8.4 Two-photon spectroscopy 165Example 8.3 Two-photon spectroscopy of the1s–2stransition in atomic
hydrogen
The 1s–2s two-photon transition in atomic hydrogen has an intrinsic
natural width of only 1 Hz because the 2s configuration is metastable.
An atom in the 2s energy level has a lifetime of 1/8 s, in the absence of
any external perturbations, since there are no p configurations of signifi-
cantly lower energy (see Fig. 2.2).^24 In contrast, the 2p configuration has^24 The microwave transition from the
2s^2 S 1 / 2 level down to the 2p^2 P 1 / 2
level has negligible spontaneous emis-
sion.
a lifetime of only 1.6 ns because of the strong Lyman-αtransition to the
ground state (with a wavelength of 121.5 nm in the vacuum ultraviolet).
This huge difference in lifetimes of the levels inn= 2 gives an indica-
tion of the relative strengths of single- and two-photon transitions. The
1s–2s transition has an intrinsic quality factor ofQ=10^15 ,calculated
from the transition frequency^34 cR∞divided by its natural width. To
excite this two-photon transition the experiments required ultraviolet
radiation at wavelengthλ= 243 nm.^2525 Such short-wavelength radiation can-
not be produced directly by tunable
dye lasers, but requires frequency dou-
bling of laser light at 486 nm by second-
harmonic generation in a nonlinear
crystal (a process that converts two
photons into one of higher energy).
Thus the frequency of the laser light
(at 486 nm) is exactly one-quarter of
the 1s–2s transition frequency (when
both the factors of 2 for the frequency-
doubling process and two-photon ab-
sorption are taken into account); thus
the laser light (at 486 nm) has a fre-
quency very close to that of the Balmer-
βline (n=2ton= 4) because the
energies are proportional to 1/n^2 in hy-
drogen.
Figure 8.11 shows a Doppler-free spectrum of the 1s–2s transition. A
resolution of 1 part in 10^15 has not yet been achieved because the various
mechanisms listed below limit the experimental line width.
(a)Transit time Two-photon absorption is a nonlinear process^26 with
(^26) In contrast, single-photon scattering
well below saturationis a linear process
proportional to the intensityI.Satu-
rated absorption spectroscopy is a non-
linear process.
a rate proportional to the square of the laser beam intensity,I^2 (see
Appendix E). Thus to give a high signal experimenters focus the
counter-propagating beams down to a small size in the sample, as
indicated in Fig. 8.8. For a beam diameter ofd=0.5mm transit-
time broadening gives a contribution to the line width of
∆ftt=
∆ωtt
2 π
u
d=
2200 m s−^1
5 × 10 −^4 m=4MHz, (8.21)whereuis a typical velocity for hydrogen atoms (see eqn 8.7).(b)Collision broadening (also called pressure broadening) Collisions
with other atoms, or molecules, in the gas perturb the atom (inter-
acting with the radiation) and lead to a broadening and frequency
shift of the observed spectral lines. This homogeneous broadening
mechanism causes an increase in the line width that depends on the
collision rate 1/τcoll,whereτcollis the average time between colli-
sions. In a simple treatment, the homogeneous width of a transition
whose natural width is Γ becomes ∆ωhomog=Γ+2/τcoll=Γ+2Nσv,
whereσis the collision cross-section andvis the mean relative ve-
locity, as described by Corney (2000)—see also Loudon (2000) or
Brooker (2003). The number density of the perturbing speciesNis
proportional to the pressure. For the 1s–2s transition frequency the
pressure broadening was measured to be 30 GHz/bar for hydrogen
atoms in a gas that is mostly hydrogen molecules (H 2 ), and this
gives a major contribution to the line width of the signal shown in
Fig. 8.11 of about 8 MHz (at the frequency of the ultraviolet radia-
tion near 243 nm).^27 Some further details are given in Exercise 8.7.
(^27) Collisions shift the 1s–2s transi-
tion frequency by −9 GHz/bar and
this pressure shift is more troublesome
for precision measurements than the
broadening of the line (see Boshieret al.
1989 and McIntyreet al.1989).
(c)Laser bandwidth The first two-photon experiments used pulsed
lasers to give high intensities and the laser bandwidth limited the