Slow lightKirk T. McDonaldJoseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544

~Received 6 July 1999; accepted 23 September 1999!

NEW PROBLEMS

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

‘‘New Problems’’ resumes this month with the first of a series of problems from Kirk McDonald ofPrinceton University. ‘‘New Problems’’ continues to solicit interesting and novel worked problems foruse in undergraduate physics courses beyond the introductory level. We seek problems that convey theexcitement and interest of current developments in physics and that are useful for teaching coursessuch as Classical Mechanics, Electricity and Magnetism, Statistical Mechanics and Thermodynamics,‘‘Modern’’ Physics, and Quantum Mechanics. We challenge physicists everywhere to create problemsthat show how their various branches of physics use the central unifying ideas of physics to advancephysical understanding. We want these problems to become an important source of ideas and infor-mation for students of physics and their teachers. All submissions are peer-reviewed prior to publica-tion. Send manuscripts directly to Christopher R. Gould,Editor.

trar

a

oft a

gn

o

.

excl-

-

t

y

I. PROBLEM

Consider a classical model of matter in which speclines are associated with oscillators. In particular, considegas with two closely spaced spectral lines,v1,25v06D,whereD!v0 . Each line has oscillator strength 1/Z, whereZis the atomic number of a gas atom, and each has the sdamping constant~and spectral width! g. For simplicity, youmay suppose thatD5g.

Ordinarily, the gas would exhibit strong absorptionlight in the vicinity of the spectral lines. But suppose thalaser of frequencyv2 ‘‘pumps’’ the second oscillator into aninverted population. Classically, this is described by assiing a negative damping constant to this oscillator:g2

52g.Deduce an expression for the group velocity of a pulse

light centered on frequencyv0 in this medium. Show alsothat frequencies very nearv0 propagate without attenuation

In a recent experiment,1 the group velocity of light wasreduced to 38 mph~17 m/s! by this technique in a sodiumvapor of densityN5531012atoms/cm3 using a pair of linesfor which 2D'107/s.

II. SOLUTION

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

k5vn

c, ~1!

wherek is the wave number andc is the speed of light. Thegroup velocity is then given by

vg5dv

dk5

1

dk/dv5

c

n1vdn

dv

. ~2!

293 Am. J. Phys.68 ~3!, March 2000

la

me

-

f

We next recall the classical oscillator model for the indof refraction. The indexn is the square root of the dielectriconstante, which is in turn related to the atomic polarizabiity a according to~in Gaussian units!

D5eE5E14pP5E~114pNa!, ~3!

whereD is the electric displacement,E is the electric field,Pis the polarization density, andN is the atomic number density. Then,

n5Ae'112pNa, ~4!

for a dilute gas with index near 1.The polarizabilitya is obtained from the dipole momen

p5ex5aE induced by electric fieldE. In the case of asingle spectral line of frequencyv0 , we say that the chargee is bound to the~fixed! nucleus by a spring of constantk5mv0

2, and the motion is subject to damping2mg x. Theequation of motion in the presence of a wave of frequencvis

x1g x1v02x5

eE

m5

eE0

meivt. ~5!

Hence,

x5eE

m

1

v022v21 igv

5eE

m

v022v22 igv

~v022v2!21g2v2 , ~6!

and so the polarizability is

a5e2

m

v022v22 igv

~v022v2!21g2v2 . ~7!

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

5v07g, both of oscillator strength 1/Z, but the populationof line 2 is inverted so thatg252g152g. In this case, thepolarizability is given by

293© 2000 American Association of Physics Teachers

rm

o

b-

cn

e

hidls

to

-

5ate.

an

a51

Z

e2

m

~v02g!22v22 igv

~~v02g!22v2!21g2v2

11

Z

e2

m

~v01g!22v21 igv

~~v01g!22v2!21g2v2

'1

Z

e2

m

v0222gv02v22 igv

~v0222gv02v2!21g2v2

11

Z

e2

m

v0212gv02v21 igv

~v0212gv02v2!21g2v2 , ~8!

where the approximation is obtained by the neglect of tein g2 compared to those ingv0 .

We now consider the issue of attenuation of a pulsefrequencyv. Sincek5vn/c'v(112pNa)/c, the spatialdependenceeikz of a pulse propagating in thez directionincludes attenuation if the imaginary part of the indexn isnonzero. However, the population inversion describedg252g1 leads to Im@a(v0)#50. Hence, there is no attenuation of a probe pulse at frequencyv0 .

In the present model, the pulse is attenuated at frequenless thanv0 , but grows~lases! at frequencies greater thav0 . In the experiment of Hauet al.,1 lasing did not occurbecause line 2 actually corresponded to a transition betwthe upper level of line 1 and a third, excited level.~In asense, the quantum mechanical level structure with oneand two low energy levels is the inverse of that assumethe classical model here, i.e., one low and two high leve!Therefore, pumping at frequencyv2 did not produce an in-

294 Am. J. Phys., Vol. 68, No. 3, March 2000

s

f

y

ies

en

ghin.

verted population that could lead to lasing; but it did leadan effective sign reversal of the damping constantg2 for anarrow range of frequencies nearv0 .

To obtain the group velocity at frequencyv0 , we need thederivative

d Re~n!

dv Uv0

52pNd Re~a!

dv Uv0

524pNe2

25Zmg2v0. ~9!

Sincea(v0)50, we haven(v0)51, and the phase velocityat v0 is exactlyc. The group velocity~2! is

vg5c

1124pNe2

25Zmg2

'25Zg2

24pNe2

mc2 c2

c'Zg2

pNr0c, ~10!

wherer 05e2/mc2'3310213cm is the classical electron radius. The group velocity is lower in a denser medium.

In the experiment of Hauet al., the medium was sodiumvapor (Z511), cooled to less than 1mK to increase thedensity. An additional increase in density by a factor ofwas obtained when the vapor formed a Bose condensPlugging in the experimental parameters,N5531012/cm3

andg553106/s, we find

vg'11•~53106!2

3•531012•3310213

•331010'2000 cm/s, ~11!

compared to the measured value of 1700 cm/s.1L. V. Hau et al., ‘‘Light speed reduction to 17 metres per second inultracold atomic gas,’’ Nature~London! 397, 594–598~1999!.

MAKE AN IMPORTANT DISCOVERY!

Make an important discovery, and you are a successful scientist in the true, elitist sense in aprofession where elitism is practiced without shame. You go into the textbooks. Nothing can takethat away; you may rest on your laurels the rest of your life. But of course you won’t. Almost noone driven enough to make an important discovery ever rests. And any discovery at all is thrilling.There is no feeling more pleasant, no drug more addictive, than setting foot on virgin soil.

Edward O. Wilson, ‘‘Scientists, Scholars, Knaves and Fools,’’ Am. Scientist86 ~1!, 6–7 ~1998!.

294New Problems