=4$, and $\tau =16$. The green vertical line is the end of the medium. The EIT component is absent and the other component is spatially broadened.

Figure 7

Field intensity exiting the medium with ${\tilde{\chi }}^{\prime }=40$, $\tilde{\alpha }=160$, $\xi =-1$, $\tilde{L}=4$, and $\tau =40$. The EIT component is now present and the other component is spatially narrowed (the dashed green curve represents the original pulse width).

Figure 8

Field intensity dynamics with ${\tilde{\chi }}^{\prime }=20$, $\tilde{\alpha }=60$, $\xi =-1$, and $\tilde{L}=4$.