25

fig. 1.4. young double-slit experiment illustrating how coherent light can interfere with itself

e+(r, t) = e+(r1, t1) + e+(r2, t2) , (1.78)
where r1 and r2 are locations of the slits and t1 and t2 are the retarded times
t1,2 = t ? s1,2/c (1.79)
s1 and s2 being the distances between the slits and the detector. from (1.76), the intensity at the detector is given by
|e+(r, t)|2 = |e+(r1, t1)|2 + |e+(r2, t2)|2
+2 re[e?(r1, t1)e+(r2, t2)] , (1.80)

where we have made use of (1.9).
in general the light source contains noise. to describe light with noise we use a statistical approach, repeating the measurement many times and averaging the results. mathematically this looks like
(|e+(r, t)2|) = (|e+(r1, t1)|2) + (|e+(r2, t2)|2)
+ 2re(e?(r1, t1)e+(r2, t2)) , (1.81)
where the brackets (•••) stands for the ensemble average. introducing the ?rst-order correlation function
g(1)(r1t1, r2t2) ? (e?(r1, t1)e+(r2, t2)) , (1.82)

we rewrite (1.79) as
(|e+(r, t)|2) = g(1)(r1t1, r1t1) + g(1)(r2t2, r2t2)
+2reg(1)(r1t1, r2t2) . (1.83)
g(1)(riti, riti) is clearly a real, positive quantity, while g(1)(riti, rj tj ) is in general complex.


with the cross-correlation function rewritten as
g(1)(r1t1, r2t2) = |g(1)(r1t1, r2t2)|ei?(r1 t1 ,r2 t2 ) , (1.84) (1.81) becomes
(|e+(r, t)|2) = g(1)(r1t1, r1t1) + g(1)(r2t2, r2t2)
2|g(1)(r1t1, r2t2)| cos ?. (1.85)

the third term in (1.83) is responsible for the appearance of interferences.
we say that the highest degree of coherence corresponds to a light ?eld that produces the maximum contrast on the screen, where contrast is de?ned as
v = imax ? imin . (1.86)
imax + imin
substituting (1.83) with cos ? = 1, we readily obtain
2|g(1)(r1t1, r2t2)|

v =
g(1)(r1t1, r1t1) + g(1)(r2t2, r2t2)

. (1.87)


the denominator in (1.85) doesn’t play an important role g(1)(riti, riti) is just the intensity on the detector due to the ith slit and the denominator acts
as a normalization constant. to maximize the contrast for a given source and geometry, we need to maximize the numerator 2|g(1)(r1t1, r2t2)|. to achieve
this goal we note that according to the schwarz inequality
g(1)(r1t1, r1t1)g(1)(r2t2, r2t2) ? |g(1)(r1t1, r2t2)|2 . (1.88)

the coherence function is maximized when equality holds, that is when
|g(1)(r1t1, r2t2)| = [g(1)(r1t1, r1t1)g(1)(r2t2, r2t2)]1/2 , (1.89)

which is the coherence condition used by born and wolf. as pointed out by glauber, it is convenient to replace this condition by the equivalent expression
g(1)(r1t1, r2t2) = e ?(r1t1)e (r2t2) , (1.90)

where the complex function e (r1t1) is some function, not necessarily the electric ?eld. if g(1)(r1t1, r2t2) may be expressed in the form (1.88), we say that g(1) factorizes. this factorization property de?nes ?rst-order coherence: when (1.88) holds, the fringe contrast v is maximum.
this de?nition of ?rst-order coherence can be readily generalized to higher orders. a ?eld is said to have nth-order coherence if its mth-order correlation
functions
g(m)(x1 ... xm, ym ... y1) = (e?(x1) ••• e?(xm)e+(ym) ••• e+(y1)) (1.91) factorize as


g(m)(x1 ... xm, ym ... y1) = e ?(x1) •••e ?(xm)e (ym) •••e (y1) (1.92)

for all m ? n. here we use the compact notation xj = (rj , tj ), yj = (rm+j , tm+j ), and g(m) is a direct generalization of (1.80).
before giving an example where second-order correlation functions play
a crucial role, we point out that although a monochromatic ?eld is coherent to all orders, a ?rst-order coherent ?eld is not necessarily monochromatic. one might be led to think otherwise because we often deal with stationary
light, such as that from stars and cw light sources. by de?nition, the two-
time properties of a stationary ?eld depend only on the time di?erence. the
corresponding ?rst-order correlation function thus has the form
g(1)(t1, t2) = g(1)(t1 ? t2) . (1.93)

if such a ?eld is ?rst-order coherent, then with (1.88), we ?nd
g(1)(t1 ? t2) = e ?(t1)e (t2) , (1.94)


which is true only if


e (t1) ? e?i?t1 , (1.95)

that is, stationary ?rst-order coherent ?elds are monochromatic!
let us now turn to the famous hanbury brown-twiss experiment fig. 1.5,
which probes higher-order coherence properties of a ?eld. in this experiment, a beam of light (from a star in the original experiment) is split into two
beams, which are detected by detectors d1 and d2. the signals are multiplied
and averaged in a correlator. this procedure di?ers from the young two-slit
experiment in that light intensities, rather than amplitudes, are compared. two absorption measurements are performed on the same ?eld, one at time
t and the other at t + ? . it can be shown [cohen-tannoudji et al. 1989] that this measures |e+(r,t + ?, )e+(r, t)|2 . dropingping the useless variable r and
averaging, we see that this is precisely the second-order correlation function
g(2)(t, t + ?, t + ?, t) = (e?(t)e?(t + ? )e+(t + ? )e+(t)) , (1.96)













fig. 1.5. diagram of hanbury brown-twiss experiment


or for a stationary process,
g(2)(? ) = (e?(0)e?(? )e+(? )e+(0)) . (1.97)

according to (1.89), the ?eld is second-order coherent if (1.92) holds and
g(2)(? ) = e ?(0)e ?(? )e (? )e (0) . (1.98)

it is convenient to normalize this second-order correlation function as


(2)
g(2)(? ) =

(? )


. (1.99)

|g(1)(0)|2

since a stationary ?rst-order coherent ?eld is monochromatic and satis?es (1.93), second-order coherence implies that
g(2)(? ) = 1 . (1.100)
that is, g(2)(? ) is independent to the delay ? .
the original experiment on hanbury brown-twiss was used to measure
the apparent diameter of stars by passing their light through a pinhole. a second-order correlation function like that in fig. 1.6 was measured. although the light was ?rst-order coherent, we see that it was not second-order coher-
ent. the energy tended to arrive on the detector in bunches, with strong
statistical correlations.
in contrast to the well-stabilized laser with a unity g(2) and the star-light with bunching, recent experiments in resonance ?uorescence show antibunch- ing, with the g(2) shown in fig. 1.7. chapter 16 discusses this phenomenon



g(2)(?)







1




0 ?

fig. 1.6. second-order correlation function (1.97) for starlight in original hanbury brown-twiss experiment


g(2)(?)

1





0 ?

fig. 1.7. second-order correlation function showing antibunching found in reso- nance ?uorescence


in detail here we point out that such behavior cannot be explained with classical ?elds. to see this, note that
(2)(0) ? |g(1)(0)|2

g(2)(0) ? 1 = g

in terms of intensities, this gives


|g(1)(0)|2

. (1.101)

g(2)(0) ? 1 = (i

2)? (i)2
(i)2


((i ? (i))2) (i)2


, (1.102)

where we do not label the times, since we consider a stationary system with
? = 0. introducing the probability distribution p (i) to describe the average
over ?uctuations, we ?nd for (1.100)


g(2)(0) ? 1 =

1 ¸
(i)2


dip (i)(i ? (i))2 . (1.103)

classically this must be positive, since (i ? (i))2 ? 0 and the probability distribution p (i) must be positive. hence g(2) cannot be less than unity, in
contradiction to the experimental result shown in fig. 1.7. at the beginning
of this chapter we say that the ?elds we use can usually be treated classically. well we didn’t say always! to use a formula like (1.101) for the antibunched
case, we need to use the concept of a quasi -probability function p (i) that
permits negative values. quantum mechanics allows just that (see sect. 13.6).


1.5 free-electron lasers

at this point we already have all the ingredients necessary to discuss the basic features of free-electron lasers (fel). they are extensions of devices such as klystrons, undulators, and ubitrons, which were well-known in the millimeter regime many years ago, long before lasers existed. in principle, at least, nothing should have prevented their invention 30 or 40 years ago.


as shown in chap. 7, conventional lasers rely on the inversion of an atomic or molecular transition. thus the wavelength at which they oper- ate is determined by the active medium they use. the fel eliminates the atomic “middle-man”, and does not rely on speci?c transitions. potentially, fel’s o?er three characteristics that are often hard to get with conventional lasers, namely, wide tunability, high power, and high e?ciency. they do this by using a relativistic beam of free electrons that interact with a periodic structure, typically in the form of a static magnetic ?eld. this structure exerts a lorentz force on the moving electrons, forcing them to oscillate, sim- ilarly to the simple harmonic oscillators of sect. 1.3. as discussed at the end of that section, oscillating electrons emit radiation with the ?eld shown in fig. 1.3. in the laboratory frame, this radiation pattern is modi?ed according to lorentz transformations. note that in contrast to the case of radiative de- cay discussed in sect. 1.3, the fel electron velocity approaches that of light
and the v×b factor in the lorentz force of (1.65) cannot be neglected.
the emitted radiation is mostly in the forward direction, within a cone of solid angle ? = 1/? (see fig. 1.8). here ? is the relativistic factor
? = [1 ? v2/c2]?1/2 , (1.104)

where v is the electron velocity. for ? = 200, which corresponds to electrons with an energy on the order of 100 mev, ? is about 5 milliradians, a very
small angle.
in general for more that one electron, each dipole radiates with its own phase, and these phases are completely random with respect to one another.
the total emitted ?eld is et = e+ + e?, where
t t
















fig. 1.8. highly directional laboratory pattern of the radiation emitted by a rela- tivistic electron in circular orbit in the x-y plane while moving along the z axis at the speed v = 0.9c. the x axis is de?ned to be that of the instantaneous accelera- tion. equation (14.44) of jackson (1999) is used for an observation direction n in the x-z plane (the azimuthal angle ? = 0). in the nonrelativistic limit (v c), this formula gives the butter?y pattern of fig. 1.3



n
t =
k=1


e+ei?k


, (1.105)


and the sum is over all electrons in the system.
the total radiated intensity it is proportional to |e+|2, which is
. n .2

it = .. e+ei?k .

. (1.106)

. .
. k .
.k=1 .

expanding the absolute value in (1.104), we obtain

n
it = . |e+|2 + . e?e+e?i(?k ??j ) . (1.107)

k
k=1

k j
kƒ=j


assuming that the amplitudes of the ?elds emitted by each electron are the same



we obtain

|ek |2 = i , (1.108)

it = ni + i . e?i(?k ??j ) . (1.109)
kƒ=j

for random phases, the second term in (1.107) averages to zero, leaving
it = ni , (1.110)

that is, the total intensity is just the sum of the individual intensities. the contributions of the electrons add incoherently with random interferences, as is the case with synchrotron radiation.
however if we could somehow force all electrons to emit with roughly the same phase, ?k c ?j for all k and j, then (1.107) would become
it = ni + n (n ? 1)i = n 2i . (1.111)

here the ?elds emitted by all electrons would add coherently, i.e., with con- structive interference, giving an intensity n times larger than with random
phases.
the basic principle of the fel is to cause all electrons to have approxi- mately the same phase, thereby producing constructive interferences (stim- ulated emission). a key feature of these lasers is that the wavelength of the emitted radiation is a function of the electron energy. to understand this, note that an observer moving along with the electrons would see a wiggler moving at a relativistic velocity with a period that is strongly lorentz con- tracted. to this observer the ?eld appears to be time-dependent, rather than static, since it ties by. in fact, the wiggler magnetic ?eld appears almost as an electromagnetic ?eld whose wavelength is the lorentz-contracted period


of the wiggler. it is well-known that an electron at rest can scatter electro- magnetic radiation. this is called thomson scattering. because the electron energy is much higher than that of the photons, at least in the visible range, we can neglect their recoil, and hence the wavelength of the scattered radia- tion equals that of the incident radiation
s = ?w . (1.112)
?r r

here we use primes to mean that we are in the electron rest frame. going back to the laboratory frame, we examine the radiation emitted in the for- ward direction. as prob. 1.16 shows, this is also lorentz contracted with the wavelength
?s = ?w /2?2 , (1.113)
where ?w is the period of the wiggler and
?z = (l ? v2/c2)?1/2 . (1.114)

here we use ?z rather than ? because the relevant velocity for the lorentz transformation is the component along the wiggler (z) axis. since v is directed primarily along this axis, ?s is to a good approximation given by ?w /2?2.
an alternative way to obtain the scattered radiation wavelength ?s of
(1.111) is to note that for constructive interference of scattered radiation, ?s + ?w must equal the distance ct the light travels in the transit time t = ?w /vz it takes for the electrons to move one wiggler wavelength. this gives ?s + ?w = c?w /vz , and (1.111) follows with the use of (1.112).
we see that two lorentz transformations are needed to determine ?s. since ?z c ? is essentially the energy of the electron divided by mc2, we can change the wavelength ?s of the fel simply by changing the energy of the
electrons. the fel is therefore a widely tunable system. in principle the fel
should be tunable continuously from the infrared to the vacuum ultraviolet.
we now return to the problem of determining how the electrons are forced of emit with approximately the same phase, so as to produce constructive interferences. we can do this with hamilton’s formalism in a straightforward way. for this we need the hamiltonian for the relativistic electron interacting with electric and magnetic ?elds. we note that the energy of a relativistic electron is
e = ,m2c4 + p2c2 (1.115)
where p is the electron momentum. for an electron at rest, p = 0, giving einstein’s famous formula e = mc2. for slow electrons (p mc), we expand the square root in (1.113) ?nding e c mc2 + p2/2m, which is just the rest en-
ergy of the electron plus the nonrelativistic kinetic energy. for the relativistic
electrons in fel, we need to use the exact formula (1.113).
to include the interaction with the magnetic and electric ?elds, we use the principle of minimum coupling, which replaces the kinetic momentum p
by the canonical momentum


p ? p ? ea . (1.116)

here a is the vector potential of the ?eld. using the prescription (1.113), we ?nd the required hamiltonian
h = c[(p ? ea)2 + m2c2]1/2 ? ?mc2 . (1.117)

hamilton’s equations of motion are

p? ?h
i
q? = ? ?h
i


(1.118)

, (1.119)


where the three components of the canonical momentum, pi, and the three electron coordinates, qi, completely describe the electron motion. to obtain
their explicit form, we need to know a. this consists of two contributions,
that of the static periodic magnetic ?eld, and that of the scattered laser ?eld.
if the transverse dimensions of the electron beam are su?ciently small compared to the transverse variations of both ?elds, we can treat the ?elds simply as plane waves. a then has the form


1
a = 2 eˆ?[aw e?


ikw z


+ ase


?i(ws t?ks z)


] + c.c. (1.120)

here aw and as are the amplitudes of the vector potential of the wiggler and the laser, respectively, and
?

eˆ? = eˆ?

= (xˆ ? iyˆ)/

2 , (1.121)

where xˆ and yˆ are the unit vectors along the transverse axes x and y, respec- tively. this form of the vector potential is appropriate for circularly polarized
magnets. also kw = 2?/?w , where ?w is the wiggler period, and ?s and ks
are the frequency and wave number of the scattered light.
with this form of the vector potential, the hamiltonian (1.115) doesn’t depend explicitly on x and y. hence from (1.116), we have
?
x = ? ?x = 0 ,
?
y = ? ?z = 0 , (1.122)
that is, the transverse canonical momentum is constant. furthermore, this constant equals zero if the electrons have zero transverse canonical momen-
tum upon entering the wiggler

pt = 0 . (1.123)

this gives the kinetic transverse momentum


pt = ?ea , (1.124)

which shows that the transverse motion of the electron is simply a circular orbit, as might be expected intuitively.
for the longitudinal motion, the hamilton equations of motion reduce to
z? = pz /m? , (1.125)
e2 ? 2
p?z = ? 2m? ?z (a ) . (1.126)
equation (1.123) just gives the usual formula pz = ?mvz for relativistic particles. equation (1.124) is more informative and states that the time rate of change of the longitudinal electron momentum is given by the spatial
derivative of the square of the vector potential. potentials proportional to a2 are common in plasma physics where they are called ponderomotive potentials. computing ?(a2)/?z explicitly, we ?nd
?(a2)



where

= 2ika? asei(kz??s t) + c.c., (1.127)
?z

k = kw + ks . (1.128)

thus in the longitudinal direction, the electron is subject to a longitudinal force moving with the high speed
v = ?s . (1.129)
s k
since according to (1.111) ks kw , vs is almost the speed of light.
in the laboratory frame, both the electrons and the potential move at
close to the speed of light. it is convenient to rewrite the equations of motion (1.123, 1.124) in a frame moving at velocity vs, that is, riding on the pondero-
motive potential. for this we use
? = z ? vst + ?0 ? ?, (1.130) which is the position of the electron relative to the potential and k? is
the phase of the electron in the potential. ?0 is determined by a? as =
|aw as| exp(ik?0) and k?0 is the phase of the electron relative to the pon- deromotive potential at z = t = 0. this gives readily
?? = vz = vs , (1.131)

which is the electron velocity relative to the potential. to transform (1.124) we have to take into account that ? is not constant. first, taking aw and as
real, we readily ?nd


p?z = ?

2ke2
m? |aw as| sin k? . (1.132)



this is a nonlinear oscillator equation that includes all odd powers of the displacement k?. noting further that p?z = m??2v?z (see prob. 1.17) and

that v?z = ?¨, we obtain




where



2k e2
? = ? 2 4 |aw as| sin k? , (1.133)
s

m = m[1 + (ea/mc)2]1/2 (1.134)


is the e?ective (or shifted) electron mass, and we have at the last stage of the derivation approximated ?z by ?s = [1 ? v2/c2]1/2. equation (1.131)

is the famous pendulum equation. thus in the frame moving at velocity

vs,

the dynamics of the electrons is the same as the motion of a particle in a sinusoidal potential. note that the shifted mass m is used rather than the electron mass m.
the pendulum equation describes the motion of particles on a corrugated
rooftop. in the moving frame, the electrons are injected at some random position (or phase) ?0 with some relative velocity ??(0). intuitively, we might
expect that if this velocity is positive, the electron will decelerate, transferring
energy to the ?eld, while if the velocity is negative, the electron will accelerate, absorbing energy from the ?eld. however as we know from the standard
pendulum problem, the relative phase ?0 with respect to the ?eld also plays a crucial role. from (1.130), we see that p?z is negative if and only if sin k?


??





?? ? ?









fig. 1.9. initial phase-space con?guration of the electrons relative to the pon- deromotive potential. the phases (plotted horizontally) are shown only between
?? ? ? ? ?. the vertical axis gives the electron energies. initially, the electron beam is assumed to have vanishing energy spread and random phase


is positive. hence the electrons initially absorb energy for 0 ? ?0 ? ?/k and give up energy for ?/k ? ?0 ? 2?/k. this is illustrated in fig. 1.9, which shows the phase space of the pendulum. the abscissa is the phase ? of the
electron relative to the potential, while the ordinate is the relative velocity
??. initially all electrons have the same velocity (or energy) ??. since there is
no way to control their initial phases, they are distributed uniformly between
?? and ?.
electrons with phases between ?? and 0 accelerate, while others decel-
erate, so that after a small time, the phase-space distribution looks like that
in fig. 1.10. three important things have occurred. first, the electrons now have di?erent energies, more or less accelerated or decelerated, depending on their initial phases. thus the initially monoenergetic electron beam now has

an energy spread. second, the average relative velocity (??

of the electrons

has decreased, giving an average energy loss by the electrons. conservation
of energy shows that the ?eld energy has increased by the same amount. the recoil of the electrons leads to gain. third, the electron distribution is no longer uniformly distributed between ? = ??/k and ?/k. the electrons are now bunched in a smaller region. instead of producing random interferences with an emitted intensity proportional to n , they are redistributed by the
ponderomotive potential to produce constructive interference as discussed for
(1.109). these three e?ects, recoil, bunching, and spread, are key to under- standing fel’s. they always occur together, and a correct fel description must treat them all.

??





?? ? ?









fig. 1.10. as in fig. 1.9, but a small time after injection into the wiggler. we see a bunching in position (horizontial axis) and spread in energy (vertical axis)


??





?? ? ?









fig. 1.11. as in fig. 1.9, but at the instant of maximum energy extraction

??





?? ? ?









fig. 1.12. as in fig. 1.9, but for longer times such that the electron absorb energy from the laser ?eld


what happens for even longer times is shown in fig. 1.11. the bunching, recoil, and spread have all increased. note that the spread increases much faster than the recoil. this is a basic feature of the fel that makes it hard to operate e?ciently. since pendulum trajectories are periodic, still longer times cause electrons that ?rst decelerated to accelerate and vice versa as shown in fig. 1.12. for such times the average electron energy increases, that


is, laser saturated. to maximize the energy transfer from the electrons to the
laser ?eld, the length of the wiggler should be chosen just short enough to
avoid this backward energy transfer. this kind of saturation is quite di?erent from that for two-level media discussed in chap. 5. for the latter, the gain is bleached toward zero, but does not turn into absorption. here the saturation results from the onset of destructive interference in a fashion analogous the phase matching discussed in sect. 1.2. to maintain the constructive interfer- ence required for (1.111) as the electrons slow down, some fel’s gradually decrease the wiggler wavelength along the propagation direction. this kind
of wiggler is called a tapered wiggler.
in this qualitative discussion, we have assumed that the initial relative
velocity of the electrons was positive, i.e., that the average electron moves faster than the ponderomotive potential. if the initial velocity is negative, the average electron initially absorbs energy. these trends are depicted in fig. 1.13, which plots the small-signal gain of the fel versus the relative electron velocity.










gain

0.6


0.3


0.0




?0.3

?0.6
?2? ?? 0
x






? 2?


fig. 1.13. fel gain function versus initial electron velocity relative to the pon- deromotive potential


this elementary discussion of the fel only consider the electrons, and uses conservation of energy to determine whether the ?eld is ampli?ed. a
more complete fel theory would be self-consistent, with the electrons and
?eld treated on the same footing. such a theory of the fel is beyond the
scope of this book and the reader is referred to the references for further discussion. a self-consistent theory of conventional lasers is given in chap. 7.


problems


derive the wave equation from microscopic maxwell’s equations that include a charge density and current. for this, (1.2, 1.4) become
?

?•e = ,
0


1 ?e

?×b = ?0j + c2 ?t ,
respectively. hint: ?rst show that the conservation of charge equation
?? + j = 0
?t
is solved by ? = ??•p + ?free, j = ?p/?t + ?×m + jfree. for our purposes, assume m = jfree = ?free = 0, and neglect a term proportional to ??.
show using the divergence theorem that

¸
dv (?• p)r =

¸
(p • d?)r ?


p dv .


given the relation ? = ??• p from prob. 1.1, show that the polarization p
can be interpreted as the dipole moment per unit volume.

derive the slowly-varying amplitude and phase equations of motion (1.31, 1.32) by substituting (1.26, 1.28) into the wave equation (1.25). specify which terms you droping and why.

derive the equations (1.56, 1.57) of motion for the classical bloch vector components u and v by substituting (1.28) into (1.44) and using the slowly-
varying approximation. calculate the evolution and magnitude of the classical
bloch vector in the absence of decay.
what are the units of ke2/2m??? in (1.54), where ? is given by (1.74). what is the value of this quantity for the 632.8 nm line of the he–ne
laser (take ? = ?)? calculate the absorption length (1/?) for a 1.06 ?m
nd:yag laser beam propagating through a resonant linear medium with
1016 dipoles/m3.
a ?eld of the form e(z, t) cos(kz??t) interacts with a medium. using the “classical bloch equations”, derive an expression for the index of refraction
of the medium. assume the oscillator frequency ? is su?ciently close to the ?eld frequency ? so that the rotating-wave approximation of (1.50) may be
made.

in both laser physics and nonlinear optics, the polarization of a medium frequently results from the interaction of several separate ?elds. if p(r, t) is
given by



p(r, t) = . pn(r, t)ei(kn •r) ,
n
solve for the polarization amplitude component pn(r, t).

problems 33

find the magnetic ?eld b corresponding to the electric ?eld

1
e(z, t) = 2 xˆe u (z)e?

i?t

+ c.c. ,

for running-wave [u (z) = e?ikz ] and standing-wave [u (z) = sin kz] ?elds. draw a “3-d” picture showing how the ?elds look in space at one instant of time.
the change of variables z ? zr and t ? tr = tr ? z/c transforms the slowly-varying maxwell’s equations from the laboratory frame to the so-called retarded frame. write the slowly-varying maxwell’s equations in this frame. discuss beer’s law in this frame.

in an optical cavity, the resonant wavelengths are determined by the constructive-interference condition that an integral number of wavelengths must occur in a round trip. the corresponding frequencies are determined by these wavelengths and the speed of light in the cavity. given a cavity with a medium having anomalous dispersion, would it be possible to have more than one frequency resonant for a single wavelength? how?

using cartesian coordinates and using spherical coordinates show that the spherical wave exp(ikr ? i?t)/r satis?es the wave equation for free space.
calculate the magnitudes of the electric and magnetic ?elds for a 3 mw
632.8 nm laser focussed down to a spot with a 2 ?m radius. assume constant intensity across the spot. how does this result scale with wavelength?
derive the index of refraction (1.40) for the case that ? = ?(t), i.e., not
?(z) as assumed in (1.39). the ?(t) case is generally more appropriate for
lasers.
solve (1.47) for x(t), i.e., as function of time.
calculate the ?rst and second-order coherence functions for the ?eld
e+(r, t) = e0 e?(?+i?)(t?r/c)?(t r/c) ,
r
where ? is the heaviside (step) function. this would be the ?eld emitted by an atom located at r = 0 and decaying spontaneously from time t = 0, if
such a ?eld could be described totally classically.
derive the fel equation (1.111) using the lorentz transformation zr =
?(z ? ?ct) and tr = ?(t ? ?z/c), where ? is given by (1.102) and ? = v/c.
show that p?z = m??2v?z and proceed to derive the pendulum equa- tion (1.130). use a personal computer to draw electron trajectories shown in figs. 1.10–1.13 and discuss the trajectories.



the kramers-kronig relations allow one to calculate the real and imag- inary parts of a linear susceptibility ?(?) as integrals over one another as
follows:

1 ¸ ?

d?r?r(?r)

?rr(?) = ? ? p.v.
??

?r ?

(1.135)
?

1
?r(?) =

p.v.

¸ ? d?r?rr(?r)

, (1.136)

? ? ?r ? ?
where p.v. means the principal value, i.e., the integral along the real axis excluding an arbitrarily small counterclockwise semicircle around the pole at
?r = ?. equations (1.133, 1.134) are based on the assumption that ? has no
poles in the upper half plane an equivalent set with a change in sign results for a ? that has no poles in the lower half plane. from (1.33, 1.51) we have
the linear susceptibility

?(?) = ?r(?) + i?rr(?) = n ex0x(?) =

n e2 1

n e2

? ? ? ? i?

?e0

? 2??m ? ? ? + i?

= ? 2??m (?

?)2 + ?2 . (1.137)

show that this ?(?) satis?es (1.133, 1.134). hint: for (1.133) use the residue theorem as follows: the desired principal part = the residue for the pole at
?r = ? + i? minus the half residue for the pole at ?r = ?. it is interesting
to note that the power-broadened version of (1.135), namely (5.29), does not
satisfy the kramers-kronig relations, since unlike (1.135), (5.29) does not reduce to a single pole in the lower half plane.

2 classical nonlinear optics











many problems of interest in optics and virtually all of those in this book in- volve nonlinear interactions that occur when the electromagnetic interaction becomes too large for the medium to continue to respond linearly. we have already seen how a nonlinearity plays an essential role in the free electron laser pendulum equation (1.133). in another example that we discuss in detail in chap. 7, the output intensity of a laser oscillator builds up until satura- tion reduces the laser gain to the point where it equals the cavity losses. in situations such as second harmonic generation, one uses the fact that non- linearities can couple electromagnetic ?eld modes, transferring energy from one to another. such processes can be used both to measure properties of the nonlinear medium and to produce useful applications such as tunable light sources.
in this chapter we extend sect. 1.3’s discussion of the simple harmonic oscillator to include quadratic and cubic nonlinearities, i.e., nonlinearities
proportional to x2 and x3, respectively. such nonlinearities allow us to un-
derstand phenomena such as sum and di?erence frequency generation, mode
coupling, and even chaos in a simple classical context. subsequent chapters treat these and related phenomena in a more realistic, but complex, quantized environment.


nonlinear dipole oscillator

section 1.3 discusses the response of a linear dipole oscillator to a monochro- matic electric ?eld. when strongly driven, most oscillators exhibit nonlinear- ities that can be described by equations of motion of the form [compare with (1.46)]

x¨(z, t) + 2?x? (z, t) + ?2x(z, t) + ax2(z, t) + bx3(z, t) + •••
= e e(z, t) . (2.1)
m

here we include a z dependence, since the polarization modeled by x(z, t) is a function of z. speci?cally, x describes the position of an electron relative to the nucleus (internal degree of freedom) while z is the location of the



dipole in the sample (external degree of freedom). such oscillators are called “anharmonic”. in many cases such as for isolated atoms, the coe?cient a vanishes, leaving bx3 as the lowest order nonlinear term.
we can determine the e?ect of the nonlinear terms on the response by a
process of iteration that generates an increasingly accurate approximation to
x(t) in the form of a power series
x(t) c x(1)(t) + x(2)(t) + ••• + x(n)(t) + ••• , (2.2) the leading term in this series is just the linear solution (1.51) itself or a
linear superposition (1.63) of such solutions. to obtain the second-order con-
tributions, we substitute the linear solution into (2.1), assume an appropriate form for the second-order contributions, and solve the resulting equations in a fashion similar to that for the linear solution. in the process, we ?nd that
new ?eld frequencies are introduced. in general, the nth order term is given
by solving the equation assuming that the nonlinear terms can be evalu- ated with the (n ? 1)th order terms. many of the phenomena in this book
require solutions that go beyond such a perturbative approach, since the cor-
responding series solution may fail to converge. nevertheless, the subjects usually considered under the heading “nonlinear optics” are very useful and are typically described by second- and third-order nonlinearities.
we consider ?rst the response of a medium with an ax2 nonlinearity subjected to a monochromatic ?eld of frequency ?n, that is, (1.62) with a single amplitude en(z). choosing x(1) to be given by the corresponding linear solution (1.51), we ?nd for x2 the approximate value


x2 c [x(1)]2 =


1 1
|x(1) 2


(1) 2


2i(kn z??n t)

4 n |

+ [ ] e
4 n

+ c.c., (2.3)

where the slowly varying amplitude x(1) = x0nxn

and xn


is given by (1.50).

we see that this nonlinear term contains both a dc contribution and one at
twice the initial frequency. the dc term gives the intensity measured by a square-law detector and is the origin of the kerr electro-optic e?ect in crys- tals, while the doubled frequency term leads to second harmonic generation. observation of the latter in quartz subjected to ruby laser light kicked o? the ?eld known as nonlinear optics [see franken et al. (1961)]. with an an- harmonic forcing term proportional to (2.3), the second-order contribution
x(2)(t) has the form
x(2)(t) = 1 x(2) + 1 x(2) ei(2kn z?2?n t) + c.c.. (2.4)

2 dc

2 2?n

according to our iteration method, we determine the second-order coe?-

cients x(2)

and x(2) by substituting (2.2) into (1.1) approximating x2 by the

?rst-order expression (2.3). by construction, the terms linear in x(1) cancel
the driving force (e/m)e, and we are left with a simple harmonic oscillator equation for the x(2) coe?cient, namely,

2.1 nonlinear dipole oscillator 37
x¨(2)(t) + 2?x? (2)(t) + ?2x(2)(t) = a[x(1)(t)]2 . (2.5)
equating coe?cients of terms with like time dependence, we ?nd

x(2)

a (1) 2

dc = ? 2?2 |xn |
a . ee n (z)/m .2

.
. 2 2
n ?

.
2i?n .

, (2.6)

x(2)

a (1) 2 1

2?n = ? 2 (xn )

?2 ? (2?n)2

? 2i(2?n)?

. (2.7)

note here that if the applied ?eld frequency ?n is approximately equal to the natural resonance frequency ?, both the dc and second-harmonic coe?cients
are divided by squares of optical frequencies. hence these terms are usually
very small, and second-order theory is a good approximation, for example, in noncentrosymmetric crystals.
now consider the response of this nonlinear oscillator to an electric ?eld given by (1.62) with two frequency components at the frequencies ?1 and ?2.
to lowest order (?rst order in the ?elds), we neglect the nonlinearities and ?nd x(t) c x(1)(t), which is given by the linear superposition (1.63) with two modes. the approximate second-order nonlinearity [x(1)]2 has the explicit
form
[x(1)]2 = [x(1)ei(k1 z??1 t) + x(1) 2 2
1 2 ei(k z?? t)]2
1 1

= x(1)

(1)

i[(k1 ?k2 )z?(?1 ??2 )t]

(1)

(1)

i[(k1 +k2 )z?(?1 +?2 )t]

2 1 [x2 ]? e

+ x x e
2 1 2

2
+ .(|x(1)|2 + [x(1)]2 e2i(kn z??n t)) + c.c.. (2.8)
4 n n n=1

as one expects from (2.4), each mode contributes a dc component and a com- ponent oscillating at its doubled frequency, 2?n. in addition, (2.8) contains components at the sum and di?erence frequencies ?1 ± ?2. hence the non-
linear dipoles can generate ?elds at these frequencies. more generally, when
higher orders of perturbation are considered, the polarization of a medium consisting of such anharmonic oscillators can generate all frequency com-
ponents of the form m?1 ± n?2, with m and n being integers. when such combinations lead to frequencies other than harmonics of ?1 or ?2, they are
called combination tones. such tones are responsible for self mode locking
in lasers (sect. 11.3) and three- and four-wave mixing (rest of the present chapter and chap. 10).
generalizing the second-order contribution (2.4) to include the sum and di?erence frequencies, we have

x(2)(t) = 1 x(2)

i[(k1 +k2 )z?(?1 +?2 )t]

1 (2)

i[(k1 ?k2 )z?(?1 ??2 )t]

2 s e

+ x e
2 d

+ 1 . x(2)

i(2kn z?2?n t) 1


(2)

2
n=1

2?n e

+ x + c.c.. (2.9)
2 dc


substituting (2.8, 2.9) into (2.5) and equating coe?cients of like time de- pendence, we ?nd that x(2) is given by the sum of two terms like (2.6), the doubled frequency terms are given by (2.7), and the di?erence and sum fre-
quency terms are given by


(1)


(1)

d = ?a ?2

x1 [x2 ]?
2

, (2.10)

? (?2 ? ?2)
(1)

? 2i(?1 ? ?2)?
(1)

s = ?a ?2

x1 x2
2

. (2.11)

? (?1 + ?2)

? 2i(?1 + ?2)?

except for a factor of 2 for the degenerate frequency case ?1 = ?2 the di?er- ence frequency term (2.10) reduces to the dc term (2.6) and the sum frequency
term (2.11) reduces to the second harmonic term (2.7).
frequency combinations like those in (2.9) also appear in quantum me- chanical descriptions of the medium, which typically involve more intricate nonlinearities. in particular, di?erence frequency generation induces pulsa- tions of the populations in a medium consisting of two-level atoms irradiated by two beams of di?erent frequencies. these pulsations play an important role in saturation spectroscopy, as discussed in chap. 9.


coupled-mode equations

so far we have obtained the steady-state response of the nonlinear dipole to second order in the ?eld and have seen how combination tones at the fre-
quencies m?1 + n?2 can be generated by such systems. the way in which
the corresponding waves evolve is readily obtained from the wave equation (1.25) giving the propagation of an electromagnetic ?eld e(z, t) inside a medium of polarization p (z, t). for a medium consisting of nonlinear os- cillators, p (z, t) = n (z)ex(z, t), where n (z) is the oscillator density and z
labels the position inside the medium. to analyze the growth of a wave at a frequency ?3, we consider three modes in the ?eld (1.62) and in the po-
larization (1.65). the slowly varying amplitude approximation allows us to
derive coupled di?erential equations for the evolution of the ?eld envelopes
en. these are called coupled-mode equations, and play an important role in
multiwave phenomena such as phase conjugation (sect. 2.4 and chap. 10) and
the generation of squeezed states (chap. 17), in which case a fully quantum mechanical version of these equations is required. in general, coupled-wave equations form an in?nite hierarchy of ordinary di?erential equations, and some kind of approximation scheme is needed to truncate them. for instance, if we are only interested in the small signal build-up of the wave at frequency
?3, we can neglect its back-action on the nonlinear polarization p (z, t) – ?rst- order theory in e3. this is the procedure used in sect. 2.1. another common approximation assumes that the waves at frequencies ?1 and ?2 are so intense
that their depletion via the nonlinear wave-mixing process can be neglected.

2.2 coupled-mode equations 39

an important feature of coupled-mode equations is phase matching, which represents the degree to which the induced mode coupling terms in the po- larization have the same phase as the ?eld modes they a?ect. to the extent that the phases di?er, the mode coupling is reduced. phase matching involves di?erences in wave vectors and amounts to conservation of momentum. this is distinct from the frequency di?erences of the last section, which amount to conservation of energy.
indeed, the resonance denominators appearing in the nonlinear suscepti- bilities can be interpreted as a consequence of energy conservation. in con- trast, the spatial phase factors are a result of momentum conservation. this is particularly apparent when the electromagnetic ?eld is quantized, as will be the case in the second part of the book, because in that case, it is easy
to show that the energy and momentum of a photon of frequency ? and mo- mentum k are k? and kk, respectively. in a vacuum, one has that ? = kc
for all frequencies, so that energy conservation automatically guarantees mo-
mentum conservation. but this ceases to be true in dispersive media, where the factor of proportionality between ? and k is frequency-dependent.
consider for instance the case of di?erence-frequency generation, where two incident waves at frequencies ?1 and ?2 combine in the nonlinear medium to generate a wave at the di?erence frequency ?d. in the slowly-varying am- plitude approximation and in steady state (we neglect the ?/?t terms), (1.43)
becomes

den = ikn

, (2.12)

dz 2?n pn
where ?n is the permittivity at the frequency ?n of the host medium in which
our oscillators are found. from (1.64, 1.65), we use the linear solutions for
modes 1 and 2

de n

n (z)e2 en

dz c ikn

2?nm


2 2
n ?

2i?n

= ??? en . (2.13)


note that in this example we assume that the host medium is purely dis- persive otherwise another absorption term would have to be included. us- ing (2.10, 2.12), we ?nd for the di?erence-frequency term at the frequency
?d = ?1 ? ?2

. (1)


(1)

i(k1 ?k2 ?kd )z .

de d = ikd n (z)e

(e/m)ed ? ax1 [x2 ]?e

. (2.14)

dz 2?d

?2 ? ?2 ? 2i?d?



the coupled-mode equations (2.13, 2.14) take a rather simple form, since we have neglected the back action of ed on e1 and e2. equation (2.13) sim- ply describes the linear absorption and dispersion of e1 and e2 due to the
nonlinear oscillators, as does the ?rst term on the right-hand side of (2.14) for ed. in many cases these are small e?ects compared to those of the host medium accounted for by ?n, and we neglect them in the following. e1 and e2



then remain constant and (2.14) (without the leading term) can be readily integrated over the length l of the nonlinear medium to give
ei?kl ? 1

ed(l) = ge1e ?

i?k , (2.15)

where

ikd an e . x(1)[x(1)]?/e1e ? .

g = ? 2? ?2 ,
d ? ?2 ? 2i?d?
and the k-vector mismatch ?k = k1 ? k2 ? kd. the resulting intensity
id = |ed|2 is

id(l) = |g|2i1i2

sin2(?kl/2)
(?k/2)2

. (2.16)

if ?k = 0, id(l) reduces to |g|2i1i2l2, but for ?k ƒ= 0, it oscillates
periodically. to achieve e?cient frequency conversion, it is thus crucial that (k1 ?k2)l be close to kdl. for ?k ƒ= 0 the maximum intensity id is reached for a medium of length l = ?/?k. for larger values of ?kl, the induced polarization at the frequency ?d and the wave propagation at that frequency
start to interfere destructively, attenuating the wave. for still larger values
l, the interference once again becomes positive, and continues to oscillate
in this fashion. since nonlinear crystals are expensive, it is worth trying to achieve the best conversion with the smallest crystal, namely for ?kl = ?.
in the plane-wave, collinear propagation model described here, perfect phase matching requires that the wave speeds un = vn/kn all be equal, as would
be the case in a dispersionless medium. more generally we have the di?erence
?k = k1 ? k2 ? kd = (n1?1 ? n2?2 ? nd?d)/c ƒ= 0 since the n’s di?er. for
noncollinear operation the vectorial phase matching condition
?kl = |k1 ? k2 ? kd|l c ? (2.17)
must be ful?lled for maximum id. there are a number of ways to achieve this, including appropriate geometry, the use of birefringent media, and tem- perature index tuning.


2.3 cubic nonlinearity

we already mentioned that quadratic nonlinearities such as described in the preceding sections to not occur in isolated atoms, for which the lowest order nonlinear e?ects are cubic in the ?elds. these can be described in our classical
model by keeping the bx3 term instead of ax2 in the nonlinear oscillator equation (2.1). in the presence of two strong pump ?elds at frequencies ?1 and ?2, the third-order polarization given by bx3 includes contributions at the frequencies ?1 and ?2 and at the sideband frequencies ?0 = ?1 ? ? and ?3 = ?2 + ? as well, where

2.3 cubic nonlinearity 41
? = ?2 ? ?1 . (2.18)
the generation of these sidebands is an example of four-wave mixing. to describe the initial growth of the sidebands, we write the anharmonic term
bx3 to third-order in x1 and x2, and ?rst-order in the small displacements x0
and x3, that is,

[x(1)]3 = 1 [x(1) ei(k1 z??1 t) + x(1)


i(k2 z??2 t)

8 1 2 e

+3x(1)

(1)

0 ei(k0 z??0 t) + 3x3 ei(k3 z??3 t) + c.c.]
,

× 2x(1)

(1)

1 x2 ei[(k1 +k2 )z?(?1 +?2 )t]

+2x(1)

(1)

1 [x2 ]? ei[(k1 ?k2 )z?(?1 +?2 )t]
2 .
+ .{[x(1)]2 e2i(kn z??n t) + |x(1) 2

n n=1

n | } + c.c.

, (2.19)


where the terms in {} are similar to those in (2.9). the factor of 3 results from the three ways of choosing the x0 and x3 from the triple product.
the curly braces in (2.19) contain two dc terms, a contribution oscillat- ing at the di?erence frequency ?, and three rapidly oscillating contributions oscillating at the frequencies ?n + ?m. these time-dependent terms are some-
times called (complex) index gratings, and the nonlinear polarization may be interpreted as the scattering of a light ?eld en from the grating produced by two ?elds em and ek . in this picture, the dc terms are “degenerate” grat- ings produced by the ?elds em and e ? . equation (2.19) readily gives the third-order contributions to the components of the polarization pn at the
frequencies of interest.
one can interpret (2.19) as the scattering of components in the [ ] of the ?rst lines o? the slowly varying terms in the {}. speci?cally the |x(1)|2 terms in (2.19) contribute nonlinear changes at the respective frequencies of the
components in the [ ]. in contrast, the scattering o? the “raman-like” term exp[i(k2 ? k1)z ? i?t] and its complex conjugate contribute corrections at frequencies shifted by ±?. taking ?2 > ?1, we see that the ?2 term in the [ ] scatters producing components at both the lower frequency ?1 (called a stokes shift) and the higher frequency ?3 = ?2 + ? (called an anti-stokes shift). similarly the ?1 term in the [ ] leads to contributions at the frequencies ?0 = ?1 ? ? and at ?2. the induced polarization components at the frequencies ?0 and ?3 are called combination tones. they are generated in the nonlinear medium from other frequencies. if the two pump beams at ?1 and ?2 are copropagating, the index grating represented by the k2 ?k1 term propagates
at approximately the velocity of light in the host medium, but if the beams
are counterpropagating, the grating propagates at the relatively slow speed v = ??/(k1 + k2). in particular, it becomes stationary for the degenerate case ?1 = ?2. (compare with the ponderomotive force acting on the electrons


in the free electron laser, (1.126)! –can you draw an analogy between the two situations?)
we are often only interested in induced polarizations near or at the fun- damental frequencies ?n,n = 0, 1, 2, 3. keeping only these in (2.19) and neglecting combination tones involving x0 and x1 in the pump-mode polar-
izations (prob. 2.7), we ?nd

3

|[x(1)]3|fund =

x(1)(|x(1)|2 + 2|x(1)|2) ei(k1 z??1 t)

8 1
+ x(1)

1

(1) 2

2

(1) 2


i(k2 z??2 t)

8 2 (|x2 |
6

+ 2|x1 | )e

+ [|x(1)|2 + |x(1)|2][x(1) ei(k0 z??0 t) + x(1) ei(k3 z??3 t)]

8 1
+ 6 x(1)


(1)

2

(1)

0 3

i[(k1 +k2 ?k3 )z??0 t]

8 1 x2 [x3 ]? e
6

+ x(1)

(1)

(1)

i[(k1 +k2 ?k0 )z??3 t]

8 1 x2 [x0 ]? e
3

+ [x(1) 2

(1)

i[(2k1 ?k2 )z??0 t]

8 1 ] [x2 ]? e
3

+ [x(1) 2

(1)

i[(2k2 ?k1 )z??3 t]

8 2 ] [x1 ]? e

+ c.c. (2.20)


combining the various terms, we ?nd that the third-order polarization com- ponents are given by


(3) 6


(1) 2


(1) 2


(1)

p0 = 8 n eb [|x1 |
6

+ |x2 | ]x0

+ n eb x(1)x(1)

(1)

i(k1 +k2 ?k3 ?k0 )z

8 1 2 [x3 ]? e
3
8 1 2

(3) 3

(1)

(1) 2

(1) 2

p1 = 8 n eb x1 [|x1 |

+ 2|x2 | ] (2.21b)

(3) 3

(1)

(1) 2

(1) 2

p2 = 8 n eb x2 [2|x1 |

+ |x2 | ] (2.21c)

(3) 6

(1) 2

(1) 2

(1)

p3 = 8 n eb [|x1 |
6

+ |x2 | ]x3

+ n eb x(1)x(1)

(1)

i(k1 +k2 ?k0 ?k3 )z

8 1 2 [x0 ]? e
3
+ n eb [x(1)]2[x(1)]? ei(2k2 ?k1 ?k3 ) . (2.21d)
8 2 1
the polarization components p(3) and p(3) are solely due to the exis-
0 3
tence of index gratings, which are also responsible for the factors of 2 in the
cross coupling terms for p(3) and p(3). this asymmetry is sometimes called
1 2
nonlinear nonreciprocity and was discovered in quantum optics by chiao


et al. (1966). it also appears in the work by van der pol (1934) on coupled vacuum-tube tank circuits. in the absence of index gratings, the factors of 2
in (2.21b, c) are replaced by 1, and |x(1)|2 and |x(1)|2 play symmetrical roles

in p(3)

1 2
(3)

1 and p2 .
the polarizations pn lead to coupled-mode equations for the ?eld en-
velopes. the procedure follows exactly the method of sect. 2.2 and we obtain
(prob. 2.2)


de0 =

[? ?

2 ? 2] + ?


2 ? ei(2k1 ?k2 ?k0 )z

dz ?e0

0 ? 01|e1| ?

02|e2|

0121e1 e2

3 e ei(k1 +k2 ?k3 ?k0 )z , (2.22a)

de1 =
dz

?e [?1 ?

?1|e1|2

? ?12|e2|

2] , (2.22b)

de2 =
dz

?e2

[?2 ?

?2|e2|2

? ?21|e1|

2] , (2.22c)

de3 =

[? ?

2 ? 2 + ?

2 ? ei(2k2 ?k1 ?k3 )z

dz ?e3

3 ? 31|e1| ?

32|e2|

3212e2 e1

0 e ei(k1 +k2 ?k0 ?k3 )z . (2.22d)
here en is the complex amplitude of the ?eld at frequency ?n, and the ??nen
terms allow for linear dispersion and absorption.
equations (2.22b, c) for the pump modes amplitudes are coupled by the cross-coupling (or cross-saturation) coe?cients ?nj . to this order of pertur- bation, they are independent of the sidemode amplitudes e0 and e3. because e1 and e2 always conspire to create an index grating of the correct phase, the
evolution of these modes is not subject to a phase matching condition. equa-
tions of this type are rather common in nonlinear optics and laser theory. in sect. 7.4, we obtain an evolution of precisely this type for the counterpropa- gating modes in a ring laser. we show that the cross-coupling between modes can lead either to the suppression of one of the modes or to their coexistence,
depending on the magnitude of the coupling parameter c = ?12?21/?1?2 and relative sizes of the ?n.
in contrast, the sidemodes e0 and e3 are coupled to the strong pump ?elds
e1 and e2 only, and not directly to each other. they have no back-action on
the pump modes dynamics, and their growth is subject to a phase-matching
condition.


four-wave mixing with degenerate pump frequencies


in many experimental situations, it is convenient to drive the nonlinear medium with two pump ?elds of the same frequency ?2, but with opposite
propagation directions given by the wave vectors k2? and k2?. the pump


waves cannot by themselves generate polarization components at sideband frequencies. however one can still take advantage of the index gratings pro- duced by the pump beams with weak waves at frequencies symmetrically
detuned from ?2 by a small amount ±? (see fig. 2.1). this procedure has
gained considerable popularity in connection with optical phase conjugation.
in optical phase conjugation, one of the sidebands is called the probe (at
?1 = ?2 ? ? ) and the other the signal (at ?3 = ?2 + ?), and we adopt this
notation here in anticipation of chaps. 9, 10.










?1 ?2 ?3
fig. 2.1. mode spectrum in four-wave mixing for optical phase conjugation


we consider the wave confguration in fig. 2.2 with two counterpropa- gating pump beams along one direction, and counterpropagating signal and
“conjugate” waves along another direction, which we call the z axis. the
electric ?eld for these four waves has the form


1
e(r, t) = 2 [e1 e


i(k1 z??1 t)


+ e2? e


i(k2? •r??2 t)


+ e2? e


i(k2? •r??2 t)

+e3 ei(?k3 z??3 t)] + c.c., (2.23)
where we take k2? = ?k2?. the ?eld fringe patterns resulting from interfer- ence between the various waves can induce index gratings. the corresponding
linear displacement x(1)(t) contains components proportional to each of the ?eld amplitudes, and the third-order nonlinear displacement x(3) consists of
the sum of all terms proportional to the products of three ?elds, each of which
can be anyone of the four waves or their complex conjugates. this gives a grand total of 8 • 8 • 8 = 512 terms. fortunately, we’re only interested in a rel-
atively small subset of these terms, namely those with the positive frequency
?1 linear in e1. this gives a third-order signal polarization p(3) proportional
to




























fig. 2.2. diagram of interaction between standing-wave pump beam (?2) with probe (?1) and conjugate (?3) beams used in phase conjugation



e1e2?e ?

+ e1e ? e2? + e2?e1e ?

+ e ? e1e2?

2? 2?

2? 2?

+e2?e ? e1? + e ? e2?e1 + e1e2?e ?

+ e1e ? e2?

2? 2?

2? 2?

+e2?e1e ?

+ e ? e1e2? + e2?e ? e1 + e ? e2?e1

2? 2?

2? 2?

+[e ?e2?e2? + e ?e2?e2? + e2?e ?e2? + e2?e ?e2?
3 3 3 3
i(k3 ?k1 )z

+e2?e2?e ? + e2?e2?e ?]e

(2.24a)

3 3
2 2 ?

i(k3 ?k1 )z

= 6e1(|e2?|

+ |e2?| ) + 6e3 e2?e2? e

. (2.24b)


the various terms in (2.24a) have simple physical interpretations. for instance, the ?rst term results from the product in [x(1)]3


e1 ei(k1 z??1 t)e2

i(k2? • r??2 t)


? ?1(k2? • r??2 t) 2?

note that the pump phase dependencies k2? • r cancel identically, as they do for all terms in (2.24). the ?rst and second terms represent contributions to
the nonlinear refraction of the ?eld e1 due the nonlinear index (kerr e?ect) induced by the pump ?eld intensity i2?. the third term can be understood as originating from the scattering of the ?eld e2? o? the grating produced by e1 and e2?, etc. its e?ect on the polarization is precisely the same as that of the


?rst two terms, but we have intentionally written it separately in anticipation of the quantum mechanical discussion of chap. 10, where the order in which the ?elds are applied to the medium matters. indeed in our classical model, the terms in the ?rst three lines in (2.24a) are all proportional to the product
of a pump ?eld intensity and the probe ?eld e1, and can be globally described
as nonlinear absorption and refraction terms.
the last two lines in (2.24a) couple the sidemode e3 to e1 via the following scattering mechanism: the ?eld e ? interferes with the pump ?elds e2? and e2? to induce two complex index gratings that scatter e2? and e2?, respec- tively, into e1. this process, which is essentially the real-time realization of
holographic writing and reading, is called phase conjugation and is discussed
in detail in chap. 10. the process retrore?ects a wavefront, sending it back along the path through which it came (see fig. 10.1). it can be used to com- pensate for poor optics. note that although the pump phase dependencies cancel one another as they do for the terms in the ?rst four lines of (2.24a),
the induced polarization has the phase exp[i(k3z ? ?1t)], while maxwell’s equations require exp[i(k1z ? ?1t)]. this gives the phase mismatch factor exp[i(k3 ? k1)z], which is important except in the degenerate frequency case ?1 = ?2 = ?3, for which k3 = k1.
neglecting the depletion of the pump beams e2? and e2?, we ?nd the coupled-mode equations for e1 and e ?

de1 =
dz
de ?


??1e1


+ ?1e ? e2i?kz , (2.25a)

3 ? ?

? ?2i?kz



where

? dz = ??3 e3 + ?3 e1 e

, (2.25b)

?k = k3 ? k1 . (2.26)
2
here ?de ?/dz appears since e ? propagates along ?z, and we use ?n for the
3 3
coupling coe?cient to agree with later usage, although it is only a part of a
susceptibility.
to solve these equations, we proceed by ?rst transforming away the phase mismatch by the substitution

e1 = a1 e2i?kz (2.27)

into (2.25). in particular, (2.25a) becomes


da1 =
dz

?(?1

+ 2i?k)a1

+ ?1e ? . (2.28)

we seek solutions of (2.25b, 2.28) of the form e?z . substituting a1 = e?z into (2.28), solving for e ?, and substituting the result into (2.25b), we ?nd the
eigenvalues


1 2 ? 1/2
?± = ? 2 (?1 ? ?? + 2i?k) ± [(?1 + ?? + 2i?k) /4 ? ?1? ]
3 3 3
= ?a ± [?2 ? ?1??]1/2 = ?a ± w. (2.29)

hence the general solutions are
a1(z) = e?az [a ewz + b e?wz ] (2.30a)


and


e ? ?az wz


?wz

3 (z) = e

[c e

+ d e

] , (2.30b)

where the coe?cients a, b, c and d are determined by the boundary condi-
tions of the problem.
we suppose here that a weak signal weak ?eld e1(0) is injected inside the nonlinear medium at z = 0, and we study the growth of the counterpropa- gating conjugate wave e ?, which is taken to be zero at z = l. this means that a1(0) = e1(0) = constant, and e ?(l) = 0, in which case one has immediately b = e1(0)?a and d = ?c ewl. matching the boundary conditions of (2.25b, 2.28) at z = l yields


1 wl
a = 2 a1(0) e?


(w ? ?)/(w cosh wl + ? sinh wl) , (2.31)

2wc ewl = ??(a sinh wl + a1(0) e?wl . (2.32)

further manipulation yields ?nally


e1(z) = e1(0) e?(a+w+2i?k)z

. (w ?) ew(z?l) sinh wz .
1 +
w cos wl + ? sinh wl


, (2.33)


3 (z) = ?3e1(0)

e?az sinh w(z ? l)


. (2.34)

e ? ?

w cosh wl + ? sinh wl

in particular the amplitude re?ection coe?cient r = e ?(0)/e1(0) is given by

r = e3 (0) =


sinh wl ??


. (2.35)

e1(0)

? 3 w cosh wl + ? sinh wl

see chap. 10 on phase conjugation for further discussion of these equations.


coupled modes and squeezing

a popular topic is the “squeezing”, i.e., deamplifying, of noise in one quadra- ture of an electromagnetic wave at the expense of amplifying the noise in the orthogonal quadrature. one way to achieve such squeezing is through the use of mode coupling mechanisms such as described by (2.25a, b). to see
under which conditions the ?n coupling factors can lead to this quadrature- dependent ampli?cation, let’s droping the ?n term in (2.25) and put the time
dependencies back in. we ?nd for example the schematic equation



{?(3)e 2

?2i?t

}[e3 e

i?t]?

? e1e

?i?t

, (2.36)


where (?3) is a third-order susceptibility. suppose that at a time t, ei?t = 1 and that {} = 1. according to (2.36), this tends to amplify e1. now wait until the orthogonal quadrature phasor exp(i?t ? i?/2) = 1. at this time, the second-harmonic (two-photon) phasor exp(?2i?t) has precessed through two times ?/2, that is, {} = –1. hence a two-photon coupling {} ?ips the sign
of the coupling between orthogonal quadratures. this is the signature of a coupling process that can lead to squeezing. it is equally possible that a ?(2) process with a single pump photon having the value 2? can cause squeezing. this ?(2) process is known as parametric ampli?cation. chapter 17 discusses
the squeezing of quantum noise by four-wave mixing.


nonlinear susceptibilities

so far we have used an anharmonic oscillator to introduce some aspects of nonlinear optics that are useful in the remainder of this book. such a simple model is surprisingly powerful and permits us to understand numerous non- linear optics e?ects intuitively. in general, however, ?rst principle quantum mechanical calculations are needed to determine the response of a medium to a strong electromagnetic ?eld. a substantial fraction of this book ad- dresses this problem under resonant or near-resonant conditions, i.e., under conditions such that the frequency (ies) of the ?eld(s) are near an atomic transition. perturbative analyzes such as sketched in this chapter are usually not su?cient to describe these situations.
in many cases, however, the incident radiation is far from resonance with any transition of interest, and/or the material relaxation rate is exceedingly fast. in such cases, perturbation theory based on the concept on nonlinear susceptibility may be of great advantage. this is the realm of conventional nonlinear optics, and the reader should consult the recent treatises by shen (1984), by hopf and stegeman (1986), and by boyd (1992), as well as the classic book by bloembergen (1965), for detailed descriptions of these ?elds. here we limit ourselves to a brief introduction to the formalism of nonlinear susceptibility.
in linear problems, the polarization of the medium is (by de?nition) a linear function of the applied electric ?elds. the most general form that it can take is given by the space-time convolution of a linear susceptibility tensor ?(1) with the electric ?eld:

¸
p(r, t) = ?0

¸ t
d3r
??


dtr?(1)(r ? rr,t ? tr) : e(rr, tr) . (2.37)


taking the four-dimensional fourier transform of this expression for a monochro- matic wave e(r, t) = e (k, ?)ei(k•r??t), we ?id

2.5 nonlinear susceptibilities 49
p(k, ?) = ?0?(1)(k, ?) : e(k, ?) , (2.38)

with

¸
?(1)(k, ?) =


¸ t
d3r
??



dtr?(1)(r, tr)ei(k • r??tr ) . (2.39)

the linear dielectric constant is related to ?(1)(k, ?) via
?(k, ?) = ?0[1 + ?(1)(k, ?)] . (2.40)

in the nonlinear case, and for electric ?elds su?ciently weak that perturbation theory is valid, one gets instead

¸
p (r, t) = ?0
¸

¸ t
d3r
??


dtr?(1)(r ? rr,t ? tr) • e(rr, tr)

+?0
¸
+?0

.

dr1dt1dr2dt2?(2)(r ? r1,t ? t1 r ? r2,t ? t2) : e(r1, t1)e(r2, t2)

dr1dt1dr2dt2dr3dt2?(3)(r ? r1,t ? t1 r ? r2,t ? t2 r ? r3,t ? t3)

. e(r1, t1)e(r2, t2)e(r3, t3) + ... , (2.41)
where ?(n) the nth-order susceptibility. if e(r, t) can be expressed as a sum of plane waves,
e(r, t) = . e(kn, ?n)ei(kn • r??n t) , (2.42)
n
then as in the linear case, the fourier transform of (2.41) gives
p(k, ?) = p(1)(k, ?) + p(2)(k, ?) + p(3)(k, ?) + ... (2.43)
with p(1)(k, ?) given by (2.38) and
p(2)(k, ?) = ?(2)(k = kn + km,? = ?n + ?m) : e(kn, ?n)e(km, ?m)
p(3)(k, ?) = ?(3)(k = kn + km + ka,? = ?n + ?m + ?a)
.e(kn, ?n)e(km, ?m)e(ka, ?a)


and



?(n)(k = k1 + k2 + ... + kn,? = ?1 + ?2 + ... + ?n)
¸

= d3r1dt1dr dt2 ... d3rndtn?(n)
×(r ? r1,t ? t1 r ? r2,t ? t2 ... r ? rn,t ? tn)
× exp[+iki•(r ? r1) ? i?1(t ? t1) + ... + ikn•(r ? rn) ? i?n(t ? tn)] .


problems

solve the couplem. ode equations

de1 =
dz


??1e1


+ ?1e ? , (2.44)

de ? =

?? ?

+ ??

, (2.45)

dz ?

3 e3

3e1

valid for phase-matched forward three-wave mixing. ans:
e1(z) = e?az [e1(0) cosh wz + (??e1(0) + ?1e ?(0)) sinh wz/w] (2.46)
e ? ?az ? ? ?

3 (z) = e

[e3 (0) cosh wz + (ae3 (0) + ?3e1(0)) sinh wz/w] , (2.47)

where a = (?1 + ??)/2,? = (?1 ? ??)/2, and w = ,?2 + ?1??.
3 3 3
derive the coe?cients in the coupled-mode equations (2.22).
calculate all wavelengths generated in a ?(3) nonlinear medium by a combination of 632.8 nm and 388 nm laser light.
calculate the coupling coe?cient ?n for four-wave mixing based on an anharmonic oscillator.

write the propagation equations for second-harmonic generation. com- ment on phase matching.

calculate the phase mismatch for a conjugate wave of frequency ?3 = ?2 + (?2 ? ?1) generated by signal and pump waves with frequencies ?1 and ?2, respectively, and propagating in the same direction. include the fact that
the indices of refraction for the three waves are in general di?erent, that is,
?(?1) = ?(?2) + ??1 and ?(?3) = ?(?2) + ??3.
show that (2.20) contains all the fundamental contributions from the third-order expression (2.19).

evaluate the re?ection coe?cient r of (2.35) in the limit of large l. answer: r = ???/(w ± ?) for re(w) ? 0.

3 quantum mechanical background











chapters 1, 2 describe the interaction of radiation with matter in terms of a phenomenological classical polarization p. the question remains as to when this approach is justi?ed and what to do when it isn’t. unexcited systems interacting with radiation far from the system resonances can often be treated purely classically. the response of the system near and at resonance often deviates substantially from the classical descriptions. since the laser itself and many applications involve systems near atomic (or molecular) resonances, we need to study them with the aid of quantum mechanics.
in preparation for this study, this chapter reviews some of the highlights of quantum mechanics paying particular attention to topics relevant to the interaction of radiation with matter. section 3.1 introduces the wave function for an abstract quantum system, discusses the wave function’s probabilistic interpretation, its role in the calculation of expectation values, and its equa- tion of motion (the schro¨dinger equation). expansions of the wave function in various bases, most notably in terms of energy eigenstates, are presented and used to convert the schro¨dinger partial di?erential equation into a set of ordinary di?erential equations. the dirac notation is reviewed and used to discuss the state vector and how the state vector is related to the wave func- tion. system time evolution is revisited with a short review of the schro¨dinger, heisenberg and interaction pictures.
in chaps. 4–12, we are concerned with the interaction of classical electro- magnetic ?elds with simple atomic systems. section 3.2 lays the foundations
for these chapters by discussing wave functions for atomic systems and study- ing their evolution under the in?uence of applied perturbations. time depen- dent perturbation theory and the rotating wave approximation are used to predict this evolution in limits for which transitions are unlikely. the fermi golden rule is derived. section 3.3 deals with a particularly simple atomic model, the two-level atom subject to a resonant or nearly resonant classi- cal ?eld. we ?rst discuss the nature of the electric-dipole interaction and
then use the fermi golden rule to derive einstein’s a and b coe?cients for
spontaneous and stimulated emission. we then relax the assumption that the
interaction is weak and derive the famous rabi solution.
in chaps. 13–19, we discuss interactions for which the electromagnetic ?eld as well as the atoms must be quantized. in particular, chap. 13 shows


that electromagnetic ?eld modes are described mathematically by simple har- monic oscillators. in addition, these oscillators can model the polarization of certain kinds of media, such as simple molecular systems. in preparation for such problems, sect. 3.4 quantizes the simple harmonic oscillator. the section writes the appropriate hamiltonian in terms of the annihilation and creation operators, and derives the corresponding energy eigenstates.
this chapter is concerned with the quantum mechanics of single systems in pure states. discussions of mixtures of systems including the decay phe- nomena and excitation mechanisms that occur in lasers and their applications are postponed to chap. 4 on the density matrix.


3.1 review of quantum mechanics


according to the postulates of quantum mechanics, the best possible knowl- edge about a quantum mechanical system is given by its wave function ?(r, t). although ?(r, t) itself has no direct physical meaning, it allows us to calcu- late the expectation values of all observables of interest. this is due to the
fact that the quantity
?(r, t)??(r, t) d3r

is the probability of ?nding the system in the volume element d3r. since the system described by ?(r, t) is assumed to exist, its probability of being
somewhere has to equal 1. this gives the normalization condition
¸
?(r, t)??(r, t) d3r = 1 , (3.1)

where the integration is taken over all space.
an observable is represented by a hermitian operator o(r) and its ex- pectation value is given in terms of ?(r, t) by
¸

(o)(t) =

d3r??(r, t) o(r)?(r, t) . (3.2)



experimentally this expectation value is given by the average value of the results of many measurements of the observable o acting on identically pre- pared systems. the accuracy of the experimental value for (o) typically de-
pends on the number of measurements performed. hence enough measure- ments should be made so that the value obtained for (o) doesn’t change
signi?cantly when still more measurements are performed. it is crucial to
note that the expectation value (3.2) predicts the average from many mea- surements in general it is unable to predict the outcome of a single event with absolute certainty. this does not mean that quantum mechanics in other ways is unable to make some predictions about single events.


the reason observables, such as position, momentum, energy, and dipole moment, are represented by hermitian operators is that the expectation val-
ues (3.2) must be real. denoting by (?, ?) the inner or scalar product of two vectors ? and ?, we say that a linear operator o is hermitian if the equality
(?, o?) = (o?, ?) . (3.3)
holds for all ? and ?. in this notation, (3.2) reads (o) = (?, o?).
an important observable in the interaction of radiation with bound elec- trons is the electric dipole er. this operator provides the bridge between
the quantum mechanical description of a system and the polarization of the
medium p used as a source in maxwell’s equations for the electromagnetic ?eld. according to (3.2) the expectation value of er is
¸

(er)(t) =

d3 r r e| ? (r, t) |2 , (3.4)


where we can move er to the left of ?(r, t)? since the two commute (an opera- tor like ?cannot be moved). here we see that the dipole-moment expectation value has the same form as the classical value if we identify ? = e|?(r, t)|2 as
the charge density.
in nonrelativistic quantum mechanics, the evolution of ?(r, t) is governed
by the schro¨dinger equation.
?
ik ?t ?(r, t) = h?(r, t) , (3.5)
where h is the hamiltonian for the system and k = 1.054 × 10?34 joule- seconds is planck’s constant divided by 2?. the hamiltonian of an unper-
turbed system, for instance an atom not interacting with light, is the sum of
its kinetic and potential energies


2 2
= ? ?
2m


+ v (r) , (3.6)

where m is its mass and v (r) the potential energy. as we shall see shortly,
in the coordinate representation we are considering here the momentum op-
erator pˆ is expressed in terms of the system’s position operator r as
pˆ = ?ik? , (3.7)

so that we recognize that the ?rst term of the hamiltonian (3.6) is nothing but the kinetic energy of the system. note also the important relationship
[xˆi, pˆj ] = ik?ij (3.8)

where xˆi and pˆj are cartesian coordinates of the operators xˆ

and pˆ, and

[aˆ, ˆb] ? aˆˆb ? aˆˆb is the commutator between the operators aˆ and ˆb. observ- ables which satisfy this commutation relation are generally called conjugate variables.


in view of (3.7), we see that the schro¨dinger equation (3.5) is a partial di?erential equation. the time and space dependencies in (3.5) separate for functions having the form
?n(r, t) = un(r)e?i?n t (3.9) for which the un(r) satisfy the energy eigenvalue equation
hun(r) ? k?nun(r) . (3.10) the eigenfunctions un(r) can be shown to be orthogonal, and we take them to be normalized according to (3.1), so that they are then orthonormal,

¸ u? 3

. 1 n = m

and complete

n(r)um(r)d r = ?n,m =

(3.11)
0 n ƒ= m

. u? r r
n(r)un(r ) = ?(r ? r ) , (3.12)
n
where ?n,m and ?(r ? rr) are the kronecker and dirac delta functions, re- spectively. the completeness relation (3.12) means that any function can be
written as a superposition of the un(r). problem 3.1 shows that this de?nition
is equivalent to saying that any wave function can be expanded in a complete
set of states.
in particular the wave function ?(r, t) itself can be written as the super- position of the ?n(r, t):
?(r, t) = . cn(t)un(r)e?i?n t . (3.13)
n
the expansion coe?cients cn(t) are actually independent of time for prob- lems described by a hamiltonian satisfying the eigenvalue equation (3.10). we have nevertheless included a time dependence in anticipation of adding an interaction energy to the hamiltonian. such a modi?ed hamiltonian wouldn’t
quite satisfy (3.10), thereby causing the cn(t) to change in time.
substituting (3.13) into the normalization condition (3.1) and using the
orthonormality condition (10), we ?nd
.
|cn|2 = 1 . (3.14)
n

the |cn|2 can be interpreted as the probability that the system is in the nth energy state. the cn are complex probability amplitudes and completely
determine the wave function. to ?nd the expectation value (3.2) in terms of the cn, we substitute (3.13) into (3.2). this gives
(o) = . cnc? omne?i?nm t , (3.15)
n,m


where the operator matrix elements omn are given by
¸

omn =

d3r u?

(r)oun(r) , (3.16)



and the frequency di?erences



?nm = ?n ? ?m . (3.17)


typically we consider the interaction of atoms with electromagnetic ?elds. to treat such interactions, we add the appropriate interaction energy to the hamiltonian, that is
h = h0 + v . (3.18)

if we expand the wave function in terms of the eigenfunctions of the “unper- turbed hamiltonian” h0, rather than those of the total hamiltonian h, the probability amplitudes cn(t) change in time. to ?nd out just how, we sub-
stitute the wave function (3.13) and hamiltonian (3.18) into schro¨dinger’s
equation (3.5) to ?nd
.(k?n + v)cnun(r)e?i?n t = .(k?ncn + ikc? n)un(r)e?i?n t . (3.19)
n n

cancelling the k?n terms, changing the summation index n to m, multiplying through by u? (r) exp(i?nt), and using the orthonormality property (3.11), we ?nd the equation of motion for the probability amplitude cn(t)
i

c? n(t) =
k

where the matrix element

(n|v|m)ei?nm tcm(t) , (3.20)
m


¸

(n|v|m) =

d3ru? (r)vum(r) . (3.21)


note that instead of the form (3.13), we can also expand the wave function
?(r, t) as
?(r, t) = . cn(t)un(r) , (3.22)
n
for which the k?n time dependence in (3.19) doesn’t cancel out. the cn(t) then obey the equation of motion

c? (t) = ?i? c (t) ? i .
k m


(n|v|m)cm(t) . (3.23)

in terms of the cn, the expectation value (3.2) becomes
(o) = . cnc? omn . (3.24)
n,m


equation (3.20) and equivalently (3.23) shows how the probability am- plitudes for the wave function written as a superposition of energy eigen- functions changes in time. they are equivalent to the original schro¨dinger equation (3.5), but are no longer concerned with the precise position depen- dence, which is already accounted for by the r-dependence of the eigenfunc-
tions un(r). in particular if we’re only concerned about how a system such
as an atom absorbs energy from a light ?eld, this development is completely described by the changes in the cn or cn.
the choice of using the relatively slowly varying cn versus using the rapidly varying cn is a matter of taste and convenience. the time dependence of the cn is due to the interaction energy v alone, while that of the cn is due to the total hamiltonian h. to distinguish between the two, we say that the cn are in the interaction picture, while the cn are in the schr¨odinger picture.
we discuss this more formally at the end of this section.
armed with (3.20) or (3.23), you can skip directly to sect. 3.2, which shows how systems evolve in time due to various interactions. before going ahead, we review the dirac notation and some other aspects of the wave
function and of its more abstract form, the state vector |?). this material is
needed for our discussions involving quantized ?elds in chaps. 13–19, and is
useful in proving various properties of the density operator in chap. 4.
up to now we have used the so-called coordinate representation, where all operators, as well as the wave function, are expressed as functions of the system’s position r. alternatively, one can work in a number of other representations, a rather common one being the momentum representation. here, operators and wave functions are expressed as functions of the system’s momentum p. as we shall see shortly, one can transform the system’s wave function from the coordinate to the momentum representation by a simple
fourier transformation of ?(r, t),


?(p, t) =

1 ¸
(2?k)3/2


d3r ?(r, t)e?ip•r/k . (3.25)

here ?(p, t) describes the same dynamical state as ?(r, t). it doesn’t make any di?erence in principle which representation we choose to use, and as we
see with the c? n, we sometimes don’t have to worry about the coordinate
representation at all.
we now turn to a discussion of a general formalism which does away with the explicit use of representations, and allows us to switch from one to another representation, when it is desirable.


dirac notation

the formalism that permits one to achieve this goal is the dirac notation. roughly speaking dirac’s formulation is analogous to using vectors instead of coordinates. the notation has an additional advantage in that one can label











fig. 3.1. a two dimensional vector written in ordinary vector notation and in dirac notation


the basis vectors much more conveniently than with ordinary vector notation. we start our discussion with a comparison between ordinary notation for a vector in a two-dimensional space and dirac’s version. as shown in fig. 3.1, a vector v can be expanded as
v = vxxˆ + vy yˆ , (3.26)
where xˆ and yˆ are unit vectors along the x and y axes, respectively. in dirac notation, this reads
|v) = vx|x) + vy |y) , (3.27)
the component vx given in ordinary vector notation by the dot product xˆ • v
is given in dirac notation by

vx = (x|v) . (3.28)

the dirac vector |v) is called a “ket” and the vector (v| a “bra”, which come from calling the inner product (3.28) a “bra c ket”. with the notation (3.28),
(3.27) reads

|v) = |x)(x|v) + |y)(y|v) . (3.29) this immediately gives the identity diadic (outer product of two vectors)
|x)(x| + |y)(y| = i . (3.30)

equations (3.27, 3.30) can be immediately generalized to many dimensions as in




n
where the {|n)} are a complete orthonormal set of vectors, i.e., a basis. the inner products (n|v) are the expansion coe?cients of the vector |v) in this basis. the bra (n| is the adjoint of the ket |n) and the expansion coe?cients
have the property


(k|v) = (v|k)? . (3.33)
unlike the real spaces of usual geometry, quantum mechanics works in a complex vector space called a hilbert space, where the expansion coe?cients are in general complex.
the basis {|n)} is discrete. alternatively, we can expand vectors in terms of the coordinate basis {|r)} which like the {|n)} forms a complete basis,
albeit a continuous one. for such a situation we need to use continuous sum-
mations in (3.31, 3.32), that is, integrals. for example, the identity operator of (3.32) can be expanded as
¸
i = d3r |r)(r| . (3.34)


one major advantage of the bra and ket notation is that you can label the vectors with as many letters as desired. for example, you could write |r??) in place of |r).
the vector of primary interest in quantum mechanics is the state vector
|?(t)). the wave function is actually the expansion coe?cient of |?) in the
coordinate basis




where the wave function

¸
|?) =

d3r|r)(r|?) , (3.35)

?(r, t) = (r|?) . (3.36)
hence the state vector |?) is equivalent to the wave function ?(r, t), but doesn’t explicitly display the coordinate dependence.
instead of using the position expansion of (3.35), we can expand the state vector in the discrete basis {|n)} as
|?) = . cn|n) . (3.37)
n

the most common basis to use consists of the eigenstates of the unperturbed hamiltonian operator h0. for this basis, the expansion coe?cients cn are just
those in (3.22), and the energy eigenfunctions are related to the eigenvectors
by
un(r) = (r|n) . (3.38)
a useful trick in transforming from one basis to another is to think of the vertical bar as an identity operator expanded either as in (3.32) or as in (3.34). using the form of (3.34) in (3.37) on both sides of the equation along with (3.38), we recover (3.22).
the expectation value of the operator o is given in terms of the state
vector by
(o)(t) = (?(t)|o|?(t)) . (3.39)


to see that this is the same as the position representation version of (3.2) is a little trickier. we substitute the identity expansion (3.34) in for the left bar
and the same expansion with r ? rr in for the right bar obtaining

¸
(o) =
¸
=

¸
d3r
¸
d3r


d3rr(?|r)(r|o|rr)(rr|?)

d3rr??(r, t)o(r, rr)?(rr, t) . (3.40)


hermitian operators in quantum mechanics often turn out to be local, and hence
o(r, rr) = o(r, r)?(r ? rr) . (3.41)
this fact reduces (3.40) to (3.2) as desired. more generally we can interpret
o?(r, t) on the right-hand side of (3.2) as
¸

o?(r, t) =

d3rro(r, rr)?(rr, t) .


similarly substituting the expansion (3.37) into (3.39), we ?nd the ex- pectation value (3.24), where the operator matrix elements written earlier as (3.16) can be written simply as
omn = (m|o|n) . (3.42)

problem 3.2 shows that this equals (3.16). it is also useful to express the operator o directly in terms of the basis set. this then reads as
o = . onm|n)(m| . (3.43)
n,m

finally, we note that the state vector version of the schro¨dinger equation (3.5) is




where

?
ik ?t |?) = h|?) , (3.44)

pˆ2
h = 2m + v (ˆr) . (3.45)

this reduces to the schr¨odinger equation (3.5) in the coordinate representa- tion.


coordinate and momentum representations

equation (3.36) introduced the coordinate representation wave function
?(r, t) = (r|?). one can in a similar fashion introduce the momentum repre-
sentation wave function
?(p, t) = (p|?) . (3.46)


in order to discuss the connection between the coordinate and momentum representations, we restrict ourselves to a one-dimensional situation where
pˆ ? pˆ and xˆ ? xˆ, with [xˆ, pˆ] = ik. we proceed by introducing the translation, or shift operator
s(?) = exp(?i?pˆ/k) . (3.47)
s(?) is unitary, with s†(?) = s?1(?) = s(??), and it has a number of interesting properties. for instance, using the commutation relations
[xˆ, f (pˆ)] = ikf r(pˆ) , (3.48)
[pˆ, g(xˆ)] = ?ikgr(xˆ) , (3.49)

which are proven in prob. 3.19, one ?nds readily that
[xˆ, s(?)] = ?s(?) , (3.50)

so that xˆs(?) = s(?)[xˆ + ?]. since the ket |x) is an eigenstate of the operator xˆ with xˆ|x) = x|x), we have therefore that xˆs(?)|x) = s(?)(x + ?)|x) = (x + ?)s(?)|x). in other words, the state s(?)|x) is also an eigenstate of xˆ, but with eigenvalue (x + ?): the action of the operator s(?) on the ket |x) is to transform it into a new eigenvector of xˆ with eigenvalue shifted by the arbitrary amount ?,
s(?)|x) = |x + ?) . (3.51)

this also proves that the spectrum of xˆ is continuous. finally, we note that the coordinate representation wave function corresponding to the ket s(?)|?)
is
(x|s(?)|?) = (x ? ?|?) = ?(x ? ?) . (3.52)

the translation operator permits one to easily determine the action of the momentum operator pˆ in the coordinate representation. considering a small shift ? such that s(??) = exp(i?pˆ/k) c 1 + i ? pˆ + o(?2), we have





so that

?
(x|s(??)|?) = ?(x) + i


(x|pˆ|?) + o(?2) = ?(x + ?) , (3.53)

(x|pˆ|?) =

k lim i ??0

?(x+ ?) ? ?(x)
?

= k d
i dx

?(x) . (3.54)

the action of pˆ in the coordinate representation is therefore

d
pˆ ? ?ik dx . (3.55)

a similar derivation shows that in the momentum representation, the action of xˆ is
d
xˆ ? ik dp . (3.56)


armed with this knowledge, it is quite easy to obtain the form of the schr¨odinger equation in the momentum representation from its coordinate
representation of (3.6). this requires evaluating (p|v (xˆ)|?(t)). introducing the identity ¸ dp |p)(p| = 1, we have

¸
(p|v (xˆ)|?(t)) =

¸
dpr(p|v (xˆ)|pr)(pr|?(t)) =


dprv (p ? pr)?(pr, t) , (3.57)


where v (p ? pr) are of course the matrix elements of the potential v in
the momentum representation. they can be obtained from the coordinate
representation by noting that

¸
(p|v (x)|pr) =


dx(p|x)(x|v (xˆ)|x)(x|pr) . (3.58)


since (x|pr) may be interpreted as the coordinate representation wave func- tion ?(x) associated with the state vector |pr) we have from (3.54)
d?(x)



so that

(x|pˆ|p) = ?ik

dx ,

?(x) =

. ? ik . d?(x)

p dx
or
1
(x|p) = ? exp(ipx/k) , (3.59) 2?k
where we have used the plane wave normalization appropriate for the one- dimensional situation at hand. introducing this result into (3.58) yields

1 ¸
v (p) = ?
2?k


dx e


?ipx/k


v (x) , (3.60)

that is, v (p) is the fourier transform of v (x). with this result, the momen- tum representation of the schro¨dinger equation is therefore

ik d?(p) =
dt

. p2 .
2m

1 ¸
?(p) +
(2?k)n/2


dnp v (p ? p )?(p ) (3.61)

r r r
here, we have extended the one-dimensional result to n dimensions in a straightforward way. note that in contrast to the coordinate representation, where the potential energy term is usually local and the kinetic energy term is not, the kinetic energy term is now local, but the potential energy term is not.


schr¨odinger, heisenberg and interaction pictures

the problem we are normally interested in quantum optics is to determine the expectation values (3.39) of observables at the time t. to do this we typ-
ically start with a system in a well de?ned state at an earlier time and follow the development up to the time t using the schro¨dinger equation (3.44). it is
possible to follow this evolution in three general ways and in many combina-
tions thereof. the one we use primarily in the ?rst part of this book is called the schro¨dinger picture and puts all of the time dependence in the state vec- tor. the interaction picture puts only the interaction-energy time dependence into the state vector, putting the unperturbed energy dependence into the operators. the heisenberg picture puts all of the time dependence into the operators, leaving the state vector stationary in time. in the remainder of this section, we review the way in which these three pictures are tied together.
the schr¨odinger equation (3.44) can be formally integrated to give
|?(t)) = u (t)|?(0)) , (3.62)
where the evolution operator u (t) for a time-independent hamiltonian is given by
u (t) = exp(?iht/k) . (3.63)
substituting (3.62) into the expectation value (3.39) we obtain
(o) = (?(0)|u †(t)o(0)u (t)|?(0)) . (3.64)

we can also ?nd this same value if we can determine the time dependent operator.
o(t) = u †(t)o(0)u (t) . (3.65)
as heisenberg ?rst showed, it is possible to follow the time evolution of the quantum mechanical operators. in fact we can obtain their equations of motion by di?erentiating (3.65). this gives


d du †

?o du



i.e.,

dt o(t) =

dt ou + u † ?t u + u †o dt ,

d i

?o

dt o(t) = k [h, o] + u

u , (3.66)
?t

where

[h, o] ? ho ? oh (3.67)

is the commutator of h with o. in deriving (3.66), we have used the fact that h commutes with u , which follows from the fact that u is a function of h only. the ?o/?t accounts for any explicit time dependence of the schro¨dinger operator o.


in general when the system evolution is determined by integrating equa- tions of motion for the observable operators, we say the heisenberg picture is being used. when the evolution is determined by integrating the schro¨dinger equation, we say that the schro¨dinger picture is being used. in either case, (3.64) shows that we get the same answers. you ask, why use one picture instead of the other? the answer is simply, use the picture that makes your life easier. typically the insights obtained with one di?er somewhat from the other, but you get the same answer with either. traditionally the schro¨dinger picture is the ?rst one taught to students and many people feel more com- fortable with it. much of this book is carried out in the schro¨dinger picture.
on the other hand, the heisenberg picture is a “natural” picture in the sense that the observables (electric ?elds, dipole moment, etc.) are time-
dependent, exactly as in classical physics. as a result, their equations of motion usually have the same form as in the classical case, although they are operator equations, which modi?es the way one can integrate and use them.
another aspect is that in the schro¨dinger picture, one has to ?nd |?(t))(or its generalization the density operator ?) before computing the desired expec- tation values. since |?(t)) contains all possible knowledge about the system,
you have to solve the complete problem, which may be more than you need.
in many cases, you only want to know one or a few observables of the system. the heisenberg picture allows you to concentrate on precisely those observ- ables, and with some luck, you may not have to solve the whole problem to get the desired answers.
in discussing (3.13, 3.22), we hinted at another way of following the time
dependence, namely, we put only the time dependence due to the interaction energy into the cn(t), while the time dependence of the total hamiltonian is contained in the cn(t). the state vector of (3.37) is the schr¨odinger-picture
state vector, while the state vector
|?i (t)) = . cn(t)|n) (3.68)
n

is said to be the interaction-picture state vector. the thought behind using the interaction picture is to take advantage of the fact that we often face situations where we already know the solutions of the problem in the absence of the interaction.
more formally, to eliminate the known part of the problem, we substitute the state vector



where

|?s (t)) = u0(t)|?i (t)) , (3.69)

u0(t) = exp(?ih0t/k) (3.70)

into the schro¨dinger equation (3.44). we include the subscript s in (3.69) to remind ourselves that |?s(t)) is the schr¨odinger-picture state vector. we
?nd

d i
dt |?i (t)) = ? k vi (t)|?i (t)) , (3.71)
where we have de?ned the interaction-picture interaction energy
vi (t) = u †(t)vs u0(t) (3.72)
and put a subscript s on the rhs to remind ourselves that vs is in the schr¨odinger picture. from (3.69, 3.39), we also immediately ?nd that the
ex-ceptation value of an operator o in the interaction picture is given by
(o(t)) = (?i (t)|oi (t)|?i (t)) , (3.73)


where


oi (t) = u †(t)os u (t)0 . (3.74)

note that since we know the solution of the unperturbed problem, oi (t) is
already known. comparing the equation of motion (3.71) achieved with the
original schro¨dinger equation (3.44), we see that we have achieved our goal, namely, that we have eliminated the part of the problem whose solution we al- ready knew. we see in chap. 4 that the interaction picture (or more precisely, an interaction picture) is particularly helpful in visualizing the response of a two-level atom to light.


3.2 time-dependent perturbation theory

to predict expectation values of operators, we need to know what the wave function is. typically, we know the initial value of the wave function, which then evolves in time according to the schro¨dinger equation, or equivalently, the operators of interest evolve according to the heisenberg equations. for some problems, these equations can be integrated exactly, giving us the values
needed to compute the expectations values and the desired time t. more gen-
erally the equations can be integrated approximately using a method called
time-dependent perturbation theory. the name comes from the introduction of a perturbation energy v as given in (3.18), which describes the interaction
of the quantum system under consideration with some other system. an atom
interacting with an electromagnetic ?eld is the combination that we consider most often in this book. the perturbation energy forces the probability am- plitudes in (3.13) or (3.22) to be time dependent. the method of time de- pendent perturbation theory consists of formally integrating the schro¨dinger equation, converting it into an integral equation, and then solving the in-
tegral equation iteratively. one way to proceed is by writing v ? ?v and
|?(t))? |?(0)(t)) + ?|?(1)(t)) + ?2|?(2)(t)) + .. .. the zeroth-order solution,
the solution in absence of perturbation, is then obtained by equating terms proportional to order ?0 on both sides of schr¨odinger’s equation. first-order perturbation theory is obtained by equating terms proportional to ?, second- order theory to terms proportional to ?2, and so on. one can then set ? = 1



at the end of the calculation. for example on the rhs of the equations of motion (3.20) for the cn, we insert the initial values of the cn and inte-
grate, obtaining better values on the lhs. this ?rst integration gives the “?rst-order” corrections to the cn. that may be accurate enough for your
purposes, and it is used in the famous fermi golden rule. if it is not accurate
enough, you substitute the improved values in on the rhs and integrate to obtain a second-order correction. one can iterate this procedure to succes- sively higher orders of perturbation. this section carries out this procedure to ?rst order in the perturbation energy, i.e., one time integration. the answer is illustrated and then used to derive the fermi golden rule. the section concludes with a general formulation of higher-order perturbation theory.
an important question is, given a quantum system initially in the state
|i), what are the probabilities that transitions occur to other states? this
question asks, for example, what the probability is that an initially unexcited
atom interacting with an electromagnetic ?eld absorbs energy from the ?eld.
the wave function (3.13) has the initial value
?(r, 0) = ui(r) ,

that is,


ci(0) = 1, cnƒ=i(0) = 0 . (3.75)

to ?nd out the ?rst-order correction to the cn(t), we use the initial values (3.75) on the rhs of the schro¨dinger equations of motion (3.20) for the cn(t).
this gives
c? n(t) c c? (1)(t) = ?ik?1(n|v|i)ei?ni t , (3.76)
where c(1) is a special case of c(k) , which means we have iterated (3.20) k
n n
times.
equation (3.76) is easy to integrate for two important kinds of perturba- tion energies: one time independent, and one sinusoidal such that
v = v0 cos ?t . (3.77)
integrating (3.76) from 0 to t for a time independent v(? = 0), we have
ei?ni t ? 1

cn(t) c c(1)(t) = ?ik?1vni

i?ni


sin (?ni t/2)

= ?ik?1vniei?ni t/2

,
?ni/2

where we write vni for (n|v0|i). the probability that a transition occurs to level n is given by

(1) 2
| n |

= |vni

|2 sin2(?ni

t/2)

. (3.78)

k2 (?ni/2)2
it’s interesting to note that we have already seen this kind of result in the phase matching discussion of sect. 2.2, for which electromagnetic ?eld am- plitudes are used instead of the probability amplitudes used here. problem



3.3 discusses the analogy between the two problems. the value of (3.78) is accurate so long as ci(t) doesn’t change appreciably from the initial value ci(0) = 1. in view of the normalization condition (3.14), this means that the
total transition probability
pt = 1 ? |c(1)|2 = . |c(1)|2 (3.79)

i


must be much less than unity.

n nƒ=i

figure 3.2 plots the probability in (3.78) at the time t as a function of the frequency di?erence ?ni. for short enough times we can expand the sine
in (3.78) to ?nd

|c(1) 2

2
|vni| 2

n | c

k2 t

(3.80)

which shows that the center of the curve increases proportionally to t2. we further see that for increasing frequency di?erences |?ni|, the probability that the interaction induces a transition to level n becomes smaller rapidly. thus
transitions are much more likely if the energy is conserved between initial
and ?nal states.
consider now the sinusoidal interaction energy (3.77), which can be used to model an atom interacting with a monochromatic electromagnetic ?eld. for such a ?eld, (3.77) is proportional to the electric ?eld amplitude, as we see in sect. 3.3. integrating (3.76) accordingly, we have

. ei(?ni +? )t 1
cn(t) c c(1)(t) = ?i

ei(?ni +?)t 1 .
+


. (3.81)

n 2k

i(?ni + ?)

i(?ni ? ?)


for the sake of de?niteness, consider the case ?ni > 0. then the denomina- tor ?ni + ? is always positive and larger than ?ni. this is not true for the


.5




.25




0
0 2? 3? 4?
(? ? ?) t

fig. 3.2. probability |cn (t)|2 of (3.78) versus (? ? ?)t/2


denominator ?ni ? ?, which vanishes if the resonance condition
? c ?ni (3.82)

is satis?ed. for interactions near resonance, the term with the relatively small denominator ?ni ? ? is much larger than that with the ?ni + ?, allowing us to
neglect the latter. for the same reason, we can probably neglect transitions to levels with energies very di?erent from k?. this observation is used to
justify the two-level atom approximation discussed in sect. 3.3. neglecting the term with the relatively large denominator ?ni + ? is called the rotating- wave approximation. it is used in much of the book.
making the rotating-wave approximation in (3.81), we ?nd the transition
probability


(1) 2
| n |

= |vni

|2 sin2[(?ni

? ?)t/2]

. (3.83)

4k2

(?ni ? ?)2/4


this result is formally the same as the dc case of (3.78), provided we substi- tute ?ni ? ? for ?ni. thus fig. 3.2 and the corresponding discussion apply to
this case as well. in particular, we see that in the course of time, transitions
are unlikely to occur unless the resonance condition (3.82) is satis?ed, that is, unless the applied ?eld frequency matches the transition frequency.
so far we have in?nitely sharp energy levels. this is not realistic, since levels can be broadened by e?ects like spontaneous emission and collisions. furthermore, there may be a continuum of levels such as in the energy bands in solid-state media. for these situations, the summation in the total transi- tion probability (3.79) can be replaced by an integral with a density of state
factor d(?) to weight the distribution correctly. for example, there are typi-
cally more states per frequency interval for higher frequencies then for lower frequencies. the total transition probability pt then has the value
¸

pt c

d(?)|c(1)(?)|2d?,


where the discrete frequency ?ni is replaced by the continuous frequency ?.
substituting (3.83), we ?nd


¸
pt =

d? d(?) |v

(?)|2

sin2[(? ? ?)t/2]


. (3.84)

4k2

[(? ? ?)t/2]2


it is interesting to evaluate the total transition probability integral (3.84) for two reasons. first, so long as it is small enough, we know that the ?rst-
order perturbation theory answer is valid. secondly, dpt /dt gives the rate at
which transitions occur. the equation for this rate is called the fermi golden
rule, and can be used to ?nd a variety of rates, such as those occurring in the photoelectric e?ect, spontaneous emission, and the planck radiation law.
equation (3.84) is a special case of the general integral

¸
j = d?f (?)g(?) . (3.85)

there are problems for which both the density of states factor d(?) and the matrix elements vni are known. but even in such situations the resulting in- tegral for (3.84) is typically hard to solve. however, we can approximate j if either f (?) or g(?) varies little over the frequency range for which the other
has an appreciable value. the extreme example is when one of the functions, say g(?) is the delta function ?(? ? ?0). then j = f (?0). more generally, suppose g(?) is sharply peaked about ?0 and that f (?0) varies little in this interval. then j c f (?0) ¸ g(?). for the purposes of this problem, g(?) is
a delta function, and it is in this way that delta functions approximate nat-
ural behavior. this is an example of what one sometimes calls an “adiabatic elimination”. this kind of elimination is equally important in the solution of coupled di?erential equations, for which one function varies slowly compared to another. typically we consider problems in which atoms coupled to an electromagnetic ?eld vary rapidly compared to the ?eld envelope. in such cases, the technique of adiabatic elimination allows us to solve the atomic equations of motion assuming that the ?eld envelope is constant, and then to substitute the resulting steady-state polarization of the medium into the correspondingly simpli?ed slowly-varying ?eld equations of motion (chap. 5).
2

the integral (3.84) has the form of (3.85) with f (?) = d(?)|v(?)|

and

g(?) = sin2[(???)t/2]/[(w??)t/2]2. hence we can solve (3.84) when either f or g varies rapidly compared to the other. in particular, for times su?ciently small that all relevant values of |? ? ?|t are much less than unity, the g = sin x/x function in (3.84) can be approximated by unity. by relevant values of |? ? ?|t, we mean those for which the density of states factor d(?) and the matrix element v(?) have appreciable values. this then gives
¸ d(?)|v(?)|2

pt c t2


4k2

d?. (3.86)

hence for such small times, the transition rate dpt /dt is proportional to time, starting up from zero. unless the density of states and the interaction energy matrix element have in?nitely wide frequency response, i.e., in?nite bandwidth, this limit implies a build-up time in the response of the system
to the applied perturbation v(?). (this means, for example, that detectors
have a nonzero response time).
after this initial small time region, we suppose that the factor d(?)|vni|2 varies little in the frequency interval for which the sin2 x/x2 function in (3.84) has appreciable values (see fig. 3.3). in this limit, we can evaluate d(?)|vni|2 at the peak ? = ? of the sin2 x/x2 function, ?nding

pt = d(?) |v

(?) 2 ¸ ?
t


sin2x dx

= ?

2 2 x2
??
2

2k2 d(?)|v(?)| t. (3.87)


1.00


0.75



sin2x x2


0.50



0.25



0.00
?2? ?? 0
x



? 2?

fig. 3.3. sin2 x/x2 term in the total transition probability integral (3.87) versus
x = (? ??)t/2. as time increases, this function peaks up like the ?-function ?(? ??)


here, we have extended the limits of the integral to ±? since in the present approximation this adds little to the integral and yields an analytic answer. equation (3.87) gives then the fermi golden rule rate


dpt

d (1) 2 ? 2

? = = c
dt dt

= d(?)|v(?)|

, (3.88)

which is a constant in time.
this constant rate proportional to the intensity of the incident radiation is what people typically observe in the photoelectric e?ect. note that the
rate vanishes if no transitions exist for the frequency ?. the photoelectric
e?ect occurs in media that have an energy gap above the ground state. to be absorbed, the applied photon energy k? must be larger than this gap.
summarizing the rate at which transitions occurs, we see that for times short compared to the reciprocal of the width of the function |v0|2d, the rate
increases linearly in time. for longer times the rate becomes constant. for still longer times, when pt does not remain small, we cannot assume that
the probability of the initial state is unity. section 14.3 shows that to a good
approximation, we can ?x up the rate by multiplying it by the initial state probability |ci(t)|2. this generalizes the fermi golden rule to


d 2
dt |ci|


= ?? |ci|2


, (3.89)

which states that the probability for being in the initial level decays exponen- tially in time. such a formula doesn’t make a ?rst-order perturbation theory


approximation. this kind of time response is typical of certain important pro- cesses in quantum optics, such as spontaneous emission, which is described in chap. 14.
note that we get the same kind of integral (3.84) for two sharp levels
interacting with a non-monochromatic ?eld with a spectral intensity distri- bution proportional to v(?). this yields a transition rate ? given by (3.88) with ? replaced by ?, since it is the atomic frequency ?, rather than the ?eld frequency ?, that determines the center of the sin2 x/x2 distribution. this
observation is important in the next section, where we derive the planck
black-body radiation formula.


higher-order perturbation theory

the iterative approach outlined at the start of this section can be written in an analytic form by using the formal solution of (3.62) in the interaction picture as
|?i (t)) = ui (t)|?i (0)) . (3.90)
here, we use the subscript i to remind ourselves that we are working in the interaction picture. taking the time rate of change of (3.90) and using the schr¨odinger equation (3.71), we obtain
dui (t)
ik = vi (t)ui (t) . (3.91)

remembering that ui (0) = 1, we integrate this equation formally to get
i ¸ t

ui (t) = 1 ? k

dt vi (t )ui (t ) . (3.92)

r r r
0

we can solve this equation by successive iterations, obtaining
i ¸ t

ui (t) = 1 ? k

dt1 vi (t1)
0

. i .2 ¸ t
+

dt1 vi (t1)

¸ t1

dt2 vi (t2) + ... . (3.93)

k 0 0
truncating this expression after the lowest order term in vi gives ?rst-order perturbation theory. keeping higher-order terms gives second-order, third- order, etc., perturbation theory. note that this iteration process implies a
time ordering such that t2 ? t1 ? t.
by way of illustration, we calculate the ?rst-order answer this way as

.
|?i (t))c

i ¸ t
1 ?
0

.
dt1 vi (t1)


|?i (0)) . (3.94)

substituting this into the equation for the transition probability to level m






we have

|cm(t)|2 = |(m|?i (t))|2 = |(m|?s (t))|2 , (3.95)

¸

|cm(t)|2 c |(m|i)|2 + k?2|(m|

dt1vi (t1)|i)|2 .


converting vi back to the schro¨dinger value using (3.72), we have

2
1 .¸ t r

|cm(t)|2 c |c(1)(t)|2 =

.
k2 . 0

dt ei?mi t (m|vs |i).
.

. (3.96)


this gives (3.83) as before. by including more terms in (3.94), we can calcu- late successively higher-order contributions.


3.3 atom-field interaction for two-level atoms

this section introduces the two-level atom, a concept we write a great deal about in this book. such later consideration merits a careful introduction. consider ?rst the simplest of all atoms, hydrogen. this atom has an in?nite number of bound levels, characterized by the energies
e r ?
en = ? 2a n2 = ? n2 , (3.97)
where n = 1, 2, 3,... , a0 is the bohr radius (a0 = 0.53 ?a), and r? = 13.6 ev is rydberg’s constant. a few of these energy levels are shown in fig. 3.4.


0.00


?0.25

?0.50
en
?0.75

?1.00


0 2 4 6 8
r

fig. 3.4. energy levels of the hydrogen atom units of r?


unlike the quantum simple harmonic oscillator of sect. 3.4, the energy levels of hydrogen and of atoms in general are not equally spaced. for example,

3 5
e2 ? e1 = 4 r? , e3 ? e2 = 36 r? .
in quantum optics and in laser spectroscopy, we often shine monochro- matic laser light of such an atom and study what happens. if the laser fre- quency almost matches a particular transition frequency, then the transition probability predicted by (3.83) for this transition is much larger than that for other transitions. the approximation is almost the same as that used in making the rotating-wave approximation: in both cases, one neglects terms with denominators large compared to the term with the resonant denomina-
tor. a particular frequency di?erence ?ni ? ? is much smaller, say 2?×108 radians/sec, than the sum ?ni + ?(c 2?×1014 radians/sec) for the antiro- tating wave or the di?erence ?mi ? ?, m ƒ= n, which might be 0.6?×1014
radians/sec for some nonresonant transition. if this is the case, the problem
reduces to two levels. since the antirotating-wave contribution is actually smaller than many nonresonant contributions, it follows that the two-level atom approximation is usually only consistent if made simultaneously with the rotating-wave approximation. if one decides to keep the antirotating wave contribution, one must also keep all the nonresonant contributions as well. this is not as hard as it might seem, since nonresonant contributions can be usually treated using ?rst-order perturbation theory. also, we can account to some degree for transitions to levels other than the principal two by including various decay and pump rates.
a famous two-level system is the spin 1/2 magnetic dipole in nuclear
magnetic resonance. this is a true two-level system with relatively simple decay mechanisms. it has a lot in common with its brother the two-level atom, but its response can di?er signi?cantly in cases where level decay rates play an important role.
in our treatment of atoms using the two-level approximation, we ignore the fact that levels usually have a number of sublevels that all can contribute
to a resonant transition. this produces complications when experiments with real atoms are used to test theories based on the two-level approximation. in such cases, optical pumping techniques can sometimes be used to produce a true two-level atom.
we emphasize the two-level atom because we can often describe its inter-
action with the electromagnetic ?eld in detail and obtain analytic solutions. it thus allows us to learn a great deal about the atom-?eld interaction, and hopefully this knowledge can be generalized to more realistic situations. note
that although the two-level atom includes the low-order ?(3) type of nonlin-
earity of sect. 2.3 as a special case, in general it provides for more complicated
nonlinear responses, such as saturation.
we label the upper level of our two-level atom by the letter a, and the lower by b as shown in fig. 3.5. the corresponding wave function is














fig. 3.5. energy level diagram of two-level atom

?(r, t) = ca(t)e?i?a tua(r) + cb(t)e?i?b tub(r) . (3.98) before we see how this wave function evolves under the in?uence of an
applied electromagnetic ?eld, let us consider the kind of charge distribution it represents. to be speci?c, suppose the lower level is the 1s ground state of
hydrogen with the energy eigenfunction
ub(r) = u100(r, ?, ?) = (?a3)?1/2e?r/a0 , (3.99) and the upper level is the 2p state with the eigenfunction
ua(r) = u210(r, ?, ?) = (32?a3)?1/2(r/a0) cos ? e?r/2a0 . (3.100) these eigenfunctions are plotted versus the z coordinate in fig. 3.6, followed
by the superposition ?(r, t) of (3.98) for two times separated by ?/?. for one
of these times, the two probability amplitudes in (3.98) add. half a period
2

later, they subtract. figure 3.6c shows the probability density |?(z, t)|

for

these two points in time. we see that this probability density, and hence the “charge density” e|?(z, t)|2, oscillates back and forth across the nucleus in
a fashion analogous to the charge on the spring in sect. 1.3. this similarity
is the underlying reason why the classical model of chap. 2 is so successful in describing the linear absorption of light by a collection of atoms. chapter 5 derives this response quantum mechanically in detail, revealing where the classical model fails in laser physics.


electric dipole interaction

we mentioned in the discussion of the free-electron laser of chap.1 that the interaction between light and charged particles is described by invoking the principle of minimum coupling , which states that the canonical momentum
p of a particle of charge q is no longer its kinetic momentum mr? , as is the
case for a free particle, but rather
p = mr? + qa(r) , (3.101)


where a(r) is the vector potential and u (r) the scalar potential and
e = ?a u ,
?t
b = ?× a . (3.102)

as discussed at length in problems (3.20) – (3.212) the classical version of this hamiltonian guarantees that charged particles are subjected to the lorentz force, as should be the case, and also that the electromagnetic ?eld is governed
by maxwell equations.1 more formally, it follows from the requirement of local gauge invariance, which states that the physical predictions of our theory
must remain unchanged under the gauge transformation
?(r, t) ? ?(r, t)ei?(r,t) .

the interaction between a light ?eld a charge q bound to an atomic nucleus by a potential v (r) is then given in the non-relativistic limit by the hamiltonian


1 2
h = 2m [p ? qa(r, t)]


+ qu (r, t) + v (r) , (3.103)

where the vector potential a and the scalar potential u are evaluated at the location r of the charge. we recognize that the ?rst term in that hamiltonian
is just the kinetic energy of the charged particle.
we are free to choose to work in the so-called radiation gauge, where
u (r, t) = 0 (3.104)


and


?• a(r, t) = 0 , (3.105)

and we do so consistently in the remainder of this book. in addition, we exploit the fact that in most problems of interest in quantum optics, the wavelength of the optical ?eld is large compared to the size of an atom, and it is justi?ed to evaluate the vector potential at the location r of the nucleus rather than at the location r of the electron. this amounts to approximating that ?eld as constant over the dimensions of the atom, and is called the
electric dipole approximation, or dipole approximation in short.
with the coordinate representation form of the canonical momentum p =
?ik?, the schro¨dinger equation becomes then

??(r,t)

k2 . iq .2

ik =
?t 2m

?? k a(r, t)

+ v (r) . (3.106)

introducing the new wave function ?(r, t) via the gauge transformation

1 theoretically inclined students are strongly encouraged to work through these problems in detail.


?(r, t) = exp[(?iqr/k) • a(r, t)]?(r, t) , (3.107)
and remembering that [p, f (x)] = ?ikf r(x) and that in the coulomb gauge the electric ?eld and the potential vector are related by
?a(r, t)

e(r, t) = ?

(3.108)
?t

we ?nd that ?(r, t) obeys the schro¨dinger equation
?(r, t) = [h0 ? qr • e(r, t)]?(r, t) . (3.109)
where h0 = p2/2m + v (r) is the unperturbed hamiltonian of the electron. this shows that in the electric dipole approximation, the interaction between the electron and the electromagnetic ?eld is described by the interaction hamiltonian
v = ?qr • e(r, t) , (3.110)
where r is the position of the center of the mass of the atom.2
typically we are also interested in plane waves, for which we write simply
e(z, t), where z is the axis of propagation. the dipole traditionally is writ-
ten as the positive charge value times the distance vector pointing from the negative to the positive charge. this gives the same answer as er, which is
the negative charge value of the electron times the distance vector pointing
from the positive charge to the negative charge. there has been substantial discussion over the years since lamb (1952) ?rst brought it up concerning
the use of (3.110) versus a hamiltonian involving a • p, where a is the ?eld
vector potential and p is the electron momentum. for our purposes, (3.101)
combines intuitive appeal with excellent accuracy.
the matrix element of the dipole operator between a level and itself [recall (3.16)]

¸
er?? =

d3r er|u?(r)|2 , (3.111)


vanishes unless the system has as permanent dipole moment (like h2o), since
|u?(r)|2 is inevitably a symmetrical function of r and r itself is antisymmetric.
matrix elements of r between di?erent states can also vanish, but we are primarily interested in two levels a and b between which the matrix element
does not vanish. we can then write the electric-dipole interaction energy
matrix element
vab = ??e(r, t) , (3.112)

where ? (pronounced “squiggle” and also used for the weierstrass elliptic function) is the component of erab along e.

2 for an excellent discussion of the hamiltonian approach to electrodynamics, the electric dipole interaction, and the the a • p vs. e • r forms of the electric dipole interaction, see cohen-tannoudji, dupont-roc and grynberg (1989).






(a)








(b)

u100 (z)








u100 (z)

+ u210 (z)

z +




? u210 (z)

z ?

= ?(z, t1)

z = z




= ?(z, t2)

z = z




|?(z, t1)|2 |?(z, t2)|2


(c)




z z

fig. 3.6. (a) z dependence of ?(r, t1) = u100(r)+ u200(r) for the time t1 = 2n?/?.
(b) z dependence of ?(r, t2) = u100(r)?u210(r) for t2 = (2n+1)?/? = t1 +?/?. (c) corresponding dependencies of the probability densities |?(z, t1)|2 and |?(z, t2)|2 (with a slightly di?erent scale)


for the sake of simplicity, we ignore the spatial dependence altogether in the remainder of this section, and use
e(t) = e0 cos ?t . (3.113) this gives the interaction energy matrix element
vab = ??e0 cos ?t . (3.114)

substituting this into (3.83) with a as the ?nal state n and b as the initial state i, we have


|c(1) 2

2 sin2[(? ? ?)t/2]

? | = |?e0/2k|

(? ?

?)2/4

. (3.115)


this is the probability that the two-level atom absorbs energy under the in?uence of a driving ?eld, a phenomenon called (stimulated) absorption. al- ternatively, by identifying the initial state as a and the ?nal state as b, we describe a process called stimulated emission. it is easy to show that |c(1)|2 in this case is the same as |c(1)|2 in the case of absorption: the probabili-
ties for stimulated emission and absorption are equal. this is an example of
microscopic reversibility.


blackbody radiation

now consider the probability of a transition due to a ?eld that is not monochromatic, but rather has a continuous spectrum such as that for black-
body radiation. for this we replace the e2d(?) that occurs in using (3.115) by 2u ( )/?0, where u (?) is the energy density per radian/sec, and sum over all ?eld frequencies ?. the 2 here comes from the fact that two polarizations
are possible for each frequency. we then ?nd the total transition probability
¸ t2 sin2[(? ? ?)t/2]

pt =

d? u (?) k2

[(?

?)t/2]2 . (3.116)


this is the same kind of integral as that encountered for the fermi golden rule in (3.84), except that for this two-level atom, we integrate over the ?eld
continuum frequency ? instead of the level continuum frequency ?. here, we factor the slowly-varying energy density u (?) outside the integral, evaluating it at the peak, ? = ?, of the sin2 x/x2 curve. we ?nd the transition rate


? b(?)u (?) = 3k2?


?2u (?) , (3.117)


where the 3 comes from replacing ?2 by ?2/3 since the radiation can come from all directions: only 1/3 of the ?eld components e?ectively couple to the
dipole. note that pt of (3.116) has the same value if the atom is initially in
the upper state, rather than in the lower state as taken for (3.1115). hence
the stimulated emission rate equals the stimulated absorption rate of (3.117).
another case in which a two-level atom interacts with a radiation con- tinuum is spontaneous emission, which can be described as a combination of radiation reaction (see end of sect. 1.3) and stimulated emission by vacuum ?uctuations. this interpretation is clari?ed in chap. 14, which derives the upper-level decay formula given by (3.89). for now just think of the radiation ?eld as consisting of a continuum of modes, each of which acts like a quan- tized simple harmonic oscillator. section 3.4 shows that such an oscillator has a zero-point energy, which is associated with ?uctuations in the displacement variable. for the case of the electric ?eld, this displacement becomes the ?eld


amplitude, which then has ?uctuations. these ?uctuations are called vacuum ?uctuations, because they exist even in the vacuum, i.e., even when no clas- sical ?eld exists. the vacuum ?eld has a continuous spectrum. when this is used with the quantized ?eld version of (3.78) (see sect. 14.3), we ?nd the
spontaneous emission rate constant called a by einstein and ? in (3.89). to ?nd the value of a intuitively, we can use the rate given by (3.117) if we can guess what the energy density u (?) of the vacuum ?eld is. we note that the number of ?eld modes per unit volume between ? and ? + d? is ?2/?2c3 [see (14.46)]. multiplying this number by the energy k? of one photon, we have uspon(?) = k?3/?2c3. using this in (3.117) gives the spontaneous emission
rate

a = b k?

= ? ?

. (3.118)

?2c3

3??0kc3

the lifetime of the upper level is 1/a. this is the same result (14.60) as
derived in detail in sect. 14.3. this section also shows that spontaneous
absorption does not occur.
we can use these facts to derive informally the planck blackbody spec- trum

k?3/?2c3
u (?) = ek?/?

, (3.119)

where kb is boltzmann’s constant and t is the absolute temperature. we
describe the response of the atoms to the blackbody radiation in terms of the number of atoms na in the upper state and the number nb in the lower state.
due to the three processes of spontaneous emission, stimulated emission, and
stimulated absorption, these numbers change according to the rate equations
n? a = ?ana ? bu (?)(na ? nb) , (3.120)
n? b = +ana + bu (?)(na ? nb) . (3.121)

we solve these equations in steady state, de?ned by n? a = n? b = 0. either equation (note that n? a = ?n? b) gives
a/b
u (?) = n /n 1 ,


which with (3.118) becomes

b a ?

k?3/?2c3

u (?) = n /n

1 , (3.122)


furthermore according to boltzmann, in thermodynamic equilibrium the ra- tio of the number of atoms na in the upper state to that nb in the lower state
is given by
na = e?k?/kb t . (3.123)
nb
substituting this into (3.120), we ?nd the planck formula (3.119).


rabi flopping

blackbody radiation is emitted by a collection of atoms in thermal equilib- rium with the radiation ?eld. on a microscopic basis the atoms constantly exchange energy with the ?eld in such a way that macroscopically no change is noticed. as we see in sect. 5.1, this limit is valid in the rate equation approximation, for which the ?eld amplitude varies slowly (here not at all) compared to the atomic response.
now let us consider the opposite extreme for which we ignore atomic damping altogether, and for simplicity we take the monochromatic ?eld
(3.113) with frequency ? approximately equal to the two-level transition fre- quency ? = ?a ? ?b. examining the interaction energy (3.77, 3.81), we see that in the rotating-wave approximation we keep the e?i?t term for ?ni > 0. in the present case, ?> 0, and hence in the rotating-wave approximation we
keep only

1
vab c ? 2 ?e0e?

i?t

. (3.124)

for vba we use ei?t. because ? may di?er somewhat from ?, it is convenient to write ?(r, t) slightly di?erently from (3.98), namely, as
. . 1 . .

?(r, t) = ca(t)exp i

2 ? ? ?a

t ua(r)

+cb(t)exp

. . 1 . .
i ? 2 ? ? ?b t

ub(r) , (3.125)


where the frequency shift ? = ? ? ?. this choice places the wave function in the rotating frame used for the bloch vector in sect. 4.3. substituting this expansion for ? into the schro¨dinger equation (3.5) and projecting onto the eigenfunctions ua and ub as in the derivation of (3.20), we ?nd

1

c? a =
c? b =

2 i(??ca + r0cb) , (3.126)
1
2 i(?cb + r?ca) , (3.127)

where |r0| is the rabi ?opping frequency de?ned by
?e0

r0 ?

(3.128)
k

after rabi (1936), who studied the similar system of a spin– 1 magnetic dipole in nuclear magnetic resonance. equations (3.126, 3.127) provide a simple ex- ample of two coupled equations, a combination we see repeatedly in phase conjugation (chaps. 2, 10) and in linear stability analysis (chap. 11). equa-
tion (3.78) solves the n-level probability amplitude to ?rst order in the in-
teraction energy. here, we solve the two-level probability amplitudes to all
orders in that energy.



before solving (3.126, 3.127) generally, we can very quickly discover the basic physics by considering exact resonance, for which ? = 0. we can then
di?erentiate (3.127) with respect to time and substitute (3.128) to ?nd

1 2
¨ = ? 4 |r0| cb ,

i.e., the di?erential equation for sines and cosines. in particular if at time
t = 0 the atom is in the lower state [cb(0) = 1, ca(0) = 0], then

1
cb(t) = cos 2 |r0|t (3.129)


which from (3.127) gives




1
ca(t) = i sin 2 |r0|t. (3.130)

the probability that the system is in the lower level |cb(t)|2 = cos2 1 |r0|t =

(1 + cos|r0|t)/2, while |ca|2 = sin2 1 |r0|t

= (1

? cos

|r0|t)/2. hence the

wave function oscillates between the lower and the upper states sinusoidally at the frequency |r0|. in total contrast with blackbody radiation, instead of
coming to an equilibrium with constant probability for being in the upper
and lower levels, here the probabilities oscillate back and forth. in this case the atoms maintain a de?nite phase relationship with the inducing electric ?eld, while for blackbody radiation any such relationship averages to zero. as we see in chap. 4 on the density matrix, rabi ?opping preserves atomic coherence, while blackbody radiation destroys it. further discussions on the irreversibility of coupling to a continuum are given in sect. 14.3 on the theory of spontaneous emission, and more generally in chap. 15.
to solve the coupled equations (3.126, 3.127) including a nonzero detuning
?, we write them as a single matrix equation
d . ca(t) . = i . ?? r0 .. ca(t) . . (3.131)

dt cb(t)

? cb(t)


this is a vector equation of the form dc/dt = 1 imc, which has solutions of the form exp( 1 i?t). accordingly substituting c(t) = c(0) exp( 1 i?t) into
2 2
(3.131), we ?nd that det(m ? ?i) = 0. this yields the eigenvalues ? = ±r,
where r is the generalized rabi ?opping frequency
r? ,?2 + |r0|2 . (3.132)

equation (3.131) has simple sinusoidal solutions of the form

1 1
ca(t) = ca(0) cos 2 rt + a sin 2 rt,


1 1
cb(t) = cb(0) cos 2 rt + b sin 2 rt.
substituting these values into (3.126, 3.127) and setting t = 0, we immedi- ately ?nd the constants a and b. collecting the results in matrix form, we
have the general undamped solution

. ca(t) . . cos 1 rt ? i?r?1

sinrt ir0r?1

sin 1 rt

.. ca(0) . .

cb(t)

ir? ?1 1

1 ?1 1

c (0)

0 r sin 2 rt cos 2 rt + i?r

sin 2 rt

b
(3.133)

the 2×2 matrix in this equation is precisely the schro¨dinger evolution matrix
u (t) of (3.63) for the problem at hand. this u -matrix solution is valuable
for the discussion of coherent transients in chap. 12 and in general whenever
damping can be neglected. it yields the ?rst-order perturbation result (3.115) in the limit of a weak ?eld (r ? ?). section 4.1 shows how to account for
possibly unequal level decay from both levels. more general decay schemes
require the use of a density matrix as discussed in sect. 4.2. note that the matrix in (3.133) is a u matrix [see (3.62)]. for further discussion, see probs.
3.14–3.16 and sect. 14.1.


pauli spin matrices

in treating two-level atoms it is often handy to use a 2 × 2 matrix notation. the eigenfunctions ua and ub are then represented by the column vectors


ua ??

. 1 .
0


ub ??

. 0 .
1


, (3.134)



and the wave function by the column vector
. ca .




(3.135)

? ?? cb .

the energy and electric dipole operators are conveniently written in terms of the pauli spin matrices


?x =

. 0 1 .
1 0


?y =

. 0 ?i .
i 0 z

. 1 0
= 0 ?1

.
. (3.136)


while these matrices are hermitian, the “spin-?ip” operators


?? =

. 0 0 .
1 0


?+ =

. 0 1 .
0 0


(3.137)


are not hermitian. ?? ?ips the system from the upper level to the lower level

. 1 .
?? 0

. 0 .
= 1 ,


while ?+ ?ips from lower to upper. this property is handy for representing
transitions caused by the interaction energy.
by choosing the energy zero to be half way between the upper and lower levels, we can write the schr¨odinger picture hamiltonian (3.18) with the interaction energy (3.114) in the rotating wave approximation as
1 . k? ??e0e?i?t .

h = 2

or

??e0ei?t ?k?

k? 1

i?t

h = 2 ?z ? 2 [?e0?+e?

+ adjoint] . (3.138)

the pair of interaction-picture coupled equations (3.126, 3.127) can be writ- ten on resonance (? = 0) as
d . ca . = i . 0 r0 .. ca . . (3.139)

dt cb

2 r0 0 cb


section 4.1 uses such a matrix representation to solve a generalization of these equations including ?eld detuning and atomic damping.


3.4 simple harmonic oscillator

the simple harmonic oscillator plays a central role in quantum optics. chap- ter 13 discusses its connection with the quantum theory of radiation in detail. it is also used in simple models of vibrational states of molecules, comes into the description of large ensemble of two-level atoms, etc....
the classical energy of a harmonic oscillator of frequency ? and unit mass
is
hc = p2/2 + ?2q2/2 . (3.140)
where p is the momentum and q the position operator, satisfying the canonical commutation relation |q, p| = ik. (note that for notational clarity we omit
the “hat” on the operators in the following when no ambiguity is possible.)
using the coordinate representation correspondence

d
p = ?ik dq , (3.141)

we readily obtain the corresponding quantum-mechanical hamiltonian

k2 d2 2 2
h = ? 2 dq2 + ? q /2 . (3.142)

the eigenfunctions of this system are well known and may be expressed in terms of hermite polynomials [hn(?q)]

. ? 2 2

un(q) =

??2nn! hn(?q) exp(?? q /2) , (3.143)

where ? = ,?/k, with corresponding eigenenergies
k?n = k?(n + 1/2) ,n = 0, 1, 2,... (3.144)

although this summarizes in principle the theory of the harmonic oscillator, it is useful to look at it from another point of view which provides further
physical insight into its properties. we introduce two new operators a and a†
de?ned as
?

a = 1/

2k?(?q + ip) (3.145)

?

a† = 1/

2k?(?q ? ip) . (3.146)


inverting these expressions, we ?nd the position and momentum
q = ,k/2?(a + a†) (3.147)
p = i,k?/2(a ? a†) . (3.148)

right now this is a purely mathematical exercise, but we shall see that a and a† have important and simple physical interpretations. with the com- mutation relation [q, p] = ik, we readily ?nd that a and a† obey the boson
commutation relation
[a, a†] = 1 . (3.149)
substituting (3.147, 3.148) into (3.140), we ?nd the hamiltonian in terms of
a and a† as

h = k?

. 1 .
a†a + 2

. (3.150)

in the heisenberg picture, the time evolution of a and a† is given by (prob. 3.13)

da = i [

, a] =

i?a . (3.151)


this has the solution

dt k h ?



similarly we ?nd that

a(t) = a(0) e?i?t . (3.152)

a†(t) = a†(0) ei?t . (3.153)

consider now an energy eigenstate |h) of the harmonic oscillator with eigenvalue k?
h|h) = k?|h) , (3.154)
and evaluate the energy of the state |hr) = a|h). from (3.151), we have
ha = ah? k?a, so that
ha|h) = ah|h) ? k?a|h) = k(? ? ?)a|h) . (3.155)


that is, a|h) is again an eigenstate of the hamiltonian, but of eigenenergy k? lower than |h). because a lowers the energy, it is called an annihilation operator. repeating the operation m times, we ?nd
ham|h) = k(? ? m?)am|h) . (3.156) we can see that the lowest of these eigenvalues is positive as follows. for an
arbitrary vector |?), the expectation value of h is


(?|k?

. 1 .
a†a + 2


|?) = k?(?r|?r) + k?/2 > 0 ,


where |?r) = a|?). calling the lowest eigenvalue k?0 with eigenstate |0), we
have



and from (3.150)

a|0) = 0 (3.157)


h|0) = k?(a† a + 1/2)|0) = k?0|0) , (3.158)

that is, the lowest-energy eigenvalue k?0 = k?/2.
using the commutation relation (3.149), we ?nd
ha†|0) = [a†h + k?a†]|0) = k?(1 + 1/2)a†|0) ,

i.e., the eigenstate |1) has eigenvalue k?(1+1/2). because a† raises the energy, it is called a creation operator. substituting successively higher eigenstates
into equation, we ?nd
h(a†)n|0) = k?(n + 1/2)(a†)n|0) . (3.159) and hence the eigenstates |n)? (a†)n|0) has the eigenvalue (3.144). to ?nd
the constant of proportionality, we note that
a|n) = sn|n ? 1) , (3.160) where sn is some scalar. this implies
(n|a†a|n) = |sn|2(n ? 1|n ? 1) = |sn|2 . (3.161)
since a†a|n) = n|n), this gives sn = ?n. thus
a|n) = ?n|n ? 1) (3.162)
a†|n) = ?n + 1|n + 1) (3.163)

with (3.159), this yields the normalized eigenstates

1
|n) = ? (a†)n|0) . (3.164)
n


since a†a|n) = n|n), a†a is called the number operator. it gives the number
of quanta of excitation of the harmonic oscillator.
we can obtain the coordinate representation u0(q) of the ground state |0)
by substituting (3.145) for a into (3.157) to ?nd
(?q + ip)u0(q) = 0 (3.165)
using (3.7) in one dimension (p = ?ikd/dq), we ?nd
d ?
dq u0(q) = ? k qu0(q) ,

which has the normalized solution

u0(q) = (?/?k)1/4e?(?/2k)q

, (3.166)


in agreement with (3.143). similarly substituting (3.146) into (3.164) and using (3.166), we have


1 n
un(q) = ?n! (a†)

1 .
u0(q) = ,n!(2k?)2

d .n
?q ? k dq


u0(q) , (3.167)


which yields (3.143).
we mentioned that the commutation relation [a, a†] = 1 is called a bo- son commutation relation. as is well known, there are two kinds of quantum
particles, bosons and fermions. in the context of quantum optics, the most
famous kind of boson is the photon, which is introduced in some detail in chapter 13 on ?eld quantization. other types of bosons include a number of atomic isotopes, such as 87rb, 23na and 7li, to mention just three isotopes of alkali atoms of particular importance in the context of bose-einstein con- densation experiments. some other atomic isotopes, including for example 6li, as well as electrons and protons, are fermions. as we discuss in sect. 13.7, there are circumstances where massive particles such as electrons and atoms are conveniently described in terms of a matter-wave ?eld that is in many respects analogous to the optical ?eld. in the case of fermions, the an-
nihilation and creation operators c and c† that describe that ?eld obey the
anticommutation relations
[c, c†]+ = cc† + c†c = 1, (3.168)
[c, c]+ = [c†, c†]+ = 0. (3.169)
section 13.7 shows how these anticommutation relations imply pauli’s exclu- sion principle.


problems


show that with (3.11), the completeness relation (3.12) is equivalent to the alternative de?nition that any function f (r) can be expanded as
f (r) = . dnun(r) .
n

show that the operator matrix element of (3.16) written in terms of eigenfunctions has the same value as that of (3.42) written in terms of eigen- vectors.
compare and contrast (3.78) for the transition probability |c(1)(t)|2 with (2.16) for the phase matching of the generation of a di?erence frequency in a nonlinear medium.
what is the expectation value of the electric dipole operator er for a system in an energy eigenstate? why?
calculate the expectation value of the dipole moment operator er for an atom with the wave function
?(r, t) = c210u210(r)e?i?210 t + c100u100(r)e?i?100 t .
hint: use spherical coordinates and write the position vector in the form 1
r = r sin?[(xˆ iyˆ)ei? + (xˆ + iyˆ)e?i?] + r cos ?zˆ .
2

what is the expectation value of the energy (3.6) for the wave function (3.13)? is this value ever actually measured?

derive the wave-function schr¨odinger equation (3.5) from the state vector version (3.44) by appropriate projections.
starting with the initial conditions ca(0) = 1 and cb(0) = 0, solve the equations of motion (3.126, 3.127) to third-order in the electric-dipole interaction energy.

given (3.133), (a) what is the free evolution matrix? (b) what are the matrices for ? and ?/2 pulses? (c) how do you describe photon echo (a pulse,
a free evolution, a second pulse, and a second free evolution) in terms of these
matrices? (don’t solve, just set up) answer: see sect. 12.3.

draw the level diagram for and write the general solution to the three- level equations of motion (taking r0 real)


c? 1 =
c? 2 =
c? 3 =


1
2 ir0c2 ,
1
2 ir0(c1 + c3) ,
1
2 ir0c2 .


verify that the pauli spin operator communication relations

[?x, ?y ] ? ?x?y ? ?y ?x = 2i?z ,
[?y , ?z ] = 2i?x [?z , ?x] = 2i?y .

calculate the simple harmonic oscillator eigenfunction u1(q) using (3.167). show that it is orthogonal to the ground-state eigenfunction u0(q).

show that [x, f (p)] = ikdf /dp. hint: use the momentum representation. also show that [p, e?kk?/?p] = ?kk.

a useful alternative basis for the two-level system consists of dressed states, which are the eigenvectors of the matrix m in the equation of mo-
tion (3.122). speci?cally, show that the eigenvectors satisfying the eigenvalue equations (taking r0 real)
. u . = . ?? r0 .. u . = ? . u . , (3.170)




are given by

mr v

r0 ? v v

. u2 . = 1

. r? ? . = . cos ? . , (3.171)

v2 ,(r ? ?)2 + r2 r0

sin ?

for the eigenvalue ? = r and
. u1 . = . ? sin ? . (3.172)
v1 cos ?
for ? = ?r. hint: to ?nd (3.160), in the bottom component equation given by (3.161) with ? = r, equate u2 to the coe?cient of v2 and vice versa, and normalize the resulting vector. show that cos 2? = ??/r and sin 2? = r0/r.

the dressed states of (3.160, 3.161) de?ne the transformation matrix
. cos ? sin ? .

u = ? sin ? cos ?

, (3.173)


which diagonalizes the matrix m by


u mu ?1 =

. r 0 . .
0 ?r


this transformation matrix relates the dressed-state probability amplitudes to the “bare-state” amplitudes by
. c2(t) . = u . ca(t) . , (3.174)

c1(t)

cb(t)

where the probability amplitudes c2 and c1 obey the equations of motion 1

c? 2 =

2 irc2 , (3.175)
1

c? 1 = ? 2 irc1 . (3.176)

derive (3.124) by solving (3.164, 3.165), writing the initial values c1(0) and c2(0) in terms of ca(0) and cb(0) using (3.163), and then using the inverse of (3.163) to ?nd ca(t) and cb(t). 3.16 using the transformation matrix of

(3.162), calculate the pauli spin ?ip operators of (3.128) in the dressed-atom basis. answer for ?+:


?+ =

. cos ? sin ? cos2 ? .
? sin2 ? ? cos ? sin ?


. (3.177)


show by mathematical induction that
. a, a†m . = m(a†)m?1 , (3.178)
. a†, am . = ?mam?1 . (3.179)

hint: show validity for m = 1, then write out commutator, assume relation is true for m ? 1 and use (3.140).
prove the operator identity

eb xe?b = x + [b, x]+ 1 [b, [b, x]] + ... + 1 [b, [b,... [b, x] .. .]] + ... .

2! n!


(3.180)

in particular, prove the baker-hausdor? relation
ea+b = eaeb e? 2 [a,b] (3.181)

provided [a, [a, b]] = [b, [a, b]] = 0. alternative method: show that the derivative of the operator f (?) = e?b x e??b is f r(?) = [b, f (?)]. using this derivative and its derivatives in turn, expand f (?) in a maclaurin series. setting ? = 1 in this series yields (3.169).

show that for any pair of operators p and q satisfying the canonical commutation relation [q, p] = ik one has





and


[q, f (p)] = ik

df (p) (3.182)
dp

dg(q)

[p, g(q)] = ?ik

dq . (3.183)


this and the next two problems discuss important aspects of the la- grangian and hamiltonian formulation of the interaction between charges and electromagnetic ?elds, and their connection to the minimum coupling
hamiltonian. this topic is discussed pedagogically in great detail in cohen- tannoudji, dupont-roc and g rynberg (1989).
the classical lagrangian describing the coupling of the electromagnetic ?eld to a collection of charges {q?} at locations r? and with velocities r? ? is

l = . , m? r? 2 + q r


• a(r


, t) ? q


u (r


, t)

2 ? ? ? ? ? ? ?
+ s0 . .,
(??u (r?, t) ? a? (r?, t))2 ? c2(?× a(r?, t))2

= . m? r? 2 + ¸ d3r

j(r, t)

a(r, t)

?(r, t)u (r, t)

2 ? { • ?
?
+ s0 .e(r, t)2 c2b(r, t)2., , (3.184)
2
where the total charge is





and the current is

? = . q??(r ? r?(t)) (3.185)
?

j = . r? ?(t)?(r ? r?(t)) . (3.186)
?


show that the when applied to this lagrangian, the lagrange equations of motion for the ?eld f ,

?l ?l

d ?l


= 0 , (3.187)

?fi ? ? ?(?fi) ? dt ?f?i
yield the maxwell equations
1 ?e
?× b = c2 ?t + ?0j (3.188)
?

?• e =
0

. (3.189)

here the coordinates fi of f are the vector potential a and the scalar po-

tential u , respectively, and the corresponding velocities are a?
also that the lagrange equations of motion for a particle,

and u? . show

?l d

?l = 0 , (3.190)

?r? ? dt ?r? ?
yield the lorentz equations of motion
m?¨r? = q?[e(r?, t) + r? ? × b(r?, t)] . (3.191) hint: use the vector identity
?(a • b) = (b • ?) • a + (a • ?) • b + b × (?× a)+ a × (?× b) . (3.192)

3.22 starting from the lagrangian of problem (3.20), show that the conju- gate momenta of r? and a are
?l



and

p? = ?r?

= m?r? ? + q?a(r?, t) (3.193)

? = ?l
?a?

= s0e(r, t) . (3.194)

what is the conjugate momentum of the scalar potential u ?
from these results, show that the hamiltonian corresponding to that la-
grangian is the minimum coupling hamiltonian
h = .[p? ? q?a(r?, t)]2 + . q?u (r?, t)

?
¸
+ s0

?
d3r[e2(r, t) + c2b2(r, t)] + e(r, t) • ?u (r, t) , (3.195)


or, in the radiation gauge,
¸ . 1

h = .
?

d3r

2m?

[p? ? q?a(r?, t)]2?(r ? r?)

¸
+ s0

.
d3r[e2(r, t) + c2b2(r, t)]

. (3.196)


this is the same hamiltonian as ((3.103), except that it contains also a free ?eld part that is, as we have seen, important to obtain maxwell’s equations. the minimum coupling hamiltonian (3.103), complemented by maxwell equations, describes therefore the same physics as (min coup ham v2).

3.22 consider the form of the minimum coupling hamiltonian (3.103), and use the hamilton-jacobi equations of motion
p? = ?h = ?•h
?
r? = ? ?p (3.197)


to prove that the motion of a particle of charge q is governed by the lorentz
equation

m¨r = q

.
??u ?

?a .
?t

+ qr? × (?× a), (3.198)

or, with e = ??a/?t, b = ?× a and u = 0,
m¨r = qe + q(r? × b) . (3.199) hint: use the same vector identity as in problem (3.20).

4 mixtures and the density operator











in this chapter we generalize our treatment of two-level systems to include various kinds of damping. some of these can be incorporated directly into the equations of motion for the probability amplitudes. however, two important kinds cannot: upper to lower level decay, and more rapid decay of the electric dipole than the average level decay rate. for these two damping mechanisms, we need a more general description than can be provided by the state vector. speci?cally, we need to consider systems for which we do not possess the maximum knowledge allowed by quantum mechanics. in other words, we do not know the state vector of the system, but rather the classical probabilities for having various possible state vectors. such situations are described by
the density operator ?, which is a sum of projectors |?i)(?i| onto the possible state vectors |?i), each weighted by a classical probability pi.
we refer to a problem described by a single normalized state vector as a pure case, while a system described by a density operator consisting of an incoherent sum of pure-case contributions is a mixed case or a mixture.
such mixtures occur in particular when we consider only part of a total
system. we are often confronted with situations where an atom is coupled to a large number of modes of the electromagnetic ?eld, but are not interested in what happens to the ?eld instead we are only interested in what happens to the atom. hence we write equations for the atom alone, and ignore what happens to the ?eld. the “truncation” of the total problem automatically reduces our knowledge and usually results in a mixture. chapter 15 discusses such problems in more generality.
the projectors |?i)(?i| used in the density operator formalism involve bi-

linear combinations of probability amplitudes, such as ciac?

and ciac? for

the case of two-level atoms. although this might appear to be an added di?-
culty, it often simpli?es the mathematical analysis. to appreciate this point, note that the results of our discussions in chap. 3, such as the probability of a transition or the value of the induced dipole moment, are invariably expressed in terms of bilinear combinations of the amplitudes. in fact, the expectation value of any observable involves bilinear combinations.
the sum over all possible projectors performed in the density operator is analogous to the incoherent or partially incoherent addition of light ?elds. if two electric ?eld amplitudes are added coherently, they interfere and the


interference term is uniquely speci?ed by the amplitudes. however, if the addition is only partially coherent, the interference term is smaller than that speci?ed by the individual amplitudes. for the two-level atom, the coherence
term (electric dipole term) for the state vector |?i) is given by ciac? . the
polarization of the medium resulting from many such systems is given by a weighted sum of these individual ciac? . this sum is the matrix element ?ab = (a|?|b) of the density operator. for a number of systems with random phases between the upper and lower state probability amplitudes, ?ab tends to average to zero, even though the corresponding sum of probabilities ciac?
is una?ected by the random phases.
section 4.1 shows how simple level decay can be incorporated into the two-level probability-amplitude equations of motion, and solves these equa- tions for arbitrary tuning. this level decay causes the two-level probabil- ity amplitudes to decrease exponentially in time, thereby destroying the wave function’s normalization. such an unnormalized wave function actually describes a simple mixed case. section 4.2 introduces the density matrix for two-level and more general cases. it also gives a simple derivation of the dipole decay constant, which due to collisions is in general larger than the average level decay constant. this phenomenon is an important manifestation of the partial coherence of a mixed case. section 4.3 shows how the density matrix can be visualized in three dimensions by transforming to the bloch vector. the bloch-vector equations of motion provide an alternative to the schr¨odinger equation. they are popular in the literature and are particu- larly useful in studying coherent transients (chap. 12). the results of this chapter are needed in our treatments of lasers, optical bistability, nonlinear spectroscopy, phase conjugation, optical instabilities, and coherent transients.


4.1 level damping

we have sees how the populations of excited atomic levels decay in time because of spontaneous emission. they can also decay because of collisions and other phenomena. in fig. 4.1, we indicate one kind of such decay from
both the a and the b levels, a situation which occurs in typical laser media.
the loss of excited level probability corresponds to an increase of probability
for lower-lying levels that we do not consider explicitly. for reasons that will become apparent in chap. 15, it turns out that the ?nite level lifetimes can be described quite well by adding phenomenological decay terms to the equations of motion (3.126, 3.127). this is true provided that one is not interested in the explicit dynamics of the levels populated by these decay mechanisms. we write
1 1
c? a = ? 2 (?a + i?)ca + 2 ir0cb (4.1)
1 1
c? b = ? 2 (?b ? i?)cb + 2 ir0ca , (4.2)




fig. 4.1. energy level diagram for two-level atom, showing decay rates ?a and ?b
for the probabilities |ca|2 and |cb|2

where ? = ? ? ? and the rabi ?opping frequency r0 = ?e0/k is assumed to be real. the factors of 1 are included in the decay terms so that, for example,
the probability |ca|2 decays as exp (??at) in the absence of e0. the lifetimes are de?ned as the times at which the probabilities have decayed to 1/e of their
original values. hence they are given by the reciprocals of the decay constants
?a and ?b.
we can solve these equations by ?rst-order perturbation theory as in
sect. 3.2, or by the more exact formulations of sect. 3.3. we note here that for the simple case of equal decay constants, ?a = ?b = ?, the substitutions

b(t) = cb(t)e
cr

?t/2 ?t/2

, (4.3)

a(t) = ca(t)e ,

reduce (4.1, 4.2) to the undamped versions (3.126, 3.127). the solutions for this damped case are, then, just those for the undamped case multiplied by
the exponential decay factor exp(??t/2). in particular to lowest order in
perturbation theory, the probability that stimulated absorption takes place
changes from that given by (3.115) to the form


|ca(t)|2 c |c(1)(t)|2 =


1 2 ??t
4 r0

.s sin[(?? ?)t/2] .2
(? ? ?)/2


. (4.4)

this is illustrated in fig. 4.2. since the probability amplitudes decay away in time, the corresponding two-level wave function fails to remain normalized and as such describes a simple kind of mixed case.
we can ?nd the spectral distribution for stimulated emission by starting with the atom in the upper level and calculating the total probability that











(1) 2

e??t

|cn (t)|







t

fig. 4.2. transition probability of (4.4) with ?a = ?b = ?


it decays by spontaneous emission from the lower level to some other distant state (s). this follows because stimulated emission is needed to get from the a level to the b level in order that spontaneous emission from the b level can occur. the total probability of decay from the b level is given by
¸ ?

ps = ?b
0

dtr|cb(tr)|2 , (4.5)


since ?b|cb(t)|2 is the probability per unit time that the atom decays from the b level. for an initially excited atom (i.e., ca(0) = 1), |cb(t)|2 is given for su?ciently small values of |r0/?|2 by (4.4), since the role of the levels a and b is reversed from the case of (4.4). substituting this into (4.5), we ?nd that the pro?le is lorentzian with width ?, that is,


1
ps = 2 (?


2
?)2 + ?2 . (4.6)

of course, if the probability |cb(t)|2 fails to remain always much less than
unity, (4.6) cannot be trusted. to generalize (4.6) accordingly and in antici-
pation of future need, we now solve (4.1, 4.2) exactly.
as for (3.131), we seek solutions of the form c = c(0)ei?t and ?nd the eigenvalues
1
?1,2 = 2 (i?ab ± ?) , (4.7)
where the average decay rate constant

1
?ab = 2 (?a + ?b) , (4.8)


and where the complex rabi ?opping frequency

.. 1 .2
? = ? ? 2 i(?a ? ?b)



+ r2 . (4.9)


hence the solutions have the form
. 1 .. 1 1 .

ca(t) = exp

? 2 ?abt

ca(0) cos 2 ?t + a sin 2 ?t

. 1 .. 1 1 .

cb(t) = exp

? 2 ?abt

cb(0) cos 2 ?t + b sin 2 ?t .


substituting these values into the equations of motion (4.1, 4.2) at the time
t = 0, we ?nd
??abca(0) + ?a = ?(?a + i?)ca(0) + ir0cb(0) ,
??abcb(0) + ?b = ?(?b ? i?)cb(0) + ir0ca(0) .

this gives the integration constants ?a = ?[ 1 (?a ? ?b)+ i?]ca(0) + ir0cb(0) and ?b = [ 1 (?a ? ?b) + i?]cb(0) + ir0ca(0), that is,

. ca(t) . =
cb(t)
. cos 1 ?t ? ??1. 1 (?a ? ?b) + i?. sin 1 ?t ir0??1 sin 1 ?t .
2 2 2 2
ir0??1 sin 1 ?t cos 1 ?t + ??1. 1 (?a ? ?b) + i?. sin 1 ?t
2 2 2 2
1 . ca(0) .

× exp . ?

?abt.
2

cb(0)

. (4.10)


to ?nd the corresponding pro?le for stimulated emission, we substitute
cb(t) from (4.10) with the initial values ca(0) = 1 and cb(0) = 0 into (4.5)
and ?nd (with some algebra)


1
ps = 2 (?

2
?)2 + ?2(1 + i ) , (4.11)

? b 0
where the dimensionless intensity i0 is given by
i0 = |r0|2/?a?b . (4.12)

comparing (4.6, 4.11), we see that the frequency response of the atom to an applied ?eld is broadened not only because of decay, but also because
of saturation. this second e?ect is called power broadening. note that on resonance (? = ?) the stimulated emission pro?le of (4.11) approaches the constant value ?ab/?b as i0 becomes large.


4.2 the density matrix

the semiclassical situations discussed in the ?rst part of this book (chaps. 5–12) almost invariably use the schro¨dinger picture (see sect. 3.1). hence for the two-level system we write the wave function with the schro¨dinger picture
amplitudes ca and cb de?ned by the general expression (3.22) for two levels,
that is,

?(r, t) = ca(t)ua(r) + cb(t)ub(r) (4.13a) or equivalently by the state vector

|?(t)) = ca(t)|a) + cb(t)|b) . (4.13b) the corresponding density operator is de?ned as the projector ? = |?)(?|
onto this state, and the density matrix elements ?ij = (j|?|i) are given by
the bilinear products
?aa = cac? , probability of being in upper level
?ab = cac? , dimensionless complex dipole moment1
?ba = cbc? = ?? ,
a ab
?bb = cbc? , probability of being in lower level.
in matrix notation, the density operator ? is therefore

? = . cac?
cbc?

cac? . =
cbc?

. ?aa ?ab .
?ba ?bb


. (4.14)


this density matrix is precisely the outer product
. ca .

? =
b

(c?

c?) .


in terms of the 2 × 2 density matrix of (4.14), the expectation value (3.24) of an operator o is given by
(o) = ?aaoaa + ?aboba + ?baoab + ?bbobb . (4.15)
in particular the dipole moment is given in the ua, ub basis by
(er) = ??ab + c.c.. (4.16) equation (4.15) and, more generally, (3.24) is just the trace of the matrix
product ?o :
(o) = . . ?nmomn = .(?o)nn = tr(?o) . (4.17)
n m n

1 provided an electric-dipole transition is allowed between the a and b levels.

4.2 the density matrix 99

we show at the end of this section that this result holds in all generality.
we can derive the equations of motion for the elements of the density ma- trix from the schro¨dinger equations of motion for the probability amplitudes
ca(t) and cb(t) of (4.13). from (3.23) including phenomenological decays, we
have
c?a = ?(i?a + ?a/2)ca ? ik?1vabcb , (4.18)
c?b = ?(i?b + ?b/2)cb ? ik?1vbaca , (4.19) where vab = (a|v|b). proceeding one element at a time, we have
??aa = c?ac? + cac??
a a
= (?i?aca ? ?aca/2 ? ik?1vabcb)c?
+ca(i?ac? ? ?ac? /2 + ik?1vbac?)
a a b
= ??a?aa ? [ik?1vab?ba + c.c.] . (4.20)

it is not surprising to ?nd the complex conjugate in this equation, for prob- abilities are real. similarly, we ?nd
??bb = ??b?bb + [ik?1vab?ba + c.c.] . (4.21) apart from the decay terms, this value is equal in magnitude and opposite in
sign from that in (4.20). this expresses the fact that probability is transferred between the a and b levels by the interaction energy vab. the o?-diagonal element ?ab obeys the equation of motion
??ab = c?ac? + cac??
b b
= (?i?aca ? ?aca/2 ? ik?1vabcb)c?
+ca(i?bc? ? ?bc?/2 + ik?1vbac? )
b b a
= ?(i? + ?ab)?ab + ik?1vab(?aa ? ?bb) . (4.22)

the example treated so far can be described equally well by an unnormal- ized state vector or by a density operator. however, the unnormalized state vector description becomes usually inadequate as soon as we consider more complex situations such as those encountered in the description of many- system phenomena. the phenomenological damping factors in (4.1, 4.2) ac- tually result from the interaction of an atom with the many modes of the elec- tromagnetic ?eld. to treat more complicated cases, e.g., a decay of the dipole
term ?ab independent of the decay of the level probabilities pii the wave func-
tion becomes by itself very cumbersome to use, or even incorrect. such dipole
decay results from an incoherent superposition of simple pure-case density matrices and can be cast in terms of system-reservoir coupling, a general approach followed in chap. 15. that chapter will show that in general, it becomes necessary in such situations to abandon the idea of describing the system via a state vector. one needs to introduce instead a density opera- tor whose evolution is irreversible and governed by nonhermitian dynamics.


alternatively, it is possible to use an ensemble average over a large number of “quantum trajectories” describing the nonhermitian dynamics of an un- normalized state vector, the so-called monte carlo wave functions approach. both of these approaches, as well as a third approach involving quantum noise operator techniques, will be discussed in chap. 15. here we consider an important simple case.
elastic collisions between atoms in a gas or between phonons and atoms in a solid can cause ?ab to decay separately from the diagonal elements. specif-
ically, if during an interaction the energy levels are merely shifted slightly
without a change of state (e.g., distant van der waals interactions), the de- cay rate for ?ab is increased without much change in ?a and ?b. this is due
to the fact that the phase of the radiating atomic dipole is shifted in a some-
what random fashion, and the contributions of a collection of such dipoles tend to average to zero. we can gain a semiquantitative understanding of this process by considering the following discussion, couched in terms of phonon interactions in ruby.
the active atom in ruby is the cr3+ ion, which is surrounded with o2?
atoms. at room temperature, all atoms vibrate, with the result that the energy levels in the cr3+ ions experience random stark shifts. for simplicity we assume that this phenomenon can be expressed mathematically by adding
a random shift ??(t) to the energy di?erence ?. ignoring other perturbations
for simplicity, we can write the equation of motion for the o?-diagonal element
?ab as
??ab = ?[i? + i??(t) + ?ab]?ab . (4.23)
integrating this formally, we have


?ab(t) = ?ab(0)exp

. ¸ t
?(i? + ?ab)t ? i
0

.
dtr??(tr)


. (4.24)


we now perform a classical ensemble average of (4.24) over the random variations in ??(t). this average a?ects only the ??(t) factor. expanding the
second part of the exponential term by term, we have



exp

. ¸ t
?i
0

.
dtr??(tr)

¸ t
= 1 ?i
0

1 ¸ t
dtr??(tr) ?
0


dtr

¸ t
dtrr??(t)??(trr)
0

( )n
+
(2n)!

2n t


i=1 0


dti??(ti) + ... . (4.25)


the function ??(t) is as often positive as negative, as suggested in fig. 4.3. hence the ensemble average (??(t)) is zero (a frequency shift as well as damp- ing can occur, which would change ?). furthermore, averages of products (??(t)??(tr)) are zero as well, unless t c tr, in which case the product is mostly positive since (?1)2 = 1. assuming that variations in ??(t) are rapid compared to other changes (which occur in times like 1/?ab), we take