Muga G. Time in Quantum Mechanics

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388 R. Wynands

decreasing launch height [78]. These balls never meet in the free-ﬂight zone above

the cavity where collisions would lead to a frequency shift. But they all come

together in the detection zone – where cold collisions do not matter anymore –

to produce a strong signal. This idea has been tested at NIST [79] and is envisioned

to be implemented in NIST-F2 [80].

Another avenue is the “juggling fountain” [81] where several balls are launched

with a time separation smaller than the ﬂight time of an individual cloud. When tim-

ing and launch velocities of the individual clouds are chosen such that each time two

of them meet their relative energies fulﬁl the condition for a Ramsauer resonance,

they pass through each other basically without scattering, i.e. without additional

collisional shifts. It is straightforward to do this with just two “balls”, but amazingly

it can also be done with more than two balls [82]. The multi-ball scheme requires a

precise control of the launch times, velocities and densities of the individual balls.

Furthermore it relies on a delicate cancellation of the collisional shifts in successive

two-ball collisions, making use of the energy dependence of sign and amplitude of

the shift [82].

The Dick effect can be all but eliminated when the fountain clock is operated

in a continuous mode rather than pulsed [83, 84]. At the same time, because of the

continuous detection a lower density beam can be used, reducing the uncertainty

due to cold collisions. However, a continuous fountain poses a number of technical

and experimental challenges. First of all, since the preparation and the detection

zones have to be spatially distinct the atoms have to ﬂy along a parabolic path. This

requires a special geometry for the main cavity. Unfortunately, in this way one loses

one of the big advantages of the pulsed fountain design, where the atoms retrace

their path through the microwave ﬁeld and thus mostly cancel any end-to-end phase

shifts in the cavity. In the continuous fountain FOCS-1 a special device allows one

to rotate the cavity around the vertical axis by precisely 180

◦

, so that an effective

beam reversal occurs [85], in analogy to the procedure in thermal-beam clocks [21].

Furthermore, the suppression of stray light from the preparation zone becomes

more complicated. Unlike in the pulsed case the laser light cannot be switched off

during the free-ﬂight phase of the atoms, so for the continuous fountain one resorts

to mechanical shutters inside the vacuum vessel. A wheel with partially overlapping

ﬁlters absorbing the laser radiation is rotating rapidly through the atomic beam in

such a way that the direct line of sight from the detection zone into the free-ﬂight

zone passes through at least one ﬁlter at any one time. Not only does this chop thin

slices out of the continuous atomic beam but also does one have to have a motor

inside the ultra-high vacuum system – which also has to be nonmagnetic!

Details of all design issues can be found in the thesis by Joyet [85]. The ﬁrst such

clock, METAS-FOCS1, is now being commissioned at METAS [57, 47], with an

improved version FOCS2 being under development [86]. Design goals are a relative

short-term instability of 7 × 10

−14

(τ/s)

−1/2

(using a quartz oscillator as a local

oscillator for the 9-GHz synthesis chain) and a relative uncertainty of 10

−15

Obtaining a small frequency instability is indispensable for the evaluation of sys-

tematic uncertainty contributions at the level of 10

−15

or below. Even an instability

13 Atomic Clocks 389

of only 10

−13

(τ/s)

−1/2

still results in a 3.4 × 10

−16

statistical uncertainty after one

full day of measurement.

13.7.2.2 Systematic (Type B) Uncertainties

To a large degree the sources of frequency bias in a fountain clock are very simi-

lar to those in a thermal-beam clock. The latter are described in great quantitative

detail in the book by Vanier and Audoin [2]. The fact that now one has a state-

selected, monokinetic sample of laser-cold atoms greatly reduces some of these

biases but also introduces two new ones: the light shift and the cold-collision shift.

Here we will present just an overview and refer the reader to the review [55] and

the published formal evaluations of the individual fountain clocks, for example, in

[76, 40, 38, 42–45].

(1) Second-Order Zeeman Effect

The applied static magnetic bias ﬁeld (“C-ﬁeld”) results in a second-order

shift of the clock transition by f

= 0.0427 Hz×(B

/μT)

, where B

the magnetic ﬂux density of the C-ﬁeld, so that almost 5 × 10

−14

relative

frequency shift is obtained for the typical B

∼ 0.1 μT. One can use the

atoms themselves as probes for the mean C-ﬁeld strength, its inhomogeneity

and its temporal stability by measuring the resonance frequency f

of the

ﬁrst-order ﬁeld-sensitive transition |F = 4, m

= 1→|F = 3, m

= 1.In

contrast to a conventional beam clock, an atomic fountain provides the advan-

tageous possibility of mapping the C-ﬁeld by launching the atoms to different

heights h

max

. For a given h

max

the measured f

is an average over B

(z)

encountered by the atom cloud, weighted with the z- and h

max

-dependent

dwell time of the cloud. The series of f

max

) can be deconvoluted to

obtain B

(z)[87].

(2) Majorana Transitions

Majorana transitions (ΔF = 0,Δm

=±1) between the m

substates within

a hyperﬁne ground state |F = 4 or |F = 3 can be induced near zero cross-

ings of the magnetic ﬁeld strength [88]. In real caesium clock cavities it cannot

be avoided that due to the ﬁeld geometry some ΔF =±1,Δm

=±1

transitions occur besides the designated clock transition, albeit with a small

probability. In connection with Majorana transitions large frequency shifts can

occur [89] because atoms can end up in superposition states with the same

F quantum number but different m

quantum numbers. For these states the

hyperﬁne transition frequency is in general different from the clock transition

frequency, with a resulting overall frequency shift. For a well-controlled mag-

netic ﬁeld geometry the uncertainty estimate due to Majorana transitions is

typically stated as less than 10

−16

(3) Cavity-Related Shifts: Residual First-Order Doppler Effect and Cavity Pulling

A general advantage of an atomic fountain is that the atoms cross the same

microwave cavity twice in opposite directions. If the atomic trajectories were

390 R. Wynands

perfectly vertical, frequency shifts due to axial and radial cavity phase

variations would be perfectly cancelled as each atom would interact with the

ﬁeld ﬁrst with velocity v (upwards) and later with −v (downwards). Due to

the transverse residual thermal velocity and a possible misalignment of the

launching direction atoms in general do not retrace their upward trajecto-

ries on the way down, so a ﬁrst-order Doppler frequency shift can remain.

Uncertainty contributions of the order of at most several 10

−16

are estimated

typically [76, 40, 38, 42–45].

The frequency shift due to cavity pulling is usually negligible because atom

numbers are still rather low. Furthermore, since cavity pulling is proportional

to atom number it is corrected for automatically when the collisional shift

correction (see below) is applied.

(4) Rabi and Ramsey Frequency Pulling

Frequency shifts due to Rabi and Ramsey frequency pulling [2] can occur in

the presence of nonzero and asymmetric (with respect to m

= 0) popula-

tions of the |F, m

= 0 substates when the atoms enter the Ramsey cavity.

However, the state selection process in a fountain clock strongly reduces the

impact of these frequency pulling effects to well below 10

−16

(5) Microwave Leakage

The presence of microwave radiation outside the cavities can lead to large

frequency shifts despite the partial cancellations due to the near-symmetry of

the up-down motion of the cloud. The spread of the cloud, combined with

the fundamentally unavoidable radial microwave power variation inside the

cavity, makes the atoms see different effective microwave powers during the

two cavity passages. This greatly increases the clock’s susceptibility to stray

ﬁeld-induced frequency bias [90]. Typical uncertainties evaluated so far fall in

the low 10

−16

range once all microwave devices, cables and connectors have

been shielded carefully.

(6) Electronics and Microwave Spectral Impurities

The low total systematic uncertainty of a fountain clock requires tight spec-

iﬁcations for any electronic subsystems in order to avoid frequency bias

caused by servo offsets or microwave spectral impurities [91, 92]. Recent

improvements on other systematic uncertainties have prompted several groups

to upgrade their microwave electronics so that they no longer limit further

progress [93–96].

(7) Light Shift and Optical Pumping/Heating

All lasers must be completely blocked while the atoms are in or above the

Ramsey cavity. Otherwise, the shift of the atomic energy levels by the AC

Stark effect (light shift) [61] will lead to a frequency bias – nanowatts of

residual laser power can already be too much. Once the atoms have fallen

through the Ramsey cavity they still must be protected from stray light until

they have passed the detection zone to avoid heating of the cloud [97] and a

reduced contrast of the Ramsey fringe [98]. With combinations of large laser

frequency detuning and of acousto-optical and mechanical shutters the corre-

sponding uncertainty can be reduced to below 10

−16

13 Atomic Clocks 391

(8) Blackbody Shift

AC Stark shift of a different origin is induced by the thermal radiation of

the vacuum enclosure. Assuming that this radiation follows the spectral power

density distribution of a black body the clock transition frequency is shifted by

=−1.579 × 10

−4



300 K





1 + 0.013



300 K





(13.10)

for a vacuum enclosure at temperature T [99–103, 63]. Hence, for a cae-

sium clock at room temperature the relative frequency shift is of the order of

−17 ×10

−15

. A corresponding correction must be applied. Its relative uncer-

tainty lies below 10

−16

when the thermal environment seen by the atoms is

known within 0.2 K or so. The NIST-F2 and the NRC-FCs1 fountains under

development will all but eliminate this shift when they are operated at liquid-

nitrogen temperature [80, 52].

(9) Collisional Shift

A major source of uncertainty is the frequency shift due to collisions among

the cold atoms in the cloud [104]. The collisional cross section was found to be

strongly dependent on energy (i.e. the average temperature of the cloud) [105].

The problem is particularly serious for caesium because it was found that its

collisional cross section is unusually large at the low cloud temperatures used

in a fountain. Conceptually the simplest solution would be to choose another

element. For instance, in rubidium the collisional cross-section is almost two

orders of magnitude lower [106, 107]. Indeed, rubidium fountain clocks have

been built where the reduction of collisional shift uncertainty was one of the

motivations [108, 109].

A number of schemes have been devised to reduce the collisional shift or at

least its contribution to the uncertainty budget of the caesium fountain clock,

for instance, by lowering the density of the atomic cloud and therefore the

collision rate. A reduced number of atoms, however, reduce the detected signal

and therefore the signal-to-noise ratio and the short-term stability (Eq. 13.9).

Recent developments are helping to ease this trade-off problem for the case

of caesium. The problem of loss of signal for low-density clouds can be cir-

cumvented by controlling the preparation of the atoms such that more than

one low-density cloud at a time is travelling through the vacuum system, as

discussed above in connection with the Dick effect. The continuous fountain

also allows one to reduce the inﬂuence of collisional shift because it spreads

the atoms over the whole trajectory instead of concentrating them in a high-

density cloud.

The more traditional approach is to measure the clock’s output frequency

for two or more effective densities of the atomic cloud and then extrapolate

to zero density. Since the frequency shift due to cold collisions is linear in

effective atom density, one can extrapolate the measured frequencies obtained

with clouds of different densities to zero density using a linear regression.

This is the method currently employed by all primary fountain clocks. Since

392 R. Wynands

the actual density of the cloud is not readily accessible in the experiment, one

substitutes the number N

of detected atoms instead. This, of course, assumes

that there is a strict proportionality between N

and effective density. The

actual experimental practice in the various laboratories differs in the way that

clouds of different densities are prepared. One way is to vary the loading

time of the MOT/OM to determine the proportionality factor between atom

number and output frequency [55]. In all subsequent frequency measurements

one continuously monitors the atom number and scales it with this factor to

extrapolate to the zero density value [40]. A disadvantage of this method is

that one cannot exclude a difference in the spatial distribution of the atoms

within the cloud for the two atom number regimes, which in general could

change the proportionality factor between effective density and detected atom

number. An additional contribution therefore needs to be included into the

uncertainty budget.

As an alternative, the collisional shift can be monitored during the course

of a frequency evaluation, by switching between two different atom numbers

every few shots [43] or every hour [42]. The switching is done by varying

the microwave power or detuning for the state-selection pulse. Once again

it cannot be excluded that the density distribution of the state-selected cloud

changes between high and low atom number due to inhomogeneities of the

microwave excitation.

When a very stable local time reference, for instance, a maser ensemble,

is available, one can also run the fountain at extremely low atom number (i.e.

very low collisional shift) and just average over a longer time interval [110].

Perhaps the most elegant way is to use the technique of rapid adiabatic

passage during the preparation stage (Figs. 13.12c and 13.13c) in the state-

selection cavity [111]. This method relies on the fact that the population of

state |F = 4, m

= 0 can be transferred with 100% efﬁciency into state

|F = 3, m

= 0 when both frequency and amplitude of the microwave

radiation inside the selection cavity are ramped with just the right timing

[112]. Details of the technique can be found in [112, 111, 113]. Basically,

the microwave detuning is ramped from (ideally) minus inﬁnity to inﬁnity,

while the amplitude goes from 0 to a maximum (at detuning zero) and back to

0 again. One has to ensure that the rate of change of the microwave frequency

has to be much lower at all times than the square of the Rabi frequency (which

is proportional to microwave power).

When the pulse is switched off abruptly at detuning δ = 0 the atoms

are left in an exactly equal superposition of both states. The important fea-

ture of the rapid adiabatic passage is that this happens independently of the

actual Rabi frequency an atom sees, i.e. it does not depend on where an atom

passes through the ﬁeld inside the cavity (the ﬁeld amplitude decreases across

the aperture when going away from the centre). The pushing beam therefore

removes exactly half of all atoms, without changing the density distribution,

temperature or velocity of the cloud – in contrast to the other methods where

such changes cannot be excluded.

13 Atomic Clocks 393

At SYRTE the ratio of 1 : 2 can be prepared and maintained with an accu-

racy of 10

−3

[76], allowing for a very precise determination of the collisional

shift rate and its correction. The precise control over atomic populations in

SYRTE-FO2 has also made it possible to detect Feshbach resonances in the

dependence of the collisional shift on magnetic quantum number m

[64].

Surprisingly, these resonances occur already for ﬂux densities of 2 μTor

less.

Finally, under certain circumstances one can actually run a caesium foun-

tain clock under operating conditions where the total collisional shift experi-

enced by the cloud is 0 [114]. A typical Cs fountain standard operates with

atoms cooled down to 1–2 μK at the time of launch. As the cloud expands

during its ballistic ﬂight, correlations build up between atomic position and

velocity, an effect particularly pronounced for atoms initially trapped in a

small cloud (like in a MOT). Basically, atoms sort themselves spatially into

concentric shells of very nearly the same velocity. This means that the effec-

tive collisional energy decreases down to a few hundred nanokelvins even

before the ﬁrst Ramsey interaction [115]. At low energies the collision rate

coefﬁcients for the clock states |F = 3, m

= 0 and |F = 4, m

= 0 differ

in sign, which gives rise to a strong variation of the collisional shift if the

relative weights of the clock states in the superposition state prepared during

the ﬁrst Ramsey interaction are varied.

This could be conﬁrmed experimentally and theoretically for two inde-

pendent primary standards [114]. The fraction of the population in a given

clock state (e.g. |F = 4, m

= 0) was changed by adjusting the amplitude

0.0 0.2 0.4 0.6 0.8 1.0

Fraction of |F = 4〉 atoms

Relative collisional frequency shift

−2×10

−14

−1×10

−14

1×10

−14

Fig. 13.15 Measurement of the collisional frequency shift in PTB-CSF1 as a function of the pop-

ulation composition during the Ramsey time. The experimental data are ﬁtted by a linear function

394 R. Wynands

of the microwave ﬁeld in the Ramsey cavity. The collisional frequency shift

for atoms captured in a MOT varies linearly with the population fraction

(Fig. 13.15). For PTB-CSF1 the collisional frequency shift is cancelled for

a 30% fraction of |F = 4, m

= 0 atoms, which can be achieved experimen-

tally with only a minor reduction of the fringe contrast and short-term stability

of the fountain clock.

The practical details of this method have been considered in detail in [116];

under certain circumstances a shortening of the averaging time to reach a cer-

tain statistical uncertainty of about an order of magnitude was predicted. So

far, this method is not yet applied routinely.

(10) Background Gas Collisions

At typical vacuum pressures in the ballistic ﬂight region in the low 10

−7

range the effect of collisions of cold caesium atoms with residual gas atoms is

estimated to be well below 10

−16

(11) Time Dilatation: Relativistic Doppler Effect

Relativistic time dilatation reduces the clock frequency observed in the lab-

oratory frame by f

≈ ν

v

/(2c

), with v

 the mean quadratic veloc-

ity of the atoms above the microwave cavity and c the velocity of light.

Typically f

/ν

is of the order of 10

−17

, so that in contrast to thermal-

beam clocks time dilatation and the associated uncertainty contribution can be

neglected.

(12) Gravitational Redshift

Even though the gravitational redshift [117] of about −1.1×10

−16

per metre of

elevation above the geoid is not relevant for the realization of the proper sec-

ond of a clock and for local frequency measurements at the same gravitational

potential, its knowledge is necessary for comparing remote clocks, for instance

when contributing to international atomic time (TAI). Hence the mean height

of the atoms above the geoid during their ballistic ﬂight above the microwave

cavity centre has to be determined with a typical uncertainty of 1 m, corre-

sponding to a frequency uncertainty of 10

−16

. In principle, a limitation is given

by the accuracy with which the local gravitational potential can be determined.

Even for high-altitude laboratories like NIST in Boulder (≈ 1630 m above sea

level) the correction can be determined with a relative uncertainty of 3×10

−17

[118]. At this relative uncertainty the gravitational redshift will probably not

become a limiting factor for clocks based on microwave transitions. This is in

contrast to the case of future optical clocks.

(13) Other Systematic Effects

There are other frequency-shifting effects (dc Stark shift, Bloch–Siegert shift)

[2], which can be estimated to be less than 10

−17

in a fountain clock, or the

microwave recoil shift at around 10

−16

[119].

In conclusion it can be stated that the main contributions to the systematic uncer-

tainty are of the order of a few 10

−16

or less. As the individual systematic uncer-

tainty contributions can be assumed to be linearly independent, the resulting total

systematic uncertainty is the square root of the sum of squares of the individual

contributions.

13 Atomic Clocks 395

13.7.3 State of the Art

Worldwide, as of July 2008 there are nine primary fountain clocks in operation that

have been used to contribute to the international timescales. The individual clocks

have very different characteristics, depending on the design choices made by the

teams developing and operating them. Which of the fountains is the “best one” right

now? Well, that depends on the criterion chosen.

• Stability

In terms of stability the SYRTE fountains FO1 and FO2 with σ

(1 s) = 2×10

−14

are unsurpassed. This low instability is achieved by referencing the 9-GHz

microwave to an ultrastable cryogenic sapphire oscillator (CSO) [75, 76, 69].

So far, the other groups have shied back from the enormous technical and ﬁnan-

cial effort involved in the operation of a CSO at liquid-helium temperature. New

developments in the ﬁeld of femtosecond laser frequency combs have opened up

the possibility of using such a comb “in reverse”, by locking it to a narrow opti-

cal Fabry–Perot resonance and deriving a phase-stable 9-GHz microwave signal

from it. Performance similar to that of a CSO has been reported [77].

• Uncertainty

SYRTE-FO1, SYRTE-FO2 and NIST-F1 can now be run under conditions of

3,...,4 × 10

−16

relative uncertainty [76, 110, 120], with most other fountain

clocks currently in the range of 10

−15

or slightly below. There appears to be

room for substantial reductions in the uncertainties of collisional shift, cavity-

related effects and electronics by a more stringent control of operating parameters

and a better theoretical understanding of microwave cavities. However, it will be

difﬁcult to drive the relative uncertainty much below 1,...,2 ×10

−16

That is because reducing the uncertainty of the black-body shift appears rather

difﬁcult for the existing fountains since it would require a sufﬁciently detailed

knowledge of the effective thermal environment of the atoms during their free-

ﬂight phase. This includes not only the actual inner-wall temperature to within

better than 0.2 K but also reliable values for the emissivity of the inner surfaces

of the vacuum tube (which might change due to physisorption or chemisorption

of caesium or residual gases over the years of operation) as well as the inﬂuence

of radiation coming in through windows and other openings. Any substantial

progress on this front might require the cooling of the walls of the vacuum tube.

This is exactly what is intended for NIST-F2 and NRC-FCs1, which are currently

under construction [80, 52].

• Robustness

Although not as “sexy” as the other two criteria, the operational robustness of

a clock is important for its application in the generation of timescales, be it on

a local or on an international level, although depending on circumstances some

dead time can be tolerated. No fountain can yet run with the same round-the-

clock unattended operation as the thermal-beam clocks PTB-CS1 and PTB-CS2

have been doing for decades. The main problem here is the stability of the oper-

ating parameters of the laser sources over days and weeks, in particular slow

drifts into less favourable operational regimes or even mode hops. Using standard

396 R. Wynands

extended-cavity diode lasers with feedback from an external diffraction grating it

is possible today to run a fountain clock with better than 99.8% availability over

weeks. This is, for instance, the case with PTB-CSF1, where the small amount

of dead time is entirely due to scheduled laser parameter checks [121]; recently,

completely unattended and uninterrupted operation for about 3 weeks has been

tried successfully.

New laser sources, like the extended-cavity lasers developed for the ACES

project [122] or the high-power narrow linewidth DFB laser diodes now available

commercially, might simplify long-term operation of fountain clocks. Dead times

due to unintended laser dropouts can be minimized by automated laser relock

systems, although these cannot always deal with mode hops.

13.8 Other Types of Atomic Clocks

A discussion of atomic clocks would not be complete without at least a brief discus-

sion of some other types of atomic clocks. Out of the many possibilities only two are

selected here: the vapour-cell clock, which trades uncertainty for a smaller size, and

the hydrogen maser which trades uncertainty for superior short- and medium-term

stability.

13.8.1 Vapour-Cell Clocks

Vapour-cell atomic clocks are perhaps the most inﬂuential type of atomic clock

for everyday life – or if they are not yet, they soon will be. This is because it is

rubidium-vapour clocks that work aboard the GPS satellites, helping us navigate

unfamiliar terrain, and that are gradually ﬁnding their way into cell phone network

base stations. With the development of the chip-scale atomic clock even battery-

operated, handheld devices like the GPS receivers or the cell phones themselves

might soon contain their own atomic clock. And would not an atomic wrist watch

be ever so cool?

13.8.1.1 Microwave Vapour-Cell Clock

Caesium and rubidium have two big practical advantages that distinguish them from

almost all other chemical elements. They can easily be manipulated with read-

ily available diode laser sources, and the vapour pressure above solid caesium or

rubidium is high enough to do useful spectroscopy even at room temperature. It is

therefore possible to place a small crumb of Cs or Rb into an evacuated glass cell,

shine a light beam through the cell and observe a spectroscopic signal (typically the

transmitted power) shaped by the optical properties of the atomic vapour.

In the simplest embodiment this light actually does not come from a laser but

from a spectral lamp containing the element of interest, typically rubidium. Natural

13 Atomic Clocks 397

rubidium consists of a mixture of two isotopes,

Rb and

Rb (Fig. 13.16). Most

of the light given off by the lamp corresponds to the D

and D

lines at 794 and

780 nm wavelength, respectively. By a lucky coincidence, the Doppler-broadened

|F = 3→|F



 absorption line in

Rb overlaps the |F = 2→|F



 absorption

line in

Rb; this is true both for the D

and the D

lines. The |F = 2→|F





light component emitted by an isotopically pure

Rb lamp is therefore absorbed in

an isotopically pure

Rb “ﬁlter” cell (Fig. 13.17), leaving a transmitted light beam

that can only interact with the |F = 1→|F



 absorption line in the main

cell. Atoms in ground state |F = 1 are optically excited and then decay into either

one of the ground states. Atoms in state |F = 2 are trapped because they cannot be

excited by the ﬁltered lamp light anymore. State |F = 2 is therefore called a “dark

state” for this particular light spectrum.

F = 1

F = 2

h × 6.8 GHz

F' = 2

F' = 1

F = 2

F = 3

F' = 3

F' = 2

1/2

Fig. 13.16 Level scheme for the D

excitation line of the two naturally occurring rubidium isotopes

lamp

filter cell

main cell

Magnetic shield,

microwave cavity,

bias field solenoid

Photo diode

Fig. 13.17 Set-up of a compact optical-pumping rubidium microwave clock. The typical volume

of such a clock is 250 cm

Muga G. Time in Quantum Mechanics - Vol. 2

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