Muga G. Time in Quantum Mechanics

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378 R. Wynands

Frequency detuning (Hz)

Transition probability

–50 0 50

0.5

1.0

0.5

0.0

0.5

Rabi pedestal

Fig. 13.8 Calculated Ramsey interference fringes as a function of frequency detuning δ/2π for

Gaussian velocity distributions of different standard deviations σ

centred at a mean velocity of

¯v = 2.5 m/s (other parameters as in Fig. 13.7). Top: monokinetic beam. Centre: σ

= 0.1m/s.

Bottom: σ

= 0.4 m/s. The fringes wash out to the Rabi pedestal, which is one-half of the envelope

of the fringe pattern for monokinetic atoms

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Frequency detuning (kHz)

Transition probability

–4 –2 0 42

Fig. 13.9 Measured Ramsey fringe pattern for PTB’s CS1 thermal-beam clock

linewidth of a single microwave excitation in one of the zones, so it is much wider

than a fringe of the Ramsey curve.

A typical example of a measured Ramsey fringe pattern is shown in Fig. 13.9

for the case of PTB-CS1. The curve appears upside down compared to Fig. 13.8

because PTB-CS1 is operated in the so-called ﬂop-out conﬁguration, so that the

signal at the detector is at its minimum exactly on resonance. The thermal velocity

13 Atomic Clocks 379

spread is small enough that more than a dozen fringes can be seen on either side of

the maximum. The contrast of the Ramsey fringe is only 55% because the magnetic

state selection mostly selects for |F only, more or less regardless of the magnetic

quantum number m

. Therefore, there is always a background of |F = 4, m

= 0

atoms present at the detector.

The lowest relative uncertainty achieved for thermal-beam clocks with magnetic

state selection is 8 ×10

−15

in PTB-CS1, using a 73.6-cm separation of the Ramsey

zones [21]. Longer zones are not practical, in particular because of the increasing

inﬂuence of the exact shape of the atomic velocity distribution on potential fre-

quency biases. Some improvement can be obtained by replacing state selection and

detection by laser optical pumping and laser-induced ﬂuorescence [24–26]. Two

such clocks still are in operation today, SYRTE-JPO [24] and NICT-O1 [26], with

relative uncertainties of 6.4 and 6.8 parts in 10

, respectively.

13.7 Atomic Fountain Clocks

New developments in quantum optics, in particular the advent of laser-cooling tech-

niques [27–30], opened the door to a radically new approach to the interaction time

problem. Using a suitable arrangement of laser beams and magnetic ﬁelds one can

capture caesium atoms from a thermal vapour and at the same time cool them down

to just a few microkelvin above absolute zero temperature. Typically, in a few tenths

of a second one can trap 10 million caesium atoms in a cloud a few millimetres

in diameter and at a temperature of a microkelvin. At this temperature, the average

thermal velocity of the caesium atoms is of the order of 1 cm/s, so the cloud of atoms

stays together for a relatively long time. This cloud can be launched against gravity

using laser light. Typically, the launch velocity is chosen such that the atoms reach

a height of about 1 m above the trapping region before they turn back and fall down

the same path they came up. Operation of a fountain clock typically proceeds in a

pulsed mode, with the trapping–launching–detecting cycle taking a time T

cycle

∼ 1s.

The motion of the cloud resembles that of the water in a pulsed fountain, hence the

name “fountain clock”.

On the way up and on the way down the atoms pass through the same microwave

cavity. The two spatially separated Ramsey interactions of the thermal-beam clock

are thus replaced by two interactions in the same position but with reversed direction

of travel. The microwave power is chosen such that on each pass a π/2 pulse is

experienced by the atoms. Ideally, therefore, after the ﬁrst microwave interaction

the atoms are in a quantum-mechanical, coherent superposition of both clock states

with equal weight for each state. After the second passage through the cavity the

transition from one clock state to the other is “completed” (in the fully resonant

case).

Following the second interaction the state of the atom can be probed by laser-

induced ﬂuorescence. For a typical launch height around half a metre above the

microwave interaction zone it is possible to achieve effective interaction times of

380 R. Wynands

–100 –50 0 50 100

0.0

0.2

0.4

0.6

0.8

1.0

Frequenc

detunin

(Hz)

Transition probability

–4 0 4

Fig. 13.10 Measured Ramsey fringe pattern for PTB’s CSF2 fountain clock. Inset: the central part

enlarged. The slowly oscillating curve is the Ramsey fringe pattern of the thermal-beam clock

PTB-CS1 already shown in Fig. 13.9; its curvature is so small near detuning zero that the curve

appears almost ﬂat in the inset

more than half a second. The resulting 100-fold reduction in microwave resonance

linewidth (Fig. 13.10) is the most obvious advantage of a fountain clock over a

thermal-beam clock. Note also that the contrast of the resonance is approaching

100% because the atomic cloud can be prepared such that it consists of |m

= 0

atoms only.

Equally important is a reduction in the inﬂuence of certain systematic effects

because the atoms are so slow. Furthermore, the trajectory reversal eliminates the

frequency bias that might exist due to a small and all but unavoidable manufacture-

related phase difference between the microwave ﬁelds in the two interaction zones.

It also greatly reduces the inﬂuence of any transverse or longitudinal phase gradients

of the microwave ﬁeld inside the cavity.

Employing a laser-cooled sample is essential for a fountain clock because in

thermal samples the number of slow atoms is too small to obtain a sufﬁcient signal

strength [31] and because one would need a vacuum apparatus of several metres

in height in order for some of the slower atoms to come to a stand-still in it and

fall back [32]. The ﬁrst considerations [33] and successful realizations [34] of the

fountain principle were followed by the ﬁrst metrological fountain, FO1, at the

Observatoire de Paris/France [35]. Its design [36] became a standard for almost

all subsequently constructed fountain clocks, with some variations of course. In

the late 1990s the fountains NIST-F1 at the National Institute of Standards and

Technology (NIST) in Boulder/USA [37, 38] and CSF1 at Physikalisch-Technische

Bundesanstalt (PTB) in Braunschweig/Germany [39–41] became operational as pri-

mary standards. Recently, they were joined by the caesium fountain clocks CsF1

at the Istituto Elettrotecnico Nazionale (IEN) in Torino/Italy [42], CsF1 at the

National Physical Laboratory (NPL) in Teddington/GB [43], F1 at National Metrol-

ogy Institute of Japan (NMIJ) [44] and CsF1 at National Institute of Information and

13 Atomic Clocks 381

Communication Technology Laboratory (NICT) also in Japan [45]. With FO2 and

FOM [46] the Observatoire de Paris (SYRTE) is operating two more primary foun-

tain clocks. A number of other laboratories are currently developing fountain clocks

(see [47–54] for an incomplete list of examples). Most of these are employing cae-

sium as the active element.

Because fountain clocks currently constitute mankind’s best way to realize the

unit of time we will now discuss those clocks a little further. Many more details can

be found in a review [55] and the speciﬁc references therein.

13.7.1 Operation of a Fountain Clock

Figure 13.11 shows a simpliﬁed set-up of the vacuum subsystem of a fountain clock.

Six laser beams cross in the centre of the preparation zone, where the cold atom

cloud is produced. Above that follows the detection zone which is traversed by laser

beams for ﬂuorescence detection of the falling cloud. The microwave interactions

take place inside a magnetic shield in the presence of a well-deﬁned internal lon-

gitudinal magnetic ﬁeld (the “C-ﬁeld”). The clock is operated in a pulsed fashion,

with a cycle typically lasting about 1 s (Fig. 13.12).

Magnetic

shields

C-field coil

Vacuum tank

1 m

Ramsey cavity

State-selection

cavity

Caesium

reservoir

to pump

and window

Detection zone

Preparation

of cold atoms

.............................

Fig. 13.11 Simpliﬁed set-up of the atomic fountain clock PTB-CSF1

382 R. Wynands

(f)(c)

State-

selection

cavity

(d) (e)(a) (b)

Detection

lasers

Ramsey

cavity

Fig. 13.12 Principle of operation of the atomic fountain clock (simpliﬁed). The full cycle

(a)–(g) takes about T

cycle

≈ 1s.(a) A cloud of cold atoms in states |F = 4, m

 is trapped in the

intersection region of six laser beams. (b) The cloud is launched by frequency detuning of the ver-

tical lasers. (c) Inside the state-selection cavity certain atoms are transferred to |F = 3, m

= 0.

(d) When the cloud exits the state-selection cavity a laser pulse pushes away all atoms not in

|F = 3, m

= 0.(e) The cloud slowly expands during its ballistic ﬂight in the dark.Ontheway

up and on the way down it passes through the main microwave cavity, the Ramsey cavity, where

the transition to |F = 4, m

= 0 is driven. (f) Detection lasers are switched on; they probe the

population distribution by laser-induced ﬂuorescence

The source of caesium atoms traditionally consists of a temperature-controlled

caesium reservoir held at a suitable temperature near room temperature in order

to obtain a caesium partial pressure in the order of 10

−6

Pa in the cooling cham-

ber. Alternatively, a laser-cooled atomic beam can be directed at the main trap-

ping region, thus shortening the trapping time from a few hundred to a few tens

of milliseconds; chirp-slowed [46], LVIS [56] and 2D-MOT [57] beams have been

used.

To obtain the required low temperatures of the atom samples in an atomic foun-

tain the atoms are cooled in a magneto-optical trap (“MOT”) [58] and/or an optical

molasses (“OM”) [59, 60]. Key to both conﬁgurations is a set-up consisting of three

mutually orthogonal pairs of counter-propagating laser beams. There are either two

vertical and four horizontal beams (the (0,0,1) geometry, Fig. 13.11) or there are

three beams, arranged symmetrically around the vertical, coming from below and

the other three from above (called the (1,1,1) geometry).

Two laser frequencies are needed for efﬁcient cooling (Fig. 13.5). The main

power in the six beams is provided by light tuned slightly to the red (low-frequency)

side of the cyclic |F = 4→|F



= 5 caesium transition in order to scatter a large

number of photons in a short time. Small polarization imperfections in connection

with off-resonant excitation to the excited state |F



= 4can lead to optical pumping

into the |F = 3 state. A repumping laser beam tuned to the caesium transition

|F = 3→|F



= 4 is therefore superimposed on at least one of the six cooling

laser beams. It depletes the |F = 3level so that all atoms can continue to participate

in the cooling process. Additional laser-cooling steps are typically applied before

13 Atomic Clocks 383

and/or after the cloud has been launched towards the cavities. More details on this

are presented in [55].

Typically 10

,...,10

atoms are trapped and cooled. This atom number is

mainly limited by the available laser power and laser beam diameters. For reasons

of compactness, low power consumption, reliability and ease of use, most optical

set-ups use exclusively laser diode systems. Their light powers and frequencies need

to be controlled precisely in order to properly cool, launch and detect the atoms.

Furthermore, all laser light has to be blocked completely during the interaction of

the atoms with the microwave ﬁeld so that the atomic transition frequency is not

shifted by the ac Stark effect [61].

After trapping, the atoms are launched by the “moving molasses” technique

(Fig. 13.12b). The pattern formed by the interference of the vertical trapping beams

can be made to move at a velocity cδν/ν

when the upward-directed laser beam is

tuned to a frequency ν

+δν and the downward-directed laser beam to ν

−δν.Inthis

upward-moving interference pattern the atoms are accelerated within milliseconds

to velocities of several metres per second. At this stage in the fountain cycle the

atomic population is distributed across all ground state sublevels, mostly those with

|F = 4 (Fig. 13.13a).

The last cooling stage consists of a sub-Doppler laser-cooling phase, as described

in [55]. When the lasers are ﬁnally switched off altogether the caesium atom

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 13.13 Qualitative population of the Zeeman sublevels in the caesium ground state during key

phases of the fountain cycle. (a) After trapping and launching. (b) After post-cooling. (c)After

passage through the state-selection cavity. (d) After the laser blow-away pulse. (e) After the ﬁrst

passage through the Ramsey cavity. (f) After the second passage through the Ramsey cavity. Here

is assumed that the microwave frequency is slightly detuned from exact atomic resonance. (g)Just

before the atoms in |F = 3 are detected

384 R. Wynands

temperature is about 1 μK, corresponding to a thermal velocity of less than 1 cm/s.

When the repumping laser is switched off slightly later than the cooling laser all

atoms end up in an |F = 4 level (Fig. 13.13b).

As the next step a further state-selection process can be applied to the atoms in

order to reduce the background signal and the collisional shift due to atoms in states

with m

= 0 which do not take part in the “clock transition” between the states with

= 0. This removal of atoms with m

= 0 is a big advantage of fountain clocks

over thermal-beam clocks also because it reduces effects like Rabi and Ramsey

pulling and frequency shifts due to Majorana transitions [62], apart from improving

the contrast of the Ramsey pattern (Fig. 13.10). For the state selection the |F =

4, m

= 0 atoms are ﬁrst transferred by a microwave π pulse using the clock

transition to the state |F = 3, m

= 0. This can be done in a microwave state-

selection cavity in which the atoms pass before they enter the Ramsey cavity for the

ﬁrst time (Figs. 13.12c and 13.13c). Afterwards all the atoms which remained in the

|F = 4 state are pushed away by the strong spontaneous light force exerted by a

laser beam tuned to resonance with the |F = 4→|F



= 5 transition. This results

in a pure atomic sample of atoms all in state |F = 3, m

= 0 entering the Ramsey

cavity (Fig. 13.13d), where subsequently the clock transition |F = 3, m

= 0→

|F = 4, m

= 0 is excited.

Let us note here that this extreme nonequilibrium distribution of atomic pop-

ulation is attractive not only for time keeping. It can be used to perform precise

spectroscopic experiments with a state-selected, monokinetic, cold sample of atoms,

which can still be manipulated by laser light, microwaves or electric ﬁelds for an

observation time of about half a second. This single-populated state need not be the

|F = 3, m

= 0 state but can be almost any other of the 16 sublevels when the

microwave frequency in the state-selection cavity is tuned to a different transition.

Examples for planned or completed experiments include the measurement of the

black-body radiation shift of the clock transition when the vacuum tube is heated

[63], the investigation of state-dependent Feshbach resonances in the collisional

cross sections between cold caesium atoms [64] or the proposed search for parity

violation in atoms [65]. The fountain principle can also be used to determine the

gravitational constant G [66], the local gravitational acceleration g [67] or the ﬁne

structure constant α [68].

One of the most critical parts of a fountain clock is the microwave cavity for

the Ramsey interaction. Much work has been done on different realizations of these

delicate devices (see [55] for more details and references). Most fountain clocks

use a cylindrical microwave cavity with the ﬁeld oscillating in the TE

011

mode.

This mode (indicated in Fig. 13.14) exhibits particularly low losses which results

in a particularly small running-wave component in the cavity. The dependence of

the microwave phase on the transverse position of the atomic trajectory is therefore

small, as well. The oscillating microwave magnetic ﬁeld inside the cavity is directed

primarily along the vertical, the same direction as the static magnetic C-ﬁeld of

typically 100 nT ﬂux density. Selection rules therefore favour the Δm = 0 transi-

tions. The static ﬁeld in addition detunes all other transitions, so that even the small

curvature of the microwave ﬁeld lines near the end caps of the cavity does not lead

13 Atomic Clocks 385

Fig. 13.14 Sketch of the

microwave ﬁeld geometry

inside a typical TE

011

cylindrical microwave cavity.

Dotted line: trajectory of the

atoms

to a substantial amount of parasitic Δm

=±1 transitions. As a result, the clock

transition |F = 3, m

= 0→|F = 4, m

= 0 dominates.

After the ﬁrst passage through the Ramsey cavity the atomic population looks

like as indicated in Fig. 13.13e: a coherent superposition of the two clock states.

This atomic coherence precesses at its intrinsic rate, and during the second cavity

passage, on the way down, its phase is probed with the microwave again. In the

case of exact resonance, for instance, the microwave phase relative to that of the

atomic coherence is still the same, so the transition is completed and all atoms end

up in the upper state |F = 4, m

= 0. Figure 13.13f depicts the situation when

the microwave frequency is slightly detuned so that the transition probability is not

100%.

In a fountain clock the atoms arriving in |F = 4 and those in |F = 3 are

detected separately via laser-induced ﬂuorescence. When the falling atoms pass

through a ﬁrst transverse standing-wave light ﬁeld tuned to the |F = 4→|F



= 5

transition the ﬂuorescence light emitted by the atoms in |F = 4 is detected with

a photodiode. The time-integrated photodetector signal, N

, is proportional to the

number of atoms in the state |F = 4. From the shape of the time-dependent pho-

todetector signal one can infer the axial velocity spread of the atoms in the cloud,

which indicates the corresponding kinetic temperature. The |F = 4 atoms are

then pushed away by a second beam slightly below, a transverse travelling-wave

laser ﬁeld tuned to the |F = 4→|F



= 5 transition. Only the atoms in state

|F = 3 remain (Fig. 13.13g). These are then pumped to the state |F = 4 by

a third horizontal detection laser beam, positioned lower again and tuned to the

|F = 3→|F



= 4 transition. Superposed or slightly below another laser beam

is present, which is again tuned to the |F = 4→|F



= 5 transition. In effect,

the atoms in |F = 3 are ﬁrst pumped to the state |F = 4 and then detected

by their ﬂuorescence on the |F = 4→|F



= 5 transition with the help of a

second photodetector, giving a measure N

for the number of atoms that arrived in

the detection zone in the state |F = 3.

386 R. Wynands

In the servo system the ratio N = N

/(N

+ N

) is calculated. This ratio is

independent of the shot-to-shot ﬂuctuations in atom number, which typically lies

in the percent range, and is used as the input signal to the microwave-frequency

servo loop. As in thermal-beam clocks, the interrogation microwave is square-wave

frequency modulated, here with a period of 2T

cycle

, so that alternate “shots” are taken

at the left and the right sides of the central Ramsey fringe. From the difference of

left

and N

right

the correction signal for the microwave carrier frequency is derived.

13.7.2 Uncertainty Budget

For a precise frequency standard a detailed knowledge of all sources of uncertainty

is absolutely essential. Here we will present a rather complete list of such contri-

butions for the case of a primary fountain clock. We are choosing this clock as an

example because of its great importance today. For the optical clocks of the future

the list will be very similar in principle, but of course with some modiﬁcations in the

details. For example, the role of the cavity-related shifts in a microwave clock, dis-

cussed on Page 389, will be played by the wavefront curvature of the interrogation

laser beam(s) in an optical clock. Similarly, the Dick effect (explained below) in an

optical clock is due to the frequency noise on the interrogation laser rather than on

the noise of a microwave oscillator.

Two types of uncertainties have to be considered. Statistical uncertainties in gen-

eral decrease with longer averaging time, whereas systematic uncertainties reﬂect

our imperfect knowledge of potential frequency biases and, with the exception of

drifts of operating parameters, are independent of averaging time.

13.7.2.1 Statistical (Type A) Uncertainties

In a well-designed fountain clock the following noise contributions to the total insta-

bility have to be considered [69]:

(a) Quantum Projection Noise [70]: resulting from the fact that a fountain clock is

operated alternatingly at the left and the right sides of the central Ramsey fringe

where the transition probability is neither 0 nor 1.

(b) Photon Shot Noise: resulting from the statistical detection of a large number of

photons per atom.

typically a photodiode followed by a transimpedance ampliﬁer.

(d) Local Oscillator Noise: resulting from a downconversion process of local oscil-

lator frequency noise components because of the noncontinuous probing of

the atomic transition frequency in a pulsed fountain (“Dick effect”). Basically,

phase excursions of the local oscillator while no atoms are in or above the

microwave cavity will go undetected and therefore add some of the phase noise

of the local oscillator to the output of the clock.

13 Atomic Clocks 387

When assuming a white frequency spectrum of the noise sources Eq. (13.7) takes

the following form [69]:

(τ ) =

Δν

πν



cycle



2σ

δN

+γ



1/2

. (13.9)

The terms in the brackets of (13.9) quantify the four processes (a)–(d) listed

above. N

is the number of detected atoms, n

the average number of photons

scattered per atom at the detection and ε

is the photon collection efﬁciency. σ

δN

the uncorrelated rms ﬂuctuation of the atom number per detection channel.

With sufﬁciently high numbers of photons detected per atom and using state-

of-the-art low-noise electronic components, the noise contributions (a), (b) and (c)

can be reduced to such a level that the frequency instability is determined by the

Dick effect (d). A quantitative description of the Dick effect is beyond the scope of

this chapter; it can be found in [71–74]. With the best currently available voltage-

controlled oscillators (VCOs) the relative frequency instability is limited to the order

of 10

−13

(τ/s)

−1/2

In order to improve on that one can use a better local oscillator or reduce the

dead time of the fountain, i.e. the fraction of the fountain cycle where no atoms

are in or above the microwave cavity. However, some dead time is unavoidable

in the standard pulsed fountain described so far because one has to wait until the

detection process is ﬁnished before the next cloud can be launched. A continuous-

beam fountain clock, however, would be practically immune from the Dick effect.

Conceptually the easiest way to reduce the inﬂuence of the Dick effect on the

short-term instability is to use a more stable local oscillator. Using a cryogenic sap-

phire oscillator developed at the University of Western Australia [75] and a specially

designed low-noise microwave synthesis chain the group at SYRTE was able to

reach a fractional frequency instability of only 1.6 × 10

−14

(τ/s)

−1/2

[76], which

was only limited by quantum projection noise [69].

The additional effort of liquid-helium refrigeration required for low-noise cryo-

genic sapphire oscillators can be avoided when using a laser frequency comb (see

Sect. 13.9.2 below) to transfer the superior short-term stability of a laser stabilized

to a high-ﬁnesse optical cavity down into the microwave regime and then deriving

the 9.2-GHz interrogation signal from that phase-stable microwave. Initial results of

the phase stability of such a signal have been reported recently [77].

To speed up the loading and preparation of the cold atomic cloud the atoms can

be collected not from the residual background vapour but from a slow atomic beam

instead, as was discussed above in Sect. 13.7.1.

Loading from a slow beam is indispensable when implementing a multi-toss

scheme. The idea is to have more than one ball of atoms travelling inside the appa-

ratus simultaneously. For instance, the same number of atoms as in a single-ball

fountain can be distributed over several balls. In each of them the atomic density is

lower, and the collisional shift reduced in proportion. One implementation scheme

is to launch several balls (up to 10 or so) in quick succession but with successively

Muga G. Time in Quantum Mechanics - Vol. 2

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