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7.5 Radiative damping 135

than a derivation, but this approach does give useful physical insight.

7.5.1 The damping of a classical dipole

Damping by spontaneous emission can be introduced into the quantum
treatment of the two-level atom, in a physically reasonable way, by com-
parison with the damping of a classical system. To do this we first
review the damped harmonic oscillator and express the classical equa-
tions in a suitable form. For a harmonic oscillator of natural frequency
ω 0 , Newton’s second law leads to the equation of motion

..

x+β

.

x+ω^20 x=

F(t)

m

cosωt. (7.55)

The driving force has amplitudeF(t) that varies slowly compared to the
oscillation at the driving frequency. (The friction force isFfriction=−α

.

x
andβ=α/m,wheremis the mass.) To solve this we look for a solution
of the form
x=U(t)cosωt−V(t)sinωt. (7.56)

This anticipates that most of the time dependence of the solution is an
oscillation at frequencyω,andUis the component of the displacement
in-phase with the force, and the quadrature componentVhas a phase
lead^26 ofπ/2 with respect toFcosωt.^27 Substitution of eqn 7.56 into^26 A phase lead occurs whenV(t)>0,
since−sinωt =cos(ωt+π/2), and
V(t)<0 corresponds to a phase lag.

(^27) This method of considering the com-
ponentsUandVis equivalent to the
phasor description which is widely used
in the theory of a.c. circuits (made from
capacitors, inductors and resistors) to
represent the phase lag, or lead, be-
tweenthecurrentandanappliedvolt-
age of the formV 0 cosωt.
eqn 7.55 and equating terms that depend on sinωtand cosωtgives

.

U=(ω−ω 0 )V−

β

2

U,

.

V=−(ω−ω 0 )U−

β

2

V−

F(t)

2 mω

,

(7.57)

respectively. The amplitudesUandVchange in time as the amplitude
of the force changes, but we assume that these changes occur slowly
compared to the fast oscillation atω.Thisslowly-varying envelope ap-
proximationhas been used in the derivation of eqn 7.57, i.e.

..

Uand

..

Vhave
been neglected and

.

. VωV(see Allen and Eberly 1975). By setting

U=

.

V= 0 we find the form of the solution that is a good approxima-
tion when the amplitudes and the force change slowly compared to the
damping time of the system 1/β:

U=

ω 0 −ω

(ω−ω 0 )^2 +(β/2)^2

F

2 mω

, (7.58)

V=

−β/ 2

(ω−ω 0 )^2 +(β/2)^2

F

2 mω

. (7.59)

The approximationω^2 −ω^20 =(ω+ω 0 )(ω−ω 0 )  2 ω(ω−ω 0 )has
been used so these expressions are only valid close to resonance^28 —they

(^28) This is a very good approximation
for optical transitions since typically
β/ω 0
10 −^6. The assumption of small
damping is implicit in these equations
and therefore the resonance frequency
is very close toω 0.
give the wrong result forω0. The phase is found from tanφ=V/U
(see Exercise 7.7). The phase lies in the rangeφ=0to−πfor a force
of constant amplitude.^29
(^29) It is well known from the study of
the harmonic oscillator with damping
that the mechanical response lags be-
hind the driving. At low frequencies
the system closely follows the driving
force, but above the resonance, where
ω>ω 0 , the phase shift lies in the range
−π/ 2 <φ<−π.