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992 Part E Scattering Experiment

Phosphor screen

MCP detector

Drift tube

Velocity focusing

lens

Crossed beams

Repeller plate

product CH

product

220 m/s 235 m/s

iso

rel

iso

rel

Fig. 67.10a,b Panel (a): image of ionic products formed by H

ejection from Co

0 collision complex. Panel (b): image of

ionic products formed by CH

ejection from Co

0 collision

complex

Fig. 67.9 Schematic of crossed beam imaging instrument.

Product ﬂux extracted from the collision volume with

a single center of mass speed describes a sphere in vel-

ocity space. This ﬂux is projected onto a multichannel

plate/phosphor screen detector and recorded with a CCD

camera

that have the same lab velocity components in the plane

of the beams to be focused to a single point. Figure 67.10

shows velocity space images of the H

and CH

elimina-

tion products arising from decay of the transient collision

complex formed in collisions of Co

with iso-butane

[67.39].

This velocity space image is equivalent to the c.m.

cross section expressed in equation (67.14), and can

be appropriately averaged over initial states or summed

over ﬁnal states to yield product angular or energy distri-

butions, total cross sections, or rate constants according

to the hierarchy of Fig. 67.1. The time window that

imaging requires is ideally mated with pulsed molecu-

lar beam sources, and thus with pulsed lasers. This last

connection opens up a number of possibilities for re-

actant preparation, including electron impact ionization

via pulsed electron beams formed by laser-induced pho-

toemission from metals, pulsed IR laser excitation of

ro-vibrationally state-selected molecules, formation of

free radicals by UV laser photolysis, and multiphoton

ionization preparation of state-selected ions.

As laser-based photolysis, state-selection, and ion-

ization methods continue to advance, additional routes

for preparing atomic and molecular ions, free radi-

cals, and clusters with quantum state-speciﬁcity will

appear. These schemes will enhance all crossed beam

methodologies, but will have an especially important

impact on methods that exploit time-based snapshots

of the velocity space product distribution in coordinate

space. Point-by-point methods for reconstruction of the

three-dimensional ﬂux distribution in velocity space will

continue to play a role in the development of the crossed

beam technique, but technological advances in lasers and

in imaging systems virtually assure that the most impres-

sive gains in crossed beam technology in the next several

years will come from multichannel product detection.

References

67.1 J. J. Thomson: Rays of Positive Electricity (Longmans

Green, New York 1913)

67.2 T. Su, M. T. Bowers: Gas Phase Ion Chemistry,Vol.1,

ed. by M. T. Bowers (Academic, New York 1979) p. 84

Part E 67

Ion–Molecule Reactions References 993

67.3 R. D. Levine, R. B. Bernstein: Molecular Reaction

Dynamics and Chemical Reactivity (Oxford, New

York 1987)

67.4 Y. T. Lee: Atomic and Molecular Beam Methods,ed.

by G. Scoles, U. Buck (Oxford, New York 1988) p. 553

67.5 J. M. Farrar, W. H. Saunders, Jr.: Techniques for

the Study of Ion–Molecule Reactions (Wiley-

Interscience, New York 1988)

67.6 T. T. Warnock, R. B. Bernstein: J. Chem. Phys. 49,

1878 (1968)

67.7 G. L. Catchen, J. Husain, R. N. Zare: J. Chem. Phys.

69, 1737 (1978)

67.8 D. J. Levandier, D. F. Varley, M. A. Carpenter,

J. M. Farrar: J. Chem. Phys. 99, 148 (1993)

67.9 M. A. Carpenter, M. T. Zanni, D. J. Levandier,

D. F. Varley, J. M. Farrar: J. Chem. Phys. 72, 828

(1994)

67.10 J. E. Pollard, D. J. Trevor, J. E. Reutt, Y. T. Lee,

D. A. Shirley: J. Chem. Phys. 77, 34 (1982)

67.11 P. E. Siska: Ph. D. Thesis, Harvard University (1969),

Available from University Microﬁlms, Ann Arbor, MI

67.12 P. E. Siska: J. Chem. Phys. 59, 6052 (1973)

67.13 S. A. Safron, W. B. Miller, D. R. Herschbach: Discuss.

Faraday Soc. 44, 108 (1967)

67.14 D. M. Sonnenfroh, R. A. Curtis, J. M. Farrar: J. Chem.

Phys. 83, 3958 (1985)

67.15 J. L. Kinsey: Ann. Rev. Phys. Chem. 28, 349 (1977)

67.16 L. Hüwel, D. R. Guyer, G.-H. Lin, S. R. Leone:

J. Chem. Phys. 81, 3520 (1984)

67.17 S. L. Anderson: Adv. Chem. Phys. 82, 177 (1992)

67.18 L. A. Posey, R. D. Guettler, N. J. Kirchner, R. N. Zare:

J. Chem. Phys. 101, 3772 (1994)

67.19 R. N. Compton, J. C. Miller: Laser Applications in

Physical Chemistry, ed. by K. Evans D. (Marcel

Dekker, New York 1989) p. 221

67.20 M. K. Bullitt, C. H. Fisher, J. L. Kinsey: J. Chem. Phys.

64, 1914 (1974)

67.21 P. Tosi, O. Dmitrijev, D. Bassi: Chem. Phys. Lett.

200, 483 (1992)

67.22 A. Henglein, K. Lacmann, G. Jacobs: Ber. Bunsen-

ges. Phys. Chem. 69, 279 (1965)

67.23 W. A. Chupka, M. E. Russell: J. Chem. Phys. 4849,

1527 (1968)

67.24 W. A. Chupka, M. E. Russell: J. Chem. Phys. 49,5426

(1968)

67.25 C. Y. Ng: Adv. Chem. Phys. 82, 401 (1992)

67.26 D. Gerlich: Adv. Chem. Phys. 82, 1 (1992)

67.27 M. Tsuji: Techniques for the Study of Ion-Molecule

Reactions, ed. by J. M. Farrar, W. H. Saunders, Jr.

(Wiley-Interscience, New York 1988) p. 489

67.28 P. Tosi, O. Dmitriev, D. Bassi, O. Wick, D. Gerlich:

J. Chem. Phys. 100, 4300 (1994)

67.29 C.-L. Liao, C. Y. Ng: J. Chem. Phys. 84, 197 (1986)

67.30 P. B. Armentrout: Ann. Rev. Phys. Chem. 41,313

(1990)

67.31 Z. Dolejsek, M. Farnik, Z. Herman: Chem. Phys. Lett.

235, 99 (1995)

67.32 Y. H. Chiu, H. S. Fu, J. T. Huang, S. L. Anderson:

J. Chem. Phys. 101, 5410 (1994)

67.33 J. Zabka, O. Dutuit, Z. Dolejsek, J. Polach, Z. Her-

man: PCCP Phys. Chem. Chem. Phys. 2,781

(2000)

67.34 D. W. Chandler, P. L. Houston: J. Chem. Phys. 87,

1445 (1987)

67.35 P. L. Houston: Acct. Chem. Res. 28, 453 (1995)

67.36 A. G. Suits, R. E. Continetti (Eds.): Imaging in Chem-

ical Dynamics, Vol. 770 (American Chemical Society,

Washington DC 2001)

67.37 V. Dribinski, A. Ossadtchi, V. A. Mandelshtam,

H. Reisler: Rev. Sci. Instrum. 73, 2634 (2002)

67.38 A. Eppink, D. H. Parker: Rev. Sci. Instrum. 68,3477

(1997)

67.39 E. L. Reichert, G. Thurau, J. C. Weisshaar: J. Chem.

Phys. 117, 653 (2002)

Part E 67

995

Quantum O

Part F

Part F Quantum Optics

68 Light–Matter Interaction

Pierre Meystre, Tucson, USA

69 Absorption and Gain Spectra

Stig Stenholm, Stockholm, Sweden

70 Laser Principles

Peter W. Milonni, Los Alamos, USA

71 Types of Lasers

Richard C. Powell, Tuscon, USA

72 Nonlinear Optics

Alexander L. Gaeta, Ithaca, USA

Robert W. Boyd, Rochester, USA

73 Coherent Transients

Joseph H. Eberly, Rochester, USA

Carlos R. Stroud Jr., Rochester, USA

74 Multiphoton and Strong-Field Processes

Kenneth C. Kulander, Livermore, USA

Maciej Lewenstein, Barcelona, Spain

75 Cooling and Trapping

Juha Javanainen, Storrs, USA

76 Quantum Degenerate Gases

Juha Javanainen, Storrs, USA

77 De Broglie Optics

Carsten Henkel, Potsdam, Germany

Martin Wilkens, Potsdam, Germany

78 Quantized Field Effects

Matthias Freyberger, Ulm, Germany

Karl Vogel, Ulm, Germany

Wolfgang P. Schleich, Ulm, Germany

Robert F. O’Connell, Baton Rouge, USA

79 Entangled Atoms and Fields: Cavity QED

Dieter Meschede, Bonn, Germany

Axel Schenzle, München, Germany

80 Quantum Optical Tests of the Foundations

of Physics

Aephraim M. Steinberg, Toronto, Canada

Paul G. Kwiat, Urbana, USA

Raymond Y. Chiao, Berkeley, USA

81 Quantum Information

Sir Peter L. Knight, London, UK

Stefan Scheel, London, UK

997

Light–Matter I

68. Light–Matter Interaction

Optical physics is concerned with the dynamical

interactions of atoms and molecules with

electromagnetic to ﬁelds. Semiclassical theories,

which study the interaction of atoms with

classical ﬁelds, are often said to comprise

optical physics, while quantum optics treats the

interaction of atoms or molecules with quantized

electromagnetic ﬁelds. A signiﬁcant part of

optical physics and quantum optics is the study

of near-resonant atom-ﬁeld interactions, and

concentrates on nonperturbative dynamics, where

the effects of the optical ﬁelds have to be kept to

all orders. The atomic properties themselves are

assumed to be known.

The vast majority of problems in light-matter

interactions can be treated quite accurately within

semiclassical theories. However, an important

class of problems where this is not the case are

presented in Chapt. 78. While much of optical

physics and quantum optics ignores the effects of

the electromagnetic ﬁelds on the center-of-mass

motion of the atoms, important topics such as

atomic trapping and cooling (Chapt. 75)andde

Broglie optics (Chapt. 77) rely in an essential way

on such mechanical effects of light. The present

chapter deals with more “traditional” aspects of

optical physics, where these effects are ignored.

68.1 Multipole Expansion ............................ 997

68.1.1 Electric Dipole (E1) Interaction ..... 998

68.1.2 Electric Quadrupole (E2)

Interaction ............................... 998

68.1.3 Magnetic Dipole (M1) Interaction . 999

68.2 Lorentz Atom ...................................... 999

68.2.1 Complex Notation ..................... 999

68.2.2 Index of Refraction.................... 999

68.2.3 Beer’s Law ............................... 1000

68.2.4 Slowly-Varying Envelope

Approximation..........................1000

68.3 Two-Level Atoms ................................. 1000

68.3.1 Hamiltonian ............................. 1001

68.3.2 Rotating Wave Approximation ....1001

68.3.3 Rabi Frequency ......................... 1001

68.3.4 Dressed States .......................... 1002

68.3.5 Optical Bloch Equations ............. 1003

68.4 Relaxation Mechanisms........................1003

68.4.1 Relaxation Toward Unobserved

Levels ...................................... 1003

68.4.2 Relaxation Toward Levels

of Interest ................................1004

68.4.3 Optical Bloch Equations

with Decay ............................... 1004

68.4.4 Density Matrix Equations............ 1004

68.5 Rate Equation Approximation ............... 1005

68.5.1 Steady State ............................. 1005

68.5.2 Saturation ................................ 1005

68.5.3 Einstein A and B Coefﬁcients ......1005

68.6 Light Scattering................................... 1006

68.6.1 Rayleigh Scattering....................1006

68.6.2 Thomson Scattering ................... 1006

68.6.3 Resonant Scattering .................. 1006

References .................................................. 1007

68.1 Multipole Expansion

Consider a test charge q of mass m localized within an

atom and acted upon by an external electromagnetic ﬁeld

with electric ﬁeld E(r, t) and magnetic ﬁeld B(r, t).In

the multipole expansion formalism [68.1,2], the electric

and magnetic interaction energies between the charge

and the electromagnetic ﬁeld are

= V

(t) −q



ds· E(R+s, t), (68.1)

=−q



ds· v × B(R+s, t), (68.2)

respectively. The material of this chapter is discussed in

detail in a number of texts and review articles. We cite

such references rather than the original sources when-

ever possible. These energies correspond to the work

done by the electric and magnetic components of the

Lorentz force in ﬁrst moving the charge to a stationary

Part F 68

998 Part F Quantum Optics

origin of coordinates at a point R and then to a location

r relative to R. Here, V

(t) represents the energy of the

charge when located at the reference point R.Itmaybe

expressed in terms of the electrostatic potential φ(R, t)

as V

(t) =+q φ(R, t).

A Taylor series expansion of V

(t) and V

(t) about

r= 0 yields

(t) = V

(t) −q

∞



n=1



r ·

∂

∂ R



n−1

r · E(R, t),

(68.3)

(t) =−

∞



n=1

(n+1)!



r ·

∂

∂ R



n−1

 · B(R, t),

(68.4)

where  = r × p is the angular momentum of the

test charge relative to the coordinate origin R. Here,

use of the mechanical momentum p =m

r, instead of

the canonical momentum, neglects the electromagnetic

component of the momentum responsible for diamag-

netic effects.

In addition to the electromagnetic interaction, elec-

trons and nuclei are characterized by a spin magnetic

moment m

=(q /2m)g

s,wheres is the spin of the test

charge and g

its gyromagnetic factor, equal to 2.002 ...

for electrons. The factor q

/2m is the particle’s magne-

ton. The spin magnetic moment yields an additional term

to the magnetic energy V

, which becomes

(t) =−

∞



n=1



r ·

∂

∂ R



n−1



n +1



 +g



· B(R, t), (68.5)

where the orbital g-factor is g



= q/e (g



=−1foran

electron). For an ensemble {α} of charged particles q

in an atom, these expressions are to be summed over all

particles. Thus, the electric energy becomes

(t) =



φ(R, t) −



i=1



(α)E

(R, t)

−



i, j=1





(α)r

(α)



∂

∂R

(R, t) +···

≡ V

(t) +V

(t) +···, (68.6)

and the magnetic energy becomes

(t) =



i=1

(R, t)



[



(α)

(α) +g

(α)s

(α)

]

−



i, j=1

∂B

(R, t)

∂R







(α)

(α)r

(α)

(α)s

(α)



+···

≡ V

(t) +V

(t) +···. (68.7)

68.1.1 Electric Dipole (E1) Interaction

For optical ﬁelds whose wavelength is large compared

with the interacting atom, only the ﬁrst few terms in the

Taylor expansions of V

(t) and V

(t) need be retained.

The ﬁrst term, V

(t),ofV

(t) is the net charge of the

atom, which vanishes for neutral atoms. The second

term, V

(t), is the electric dipole interaction energy.

Introducing the electric dipole (E1) moment,

d =



r(α) , (68.8)

d =



rρ(r)r , (68.9)

for a charge distribution, this contribution to the interac-

tion energy may be re-expressed as

(t) =−d· E(R, t). (68.10)

The E1 term dominates most optical phenomena.

68.1.2 Electric Quadrupole (E2) Interaction

The V

(t) contribution to v

(t), describes electric

quadrupole (E2) interactions. In terms of the quadrupole

tensor

Q = 3



rρ(r)r

, (68.11)

(t) becomes

(t) =−



i, j=1

∂

∂R

(R, t). (68.12)

Part F 68.1

Light–Matter Interaction 68.2 Lorentz Atom 999

Alternatively, E2 interactions can be expressed in terms

of the traceless quadrupole tensor Q

(2)



rρ(r)

×(3r

−δ

). E2 interactions are typically weaker

than E1 interactions by a factor a

/λ,wherea

is the

Bohr radius and λ is the wavelength of the transition.

Since a

/λ is very small for optical transitions, E2

interactions are typically neglected in quantum optics.

68.1.3 Magnetic Dipole (M1) Interaction

The ﬁrst term in the multipole expansion the mag-

netic interaction (Sect. 68.6) is the magnetic dipole M1

interaction,

=−m· B(R, t), (68.13)

of a magnetic moment m in a magnetic ﬁeld, where

m =







[



(α)(α) +g

(α)s(α)

]

=−µ

(L+2S), (68.14)

and we have used the fact that for electrons g



=−1and

−2. The Bohr magneton µ

2mc

αea

(68.15)

where α is the ﬁne structure constant. Thus M1 interac-

tions tend to be smaller than E1 interactions by a factor

of order α/2. The connection between m and angular

momentum J is m = γ J, where γ is the gyromagnetic

ratio.

68.2 Lorentz Atom

The Lorentz atom consists of a classical electron har-

monically bound to a proton. It provides a framework

for understanding a number of elementary aspects of

the electric dipole interaction between a single atom and

light [68.3–7]. Assuming that the c.m. motion of the

atom is unaffected by the ﬁeld, and neglecting magnetic

effects, the equation of motion of the electron is



+ω



r =−

E(R, t),

(68.16)

where ω

is the electron’s natural oscillation frequency,

and Γ

represents a frictional decay rate that accounts

for the effects of radiative damping. For the classical

Lorentz atom

= 2ω

/3c , (68.17)

where r

= e

/4π

is the classical electron radius.

This damping arises physically from the radiation reac-

tion of the ﬁeld radiated by the atom on itself. In the

E1 approximation, the electric ﬁeld is evaluated at the

location R of the atomic c.m.

68.2.1 Complex Notation

The study of light-matter interactions is simpliﬁed by

the introduction of complex variables [68.8–11]. For

example, an electric ﬁeld

E(R, t) =



n,µ



cos(ω

t), (68.18)

where 

is the polarization vector of the ﬁeld Fourier

component at frequency ω

is expressed as

E(R, t) = E

(R, t) + E

−

(R, t), (68.19)

where the positive frequency part of the ﬁeld is

(R, t) =



n,µ



exp[i(k

· R−ω

t)] .

(68.20)

Due to the linearity of (68.16), it is sufﬁcient to study the

response of the Lorentz atom to a plane monochromatic

electric ﬁeld of frequency ω, complex amplitude E and

polarization . Introducing the complex dipole moment

d =−er = P exp[i(k· R−ωt)]+c.c.,

(68.21)

where P is in general complex for E real, and the com-

plex polarizability α(ω) via P = α(ω)E ,then

α(ω) =

−ω

−iγω

(68.22)

68.2.2 Index of Refraction

From the Maxwell wave equation



∇

−

∂

∂t



E(R, t) =



∂

P(R, t)

∂t

(68.23)

where P(R, t) is the electric polarization, given by the

electric dipole density of the medium P = Nd, N being

Part F 68.2

1000 Part F Quantum Optics

the atomic density, the plane wave dispersion relation is

(ω) , (68.24)

where the index of refraction n(ω) is

n(ω) =



1 +

Nα(ω)



. (68.25)

68.2.3 Beer’s Law

Since the polarizability α(ω) is normally complex, so

is the index of refraction. Its real part leads to disper-

sive effects, while its imaginary part leads to absorption.

Speciﬁcally, Re[n(ω)]−1 has the form of a standard dis-

persion curve, positive for ω −ω

< 0 and negative for

ω −ω

> 0, while Im[n(ω)]is a Lorentzian curve peaked

at ω = ω

. The intensity absorption coefﬁcient a(ω) is

a(ω) = 2Im

[

n(ω)

]

ω/c

(68.26)

2ω



1 +



m



iγω+



−ω





−ω



+γ



1/2

For atomic vapors, the corrections to the vacuum index

of refraction are normally small, so that the square root

in (68.26) can be expanded to ﬁrst order, giving

a(ω) =



N e





γω



−ω



+γ

. (68.27)

The intensity of a monochromatic ﬁeld propagating

along the z-direction through a gas of Lorentz atoms

is therefore attenuated according to Beer’s law given by

I(ω, z) = I(ω, 0) e

−a(ω)z

. (68.28)

If the index of refraction at a given frequency becomes

purely imaginary, no electromagnetic wave can prop-

agate inside the medium. This is the case for ﬁeld

frequencies smaller than the plasma frequency



m

. (68.29)

While the Lorentz atom model gives an adequate

description of absorption and dispersion in a weakly ex-

cited absorbing medium, it fails to predict the occurence

of important phenomena such as saturation and light

ampliﬁcation. This is because, in this model, the phase

of the induced atomic dipoles with respect to the inci-

dent ﬁeld is always such that the polarization ﬁeld adds

destructively to the incident ﬁeld. The description of

light ampliﬁcation requires a quantum treatment of the

medium, which gives a greater ﬂexibility to the possi-

ble relative phases between the incident and polarization

ﬁelds.

68.2.4 Slowly-Varying Envelope

Approximation

Light–matter interactions often involve quasi-mono-

chromatic ﬁelds for which the electric ﬁeld (taken to

propagate along the z-axis) can be expressed in the form

E(R, t) =

E

(R, t) e

i(kz−ωt)

+c.c., (68.30)

such that



∂E

∂t



 ω



∂E

∂z



 k



(68.31)

It is further consistent within this approximation to

assume that the polarization takes the form

P(R, t) =

P

(R, t) e

i(kz−ωt)

+c.c., (68.32)

with



∂P

∂t



 ω



(68.33)

Under these conditions, known as the slowly-varying

envelope approximation ([68.5]), Maxwell’s wave equa-

tion reduces to



∂

∂z

∂

∂t



(z, t) =−

2i 

(z, t). (68.34)

Hence, in the slowly-varying envelopes approximation

we ignore the backward propagation of the ﬁeld [68.12].

The slowly-varying amplitude and phase approxima-

tion is essentially the same, except that it expresses the

electric ﬁeld envelope in terms of a real amplitude and

phase.

68.3 Two-Level Atoms

A large number of optical phenomena can be understood

by considering the interaction between a quasi-

monochromatic ﬁeld of central frequency ω and

a two-level atom, which simulates a (dipole-allowed)

atomic transition [68.5, 7, 10, 11, 13–17]. This approx-

imation is well justiﬁed for near-resonant interactions;

Part F 68.3

Light–Matter Interaction 68.3 Two-Level Atoms 1001

i. e., ω ω

. The next three sections discuss the model

Hamiltonian for this system in the semiclassical approx-

imation where the electromagnetic ﬁeld can be described

classically. The formal results are then extended to the

case of a quantized ﬁeld, where the electric ﬁeld is

treated as an operator.

68.3.1 Hamiltonian

In the absence of dissipation mechanisms, the dipole in-

teraction between a quasi-monochromatic classical ﬁeld

and a two-level atom is

H =

|ee|+ ω

|gg|−d· E(R, t), (68.35)

where |e and |g label the upper and lower atomic

levels, of frequencies ω

and ω

, respectively, with

−ω

= ω

,andRis the location of the center of mass

of the atom. The electric dipole operator (68.8), couples

the excited and ground levels, and may be expressed as

d = 

(

|eg |+|ge|

)

, (68.36)

where 

is a unit vector in the direction of the dipole

and d the matrix element of the electric dipole operator

between the ground and excited state, which we take to

be real for simplicity. We also neglect the vector char-

acter of d and E(R, t) in the following, assuming, for

example, that both 

and  are parallel to x-axis. The

Hamiltonian (68.35) may then be expressed as

H =

|ee|+ ω

|gg|−d

(

|eg |+|ge|

)



(R, t) + E

−

(R, t)



, (68.37)

where we have generalized the notation of (68.20)in

an obvious way. One can introduce the pseudo-spin

operators

= (|ee|−|gg|)/2 ,

= s

†

−

=|eg| , (68.38)

and redeﬁne the zero of atomic energy to introduce the

commonly used form

H =

−d

(

−

)



(R, t) + E

−

(R, t)



(68.39)

68.3.2 Rotating Wave Approximation

Under the inﬂuence of a monochromatic electromag-

netic ﬁeld of frequency ω, atoms undergo transitions

between their lower and upper states by interacting

with either the positive or the negative frequency part

of the ﬁeld. The corresponding contributions to the

atomic dynamics oscillate at frequencies ω

−ω and

+ω, respectively, and their contributions to the prob-

ability amplitudes involve denominators containing this

same frequency dependence. For near-resonant atom-

ﬁeld interactions, the rapidly oscillating contributions

lead to small corrections, the ﬁrst-order one being the

Bloch–Siegert shift, whose value near resonance, is

ω  ω

[68.17].

δω

=−







4ω

(68.40)

to lowest-order in dE / ω. The neglect of these terms is

the Rotating Wave Approximation (RW A). Note that it

is normally inconsistent to regard an atom as a two-level

system and not to perform the RW A.IntheRW A,the

atomic system is described by the Hamiltonian

H =

−d



(R, t) +s

−

(R, t)



(68.41)

or, in a frame rotating at the frequency ω of the ﬁeld,

H =

∆s

−



E e

ik · R

+h.c.



, (68.42)

where ∆ =ω

−ω is the atom-light detuning. (Note that

the alternate deﬁnition δ = ω −ω

is frequently used in

the literature.) In the rest of this chapter, we consider

atoms placed at R= 0.

68.3.3 Rabi Frequency

The dynamics of the two-level atom is conveniently

expressed in terms of its density operator ρ, whose

evolution is given by the Schrödinger equation

dρ

=−

[H,ρ] , (68.43)

where ρ

=e|ρ|eand ρ

=g|ρ|gare the upper and

lower state populations P

and P

, respectively, while

the off-diagonal matrix elements ρ

=e|ρ|g=ρ



are called the atomic coherences, or simply coherences,

between levels |eand |g. These coherences play an es-

sential role in optical physics and quantum optics, since

they are proportional to the expectation value of the

electric dipole operator.

The evolution of P

(t) and P

(t) = 1 − P

(t),is

characterized by oscillations at the generalized Rabi

frequency

Ω =



Ω

+∆



1/2

, (68.44)

Part F 68.3

1002 Part F Quantum Optics

where the Rabi frequency Ω

is Ω

= dE / , (or

Ω

= dE ( 

· )/ when the vector character of the

electric ﬁeld and dipole moment are included). Specif-

ically, assuming that the atom is initially in its ground

state |g, the probability that it is in the excited state |e

at a subsequent time t is given by Rabi’s formula

(t) =

(

Ω

/Ω

)

sin

(Ωt/2). (68.45)

At resonance (∆ =0), the generalized Rabi frequency Ω

reduces to the Rabi frequency Ω

. (In addition to the

texts on quantum optics already cited, see also [68.18].)

68.3.4 Dressed States

Semiclassical Case

The atomic dynamics can alternatively be described in

terms of a dressed states basis instead of the bare states

|eand |g(see especially [68.17]). The dressed states |1

and |2 are eigenstates of the Hamiltonian (68.42), and,

by convention, the state |1 is the one with the greatest

energy. They are conveniently expressed in terms of the

bare states via the Stückelberg angle θ/2as

|1=sin θ|g+cos θ|e ,

|2=cos θ|g−sin θ|e ,

(68.46)

where sin(2θ) =−Ω

/Ω,cos(2θ) = ∆/Ω. The corres-

ponding eigenenergies are

Ω,

=−

Ω. (68.47)

These energies are illustrated in Fig. 68.1 as a function

of the ﬁeld frequency ω. The dressed levels repel each

other and form an anticrossing at resonance ω = ω

.As

the detuning ∆ varies from positive to negative values,

state |1 passes continuously from the excited state |e

to the bare ground state |g, with both bare states hav-

ing equal weights at resonance. The distances between

the perturbed levels and their asymptotes for |∆|Ω

represent the ac Stark shifts, or light shifts, of the atomic

states when coupled to the laser. From Fig. 68.1,theac

Stark shift of |g is positive for ∆ < 0 and negative for

∆ > 0, while the |estate shift is negative for ∆ < 0and

positive for ∆>0.

Quantized Field

The concept of dressed states can readily be general-

ized to a two-level atoms interacting with a single-mode

quantized ﬁeld in the dipole and rotating wave approx-

imations. The atom and its dipole interaction with the

ﬁeld are still described by the Hamiltonian (68.41), ex-

cept that the positive and negative frequency components

of the ﬁeld are now operators, and the free ﬁeld Hamil-

tonian must be included. The Hamiltonian of the total

atom-ﬁeld system becomes

H =

+ ω



†





a+a

†

−



(68.48)

where the creation and annihilation operators a

†

and a

obey the boson commutation relation



a, a

†



= 1

(Chapt. 6), and the coupling constant

g = d



2

(68.49)

is the vacuum Rabi frequency, with V being a pho-

ton normalization volume. This Hamiltonian deﬁnes the

Jaynes–Cummings model, [68.7, 19] which is discussed

in more detail in Chapt. 78.

The dressed states of the atom-ﬁeld system are the

eigenstates of the Jaynes–Cummings model. Since, in

the RW A, the dipole interaction only couples states of

same “excitation number”, e.g., |e, n and |g, n +1,

where |nis an eigenstate of the photon number operator,

†

a|n=n |n, with n an integer, the diagonalization

of the Jaynes–Cummings model reduces to that of the

semiclassical driven two-level atom in each of these

manifolds. Hence, the dressed states are

|1, n=sin θ

|g, n +1+cos θ

|e, n ,

|2, n=cos θ

|g, n +1−sin θ

|e, n , (68.50)

with

tan(2θ

) =−2g

√

n +1/∆ . (68.51)

∆

ⱍg冭

ⱍe冭

Fig. 68.1 Dressed levels of a two-level atom driven by

a classical monochromatic ﬁeld as a function of the de-

tuning ∆ = ω

−ω

Part F 68.3

Light–Matter Interaction 68.4 Relaxation Mechanisms 1003

(The factor of 2 difference between this and the semi-

classical case is due to the use of a running waves

quantization scheme, while the semiclassical discussion

was for standing waves.) The corresponding eigenener-

gies are

= (n +1)ω − R

= (n +1)ω + R

, (68.52)

where



∆

+4g

(n+1). (68.53)

Chapter 78 shows that by including the effects of

spontaneous emission, this picture yields a straightfor-

ward interpretation of a number of effects, including

the Burshtein–Mollow resonance ﬂuorescence spec-

trum. Dressed states also help to elucidate the interaction

between two-level atoms and quantized single-mode

ﬁelds, as occur for example in cavity QED ([68.19]

and Chapt. 79). Their generalization to the case of

moving atoms offers simple physical interpretations

of several aspects of laser cooling, see Chapt. 75

and [68.20].

68.3.5 Optical Bloch Equations

Introducing the density operator matrix elements ρ

a|ρ|b,wherea, b can be either e or g,aswellasthe

real quantities

U = ρ

iωt

+c.c.,

V = iρ

iωt

+c.c.,

W = ρ

−ρ

, (68.54)

the equations of motion for the density matrix elements

=i|ρ|j may be e xpressed, with (68.43), as

=−∆V ,

= ∆U +Ω

W ,

=−Ω

V . (68.55)

These are the optical Bloch equations, as discussed

extensively in [68.5, 17]. Physically, U describes the

component of the atomic coherence in phase with the

driving ﬁeld, V the component in quadrature with the

ﬁeld, and W the atomic inversion. The optical Bloch

equations have a simple geometrical interpretation of-

fered by thinking of U, V and W as the three components

of a vector called the Bloch vector U, whose equation

of motion is

= Ω × U ,

(68.56)

where Ω = (−Ω

, 0, ∆). Thus U precesses about Ω,

of length Ω, while conserving its length. The evolu-

tion of a two-level atom driven by a monochromatic

ﬁeld is thus mathematically equivalent to that of

a spin-

system in two magnetic ﬁelds B

and

cos ωt which are parallel to the z-andx-axis,

respectively, and whose amplitudes are such that

the Larmor spin precession frequencies around them

are ω and 2Ω

cos ωt, respectively. In optics, this

vectorial picture is often referred to as the Feynman–

Vernon–Hellwarth picture [68.21]. It is very useful in

discussing the coherent transient phenomena discussed

in Chapt. 73.

68.4 Relaxation Mechanisms

In addition to their coherent interaction with light

ﬁelds, atoms suffer incoherent relaxation mechanisms,

whose origin can be as diverse as elastic and inelas-

tic collisions and spontaneous emission. Collisional

broadening is discussed in Chapt. 59, while a QED

microscopic discussion of spontaneous emission is

described in Chapt. 78 in terms of reservoir the-

ory. One advantage of describing the atomic state in

terms of a density operator ρ is that the physical

interpretation of its elements allows us phenomeno-

logically to add various relaxation terms directly to its

elements.

68.4.1 Relaxation

Toward Unobserved Levels

If the relaxation mechanisms transfer populations or

atomic coherences toward uninteresting or unobserved

levels, their description can normally be given in terms

of a Schrödinger equation, but with a complex Hamilto-

nian. In contrast, if all levels involved in the relaxation

mechanism are observed, a more careful description, e.g.

in terms of a master equation, is required. Speciﬁcally,

in the case of relaxation to unobserved levels, the evo-

lution of the atomic density operator, restricted to the

Part F 68.4