Adiabatic pulse propagation in coherent atomic media with the tripod level configuration

We investigate the problem of propagation of three-component resonant light pulses with adiabatically varying amplitudes through a medium consisting of atoms with the tripod level configuration. By means of both analytic and numerical methods we find the two modes of shape-preserving pulse propagation. The pulse propagation velocity of the fast mode is equal to the speed of light in vacuum, whereas the group velocity of the other (slow) mode is significantly slowed down. These two modes represent a general asymptotic solution of the problem of adiabatic pulse propagation, i.e., a pulse of any shape, which is consistent with the adiabaticity conditions, and a finite duration evolves at large propagation distances (and, correspondingly, at large times of interaction with the medium) to a well-separated pair of fast and slow pulses. The experimental requirements for adiabatic pulse propagation in a tripod medium are similar to that needed for observation of slow light propagation in a medium with the $\Lambda$ configuration of levels. However, the tripod scheme offers a different possibility, which is absent in the $\Lambda$-medium case: collisions of fast and slow pulses. It is found numerically that after such a collision the shapes of the pulses change, so these pulses do not match the classical definition of a soliton.

Figure 1

Tripod scheme of levels driven by resonant electromagnetic fields.

Figure 2

Splitting of the incident pulse into the fast (F) and slow (S) pulses. The normalized pulse amplitudes ${\Omega }_j∕\Omega$ for $j=1$ (solid line), 2 (dashed line), and 3 (dotted line) versus scaled nonlinear time, $w∕w_0$ (the same line styles are reserved for the pulse amplitudes in the subsequent figures) are displayed for (a) $\zeta =0$ (the boundary conditions at the medium entrance) and (b) $\zeta =20w_0$ (numerical simulation results). (c) Sine of the mixing angle $\nu$ vs $w∕w_0$ for $\zeta =0$ (dashed line) and $20w_0$ (dotted line).

Figure 3

Collision of pulses. (a) The normalized pulse amplitudes vs $w∕w_0$ at (a) $\zeta =0$ (the boundary conditions), (b) $\zeta =10w_0$ (numerical results), and (c) $\zeta =30w_0$ (numerical results). See the text for more detailed description.

Figure 4

Adiabatic pulse propagation for the particular initial condition $\beta =0$. The boundary conditions $(\zeta =0)$: (a) the normalized pulse amplitudes and (b) sine functions of the angles $\alpha =\vartheta$ and $\phi$ vs $w∕w_0$. The numerical results at $\zeta =20w_0$ for (c) ${\Omega }_j∕\Omega$ and (d) $\sin \vartheta$, $\sin \phi$.