13

(a)
e


?p ? g


(b)



?



r(t)

g


?p
s g

fig. 5.4. single-photon source via intracavity stirap. (a) a three-level atom is pumped by a classical ?eld of rabi frequency ?p(t) and is coupled to the cavity ?eld with vacuum rabi frequency g. (b) the compound system atom + cavity ?eld initially in state |s, 0) undergoes stirap transfer into the state |g, 1) with a single photon in the cavity mode which escapes the cavity through a partially transmitting mirror with the rate ?, forming a single-photon wavepacket with temporal shape r(t).


we need, however, to consider the e?ects of atomic and cavity ?eld relax- ation in more detail. assuming that the lower atomic levels |s) and |g) are metastable (long-lived), the state |e, 0) of the compound system decays via the atomic spontaneous emission with the rate ? , while the state |g, 1) decays due to the cavity-?eld relaxation with the rate ?. thus, the eigenstates |d)
and |b±), whose energy separation |?± ? ?0| is characterized by ?¯, acquire
certain widths determined by ? and ?. as a result, the nonadiabatic coupling between the dark |d) and bright |b±) eigenstates is small if they do not over- lap, which requires that the condition ?¯ > ? +? be satis?ed at all times. since
the vacuum rabi frequency g is constant, while ?p(t) is time-dependent, so that ?p(ti) g and ?p(tf ) > g, at resonance ? = 0, the above requirement translates to the strong coupling condition g > ?, ?. in addition, the adiabatic following condition (3.135) requires that ?p(t) changes su?ciently slowly for the condition max(?p) ?p 1, with ?p being a characteristic raising time of ?p(t), to be satis?ed.
to verify the possibility of intracavity stirap, we thus have to solve the
liouville equation for the density matrix ? of the compound system,

?
?t ? = ?

i
k [h, ?]+ lat? + lcav?, (5.94)

where h is the hamiltonian of (5.89), while lat? and lcav? describe, respec- tively, the atomic and cavity mode relaxations. the cavity liouvillian lcav? is that of (5.15), and the atomic liouvillian lat? is given by
lat? = . 1 ?el.2 ?le? ?el ? ?ee? ? ? ?ee.
l
= . ?el ?le? ?el ? 1 ? .?ee? + ? ?ee. , (5.95)
l


where the index l = s, g,... runs over all the lower levels of the atom to which the upper level |e) can decay. hence, ?es and ?eg represent the spontaneous
decay rates from |e) to levels |s) and |g), respectively, while ? = .l ?el is the
total decay rate of |e). for the initial state |s, 0), the hamiltonian h acts in the hilbert space h = { |s, 0), |e, 0), |g, 1)}. then the decay of the cavity ?eld takes the system outside of space h, to state |g, 0) ?/ h. on the other hand, the decay of state |e, 0) due to the atomic relaxation can take the system to one of the states |s, 0), |g, 0) or |l, 0), where |l) denotes any lower atomic state other than |g) or |s). the states |g, 0) and |l, 0) are not in space h, while |s, 0) ? h. this necessitates our use of the density matrix approach, since with the amplitude equations we could not properly take into account the decay channel |e, 0)? |s, 0).


8



4



0
1



0.5



0
1


(a)







(b)







(c)




g






?s,0 s,0


?g,1 g,1
?e,0 e,0


?p(t)















pemit(t)



0.5

r(t)



0
?5 0 5
time

fig. 5.5. dynamics of the intracavity stirap with g = ? = 4? . (a) rabi fre- quencies g and ?p(t) (b) populations ?s,0 s,0(t), ?e,0 e,0 (t) and ?g,1 g,1(t) and (c) emission rate r(t) and probability pemit (t). the light-gray curves in (c) correspond to r(t) and pemit (t) for an initially excited two-level atom with the parameters of fig. 5.2, i.e., g = ? and ? = 4? . time is measured in units of ? ?1.


in fig. 5.5, we plot the results of the numerical solution of the equations for all the relevant elements of the density matrix ? (see prob. 5.6). we ?nd that, when the strong coupling condition is not quite satis?ed, during the


evolution the intermediate excited state |e, 0) acquires small but ?nite pop- ulation ?e,0 e,0, which is due to nonadiabatic transitions. on the other hand, assuming the absorption in the cavity mirrors to be negligible, ?abs ?tr c ?, the probability ?g,1 g,1 of the single-photon state |g, 1) decays due to the leak- age of the cavity ?eld through the partially transparent mirror, the outgoing photon pulse having the temporal shape r(t) = ?tr?g,1 g,1(t). yet, for the parameters of fig. 5.5, at the end of the process the photon emission prob- ability pemit(tf ) attains the value 0.93, which is close to the ideal. when we take g = 8? , with the other parameters unchanged, so as to better satisfy the strong coupling condition, we obtain completely adiabatic evolution of the system with negligible population of the excited state |e, 0), achieving pemit(tf ) ” 0.98.
under these conditions, using simple arguments, we can derive an analytic
expression for the shape of the outgoing single-photon pulse r(t). as the system adiabatically follows the dark state |d(t)) of (5.92), we expect that at any time t ? [ti, tf ] the ratio of populations of states |s, 0) and |g, 1) is given by
?s,0 s,0(t) = cot2 ?(t) , (5.96)
?g,1 g,1(t)
where the mixing angle ?(t) is de?ned through cot ?(t) = g/?p(t). next, since under adiabatic evolution, the excited state |e, 0) is never signi?cantly populated, the sum of populations of the initial |s, 0) and ?nal |g, 1) states decays only via the cavity ?eld relaxation,

? .?


(t)+ ?

(t). = ???


(t) . (5.97)

?t s,0 s,0

g,1 g,1

g,1 g,1

using (5.96) with ?p(t) ƒ= 0 for t ? [ti, tf ], one can now derive the rate equation for the population of state |g, 1) (see prob. 5.7), whose solution is


?g,1 g,1(t)= ?g,1 g,1(ti) exp

. ¸ t
?
ti


dtr

? + ?tr cot2 ?(tr) .
1+ cot2 ?(tr)


. (5.98)


at the initial time ti, we assume that g ?p(ti) ƒ= 0, and ?s,0 s,0(ti) = 1. from (5.96) we then have ?g,1 g,1(ti) = tan2 ?(ti), which should be used in the above solution. it turns out that under the strong coupling condition g > ?, ? , the analytic solution (5.98) is practically indistinguishable from the exact numerical solution of the full set of density matrix equations. this con?rms the validity of the adiabatic approximation that led to our starting equations (5.96) and (5.97).
we can now write the expression for the pulse shape in the explicit form
?tr ?p (tr )

?2(ti)

?¸ t

2g2 r

2 r ?

r(t)= ?tr p
g2

exp ?

dtr
ti

?p (t ) ? ??p (t )
g2 + ?2(tr) ?

, (5.99)


which shows that, by carefully changing the pump ?eld rabi frequency ?p(t), and thereby the mixing angle ?(t), we can manipulate at will the temporal characteristics of the outgoing pulse. in particular, we can fully control the timing and the temporal shape, or the bandwidth, of the single-photon pulse. once the photon has left the cavity, we may recycle the system by switching o? the pump ?eld and preparing the atom in the initial state |s). we could then repeat the process to generate another photon with precise timing and pulse shape. this system can thus serve as a deterministic and e?cient source of tailored single-photon pulses.


problems

verify the derivation of equations (5.11)–(5.14).

using the master equation for the cavity ?eld (5.14), derive the equations of motion for the expectation values of the ?eld amplitude (5.16) and mean photon number (5.17).

a single-mode electromagnetic ?eld (f) is coupled to a two-level system (atom a) through the master equation

? ?f = g?a .2a?fa† ? a†a?f ? ?fa†a. + g?a .2a†?fa ? aa†?f ? ?faa†. ,

?t gg

ee
(5.100)

where g is the coupling constant, and ?a

and ?a

are the populations of the

lower and upper levels of the two-level system, respectively. assume that ?a
and ?a are kept constant. derive di?erential equations governing the evolution of the expectation values (a) and (a†a) and solve them.

an electromagnetic ?eld consisting of two modes 1 and 2 is coupled to a two-level system through the master equation
?
?f = g?a .2a1a2?fa† a† ? a† a† a1a2?f ? ?fa† a† a1a2.

?t gg

1 2 1 2 1 2

+g?a

2a† a† ?fa1a2 ? a1a2a† a† ?f ? ?fa1a2a† a†

, (5.101)

ee . 1 2

1 2 1 2.


where g is the e?ective coupling constant, aj and a† are creation and annihila-

tion operators of the ?eld mode j = 1, 2, and ?a

and ?a

are the populations

of the lower and upper levels of the two-level system, respectively. assume

that ?a

and ?a

are kept constant.


(a) derive di?erential equations that govern the evolution of the expectation values (aj ) and (a†aj ) and discuss their solvability.
(b) perform the factorization of modes, i.e., take ?f ? ?(1) ? ?(2), and solve the resulting equations for the expectation values (aj ) and (a†aj ).


(c) obtain di?erential equations for (a†aj ) without factorization, but for ?a =
j ee
0 and discuss their meaning.

verify that p (?, t) given by (5.42) is the solution of the fokker–planck equation (5.40).
using (5.94), derive the equations for all the elements of density matrix
? in the basis { |s, 0), |g, 0), |l, 0), |e, 0), |g, 1)}.

using (5.96) and (5.97), derive the rate equation for population ?g,1 g,1(t) and its solution (5.98).


6

field propagation in atomic media











in this chapter, we consider several illustrative examples of weak ?eld prop- agation in atomic media. we derive the coupled evolution equations for the ?eld and the atoms, which in general should be solved self consistently. phys- ically, depending on the amplitude and the pulse shape, the ?eld interacts with the atoms in a certain linear or nonlinear way, which in turn in?uence the evolution of the ?eld upon propagating in the medium. thus the subject of this chapter contains elements of both quantum and nonlinear optics. since in quantum information applications one typically considers weak quantum ?elds, such as single-photon or weak coherent pulses, our discussion will focus mainly on weak ?eld propagation and interaction with optically dense atomic ensembles.


propagation equation
for slowly varying electric field

in chap. 3 we have seen that, when considering the interaction of an atom with a radiation ?eld whose wavelength is large compared to the size of the atom, such as optical or microwave ?eld, the dipole approximation gives the dominant contribution to the atom–?eld coupling. the resulting interaction hamiltonian (3.30) is given by the scalar product of the atomic dipole moment and the electric ?eld. we thus begin with the derivation of the propagation equation for the electric ?eld, whose envelope varies slowly in space and time as compared to its wavelength and oscillation period, respectively.
the maxwell equations in a macroscopic medium are given by (2.1). in a dielectric medium with no magnetization, the densities of currents j and free charges ? are zero, the magnetic ?eld is b = µ0h while the displacement electric ?eld is given by d = ?e + p , where the permittivity ? may contain the contribution of a passive host material, if any, and p is the macroscopic polarization of the active medium. in what follows, we will be concerned with the situation in which the medium is represented by near-resonant atoms


whose response determines the polarization p and assume ? = ?0. taking the curl of (2.1a), exchanging the order of di?erentiation ? × (?tb)= ?t(? × b), and using (2.1b) together with the expressions for b and d above, we obtain

?2e

?2p

? × (? × e)+ µ0?0 ?t2 = ?µ0 ?t2 . (6.1)
using the vector identity ? × ? × e = ?(? • e) ?? e and assuming that the electric ?eld varies slowly in the plane transverse to the propagation direction, ? • e c 0, we arrive at the wave equation

2 1 ?2e

?2p

? e ? c2

?t2 = µ0 ?t2 . (6.2)

as is typical in optics, we consider a unidirectional propagation of the ?eld along the z axis, in which case the electric ?eld and the induced polarization vectors can be expressed as
e(r, t)= eˆe(z, t) , p (r, t)= eˆp (z, t) , (6.3) where eˆ is the unit polarization vector normal to the ?eld propagation direc-
tion. the wave equation (6.2) then reduces to the 1d equation

?2e

1 ?2e

?2p

?z2 ? c2

?t2 = µ0 ?t2 . (6.4)

let us now consider a classical quasi-monochromatic electric ?eld with carrier frequency ? and wavevector k = ?/c,
e(z, t)= e (z, t)ei(kz??t) + e ?(z, t)e?i(kz??t) , (6.5) where e (z, t) is a slowly varying in time and space envelope of the ?eld, which
in general is a complex function, e = e ei? with e the real amplitude and ?
the phase. this ?eld induces the medium polarization
p (z, t)= p(z, t)ei(kz??t) + p?(z, t)e?i(kz??t) , (6.6) where again p(z, t) is a slowly varying in time and space complex function,
p = pei?. substituting (6.5) and (6.6) into the 1d wave equation (6.4), and
adopting the slowly varying envelope approximation,

. ?e .

. ?e .

. ?? .

. ?? .

. . . .

. . . .

. ?t . ?|e| ,

. ?z . k|e| ,

. ?t . ?,

. ?z . k (6.7a)

. . . .

. . . .

. ?p .

. ?p .

. . . .

. ?t . ?|p| ,

. ?z . k|p| , (6.7b)

. . . .

which amounts to assuming that the ?eld variation is small on the scale of both the optical period ??1 and the wavelength k?1, we obtain the following propagation equation

?e + 1 ?e =i k


. (6.8)

?z c ?t

2?0 p

in terms of real quantities, this equation reads

. ? 1 ? .
+ e = ?

k . ?
imp , e

1 ? .
+ ? =

k
rep . (6.9)

?z c ?t

2?0

?z c ?t

2?0


in many problems of quantum and nonlinear optics involving the propagation of slowly varying optical ?elds in near-resonant media, equations (6.8) or (6.9) constitute the starting point of the discussion. in the following sections of this chapter, we discuss several aspects of weak pulse propagation in two and three-level atomic media.
in the remainder of this section, we outline a very useful in optics for- malism of susceptibilities through which the polarization of the medium can be related to the applied ?eld. in general, the induced polarization can be a very complicated nonlinear function of the ?eld. here, however, we will be interested in the case of weak ?eld propagation in an isotropic medium, for which, to a good approximation, the polarization is a linear function of the ?eld,

¸ ?
p (z, t)= ?0
??

?(tr)e(z, t ? tr)dtr , (6.10)

where ?(t) is the linear susceptibility. for a monochromatic probe ?eld of fre- quency ?, substituting e(z, t)= e (?)ei(kz??t) +c.c. into (6.10) and comparing the result with (6.6) we obtain the familiar relation
p = ?0?(?)e (?) , (6.11)
where ?(?) is the fourier transform of ?(tr), ?(?) = ¸ ?(tr)ei?t dtr. substi- tuting this into (6.8) and taking into account that by de?nition e is time- independent, we have
? k
=i ?(?)e , (6.12)


with the solution

?z 2

e (z)= e (0) ei?(z)e?az , (6.13)
where ?(z)= 1 k re?(?)z is the phase shift and a ? 1 k im?(?) is the linear
2 2
amplitude attenuation (for a ? 0) coe?cient. thus, the real and imaginary
parts of the linear susceptibility ?(?) describe, respectively, the dispersive and absorptive properties of the medium. upon propagation in the medium, the intensity of the ?eld i = ?0 c |e|2 is attenuated according to i(z)= i(0) e?2az which is known as beer’s law of absorption.
we have just seen that when a is positive the amplitude and the inten- sity of the ?eld are attenuated in the medium. it is possible though that a is negative, meaning that the absorption is replaced by ampli?cation. clearly, if such a situation is realized, the energy of the ?eld will increase at the ex- pense of energy stored in the medium. this in turn means that there should


be some mechanism in place that will pump the energy into the medium, which is in fact what happens in lasers. this pumping mechanism can provide the energy into the system with a certain rate, however high but bound. on the other hand, if the linear regime discussed above were valid for arbitrary propagation distances, the intensity of the ?eld, or its energy for that matter, would grow exponentially and become arbitrarily large for su?ciently long distances. this of course can not happen in any realistic situation, as satura- tion e?ects will come into play once certain intensity of the propagating ?eld is reached. therefore, while the linear regime can adequately describe many practical situations involving ?eld attenuation (or energy absorption) in the medium, rigorous treatment of ampli?cation problems will necessarily require the consideration of the nonlinear response of the system which will certainly include the saturation.
consider now a quasi-monochromatic electric ?eld with the carrier fre- quency ? as in (6.5). we can express the envelope function e (t) quite generally through the fourier integral

¸
e (t)=


e (? + ?)e?i?td? , (6.14)


where e (? + ?) is the amplitude of the frequency component (? + ?) of the probe ?eld. the corresponding expression for the polarization is
¸

p(t)= ?0

?(? + ?)e (? + ?)e?i?td? . (6.15)


assuming the susceptibility is a smooth function of frequency in the vicinity of ?, to ?rst order in ? it is given by


?(? + ?) c ?(?)+

??(?)
? + o(?
??

2) ,

which after the substitution into (6.15) yields
??(?) ?e (t)

p(t)= ?0?(?)e (t)+ i?0 ??

+ ... , (6.16)
?t

where we have used (6.14) and the resulting from it relation ¸ ?e (? + ?)e?i?td? =
i?te (t). with the polarization expressed as in (6.16), the propagation equation (6.8) becomes

?e + 1 ?e =i

?(?)e , (6.17)

?z vg ?t 2
where the group velocity vg is given by
. 1 k ?? .?1 c

vg =

+
c 2 ??

= . (6.18)
?
2 ??


as will be seen shortly, vg is equal to the velocity with which the peak of the probe pulse propagates in the medium. if in (6.16) we kept terms of higher order in ?, on the right-hand-side of (6.17) we would have had terms containing
2 2

the second and higher derivatives of e , such as ? ? ? e

known as the group

velocity dispersion, which determine the pulse distortion. on the other hand,
in the frequency region where ?(?) is approximately a linear function of ?, these terms vanish and the pulse whose fourier bandwidth lies within this frequency region propagates in the medium without much distortion of its shape.
to solve the propagation equation (6.8), we introduce new variables ? = z and ? = t ? z/vg . obviously, the old variables z and t are expressed through the new ones as z = ? and t = ? + ?/vg and we have


? ?z ?
=

?t ? ? 1 ?
+ = + .

?? ?? ?z

?? ?t

?z vg ?t

in terms of the new variables, (6.17) can be written as
? k

?? e (?)=i

?e (?) , (6.19)

where e (?) ? e (?, ? + ?/vg ). its solution is e (?)= e (0) exp[ i k??], from which

we easily obtain

e (z, t)= e (0,? ) ei?(z)e?az , (6.20)

where the phase shift ? and absorption coe?cient a are the same as in (6.13). this equation indicates that, given the boundary condition for the ?eld at z = 0 and all times tr, e (0, tr), inside the medium, at coordinate z ? 0 and time t, the envelope of the pulse is related to that at z = 0 but an earlier time tr = ? = t ? z/vg (for vg > 0) which is called the retarded time. in addition, the ?eld undergoes a phase shift and absorption with the propagation distance z as per (6.20).
what if instead of the boundary value problem, i.e., given the probe ?eld envelope e at z = 0, we need to solve the initial value problem, meaning we know the ?eld at time t =0 for all zr, e (zr, 0)? similarly to the above, we can introduce another set of variables ? = z ? vg t and ? = t, in terms of which (6.17) becomes
? k

?? e (? )=i

?vg e (? ) , (6.21)

with the solution e (? ) = e (0) exp[ i k?vg ? ], where e (? ) ? e (? + vg ?, ? ). re- turning back to the old variables t and z, we have
e (z, t)= e (?, 0) ei?(vg t)e?avg t . (6.22) thus inside the medium, at coordinate z and time t> 0, the envelope of the
pulse is related to that at the initial time t = 0 but at a retarded point in


space zr = ? = z ? vg t. the phase-shift and absorption are now functions of time. note that the physical equivalence of solutions (6.20) and (6.22) stems from the mathematical equivalence of the expansions of e (0,t ? z/vg ) and e (z ? vg t, 0) is terms of temporal and spatial eigenmodes, respectively,
e (0,t ? z/vg )= . e (?)e?i?(t?z/vg ) = . e (?)ei?/vg (z?vg t)
? ?
= . e (q)eiq(z?vg t) = e (z ? vg t, 0) . (6.23)
q

in the discussion above, we have tacitly assumed that ??(?)

? 0, which

means that the group velocity of (6.18) can take values 0 ? vg ? c. it is pos-
sible, however, that the derivative of the medium susceptibility with respect to the frequency of the applied ?eld is negative. then the group velocity may exceed the speed of light or even become negative. considering such a medium of ?nite length l, the group velocity exceeding c means that the peak of the pulse appears at the exit from the medium at a time t = l/vg which is shorter than the time l/c that the light pulse needs to propagate the distance l in free space. even more dramatic is the case of the negative group velocity, for which the peak of the pulse appears at the exit from the medium even before the peak of the incident pulse enters the medium at z = 0. these observa- tions are however not as mysterious as they may seem, if one realizes that such a “superluminal” pulse propagation is possible only in amplifying me- dia, because in conventional atomic media the anomalous dispersion ??(?) < 0 around the atomic resonance frequency is accompanied by a strong absorption of the pulse, as discussed in the following section. in an amplifying medium, however, it is possible that as the pulse enters the medium, its leading edge is ampli?ed more strongly (or absorbed more weakly) than the rest of the pulse. then the pulse is getting reshaped in the medium and its peak leaving the medium is nothing else than the ampli?ed front of the incident pulse. it may therefore reach z = l even before the peak of the incident pulse has entered the medium at z = 0. of course, no information travels faster then light, since signal velocity can not exceed the lesser of the group velocity of an informa- tion carrying pulse and the phase velocity of all the frequency components of that pulse.


field propagation in a two-level atomic medium

in this section we employ the above formalism to describe the propagation of a weak probe ?eld e(z, t) through a near-resonant two-level atomic medium. the macroscopic polarization of the medium p (z, t) of (6.6), induced by the applied electric ?eld (6.5), is given by the expectation value of the dipole moment of all the atoms at position z and time t,
p (z, t)= qa(z) tr.??(z, t). , (6.24)


where ? = ? • eˆ is the projection of the dipole moment onto the ?eld po- larization direction eˆ, and qa(z) is the number density of atoms which in the following will be assumed uniform over the entire interaction volume, qa(z) = qa. expanding the trace and taking into account the fact that the diagonal matrix elements of ? are zero, we have
p (z, t)= qa.?ge?eg (z, t)+ ?eg ?ge(z, t)] . (6.25)

recall that in sect. 4.1.3 we have studied the interaction of a two-level atom with a monochromatic ?eld e employing the density matrix equations in the frame rotating with the frequency ? of the ?eld. since there we were dealing with a single atom at a ?xed position, its spatial coordinate was taken as the origin, with the consequence that the spatial dependence of the ?eld disap- peared. here we consider the ?eld propagation and interaction with the atoms at various positions z. therefore the o?-diagonal density matrix elements in (6.25) are related to the corresponding slowly varying (sv) matrix elements of (4.42), by the transformation

?eg = ?(sv) i(kz??t)

? (sv) ?i(kz??t)

eg e

, ?ge = ?eg = ?ge e .


substituting this into (6.25) and comparing it with (6.6), for the slowly varying complex polarization p we obtain
p(z, t)= qa?ge?(sv)(z, t) , (6.26)

and similarly for its complex conjugate p ?(z, t) = qa?eg ?(sv)(z, t). the ?eld propagation equation (6.8) with the polarization given by (6.26), together with the density matrix equations (4.42) with ? = ?eg e constitute the so-called maxwell–bloch equations, which in general require a self-consistent solution. typically, for strong time-dependent ?elds e(z, t), when the atomic saturation and dynamic e?ects are important, only numerical solutions of these equations are feasible. simple analytic solutions of the maxwell–bloch equations can be obtained in the two opposite limiting cases: weak, long-pulsed or continuous- wave ?eld propagation which is discussed below in some detail, and strong and short pulse propagation brie?y outlined at the end of this section.
we thus consider the propagation and near-resonant interaction of the electromagnetic ?eld with a medium of two-level atoms. the spatio-temporal evolution of the ?eld is governed by the equation

. ?
+
?z

1 ? .
c ?t

qa??ge
e (z, t)=i 2? c


?(sv)(z, t) . (6.27)

in turn, the atomic coherence ?(sv) obeys the equation (dropingping the super-

script (sv))


?
?t ?eg (z, t)= (i? ? ?eg )?eg (z, t) ? i


?eg
k



e (z, t)d(z, t) , (6.28)


where d = ?ee ??gg is the population inversion and ? = ? ??eg the detuning. as we are interested in weak ?eld propagation, we may solve (6.28) to lowest (?rst) order in e . to that end, we neglect saturation and take d(z, t) = ?1 for all z and t, obtaining


eg = ?

?eg
k

e
? + i?eg


. (6.29)


this is obviously analogous to the solution obtained in sect. 4.1.3 through the rate-equation approximation, which is valid for ?elds whose amplitudes vary little on a time-scale of ??1. consistently with this approximation, we neglect
the time derivative in (6.27), which upon the substitution of ?(1) from (6.29)

takes the form

its solution is where

?
?z e = ??e . (6.30)

e (z)= e (0) e??z , (6.31)

qa?|?ge|2
2?0ck

1
?eg ? i?

(6.32)

is the complex linear absorption coe?cient, whose real and imaginary parts determine the medium absorption a = re(?) and dispersion ?/z = im(?), respectively. near the resonance ? ? ?eg ?, we can rewrite (6.32) as


?eg

?eg |?ge|2

? = a0
eg

, a0 =
? i? 2?0ck?eg

qa , (6.33)


where in the de?nition of the resonant absorption coe?cient a0 we have re- placed ? by ?eg . in the absence of atomic collisions and other additional sources of coherence relaxation, such that ?eg = 1 ? , we can express a0 through the spontaneous decay rate of the excited atomic state





obtaining

1
? =
4??0

4?3 |?ge|2
3kc3 ,

a0 =

3?c2
?2 qa ? ?0qa . (6.34)

thus the resonant absorption coe?cient a0 is given by the product of the absorption cross-section ?0 and the atomic density qa.
comparing (6.30) with (6.12), we see that the linear susceptibility for the two-level atomic medium is given by
2a0 i?eg

?(?)=
k

?eg ? i?

. (6.35)


1.0

0.8

0.6

0.4

0.2

0.0
0.6

0.4

0.2

0.0

?0.2

?0.4


?0.6


?4 ?2 0 2 4
detuning ?/?eg


fig. 6.1. absorption and dispersion spectra of the two-level atomic medium for weak probe ?eld e in units of resonant absorption coe?cient a0.


the corresponding absorption and dispersion spectra are shown in fig. 6.1. at exact resonance ? = 0, the ?eld is strongly attenuated, e (z)= e (0) e?a0 z , and its intensity i ? |e |2 is depleted with the propagation distance according to
i(z)= i(0) e?2a0 z . (6.36)
the quantity 2a0z = 2?0qaz, is called optical depth of the medium, since it determines the fraction of the energy dissipated in a medium of number density qa and length z. away from the resonance, the absorption spectrum is lorentzian, given by


a = 1 k im?(?)= a0


2
eg =
eg

a0
1+ . ? .2


. (6.37)

2 ?2 + ?2

?eg


let us note parenthetically that, if one were to consider a short pulse propagation in the two-level medium, the rate-equation approximation made in (6.29) would have been inconsistent with keeping the time derivative in the propagation equation (6.27). the perturbative solution for ?eg should then be modi?ed as
1 ?

?eg = ?(1)

(1)

eg + i? ? ?

?eg ,
?t


where ?(1) is given by (6.29). upon substitution into (6.27), this would result in a modi?ed group velocity, which in the frequency region around resonance is given by

c
vg = a0 ,
?eg

corresponding to the anomalous dispersion ??(?) < 0 around ? = ?eg , as seen in fig. 6.1. this dispersion, however, is accompanied by strong absorption and the resulting “superluminal” group velocity is of little physical interest, as we have noted at the end of the previous section.
in the above discussion, we assumed a homogeneously broadened atomic medium, meaning that all of the atoms have a common resonance frequency ?eg and therefore their response to the applied probe ?eld is homogeneous, given by (6.29). as noted before, the homogeneous width of the atomic reso- nance ?eg consists of contributions from the atomic spontaneous decay (nat- ural width) and other phase-relaxation processes, such as atomic collisions and laser phase ?uctuations. often, however, one encounters a situation in which various atoms respond to the applied ?eld di?erently, primarily due to the variations of their resonant frequencies. for example, optically ac- tive dopants—atoms—in solid state host material typically experience dif- ferent level-shifts due to the variations in the local environment, i.e., inho- mogeneities in the crystal structure. another example often encountered in quantum optics is the thermal atomic vapor. there the atoms moving with various velocities v see di?erent doppler-shifted frequencies ?r of the applied ?eld, ?r = ? ? k • v with k being the ?eld wavevector. in turn, the e?ective resonant frequency of the moving atom, as it appears to the ?eld, is given

by ?r

= ?eg /(1 ? v/c) c ?eg + kv, with |v| c. in general, to calculate

the medium polarization (6.26), one has to sum up the contributions of all

the atoms weighed by the appropriate distribution function w (?r

) for the

atomic resonant frequencies,
¸
p(z, t)= qa?ge



r w (?r



) ?eg (?r



) . (6.38)


in particular, in the case of doppler broadening of thermal atomic ensem- ble, the dependence of ?eg on the atomic velocity can be obtained from

(6.29) by replacing ? with the e?ective detuning ?r = ? ? ?r

= ? ? kv.

the corresponding maxwellian velocity distribution function is w (v) = (u??)?1 exp(?v2/u2) with u = ,2kbt /ma being the most probable veloc-
ity at temperature t . consequently, the linear susceptibility for the doppler- broadened two-level atomic medium becomes


?(?)=

2a0 ¸ ?

dv i?eg w(v)


, (6.39)

k ??

?eg ? i(? ? kv)


and the resulting absorption spectrum is given by the convolution of the gaus- sian and lorentzian functions

a0 ¸ ?
a = ? dv


2
u , (6.40)

u ? ??

1+ . ?? kv .
eg


known as the voigt pro?le, which in general does not yield simple analytical expressions. however, when the doppler width is much larger than the ho- mogeneous width, ku ?eg , we can evaluate the gaussian at the line-center of the lorentzian, v = ?/k and pull it out of the integral. the remaining lorentzian is then easily integrated, with the result


a0?eg ?? a =
ku


exp

. ?2 .
? (ku)2


. (6.41)


in the opposite limit of a cold atomic gas, such that ?eg ku, the above expression (6.40) obviously reduces to (6.37).
before closing this section, let us brie?y address the case of strong and short input pulse, when the atomic relaxation can be neglected on the time- scale of pulse duration. mccall and hahn have found that the pulse area, de?ned by

?(z)=

2?eg ¸ ?

e (z, t) dt, (6.42)

k ??
obeys the propagation equation
?
?z ?(z)= ?a sin .?(z). , (6.43)

which is known as the pulse area theorem (see prob. 6.1). obviously, in the limit of small area pulses, so that ?(z) 1 and therefore sin .?(z). c ?(z), this
equation leads to the exponential absorption of the pulse according to beer’s law (6.13) or (6.31). in the case of strong pulse, (6.43) predicts that upon propagation the area of the pulse evolves towards the nearest even multiple of ?, i.e., ?(z) ? 2n?, where n is an integer. however, pulses with n > 1 are not stable and tend to break up into pulses with area ? = 2? which can propagate in the medium over long distances preserving their spatio-temporal shape given by

k

. t ? z/vg .

e (z, t)=
w

?eg

sech
?w

, (6.44)

where ?w is the temporal width of the pulse and vg = c/(1 + ca?w ) c (a?w )?1 is the corresponding group velocity. this e?ect is called self-induced trans- parency, which is a manifestation of optical soliton propagation in resonant atomic media.


field propagation in a three-level atomic medium

we discuss now weak ?eld propagation in a three level atomic medium. we consider the scheme of fig. 6.2 in which the probe ?eld e interacts with the atoms on the |g) ? |e) transition, while the second strong coherent ?eld drives the atomic transition |s) ? |e) with rabi frequency ?d. we


will see that under the conditions of two-photon raman resonance, the probe propagates in the medium without much attenuation and with the reduced group velocity, which is a consequence of the coherent population trapping (cpt) of atoms in the dark state |d) discussed in sect. 3.6. such absorption- free propagation of probe ?eld in coherently–driven atomic media is called electromagnetically induced transparency (eit). although eit can and has been observed in atomic media with ?, v and ? level con?gurations, here we focus upon the ? con?guration, as in this case the absorption is particularly low, owing to the long relaxation times of the atomic ground state coherence.


(a) e
?


?
e



g





?d


? r
s

(b)

?d
vg

e


fig. 6.2. electromagnetically induced transparency in an atomic medium. (a) level scheme of three-level ?-atoms interacting with a strong cw driving ?eld with rabi frequency ?d on the transition |s) ? |e) and a weak probe ?eld e acting on the transition |g) ? |e). the lower states |g) and |s) are long-lived (metastable), while the excited state |e) decays fast with the rate ? . (b) collinear geometry of the probe and driving ?eld propagation in the atomic medium for doppler-free eit.


the hamiltonian for the three-level atom interacting with two classical ?elds in the ? con?guration is given by (3.121), which in the notations of fig. 6.2(a) becomes
h?(z)= ?k.??ee + ?r?ss. ? .?eg e eikz ?eg + k?d eikd z ?es + h.c.. , (6.45)

where ? = ? ? ?eg is the detuning of the probe ?eld from the |g) ? |e) transition, and ?r = ? ? ?d = ? ? ?d ? ?sg is the two-photon raman detuning, with ?d = ?d ? ?es being the driving ?eld detuning from the
|s) ? |e) transition. in (6.45) we have explicitly shown the dependence of
h? on the atomic position z, with kd being the projection of the driving ?eld wavevector onto the probe ?eld propagation direction z. using the liouville equation

?
?t ? = ?

i
k [h?, ?]+ l??, (6.46)

which includes the relaxation matrix l?? appropriate to the ? con?guration of atomic levels (see (5.95)), we obtain the following set of density matrix equations,

?
?t ?gg = ?eg ?ee +
?
?ee = ?? ?ee +

(?ge e ?e?ikz ?eg ? c.c.) , (6.47a) (?eg e eikz ?ge ? c.c.)+ i(?d eikd z ?se ? c.c.) , (6.47b)

?t k
?ss = ?es?ee + i(?? e?ikd z ?es ? c.c.) , (6.47c)
? i

?t ?eg = (i? ? ?eg )?eg +
?

?eg e eikz (?gg ? ?ee)+ i?d eikd z ?sg , (6.47d)
i

?t ?sg = (i?r ? ?sg )?sg ?
?

?eg e eikz ?se + i??e?ikd z ?eg , (6.47e)
i

?t ?es = (i?d ? ?es)?es +

?eg e eikz ?gs + i?d eikd z (?ss ? ?ee) , (6.47f)

where ?eg and ?es are the spontaneous decay rates from level |e) to levels |g) and |s), respectively, while ? is the total spontaneous decay rate of |e), which in addition to ?eg and ?es may also include the decay to other atomic levels. finally, ?eg , ?es and ?sg are the corresponding coherence relaxation rates, with ?sg typically being much smaller than ?eg and ?es because the lower states |g) and |s) are long-lived (metastable). since the hamiltonian (6.45) corresponds to the frame rotating with the frequencies of the optical ?elds, the o?-diagonal density matrix elements in (6.47) are slowly oscillating func- tions of time. yet, their spatial oscillations are rapid, corresponding to the wavelengths of the optical ?elds. these fast spatial oscillations are removed via the transformations

?eg = ?(sv) ikz

, ?es = ?(sv)eikd z

, ?sg = ?(sv)ei(k?kd )z

, (6.48)


which results in a set of equations identical to (6.47) but without the expo- nential factors eikz and eikd z .
as discussed in the previous section, the propagation equation for the slowly varying in time and space amplitude e (z, t) of the probe ?eld is given by (6.28). we are interested in the weak probe ?eld interaction with the atoms initially prepared by the strong cw driving ?eld in the ground state |g). more precisely, we assume that the driving ?eld with rabi frequency ?d is switched on long before the probe ?eld arrives. then, as the driving ?eld saturates the transition |s)? |e), level |s) is being depleted due to the spontaneous decay from |e) to |g) with the rate ?eg , and eventually all atoms accumulate on level |g). this is the essence of optical pumping of atomic level |g). once the process of optical pumping is over, the absorption of the driving ?eld becomes negligible. with atoms so prepared, we can solve (6.47) to the lowest (?rst)
order in the weak probe ?eld e , to obtain the expression for ?(sv) (we will droping from now on the superscript (sv)). in the stationary regime, from (6.47e) we have

?
?sg c? d

?eg ,

?r + i?sg


where the term containing e ?se has been neglected due to the smallness of both e and ?se. substituting this into (6.47d), dropingping the time derivative, and taking ?gg ? ?ee c 1 as the probe is assumed too weak to cause depletion of ?gg , we obtain


eg = ?

?eg
k


? + i?eg

e
? |?d|2(?r


+ i?sg

)?1 . (6.49)


similarly to the previous section, we substitute ?(1) into the propagation equa- tion (6.27) without the time-derivative, and after comparing it with (6.12) we ?nd the complex susceptibility for the probe ?eld, which now takes the form
2a0 i?eg

?(?)=
k

?eg ? i? + |?d|2(?sg ? i?r)?1

. (6.50)


obviously, in the limit of ?d ? 0, this susceptibility reduces to that for the two-level atom (6.35). the absorption and dispersion spectra corresponding to the susceptibility of (6.50) are shown in fig. 6.3 for the case of ?d = ?ge and ?d = 0, i.e., driving ?eld is resonant with the transition |s) ? |e) and therefore ?r = ?. as seen, the interaction with the driving ?eld results in a splitting of the absorption spectrum into two peaks separated by 2?d, which is known as the autler–towns splitting. meanwhile, at the line center the medium becomes transparent to the resonant ?eld, provided the ground state coherence relaxation rate ?sg is su?ciently small, ?sg |?d|2/?eg , this is the essence of electromagnetically induced transparency (eit).
at the exit from the optically dense medium of length l (optical depth 2a0l> 1), the intensity transmission coe?cient, de?ned as ti ? i(l)/i(0) = e?2al, is given by
ti (?)= exp[?k im?(?)l] .
to determine the width of the transparency window ??tw, we expand im?(?) in a power series in the vicinity of maximum transmission ?r = 0. under the eit conditions (?eg , ?d, ?2 /?eg )?sg |?d|2, to lowest non-vanishing order

in ?r, we then obtain
. 2 .
ti (?) c exp ? r



2
, ??tw = |? |



. (6.51)

2
tw eg

2a0l


considering next the dispersive properties of eit, as shown in fig. 6.3, the dispersion exhibits a steep and approximately linear slope in the vicinity of absorption minimum ?r = 0. therefore, a probe ?eld slightly detuned from resonance by ?r < ??tw, during the propagation would acquire a large phase-

shift


?(l) c

a0?eg
|?d|2


?rl, (6.52)

while su?ering only little absorption, as per equation (6.51).


1.0

0.8

0.6

0.4

0.2

0.0
0.6

0.4

0.2

0.0

?0.2

?0.4


?0.6


?4 ?2 0 2 4
detuning ?r /?eg


fig. 6.3. absorption and dispersion spectra (?r = ?) of a three-level atomic medium for weak probe ?eld e in units of a0, for ?d/?eg =1 and ?sg /?eg = 10?3. the light-gray curves correspond to the case of ?d = 0 (two-level atom).


let us now brie?y address the consequences of inhomogeneous broadening in a thermal atomic ensemble. as we have established in the previous section, in the expression for the susceptibility (6.50), the detunings of the optical ?elds ? and ?d should then be replaced by the e?ective doppler-shifted

detunings ?r = ? ? kv and ?r

= ?d ? kdv (recall that kd is the projection

of the driving ?eld wavevector onto the z direction). consequently, the two-

photon raman detuning becomes ?r

= ?r ? (k ? kp)v. it is thus obvious

that when k c

kp the raman detuning is practically una?ected by the atomic

thermal motion. such a situation can be realized when the probe and driving
?elds have similar frequencies and propagation directions, i.e., are collinear as shown in fig. 6.2(b). since both ?elds couple to the same upper level |e), they can have similar frequencies if the lower levels |g) and |s) are closely spaced in energy, i.e., nearly degenerate. this is in fact the case for most of the alkali atoms—the workhorse of experimental quantum optics—whose electronic ground state contains a manifold of hyper?ne and zeeman levels. a pair of such levels is then selected by properly adjusting the frequencies and polarizations of the probe and driving ?elds to serve as the lower metastable levels |g) and |s). then, in the vicinity of raman resonance ?r = 0, the eit is immune to the atomic thermal motion, provided (k ? kp)v¯ < ??tw
and (kv¯, ?dkv¯/?eg )?sg |?d|2, where v¯ = ,3kbt /ma is the mean thermal
atomic velocity.


next, we discuss a pulsed ?eld propagation in the eit medium. equa- tion (6.51) implies that for the absorption-free propagation, the bandwidth ?? of a near-resonant probe ?eld should be within the transparency window, ?? < ??tw. alternatively, the temporal width ?w of a fourier-limited probe pulse should satisfy ?w ” ???1. as we know from sect. 6.1, due to the steep slope of the dispersion in the vicinity of the absorption minimum ?r = 0, a near-resonant probe pulse e (z, t) propagates in the eit medium with greatly reduced group velocity

c
vg =

c
= c

|?d|2


c. (6.53)

1+ ?

? [re?(?)]

1+ c a0 ?ge

a0?ge

2 ??

|?d |2


therefore, upon entering the medium, the spatial envelope of the pulse is compressed by a factor of vg /c 1, while its peak amplitude remains un- changed. since the dispersion slope is approximately linear around ?r = 0, during propagation the shape of the pulse experiences little distortion.
the physical origin of this behavior is the coherent populations trapping of the atoms in the dark state |d) discussed in sect. 3.6. here, for the atoms lo- cated at various spatial coordinates z the dark state of the hamiltonian (6.45) is given by


where

|d(z, t)) = cos ? |g)? ei(k?kp )z sin ? |s) , (6.54)

?d

?p

cos ? = . , sin ? = . ,
?2 2 2 2

d + ?p

?d + ?p

and ?p = ?eg e is the rabi frequency of the probe, which is a function of space and time since e = e (z, t). before the probe pulse arrives, all atoms have been prepared in state |g) and the driving ?eld is on, which means that ? = 0 and the atoms are in the dark state, |d) = |g). as the probe pulse enters and propagates in the medium, every atom adiabatically follows the dark state (6.54), provided the ?eld envelope changes in time su?ciently slowly, as required by the adiabatic criterion (3.135). stated otherwise, the fourier bandwidth of the pulse should be smaller than the autler–towns splitting of the atomic resonance, which determines the eit transparency window ??tw. upon propagation through the medium, as the probe pulse approaches some position z, the mixing angle for the atom at that position slightly rotates to adjust to the value ?(z, t) = arctan |?p(z, t)/?d|. a small fraction of atomic population, proportional to |?p/?d|2, is therefore transfered to state |s) by coherent raman scattering, i.e., absorbing a photon from the leading edge of the pulse and re-emitting it into the driving ?eld. at the end of the pulse, this population is transfered back to state |g) by the reverse process, absorbing a photon from the driving ?eld and re-emitting it into the trailing edge of the probe pulse. thus, upon propagation, photons are continuously “borrowed” by the atoms from the leading edge of the probe pulse, to be added later on to its tale. as a result, the pulse propagates without attenuation as a