The interaction of atoms

with radiation

7

7.1 Setting up the equations 123

7.2 The EinsteinB

coefficients 126

7.3 Interaction with

monochromatic radiation 127

7.4 Ramsey fringes 132

7.5 Radiative damping 134

7.6 The optical absorption

cross-section 138

7.7 The a.c. Stark effect or

light shift 144

7.8 Comment on

semiclassical theory 145

7.9 Conclusions 146

Further reading 147

Exercises 148

To describe the interaction of a two-level atom with radiation we shall
use asemiclassicaltreatment, i.e. the radiation is treated as a classical
electric field but we use quantum mechanics to treat the atom. We shall
calculate the effect of an oscillating electric field on the atom from first
principles and show that this is equivalent to the usual time-dependent
perturbation theory (TDPT) summarised by the golden rule (as men-
tioned in Section 2.2). The golden rule only gives the steady-state transi-
tion rate and therefore does not describe adequately spectroscopy exper-
iments with highly monochromatic radiation, e.g. radio-frequency radia-
tion, microwaves or laser light, in which the amplitudes of the quantum
states evolve coherently in time. In such experiments the damping time
may be less than the total measurement time so that the atoms never
reach the steady state.
From the theory of the interaction with radiation, we will be able to
find the conditions for which the equations reduce to a set of rate equa-
tions that describe the populations of the atomic energy levels (with a
steady-state solution). In particular, for an atom illuminated by broad-
band radiation, this approach allows us to make a connection with Ein-
stein’s treatment of radiation that was presented in Chapter 1; we shall
find the EinsteinBcoefficient in terms of the matrix element for the
transition. Then we can use the relation betweenA 21 andB 21 to cal-
culate the spontaneous decay rate of the upper level. Finally, we shall
study the roles of natural broadening and Doppler broadening in the ab-
sorption of radiation by atoms, and derive some results needed in later
chapters such as the a.c. Stark shift.

7.1 Setting up the equations

We start from the time-dependent Schr ̈odinger equation^11 The operators do not have hats so
H≡Hˆ,aspreviously.
i

∂Ψ

∂t

=HΨ. (7.1)

The Hamiltonian has two parts,

H=H 0 +HI(t). (7.2)

That part of the Hamiltonian that depends on time,HI(t), describes
the interaction with the oscillating electric field that perturbs the eigen-