7.5 Radiative damping 1377.5.2 The optical Bloch equations
A two-level atom has an energy proportional to the excited-state pop-
ulation,E =ρ 22 ω 0. By analogy with eqn 7.60 for the energy of a
classical oscillator, we introduce a damping term into eqn 7.44 to give
.
ρ 22 =−Γρ 22 +Ω
2
v. (7.66)In the absence of the driving term (Ω = 0) this gives exponential decay of
the population in level 2, i.e.ρ 22 (t)=ρ 22 (0) exp (−Γt). In this analogy,
between the quantum system and a classical oscillator, Γ corresponds to
β. From eqns 7.57 we see that the coherencesuandvhave a damping
factor of Γ/2 and eqns 7.46 become theoptical Bloch equations^34
(^34) The main purpose of the rather
lengthy discussion of the classical case
was to highlight the correspondence be-
tweenu, vandU, Vto make this step
seem reasonable. An auxiliary feature
of this approach is to remind the reader
of the classical electron oscillator model
of absorption and dispersion (which is
u.=δv−Γ important in atomic physics).
2
u,.
v=−δu+Ωw−Γ
2
v,.
w=−Ωv−Γ(w−1).(7.67)
For Ω = 0 the population differencew→1. These optical Bloch equa-
tions describe the excitation of a two-level atom by radiation close to
resonance for a transition that decays by spontaneous emission. There
is not room here to explore all of the features of these equations and
their many diverse and interesting applications; we shall concentrate
on the steady-state solution that is established at times which are long
compared to the lifetime of the upper level (tΓ−^1 ), namely^35
(^35) The steady-state solution is obtained
by settingu.=.v=w.= 0 in eqns 7.67
to give three simultaneous equations.
u
v
w
=^1
δ^2 +Ω^2 /2+Γ^2 / 4
Ωδ
ΩΓ/ 2
δ^2 +Γ^2 / 4
. (7.68)
These show that a strong driving field (Ω→∞) tends to equalise the
populations, i.e.w→0. Equivalently, the upper level has a steady-state
population of
ρ 22 =1 −w
2=
Ω^2 / 4
δ^2 +Ω^2 /2+Γ^2 / 4, (7.69)
andρ 22 → 1 /2 as the intensity increases. This key result is used in
Chapter 9 on radiation forces.
In the above, the optical Bloch equations have been justified by anal-
ogy with a damped classical oscillator but they also closely resemble the
Bloch equations that describe the behaviour of a spin-1/2 particle in a
combination of static and oscillating magnetic fields.^36 The reader fa-
(^36) The Zeeman effect leads to a split-
ting between states withms=± 1 / 2
to give a two-level system, and the os-
cillating magnetic field drivesmagnetic
dipoletransitions between the levels. In
atomic physics such transitions occur
between Zeeman states and hyperfine
levels (Chapter 6).
miliar with magnetic resonance techniques may find it useful to make an
analogy with that historically important case.^37 For times much shorter
(^37) The Bloch equations were well known
from magnetic resonance techniques
before lasers allowed the observation
of coherent phenomena in optical tran-
sitions. Radio-frequency transitions
have negligible spontaneous emission
and the magnetic dipole of the whole
sample decays by other mechanisms.
Where the optical Bloch equations
(eqns 7.67) have decay constants of Γ
and Γ/2 for the population and co-
herences, respectively, the Bloch equa-
tions have 1/T 1 and 1/T 2. The de-
cay rates 1/T 1 and 1/T 2 in magnetic
resonance techniques are expressed in
terms ofT 1 andT 2 , the longitudinal
and transverse relaxation times, respec-
tively. Under some conditions the two
relaxation times are similar, but in
other casesT 2 T 1. T 1 describes
the relaxation of the component of the
magnetic moment parallel to the ap-
plied fieldBwhich requires exchange
of energy (e.g. with the phonons in a
solid).T 2 arises from the dephasing of
individual magnetic moments (spins) so
that the magnetisation of the sample
perpendicular toBdecays. For further
details see condensed matter texts, e.g.
Kittel (2004).
than any damping or relaxation time, the two-level atom and spin-1/2
system behave in the same way, i.e. they have a coherent evolution such
asπ-andπ/2-pulses, etc. A steady-state solution of the optical Bloch
equations has been presented (and nothing has been said about the dif-
ferent result for a spin-1/2 system).