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# Negative group vel

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### Negative group vel

2. 2. at xϭ0, the wave function may have a noncontinuous deriva- tive. By integrating the Schro¨dinger equation in a neighbor- hood of the origin, the derivative difference is6 ប2 2m ͓⌽2Ј͑0͒Ϫ⌽1Ј͑0͔͒ϩ␣⌽͑0͒ϭ0, ͑1͒ where, as shown in Fig. 4, ␣ represents the strength of the ␦ function at the origin, and is, for a thin layer, the product of the potential well and its spatial extension. We are interested in charge accumulation at the interface. Then, we look for electron bound eigenstates. These eigen- states ͑in this particular problem there is only one͒ must have negative total energy E. If they did not, they would be delo- calized states. As in the case of transmission and reﬂection through a potential step, if 0ϽEϽV0 then the electron is delocalized for xϽ0 and conﬁned within a small region for xϾ0. If, on the other hand, V0ϽE, then the electron state is delocalized for all x. In the case of a potential step ͑that is, our problem with ␣ϭ0͒, there cannot be states completely conﬁned around xϭ0, since for EϽ0 the only solution to the Schro¨dinger equation would be ⌽ϵ0. However, nontrivial solutions can exist when ␣ 0. Thus we search for exponen- tial solutions decaying away from the origin and with EϽ0: ⌽1͑x͒ϭAe␯x , ͑2͒ ⌽2͑x͒ϭAeϪ␮x , ͑3͒ EϭϪ ប2 ␯2 2m , ͑4͒ EϭVϪ ប2 ␮2 2m , ͑5͒ where ␮ and ␯ are real constants. We next deﬁne a new parameter, ␤, to simplify the nota- tion: ␯ϭͱ2mV ប2 sinh ␤, ͑6͒ Fig. 1. Typical differential interfacial capacitance of mercury in contact with aqueous solution of guanidinium nitrate for concentrations below 0.3 M. Fig. 2. Typical differential interfacial capacitance of mercury in contact with aqueous solution of guanidinium nitrate for concentrations above 0.3 M. Fig. 3. Diagram of the system show- ing the metal, the electrolyte, the thin layer, and the electron wave function. 602 602Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
3. 3. ␮ϭͱ2mV ប2 cosh ␤. ͑7͒ From these deﬁnitions and Eqs. ͑4͒ and ͑5͒, one obtains the state given by ␤ϭ 1 2 log 2m␣2 ប2 V . ͑8͒ Let Q1 and Q2 be the net charges to the left and to the right of the interface, respectively. Then Q1ϭe ͵Ϫϱ 0 ͉⌽1͑x͉͒2 dxϭ eA2 2␯ , ͑9͒ where the normalization constant A is obtained by requiring that ͵Ϫϱ 0 ͉⌽1͑x͉͒2 dxϩ ͵0 ϩϱ ͉⌽2͑x͉͒2 dxϭ1: A2 ϭ2 V ␣ cosh ␤ sinh ␤. ͑10͒ In the previous expression, we have made use of the fact that Eq. ͑1͒ is equivalent to ␮ϩ␯ϭ2ma/ប2 . Then Q1ϭ e␮ ␮ϩ␯ . ͑11͒ And, similarly Q2ϭ e␯ ␮ϩ␯ . ͑12͒ The charge at the interface capacitor can now be evaluated QcϭQ1ϪQ2ϭe ␮Ϫ␯ ␮ϩ␯ ϭe ប2 V 2m␣2 . ͑13͒ The capacitance, CϭdQc /dV, is Cϭ eប2 2m␣2 , ͑14͒ which is a constant that depends only on the properties of the layer through the parameter ␣. Thus by simply adding a small conﬁning layer at the in- terface, it is possible to explain the ﬂat characteristics of the total capacitance. As the external voltage in the electrolytic cell changes, the thin layer may appear or disappear, thus creating ‘‘normal’’ C–V regions, and ﬂat ones. ACKNOWLEDGMENTS I would like to thank Dr. Steven J. Eppell and Dr. Lesser Blum for useful comments. The National Cancer Institute through Grant No. CA77796-01 has ﬁnancially supported this work. a͒ Electronic mail: zypman@ymail.yu.edu 1 E. Budevski, G. Staikov, and W. J. Lorenz, Electrochemical Phase For- mation and Growth ͑Wiley, New York, 1996͒, pp. 200–210. 2 Wolfgang Lorenz, ‘‘The rate of absorption and of two-dimensional asso- ciation of fatty acids at the mercury-electrolyte interface,’’ Z. Elektro- chem. 62 ͑1͒, 192–199 ͑1958͒. 3 M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms for neutral organic-compounds—A hierarchy in modeling,’’ J. Electroanal. Chem. Interfacial Electrochem. 176 ͑1-2͒, 45–64 ͑1984͒. 4 M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isotherms— microscopic modeling,’’ J. Electroanal. Chem. Interfacial Electrochem. 176 ͑1-2͒, 119–137 ͑1984͒. 5 T. Wandlowski, G. Jameson, and R. De Levie, ‘‘Two-Dimensional Con- densation of Guadinidium Nitrate at the Mercury-Water Interface,’’ J. Phys. Chem. 97 ͑39͒, 10119–10126 ͑1993͒. 6 C. Cohen-Tannoudji, Bernard Diu, and Frank Laloe¨, Quantum Mechanics ͑Wiley, New York, Paris, 1977͒, Vol. 1, p. 87. Fig. 4. Potential function representing the metal, the interface, and the elec- trolyte. 603 603Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
4. 4. Electromagnetically induced transparency Tao Panga) Department of Physics, University of Nevada, Las Vegas, Nevada 89154-4002 ͑Received 6 June 2000; accepted 21 September 2000͒ ͓DOI: 10.1119/1.1331303͔ I. SCOPE Imagine that you are taking a quick walk and you are in fact traveling faster than light. This is what has been achieved in a recent experiment by Lene Hau and her co-workers.1 In that experiment, the group velocity of a pulsed laser was effectively reduced to about 1 mile per hour ͑0.45 m/s͒ in a cold, laser-dressed sodium atom cloud. In an earlier experiment, the same group had successfully slowed the group velocity of light to 38 miles per hour ͑17 m/s͒ in a similar system.2 Propagation of light in a medium is a well-studied subject even though there have been some very tough questions, such as the group velocity exceeding the speed of light in vacuum.3 Five years ago, it was demonstrated for the ﬁrst time that the speed of light can be reduced signiﬁcantly4 in a cold atom cloud that is in a state called electromagnetically induced transparency.5,6 What has made the experimental work of Hau et al. so unique is that the orders of magnitude in the reduction of speed of light, which cannot commonly be accomplished by increasing the index of refraction, is achieved instead by the extremely rapid variation of the in- dex with the frequency, and the elimination of light absorp- tion at resonance frequency—a quantum phenomenon result- ing from the coupling and interaction between lasers and electrons at different atomic levels. Here we would like to highlight some basic understanding of this exciting phenom- enon. From the Maxwell equations for a propagating electro- magnetic wave with angular frequency ␻ and complex wave vector ␬ in a nonconducting medium, we have7 ␬2 ϭ␮⑀␻2 , ͑1͒ where ␮ is the magnetic permeability and ⑀ is the electric permittivity of the medium. If we assume that ␻ is real and ␬ϭkϩi ␣ 2 , ͑2͒ where k is the real propagating wave vector and ␣ is the absorption coefﬁcient, we have nϭ ck ␻ ϭReͱ ␮⑀ ␮0⑀0 , ͑3͒ ␣ϭ2␻ Im ͱ␮⑀, ͑4͒ with cϭ1/ͱ␮0⑀0 being the speed of light in vacuum and ␮0 and ⑀0 the vacuum permeability and permittivity, respec- tively. In most cases, we have ␮Ӎ␮0 , a condition that is assumed here. From the above relation, we ﬁnd the phase velocity of the wave, vpϭ ␻ k ϭ c n , ͑5͒ which characterizes how fast the wave changes its phase. So the ﬁeld propagating along the z direction is proportional to eikzϪ␣z/2Ϫi␻t . Now let us consider that the electric ﬁeld is a nondecaying (␣ϭ0) wave packet with a range of frequencies ␻ϭ␻(k): E͑z,t͒ϭ ͵dk A͑k͒eikzϪi␻t , ͑6͒ where A(k) is a narrow function peaked at kϭk0 . We can then expand ␻(k) as ␻͑k͒ϭ␻͑k0͒ϩ͑kϪk0͒ d␻ dk ͯkϭk0 ϩO͓͑kϪk0͒2 ͔, ͑7͒ if it is well behaved, that is, a smooth function around the given wave vector k0 . If only the zeroth- and ﬁrst-order terms are kept in the expansion, we have E͑z,t͒Ӎei͓k0vgϪ␻͑k0͔͒t ͵dk A͑k͒eikzϪikvgt ϭei͓k0vgϪ␻͑k0͔͒t E͑zϪvgt,0͒, ͑8͒ where vgϭ d␻ dk ͯkϭk0 ͑9͒ is termed the group velocity of the wave at kϭk0 because the packet acts if it is traveling in space with such a velocity without changing its shape, with an overall phase change. Using Eq. ͑3͒, we obtain vgϭ vp 1ϩ͑␻/n͒͑dn/d␻͒ . ͑10͒ Note that we have assumed that n is a function of ␻ through k. We can consider the group velocity to be the velocity of the wave packet, that is, the velocity of the energy and in- formation contained in the packet, if the linear term in the above Taylor expansion is the dominant term. However, the meaning of the group velocity can change if the wave packet becomes incoherent. Care must be taken when the angular frequency of the wave is near a resonance or dn/d␻Ͻ0.3 Quantum mechanically, the resonance occurs when the frequency of light matches the energy difference between two allowed quantum levels in the system and is typically accompanied by strong absorption under normal circum- stances. This is why normal matter that can be well approxi- mated by a two-level model can never slow the light very much. For laser-dressed atom clouds, the coupling and inter- action between a three-level atom and two lasers can drasti- cally alter the behavior of the system, including effectively 604 604Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers
5. 5. eliminating the absorption at the resonance frequency and therefore creating electromagnetically induced transparency, as shown in the problems given here. II. PROBLEMS A. Coherent population trapping The key to keeping the group velocity at the vicinity of a resonance frequency meaningful lies in the properties of the laser-dressed atomic cloud. Without such an effect, absorp- tion would be too strong to have any transmitted light. Consider that each atom in the medium has three levels. The presence of a coupling ͑dressing͒ laser (␻cӍ␻2Ϫ␻1) and a probe laser (␻Ӎ␻2Ϫ␻0) causes a mixing of the three levels, ͉0͘, ͉1͘, and ͉2͘. The Hamiltonian of such a system is HϭH0ϩH1 . ͑11͒ Here the unperturbed Hamiltonian H0 is given by ͗l͉H0͉lЈ͘ϭប␻l␦llЈ , ͑12͒ with l, lЈϭ0,1, 2. The perturbation H1 is restricted to be ͗l͉H1͉lЈ͘ϭ͗lЈ͉H1͉l͘*ϭប⍀llЈeϪi␻llЈt , ͑13͒ with ␻llЈϭ␻lϪ␻lЈ and ⍀llϭ⍀01ϭ⍀10ϭ0. Note that ␻2 Ͼ␻1Ͼ␻0ϭ0 and ␻l0ϭ␻l . This is a so-called ‘‘⌳’’ system with the highest level coupled to two lower levels. For the Hamiltonian given, ﬁnd the time-dependent wave function ͉␺͑t͒͘ϭ͚lϭ0 2 cl͑t͉͒l͘, ͑14͒ if ͉␺(0)͘ϭc0(0)͉0͘ϩc1(0)͉1͘ with ͉c0(0)͉2 ϩ͉c1(0)͉2 ϭ1. Discuss the condition for c2(t)ϵ0 and its implication. B. Electromagnetically induced transparency If we deﬁne a density matrix ␳͑t͒ϭ͉␺͑t͒͗͘␺͑t͉͒, ͑15͒ whose diagonal elements are the probabilities of occupying speciﬁc states and off-diagonal elements represent the tran- sition rates between two given states, we have iប ‫␳ץ‬ ‫ץ‬t ϭ͓H,␳͔, ͑16͒ from the Schro¨dinger equation. The interactions between at- oms in the cloud can cause a ﬁnite linewidth and decay of each level, which can be accounted for by a relaxation ma- trix: ͗l͉⌫͉lЈ͘ϭ2␥l␦llЈ , ͑17͒ and change Eq. ͑16͒ into iប ‫␳ץ‬ ‫ץ‬t ϭ͓H,␳͔Ϫ iប 2 ͑⌫␳ϩ␳⌫͒. ͑18͒ Assuming that only the dominant decaying factor is nonzero, that is, ␥2ϭ␥ and ␥0,1ϭ0, and that the atom is in the ground state at tϭ0, show that ⑀͑␻͒ϭ⑀0ͫ1ϩ ␻d͑␻Ϫ␻2͒ ␻R 2 /4Ϫ͑␻Ϫ␻2͒2 Ϫi␥͑␻Ϫ␻2͒ͬ, ͑19͒ where ␻dϭna͉p20͉2 /ប⑀0 with ͉p20͉ being the coupling dipole strength between ͉2͘ and ͉0͘ and ␻Rϭ2͉⍀21͉ the Rabi angu- lar frequency between ͉2͘ and ͉1͘. C. The slowest light In the recent experiment, Hau and co-workers have suc- cessfully reduced the group velocity of light in a cold, laser- dressed sodium atom cloud to 1 mile per hour ͑0.45 m/s͒.1 Each sodium atom can be approximated well by a three-level system. Assume that the permittivity of such a laser-dressed atom cloud is given by Eq. ͑19͒ and the frequency of the probe laser (␻/2␲) is near the resonance frequency ͑␻2/2␲ Ӎ5.1ϫ1014 Hz for sodium atom͒. Estimate the number den- sity of the atom cloud needed in order to have vg Ӎ0.45 m/s. Assume that the Rabi angular frequency is about ␻Rϭ3.5ϫ107 rad/s and the coupling dipole strength is about ͉p20͉Ӎ2.5ϫ10Ϫ29 C m. III. SOLUTIONS A. Coherent population trapping From the time-dependent Schro¨dinger equation iប ‫ץ‬͉␺͑t͒͘ ‫ץ‬t ϭH͉␺͑t͒͘, ͑20͒ we have ic˙0͑t͒ϭ␻0c0͑t͒ϩ⍀20ei␻2t c2͑t͒, ͑21͒ ic˙1͑t͒ϭ␻1c1͑t͒ϩ⍀21ei␻21t c2͑t͒, ͑22͒ ic˙2͑t͒ϭ␻2c2͑t͒ϩ⍀02eϪi␻2t c0͑t͒ϩ⍀12eϪi␻21t c1͑t͒. ͑23͒ If we redeﬁne the coefﬁcients by cl͑t͒ϭeϪi␻lt bl͑t͒, ͑24͒ the equation set is simpliﬁed to ib˙ 0͑t͒ϭ⍀20b2͑t͒, ͑25͒ ib˙ 1͑t͒ϭ⍀21b2͑t͒, ͑26͒ ib˙ 2͑t͒ϭ⍀02b0͑t͒ϩ⍀12b1͑t͒. ͑27͒ Multiplying Eq. ͑25͒ with ⍀02 and Eq. ͑26͒ with ⍀12 and adding them together, and substituting the resulting equation into Eq. ͑27͒ after taking one more time derivative, we ob- tain b¨ 2͑t͒ϭϪ͉͑⍀20͉2 ϩ͉⍀21͉2 ͒b2͑t͒. ͑28͒ We have used ⍀02ϭ⍀20* and ⍀12ϭ⍀21* . So we have b2͑t͒ϭAei⍀t ϩBeϪi⍀t , ͑29͒ with ⍀ϭͱ͉⍀20͉2 ϩ͉⍀21͉2 . Taking the initial condition b2(0)ϭc2(0)ϭ0, we arrive at b2͑t͒ϭC sin ⍀t, ͑30͒ with C being a constant. Substituting this result back into Eqs. ͑25͒ and ͑26͒, we have b0͑t͒ϭ͓c0͑0͒Ϫ␣͔cos ⍀tϩ␣, ͑31͒ b1͑t͒ϭ͓c1͑0͒Ϫ␤͔cos ⍀tϩ␤, ͑32͒ where ␣ and ␤ are constants constrained by 605 605Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
6. 6. ⍀02␣ϩ⍀12␤ϭ0. ͑33͒ We have used the initial conditions b0(0)ϭc0(0) and b1(0)ϭc1(0). The coefﬁcient C is given by Cϭ i ⍀ ͓⍀02c0͑0͒ϩ⍀12c1͑0͔͒. ͑34͒ If c0(0) and c1(0) are such that Cϵ0, we have c2(t)ϵ0 all the time. A typical case is ͉⍀02͉ϭ͉⍀12͉ and ͉c0(0)͉ ϭ͉c1(0)͉ϭ1/&, with the total phase difference between the two terms being ␲. So the state ͉2͘ will stay empty and the atoms are trapped in the lower states. The effect of such a coherent population trapping is that the absorption or emis- sion of light is completely eliminated. B. Electromagnetically induced transparency Consider that the traveling ͑probing͒ laser is described by a time-dependent electric ﬁeld E(t)ϭE0eϪi␻t with ␻ very close to ␻2 . The perturbation from such a ﬁeld is ͗2͉H1͉0͘ϭϪ͗2͉p͉0͘E0eϪi␻t ϭប⍀20eϪi␻t , ͑35͒ where p is the dipole moment induced by the ﬁeld. Now if we examine the density matrix elements between two states, ␳llЈϭ͗l͉␳͉lЈ͘, we have i ‫␳ץ‬20 ‫ץ‬t ϭ͑␻2Ϫi␥͒␳20ϩ⍀21eϪi␻21t ␳10 ϩ⍀20eϪi␻t ͑␳00Ϫ␳22͒, ͑36͒ i ‫␳ץ‬10 ‫ץ‬t ϭ␻1␳10ϩ⍀12eϪi␻12t ␳20Ϫ⍀20eϪi␻2t ␳12 . ͑37͒ We have used ͚lϭ0 2 ͉l͗͘l͉ϭ1 ͑38͒ in deriving the above equations. We can then replace ␳00 , ␳22 , and ␳12 by their values at tϭ0, that is, ␳00ϭ1, ␳22ϭ0, and ␳12ϭ0, and change a variable with ␨10ϭ␳10eϪi␻21t , be- cause we are only looking for the linear solution. Then we have i ‫␳ץ‬20 ‫ץ‬t ϭ͑␻2Ϫi␥͒␳20ϩ⍀21␨10ϩ⍀20eϪi␻t , ͑39͒ i ‫␨ץ‬10 ‫ץ‬t ϭ␻2␨10ϩ⍀12␳20 . ͑40͒ This equation set resembles a harmonic oscillator under damping and driving forces. The steady solutions are there- fore given by ␳20͑t͒ϭAeϪi␻t , ͑41͒ ␨10͑t͒ϭBeϪi␻t . ͑42͒ Substituting the above solutions into the equations, we obtain Aϭ ⍀20͑␻Ϫ␻2͒ ͑␻Ϫ␻2ϩi␥͒͑␻Ϫ␻2͒Ϫ͉⍀21͉2 . ͑43͒ Because ␳20 represents the dipole transition rate between ͉2͘ and 0͘, the polarization of the system is given by P ϭna␳20p02ϭ(⑀Ϫ⑀0)E(t) with p02ϭ͗0͉p͉2͘ϭp20* . Then we reach Eq. ͑19͒. C. The slowest light We know that the group velocity is given by vgϭ d␻ dk ϭ c nϩ␻͑dn/d␻͒ . ͑44͒ For all known materials, nϳO(1). So if vgӶc, we must have vgӍ c ␻͑dn/d␻͒ . ͑45͒ For vgϭ0.45 m/s, as observed in the experiment by Hau’s group,1 one must have ␻͑dn/d␻͒Ӎ6.7ϫ108 . ͑46͒ From the given permittivity, we have nϩi c␣ 2␻ Ӎ1ϩ 1 2 ␻d͑␻Ϫ␻2͒ ␻R 2 /4Ϫ͑␻Ϫ␻2͒2 Ϫi␥͑␻Ϫ␻2͒ . ͑47͒ Considering that ␻ is very close to ␻2 , we have nϩi c␣ 2␻ Ӎ1ϩ 2␻d͑␻Ϫ␻2͒ ␻R 2 ϫͫ1ϩ 4͑␻Ϫ␻2͒2 ␻R 2 ϩ i4␥͑␻Ϫ␻2͒ ␻R 2 ϩ¯ͬ, ͑48͒ which gives ␻͑dn/d␻͒Ӎ 2␻ ប⑀0 na͉p20͉2 ␻R 2 . ͑49͒ We have used ␻dϭna͉p20͉2 /ប⑀0 . With the numerical values of the quantities given, we then obtain naӍ2ϫ1020 mϪ3 , a density quite difﬁcult to achieve experimentally. Note that the absorption coefﬁcient ␣ is zero at the reso- nance frequency. This is the essence of the electromagneti- cally induced transparency, a condition that must be met in order to have a signiﬁcant light transmission at the resonance frequency. Otherwise, the drastically slowed group velocity of light observed by Hau’s group would not have been pos- sible. a͒ Electronic mail: pang@nevada.edu 1 L. V. Hau, presentation at the American Association for the Advancement of Science, February 2000, Washington, DC. 2 L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, ‘‘Light speed reduction to 17 meters per second in an ultracold atomic gas,’’ Nature ͑London͒ 397, 594–598 ͑1999͒. 3 For a recent review, see R. Y. Chiao and A. M. Steinberg, Tunneling Times and Superluminality, Progress in Optics Vol. 37, edited by E. Wolf ͑Elsevier, Amsterdam, 1997͒, pp. 347–405. 4 A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, ‘‘Electromagnetically introduced transparency: Propagation dynamics,’’ Phys. Rev. Lett. 74, 2447–2450 ͑1995͒. 5 S. E. Harris, ‘‘Electromagnetically induced transparency,’’ Phys. Today 50 ͑7͒, 36–42 ͑1997͒. 6 M. O. Scully and M. S. Zubairy, Quantum Optics ͑Cambridge U.P., Cam- bridge, 1997͒, Secs. 7.2 and 7.3. 7 D. J. Jackson, Classical Electrodynamics ͑Wiley, New York, 1999͒, 3rd ed., Secs. 7.5 and 7.8. 606 606Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
7. 7. Negative group velocity Kirk T. McDonalda) Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 ͑Received 21 August 2000; accepted 21 September 2000͒ ͓DOI: 10.1119/1.1331304͔ I. PROBLEM Consider a variant on the physical situation of ‘‘slow light’’ 1,2 in which two closely spaced spectral lines are now both optically pumped to show that the group velocity can be negative at the central frequency, which leads to apparent superluminal behavior. A. Negative group velocity In more detail, consider a classical model of matter in which spectral lines are associated with oscillators. In par- ticular, consider a gas with two closely spaced spectral lines of angular frequencies ␻1,2ϭ␻0Ϯ⌬/2, where ⌬Ӷ␻0 . Each line has the same damping constant ͑and spectral width͒ ␥. Ordinarily, the gas would exhibit strong absorption of light in the vicinity of the spectral lines. But suppose that lasers of frequencies ␻1 and ␻2 pump both oscillators into inverted populations. This can be described classically by assigning negative oscillator strengths to these oscillators.3 Deduce an expression for the group velocity vg(␻0) of a pulse of light centered on frequency ␻0 in terms of the ͑uni- valent͒ plasma frequency ␻p of the medium, given by ␻p 2 ϭ 4␲Ne2 m , ͑1͒ where N is the number density of atoms, and e and m are the charge and mass of an electron. Give a condition on the line separation ⌬ compared to the linewidth ␥ such that the group velocity vg(␻0) is negative. In a recent experiment by Wang et al.,4 a group velocity of vgϭϪc/310, where c is the speed of light in vacuum, was demonstrated in cesium vapor using a pair of spectral lines with separation ⌬/2␲Ϸ2 MHz and linewidth ␥/2␲ Ϸ0.8 MHz. B. Propagation of a monochromatic plane wave Consider a wave with electric ﬁeld E0ei␻(z/cϪt) that is incident from zϽ0 on a medium that extends from zϭ0 to a. Ignore reﬂection at the boundaries, as is reasonable if the index of refraction n(␻) is near unity. Particularly simple results can be obtained when you make the ͑unphysical͒ as- sumption that the ␻n(␻) varies linearly with frequency about a central frequency ␻0 . Deduce a transformation that has a frequency-dependent part and a frequency-independent part between the phase of the wave for zϽ0 to that of the wave inside the medium, and to that of the wave in the region aϽz. C. Fourier analysis Apply the transformations between an incident monochro- matic wave and the wave in and beyond the medium to the Fourier analysis of an incident pulse of form f(z/cϪt). D. Propagation of a sharp wave front In the approximation that ␻n varies linearly with ␻, de- duce the waveforms in the regions 0ϽzϽa and aϽz for an incident pulse ␦(z/cϪt), where ␦ is the Dirac delta function. Show that the pulse emerges out of the gain region at zϭa at time tϭa/vg , which appears to be earlier than when it enters this region if the group velocity is negative. Show also that inside the negative group velocity medium a pulse propa- gates backwards from zϭa at time tϭa/vgϽ0 to zϭ0 at t ϭ0, at which time it appears to annihilate the incident pulse. E. Propagation of a Gaussian pulse As a more physical example, deduce the waveforms in the regions 0ϽzϽa and aϽz for a Gaussian incident pulse E0eϪ(z/cϪt)2/2␶2 ei␻0(z/cϪt) . Carry the frequency expansion of ␻n(␻) to second order to obtain conditions of validity of the analysis such as maximum pulse width ␶, maximum length a of the gain region, and maximum time of advance of the emerging pulse. Consider the time required to generate a pulse of rise time ␶ when assessing whether the time advance in a negative group velocity medium can lead to superlumi- nal signal propagation. II. SOLUTION The concept of group velocity appears to have been ﬁrst enunciated by Hamilton in 1839 in lectures of which only abstracts were published.5 The ﬁrst recorded observation of the group velocity of a ͑water͒ wave is due to Russell in 1844.6 However, widespread awareness of the group velocity dates from 1876 when Stokes used it as the topic of a Smith’s Prize examination paper.7 The early history of group velocity has been reviewed by Havelock.8 H. Lamb9 credits A. Schuster with noting in 1904 that a negative group velocity, i.e., a group velocity of opposite sign to that of the phase velocity, is possible due to anoma- lous dispersion. Von Laue10 made a similar comment in 1905. Lamb gave two examples of strings subject to external potentials that exhibit negative group velocities. These early considerations assumed that in case of a wave with positive group and phase velocities incident on the anomalous me- dium, energy would be transported into the medium with a positive group velocity, and so there would be waves with negative phase velocity inside the medium. Such negative phase velocity waves are formally consistent with Snell’s 607 607Am. J. Phys. 69 ͑5͒, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers
8. 8. law11 ͑since ␪tϭsinϪ1 ͓(ni /nt)sin ␪i͔ can be in either the ﬁrst or second quadrant͒, but they seemed physically implausible and the topic was largely dropped. Present interest in negative group velocity is based on anomalous dispersion in a gain medium, where the sign of the phase velocity is the same for incident and transmitted waves, and energy ﬂows inside the gain medium in the op- posite direction to the incident energy ﬂow in vacuum. The propagation of electromagnetic waves at frequencies near those of spectral lines of a medium was ﬁrst extensively discussed by Sommerfeld and Brillouin,12 with emphasis on the distinction between signal velocity and group velocity when the latter exceeds c. The solution presented here is based on the work of Garrett and McCumber,13 as extended by Chiao et al.14,15 A discussion of negative group velocity in electronic circuits has been given by Mitchell and Chiao.16 A. Negative group velocity In a medium of index of refraction n(␻), the dispersion relation can be written kϭ ␻n c , ͑2͒ where k is the wave number. The group velocity is then given by vgϭReͫd␻ dk ͬϭ 1 Re͓dk/d␻͔ ϭ c Re͓d͑␻n͒/d␻͔ ϭ c nϩ␻ Re͓dn/d␻͔ . ͑3͒ We see from Eq. ͑3͒ that if the index of refraction de- creases rapidly enough with frequency, the group velocity can be negative. It is well known that the index of refraction decreases rapidly with frequency near an absorption line, where ‘‘anomalous’’ wave propagation effects can occur.12 However, the absorption makes it difﬁcult to study these effects. The insight of Garrett and McCumber13 and of Chiao et al.14,15,17–19 is that demonstrations of negative group ve- locity are possible in media with inverted populations, so that gain rather than absorption occurs at the frequencies of interest. This was dramatically realized in the experiment of Wang et al.4 by use of a closely spaced pair of gain lines, as perhaps ﬁrst suggested by Steinberg and Chiao.17 We use a classical oscillator model for the index of refrac- tion. The index n is the square root of the dielectric constant ⑀, which is in turn related to the atomic polarizability ␣ ac- cording to Dϭ⑀EϭEϩ4␲PϭE͑1ϩ4␲N␣͒ ͑4͒ ͑in Gaussian units͒, where D is the electric displacement, E is the electric ﬁeld, and P is the polarization density. Then, the index of refraction of a dilute gas is nϭͱ⑀Ϸ1ϩ2␲N␣. ͑5͒ The polarizability ␣ is obtained from the electric dipole moment pϭexϭ␣E induced by electric ﬁeld E. In the case of a single spectral line of frequency ␻j , we say that an electron is bound to the ͑ﬁxed͒ nucleus by a spring of con- stant Kϭm␻j 2 , and that the motion is subject to the damping force Ϫm␥jx˙, where the dot indicates differentiation with respect to time. The equation of motion in the presence of an electromagnetic wave of frequency ␻ is x¨ϩ␥jx˙ϩ␻j 2 xϭ eE m ϭ eE0 m ei␻t . ͑6͒ Hence, xϭ eE m 1 ␻j 2 Ϫ␻2 Ϫi␥j␻ ϭ eE m ␻j 2 Ϫ␻2 ϩi␥j␻ ͑␻j 2 Ϫ␻2 ͒2 ϩ␥j 2 ␻2 , ͑7͒ and the polarizability is ␣ϭ e2 m ␻j 2 Ϫ␻2 ϩi␥j␻ ͑␻j 2 Ϫ␻2 ͒2 ϩ␥j 2 ␻2 . ͑8͒ In the present problem we have two spectral lines, ␻1,2 ϭ␻0Ϯ⌬/2, both of oscillator strength Ϫ1 to indicate that the populations of both lines are inverted, with damping con- stants ␥1ϭ␥2ϭ␥. In this case, the polarizability is given by ␣ϭϪ e2 m ͑␻0Ϫ⌬/2͒2 Ϫ␻2 ϩi␥␻ ͑͑␻0Ϫ⌬/2͒2 Ϫ␻2 ͒2 ϩ␥2 ␻2 Ϫ e2 m ͑␻0ϩ⌬/2͒2 Ϫ␻2 ϩi␥␻ ͑͑␻0ϩ⌬/2͒2 Ϫ␻2 ͒2 ϩ␥2 ␻2 ϷϪ e2 m ␻0 2 Ϫ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 Ϫ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 Ϫ e2 m ␻0 2 ϩ2⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 ϩ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 , ͑9͒ where the approximation is obtained by the neglect of terms in ⌬2 compared to those in ⌬␻0 . For a probe beam at frequency ␻, the index of refraction ͑5͒ has the form n͑␻͒Ϸ1Ϫ ␻p 2 2 ͫ ␻0 2 Ϫ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 Ϫ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2 ϩ ␻0 2 ϩ⌬␻0Ϫ␻2 ϩi␥␻ ͑␻0 2 ϩ⌬␻0Ϫ␻2 ͒2 ϩ␥2 ␻2ͬ, ͑10͒ where ␻p is the plasma frequency given by Eq. ͑1͒. This is illustrated in Fig. 1. The index at the central frequency ␻0 is Fig. 1. The real and imaginary parts of the index of refraction in a medium with two spectral lines that have been pumped to inverted populations. The lines are separated by angular frequency ⌬ and have widths ␥ϭ0.4⌬. 608 608Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
9. 9. n͑␻0͒Ϸ1Ϫi ␻p 2 ␥ ͑⌬2 ϩ␥2 ͒␻0 Ϸ1Ϫi ␻p 2 ⌬2 ␥ ␻0 , ͑11͒ where the second approximation holds when ␥Ӷ⌬. The electric ﬁeld of a continuous probe wave then propagates according to E͑z,t͒ϭei͑kzϪ␻0t͒ ϭei␻͑n͑␻0͒z/cϪt͒ Ϸez/͓⌬2c/␥␻͑2/p͔͒ ei␻0͑z/cϪt͒ . ͑12͒ From this we see that at frequency ␻0 the phase velocity is c, and the medium has an amplitude gain length ⌬2 c/␥␻p 2 . To obtain the group velocity ͑3͒ at frequency ␻0 , we need the derivative d͑␻n͒ d␻ ͯ␻0 Ϸ1Ϫ 2␻p 2 ͑⌬2 Ϫ␥2 ͒ ͑⌬2 ϩ␥2 ͒2 , ͑13͒ where we have neglected terms in ⌬ and ␥ compared to ␻0 . From Eq. ͑3͒, we see that the group velocity can be negative if ⌬2 ␻p 2Ϫ ␥2 ␻p 2 у 1 2 ͩ⌬2 ␻p 2 ϩ ␥2 ␻p 2 ͪ2 . ͑14͒ The boundary of the allowed region ͑14͒ in (⌬2 ,␥2 ) space is a parabola whose axis is along the line ␥2 ϭϪ⌬2 , as shown in Fig. 2. For the physical region ␥2 у0, the boundary is given by ␥2 ␻p 2 ϭͱ1ϩ4 ⌬2 ␻p 2Ϫ1Ϫ ⌬2 ␻p 2 . ͑15͒ Thus, to have a negative group velocity, we must have ⌬р&␻p , ͑16͒ which limit is achieved when ␥ϭ0; the maximum value of ␥ is 0.5␻p when ⌬ϭ0.866␻p . Near the boundary of the negative group velocity region, ͉vg͉ exceeds c, which alerts us to concerns of superluminal behavior. However, as will be seen in the following sections, the effect of a negative group velocity is more dramatic when ͉vg͉ is small rather than large. The region of recent experimental interest is ␥Ӷ⌬Ӷ␻p , for which Eqs. ͑3͒ and ͑13͒ predict that vgϷϪ c 2 ⌬2 ␻p 2 . ͑17͒ A value of vgϷϪc/310 as in the experiment of Wang cor- responds to ⌬/␻pϷ1/12. In this case, the gain length ⌬2 c/␥␻p 2 was approximately 40 cm. For later use we record the second derivative, d2 ͑␻n͒ d␻2 ͯ␻0 Ϸ8i ␻p 2 ␥͑3⌬2 Ϫ␥2 ͒ ͑⌬2 ϩ␥2 ͒3 Ϸ24i ␻p 2 ⌬2 ␥ ⌬2 , ͑18͒ where the second approximation holds if ␥Ӷ⌬. B. Propagation of a monochromatic plane wave To illustrate the optical properties of a medium with nega- tive group velocity, we consider the propagation of an elec- tromagnetic wave through it. The medium extends from z ϭ0 to a, and is surrounded by vacuum. Because the index of refraction ͑10͒ is near unity in the frequency range of inter- est, we ignore reﬂections at the boundaries of the medium. A monochromatic plane wave of frequency ␻ and incident from zϽ0 propagates with phase velocity c in vacuum. Its electric ﬁeld can be written E␻͑z,t͒ϭE0ei␻z/c eϪi␻t ͑zϽ0͒. ͑19͒ Inside the medium this wave propagates with phase velocity c/n(␻) according to E␻͑z,t͒ϭE0ei␻nz/c eϪi␻t ͑0ϽzϽa͒, ͑20͒ where the amplitude is unchanged since we neglect the small reﬂection at the boundary zϭ0. When the wave emerges into vacuum at zϭa, the phase velocity is again c, but it has accumulated a phase lag of (␻/c)(nϪ1)a, and so appears as E␻͑z,t͒ϭE0ei␻a͑nϪ1͒/c ei␻z/c eϪi␻t ϭE0ei␻an/c eϪi␻͑tϪ͑zϪa͒/c͒ ͑aϽz͒. ͑21͒ It is noteworthy that a monochromatic wave for zϾa has the same form as that inside the medium if we make the frequency-independent substitutions z→a, t→tϪ zϪa c . ͑22͒ Since an arbitrary waveform can be expressed in terms of monochromatic plane waves via Fourier analysis, we can use these substitutions to convert any wave in the region 0Ͻz Ͻa to its continuation in the region aϽz. A general relation can be deduced in the case where the second and higher derivatives of ␻n(␻) are very small. We can then write ␻n͑␻͒Ϸ␻0n͑␻0͒ϩ c vg ͑␻Ϫ␻0͒, ͑23͒ where vg is the group velocity for a pulse with central fre- quency ␻0 . Using this in Eq. ͑20͒, we have E␻͑z,t͒ϷE0ei␻0z͑n͑␻0͒/cϪ1/vg͒ ei␻z/vgeϪi␻t ͑0ϽzϽa͒. ͑24͒ In this approximation, the Fourier component E␻(z) at fre- quency ␻ of a wave inside the gain medium is related to that of the incident wave by replacing the frequency dependence Fig. 2. The allowed region ͑14͒ in (⌬2 ,␥2 ) space such that the group ve- locity is negative. 609 609Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
10. 10. ei␻z/c by ei␻z/vg, i.e., by replacing z/c by z/vg , and multi- plying by the frequency-independent phase factor ei␻0z(n(␻0)/cϪ1/vg) . Then, using transformation ͑22͒, the wave that emerges into vacuum beyond the medium is E␻͑z,t͒ϷE0ei␻0a͑n͑␻0͒/cϪ1/vg͒ ϫei␻͑z/cϪa͑1/cϪ1/vg͒͒ eϪi␻t ͑aϽz͒. ͑25͒ The wave beyond the medium is related to the incident wave by multiplying by a frequency-independent phase, and by replacing z/c by z/cϪa(1/cϪ1/vg) in the frequency- dependent part of the phase. The effect of the medium on the wave as described by Eqs. ͑24͒ and ͑25͒ has been called ‘‘rephasing.’’ 4 C. Fourier analysis and ‘‘rephasing’’ The transformations between the monochromatic incident wave ͑19͒ and its continuation in and beyond the medium, ͑24͒ and ͑25͒, imply that an incident wave E͑z,t͒ϭf͑z/cϪt͒ϭ ͵Ϫϱ ϱ E␻͑z͒eϪi␻t d␻ ͑zϽ0͒, ͑26͒ whose Fourier components are given by E␻͑z͒ϭ 1 2␲ ͵Ϫϱ ϱ E͑z,t͒ei␻t dt, ͑27͒ propagates as E͑z,t͒Ϸ Ά f͑z/cϪt͒ ͑zϽ0͒ ei␻0z͑n͑␻0͒/cϪ1/vg͒ f͑z/vgϪt͒ ͑0ϽzϽa͒ ei␻0a͑n͑␻0͒/cϪ1/vg͒ f͑z/cϪtϪa͑1/cϪ1/vg͒͒ ͑aϽz͒. ͑28͒ An interpretation of Eq. ͑28͒ in terms of ‘‘rephasing’’ is as follows. Fourier analysis tells us that the maximum ampli- tude of a pulse made of waves of many frequencies, each of the form E␻(z,t)ϭE0(␻)ei␾(␻) ϭE0(␻)ei(k(␻)zϪ␻tϩ␾0(␻)) with E0у0, is determined by adding the amplitudes E0(␻). This maximum is achieved only if there exist points ͑z,t͒ such that all phases ␾͑␻͒ have the same value. For example, we consider a pulse in the region zϽ0 whose maximum occurs when the phases of all component frequencies vanish, as shown at the left of Fig. 3. Referring to Eq. ͑19͒, we see that the peak occurs when zϭct. As usual, we say that the group velocity of this wave is c in vacuum. Inside the medium, Eq. ͑24͒ describes the phases of the components, which all have a common frequency- independent phase ␻0z(n(␻0)/cϪ1/vg) at a given z, as well as a frequency-dependent part ␻(z/vgϪt). The peak of the pulse occurs when all the frequency-dependent phases van- ish; the overall frequency-independent phase does not affect the pulse size. Thus, the peak of the pulse propagates within the medium according to zϭvgt. The velocity of the peak is vg , the group velocity of the medium, which can be nega- tive. The ‘‘rephasing’’ ͑24͒ within the medium changes the wavelengths of the component waves. Typically the wave- length increases, and by greater amounts at longer wave- lengths. A longer time is required before the phases of the waves all become the same at some point z inside the me- dium, so in a normal medium the velocity of the peak ap- pears to be slowed down. But in a negative group velocity medium, wavelengths short compared to ␭0 are lengthened, long waves are shortened, and the velocity of the peak ap- pears to be reversed. By a similar argument, Eq. ͑25͒ tells us that in the vacuum region beyond the medium the peak of the pulse propagates according to zϭctϩa(1/cϪ1/vg). The group velocity is again c, but the ‘‘rephasing’’ within the medium results in a shift of the position of the peak by the amount a(1/c Ϫ1/vg). In a normal medium where 0Ͻvgрc the shift is negative; the pulse appears to have been delayed during its passage through the medium. But after a negative group ve- locity medium, the pulse appears to have advanced! This advance is possible because, in the Fourier view, each component wave extends over all space, even if the pulse appears to be restricted. The unusual ‘‘rephasing’’ in a negative group velocity medium shifts the phases of the fre- quency components of the wave train in the region ahead of the nominal peak such that the phases all coincide, and a peak is observed, at times earlier than expected at points beyond the medium. As shown in Fig. 3 and further illustrated in the examples in the following, the ‘‘rephasing’’ can result in the simulta- neous appearance of peaks in all three regions. Fig. 3. A snapshot of three Fourier components of a pulse in the vicinity of a negative group velocity medium. The component at the central wavelength ␭0 is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wavelength component is lengthened. Then, even when the incident pulse has not yet reached the medium, there can be a point inside the medium at which all components have the same phase, and a peak appears. Simultaneously, there can be a point in the vacuum region beyond the medium at which the Fourier components are again all in phase, and a third peak appears. The peaks in the vacuum regions move with group velocity vgϭc, but the peak inside the medium moves with a negative group velocity, shown as vgϭϪc/2. The phase velocity vp is c in vacuum, and close to c in the medium. 610 610Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems
11. 11. D. Propagation of a sharp wave front To assess the effect of a medium with negative group ve- locity on the propagation of a signal, we ﬁrst consider a waveform with a sharp front, as recommended by Sommer- feld and Brillouin.12 As an extreme but convenient example, we take the inci- dent pulse to be a Dirac delta function, E(z,t)ϭE0␦(z/c Ϫt). Inserting this in Eq. ͑28͒, which is based on the linear approximation ͑23͒, we ﬁnd E͑z,t͒Ϸ Ά E0␦͑z/cϪt͒ ͑zϽ0͒ E0ei␻0z͑n͑␻0͒/cϪ1/vg͒ ␦͑z/vgϪt͒ ͑0ϽzϽa͒ E0ei␻0a͑n͑␻0͒/cϪ1/vg͒ ␦͑z/cϪtϪa͑1/cϪ1/vg͒͒ ͑aϽz͒. ͑29͒ According to Eq. ͑29͒, the delta-function pulse emerges from the medium at zϭa at time tϭa/vg . If the group ve- locity is negative, the pulse emerges from the medium before it enters at tϭ0! A sample history of ͑Gaussian͒ pulse propagation is illus- trated in Fig. 4. Inside the negative group velocity medium, an ͑anti͒pulse propagates backwards in space from zϭa at time tϭa/vgϽ0 to zϭ0 at time tϭ0, at which point it ap- pears to annihilate the incident pulse. This behavior is analogous to barrier penetration by a rela- tivistic electron20 in which an electron can emerge from the far side of the barrier earlier than it hits the near side, if the electron emission at the far side is accompanied by positron emission, and the positron propagates within the barrier so as to annihilate the incident electron at the near side. In the Wheeler–Feynman view, this process involves only a single electron which propagates backwards in time when inside the barrier. In this spirit, we might say that pulses propagate backwards in time ͑but forward in space͒ inside a negative group velocity medium. The Fourier components of the delta function are indepen- dent of frequency, so the advanced appearance of the sharp wave front as described by Eq. ͑29͒ can occur only for a gain medium such that the index of refraction varies linearly at all frequencies. If such a medium existed with negative slope dn/d␻, then Eq. ͑29͒ would constitute superluminal signal propagation. However, from Fig. 1 we see that a linear approximation to the index of refraction is reasonable in the negative group velocity medium only for ͉␻Ϫ␻0͉Շ⌬/2. The sharpest wave front that can be supported within this bandwidth has char- acteristic rise time ␶Ϸ1/⌬. For the experiment of Wang et al. where ⌬/2␲Ϸ106 Hz, an analysis based on Eq. ͑23͒ would be valid only for pulses with ␶տ0.1 ␮s. Wang et al. used a pulse with ␶Ϸ1 ␮s, close to the minimum value for which Eq. ͑23͒ is a reason- able approximation. Since a negative group velocity can only be experienced over a limited bandwidth, very sharp wave fronts must be excluded from the discussion of signal propagation. How- ever, it is well known12 that great care must be taken when discussing the signal velocity if the waveform is not sharp. E. Propagation of a Gaussian pulse We now consider a Gaussian pulse of temporal length ␶ centered on frequency ␻0 ͑the carrier frequency͒, for which the incident waveform is Fig. 4. Ten ‘‘snapshots’’ of a Gaussian pulse as it traverses a negative group velocity region (0ϽzϽ50), according to Eq. ͑31͒. The group velocity in the gain medium is vgϭϪc/2, and c has been set to 1. 611 611Am. J. Phys., Vol. 69, No. 5, May 2001 New Problems