pulse width is narrower than the gain region [in violation of condition(39)), as shown in Fig. 4. Here, the 0<2<50, the group velocity is taken to be -c/2 defined to be unity. The behavior illustrated in Fig. 4 is per- haps less surprising when the pulse amplitude is plotted on a logarithmic scale, as in Fig. 5. Although the overall gain of the system is near unity, the leading edge of the pulse is plified by about 70 magnitude in this [the implausibility of which underscores that condition(39) Gain cannot be evaded], while the trailing edge of the pulse is attenuated by the same amount. The gain medium has tem porarily loaned some of its energy to the pulse permitting the E|t-125 leading edge of the pulse to appear to advance faster than the Our discussion of the pulse has been based on a classical analysis of interference, but, as remarked by Dirac, classi cal optical interference describes the behavior of the wave functions of individual photons, not of interference between photons. Therefore, we expect that the behavior discussed above will soon be demonstrated for apulse consisting of a single photon with a Gaussian wave packet ACKNOWLEDGMENTS E↓t=-75 The author thanks Lijun Wang for discussions of his ex- Gain periment, and Alex Granik for references to the early history of negative group velocity and for the analysis contained in Eqs.(14)-(16) Electronic mail: mcdonald @puphed. princeton.edu ond in an T. McDonald, ""Slow light, Am. J. Phys. 68, (2000.A figure to be compared with Fig. I of the present paper has been added in 'This is in contrast to the "A"configuration of the three-level atomic Ga system required for slow light (Ref. 2)where the pump laser does not pumped c an inverted population, in which case an adequate classical de- is simply to reverse the sign of the damping constant for the J. Wang, A. Kuzmich, and A, Dogariu, ""Gain-assisted superluminal light propagation, Nature(London)406, 277-279(2000). Their website, http://www.neci.nj.nec.com/homepages/lwan/gas.htmcontainsadditional material, including an animation much like Fig. 4 of the present paper. w.R. Hamilton,""Researches respecting vibration, connected with the theory of light, Proc. R. Ir. Acad. 1, 267, 341(1839) J. S. Russell, ""Report on waves, Br. Assoc Reports(1844), pp. 311 390. This report features the first recorded observations of solitary waves G. G. Stokes, Problem 1l of the Smiths Prize examination papers (2 February 1876), in Mathematical and Physical Papers (Johnson Reprint Co., New York, 1966), Vol. 5, p. 362. Et=50 ST. H. Havelock, The Propagation of Disturbances in Dispersive Media (Cambridge U P, Cambridge, 1914). H. Lamb,""On Group-Velocity, Proc. London Math. Soc. 1, 473-479 Gain (1904) 200-150-100-50050100150200250 ee p. 551 of M, Laue, "Die Fortpfianzung der Strahlung in enden und Absorbierenden Medien, Ann. Phys. (Leipzig) (1905) Fig. 5. The same as Fig. 4, but with the electric field plotted on a logarith- L. Mandelstam, Lectures on Optics, Relativity and Quantum Mechanics mic scale from I to 10-65 Nauka, Moscow, 1972); in Russian. L. Brillouin, Wave Propagation and Group Velocity(Academic, New York, 1960). That the group velocity can be negative is mentioned elocities. However, the medium is merely using information stored in the early part of the pulse during its(lengthy)time C.G. B. Garrett and D. E McCumber, "Propagation of a gaussian Light of generation to bring the apparent velocity of the pulse R. Y Chiao, "Superluminal( but causal) propagation of w The effect of the negative group velocity medium can be transparent media with inverted atomic populations, Phys. Rev. A 48 matized in a calculation based on Eq (31)in which the R34-R37(1993) 613 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problemsvelocities. However, the medium is merely using information stored in the early part of the pulse during its ~lengthy! time of generation to bring the apparent velocity of the pulse closer to c. The effect of the negative group velocity medium can be dramatized in a calculation based on Eq. ~31! in which the pulse width is narrower than the gain region @in violation of condition ~39!#, as shown in Fig. 4. Here, the gain region is 0,z,50, the group velocity is taken to be 2c/2, and c is defined to be unity. The behavior illustrated in Fig. 4 is per￾haps less surprising when the pulse amplitude is plotted on a logarithmic scale, as in Fig. 5. Although the overall gain of the system is near unity, the leading edge of the pulse is amplified by about 70 orders of magnitude in this example @the implausibility of which underscores that condition ~39! cannot be evaded#, while the trailing edge of the pulse is attenuated by the same amount. The gain medium has tem￾porarily loaned some of its energy to the pulse permitting the leading edge of the pulse to appear to advance faster than the speed of light. Our discussion of the pulse has been based on a classical analysis of interference, but, as remarked by Dirac,21 classi￾cal optical interference describes the behavior of the wave functions of individual photons, not of interference between photons. Therefore, we expect that the behavior discussed above will soon be demonstrated for a ‘‘pulse’’ consisting of a single photon with a Gaussian wave packet. ACKNOWLEDGMENTS The author thanks Lijun Wang for discussions of his ex￾periment, and Alex Granik for references to the early history of negative group velocity and for the analysis contained in Eqs. ~14!–~16!. a! Electronic mail: mcdonald@puphed.princeton.edu 1 L. V. Hau et al., ‘‘Light speed reduction to 17 metres per second in an ultracold atomic gas,’’ Nature ~London! 397, 594–598 ~1999!. 2 K. T. McDonald, ‘‘Slow light,’’ Am. J. Phys. 68, 293–294 ~2000!. A figure to be compared with Fig. 1 of the present paper has been added in the version at http://arxiv.org/abs/physics/0007097 3 This is in contrast to the ‘‘L’’ configuration of the three-level atomic system required for slow light ~Ref. 2! where the pump laser does not produce an inverted population, in which case an adequate classical de￾scription is simply to reverse the sign of the damping constant for the pumped oscillator. 4 L. J. Wang, A. Kuzmich, and A. Dogariu, ‘‘Gain-assisted superluminal light propagation,’’ Nature ~London! 406, 277–279 ~2000!. Their website, http://www.neci.nj.nec.com/homepages/lwan/gas.htm, contains additional material, including an animation much like Fig. 4 of the present paper. 5 W. R. Hamilton, ‘‘Researches respecting vibration, connected with the theory of light,’’ Proc. R. Ir. Acad. 1, 267,341 ~1839!. 6 J. S. Russell, ‘‘Report on waves,’’ Br. Assoc. Reports ~1844!, pp. 311– 390. This report features the first recorded observations of solitary waves ~p. 321! and of group velocity ~p. 369!. 7 G. G. Stokes, Problem 11 of the Smith’s Prize examination papers ~2 February 1876!, in Mathematical and Physical Papers ~Johnson Reprint Co., New York, 1966!, Vol. 5, p. 362. 8 T. H. Havelock, The Propagation of Disturbances in Dispersive Media ~Cambridge U.P., Cambridge, 1914!. 9 H. Lamb, ‘‘On Group-Velocity,’’ Proc. London Math. Soc. 1, 473–479 ~1904!. 10See p. 551 of M. Laue, ‘‘Die Fortpflanzung der Strahlung in Dispergier￾enden und Absorbierenden Medien,’’ Ann. Phys. ~Leipzig! 18, 523–566 ~1905!. 11L. Mandelstam, Lectures on Optics, Relativity and Quantum Mechanics ~Nauka, Moscow, 1972!; in Russian. 12L. Brillouin, Wave Propagation and Group Velocity ~Academic, New York, 1960!. That the group velocity can be negative is mentioned on p. 122. 13C. G. B. Garrett and D. E. McCumber, ‘‘Propagation of a Gaussian Light Pulse through an Anomalous Dispersion Medium,’’ Phys. Rev. A 1, 305– 313 ~1970!. 14R. Y. Chiao, ‘‘Superluminal ~but causal! propagation of wave packets in transparent media with inverted atomic populations,’’ Phys. Rev. A 48, R34–R37 ~1993!. Fig. 5. The same as Fig. 4, but with the electric field plotted on a logarith￾mic scale from 1 to 10265. 613 Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems 613