Atomic physics (2005)

146 The interaction of atoms with radiation

55The prompt emission of electrons after the light hits the surface can also be explained semiclassically.

face only emits electrons if ω > Φ. Semiclassical theory explains this observation if the least tightly bound energy level, or energy band, of electrons in the metal has a binding energy of Φ (cf. the ionization energy of atoms). Electrons come o the surface with a maximum kinetic energy of ω − Φ because the oscillating field resonantly drives transitions that have angular frequencies close to ω (see eqn 7.15), from a lower level in the metal to an upper unbound ‘level’; this explanation does not require quantisation of the light into photons, as often implied in elementary quantum physics. A common line of argument is that a purely classical theory cannot explain various aspects of the photoelectric e ect and therefore the light must be quantised. The above discussion shows that this e ect can be explained by quantisation of the atoms.55

However, there are phenomena that do require quantisation of the radiation field, e.g. in the fluorescence from a single trapped ion it is found that two photons have a lower probability of arriving at the detector together (within the short time period of a measurement) than predicted for a totally random source of photons. This spreading out of the photons, or anti-bunching, occurs because it takes a time to excite the ion again after spontaneous emission. Such correlation of photons, or anticorrelation in this case, goes beyond the semiclassical theory presented here. Quantitative calculations of photon statistics require a fully quantum theory called quantum optics. The book on this subject by Loudon (2000) contains more fascinating examples, and also gives rigorous derivations of many results used in this chapter.

56In the radio-frequency region the damping time can easily be longer than the time of measurement (even if it lasts several seconds). For allowed optical transitions the coherent evolution lasts for only a few nanoseconds, but it can be observed using short pulses of laser radiation. Optical experiments have also been carried out with exceptionally long-lived two-photon transitions (Demtr¨oder 1996).

7.9Conclusions

The treatment of the interaction of atoms with radiation that has been described in this chapter forms the foundation of spectroscopy, e.g. laser spectroscopy as described in Chapter 8 (and laser physics); it is also important in quantum optics. There are a variety of approaches to this subject and it is worthwhile summarising the particular route that has been followed here.

The introductory sections closely follow Loudon (2000), and Section 7.3 gave a terse account of the coherent evolution of a two-level system interacting with single-frequency radiation—the atom undergoes Rabi oscillations and we saw that the Bloch sphere gives a useful way of thinking about the e ect of sequences of π- and π/2-pulses of the atom. In Section 7.5 the introduction of damping terms in the equations was justified by analogy with a behaviour of a classical dipole oscillator and this led to the optical Bloch equations. These equations give a complete description of the system and show how the behaviour at short times where damping has a negligible e ect56 is connected to what happens at longer times (much greater than the damping time) when a steady state has been established. We found that for radiative damping the system settles down to a steady-state solution described by a set of rate equations for the populations of the levels—this turns out to be a general

feature for all broadening mechanisms.57 In particular, this means that the steady-state populations for an atom illuminated by monochromatic (laser) radiation can be related to the case of illumination by broadband radiation and the rate equations that Einstein wrote down. The theory of the interaction between radiation and matter can also be developed by working ‘backwards’ from the rate equations in Einstein’s treatment of broadband radiation to find the rate equations for the case of monochromatic radiation by making ‘reasonable’ assumptions about the atomic line shape, etc. This approach avoids perturbation theory and is commonly adopted in laser physics; however, it gives no information about coherent phenomena (Rabi oscillations, etc.).58

Section 7.6 introduced the concept of an absorption cross-section and its use in the calculation of the absorption of radiation propagating through a gas with a certain number density of atoms.59 The discussion of the cross-section provides a link between two di erent perspectives on the interaction of radiation with matter, namely (a) the e ect of the radiation on the individual atoms, and (b) the e ect of the atomic gas (medium) on the radiation, e.g. absorption.60 The saturation of absorption as characterised by the saturation intensity forms the basis for a method of Doppler-free spectroscopy described in Chapter 8. From viewpoint (a), saturation arises because there is a maximum rate at which an atom can scatter radiation and this result for individual atoms is important in the discussion of radiation forces in Chapter 9. The formalism developed in this chapter allowed a straightforward derivation of the a.c. Stark e ect on the atomic energy levels; this light shift is used in some of the methods of trapping and cooling atoms with laser radiation (also described in Chapter 9).

7.9Conclusions 147

57Both for mechanisms that are homogeneous like radiative broadening, e.g. collisions as shown in Loudon (2000), and also for inhomogeneous mechanisms such as Doppler broadening (Exercise 7.9).

58Using time-dependent perturbation theory to describe the underlying behaviour of the two-level atom and finding the rate equations from them gives a clear understanding of the conditions under which these rate equations for the populations are valid.

59The inverse of absorption is gain; this is a critical parameter in laser systems that is calculated in terms of an optical cross-section in a very similar way to absorption. The gain also exhibits saturation.

60An important objective of this chapter was to show that both of these viewpoints embody the same physics.

Further reading

Loudon’s book on quantum optics gives more rigorous derivations of many formulae in this chapter.61 Further properties of the optical Bloch equations are discussed by Cohen-Tannoudji et al. (1992) and Barnett and Radmore (1997). The treatment of the optical absorption crosssection of a gas closely resembles the calculation of the gain cross-section for a laser and further details can be found in books on laser physics including detailed discussion of broadening mechanisms, e.g. Davis (1996) and Corney (2000).

61I have used similar notation, except Γ for the full width at half maximum (FWHM), whereas Loudon uses the half width γ = Γ/2.

148 The interaction of atoms with radiation

Exercises

(7.1) Averaging over spatial orientations of the atom

(a) Light linearly polarized along the x-axis gives a dipole matrix element of X12 =2| r |1 cos φ sin θ. Show that the average over all solid angles gives a factor of 1/3, as in eqn 7.21.

(b)Either show explicitly that the same factor of 1/3 arises for light linearly polarized along the z-axis, E = E0ez cos ωt, or prove this by a general argument.

(7.2) Rabi oscillations

(a)Prove that eqns 7.25 lead to eqn 7.26 and that this second-order di erential equation has a solution consistent with eqn 7.27.

(b)Plot |c2 (t)|2 for the cases of ω − ω0 = 0, Ω and 3Ω.

(7.3) π- and π/2-pulses

(a)For zero detuning, ω = ω0, and the initial conditions c1 (0) = 1 and c2 (0) = 0, solve eqns 7.25 to find both c1 (t) and c2 (t).

(b)Prove that a π-pulse gives the operation in eqn 7.30.

(c)What is the overall e ect of two π-pulses acting on |1 ?

(d) Show that √a π/2-pulse gives |1 → {|1 − i|2 } / 2.

(e)What is the overall e ect of two π/2-pulses acting on |1 ? When state |2 experiences a phase shift of φ, between the two pulses, show

that the probabilities of ending up in states |1 and |2 are sin2(φ/2) and cos2(φ/2), respectively.

(f)Calculate the e ect of the three-pulse sequence π/2–π–π/2, with a phase shift of φ between the second and third pulses. (The operators can be written as 2 × 2 unitary matrices, although this is not really necessary for this simple case.)

Comment. Without the factors of −i the signals in the two output ports of the interferometer are not complementary. The fact that the identity operation is a 4π-pulse rather than 2π stems from

the isomorphism between the two-level atom and a spin-1/2 system.

(7.4) The steady-state excitation rate with radiative broadening

An alternative treatment of radiative decay simply introduces decay terms into eqn 7.25 to give

.		Ω		i(ω−ω0)t			Γ
ic1	= c2	2	e		+ i		2	c2		,	(7.94)
.		Ω		−i(ω−ω0)t		− i		Γ
ic2	= c1	2	e					2	c2 .		(7.95)

This is a phenomenological model, i.e. a guess that works. The integrating factor exp (Γt/2) allows eqn 7.95 to be written as

d

c2 exp

Γt

dt

2

Ω

iΓ

= −ic1

exp −i ω − ω0 +

t .

2

2

(a)

Show that for Ω = 0 eqn 7.95 predicts that

|c2 (t)|2 = |c2 (t = 0)|2 e−Γt .

(b)

For the initial

conditions c1 (0) = 1 and

c2 (0) = 0, integration of eqn 7.94 gives c1

1. For these conditions and weak excitation

(Ω Γ) show that, after a time which is long

compared to the radiative lifetime, level 2 has

a steady-state population given by

|c2|

2

Ω2/4

=

.

(ω − ω0)2 + Γ2/4

(7.5) Saturation of absorption

The 3s–3p resonance line of sodium has a wavelength of λ = 589 nm.

(a)Sodium atoms in a magnetic trap form a spherical cloud of diameter 1 mm. The Doppler shift and the Zeeman e ect of the field are both small compared to Γ. Calculate

the number of atoms that gives a transmission of e−1 = 0.37 for a weak resonant laser beam.

(b)Determine the absorption of a beam with intensity I = Isat.

Exercises for Chapter 7

149

(7.6) The properties of some transitions in hydrogen

Show that the angular frequency ω at which

The table gives the values of A21 for transitions

this amplitude is maximum is very close to ω0

from the n = 3 shell of hydrogen to lower levels.

for a narrow resonance.

(Spin and fine structure are ignored.)

(c)

Show that the phase is given by

tan φ =

V

=

βω

.

Transition

A21 (s−1)

U

ω2 − ω02

6

How does this phase vary as the angular fre-

2p–3s

6.3 × 108

quency ω increases from ω ω0 to ω ω0?

1s–3p

1.7 × 107

2s–3p

2.2 × 107

(d)

For frequencies close to the atomic resonance

2p–3d

6.5 × 10

(ω ω0), show that your expressions for U

and V can be written in an approximate form

that agrees with eqns 7.58 and 7.59 (that were

(a)

Draw an energy-level diagram for the n = 1, 2

derived using the slowly-varying envelope ap-

and 3 shells in hydrogen that shows the al-

proximation in which the amplitude of the

lowed electric dipole transitions between the

driving force may change slowly over time).

orbital

angular

momentum

levels.

(Neglect

(e)

Show that in the steady state the power P ab-

transitions for which n does not change, e.g.

sorbed by the electron is a Lorentzian function

2s–2p.)

of ω:

1

(b)

Calculate the lifetimes of the 3s, 3p and 3d

P (ω − ω0)2 + (β/2)2 .

configurations.

What fraction of atoms that

start in 3p end up in the 2s configuration?

(7.8)

Oscillator strength

(c)

An electron in the 2p configuration has a life-

This question

shows the

usefulness

of a

dimen-

time of only 1.6 ns. Why is this shorter than

sionless parameter called the absorption oscillator

for the 3p configuration?

strength, denoted by f12.

(d)

Calculate the radial matrix elements D12

in

(a)

Show that for the cross-section in eqn 7.79 we

units of a0 for the transitions in the table, and

have

for 1s–2p.

∞

(e)

Calculate I

for the 2p–3s and 1s–3p transi-

sat

−∞ σ (ω) dω = 2π

2

r0c f12

,

(7.96)

tions.

(7.7) The classical model of atomic absorption

where r

= 2.8

×

10−15 m

and

f

=

20

12

(a)

A simple classical model of absorption as-

2meωD12/(3 ).

sumes that an electron (in an atom) behaves

(b)

From the simple model of the atom as an os-

like a damped simple harmonic oscillator of

cillating electron in Exercise 7.7, find the clas-

charge −e and mass me driven by the oscil-

sical absorption cross-section σcl (ω) in terms

lating electric field of the radiation: E0 cos ωt.

of β, ω0 and fundamental constants.

The electron’s equation of motion has the form

(c)

Without the driving electric field, the oscilla-

of eqn 7.55 with a driving force of constant

tor undergoes damped harmonic motion x =

amplitude F0 = −eE0. Find a solution of this

x0e−βt/2 cos (ω t

−

ϕ). The power radiated by

equation in the form x =

U

cos ωt

− V

sin ωt

an oscillating dipole leads to a decay rate given

(U and V are not functions of time in the case

by eqn 1.23 (from classical electromagnetism).

considered here).

Determine β.

(b)

Show that the displacement of the electron has

(d)

Show that σcl (ω) integrated over all frequen-

an amplitude

F0/m

cies gives 2π2r0c.

U

2

+ V

2

Comment. This classical

value is the maxi-

=

0

(ω2

−

ω02)2

+ (βω)2

mum value for any transition, so f12 1. The

2

−1/2

absorption oscillator strength is a fraction of

F0

2

β

(ω − ω0)

+

.

the integrated cross-section associated with a

2mω

4

given transition.

150 The interaction of atoms with radiation

(e) Calculate f12 for the 3s–3p transition of

Comment. An electric dipole does not radiate uni-

sodium. (A21 = Γ = 2π × 107 s−1.)

formly in all directions but this does not matter

(f) Calculate

the absorption oscillator

strength

here; only a tiny fraction of the spontaneous emis-

sion goes along the direction of the incident beam.

for the 1s–2p and 1s–3p transitions in hydro-

For example, in an experiment to measure the at-

gen using the data from Exercise 7.6(d).

tenuation of a laser beam as it passes through a

(7.9)

Doppler broadening

gas cell, a negligible fraction of the light scattered

The Maxwell–Boltzmann distribution of the veloc-

out of the beam falls on the photodetector that

ities in a gas is a Gaussian function f (v), as de-

measures the power after the sample.

fined in eqn 8.3.

Explain why for excitation by

(7.12)

The d.c. Stark e ect

monochromatic radiation of angular frequency ω

This exercise goes through a treatment of the d.c.

the population in the upper level is given by

Stark e ect for comparison with the a.c. Stark ef-

e

2

2

fect.

2

X12

2

|c2 (t)|

=

|E (ω)|

2

(a)

An atom with two levels of energies E2 > E1,

∞

sin2 {(ω − ω0 + kv)t/2}

f (v) dv .

and a separation of = E2

−

E1 is placed in

× −∞

(ω − ω0 + kv)

2

a static electric field. Show that the Hamilto-

Assuming that the sinc2 in the integrand acts like a

nian for the system has the form

Dirac delta function (as explained in Section 7.2),

/2

V

show that |c2|2 is proportional to gD (ω) in eqn 8.4.

H = V

/2 ,

Comment.

Doppler broadening washes out the

−

Rabi oscillations because their frequency depends

where the matrix element for the perturbation

V is proportional to the magnitude of the elec-

on the velocity, giving an equation similar to that

tric field. Find the energy eigenvalues. The

for broadband radiation. For all broadening mech-

two levels move apart as shown in Fig. 7.9—

anisms Rabi oscillations, and other coherent phe-

this is a general feature of systems where a

nomena, are only seen on time-scales shorter than

perturbation mixes the wavefunctions.

the reciprocal of the line width.

(7.10)

An example of the use of Fourier transforms

(b)

Show that ‘weak’ fields produce a quadratic

Stark e ect on

the atom,

equivalent to the

Show that an oscillator whose amplitude decays

usual second-order perturbation-theory ex-

exponentially according to x0e−βt/2 cos (ωt) radi-

ates with a Lorentzian power spectrum.

pression for a perturbation HI:

(7.11)

The balance between absorption and spontaneous

2

∆E1

=

−

| 2| HI |1 |

.

emission

E2

−

E1

Explain why absorption and the population in the

A similar expression can be found for the en-

upper level are related by

ergy shift ∆E2 of the other level (in the oppo-

κ (ω, I) I = N2A21 ω = N ρ22A21 ω .

(7.97)

site direction).

Show that this is consistent with eqns 7.87 and

(c)

Estimate the Stark shift for the ground state

of a sodium atom in a field of 106 V m−1 (e.g.

7.69 for κ(ω, I) and ρ22, respectively.

104 V between plates 1 cm apart).

Web site:

http://www.physics.ox.ac.uk/users/foot

This site has answers to some of the exercises, corrections and other supplementary information.

Doppler-free laser

spectroscopy	8

Doppler broadening is usually the dominant contribution to the observed width of lines in atomic spectra, at room temperature. The techniques of Doppler-free laser spectroscopy overcome this limitation to give much higher resolution than, for example, a Fabry–Perot ´etalon analyzing the light from a discharge lamp, as shown in Fig. 1.7(a). This chapter describes three examples that illustrate the principles of Doppler-free techniques: the crossed-beam method, saturated absorption spectroscopy and two-photon spectroscopy. To use these high resolution techniques for precision measurements of atomic transition frequencies the laser frequency must be determined accurately. Thus the calibration is a crucial part of laser spectroscopy experiments, as discussed at the end of this chapter. Since it is important to understand the problem before looking at the solution, the chapter starts with an outline of Doppler broadening of spectral lines in gases.

8.1	Doppler broadening of
	spectral lines	151
8.2	The crossed-beam
	method	153
8.3	Saturated absorption
	spectroscopy	155

8.4Two-photon spectroscopy 163

8.5Calibration in laser

spectroscopy	168
Further reading	175
Exercises	175

8.1Doppler broadening of spectral lines

The relationship between the angular frequency ω of radiation in the laboratory frame of reference and the angular frequency seen in a frame of reference moving at velocity v, as shown in Fig. 8.1, is

ω = ω	−	kv ,	(8.1)

where the wavevector of the radiation has magnitude k = ω/c = 2π/λ. It is the component of the velocity along k that leads to the Doppler e ect and here it has been assumed that k · v = kv.1

1If necessary, this and other equations in this chapter could be generalised by replacing kv with the scalar product k · v.

Atom

Fig. 8.1 The Doppler e ect on the observed frequency of radiation. Radiation that has an angular frequency of ω in the laboratory frame of reference has the frequencies indicated in a reference frame moving with a speed v, e.g. the rest frame of an atom. Only the component of the velocity along the wavevector k contributes to the firstorder Doppler shift.

= ln 2 .

4Note that

152 Doppler-free laser spectroscopy

2Section 8.3 describes what happens when the atoms absorb a range of frequencies, given by the homogeneous width, in addition to any Doppler broadening. Absorption is considered here because of its relevance to laser spectroscopy, but Doppler broadening of an emission line arises in the same way—atoms emit at ω0 in their rest frame and we see a frequency shift in the laboratory.

3This can easily be shown by di erentiating the Maxwell speed distribution which is proportional to v2 times the velocity distribution, see Table 8.1.

−∞∞ g (ω) dω = 1.

5The simple estimate of the FWHM as2u/c that leads to eqn 6.38 turns out to be quite accurate.

In this section, we consider the Doppler e ect on the absorption by a gas where each atom absorbs radiation at frequency ω0 in its rest frame, i.e. when ω = ω0.2 Thus atoms moving with velocity v absorb radiation

when δ = ω − ω0 = kv, or equivalently
	δ	=	v	.	(8.2)
	ω0
			c

In a gas the fraction of atoms with velocity in the range v to v + dv is

				M v2	1				v2
	f (v) dv =	M	exp −					exp −		dv . (8.3)
					dv ≡	u√
		π2kBT		2kBT					u2
							π

Here u = 2kBT /M is the most probable speed for atoms of mass M

at temperature T .3

Relating v to the frequency via eqn 8.2, we find that

the absorption has the Gaussian line shape function4

c

c2

ω − ω0

2

g

(ω) =

exp

.

(8.4)

−u2

D

uω0√π

ω0

The maximum value occurs at ω = ω0 and the function falls to half its maximum value at ω − ω0 = δ1/2, where

c δ1/2 2

u ω0

(8.5)

The Doppler-broadened line has a full width at half maximum (FWHM) of ∆ωD = 2δ1/2 given by5

∆ωD

√

u

u

= 2

ln 2

1.7

.

(8.6)

ω0

c

c

Kinetic theory gives the most probable speed in a gas as

u = 2230 m s−1 ×

T

1 a.m.u.

(8.7)

×

.

300 K

M

Table 8.1 The characteristic velocities in a gas with a Maxwellian distribution of speeds and in an e usive atomic beam; u = 2kBT /M , where T is the temperature and M is the mass. The extra factor of v in the distribution for a beam, as compared to that of a gas, arises from the way that atoms e use through a small hole of area A. Atoms with speed v are incident on a surface of area A at a rate of N (v) vA/4, where N (v) is the number density of atoms with speeds in the range v to v + dv— faster atoms are more likely to pass through the hole. Integration over v leads to the well-known kinetic theory result N vA/4 for the flux that arrives at the surface, where N is the total number density. The mean speed v has a value between the most probable and the root-mean-square velocities.

	Gas				Beam
Distribution	v2 exp −v2/u2				v3 exp		−v2/u2
	u
Most probable v						3/2 u
					√
Root-mean-square velocity		3/2 u					2 u

8.2 The crossed-beam method 153

In the formula, the atomic mass M must be expressed in atom mass units, e.g. M = 1 a.m.u. for atomic hydrogen. Numerical values of u are given below for hydrogen and a vapour of caesium, both at a temperature

of T = 300 K.6							6In this table ∆fD = 1.7u/λ.
		M (a.m.u.)	u (m s−1)	∆ωD/ω0		∆fD (GHz), for 600 nm
					−5
	H	1	2230	1	× 10 6	6
	Cs	133	200	1	× 10−	0.5

The values given for the fractional width ∆ωD/ω0 show that heavy elements have an order of magnitude smaller Doppler width than hydrogen. The Doppler shift of the frequency ∆fD is also given for a wavelength of 600 nm. (This wavelength does not correspond to actual transitions.7) These calculations show that Doppler broadening limits optical spectroscopy to a resolution of 106 even for heavy elements.8

The Doppler e ect on the absorption of a gas is an example of an inhomogeneous broadening mechanism; each atom interacts with the radiation in a di erent way because the frequency detuning, and hence absorption and emission, depend on the velocity of the individual atom. In contrast, the radiative broadening by spontaneous decay of the excited level gives the same natural width for all atoms of the same species in a gas—this is a homogeneous broadening mechanism.9 The di erence between homogeneous and inhomogeneous broadening is crucially important in laser physics and an extensive discussion and further examples can be found in Davis (1996) and Corney (2000).10

8.2The crossed-beam method

Figure 8.2 shows a simple way to reduce the Doppler e ect on a transition. The laser beam intersects the atomic beam at right angles. A thin vertical slit collimates the atomic beam to give a small angular spread α. This gives a spread in the component of the atomic velocity along the direction of the light of approximately αvbeam. Atoms in the beam have slightly higher characteristic velocities than in a gas at the same temperature, as shown in Table 8.1, because faster atoms have a higher probability of e using out of the oven. Collimation reduces the Doppler broadening to

∆f	αvbeam	α∆fD ,	(8.8)
	λ

7The Doppler widths of optical transitions in other elements normally lie between the values for H and Cs. A useful way to remember the correct order of magnitude is as follows. The speed of sound in air is 330 m s−1 (at 0 ◦C), slightly less than the speed of the air molecules. The speed of sound divided by the speed of light equals 10−6. Multiplication by a factor of 2 converts the half-width to a FWHM of ∆ωD/ω0 2 ×10−6, which gives a reasonable estimate of the fractional Doppler shift of medium-heavy elements.

8The resolving power of a Fabry–Perot ´etalon can easily exceed 106 (Brooker 2003), so that normally the instrumental width does not limit the resolution, in the visible region.

9The di erent characteristics of the two types of broadening mechanism are discussed further in this chapter, e.g. see Fig. 8.3.

10The treatment of the saturation of gain in di erent classes of laser system is closely related to the discussion of saturation of absorption, both in principle and also in the historical development of these subjects.

where ∆fD is the Doppler width of a gas at the same temperature as the beam.11

Example 8.1 Calculation of the collimation angle for a beam of sodium that gives a residual Doppler broadening comparable with the natural width ∆fN = 10 MHz (for the resonance transition at λ = 589 nm)

11A numerical factor of 0.7 1.2/1.7 has been dropped. To obtain a precise formula we would have to consider the velocity distribution in the beam and its collimation—usually the exit slit of the oven and the collimation slit have comparable widths.

154 Doppler-free laser spectroscopy

(a)	Oven	Detector

Laser

beam

Slit

Laser

Atomic

beam

	(b)						Laser
Fig. 8.2 Laser spectroscopy of a colli-							beam
mated atomic beam. The component	Oven			Slit			Atoms
of atomic velocity along the laser beam
has a small spread αvbeam, where α is
the collimation angle shown on the plan
view in (b).

12From eqn 7.50, ∆f 1/∆t.

Sodium vapour at 1000 K has a Doppler width of ∆fD = 2.5 GHz and the most probably velocity in the beam is vbeam 1000 m s−1. Thus a suitable collimation angle for a beam of sodium e using from an oven at this temperature is

α =	∆fN	=	10		= 4 × 10−3 rad .	(8.9)
	∆fD		2500

This angle corresponds to a slit 1 mm wide positioned 0.25 m from the oven. Collimation of the beam to a smaller angular spread would just throw away more of the atomic flux to give a weaker signal without reducing the observed line width.

In this experiment, atoms interact with the light for a time ∆t d/vbeam, where d is the laser beam diameter. The finite interaction time leads to a spread in frequencies called transit-time broadening.12 For a laser beam of diameter 1 mm we find

∆ftt =	vbeam	=	1000	1 MHz .	(8.10)
	d		10−3

Thus this broadening mechanism does not have a significant e ect in this experiment, compared to the natural width of the optical transition. (Transit time is an important consideration for the radio-frequency measurements with atomic beams described in Section 6.4.2.) Collision broadening has a negligible e ect in the experiment shown in Fig 8.2 because of the low density of atoms in the atomic beam and also in the background gas in the vacuum chamber. An atomic-beam apparatus must have a high vacuum since even a glancing collision with a

8.3 Saturated absorption spectroscopy 155

background gas molecule deflects an atom out of the highly-collimated beam.

Example 8.2 Figure 6.12 in the chapter on hyperfine structure showed a spectrum of tin (Sn) obtained by the crossed-beam technique that illustrated isotope and hyperfine structure. Comparison with the Dopplerbroadened spectrum emitted by a cadmium lamp clearly showed the advantages of the crossed-beam technique.13

Experimenters used highly-monochromatic light sources to demonstrate the principle of the crossed-beam method before the advent of lasers, but the two other techniques described in this chapter rely on the high-intensity and narrow-frequency bandwidth of laser light.

13The spacings between the lines from di erent isotopes does not depend on the angle between the laser beam and the atomic beam, but for absolute measurements of transition frequencies the angle must be accurately set to 90◦.

8.3Saturated absorption spectroscopy

We derived the line shape for Doppler broadening in Section 8.1 on the assumption that an atom at rest absorbs radiation exactly at ω0. In reality the atoms absorb radiation over a range of frequencies given by the homogeneous width of the transition, e.g. the line width Γ caused by radiative broadening. In this section we shall reconsider absorption of monochromatic radiation in a way that includes homogeneous broadening together with the inhomogeneous broadening caused by the atom’s motion. This approach leads naturally into a discussion of saturated absorption spectroscopy.

We consider a laser beam of intensity I (ω) that travels through a sample of atoms, as shown in Fig. 7.4. In this chapter we consider the atoms as moving, whereas previously they were taken to be stationary.14 Atoms with velocities in the velocity class v to v + dv see radiation with an e ective frequency of ω − kv in their rest frame, and for those atoms the absorption cross-section is σ(ω − kv), defined in eqn 7.76. The number density of atoms in this velocity class is N (v) = N f (v), where N is the total number density of the gas (in units of atoms m−3) and the distribution f (v) is given in eqn 8.3. Integration of the contributions from all the velocity classes gives the absorption coe cient as

κ (ω) = N (v)

=g2 π2c2

g1 ω02

=g2 π2c2

g1 ω02

σ (ω − kv) dv
A21	×	N (v) gH (ω − kv) dv						(8.11)
	× N						(2π)
A21			f (v)				Γ/		dv .
				(ω	−	ω0	kv)2	+ Γ2/4
							−

The integral is the convolution of the Lorentzian function gH (ω − kv) and the Gaussian function f (v).15 Except at very low temperatures the homogenous width is much less than the Doppler broadening, Γ ∆ωD, so that the Lorentzian is sharply peaked and acts like a delta function

14At room temperature, the Doppler width usually exceeds natural and other homogeneous broadening mechanisms. Very cold atomic vapours in which the Doppler shifts are smaller than the natural width of allowed transitions can be prepared by the laser cooling techniques described in Chapter 9.

15In general, the convolution leads to a Voigt function that needs to be calculated numerically (Corney 2000 and Loudon 2000).