164 Doppler-free laser spectroscopy
When twice the laser frequencyωequals the atomic resonance frequency
2 ω=ω 12 all the atoms can absorb two photons; whereas in saturation
spectroscopy the Doppler-free signal comes only from those atoms with
zero velocity.
For the energy-level structure shown in Fig. 8.9(b) the atom decays
in two steps that each emit a single photon (following the two-photon
absorption). Some of these photons end up at the detector. A brief con-
sideration of this cascade process illustrates the distinction between a
two-photon process and two single-photon transitions. It would be pos-
sibletoexciteatomsfrom1to2usingtwolaser beams with frequencies
ωL1=ω 1 iandωL2=ωi 2 resonant with the two electric dipole transi-
tions, but this two-step excitation has a completely different nature to
the direct two-photon transition. The transfer of population via the in-
termediate levelioccurs at the rate determined by the rates of the two
individual steps, whereas the two-photon transition has a virtual inter-
mediate level with no transitory population ini. (Equation 8.20 shows
that to get a Doppler-free signal the two counter-propagating beams
must have the same frequency.) This distinction between single- and
two-photon transitions shows up clearly in the theory of these processes
(see Section E.2 of Appendix E) and it is worthwhile to summarise some
of the results here. Time-dependent perturbation theory gives the rate
of transitions to the upper level induced by an oscillating electric field
E 0 cosωt. The calculation of the rate of two-photon transitions requires
second-order time-dependent perturbation theory. Resonant enhance-
ment of the second-order process occurs when 2ω=ω 12 but this still
gives a rate which is small compared to an allowed single-photon tran-
sition. Therefore, to see any second-order effects, the first-order terms
must be far off resonance; the frequency detuning from the intermediate
levelω−ω 1 imust remain large (of the same order of magnitude asω 1 i
itself, as drawn in the Fig. 8.9(b)). Two-photon absorption has many
similarities with stimulated Raman scattering—a process of simultane-
ous absorption and stimulated emission of two photons via a virtual
intermediate level, as shown in Fig. 8.10 (see Appendix E).
2
1Fig. 8.10A stimulated Raman transi-
tion between levels 1 and 2, via a virtual
level. Leveliis not resonantly excited
in this coherent process.
Finally, although the difference between two sequential electric dipole
transitions (E1) and a two-photon transition has been strongly empha-
sised above, these processes do link the same levels. So from the E1
selection rules (∆l=±1 between levels of opposite parity) we deduce
the two-photon selection rules: ∆l=0,±2 and no change of parity, e.g.
s–s or s–d transitions.
Two-photon spectroscopy was first demonstrated on the 3s–4d transi-
tion of atomic sodium which has a line width dominated by the natural
width of the upper level. The 1s–2s transition in atomic hydrogen has
an extremely narrow two-photon resonance, and the line width observed
in experiments arises from the various broadening mechanisms that we
study in the next section.