128 The interaction of atoms with radiation
|c 2 (t)|^2 =Ω^2
W^2
sin^2(
Wt
2)
, (7.27)
where
W^2 =Ω^2 +(ω−ω 0 )^2. (7.28)
At resonanceω=ω 0 andW=Ω,so|c 2 (t)|^2 =sin^2(
Ωt
2)
. (7.29)
The population oscillates between the two levels. When Ωt=πall the
population has gone from level 1 into the upper state,|c 2 (t)|^2 =1,and
when Ωt=2πthe atom has returned to the lower state. This behaviour
is completely different from that of a two-level system governed by rate
equations where the populations tend to become equal as the excitation
rate increases and population inversion cannot occur. TheseRabi os-
cillationsbetween the two levels are readily observed in radio-frequency
spectroscopy, e.g. for transitions between Zeeman or hyperfine states.
Radio-frequency and microwave transitions have negligible spontaneous
emission so that, in most cases, the atoms evolve coherently.^8(^8) This is partly a consequence of the
dependence onω^3 in eqn 7.23, but
also because themagnetic dipoletransi-
tions have smaller matrix elements than
electric dipole transitions. For electric
dipole transitions in the optical region
spontaneous emission washes out the
Rabi oscillations on a time-scale of tens
of nanoseconds (τ=1/A 21 , assuming
that the predominant decay is from 2
to 1, and we estimatedA 12 above).
Nevertheless, experimenters have ob-
served coherent oscillations by driving
the transition with intense laser radi-
ation to give a high Rabi frequency
(Ωτ>1).
7.3.1 The concepts ofπ-pulses andπ/2-pulses
A pulse of resonant radiation that has a duration oftπ=π/Ω is called
aπ-pulse and from eqn 7.29 we see that Ωt=πresults in the complete
transfer of population from one state to the other, e.g. an atom initially
in| 1 〉ends up in| 2 〉after the pulse. This contrasts with illumination by
broadband radiation where the populations (per state) become equal as
the energy densityρ(ω) increases. More precisely, aπ-pulse swaps the(^9) This can be shown by solving states in a superposition: 9
eqns 7.25 (and 7.26) forω=ω 0.
c 1 | 1 〉+c 2 | 2 〉→−i{c 1 | 2 〉+c 2 | 1 〉}. (7.30)
This swap operation is sometimes also expressed as| 1 〉↔| 2 〉,but the
factor of−i is important in atom interferometry, as shown in Exer-
cise 7.3.
Interferometry experiments also useπ/2-pulses that have half the du-
ration of aπ-pulse (for the same Rabi frequency Ω). For an atom initially
in state| 1 〉,theπ/2-pulse puts its wavefunction into a superposition of
states| 1 〉and| 2 〉with equal amplitudes (see Exercise 7.3).
7.3.2 The Bloch vector and Bloch sphere
In this section we find the electric dipole moment induced on a atom by
radiation, and introduce a very powerful way of describing the behaviour
of two-level systems by the Bloch vector. We assume that the electric
field is alonĝex, as in eqn 7.12. The component of the dipole along this
direction is given by the expectation value−eDx(t)=−∫
Ψ†(t)exΨ(t)d^3 r. (7.31)