Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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Parity Nonconserving Effects in Atoms 29.3 Many-Body Perturbation Theory 451

29.2 PNC in Cesium

We concentrate on PNC in cesium because it is the sim-

plest atom in which the effect has been measured. This is

because it consists of a single electron outside a closed

xenonlike core which is relatively unpolarizable. This

should be contrasted with, for example, thallium. While

thallium nominally also consists of one 6p

1/2

electron

outside a closed core, part of that core is a ﬁlled 6s

1/2

subshell. It is quite easy to polarize the outer subshell,

so that one really has three electrons outside a closed

core. This leads to distinctly poorer convergence prop-

erties of many-body perturbation theory, the theoretical

method used for these calculations, and consequently

less accurate atomic theory predictions. Similar consid-

erations apply to lead and bismuth. The PNC transition

in cesium that has been studied is 6s

1/2

→ 7s

1/2

,and

accurate measurements have been made in Paris [29.2]

and Boulder [29.3, 4]. Cesium is a 55 electron atom

with a nucleus consisting of 78 neutrons and 55 protons

with nuclear spin I = 7/2. The total angular momen-

tum of atomic s-states is then F = 3orF = 4. Both

of the transitions, 6s

1/2

(F = 4) → 7s

1/2

(F = 3) and

1/2

(F = 3) → 7s

1/2

(F = 4), have been measured, al-

lowing the isolation of PNC effects that depend on the

spin of the nucleus. The bulk of atomic PNC comes

from the timelike contribution of the (V

) exchange

discussed in the previous section, and can be described

by the effective Hamiltonian

√

nuc

(r)γ

. (29.5)

Here ρ

nuc

(r) is a weighted average of the neutron and

proton distributions in the nucleus, which leads to nu-

clear structure uncertainties that will be discussed below.

Using this Hamiltonian, a large scale calculation [29.10]

leads to the prediction for the nuclear-spin-independent

part of the PNC transition

PNC

=−0.905(9) ×10

−11

i|e|a

(−Q

/N).

(29.6)

Here the unknown Q

has been factored out and di-

vided by its approximate value −N. This result is

in good agreement with independent MBPT calcula-

tions of the Novosibirsk [29.12] and Göteborg [29.13]

groups. When this is compared with the experimental

measurement [29.3,4]

exp

PNC

=−0.8374(67) ×10

−11

i|e|a

(29.7)

there results a prediction for Q

=−72.17(0.58)[0.72], (29.8)

where the ﬁrst error is experimental and the second the-

oretical. The spacelike part of (V

) exchange and the

timelike part of (A

) exchange are negligible, but the

spacelike part of the latter gives a nuclear spin-dependent

effect that will be discussed below. Also discussed below

is an interesting nuclear physics source of PNC known as

the anapole moment [29.14], which arises from photon

exchange with weak radiative corrections on the nuclear

vertex. This effect enters at the several percent level, but

in a way that can be subtracted out as will be described

below.

29.3 Many-Body Perturbation Theory

While there are a variety of ways of solving the

many-electron Schrödinger equation, the most accurate

treatments of cesium PNC are based on many-body per-

turbation theory (MBPT). In MBPT, the Hamiltonian is

broken up into H = H

,where



i=1



· p

+β

m +U(r

)



(29.9)

and



−r

−



i=1



Zα

+U(r

)



(29.10)

The starting potential U(r) is generally chosen to be

a frozen core Hartree–Fock (HF) potential. This model

of the atom is rather inaccurate: valence removal ener-

gies disagree with experiment by about 10%, and matrix

elements of the hyperﬁne operator by about 40%. Thus

it is essential for accurate calculations to include the

effects of V

as fully as possible. MBPT proceeds by

expanding the many-body wave function and the energy

in powers of V

. While going to second order in V

im-

proves agreement with experiment to the percent level

for energies and the few percent level for matrix ele-

ments [29.15], more powerful methods that sum inﬁnite

classes of MBPT contributions are needed for higher ac-

Part B 29.3

452 Part B Atoms

curacy, and we turn to a description of these all order

methods.

If the frozen-core HF wave function is denoted by

, all-order singles–doubles methods add to it a correc-

tion ∆Ψ with either one or two orbitals in the HF wav e

function excited:

δΨ =





†



abmn

mnab

†



†



amn

mnav

†



(29.11)

The terms on the ﬁrst line of (29.11) describe single and

double excitations of the closed core, while those on

the second line describe single and double excitations

of the atom where the valence orbital is also excited.

Substituting (29.11) into the Schrödinger equation, one

obtains a set of coupled equations for the expansion co-

efﬁcients that can be found in [29.16]. The ﬁrst and

second iterations of the equations for the expansion co-

efﬁcients leads to results that are identical to ﬁrst- and

second-order perturbation theory. In third-order pertur-

bation theory, terms associated with triple excitations

contribute to the energy. These terms have no coun-

terpart in the iterative solution to the equations under

consideration.

To account for such terms, it is necessary to add

to ∆Ψ a triple-excitation correction of the speciﬁc

form:

∆Ψ =





abcmnr

mnrabc

†



abmnr

mnrabv

†



. (29.12)

Such a term enters in two ways. First, there are a set

of equations giving the triple-excitation coefﬁcients in

terms of the single-, double-, and triple-excitation coef-

ﬁcients. Second, the triple-excitation coefﬁcients enter

on the right-hand side of the equations for the single-

and double-excitation coefﬁcients. Solving for the triple-

excitation coefﬁcients in terms of the singles and doubles

(ignoring the triples on the right-hand sides of these

equations), one can use them on the right-hand sides

of the equations for the singles. This procedure leads

to equations which, when iterated to third order, in-

clude all of the terms from MBPT. However, the triples

also modify the the doubles equation, and this more

computationally demanding step has not yet been imple-

mented. As the effects of this modiﬁcation enter ﬁrst in

fourth-order MBPT, the calculation is complete through

third order, but still misses some fourth-order contribu-

tions. Nevertheless, the method is accurate enough to

predict PNC to 1%. Greater accuracy is expected when

the doubles equation is modiﬁed, which should lead to

a calculation complete through fourth order, and work

on this problem is in progress.

29.4 PNC Calculations

The above discussion of MBPT allows the calculation

on cesium to be accurate to a few tenths of percent for

energies and under one percent for ‘ordinary’ matrix

elements [29.16], and thus should allow a calculation of

similar accuracy for PNC. To calculate such transitions,

one modiﬁes H

by adding the weak-interaction h

the HF potential. This approach leads to a generalization

of the single-particle states in which each state acquires

an opposite-parity admixture, that is,

→ φ

. (29.13)

Thus, for example, each s

1/2

orbital will pick up a small

1/2

state admixture. One can then calculate a PNC

transition along the lines of a parity allowed transi-

tion calculation, simply replacing each orbital occuring

in the MBPT formulas in turn with its opposite par-

ity admixture. Details of this approach can be found

in [29.10].

There are a number of smaller effects that must be

considered when the 1% level of accuracy is reached.

The ﬁrst stems from the fact that the Z couples to the

density of up and down quarks, which in turn is related

to the density of protons and neutrons in the cesium

nucleus. While experimental data is available on the

charge distribution [29.17], the neutron distribution is

available only theoretically. While this uncertainty can

be shown to be unimportant for Cs

133

[29.10], there

has been interest in looking at PNC in several different

isotopes with the aim of taking ratios and greatly reduc-

ing atomic physics uncertainties. However, the neutron

distribution in different nuclei is more uncertain, and tak-

ing the ratio enhances the nuclear physics uncertainty.

This issue has been addressed recently by Chen and

Vogel [29.18], who estimate the nuclear physics uncer-

tainties to be smaller than the anticipated experimental

error.

Part B 29.4

Parity Nonconserving Effects in Atoms 29.6 Comparison with Experiment 453

The next small effect is the interaction between the

nuclear axial-vector current and the electron vector cur-

rent from Z exchange. In the limit of nonrelativistic

nucleon motion, this interaction is given by the spin-

dependent Hamiltonian

(2)

=−

√

κ −1/2

I(I +1)

α · Iρ(r).

(29.14)

Here, κ = 4, I = 7/2andK

−0.05 for the va-

lence proton of

133

Cs. Additionally, parity violation

in the nucleus leads to a parity-violating nuclear mo-

ment, the anapole moment mentioned above, that

couples electromagnetically to the atomic electrons. The

anapole–electron interaction is described by a Hamilto-

nian similar to (29.14), with K

replaced by −κ/(κ −

1/2)K

,whereK

= 0.24 −0.33 is determined from

nuclear model calculations [29.19]. Linear combina-

tions of amplitudes for different F → F



transitions

can be used to isolate either the spin-dependent or

spin-independent parts of the interaction. The ﬁnal

smaller effects are Z exchange between the electrons,

which as mentioned above turns out to be negligi-

ble, and the inclusion of the Breit interaction. While

the original estimate of the latter was a small 0.2%

effect, included in (29.6), this turned out to be an un-

derestimate, one of a number of unexpectedly large

contributions found in recent years, as discussed in the

next section.

29.5 Recent Developments

The Breit interaction was reexamined by Dere-

vianko [29.20] and found to be signiﬁcantly larger than

the 0.2% mentioned above . The 0.6% includes the ef-

fects of negative energy states, which are enhanced in

the context of PNC. This was followed by the dis-

covery of a number of effects that contributed at the

several tenths of a percent level, with theory and ex-

periment going into and out of agreement with one

another as each effect was found. At present the sit-

uation has stabilized, and, as will be discussed in the

next section, no discrepancy with the standard model

is present. In the past eight years, the principal de-

velopments have been the discovery of a surprisingly

large two-loop radiative correction associated with vac-

uum polarziation [29.21,22], recalculation of the atomic

physics part of the calculation [29.23], which, however,

only reduced the theoretical error bar, an independent

measurement of the polarizability of cesium [29.24],

and the discovery that the self-energy radiative correc-

tion, which in lowest order is −α/2π, and is included in

the radiative corrections to Q

presented in [29.11],

is strongly enhanced by binding corrections starting

at order Zα [29.25–27]. When taken together, these

different effects tend to cancel, so the qualitative pic-

ture of agreement with the standard model remains

correct.

29.6 Comparison with Experiment

We can now make use of the above analysis to extract

the value of the weak charge Q

from experiment. The

PNC amplitudes measured by Wieman et al. [29.3, 4]

are

(E

PNC

)







−1.6349(80) F = 4 → 3

−1.5576(77) F = 3 → 4

(29.15)

in units of mV/cm. The quantity β is the vector part of

the Stark induced polarizability for the 6s → 7s transi-

tion in cesium. This quantity has also been calculated

with an accuracy of better than 1% using the all-order

techniques outlined above, giving β =27.00(20)a

.We

note that this quantity can also be obtained from ex-

periment, but at present two recent experiments [29.3,

4, 24] are in disagreement, so we use the number ob-

tained from theory. Eliminating the spin-dependence

from (29.15), by taking an appropriate linear com-

bination and using the theoretical value for β, one

ﬁnds





exp

PNC



=−0.8252(184)[61]10

−11

|e|a

(29.16)

where the the ﬁrst error is from experiment and the

second from theory. Combining this result with our

calculation of the spin-independent amplitude given

in (29.6), we obtain

=−72.17(0.58)[0.72]. (29.17)

Alternatively, taking the opposite linear combination

to eliminate the spin-independent terms in (29.15), we

Part B 29.6

454 Part B Atoms

obtain the value

−(κ −1/2)/κK

= 0.72 ±0.10 (29.18)

for the constant governing the spin-dependent interac-

tion.

Radiative corrections to the weak charge Q

incor-

porating a parameterization of new physics beyond the

standard model have been worked out by Marciano and

Rosner [29.11], who ﬁnd



133



=−73.20 −0.8S−0.005T ±0.13 ,

(29.19)

assuming the values m

= 140 GeV for the top quark

mass and m

= 100 GeV for the Higgs particle mass.

The parameters S and T in (29.19) are associated partly

with deviations of the top quark and Higgs masses from

their assumed values and partly with new physics beyond

the standard model. In the absence of new physics, the

small factor multiplying T makes this prediction very

insensitive to the top quark mass. Unfortunately, both

the experimental and theoretical errors are presently

too large to make atomic PNC in cesium a precision

test of the standard model. However, there are two

features of cesium PNC that even at the present ac-

curacy lead to particle physics implications. The ﬁrst

is the fact that large positive values of S, such as can

arise in technicolor theories [29.28], will lead to dis-

agreement of theory and experiment in cesium PNC.

The second is the effect of extra Z bosons, which is

not accounted for in (29.19). Exchange of new Z’s

can be shown to be strongly constrained by atomic

PNC [29.29]. Perhaps more interesting is the possibil-

ity of having entirely new physics that has not been

thought of. Since new physics affects different weak

interaction tests differently, it is important to have

as many such tests as possible. The value of atomic

PNC tests will increase when the next stage of ac-

curacy is reached, at which time atomic physics will

have a signiﬁcant role in precision tests of the standard

model.

References

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29.2 M. A. Bouchiat et al.: J. Phys. (Paris) 47, 1709 (1986)

29.3 S. C. Bennett, C. E. Wieman: Phys. Rev. Lett. 82,

2484 (1999)

29.4 C. S. Wood et al.: Science 275, 1759 (1997)

29.5 P. S. Drell, E. D. Commins: Phys. Rev. Lett. 53,968

(1984)

29.6 T. M. Wolfenden, P. E. G. Baird, P. G. H. Sandars:

Europhys. Lett. 15, 731 (1991)

29.7 T. P. Emmons, J. M. Reeves, E. N. Fortson: Phys.

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29.8 M. J. D. Macpherson, K. P. Zetie, R. B. Warrington,

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29.9 W. Bernreuther, M. Suzuki: Rev. Mod. Phys. 63,313

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29.10 S. A. Blundell, J. Sapirstein, W. R. Johnson: Phys.

Rev. D 45, 1602 (1992)

29.11 W. Marciano, J. Rosner: Phys. Rev. Lett. 65,2963

(1990)

29.12 V. A. Dzuba, V. V. Flambaum, O. P. Sushkov: Phys.

Lett. A 141, 147 (1989)

29.13 A. C. Hartley, E. Lindroth, A.-M. Mårtensson-

Pendrill: J. Phys. B 23, 3417 (1990)

29.14 Ya. Zel’dovich: Zh. Eksp. Teor. Fiz. 33,1531

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29.15 W. R. Johnson, M. Idrees, J. Sapirstein: Phys. Rev.

A 35, 3218 (1987)

29.16 S. A. Blundell, W. R. Johnson, J. Sapirstein: Phys.

Rev. A 43, 3407 (1991)

29.17 R. Engfer et al.: At. Data Nucl. Data Tables 14,479

(1974)

29.18 B. Q. Chen, P. Vogel: Phys. Rev. C 48, 1392 (1993)

29.19 V. V. Flambaum, I. B. Khriplovich, O. P. Sushkov:

Phys. Lett. 146B,367(1984)

29.20 A. Derevianko: Phys. Rev. Lett. 85, 1618 (2000)

29.21 W. R. Johnson, I. Bednyakov, G. Soff: Phys. Rev.

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29.22 W. R. Johnson, I. Bednyakov, G. Soff: Phys. Rev.

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H. G. Berry: Phys. Rev. A 66, 020101 (2002)

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Part B 29

455

Atomic Clocks

30. Atomic Clocks and Constraints on Variations

of Fundamental Constants

Fundamental constants play an important role in

modern physics, being landmarks that designate

different areas. We call them constants, however,

as long as we only consider minor variations

with the cosmological time/space scale, their

constancy is an experimental fact rather than

a basic theoretical principle. Modern theories

unifying gravity with electromagnetic, weak,

and strong interactions, or even the developing

quantum gravity itself often suggest such

variations.

Many parameters that we call fundamental

constants, such as the electron charge and

mass [30.1, 2], are actually not truly fundamental

constants but effective parameters which are

affected by renormalization or the presence of

matter [30.3]. Living in a changing universe

we cannot expect that matter will affect these

parameters the same way during any given

cosmological epoch. An example is the inﬂationary

model of the universe which states that in

a very early epoch the universe experienced

a phase transition which, in particular, changed

a vacuum average of the so-called Higgs ﬁeld

which determines the electron mass. The latter

was zero before this transition and reached

a value close or equal to the present value after

the transition.

The problem of variations of constants has

many facets and here we discuss aspects related

to atomic clocks and precision frequency

30.1 Atomic Clocks and Frequency Standards. 456

30.1.1 Caesium Atomic Fountain ........... 456

30.1.2 Single-Ion Trap ......................... 457

30.1.3 Laser-Cooled Neutral Atoms ....... 457

30.1.4 Two-Photon Transitions

and Doppler-Free Spectroscopy... 458

30.1.5 Optical Frequency Measurements 458

30.1.6 Limitations on Frequency

Variations ................................ 458

30.2 Atomic Spectra and their Dependence on

the Fundamental Constants.................. 459

30.2.1 The Spectrum of Hydrogen

and Nonrelativistic Atoms .......... 459

30.2.2 Hyperﬁne Structure

and the Schmidt Model.............. 459

30.2.3 Atomic Spectra:

Relativistic Corrections ............... 460

30.3 Laboratory Constraints

on Time the Variations

of the Fundamental Constants.............. 460

30.3.1 Constraints from Absolute Optical

Measurements .......................... 460

30.3.2 Constraints from Microwave

Clocks ...................................... 461

30.3.3 Model-Dependent Constraints .... 461

30.4 Summary ............................................ 462

References .................................................. 462

measurements. Other related topics may be found

in [30.4].

Laboratory searches for a possible time variation of fun-

damental physical constants currently consist of two

important parts: (i) one has to measure a certain phys-

ical quantity at two different moments of time that are

separated by at least a few years; (ii) one has to be able

to interpret the result in terms of fundamental constants.

The latter is a strong requirement for a cross comparison

of different results.

The measurements which may be performed most

accurately are frequency measurements; and thus,

frequency standards or atomic clocks will be in-

volved in most of the laboratory searches. Frequency

metrology has shown great progress in the last

decade and will continue to do so for some time.

The constraints on the variations of the fundamen-

tal constants obtained in this manner are, so far,

somewhat weaker than those from other methods (as-

trophysics, geochemistry), but still competitive with

them. In contrast to other methods, however, fre-

quency measurements allow a very clear interpretation

of the ﬁnal results and a transparent evaluation pro-

cedure, making them less vulnerable to systematic

Part B 30

456 Part B Atoms

errors. While there is still potential for improvement,

the basic details of the method have been recently

ﬁxed.

The most advanced atomic clocks are discussed

in Sect. 30.1. They are realized with many-electron

atoms and their frequency cannot be interpreted in terms

of fundamental constants. However, a much simpler

problem needs to be solved: to interpret their variation in

terms of fundamental constants. This idea is discussed

in Sect. 30.2. The current laboratory constraints on the

variations of the fundamental constants are summarized

in Sect. 30.3.

30.1 Atomic Clocks and Frequency Standards

Frequency standards are important tools for precision

measurements and serve various purposes which, in

turn, have different requirements that must be satis-

ﬁed. In particular, it is not necessary for a frequency

standard to reproduce a frequency which is related to

a certain atomic transition although it may be expressed

in its terms. A well known example is the hydrogen

maser, where the frequency is affected by the wall shift

which may vary with time [30.5]. For the study of time

variations of fundamental constants it is necessary to

use standards similar to a primary caesium clock. In

this case, any deviation of its frequency from the un-

perturbed atomic transition frequency should be known

(within a known uncertainty) because this is a necessary

requirement for being a ‘primary’ standard.

From the point of view of fundamental physics, the

hydrogen maser is an artefact quite similar, in a sense,

to the prototype of the kilogram held at the Bureau

International des Poids et Mesures (BIPM) in Paris. Both

artefacts are somehow related to fundamental constants

(e.g., the mass of the prototype can be expressed in

terms of the nucleon masses and their number) but they

also have a kind of residual classical-physics ﬂexibility

which allows their properties to change. In contrast, stan-

dards similar to the caesium clock have a frequency (or

other property) that is determined by a certain natural

constant which is not ﬂexible, being of pure quantum

origin. It may change only if the fundamental constants

are changing.

In Sect. 30.3, results obtained with caesium and ru-

bidium fountains, a hydrogen beam, ultracold calcium

clouds, and trapped ions of ytterbium and mercury are

discussed. While caesium and rubidium clocks oper-

ate in the radio frequency domain, most of the other

standards listed above rely on optical transitions.

30.1.1 Caesium Atomic Fountain

Caesium clocks are the most accurate primary stan-

dards for time and frequency [30.6]. The hyperﬁne

splitting frequency between the F = 3andF = 4levels

of the

1/2

ground state of the

133

Cs atom at 9.192 GHz

has been used for the deﬁnition of the SI second since

1967. In a so-called caesium fountain (Fig. 30.1), a di-

lute cloud of laser cooled caesium atoms at a temperature

of about 1 µK is launched upwards to initiate a free

parabolic ﬂight with an apogee at about 1 m above the

cooling zone. A microwave cavity is mounted near the

lower endpoints of the parabola and is traversed by the

atoms twice – once during ascent, once during descent

– so that Ramsey’s method of interrogation with sepa-

rated oscillatory ﬁelds [30.5] can be realized. The total

interrogation time being on the order of 0.5s, a res-

onance linewidth of 1 Hz is achieved, about a factor

of 100 narrower than in traditional devices using a ther-

mal atomic beam from an oven. Selection and detection

ν = 9 192 631 770 Hz ⫾ ∆

1 pair of vertical

laser beams

Clock

output

Quartz

oscillator

Microwave

synthesis

Electronics for

detection and control

Photodetector

2 pairs of horizontal

laser beams

Cold atoms:

T = 1–2 µΚ

Launching velocity = 4 m/s

Microwave

cavity

Detection

laser

Launching

height

Fig. 30.1 Schematic of an atomic fountain clock

Part B 30.1

Atomic Clocks and Constraints on Variations of Fundamental Constants 30.1 Atomic Clocks and Frequency Standards 457

of the hyperﬁne state is performed via optical pumping

and laser induced resonance ﬂuorescence. In a care-

fully controlled setup, a relative uncertainty slightly

below 1 × 10

−15

can be reached in the realization of

the resonance frequency of the unperturbed Cs atom.

The averaging time that is required to reach this level

of uncertainty is on the order of 10

s. One limiting

effect that contributes signiﬁcantly to the systematic un-

certainty of the caesium fountain is the frequency shift

due to cold collisions between the atoms. In this re-

spect, a fountain frequency standard based on the ground

state hyperﬁne frequency of the

Rb atom at about

6.835 GHz is more favorable, since its collisional shift

is lower by more than a factor of 50 for the same atomic

density. With the caesium frequency being ﬁxed by def-

inition in the SI system, the

Rb frequency is therefore

presently the most precisely measured atomic transition

frequency [30.7].

30.1.2 Single-Ion Trap

An alternative to interrogating atoms in free ﬂight, and

a possibility to obtain practically unlimited interaction

time, is to store them in a trap. Ions are well suited be-

cause they carry electric charge and can be trapped in

radio frequency ion traps (Paul traps [30.8]) that provide

conﬁnement around a ﬁeld-free saddle point of an elec-

tric quadrupole potential. This ensures that the internal

level structure is only minimally perturbed by the trap.

Combined with laser cooling it is possible to reach the

so-called Lamb–Dicke regime where the linear Doppler

shift is eliminated. A single ion, trapped in an ultrahigh

vacuum is conceptually a very simple system that allows

good control of systematic frequency shifts [30.9]. The

use of the much higher, optical reference frequency al-

lows one to obtain a stability that is superior to micro-

wave frequency standards, although only a single ion

is used to obtain a correction signal for the reference

oscillator.

Metastable level

“Forbidden”

reference transition

Cooling transition

(dipole allowed)

Fig. 30.2 Double resonance scheme applied in single-ion-

trap frequency standards

A number of possible reference optical transitions

with a natural linewidth of the order of 1 Hz and be-

low are available in different ions, such as Yb

[30.10]

and Hg

[30.11, 12]. These ions possess a useful level

system, where both a dipole-allowed transition and a for-

bidden reference transition of the optical clock can be

driven with two different lasers from the ground state

(Fig. 30.2). The dipole transition is used for laser cooling

and for the optical detection of the ion via its reso-

nance ﬂuorescence. If a second laser excites the ion to

the metastable upper level of the reference transition,

the ﬂuorescence disappears and every single excitation

can thus be detected with practically hundred percent

efﬁciency as a dark period in the ﬂuorescence signal.

Using these techniques and a femtosecond laser

frequency comb generator (see Sect. 30.1.5) for the link

to primary caesium clocks, the absolute frequencies

of the transitions

1/2

→

5/2

199

at 1065 THz

and

1/2

→

3/2

171

at 688 THz have been

measured with relative uncertainties of only 9 × 10

−15

It is believed that single-ion optical frequency standards

offer the potential to ultimately reach the 10

−18

level of

relative accuracy.

A similar double resonance technique can be em-

ployed if the reference transition is in the microwave

domain and a number of accurate measurements of

hyperﬁne structure intervals in trapped ions has been

performed. In particular, the HFS interval in

171

has

been measured several times [30.13–15] and can be used

to obtain constraints on temporal variations.

30.1.3 Laser-Cooled Neutral Atoms

Optical frequency standards have been developed

with free laser-cooled neutral atoms, most notably of

the alkaline-earth elements that possess narrow inter-

combination transitions. The atoms are collected in

a magneto-optical trap, are then released and in-

terogated by a sequence of laser pulses to realize

a frequency-sensitive Ramsey–Bordé atom interferom-

eter [30.16]. Of these systems, the one based on the

→

intercombination line of

Ca at 657 nm has

reached the lowest relative uncertainty so far (about

2×10

−14

) [30.11, 12, 17, 18]. Limiting factors in the

uncertainty of these standards are the residual lin-

ear Doppler effect and phase front curvature of the

laser beams that excite the ballistically expanding atom

cloud. It has therefore been proposed to conﬁne the

atoms in an optical lattice, i. e., in the array of in-

terference maxima produced by several intersecting,

red-detuned laser beams [30.19]. The detuning of the

Part B 30.1

458 Part B Atoms

trapping laser could be chosen such that the light shift

it produces in the ground and excited state of the

reference transition are equal, and therefore it would

produce no shift of the reference frequency. This ap-

proach is presently being investigated and may be

applied to the very narrow (mHz natural linewidth)

→

transitions in neutral strontium, ytterbium,

or mercury.

30.1.4 Two-Photon Transitions

and Doppler-Free Spectroscopy

The linear Doppler shift of an absorption resonance can

also be avoided if a two-photon excitation is induced

by two counterpropagating laser beams. A prominent

example that has been studied with high precision is

the two-photon excitation of the

S →

S transition in

atomic hydrogen. The precise measurement of this fre-

quency is of importance for the determination of the

Rydberg constant and as a test of quantum electrody-

namics (QED). Hydrogen atoms are cooled by collisions

in a cryogenic nozzle and interact with a standing laser-

wave of 243 nm wavelength inside a resonator. Since

the atoms are not as cold as in laser cooled samples,

a correction for the second order Doppler effect is per-

formed. The laser excitation is interrupted periodically

and the excited atoms are detected in a time resolved

manner so that their velocity can be examined. An ac-

curacy of about 2 × 10

−14

has been obtained in absolute

frequency measurements with a transportable caesium

fountain [30.20, 21].

30.1.5 Optical Frequency Measurements

In recent years, the progress in stability and accuracy

of optical frequency standards has been impressive; and

there is belief that in the future an optical clock may

supersede the microwave clocks because the optical os-

cillators offer a much higher number of periods in a given

time. In addition, some systematic effects, such as the

Zeeman effect, have an absolute order of magnitude

that does not scale with the transition frequency, and

consequently is relatively less important at higher tran-

sition frequencies. A long-standing problem, however,

was the precise conversion of an optical frequency to the

microwave domain, where frequencies can be counted

electronically in order to establish a time scale or can

easily be compared in a phase coherent way.

This problem has recently been solved by the

so-called femtosecond laser frequency comb gener-

ator [30.22]. Brieﬂy, a mode-locked femtosecond laser

Time

∆t = 1/f

Frequency

f = 0

ceo

Fig. 30.3 Frequency comb generated from femtosecond

laser pulses

produces, in the frequency domain, a comb of equally

spaced optical frequencies f

(Fig. 30.3) that can be

written as f

= nf

+ f

ceo

(with f

ceo

< f

), where f

the pulse repetition rate of the laser, the mode number

n is a large integer (of order 10

), and f

ceo

(carrier-

envelope-offset) is a shift of the whole comb that is

produced by group velocity dispersion in the laser. The

repetition rate f

can easily be measured with a fast

photodiode. In order to determine f

ceo

,thecombis

broadened in a nonlinear medium so that it covers at

least one octave. Now the second harmonic of mode

n from the “red” wing of the spectrum, at frequency

2(nf

+ f

ceo

), can be mixed with mode 2n from the

“blue” wing, at frequency 2nf

+ f

ceo

,and f

ceo

is ob-

tained as a difference frequency. In this way, the precise

relation between the two microwave frequencies f

and

ceo

and the numerous optical frequencies f

is known.

The setup can now be used for an absolute optical

frequency measurement by referencing f

and f

ceo

a microwave standard and recording the beat note be-

tween the optical frequency f

to be measured and the

closest comb frequency f

.Viceversa,thesetupmay

work as an optical clockwork, for example, by adjusting

ceo

to zero and by stabilizing one comb line f

to f

so that f

is now an exact subharmonic to order n of f

The precision of these transfer schemes has been investi-

gated and was found to be so high that it will not limit the

performance of optical clocks for the foreseeable future.

30.1.6 Limitations on Frequency Variations

The frequency standards described above have been suc-

cesfully developed and their accuracy has been improved

Part B 30.1

Atomic Clocks and Constraints on Variations of Fundamental Constants Atomic Spectra and Fundamental Constants 459

Table 30.1 Limits on possible time variation of frequencies

of different transitions in SI units. Here δ f/ f is the frac-

tional uncertainty of the most accurate measurement of the

frequency f

Atom, f δ f/ f  f/t Ref.

transition

GHz (10

−15

) (Hz/yr)

H, Opt 2 466 061 14 −8 ± 16 [30.20, 21]

Ca, Opt 455 986 13 −4 ± 5 [30.17, 18]

Rb, HFS 6.835 1 (0 ±5) × 10

−6

[30.7]

,Opt 688 359 9 −1 ± 3 [30.23]

,HFS 12.642 73 (4 ± 4) × 10

−4

[30.13–15]

,Opt 1 064 721 9 0 ± 7 [30.11, 12]

in the last decade. This progress, as a consequence, has

led to certain constraints on the possible variations of

the fundamental constants. Considering frequency varia-

tions, one has to have in mind that not only the numerical

value but also the units may vary. For this reason, one

needs to deal with dimensionless quantities which are

unit-independent. During the last decade, a number of

transition frequencies were measured in the correspond-

ing SI unit, the hertz (see Table 30.1). These dimensional

results are actually related to dimensionless quantities

since a frequency measurement in SI is a measurement

with respect to the caesium hyperﬁne interval





= 9 192 631 770 ·

HFS

(Cs)

(30.1)

where





stands for the numerical value of the fre-

quency f . (Most absolute frequency measurements have

been realized as a direct comparison with a primary

caesium standard.) In Sect. 30.3, in order to simplify

notation, this symbol for the numerical value is dropped.

30.2 Atomic Spectra and their Dependence

on the Fundamental Constants

30.2.1 The Spectrum of Hydrogen

and Nonrelativistic Atoms

The hydrogen atom is the simplest atom and one can eas-

ily calculate the leading contribution to different kinds

of transitions in its spectrum [30.24, 25], such as the

gross, ﬁne, and hyperﬁne structure. The scaling behav-

ior of these contributions with the values of the Rydberg

constant R

∞

, the ﬁne structure constant α,andthemag-

netic moments of proton and Bohr magneton is clear.

The results for some typical hydrogenic transitions are

f(2p → 1s) 

∞



3/2

− 2p

1/2





∞

HFS

(1s) 

∞

. (30.2)

In the nonrelativistic approximation, the basic fre-

quencies and the ﬁne and hyperﬁne structure intervals

of all atomic spectra have a similar dependence on the

fundamental constants. The presence of a few elec-

trons and a nuclear charge of Z = 1 makes theory more

complicated and introduces certain multiplicative num-

bers but involves no new parameters. The importance

of this scaling for a search for the variations was ﬁrst

pointed out in [30.26] and was applied to astrophysical

data. Similar results may be presented for molecular

transitions (electronic, vibrational, rotational and hyper-

ﬁne) [30.27], however, up to now no measurement with

molecules has been performed at a level of accuracy

that is competitive with atomic transitions. They have

been used only in a search for variations of constants in

astrophysical observations [30.28].

30.2.2 Hyperﬁne Structure

and the Schmidt Model

The atomic hyperﬁne structure

(HFS) = const α

∞

(30.3)

involves nuclear magnetic moments µ which are dif-

ferent for different nuclei; thus, a comparison of the

constraints on the variations of nuclear magnetic mo-

ments has a reduced value. To compare them, one may

apply the Schmidt model [30.3, 29], which predicts all

the magnetic moments of nuclei with an odd number of

nucleons (odd value of atomic number A)intermsof

the proton and neutron g-factors, g

and g

, respectively,

and the nuclear magneton only. Unfortunately, the un-

certainty of the calculation within the Schmidt model

is quite high (usually from 10% to 50%). The Schmidt

model, being a kind of ab initio model, only allows for

improvements which, unfortunately, involve some ef-

fective phenomenological parameters. This would not

Part B 30.2

460 Part B Atoms

really improve the situation, but return us to the case

where there are too many possibly varying independ-

ent parameters. A comparison of the Schmidt values to

the actual data is presented for caesium, rubidium, and

ytterbium in Table 30.2.

30.2.3 Atomic Spectra:

Relativistic Corrections

A theory based on the leading nonrelativistic approxima-

tion may not be accurate enough. Any atomic frequency

can be presented as

f = f

rel

(α) , (30.4)

where the ﬁrst (nonrelativistic) factor is determined by

a scaling similar to the hydrogenic transitions (30.2).

The second factor stands for relativistic corrections

which vanish at α = 0; and thus, F

rel

(0) = 1.

The importance of relativistic corrections for the

hyperﬁne structure was ﬁrst emphasized in [30.30].

Relativistic many-body calculations for various tran-

sitions have been performed in [30.31–36]. A typical

accuracy is about 10%. Some results are summarized

in Tables 30.2 and 30.3, where we list the relative

sensitivity of the relativistic factors F

rel

to changes in α,

κ =

∂lnF

rel

∂lnα

(30.5)

Note that the relativistic corrections in heavy atoms are

proportional to (Zα)

because of the singularity of rela-

tivistic operators. Due to this, the corrections rapidly

increase with the nuclear charge Z.

The signs and magnitudes of κ are explained by

a simple estimate of the relativistic correction. For

example, an approximate expression for the relativis-

tic correction factor for the hyperﬁne structure of an

s-wave electron in an alkali-like atom is [30.30]

rel

(α) =



1 − (Zα)

1 − (4/3)(Zα)

 1 +

(Zα)

Table 30.2 Magnetic moments and relativistic corrections

for atoms involved in microwave standards. The relativis-

tic sensitivity κ is deﬁned in Sect. 30.2.3.Hereµ is an

actual value of the nuclear magnetic moment, µ

is the nu-

clear magneton, and µ

stands for the Schmidt value of

the nuclear magnetic moment; the nucleon g factors are

/2  2.79 and g

/2 −1.91

Z Atom µ/µ

/µ

µ/µ

Rb 2.75 g

/2 + 1 0.74 0.34

133

Cs 2.58 7/18 · (10 − g

) 1.50 0.83

171

0.49 −g

/6 0.77 1.5

Table 30.3 Limits on possible time variation of the fre-

quencies of different transitions and their sensitivity to

variations in α due to relativistic corrections

Atom, transition ∂ f/ f∂t κ

H, 1s − 2s − 3.2(63) ×10

−15

−1

0.00

Ca,

−

−8(11) ×10

−15

−1

0.03

171

1/2

−

3/2

− 1.2(44) ×10

−15

−1

0.9

199

1/2

−

5/2

− 0.2(70) ×10

−15

−1

−3.2

A similar rough estimation for the energy levels may be

performed for the gross structure:

E =−

∗



1 +

(Zα)

∗

j + 1/2



(30.6)

Here j is the electron angular momentum, n

∗

is the

effective value of the principle quantum number (which

determines the nonrelativistic energy of the electron),

and Z

is the charge “seen” by the valence electron – it

is 1 for neutral atoms, 2 for singly charged ions, etc. This

equation tells us that κ, for the excitation of the electron

from the orbital j to the orbital j



, has a different sign

for j > j



and j < j



. The difference of sign between the

sensitivities of the ytterbium and mercury transitions in

Table 30.3 reﬂects the fact that in Yb

a 6s-electron is

excited to the empty 5d-shell, while in Hg

a hole is

created in the ﬁlled 5d-shell if the electron is excited to

the 6s-shell.

30.3 Laboratory Constraints on Time the Variations

of the Fundamental Constants

Logarithmic derivatives (30.5) appear since we are look-

ing for a variation of the constants in relative units.

In other words, we are interested in a determination of,

e.g., ∆α/α∆t while the input data of interest are related

to ∆ f/ f∆t. Their relation takes the form

∂ln f

∂t

∂ln f

∂t

+ κ

∂lnα

∂t

(30.7)

Part B 30.3