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Physical Review A

Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations

R. Y. Chiao

Published Jul 1, 1993• 249 citations
Publisher

Abstract

The propagation of limited-bandwidth signals, such as Gaussian wave packets, tuned to a transparent spectral region far below the resonance of an inverted two-level atomic medium, can be superluminal, i.e., with phase, group, and energy velocities all exceeding the vacuum speed of light c. Causality is not violated, however. Little distortion and gain can accompany this propagation.

DOI
10.1103/physreva.48.r34

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The propagation of limited-bandwidth signals, such as Gaussian wave packets, tuned to a transparent spectral region far below the resonance of an inverted two-level atomic medium, can be superluminal, i.e., with phase, group, and energy velocities all exceeding the vacuum speed of light c. Causality is not violated, however. Little distortion and gain can accompany this propagation.
Analytic, limited-bandwidth signals, e.g. Gaussian wavepackets, whose frequencies lie in a transparent spectral window far below the resonance of an amplifying atomic medium, can propagate with phase, group, energy and 'signal' velocities (as defined by Sommerfeld and Brillouin) all exceeding the vacuum speed of light c. This 'superluminal' propagation may occur without significant distortion or gain. The front velocity, however, is not superluminal, and therefore Einstein causality is not violated. Applications will be discussed, including the possibility of using superluminal atomic coherence in condensed matter (e.g. macroscopic mass currents in superfluid helium which arise from vortex motions) for efficient gravitational antennae.
Abstract This paper considers the question of superluminal (or negative) group velocities of signals in dispersive media, as well as negative group delay times of signals passing through linear filters. It is shown that information can be transmitted only with the help of jumps in the time dependence of a function or its derivatives and, therefore, the information propagation velocity in any (including dispersive, absorbing, or amplifying) medium exactly coincides with the speed of light in vacuum. As for the group velocity of a wave packet, it is only the propagation velocity of its infinitely differentiable envelope, by means of which the transmission of information, strictly speaking, is impossible. Therefore, in regions with anomalous dispersion, the group velocity can be superluminal or negative without violating the “light limitation” of the theory of relativity or the principle of causality. Examples of situations in which superluminal or negative group velocity occurs are given.
In the region between two lines of a doublet of an inverted medium, there exists a point of zero group-velocity dispersion, where highly superluminal effects may be observable without significant gain, loss, distortion, or broadening. The results of this group-velocity analysis hold for sufficiently narrow-band, analytic pulses, and do not constitute a violation of causality, although the group, ``signal,'' and energy velocities as defined by Sommerfeld and Brillouin may all exceed c or even become negative. No sharp disturbance in the pulse (a real signal) could propagate faster than light, but the scheme offers an unusual noiseless amplification scheme for the leading edge of a pulse, both at the classical and at the single-photon level.
Experiments at Berkeley and elsewhere which show that the process of tunneling is apparently superluminal will be reviewed. Conflicting theories for the tunneling time will be compared with experiment. The tunneling particle in the Berkeley experiments was the photon. The measurement of the tunneling time utilized a two-photon light source (spontaneous parametric down-conversion), a Hong-Ou-Mandel interferometer, and a coincidence counter of photon pairs. The tunnel barrier consisted of a photonic-bandgap medium excited at midgap. We find that the peak of a tunneling single-photon wave packet appeared on the far side of the barrier 1.47 ± 0.21 fs earlier than the peak of a wave packet which traveled an equal distance in air. Group velocities which exceed c can also occur in transparent media with inverted atomic populations. Tachyon-like excitations, i.e., collective normal modes of atoms coupled to electromagnetic radiation which possess tachyon-like dispersion relations, occur in these media. However, relativistic causality is not violated by any of these superluminal phenomena.