negative group vel

Quantum capacitor at a metal–liquid interfaceFredy R. Zypmana)

Department of Physics, Yeshiva University, New York, New York 10033

~Received 17 March 2000; accepted 4 September 2000!

@DOI: 10.1119/1.1328352#

NEW PROBLEMS

Christopher R. Gould,EditorPhysics Department, Box 8202North Carolina State University, Raleigh, North Carolina 27695

‘‘New Problems’’ solicits interesting and novel worked problems for use in undergraduate physicscourses beyond the introductory level. We seek problems that convey the excitement and interest ofcurrent developments in physics and that are useful for teaching courses such as Classical Mechanics,Electricity and Magnetism, Statistical Mechanics and Thermodynamics, ‘‘Modern’’ Physics, andQuantum Mechanics. We challenge physicists everywhere to create problems that show how contem-porary research in their various branches of physics uses the central unifying ideas of physics toadvance physical understanding. We want these problems to become an important source of ideas andinformation for students of physics and their teachers. All submissions are peer-reviewed prior topublication. Send manuscripts directly to Christopher R. Gould,Editor.

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I. PROBLEM

Consider a metal immersed in an electrolyte. As the melectrostatic potential,V, is externally varied, a layer of solute deposits on the metal surface. The charge,Q, depositedon the metal surface provides topographic information. Eperimentalists typically report thedifferential capacitanceC5dQ/dV as a function ofV, C(V). ‘‘Normally’’ C is anonconstant function ofV. However, for large solute concentrations, theC(V) curves show plateaus, that is, voltaregions in which the capacitance does not change. T‘‘anomalous’’ behavior is presented and explained in tproblem.

The complete understanding of metal–electrolyte intfaces is of great relevance in physics applied to biomedicFor example in electrocardiograms and electroencephgrams, the electrodes are in contact with the skin, whplays the role of an electrolyte~dehydrated skin is an insulator!. Another example is the connection of nerves by mtallic wires. The nerve–wire interface is of an electrolytemetal kind. In orthopedic implants, metal–metal interfacare a concern. An artificial hip must be soft and squishythe region close to the femur so as not to abrade the msofter bone, but hard everywhere else. Thus, a typical arcial hip is made of ‘‘soft’’ titanium in contact with hard alloyCo–Cr–Mo. Due to differences in chemical potential, thmetal interface is prone to corrosion.

In the formation of adlayers of mostly organic adsorbaon metals1 such as Ag, Au, and particularly liquid Hg ainteresting ‘‘cut’’ appears in the differential capacitanceCd

curve as a function of potential. The first such observatwas made by Lorenz, who studied the capacitance of a stion of nanoic acid near a mercury electrode.2 The effect wasinitially attributed to film formation via lateral interactionsand theoretically studied by Rangarajan and co-workersing mean field techniques3,4 and a lattice gas model for th

601 Am. J. Phys.69 ~5!, May 2001 http://ojps.aip.org/aj

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surface. However, more recent experiments by Wandlowand De Levie5 show a rather sharp cut in the differenticapacity curve that is very difficult to understand in termsa classical model of adlayer formation.

Figure 1 shows a typical plot of differential capacitanversus voltage for an aqueous solution–metal interface,concentrations below a critical value. The height and peposition changes with solute concentration, but the genshape remains the same.

However, above the critical concentration, there appplateaus in theC–V curve, as shown in Fig. 2.

The problem in understanding this phenomenon is tindependently of the suddenness of the formation ofmonolayer, there is no reasonable classical explanationthe almost perfectly flat region inside the cut.

Problem. Explain this behavior.Hint. Postulate the existence of a surface state of trap

electrons. A very simple model shows that indeedC5constant by invoking surface trapped electrons in a cducting state, which completely shield the metal charOnce this band is filled then the excess charge is screenethe solution and theC–V curve recovers its normal appeaance. The cuts appear when the adlayer undergoes a minsulator transition~as, for example, the Cs–Au alloy!,which is related to a percolation transition.

II. SOLUTION

We consider the electrons at the interface to be immerin the step potential due to a potential difference betweenelectrolyte and the electrode, and a thin, confining film, dposited on the surface~Fig. 3!.

We defineF1(x) as the wave function to the left of thinterface, andF2(x) as that to the right of it~Fig. 4!. Thetotal wave function is continuous across the interface, thaF1(0)5F2(0)[F(0), wherex50 is the coordinate defining the plane of the interface. Since there is a delta funct

601p/ © 2001 American Association of Physics Teachers

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atx50, the wave function may have a noncontinuous deritive. By integrating the Schro¨dinger equation in a neighborhood of the origin, the derivative difference is6

\2

2m@F28~0!2F18~0!#1aF~0!50, ~1!

where, as shown in Fig. 4,a represents the strength of thedfunction at the origin, and is, for a thin layer, the productthe potential well and its spatial extension.

We are interested in charge accumulation at the interfaThen, we look for electron bound eigenstates. These eigstates~in this particular problem there is only one! must havenegative total energyE. If they did not, they would be delocalized states. As in the case of transmission and reflecthrough a potential step, if 0,E,V0 then the electron isdelocalized forx,0 and confined within a small region fox.0. If, on the other hand,V0,E, then the electron state idelocalized for allx. In the case of a potential step~that is,our problem witha50!, there cannot be states completeconfined aroundx50, since forE,0 the only solution to theSchrodinger equation would beF[0. However, nontrivial

Fig. 1. Typical differential interfacial capacitance of mercury in contwith aqueous solution of guanidinium nitrate for concentrations be0.3 M.

602 Am. J. Phys., Vol. 69, No. 5, May 2001

-

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on

solutions can exist whenaÞ0. Thus we search for exponential solutions decaying away from the origin and wiE,0:

F1~x!5Aenx, ~2!

F2~x!5Ae2mx, ~3!

E52\2n2

2m, ~4!

E5V2\2m2

2m, ~5!

wherem andn are real constants.We next define a new parameter,b, to simplify the nota-

tion:

n5A2mV

\2 sinhb, ~6!

tFig. 2. Typical differential interfacial capacitance of mercury in contawith aqueous solution of guanidinium nitrate for concentrations ab0.3 M.

-

.

Fig. 3. Diagram of the system showing the metal, the electrolyte, the thinlayer, and the electron wave function

602New Problems

gc-

Fig. 4. Potential function representinthe metal, the interface, and the eletrolyte.

e

fa

at

the

n-thetic

thus

erteed

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foral.

s—m.

n-J.

m5A2mV

\2 coshb. ~7!

From these definitions and Eqs.~4! and ~5!, one obtains thestate given by

b51

2log

2ma2

\2V. ~8!

Let Q1 and Q2 be the net charges to the left and to thright of the interface, respectively.

Then

Q15eE2`

0

uF1~x!u2dx5eA2

2n, ~9!

where the normalization constantA is obtained by requiringthat

E2`

0

uF1~x!u2dx1E0

1`

uF2~x!u2dx51:

A252V

acoshb sinhb. ~10!

In the previous expression, we have made use of thethat Eq.~1! is equivalent tom1n52ma/\2.

Then

Q15em

m1n. ~11!

And, similarly

Q25en

m1n. ~12!

The charge at the interface capacitor can now be evalu


ct

ed

Qc5Q12Q25em2n

m1n5e

\2V

2ma2 . ~13!

The capacitance,C5dQc /dV, is

C5e\2

2ma2 , ~14!

which is a constant that depends only on the properties oflayer through the parametera.

Thus by simply adding a small confining layer at the iterface, it is possible to explain the flat characteristics oftotal capacitance. As the external voltage in the electrolycell changes, the thin layer may appear or disappear,creating ‘‘normal’’ C–V regions, and flat ones.

ACKNOWLEDGMENTS

I would like to thank Dr. Steven J. Eppell and Dr. LessBlum for useful comments. The National Cancer Instituthrough Grant No. CA77796-01 has financially supportthis work.

a!Electronic mail: [email protected]. Budevski, G. Staikov, and W. J. Lorenz,Electrochemical Phase For-mation and Growth~Wiley, New York, 1996!, pp. 200–210.

2Wolfgang Lorenz, ‘‘The rate of absorption and of two-dimensional asciation of fatty acids at the mercury-electrolyte interface,’’ Z. Elektrchem.62 ~1!, 192–199~1958!.

3M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isothermsneutral organic-compounds—A hierarchy in modeling,’’ J. ElectroanChem. Interfacial Electrochem.176 ~1-2!, 45–64~1984!.

4M. V. Sangaranarayanan and S. K. Rangarajan, ‘‘Adsorption-isothermmicroscopic modeling,’’ J. Electroanal. Chem. Interfacial Electroche176 ~1-2!, 119–137~1984!.

5T. Wandlowski, G. Jameson, and R. De Levie, ‘‘Two-Dimensional Codensation of Guadinidium Nitrate at the Mercury-Water Interface,’’Phys. Chem.97 ~39!, 10119–10126~1993!.

6C. Cohen-Tannoudji, Bernard Diu, and Frank Laloe¨, Quantum Mechanics~Wiley, New York, Paris, 1977!, Vol. 1, p. 87.

603New Problems

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I. SCOPE

Imagine that you are taking a quick walk and you arefact traveling faster than light. This is what has beachieved in a recent experiment by Lene Hau andco-workers.1 In that experiment, the group velocity ofpulsed laser was effectively reduced to about 1 mile per h~0.45 m/s! in a cold, laser-dressed sodium atom cloud. Inearlier experiment, the same group had successfully slothe group velocity of light to 38 miles per hour~17 m/s! in asimilar system.2

Propagation of light in a medium is a well-studied subjeeven though there have been some very tough questsuch as the group velocity exceeding the speed of lighvacuum.3 Five years ago, it was demonstrated for the fitime that the speed of light can be reduced significantly4 in acold atom cloud that is in a state called electromagneticinduced transparency.5,6 What has made the experimentwork of Hauet al. so unique is that the orders of magnituin the reduction of speed of light, which cannot commonbe accomplished by increasing the index of refraction,achieved instead by the extremely rapid variation of thedex with the frequency, and the elimination of light absotion at resonance frequency—a quantum phenomenon reing from the coupling and interaction between lasers aelectrons at different atomic levels. Here we would likehighlight some basic understanding of this exciting phenoenon.

From the Maxwell equations for a propagating electmagnetic wave with angular frequencyv and complex wavevectork in a nonconducting medium, we have7

k25mev2, ~1!

wherem is the magnetic permeability ande is the electricpermittivity of the medium. If we assume thatv is real and

k5k1 ia

2, ~2!

where k is the real propagating wave vector anda is theabsorption coefficient, we have

n5ck

v5ReA me

m0e0, ~3!

a52v Im Ame, ~4!

with c51/Am0e0 being the speed of light in vacuum andm0

and e0 the vacuum permeability and permittivity, respetively. In most cases, we havem.m0 , a condition that isassumed here. From the above relation, we find the phvelocity of the wave,

vp5v

k5

c

n, ~5!


r

urned

tns,int

ly

s--lt-d

-

-

se

which characterizes how fast the wave changes its phasethe field propagating along thez direction is proportional toeikz2az/22 ivt.

Now let us consider that the electric field is a nondecay(a50) wave packet with a range of frequenciesv5v(k):

E~z,t !5E dk A~k!eikz2 ivt, ~6!

whereA(k) is a narrow function peaked atk5k0 . We canthen expandv(k) as

v~k!5v~k0!1~k2k0!dv

dkUk5k0

1O@~k2k0!2#, ~7!

if it is well behaved, that is, a smooth function around tgiven wave vectork0 . If only the zeroth- and first-ordeterms are kept in the expansion, we have

E~z,t !.ei @k0vg2v~k0!#tE dk A~k!eikz2 ikvgt

5ei @k0vg2v~k0!#tE~z2vgt,0!, ~8!

where

vg5dv

dkUk5k0

~9!

is termed the group velocity of the wave atk5k0 becausethe packet acts if it is traveling in space with such a velocwithout changing its shape, with an overall phase chanUsing Eq.~3!, we obtain

vg5vp

11~v/n!~dn/dv!. ~10!

Note that we have assumed thatn is a function ofv throughk. We can consider the group velocity to be the velocitythe wave packet, that is, the velocity of the energy andformation contained in the packet, if the linear term in tabove Taylor expansion is the dominant term. However,meaning of the group velocity can change if the wave pacbecomes incoherent. Care must be taken when the angfrequency of the wave is near a resonance ordn/dv,0.3

Quantum mechanically, the resonance occurs whenfrequency of light matches the energy difference betwetwo allowed quantum levels in the system and is typicaaccompanied by strong absorption under normal circustances. This is why normal matter that can be well appromated by a two-level model can never slow the light vemuch. For laser-dressed atom clouds, the coupling and inaction between a three-level atom and two lasers can drcally alter the behavior of the system, including effective

Electromagnetically induced transparencyTao Panga)

Department of Physics, University of Nevada, Las Vegas, Nevada 89154-40

~Received 6 June 2000; accepted 21 September 2000!

@DOI: 10.1119/1.1331303#


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eliminating the absorption at the resonance frequencytherefore creating electromagnetically induced transpareas shown in the problems given here.

II. PROBLEMS

A. Coherent population trapping

The key to keeping the group velocity at the vicinity ofresonance frequency meaningful lies in the properties oflaser-dressed atomic cloud. Without such an effect, abstion would be too strong to have any transmitted light.

Consider that each atom in the medium has three levThe presence of a coupling~dressing! laser (vc.v22v1)and a probe laser (v.v22v0) causes a mixing of the threlevels, u0&, u1&, and u2&.

The Hamiltonian of such a system is

H5H01H1 . ~11!

Here the unperturbed HamiltonianH0 is given by

^ l uH0u l 8&5\v ld l l 8 , ~12!

with l, l 850,1, 2. The perturbationH1 is restricted to be

^ l uH1u l 8&5^ l 8uH1u l &* 5\V l l 8e2 iv l l 8t, ~13!

with v l l 85v l2v l 8 and V l l 5V015V1050. Note thatv2

.v1.v050 andv l05v l . This is a so-called ‘‘L’’ systemwith the highest level coupled to two lower levels.

For the Hamiltonian given, find the time-dependent wafunction

uc~ t !&5(l 50

2

cl~ t !u l &, ~14!

if uc(0)&5c0(0)u0&1c1(0)u1& with uc0(0)u21uc1(0)u251.Discuss the condition forc2(t)[0 and its implication.

B. Electromagnetically induced transparency

If we define a density matrix

r~ t !5uc~ t !&^c~ t !u, ~15!

whose diagonal elements are the probabilities of occupyspecific states and off-diagonal elements represent thesition rates between two given states, we have

i\]r

]t5@H,r#, ~16!

from the Schro¨dinger equation. The interactions betweenoms in the cloud can cause a finite linewidth and decayeach level, which can be accounted for by a relaxation mtrix:

^ l uGu l 8&52g ld l l 8 , ~17!

and change Eq.~16! into

i\]r

]t5@H,r#2

i\

2~Gr1rG!. ~18!

Assuming that only the dominant decaying factor is nonzethat is,g25g andg0,150, and that the atom is in the grounstate att50, show that

e~v!5e0F11vd~v2v2!

vR2/42~v2v2!22 ig~v2v2!G , ~19!


dy,

ep-

ls.

e

gn-

-f

a-

,

wherevd5naup20u2/\e0 with up20u being the coupling dipolestrength betweenu2& and u0& andvR52uV21u the Rabi angu-lar frequency betweenu2& and u1&.

C. The slowest light

In the recent experiment, Hau and co-workers have scessfully reduced the group velocity of light in a cold, lasedressed sodium atom cloud to 1 mile per hour~0.45 m/s!.1

Each sodium atom can be approximated well by a three-lesystem. Assume that the permittivity of such a laser-dresatom cloud is given by Eq.~19! and the frequency of theprobe laser (v/2p) is near the resonance frequency~v2/2p.5.131014Hz for sodium atom!. Estimate the number density of the atom cloud needed in order to havevg

.0.45 m/s. Assume that the Rabi angular frequency is abvR53.53107 rad/s and the coupling dipole strength is aboup20u.2.5310229C m.

III. SOLUTIONS

A. Coherent population trapping

From the time-dependent Schro¨dinger equation

i\]uc~ t !&

]t5Huc~ t !&, ~20!

we have

i c0~ t !5v0c0~ t !1V20eiv2tc2~ t !, ~21!

i c1~ t !5v1c1~ t !1V21eiv21tc2~ t !, ~22!

i c2~ t !5v2c2~ t !1V02e2 iv2tc0~ t !1V12e

2 iv21tc1~ t !.~23!

If we redefine the coefficients by

cl~ t !5e2 iv l tbl~ t !, ~24!

the equation set is simplified to

i b0~ t !5V20b2~ t !, ~25!

i b1~ t !5V21b2~ t !, ~26!

i b2~ t !5V02b0~ t !1V12b1~ t !. ~27!

Multiplying Eq. ~25! with V02 and Eq.~26! with V12 andadding them together, and substituting the resulting equainto Eq. ~27! after taking one more time derivative, we obtain

b2~ t !52~ uV20u21uV21u2!b2~ t !. ~28!

We have usedV025V20* andV125V21* . So we have

b2~ t !5AeiVt1Be2 iVt, ~29!

with V5AuV20u21uV21u2. Taking the initial conditionb2(0)5c2(0)50, we arrive at

b2~ t !5C sinVt, ~30!

with C being a constant. Substituting this result back inEqs.~25! and ~26!, we have

b0~ t !5@c0~0!2a#cosVt1a, ~31!

b1~ t !5@c1~0!2b#cosVt1b, ~32!

wherea andb are constants constrained by

605New Problems

he

hi

ifte

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dture

lf

ly

ay

V02a1V12b50. ~33!

We have used the initial conditionsb0(0)5c0(0) andb1(0)5c1(0). ThecoefficientC is given by

C5i

V@V02c0~0!1V12c1~0!#. ~34!

If c0(0) andc1(0) are such thatC[0, we havec2(t)[0 allthe time. A typical case isuV02u5uV12u and uc0(0)u5uc1(0)u51/&, with the total phase difference between ttwo terms beingp. So the stateu2& will stay empty and theatoms are trapped in the lower states. The effect of succoherent population trapping is that the absorption or emsion of light is completely eliminated.

B. Electromagnetically induced transparency

Consider that the traveling~probing! laser is described bya time-dependent electric fieldE(t)5E0e2 ivt with v veryclose tov2 . The perturbation from such a field is

^2uH1u0&52^2upu0&E0e2 ivt5\V20e2 ivt, ~35!

wherep is the dipole moment induced by the field. Nowwe examine the density matrix elements between two star l l 85^ l uru l 8&, we have

i]r20

]t5~v22 ig!r201V21e

2 iv21tr10

1V20e2 ivt~r002r22!, ~36!

i]r10

]t5v1r101V12e

2 iv12tr202V20e2 iv2tr12. ~37!

We have used

(l 50

2

u l &^ l u51 ~38!

in deriving the above equations. We can then replacer00,r22, andr12 by their values att50, that is,r0051, r2250,andr1250, and change a variable withz105r10e

2 iv21t, be-cause we are only looking for the linear solution. Thenhave

i]r20

]t5~v22 ig!r201V21z101V20e

2 ivt, ~39!

i]z10

]t5v2z101V12r20. ~40!

This equation set resembles a harmonic oscillator undamping and driving forces. The steady solutions are thfore given by

r20~ t !5Ae2 ivt, ~41!

z10~ t !5Be2 ivt. ~42!

Substituting the above solutions into the equations, we ob

A5V20~v2v2!

~v2v21 ig!~v2v2!2uV21u2. ~43!

Becauser20 represents the dipole transition rate betweenu2&and 0&, the polarization of the system is given byP


as-

s,

e

ere-

in

5nar20p025(e2e0)E(t) with p025^0upu2&5p20* . Then wereach Eq.~19!.

C. The slowest light

We know that the group velocity is given by

vg5dv

dk5

c

n1v~dn/dv!. ~44!

For all known materials,n;O(1). So if vg!c, we musthave

vg.c

v~dn/dv!. ~45!

For vg50.45 m/s, as observed in the experiment by Hagroup,1 one must have

v~dn/dv!.6.73108. ~46!

From the given permittivity, we have

n1 ica

2v.11

1

2

vd~v2v2!

vR2/42~v2v2!22 ig~v2v2!

. ~47!

Considering thatv is very close tov2 , we have

n1 ica

2v.11

2vd~v2v2!

vR2

3F114~v2v2!2

vR2 1

i4g~v2v2!

vR2 1¯G ,

~48!

which gives

v~dn/dv!.2v

\e0

naup20u2

vR2 . ~49!

We have usedvd5naup20u2/\e0 . With the numerical valuesof the quantities given, we then obtainna.231020m23, adensity quite difficult to achieve experimentally.

Note that the absorption coefficienta is zero at the reso-nance frequency. This is the essence of the electromagcally induced transparency, a condition that must be meorder to have a significant light transmission at the resonafrequency. Otherwise, the drastically slowed group velocof light observed by Hau’s group would not have been psible.

a!Electronic mail: [email protected]. V. Hau, presentation at the American Association for the Advancemof Science, February 2000, Washington, DC.

2L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, ‘‘Light speereduction to 17 meters per second in an ultracold atomic gas,’’ Na~London! 397, 594–598~1999!.

3For a recent review, see R. Y. Chiao and A. M. Steinberg,TunnelingTimes and Superluminality, Progress in Optics Vol. 37, edited by E. Wo~Elsevier, Amsterdam, 1997!, pp. 347–405.

4A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, ‘‘Electromagneticalintroduced transparency: Propagation dynamics,’’ Phys. Rev. Lett.74,2447–2450~1995!.

5S. E. Harris, ‘‘Electromagnetically induced transparency,’’ Phys. Tod50 ~7!, 36–42~1997!.

6M. O. Scully and M. S. Zubairy,Quantum Optics~Cambridge U.P., Cam-bridge, 1997!, Secs. 7.2 and 7.3.

7D. J. Jackson,Classical Electrodynamics~Wiley, New York, 1999!, 3rded., Secs. 7.5 and 7.8.

606New Problems

4

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I. PROBLEM

Consider a variant on the physical situation of ‘‘slolight’’ 1,2 in which two closely spaced spectral lines are nboth optically pumped to show that the group velocity cannegative at the central frequency, which leads to appasuperluminal behavior.

A. Negative group velocity

In more detail, consider a classical model of matterwhich spectral lines are associated with oscillators. In pticular, consider a gas with two closely spaced spectral liof angular frequenciesv1,25v06D/2, whereD!v0 . Eachline has the same damping constant~and spectral width! g.

Ordinarily, the gas would exhibit strong absorptionlight in the vicinity of the spectral lines. But suppose thlasers of frequenciesv1 and v2 pump both oscillators intoinverted populations. This can be described classicallyassigning negative oscillator strengths to these oscillator3

Deduce an expression for the group velocityvg(v0) of apulse of light centered on frequencyv0 in terms of the~uni-valent! plasma frequencyvp of the medium, given by

vp25

4pNe2

m, ~1!

whereN is the number density of atoms, ande andm are thecharge and mass of an electron. Give a condition on theseparationD compared to the linewidthg such that the groupvelocity vg(v0) is negative.

In a recent experiment by Wanget al.,4 a group velocity ofvg52c/310, wherec is the speed of light in vacuum, wademonstrated in cesium vapor using a pair of spectral liwith separation D/2p'2 MHz and linewidth g/2p'0.8 MHz.

B. Propagation of a monochromatic plane wave

Consider a wave with electric fieldE0eiv(z/c2t) that isincident fromz,0 on a medium that extends fromz50 to a.Ignore reflection at the boundaries, as is reasonable ifindex of refractionn(v) is near unity. Particularly simpleresults can be obtained when you make the~unphysical! as-sumption that thevn(v) varies linearly with frequencyabout a central frequencyv0 . Deduce a transformation thahas a frequency-dependent part and a frequency-indepenpart between the phase of the wave forz,0 to that of thewave inside the medium, and to that of the wave inregiona,z.


ent

r-s

t

y

e

s

e

ent

e

C. Fourier analysis

Apply the transformations between an incident monochmatic wave and the wave in and beyond the medium toFourier analysis of an incident pulse of formf (z/c2t).

D. Propagation of a sharp wave front

In the approximation thatvn varies linearly withv, de-duce the waveforms in the regions 0,z,a anda,z for anincident pulsed(z/c2t), whered is the Dirac delta function.Show that the pulse emerges out of the gain region atz5a attime t5a/vg , which appears to be earlier than when it entethis region if the group velocity is negative. Show also thinside the negative group velocity medium a pulse progates backwards fromz5a at time t5a/vg,0 to z50 at t50, at which time it appears to annihilate the incident pul

E. Propagation of a Gaussian pulse

As a more physical example, deduce the waveforms inregions 0,z,a and a,z for a Gaussian incident puls

E0e2(z/c2t)2/2t2eiv0(z/c2t). Carry the frequency expansion o

vn(v) to second order to obtain conditions of validity of thanalysis such as maximum pulse widtht, maximum lengthaof the gain region, and maximum time of advance of temerging pulse. Consider the time required to generapulse of rise timet when assessing whether the time advanin a negative group velocity medium can lead to superlunal signal propagation.

II. SOLUTION

The concept of group velocity appears to have beenenunciated by Hamilton in 1839 in lectures of which onabstracts were published.5 The first recorded observation othe group velocity of a~water! wave is due to Russell in1844.6 However, widespread awareness of the group velodates from 1876 when Stokes used it as the topic oSmith’s Prize examination paper.7 The early history of groupvelocity has been reviewed by Havelock.8

H. Lamb9 credits A. Schuster with noting in 1904 thatnegative group velocity, i.e., a group velocity of oppossign to that of the phase velocity, is possible due to anomlous dispersion. Von Laue10 made a similar comment in1905. Lamb gave two examples of strings subject to extepotentials that exhibit negative group velocities. These eaconsiderations assumed that in case of a wave with posgroup and phase velocities incident on the anomalousdium, energy would be transported into the medium withpositive group velocity, and so there would be waves wnegative phase velocity inside the medium. Such negaphase velocity waves are formally consistent with Sne

Negative group velocityKirk T. McDonalda)

Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 0854

~Received 21 August 2000; accepted 21 September 2000!

@DOI: 10.1119/1.1331304#


tle

o

tteop

ciee

ityis

y.

n

-citoninr.s

esost oa

acn

e

e

-ng

hf an

n-y

ms

umThe

law11 ~sinceu t5sin21@(ni /nt)sinui# can be in either the firsor second quadrant!, but they seemed physically implausiband the topic was largely dropped.

Present interest in negative group velocity is basedanomalous dispersion in a gain medium, where the signthe phase velocity is the same for incident and transmiwaves, and energy flows inside the gain medium in theposite direction to the incident energy flow in vacuum.

The propagation of electromagnetic waves at frequennear those of spectral lines of a medium was first extensivdiscussed by Sommerfeld and Brillouin,12 with emphasis onthe distinction between signal velocity and group velocwhen the latter exceedsc. The solution presented herebased on the work of Garrett and McCumber,13 as extendedby Chiaoet al.14,15 A discussion of negative group velocitin electronic circuits has been given by Mitchell and Chiao16

A. Negative group velocity

In a medium of index of refractionn(v), the dispersionrelation can be written

k5vn

c, ~2!

where k is the wave number. The group velocity is thegiven by

vg5ReFdv

dkG51

Re@dk/dv#

5c

Re@d~vn!/dv#5

c

n1v Re@dn/dv#.

~3!

We see from Eq.~3! that if the index of refraction decreases rapidly enough with frequency, the group velocan be negative. It is well known that the index of refractidecreases rapidly with frequency near an absorption lwhere ‘‘anomalous’’ wave propagation effects can occu12

However, the absorption makes it difficult to study theeffects. The insight of Garrett and McCumber13 and of Chiaoet al.14,15,17–19is that demonstrations of negative group vlocity are possible in media with inverted populations,that gain rather than absorption occurs at the frequencieinterest. This was dramatically realized in the experimenWanget al.4 by use of a closely spaced pair of gain lines,perhaps first suggested by Steinberg and Chiao.17

We use a classical oscillator model for the index of refrtion. The indexn is the square root of the dielectric constae, which is in turn related to the atomic polarizabilitya ac-cording to

D5eE5E14pP5E~114pNa! ~4!

~in Gaussian units!, whereD is the electric displacement,E isthe electric field, andP is the polarization density. Then, thindex of refraction of a dilute gas is

n5Ae'112pNa. ~5!

The polarizabilitya is obtained from the electric dipolmomentp5ex5aE induced by electric fieldE. In the caseof a single spectral line of frequencyv j , we say that anelectron is bound to the~fixed! nucleus by a spring of constantK5mv j

2, and that the motion is subject to the dampi


nofd-

sly

y

e,

e

-

off

s

-t

force 2mg j x, where the dot indicates differentiation witrespect to time. The equation of motion in the presence oelectromagnetic wave of frequencyv is

x1g j x1v j2x5

eE

m5

eE0

meivt. ~6!

Hence,

x5eE

m

1

v j22v22 ig jv

5eE

m

v j22v21 ig jv

~v j22v2!21g j

2v2 , ~7!

and the polarizability is

a5e2

m

v j22v21 ig jv

~v j22v2!21g j

2v2 . ~8!

In the present problem we have two spectral lines,v1,2

5v06D/2, both of oscillator strength21 to indicate that thepopulations of both lines are inverted, with damping costantsg15g25g. In this case, the polarizability is given b

a52e2

m

~v02D/2!22v21 igv

~~v02D/2!22v2!21g2v2

2e2

m

~v01D/2!22v21 igv

~~v01D/2!22v2!21g2v2

'2e2

m

v022Dv02v21 igv

~v022Dv02v2!21g2v2

2e2

m

v0212Dv02v21 igv

~v021Dv02v2!21g2v2 , ~9!

where the approximation is obtained by the neglect of terin D2 compared to those inDv0 .

For a probe beam at frequencyv, the index of refraction~5! has the form

n~v!'12vp

2

2 F v022Dv02v21 igv

~v022Dv02v2!21g2v2

1v0

21Dv02v21 igv

~v021Dv02v2!21g2v2G , ~10!

wherevp is the plasma frequency given by Eq.~1!. This isillustrated in Fig. 1.

The index at the central frequencyv0 is

Fig. 1. The real and imaginary parts of the index of refraction in a mediwith two spectral lines that have been pumped to inverted populations.lines are separated by angular frequencyD and have widthsg50.4D.

608New Problems

te

iv

onanse

r-h

a-ec-

x ofr-m.

city

mallo

s

the

s ofuse

the

-

atnce

-

n~v0!'12 ivp

2g

~D21g2!v0'12 i

vp2

D2

g

v0, ~11!

where the second approximation holds wheng!D. Theelectric field of a continuous probe wave then propagaaccording to

E~z,t !5ei ~kz2v0t !5eiv~n~v0!z/c2t !

'ez/@D2c/gv~2/p!#eiv0~z/c2t !. ~12!

From this we see that at frequencyv0 the phase velocity isc,and the medium has an amplitude gain lengthD2c/gvp

2.To obtain the group velocity~3! at frequencyv0 , we need

the derivative

d~vn!

dv Uv0

'122vp

2~D22g2!

~D21g2!2 , ~13!

where we have neglected terms inD andg compared tov0 .From Eq.~3!, we see that the group velocity can be negatif

D2

vp22

g2

vp2 >

1

2 S D2

vp2 1

g2

vp2D 2

. ~14!

The boundary of the allowed region~14! in (D2,g2) space isa parabola whose axis is along the lineg252D2, as shownin Fig. 2. For the physical regiong2>0, the boundary isgiven by

g2

vp2 5A114

D2

vp2212

D2

vp2 . ~15!

Thus, to have a negative group velocity, we must have

D<&vp , ~16!

which limit is achieved wheng50; the maximum value ofgis 0.5vp whenD50.866vp .

Near the boundary of the negative group velocity regiuvgu exceedsc, which alerts us to concerns of superluminbehavior. However, as will be seen in the following sectiothe effect of a negative group velocity is more dramatic whuvgu is small rather than large.

The region of recent experimental interest isg!D!vp ,for which Eqs.~3! and ~13! predict that

Fig. 2. The allowed region~14! in (D2,g2) space such that the group velocity is negative.


s

e

,l,n

vg'2c

2

D2

vp2 . ~17!

A value of vg'2c/310 as in the experiment of Wang coresponds toD/vp'1/12. In this case, the gain lengtD2c/gvp

2 was approximately 40 cm.For later use we record the second derivative,

d2~vn!

dv2 Uv0

'8ivp

2g~3D22g2!

~D21g2!3 '24ivp

2

D2

g

D2 , ~18!

where the second approximation holds ifg!D.

B. Propagation of a monochromatic plane wave

To illustrate the optical properties of a medium with negtive group velocity, we consider the propagation of an eltromagnetic wave through it. The medium extends fromz50 to a, and is surrounded by vacuum. Because the inderefraction~10! is near unity in the frequency range of inteest, we ignore reflections at the boundaries of the mediu

A monochromatic plane wave of frequencyv and incidentfrom z,0 propagates with phase velocityc in vacuum. Itselectric field can be written

Ev~z,t !5E0eivz/ce2 ivt ~z,0!. ~19!

Inside the medium this wave propagates with phase veloc/n(v) according to

Ev~z,t !5E0eivnz/ce2 ivt ~0,z,a!, ~20!

where the amplitude is unchanged since we neglect the sreflection at the boundaryz50. When the wave emerges intvacuum atz5a, the phase velocity is againc, but it hasaccumulated a phase lag of (v/c)(n21)a, and so appears a

Ev~z,t !5E0eiva~n21!/ceivz/ce2 ivt

5E0eivan/ce2 iv~ t2~z2a!/c! ~a,z!. ~21!

It is noteworthy that a monochromatic wave forz.a has thesame form as that inside the medium if we makefrequency-independent substitutions

z→a, t→t2z2a

c. ~22!

Since an arbitrary waveform can be expressed in termmonochromatic plane waves via Fourier analysis, we canthese substitutions to convert any wave in the region 0,z,a to its continuation in the regiona,z.

A general relation can be deduced in the case wheresecond and higher derivatives ofvn(v) are very small. Wecan then write

vn~v!'v0n~v0!1c

vg~v2v0!, ~23!

wherevg is the group velocity for a pulse with central frequencyv0 . Using this in Eq.~20!, we have

Ev~z,t !'E0eiv0z~n~v0!/c21/vg!eivz/vge2 ivt ~0,z,a!.~24!

In this approximation, the Fourier componentEv(z) at fre-quencyv of a wave inside the gain medium is related to thof the incident wave by replacing the frequency depende

609New Problems

lengthlengthenedsame phase,all in phase,up

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 wavel0

is unaltered by the medium, but the wavelength of a longer wavelength component is shortened, and that of a shorter wavelength component is.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 theand a peak appears. Simultaneously, there can be a point in the vacuum region beyond the medium at which the Fourier components are againand a third peak appears. The peaks in the vacuum regions move with group velocityvg5c, but the peak inside the medium moves with a negative grovelocity, shown asvg52c/2. The phase velocityvp is c in vacuum, and close toc in the medium.

to

avb

b

em

lih

ein

ey-

van-ffectithin

a-

eve-ave-the

p-ity

ap-

ates

a

g itsve-

ew,the

in afre-d ofd aints

es-

eivz/c by eivz/vg, i.e., by replacingz/c by z/vg , and multi-plying by the frequency-independent phase faceiv0z(n(v0)/c21/vg). Then, using transformation~22!, the wavethat emerges into vacuum beyond the medium is

Ev~z,t !'E0eiv0a~n~v0!/c21/vg!

3eiv~z/c2a~1/c21/vg!!e2 ivt ~a,z!. ~25!

The wave beyond the medium is related to the incident wby multiplying by a frequency-independent phase, andreplacing z/c by z/c2a(1/c21/vg) in the frequency-dependent part of the phase.

The effect of the medium on the wave as describedEqs.~24! and ~25! has been called ‘‘rephasing.’’4

C. Fourier analysis and ‘‘rephasing’’

The transformations between the monochromatic incidwave ~19! and its continuation in and beyond the mediu~24! and ~25!, imply that an incident wave

E~z,t !5 f ~z/c2t !5E2`

`

Ev~z!e2 ivt dv ~z,0!, ~26!

whose Fourier components are given by

Ev~z!51

2p E2`

`

E~z,t !eivt dt, ~27!

propagates as

E~z,t !'5f ~z/c2t ! ~z,0!

eiv0z~n~v0!/c21/vg! f ~z/vg2t ! ~0,z,a!

eiv0a~n~v0!/c21/vg! f ~z/c2t2a~1/c21/vg!!

~a,z!.

~28!

An interpretation of Eq.~28! in terms of ‘‘rephasing’’ is asfollows. Fourier analysis tells us that the maximum amptude of a pulse made of waves of many frequencies, eacthe form Ev(z,t)5E0(v)eif(v)5E0(v)ei (k(v)z2vt1f0(v))

with E0>0, is determined by adding the amplitudesE0(v).This maximum is achieved only if there exist points~z,t!such that all phasesf~v! have the same value.

For example, we consider a pulse in the regionz,0whose maximum occurs when the phases of all componfrequencies vanish, as shown at the left of Fig. 3. Referr


r

ey

y

nt,

-of

ntg

to Eq. ~19!, we see that the peak occurs whenz5ct. Asusual, we say that the group velocity of this wave isc invacuum.

Inside the medium, Eq.~24! describes the phases of thcomponents, which all have a common frequencindependent phasev0z(n(v0)/c21/vg) at a givenz, as wellas a frequency-dependent partv(z/vg2t). The peak of thepulse occurs when all the frequency-dependent phasesish; the overall frequency-independent phase does not athe pulse size. Thus, the peak of the pulse propagates wthe medium according toz5vgt. The velocity of the peak isvg , the group velocity of the medium, which can be negtive.

The ‘‘rephasing’’ ~24! within the medium changes thwavelengths of the component waves. Typically the walength increases, and by greater amounts at longer wlengths. A longer time is required before the phases ofwaves all become the same at some pointz inside the me-dium, so in a normal medium the velocity of the peak apears to be slowed down. But in a negative group velocmedium, wavelengths short compared tol0 are lengthened,long waves are shortened, and the velocity of the peakpears to be reversed.

By a similar argument, Eq.~25! tells us that in the vacuumregion beyond the medium the peak of the pulse propagaccording toz5ct1a(1/c21/vg). The group velocity isagainc, but the ‘‘rephasing’’ within the medium results inshift of the position of the peak by the amounta(1/c21/vg). In a normal medium where 0,vg<c the shift isnegative; the pulse appears to have been delayed durinpassage through the medium. But after a negative grouplocity medium, the pulse appears to have advanced!

This advance is possible because, in the Fourier vieach component wave extends over all space, even ifpulse appears to be restricted. The unusual ‘‘rephasing’’negative group velocity medium shifts the phases of thequency components of the wave train in the region aheathe nominal peak such that the phases all coincide, anpeak is observed, at times earlier than expected at pobeyond the medium.

As shown in Fig. 3 and further illustrated in the examplin the following, the ‘‘rephasing’’ can result in the simultaneous appearance of peaks in all three regions.

610New Problems

vea

e

c

r

s

fo

-m

elae

thtroothgida

enh

apl

ionou

a

cebw

enp.

h

roup

D. Propagation of a sharp wave front

To assess the effect of a medium with negative grouplocity on the propagation of a signal, we first considerwaveform with a sharp front, as recommended by Sommfeld and Brillouin.12

As an extreme but convenient example, we take the indent pulse to be a Dirac delta function,E(z,t)5E0d(z/c2t). Inserting this in Eq.~28!, which is based on the lineaapproximation~23!, we find

E~z,t !'5E0d~z/c2t ! ~z,0!

E0eiv0z~n~v0!/c21/vg!d~z/vg2t ! ~0,z,a!

E0eiv0a~n~v0!/c21/vg!d~z/c2t2a~1/c21/vg!!

~a,z!.~29!

According to Eq.~29!, the delta-function pulse emergefrom the medium atz5a at time t5a/vg . If the group ve-locity is negative, the pulse emerges from the medium beit enters att50!

A sample history of~Gaussian! pulse propagation is illustrated in Fig. 4. Inside the negative group velocity mediuan ~anti!pulse propagates backwards in space fromz5a attime t5a/vg,0 to z50 at timet50, at which point it ap-pears to annihilate the incident pulse.

This behavior is analogous to barrier penetration by a rtivistic electron20 in which an electron can emerge from thfar side of the barrier earlier than it hits the near side, ifelectron emission at the far side is accompanied by posiemission, and the positron propagates within the barrier sto annihilate the incident electron at the near side. InWheeler–Feynman view, this process involves only a sinelectron which propagates backwards in time when insthe barrier. In this spirit, we might say that pulses propagbackwards in time~but forward in space! inside a negativegroup velocity medium.

The Fourier components of the delta function are indepdent of frequency, so the advanced appearance of the swave front as described by Eq.~29! can occur only for a gainmedium such that the index of refraction varies linearly atfrequencies. If such a medium existed with negative slodn/dv, then Eq.~29! would constitute superluminal signapropagation.

However, from Fig. 1 we see that a linear approximatto the index of refraction is reasonable in the negative grvelocity medium only foruv2v0u&D/2. The sharpest wavefront that can be supported within this bandwidth has chacteristic rise timet'1/D.

For the experiment of Wanget al. whereD/2p'106 Hz,an analysis based on Eq.~23! would be valid only for pulseswith t*0.1ms. Wang et al. used a pulse witht'1 ms,close to the minimum value for which Eq.~23! is a reason-able approximation.

Since a negative group velocity can only be experienover a limited bandwidth, very sharp wave fronts mustexcluded from the discussion of signal propagation. Hoever, it is well known12 that great care must be taken whdiscussing the signal velocity if the waveform is not shar


-

r-

i-

re

,

-

en

aselee

te

-arp

lle

p

r-

de-

E. Propagation of a Gaussian pulse

We now consider a Gaussian pulse of temporal lengttcentered on frequencyv0 ~the carrier frequency!, for whichthe incident waveform is

Fig. 4. Ten ‘‘snapshots’’ of a Gaussian pulse as it traverses a negative gvelocity region (0,z,50), according to Eq.~31!. The group velocity in thegain medium isvg52c/2, andc has been set to 1.

611New Problems

toer

th

s

r

-

y

s

dis-

ame-

s

ityer-de-tiveate

citytthe

ve-

e to

bea-

edith

riseont.loc-alnedidednal

aothak.ainlse

E~z,t !5E0e2~z/c2t !2/2t2eiv0z/ce2 iv0t ~z,0!. ~30!

Inserting this in Eq.~28! we find

E~z,t !55E0e2~z/c2t !2/2t2

eiv0~z/c2t ! ~z,0!

E0e2~z/vg2t !2/2t2eiv0~n~v0!z/c2t ! ~0,z,a!

E0eiv0a~n~v0!21!/ce2~z/c2a~1/c21/vg!2t !2/2t2

3eiv0~z/c2t ! ~a,z!.~31!

The factoreiv0a(n(v0)21)/c in Eq. ~31! for a,z becomes

evp2ga/D2c using Eq.~11!, and represents a small gain due

traversing the negative group velocity medium. In the expment of Wanget al., this factor was only 1.16.

We have already noted in the previous section thatlinear approximation tovn(v) is only good over a fre-quency interval aboutv0 of orderD, and so Eq.~31! for thepulse after the gain medium applies only for pulse width

t*1

D. ~32!

Further constraints on the validity of Eq.~31! can be ob-tained using the expansion ofvn(v) to second order. Fothis, we repeat the derivation of Eq.~31! in slightly moredetail. The incident Gaussian pulse~30! has the Fourier decomposition~27!,

Ev~z!5t

A2pE0e2t2~v2v0!2/2eivz/c ~z,0!. ~33!

We again extrapolate the Fourier component at frequencvinto the regionz.0 using Eq.~20!, which yields

Ev~z!5t

A2pE0e2t2~v2v0!2/2eivnz/c ~0,z,a!. ~34!

We now approximate the factorvn(v) by its Taylor ex-pansion through second order:

vn~v!'v0n~v0!1c

vg~v2v0!

11

2

d2~vn!

dv2 Uv0

~v2v0!2. ~35!

With this, we find from Eqs.~26! and ~34! that

E~z,t !5E0

Ae2~z/vg2t !2/2A2t2

eiv0n~v0!z/ce2 iv0t

~0,z,a!. ~36!

where

A2~z!512 iz

ct2

d2~vn!

dv2 Uv0

. ~37!

The waveform forz.a is obtained from that for 0,z,a bythe substitutions~22! with the result

E~z,t !5E0

Aeiv0a~n~v0!21!/ce2~z/c2a~1/c21/vg!2t !2/2A2t2

3eiv0z/ce2 iv0t ~a,z!, ~38!


i-

e

whereA is evaluated atz5a here. As expected, the form~36! and ~38! revert to those of Eq. ~31! whend2(vn(v0))/dv250.

So long as the factorA(a) is not greatly different fromunity, the pulse emerges from the medium essentially untorted, which requires

a

ct!

1

24

D2

vp2

D

gDt, ~39!

using Eqs.~18! and ~37!. In the experiment of Wanget al.,this condition is thata/ct!1/120, which was well satisfiedwith a56 cm andct5300 m.

As in the case of the delta function, the centroid ofGaussian pulse emerges from a negative group velocitydium at time

t5a

vg,0, ~40!

which is earlier than the timet50 when the centroid enterthe medium. In the experiment of Wanget al., the time ad-vance of the pulse wasa/uvgu'300a/c'631028 s'0.06t.

If one attempts to observe the negative group velocpulse inside the medium, the incident wave would be pturbed and the backwards-moving pulse would not betected. Rather, one must deduce the effect of the negagroup velocity medium by observation of the pulse themerges into the regionz.a beyond that medium, where thsignificance of the time advance~40! is the main issue.

The time advance caused by a negative group velomedium is larger whenuvgu is smaller. It is possible thauvgu.c, but this gives a smaller time advance than whennegative group velocity is such thatuvgu,c. Hence, there isno special concern as to the meaning of negative grouplocity when uvgu.c.

The maximum possible time advancetmax by this tech-nique can be estimated from Eqs.~17!, ~39!, and~40! as

tmax

t'

1

12

D

gDt'1. ~41!

The pulse can advance by at most a few rise times dupassage through the negative group velocity medium.

While this aspect of the pulse propagation appears tosuperluminal, it does not imply superluminal signal propagtion.

In accounting for signal propagation time, the time needto generate the signal must be included as well. A pulse wa finite frequency bandwidthD takes at least timet'1/D tobe generated, and so is delayed by a time of order of itstime t compared to the case of an idealized sharp wave frThus, the advance of a pulse front in a negative group veity medium by&t can at most compensate for the origindelay in generating that pulse. The signal velocity, as defiby the path length between the source and detector divby the overall time from onset of signal generation to sigdetection, remains bounded byc.

As has been emphasized by Garrett and McCumber13 andby Chiao,18,19 the time advance of a pulse emerging fromgain medium is possible because the forward tail of a smopulse gives advance warning of the later arrival of the peThe leading edge of the pulse can be amplified by the gmedium, which gives the appearance of superluminal pu

612New Problems

on

ls

b

is

er-n aof

isple

stem-then the

sical

aveensedof

x-ryin

an

ed in

tl de-the

al

onalr.the

ves

ia

er-

ics

p.

ght

ith

velocities. However, the medium is merely using informatistored in the early part of the pulse during its~lengthy! timeof generation to bring the apparent velocity of the pucloser toc.

The effect of the negative group velocity medium candramatized in a calculation based on Eq.~31! in which the

Fig. 5. The same as Fig. 4, but with the electric field plotted on a logarmic scale from 1 to 10265.


e

e

pulse width is narrower than the gain region@in violation ofcondition ~39!#, as shown in Fig. 4. Here, the gain region0,z,50, the group velocity is taken to be2c/2, andc isdefined to be unity. The behavior illustrated in Fig. 4 is phaps less surprising when the pulse amplitude is plotted ologarithmic scale, as in Fig. 5. Although the overall gainthe system is near unity, the leading edge of the pulseamplified by about 70 orders of magnitude in this exam@the implausibility of which underscores that condition~39!cannot be evaded#, while the trailing edge of the pulse iattenuated by the same amount. The gain medium hasporarily loaned some of its energy to the pulse permittingleading edge of the pulse to appear to advance faster thaspeed of light.

Our discussion of the pulse has been based on a clasanalysis of interference, but, as remarked by Dirac,21 classi-cal optical interference describes the behavior of the wfunctions of individual photons, not of interference betwephotons. Therefore, we expect that the behavior discusabove will soon be demonstrated for a ‘‘pulse’’ consistinga single photon with a Gaussian wave packet.

ACKNOWLEDGMENTS

The author thanks Lijun Wang for discussions of his eperiment, and Alex Granik for references to the early histoof negative group velocity and for the analysis containedEqs.~14!–~16!.

a!Electronic mail: [email protected]. V. Hau et al., ‘‘Light speed reduction to 17 metres per second inultracold atomic gas,’’ Nature~London! 397, 594–598~1999!.

2K. T. McDonald, ‘‘Slow light,’’ Am. J. Phys.68, 293–294~2000!. Afigure to be compared with Fig. 1 of the present paper has been addthe version at http://arxiv.org/abs/physics/0007097

3This is in contrast to the ‘‘L’’ configuration of the three-level atomicsystem required for slow light~Ref. 2! where the pump laser does noproduce an inverted population, in which case an adequate classicascription is simply to reverse the sign of the damping constant forpumped oscillator.

4L. J. Wang, A. Kuzmich, and A. Dogariu, ‘‘Gain-assisted superluminlight propagation,’’ Nature~London! 406, 277–279~2000!. Their website,http://www.neci.nj.nec.com/homepages/lwan/gas.htm, contains additimaterial, including an animation much like Fig. 4 of the present pape

5W. R. Hamilton, ‘‘Researches respecting vibration, connected withtheory of light,’’ Proc. R. Ir. Acad.1, 267,341~1839!.

6J. S. Russell, ‘‘Report on waves,’’ Br. Assoc. Reports~1844!, pp. 311–390. This report features the first recorded observations of solitary wa~p. 321! and of group velocity~p. 369!.

7G. G. Stokes, Problem 11 of the Smith’s Prize examination papers~2February 1876!, in Mathematical and Physical Papers~Johnson ReprintCo., New York, 1966!, Vol. 5, p. 362.

8T. H. Havelock,The Propagation of Disturbances in Dispersive Med~Cambridge U.P., Cambridge, 1914!.

9H. Lamb, ‘‘On Group-Velocity,’’ Proc. London Math. Soc.1, 473–479~1904!.

10See p. 551 of M. Laue, ‘‘Die Fortpflanzung der Strahlung in Dispergienden und Absorbierenden Medien,’’ Ann. Phys.~Leipzig! 18, 523–566~1905!.

11L. Mandelstam,Lectures on Optics, Relativity and Quantum Mechan~Nauka, Moscow, 1972!; in Russian.

12L. Brillouin, Wave Propagation and Group Velocity~Academic, NewYork, 1960!. That the group velocity can be negative is mentioned on122.

13C. G. B. Garrett and D. E. McCumber, ‘‘Propagation of a Gaussian LiPulse through an Anomalous Dispersion Medium,’’ Phys. Rev. A1, 305–313 ~1970!.

14R. Y. Chiao, ‘‘Superluminal~but causal! propagation of wave packets intransparent media with inverted atomic populations,’’ Phys. Rev. A48,R34–R37~1993!.

-

613New Problems

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c-

15E. L. Bolda, J. C. Garrison, and R. Y. Chiao, ‘‘Optical pulse propagatat negative group velocities due to a nearby gain line,’’ Phys. Rev. A49,2938–2947~1994!.

16M. W. Mitchell and R. Y. Chiao, ‘‘Causality and negative group delaysa simple bandpass amplifier,’’ Am. J. Phys.68, 14–19~1998!.

17A. M. Steinberg and R. Y. Chiao, ‘‘Dispersionless, highly superluminpropagation in a medium with a gain doublet,’’ Phys. Rev. A49, 2071–2075 ~1994!.

18R. Y. Chiao, ‘‘Population Inversion and Superluminality,’’ inAmazing

leigraaeao

or

inar

otr

ar

heuararus

pn

in

w

sds

to


l

Light, edited by R. Y. Chiao~Springer-Verlag, New York, 1996!, pp.91–108.

19R. Y. Chiao and A. M. Steinberg, ‘‘Tunneling Times and Superluminity,’’ in Progress in Optics, edited by E. Wolf~Elsevier, Amsterdam,1997!, Vol. 37, pp. 347–405.

20See p. 943 of R. P. Feynman, ‘‘A Relativistic Cut-Off for Classical Eletrodynamics,’’ Phys. Rev.74, 939–946~1948!.

21P. A. M. Dirac, The Principles of Quantum Mechanics~Clarendon, Ox-ford, 1958!, 4th ed., Sec. 4.

72

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e of. A-g-

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le.

I. PROBLEM

Ferrofluids1 are stable suspensions of magnetic partichaving linear dimension on the order of 10 nm. Due to vorous Brownian motion the magnetic particles assumedom orientations rendering the suspension as a whole pmagnetic. These complex fluids show a variety of phnomena and instabilities that amuse and delight studentsteachers alike.2 Because these fluids are used in a varietyapplications including rotary seals, sensors, and actuat3

they are commercially available.4

Figure 1 shows the experimental apparatus for viewand recording the response of a ferrofluid film trapped atair–water interface. Figure 2 shows recorded images fodrop ~;40 ml! of mineral-oil-based ferrofluid5 introduced tothe surface of clean, filtered, de-ionized~18 MV! water. Thehydrophobic ferrofluid spreads uniformly over the surfacewater contained in a Petri dish. We gently stir the surfaceemulsify the film, creating a collection of dark flat circuladrops of ferrofluid as recorded in Fig. 2~a!. Figure 2~b!shows the film 1 min after a cylindrical magnet havingradius of 1 cm is introduced with the axis of symmetry vetical and the lower end 3.3 cm above the ferrofluid film. Tferrofluid film clears from directly beneath the magnet bmoves radially inward at large distances, forming teshaped drops with the clearer regions streaming outwThe ferrofluid collects in a ring structure at a finite radi~which is most dense at radius;1.0 cm! from the center ofthe magnetic field symmetry axis. As the ferrofluid builds uclumps or cone-shaped structures develop. As the cogrow, they become unstable and migrate one at a timethe central region. Figure 2~c! taken at 314 min shows theclumping in the ring-shaped structure with one cone at to’clock escaping to the central region. Finally in Fig. 2~d!,taken 21 min after introducing the magnet, a regular ‘‘crytalline’’ array of well-separated ferrofluid cones has formeYet there remains a ferrofluid film ring surrounding this crytalline structure.

How is it possible that the ferrofluid is both attracted~cones! and repelled from~film! the region directly below thecylindrical magnet?

s-n-ra--ndfs,

gna

fo

-

t-d.

,esto

o

-.-

II. SOLUTION

Magnetic body forces, surface tension, viscous drag,gravity combine to produce the peculiar behavior obserhere. We focus on the magnetic body forces as the primexplanation for the question posed above. Figure 3 showside view of one of the ‘‘cones’’ of ferrofluid which form thetwo-dimensional crystalline array beneath the magnet. Thclumps have a nearly ellipsoidal shape above the waterface with the symmetry axis parallel to the applied fieOther studies show that ferrofluid droplets submerged inimmiscible fluid deform into ellipsoids and align parallelthe direction of a uniform external magnetic field.6 Solutionshave long existed for the magnetic~electric! field of an el-lipsoid of permeabilitym ~dielectric constante! subjected toa uniform external field in a surrounding medium of permability m0 ~dielectric constante0!.7 When the symmetry axisof the ellipse is aligned parallel to the external field directiothe field inside the ellipse is uniform and parallel to the aplied field ~at large distances!. Thus, to first order the ‘‘ellip-soidal cones’’ behave like little magnets oriented parallelthe external field and so move along the water surface tostrongest field regions located directly beneath the cylincal magnet. However, the magnetic field induced in eachthese cones is aligned parallel with the neighboring cofields; therefore, the cones repel one another in the planthe interface just like parallel oriented permanent magnetscrystal lattice results.8 These results are qualitative, but intuitive, given our experience playing with permanent manets.

How do we understand the quite different behavior of tfilm? Rosensweig1 gives a general derivation of the bodforce f or force per unit volume, which reduces for ferroflususpensions to

f5m0~M "¹!H5m0M“H, ~1!

wherem0 is the vacuum permeability,H is the magnetic fieldstrength, andM is the magnetization in the film volume eement. This functional form suggests the Kelvin force desity on an isolated body, except that the local fieldH re-places the applied fieldH0 . Intuitively, we understand thisbody force to be like the force acting on a magnetic dipo

Forces in complex fluidsBruce J. Ackersona) and Anitra N. Novyb)

Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078-30

~Received 25 August 2000; accepted 18 December 2000!

@DOI: 10.1119/1.1351151#


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eadnceeticax-ns:

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.

The force depends on the orientation of the dipole~repre-sented byM ! in the local field~represented byH! and on thegradient of the local field in the vicinity of the dipole. Theis no force, only a torque, acting on a dipole in a unifofield. The derivation of the last equality in Eq.~1! reasonablyassumes that“ÃH50 and that the magnetic flux densityBis linearly related to the magnetic field strength and tomagnetization as follows:

B5mH5m0~H1M ! ~2!

with m the permeability of the film. Using Eq.~2! to repre-sent the body force in terms of the magnetic flux densitythe second equality of Eq.~1! gives

f51

m S 12m0

m DB“B51

2m S 12m0

m D“B2. ~3!

Consequently, the negative square of the magnetic flux dsity within the film corresponds to a ‘‘potential field’’ towhich the film responds.

Fig. 1. Experimental setup for viewing the dynamics of a ferrofluid film


e

n

n-

Consider the film to be of constant thickness and spruniformly over the surface of the water when in the preseof the external magnetic field. In the absence of magncharge and surface current density, consideration of the Mwell equations leads to the following boundary conditiofor the magnetic field strength and magnetic flux density

~B2B0!"n50, ~H2H0!Ãn50, ~4!

where the terms with a subscript refer to the air~or vacuum!side of the film boundary and the terms without a subscripthe ferrofluid. The vectorn is a unit normal to the film sur-face. Note that these are the same boundary conditionsfield relationships used to find the field inside an ellipsosubjected to a uniform external field. For the film problewe take the normal to the film, thez direction, to be parallelto the magnet axis and assume azimuthal symmetry. Thevector r designates the radial direction with respect to t

Fig. 3. Ellipsoidal ‘‘clump’’ or cone of ferrofluid.

he
Fig. 2. Ferrofluid film before the magnet was introduced~a!, after 1 min~b!, after 314 min ~c!, and after 21 min~d!. Note that any dark spots that appear t
same in all frames are due to imperfections in the optical train.

615New Problems

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magnet axis. Using the relationships given in Eqs.~2! and~4!, the magnetic flux density and magnetic field strength jinside the film are represented in terms of the magneticdensity just outside the film as

B5B0zz1m

m0B0rr,

~5!

H51

mB0zz1

1

m0B0rr.

Since the film is thin, we assume these boundary resultrepresent the field within the film.

Next we approximate the external field at the surfacethe film by a dipole field with momentm directed parallelto z,

B05m0

4pr 3 ~3~m"r ! r2m!

5m0m

4p F 3rz

~r21z2!5/2 r12z22r2

~r21z2!5/2 zG , ~6!

where z measures the distance from the film to the dipsource~cylindrical magnet!, r is the radial position in thefilm, r 5Ar21z2 is the position vector magnitude in sphecal coordinates, and the vectors with a caret are unit vecThis magnetic flux density, when plugged into the right-haside of Eq.~5!, gives an estimate of the magnetic flux densinside the film. Directly beneath the dipole the magnetic fldensity is normal to the film and has the same magnitudeboth sides of the film surface. Asr increases from zerohowever, there is a component of the magnetic flux denwhich is parallel to the surface and larger inside the film thoutside form/m0.1. While the magnitude of the externamagnetic flux density decreases monotonically with increing r for this dipole field, the magnetic flux density insidthe film will increase for sufficiently large permeability ratiom/m0 , before decreasing to zero. The increasing magnflux density results in a force@see Eq.~3!# that pushes thefilm outward radially to collect in a ring where the flux desity reaches a maximum. Note that we neglected contrtions from the film to the external magnetic field on tassumption that this field is small for a very thin film. Whian exact solution to this problem is beyond the scope o

Fig. 4. Reduced potential for an element of ferrofluid film forz51 as afunction of the distancer from the magnet axis and shown for differevalues of the permeability ratiom/m051, 2, 2.9, and 5 from top to bottom


tx

to

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rs.d

xn

tyn

s-

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u-

a

typical undergraduate course in electromagnetism,checked our results using the method of images to findmagnetic flux density produced by a dipole near a thin plaof material with permeabilitym. In the limit that the thick-ness goes to zero, the fields inside and outside the filmcome identical to those approximated here. There is oneveat. The body force must be calculated using the secequality in Eq.~1!. The first equality involves a derivativewith respect toz, which must be performed before the filmthickness is taken to zero.

Figure 4 shows the reduced potential, estimated forferrofluid film using the dipole field given in Eq.~6!, as afunction ofr for z51, and different values of the permeabity ratio m/m0 . The functional form of this reduced potentiais

F52B2

~m0m/4p!2

52S r41~2419~m/m0!2!r2z214z4

~r21z2!5 D . ~7!

For permeability ratios greater thanA8/3'1.63 a potentialminimum obtains at finite radius. The minimum positionrminobeys the following linear relationship with respect toz:

rmin /z5A26~m/m0!2131A36~m/m0!4233~m/m0!211.~8!

This ratio ranges between zero and one half as the perability ratio increases from 1.63 to infinity. Since the initimagnetic susceptibility of our ferrofluid isx51.9, the per-meability ratio is 2.9, and Eq.~8! predictsrmin /z50.43. Ap-proximating the magnet by a dipole placed at the lower eof the magnet, the end closest to the ferrofluid, givesrmin /z;1.0 cm/3.3 cm;0.30 or placing the dipole at the physiccenter of the magnet givesrmin /z;1.0 cm/5.8 cm;0.17.Considering that the magnet is an extended source rathan a point dipole, the agreement is quite good.

The response of a ferrofluid film to a nonuniform externfield is complex, but a fairly straightforward undergraduaboundary value calculation explains puzzling observatioThe large susceptibility of the magnetic fluid and fluid suface orientation beneath a magnet determines the net foon the ferrofluid, giving attraction for the cones and repusion for the film.

a!Author to whom correspondence should be addressed; electronic [email protected]

b!This communication originated as an honors thesis project of A.N., whpresently a graduate student in physics at Colorado State University

1R. E. Rosensweig,Ferrohydrodynamics~Cambridge U.P., Cambridge1985!, p. 110 ff.

2R. E. Rosensweig, ‘‘Magnetic Fluids,’’ Sci. Am.247~4!, 136–145~1982!.3B. M. Berkovsky, V. F. Medvedev, and M. S. Krakov,Magnetic FluidsEngineering Applications~Oxford U.P., Oxford, 1993!, Chap. 6.

4Ferrofluidics Corporation, 40 Simon Street, Nashua, NH 03060-3075.5Ferrofluidics: Catalog No. EMG 905, Lot No. F8193A.6V. I. Arkhipenko, Yu. D. Barkov, and V. G. Bashtovoi, ‘‘Study of amagnetized fluid drop shape in a homogeneous magnetic field,’’ MaGidrodin. ~3!, 131–134~1978!.

7G. Arfken, Mathematical Methods for Physicists~Academic, New York,1970!, p. 603; S. D. Poisson, ‘‘Seconde me´moire sur la the´orie du mag-netisme,’’ Mem. Acad. R. Sci. Inst. France5, 488–533~1821–1822!.

8A. T. Skjeltorp, ‘‘One- and two-dimensional crystallization of magneholes,’’ Phys. Rev. Lett.51 ~25!, 2306–2309~1983!.

616New Problems

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-

at-

s

,

I. PROBLEM

A popular model at science museums~and also a sciencetoy1! that illustrates how curvature can be associated wgravity consists of a surface of revolutionr 52k/z with z,0 about a vertical axisz. The curvature of the surfacecombined with the vertical force of Earth’s gravity, leadsan inward horizontal acceleration ofkg/r 2 for a particle thatslides freely on the surface in a circular, horizontal orbit.

Consider the motion of a particle that slides freely onarbitrary surface of revolution,r 5r (z)>0, defined by a con-tinuous and differentiable function on some interval ofz. Thesurface may have a nonzero minimum radiusR at which theslopedr/dz is infinite. Discuss the character of oscillationof the particle about circular orbits to deduce a condition tthere be a critical radiusr crit.R, below which the orbits areunstable. That is, the motion of a particle withr ,r crit rapidlyleads to excursions to the minimum radiusR, after which theparticle falls off the surface.

Give one or more examples of analytic functionsr (z) thatexhibit a critical radius as defined above. These examprovide a mechanical analogy as to how departures of grtational curvature from that associated with a 1/r 2 force canlead to a characteristic radius inside which all motion tentoward a singularity.

II. SOLUTION

We work in a cylindrical coordinate system (r ,u,z) withthe z axis vertical. It suffices to consider a particle of unmass.

In the absence of friction, there is no torque on a partiabout thez axis, so the angular momentum componenJ

5r 2u about that axis is a constant of the motion, wheredot ~•! indicates differentiation with respect to time.

For motion on a surface of revolutionr 5r (z), we haver 5r 8z, where the prime~8! indicates differentiation withrespect toz. Hence, the kinetic energy can be written

T5 12~ r 21r 2u21 z2!5 1

2@ z2~11r 82!1r 2u2#. ~1!

The potential energy isV5gz. Using Lagrange’s methodthe equation of motion associated with thez coordinate is

z~11r 82!1 z2rr 952g1Jr8

r 3 . ~2!

For a circular orbit at radiusr 0 , we have

r 035

J2r 08

g. ~3!

We write u05V, so thatJ5r 02V.

For a perturbation about this orbit of the form


h

n

t

esi-

s

e

e

z5z01e sinvt, ~4!

we have, to ordere,

r ~z!'r ~z0!1r 8~z0!~z2z0!5r 01er 08 sinvt, ~5!

r 8'r 081er 09 sinvt, ~6!

1

r 3 '1

r 03 S 123e

r 08

r 0sinvt D . ~7!

Inserting~4!–~7! into ~2! and keeping terms only to ordee, we obtain

2ev2 ~11r 082!sin vt

'2g1J2

r 03 S r 0823e

r 082

r 0sinvt1e r 09 sinvt D . ~8!

From the zero’th-order terms we recover~3!, and from theorder-e terms we find that

v25V23r 08

22r 0r 09

11r 082 . ~9!

The orbit is unstable whenv2,0, i.e., when

r 0r 09.3r 082. ~10!

This condition has the interesting geometrical interpretat~noted by a referee! that the orbit is unstable wherever

~1/r 2!9,0, ~11!

i.e., where the function 1/r 2 is concave inwards.For example, ifr 52k/z, then 1/r 25z2/k2 is concave

outwards,v25J2/(k21r 04), and there is no regime of insta

bility.We give three examples of surfaces of revolution that s

isfy condition ~11!.First, the hyperboloid of revolution defined by

r 22z25R2, ~12!

whereR is a constant. Here,r 085z0 /r 0 , r 095R2/r 03, and

v25V23z0

22R2

2z021R2 5V2

3r 0224R2

2r 022R2 . ~13!

The orbits are unstable for

z0,)R, ~14!

or equivalently, for

r 0,2)

3R51.1547R[r crit . ~15!

As r 0 approachesR, the instability growth time approachean orbital period.

Another example is the Gaussian surface of revolution

A mechanical model that exhibits a gravitational critical radiusKirk T. McDonalda)

Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544

~Received 7 July 1999; accepted 21 December 2000!

@DOI: 10.1119/1.1351152#


-f

t

dcleom-

al

.

r 25R2ez2, ~16!

which has a minimum radiusR, and a critical radiusr crit

5RA4 e51.28R.Our final example is the surface

r 52k

zA12z2~21,z,0!, ~17!

which has a minimum radius ofR52k, approaches the surfacer 52k/z at larger ~smallz!, and has a critical radius or crit56k/A551.34R.

These examples arise in a 211 geometry with curvedspace but flat time. As such, they are not fully analogous


o

black holes in 311 geometry with both curved space ancurved time. Still, they provide a glimpse as to how a partiin curved space–time can undergo considerably more cplex motion than in flat space–time.

ACKNOWLEDGMENTS

The author wishes to thank Ori Ganor and Vipul Periwfor discussions of this problem.

a!Electronic mail: [email protected] Vortx~tm! Miniature Wishing Well, Divnick International, Inc., 321 SAlexander Road, Miamisburg, OH 45342, http://www.divnick.com/

AWESTRUCK SCIENTISTS

The second feature of science is that it shows that the world is simple. Even many scientists donot appreciate that they are hewers of simplicity from complexity. They are often more deludedthan those they aim to tell. Scientists are often overawed by the complexity of detecting simplicity.They look at the latest fundamental particle experiment, see that it involves a thousand kilogramsof apparatus and a discernible percentage of a gross national product, and become thunderstruck.They see the complexity of the apparatus and the intensity of the effort needed to construct andoperate it, and confuse that with the simplicity that the experiment, if successful, will expose.Some scientists are so awestruck that they even turn to religion! Others keep a cool head, andmarvel not at an implied design but at the richness of simplicity.

P. W. Atkins, ‘‘The Limitless Power of Science,’’ inNature’s Imagination—The Frontiers of Scientific Vision, edited byJohn Cornwell~Oxford University Press, New York, 1995!.

618New Problems

negative group vel

Technology

electromagnetically

joseph henry

coherent population

magnetic ux

ultracold

magnetic eld

external magnetic

negative group