Superluminal nonlinear anti-echos

Abstract

We investigate the possibility of a superluminal photon anti-echo in a degenerate four-wave mixing configuration when two pulses vvith different carrier frequencies collide in a nonlinear sample. An experiment recently performed at !\'EC Princeton by Wang et al. (1) has shown that laser pulses can travel through a cesium vapor cell with group velocities so much faster than the vacuum speed of light c that they become negative. The meaning of negative group velocities is that the peak of the output pulse leaves the exit face of the cell before the peak of the input pulse arrives at the entrance face(2)_ These results do not imply that the theory of special relativity has been overthrown, but only that special relativity does not imply that group velocities can never exceed c. contrary to common belief. In an earlier nonlinea1' optics experiment(3). it was observed that a sequence of two strong laser pulses, peaking at times tl -patial sidebands were converted back to temporal sidebands by means of a second grating. However, it was pointed out that the Kramers-Kronig relations could not be applied to this nonlinear situation, and they therefore called causality into question. We want to show that causality is not violated, analyzing a simplified version of their experiment in which two laser pulses directly propagate in a dispersive and nonlinear medium. These pulses directly generate echoes and anti-echoes from a temporal index grating created by the collision of the two pulses inside the nonlinear sample. Thus ollr idE'.a has no need of any temporal-~spatial conversions and back-conversions by means of gratings.