Muga G. Time in Quantum Mechanics

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368 R. Wynands

1960 1970 1980 1990 2000 2010

−30

−20

−10

Year

Time difference (s)

UT1 − TAI

UTC − TAI

)

(

Fig. 13.3 The difference between UTC and TAI over the last decades (step-like curve)andthe

time lag of Earth’s rotation phase UT1 with respect to TAI (smooth curve). The data beyond July

2008 are an extrapolation calculated by IERS

accuracy over more than a few months. A leap second is announced about half a year

in advance by a bulletin of the International Earth Rotation and Reference Systems

Service (IERS) [8]. For instance, Issue 36 of Bulletin-C of the IERS published on

04 July 2008 announces another leap second at 23:59:60 h UTC on 31 December

2008 [14]. From then on UTC will lag TAI by 34 s – until the next leap second.

More information on the history and present situation regarding timescales can

be found in a review [15].

13.3 What Is a Clock?

From the clock-maker’s perspective, which we are assuming in this chapter, a prac-

tical description of a clock is this: It is a combination of a frequency standard and

a counter. The frequency standard (think of the pendulum in a pendulum clock)

provides a periodic sequence of events with a period of known length. The counter

(the clockwork consisting of cogs and wheels in the case of a pendulum clock)

counts the number of oscillation periods that have passed since some initial moment

in time and translates this into a suitable display of the passing of time, for instance,

by moving clock hands around a dial.

In the example of a quartz wrist watch the frequency standard is formed by a

specially shaped quartz crystal oscillating at 32,768 Hz. The counter is an electronic

circuit that counts off 32,768 periods and then advances the display by 1 s. Even

the good old hour glass ﬁts the above deﬁnition. The period is given by the time

the grains of sand or egg shell need to trickle through into the lower compartment,

while the counting is done by a human operator reversing the hour glass once all

grains have trickled through.

13 Atomic Clocks 369

All the clockmaker’s expertise goes into ensuring that the period of the frequency

standard remains constant, not only when the clock itself ages but also when the

conditions around it change. For instance, in an ill-constructed pendulum clock the

oscillation period might increase when a temperature rise leads to a lengthening of

the pendulum. Issues of air pressure, humidity and also aging properties of materials

and greases need to be taken into account and compensated. In an atomic clock,

other effects can lead to a change of the period, like changes in magnetic ﬁelds or

collisions of atoms among themselves or with the walls of a container.

Let us deﬁne a few technical terms here for use in the remainder of this chapter

(for the full formal deﬁnitions of metrological terms, see [16]). When the clock

period is different from its nominal value, this is called a frequency bias. Such a

bias is typically caused by a systematic effect, for instance, by the Zeeman effect

shifting an atomic energy level in the presence of a magnetic ﬁeld. The uncertainty

of a frequency standard describes how sure we can be that the period matches its

expected or nominal value, for instance, how well a clock reproduces the length

of the SI second. Of course there will always be a certain amount of period-to-

period ﬂuctuation (noise) in the length of the period, as characterized by the standard

deviation of the noise; this is called the instability of the standard. If a clock is

switched off and switched on again at a later time the reproducibility speciﬁes how

well the new clock output frequency matches the previous one; this last parameter

is sometimes called frequency retrace in technical speciﬁcations of commercially

available atomic clocks.

The terms primary and secondary frequency standards both denote a device that

has been thoroughly characterized such that all potential frequency biases are either

excluded or, if they cannot be avoided, at least understood so well that one can

correct the standard’s output by a corresponding amount. If such a standard is based

on the very transition used to deﬁne the length of the SI second, it is called a primary

standard.

Note that from a pragmatic point of view a clock with a bias is just as usable for

precise timing as a clock without bias, provided the bias is known with sufﬁcient

precision. All one needs to do is to correct the clock’s display by the daily bias

multiplied by the number of days that have passed. This only works as well as the

value of the bias is known – as quantiﬁed by the uncertainty of the clock. Deter-

mining and minimizing this uncertainty is perhaps the major task of the atomic

clock-maker: thinking about effects that could lead to a frequency bias, avoiding

them or correcting for them, and ensuring that operating conditions and corrected

frequency biases remain constant.

13.4 How an Atomic Clock Works

Simply put: Just like any other clock! The underlying phenomenon of known

duration in this case is the period of oscillation of an atomic transition moment

between two quantum states or of the electromagnetic wave needed to induce it. The

clockwork in general consists of sophisticated electronic circuitry, although lasers

are coming more and more into play.

370 R. Wynands

oscillator

output

signal

resonance

signal

oscillator

frequency

spectroscopy

detector

servo system

Fig. 13.4 Principle of operation of the atomic frequency standard. The output of an oscillator

probes a spectral feature and a detector registers the response of the system. A servo loop uses this

signal to retune the oscillator to the desired working point along the spectral feature. The external

output of the oscillator is thus referenced to an internal feature of the atom (or molecule or ion)

In an atomic clock the frequency standard is based on a transition between

energy levels in a quantum system, for instance, an atom, an ion or a molecule.

By stabilizing the frequency of an oscillator to this transition frequency one can

counteract technical noise, aging or other sources of drift of the oscillator fre-

quency. Figure 13.4 shows the main principle of operation of such an atomic fre-

quency standard. The oscillator not only provides the output signal of the stan-

dard but also is used to interrogate a spectral feature in the quantum system.

This spectral line is used as a frequency discriminator. A frequency excursion of

the oscillator leads to a change in detected signal. A servo loop then tunes the

oscillator back to the desired working point along the proﬁle of the spectral fea-

ture, for instance an atomic absorption line. The output of the frequency standard

therefore serves as a macroscopic representation of a microscopic property of the

atom in question, for instance, the energy difference between two particular energy

levels.

In order to characterize and quantify the frequency instability of a clock one

could in principle just compute the standard deviation of a sequence of clock read-

ings relative to a perfect (or at least a much better) clock. The smaller the number,

the more stable the clock is. However, using this deﬁnition a clock with a very stable

bias would erroneously be assigned a high instability. Even worse, this standard

deviation grows with time!

Instead, the time-and-frequency community uses the two-sample relative Allan

standard deviation σ

, or its square, the Allan variance σ

[17], which will be

described in the following. Consider the sequence of time intervals x

between the

“tick” of the clock under test and the “tick” of a perfect clock, taken at regular time

intervals τ . Next compute the normalized instantaneous frequency

= (x

i+1

− x

)/(2πν

τ ) , (13.1)

where ν

is the clock’s nominal tick frequency. Normalizing the instantaneous fre-

quency by the nominal one has the advantage that the relative stabilities of oscilla-

tors running at very different frequencies can be compared in a meaningful way.

13 Atomic Clocks 371

The variance of the sequence of y

estimated from a sample of size N –the

N-sample variance – is

N −1



i=1

−

, (13.2)

where

y = (



i=1

)/N is the average value of the instantaneous frequency. For

the characterization of frequency stability it has turned out [17] that the two-sample

variance possesses particularly useful properties. It is obtained by setting N = 2in

Eq. (13.2):

(τ ) =

− y

)



, (13.3)

where the τ dependence is implicit in the y

and the angle brackets indicate the

ensemble average.

Equation (13.3) basically represents (except for the factor of 1/2) the average

value of the squared relative frequency changes from sample point to sample point.

A convenient feature of this deﬁnition is that σ

(τ ), called the Allan variance in

the time-and-frequency community [17], is insensitive to a constant frequency bias

between the reference oscillator and the oscillator under test. In that sense the Allan

variance is a representative measure of the ﬂuctuations of a frequency around its

mean value. The square root of this variance is called the relative Allan standard

deviation (or “Allan deviation” for short). Note that in the special case of pure white

frequency noise σ

(τ ) coincides with the standard deviation of the Gaussian distri-

bution describing the frequency ﬂuctuations.

In practice, of course, one does not have an inﬁnite series of measurements x

at one’s disposal, so a ﬁnite approximation is used:

(τ ) =

2(k −1)

k−1



i=1

i+1

− y

)

, (13.4)

with k sufﬁciently large to get a meaningful estimate. Lesage and Audoin have pub-

lished a series of papers dealing with the question of what the uncertainty of that

estimate is [18, 19]. For instance, the relative uncertainty in the case of pure white

frequency noise is

Δ(σ

)

3k − 4

(k − 1)

(13.5)

Δ(σ

)

√

3k − 4

2(k −1)

(13.6)

372 R. Wynands

for the Allan variance and the Allan deviation, respectively. So typically one should

have k ≥ 19 to reach a 20% relative uncertainty for the Allan deviation, which

means a measurement of total duration 20τ .

By looking at the changes in instantaneous frequency from data point to data

point one removes the inﬂuence of a constant bias from the estimate σ

of the clock’s

instability. Of course, the Allan variance is still inﬂuenced by changes of a bias,

for instance, so in general σ

depends on averaging time τ . For white noise, this

dependence takes the form of τ

−1

for the Allan variance or τ

−1/2

for the Allan

standard deviation . When 1/ f noise is present the Allan deviation will level off at

some higher τ or even start to increase in the case of drift.

For the case of white frequency noise, which should govern the behaviour of a

good primary or secondary frequency standard, the Allan standard deviation can be

written as

(τ ) =

Δν

S/N



cycle

. (13.7)

Here Δν is the width of the atomic transition line of resonance frequency ν

, S/N

is the signal-to-noise ratio of the detection process, T

cycle

is the period at which any

corrections to the clock frequency are applied and τ is the averaging time.

It is clear from Eq. (13.7) that it only makes sense to build an atomic clock when

the frequency discriminator has a narrow linewidth Δν. This implies a steep slope at

the side of the resonance (or a sharp extremum at the centre of the resonance when

a modulation technique is used), so that already a very small frequency excursion of

the oscillator gives a detectable signal change.

One also sees that the actual resonance frequency ν

should be as high as possi-

ble. The choice of caesium is a good one in this respect because not many elements

provide a conveniently accessible microwave transition at a frequency as high as

9.2 GHz. The inﬂuence of ν

is one important reason why the clocks of the future

in all probability will be optical clocks. With a ν

of hundreds of THz one easily

gains several orders of magnitude compared to caesium. Of course, ν

is not the

only factor determining the instability of a clock, and the other factors can work for

or against the various candidate optical clocks. This will be discussed in more detail

in Sect. 13.9.

13.5 The “Classic” Caesium Clock

The publication in 1955 by Essen and Parry [20] is regarded as the birth of the

caesium atomic clock. Performance has greatly increased since then, but the work-

ing principle has remained the same even in today’s commercial caesium-beam

clocks.

13 Atomic Clocks 373

13.5.1 Why Caesium?

Choosing the alkali metal caesium for an atomic clock offers a number of advan-

tages. All radioactive isotopes of caesium have rather short half-lives, so naturally

occurring caesium only contains the only stable isotope,

133

Cs. As an alkali metal,

it has a

1/2

ground state, which simpliﬁes the hyperﬁne structure, even though

the nuclear spin is characterized by spin quantum number I = 7/2 (Fig. 13.5).

Since

133

Cs is bosonic there exists a pair of ground states with magnetic quantum

number m

= 0. Those two states show a Zeeman frequency shift in response to an

external quasi-static magnetic ﬁeld only in second order, which allows an essential

relaxation of the experimental requirements for the stability and homogeneity of

such an external ﬁeld. Due to selection rules this “clock transition” can be induced

only by an alternating magnetic ﬁeld polarized along an external static magnetic

ﬁeld.

From Eq. (13.7) we have seen that one wants a transition with a high-resonance

frequency. Caesium is well suited in this regard, because its 9-GHz ground state

hyperﬁne splitting is among the largest of all elements and not too large to be con-

veniently reached by microwave equipment available in the 1950s when the caesium

clock was invented. As an alkali atom, caesium easily parts with its outer electron, so

a hot-wire ionization detector can provide a high detection efﬁciency. Other prac-

tical advantages are that caesium can be evaporated at a sufﬁcient rate already at

rather modest temperatures, that caesium is not poisonous and it is readily available,

too.

In retrospect, the choice of caesium was an extremely fortunate one. Cae-

sium arguably is the chemical element that is most easily laser-cooled and laser-

manipulated, due to the fact that its level scheme is rather simple and that the 852-nm

wavelength of its D

line falls right into the range where silicon photodiodes are

most efﬁcient and laser diode is available commercially at high powers and with a

F = 3

F = 4

F' = 5

F' = 4

F' = 3

F' = 2

repumping

cooling

3/2

1/2

m = 0

clock transition (M1)

m = +3m = −3

m = +4m = −4

Fig. 13.5 Right: Simpliﬁed

133

Cs energy level diagram (not to scale) showing the hyperﬁne split-

ting of the 6

1/2

ground state and the 6

3/2

excited state. The excitation lines for laser cooling

and repumping, used in the caesium fountain clock, are indicated by arrows. Left: Blow-up of the

ground states, indicating the 16 Zeeman sublevels and the clock transition, a magnetic dipole (M1)

transition

374 R. Wynands

narrow linewidth. This has made it possible to switch from the thermal-beam clocks

to be described next to the fountain clocks described further below. The introduction

of fountain clocks reduced the uncertainty in the realization of the length of the SI

second by more than an order of magnitude over the very best thermal-beam clocks,

and perhaps another order of magnitude in the near future. It is hard to imagine a

similar success and a similar improvement in the realization of the unit of time had

Essen and Parry chosen another chemical element for their atomic clock.

13.5.2 The Thermal-Beam Caesium Clock

Caesium atoms emanate from an oven at typically 170

◦

C and cross a ﬁrst inhomo-

geneous magnetic ﬁeld (Fig. 13.6). In PTB-CS1 and PTB-CS2, for instance, this

ﬁeld can be produced by a cylindrical quadrupole or a hexapole or a combination of

both. Roughly speaking, atoms in state |F = 4 are collimated by the magnet and

those in |F = 3 defocused and lost from the beam. There are some tricky details

here which are described in the literature (see [21] and references therein).

microwave

oscillator

output

signal

microwave

cavity

oven

detector

polarizer

magnet

analyzer

magnet

servo system

Fig. 13.6 Principle of operation of the “classical” caesium atomic clock making use of a thermal

beam of atoms emanating from an oven. The atoms are spin selected by a multipole magnet, then

traverse a microwave resonator structure, before a second magnet separates the atoms that did make

the transition from those that did not. A servo loop adjusts the microwave frequency such that an

optimum number of atoms arrive at a hot-wire detector. In later versions the magnets and detector

have been replaced by suitable laser excitation zones, the so-called optically pumped thermal-beam

clocks

The net result is that after the magnet the caesium beam is spin polarized and

then crosses the microwave interaction region. The peculiar two-pronged shape of

the microwave cavity is explained in the next section. Note that the microwave inter-

action has to take place inside a resonant cavity because only in this way the phase

13 Atomic Clocks 375

of the microwave radiation can stably be controlled over the whole cross section and

interaction length of the atomic beam.

A second magnet deﬂects atoms that made the transition to |F =3 onto a hot-

wire detector (“ﬂop-in conﬁguration”), where the atoms are ionized and a small

current (≈ 10 pA) is registered. So the fact that atoms are arriving at the detector is

an indication that the microwave frequency is approximately correct. An electronic

servo loop then adjusts the frequency to the optimum value, as registered in the

detected signal.

The output signal of the clock consists of a microwave signal of frequency 9,

192, 631, 770 Hz. In other words, 1 s is over once the microwave has completed 9,

192, 631, 770 periods of its oscillation – just as the deﬁnition of the SI second says.

The microwave signal feeding the cavity for probing of the atomic transition

is typically synthesized from a low-noise voltage-controlled quartz crystal oscilla-

tor (VCO). Usually the VCO is weakly phase-locked to a hydrogen maser which

serves as a frequency reference for the caesium clock. By proper multiplication and

mixing of the VCO frequency with the frequency of a radio frequency synthesizer

the 9.2-GHz microwave signal for the interrogation is generated. By square-wave

frequency modulation of the synthesizer output frequency with an amplitude of half

the linewidth, the atoms are probed alternatively at the left and the right sides of the

resonance line. The difference Δp of the two transition probabilities for each side is

a measure of the offset of the microwave carrier frequency from the centre of the res-

onance line. If it is not 0, the synthesizer frequency is corrected accordingly. In this

case the series of correction values gives the relative frequency difference between

the clock and the frequency reference. Alternatively Δp can be used directly to

control the frequency of a VCO in a servo loop. In this case the VCO output directly

represents the clock frequency.

The measured clock frequency differs from the unperturbed caesium transition

frequency ν

= 9, 192, 631, 770 Hz by the sum of all frequency biases. For exam-

ple, this sum is of the order of 2 Hz in the case of PTB-CS2, a thermal-beam clock

in operation since 1985. The correction amounts to about 1 mHz in the case of PTB-

CSF1, a fountain primary clock in operation since 1999.

13.6 The Ramsey Technique

The microwave cavity of a caesium atomic beam clock is not a simple rectangular or

cylindrical box (Fig. 13.6). Instead, the microwave cavity is split into two interaction

zones, so that the Method of Separated Oscillatory Fields can be employed. This

method is usually called the Ramsey method, after Norman Ramsey who invented

it in 1949 [22] and received the 1989 Nobel Prize in Physics for it (and for his

contributions to the hydrogen maser, to be discussed in Sect. 13.8.2).

A big problem facing the atomic clock-maker is posed by the uncertainty relation

between energy (or frequency) and time [23]. If we want to determine the exact

transition frequency in an atom with ever higher accuracy, we need to achieve ever

376 R. Wynands

smaller linewidths Δν. This means we need to observe the unperturbed transition

moment over ever longer time Δt. For the simplest microwave cavity, a cylindrical

box roughly one-half wavelength long, the interaction time for caesium atoms with

a velocity selected around v ≈ 100 m/s (as in PTB-CS1) would lead to an observed

linewidth Δν of about 10 kHz.

It was found by Ramsey, however, that one could split the interaction zone into

two parts separated by some longer distance L and obtain a resonance feature with

a width inversely proportional to the transit time L/v. For a thermal-beam clock the

Ramsey method provides a linewidth reduced by l/L, which could amount to one

or two orders of magnitude. In the following we will look at the resulting line shape

in more detail.

If we assume for simplicity that the microwave, as seen by the atom, is switched

on and off instantaneously when the atom enters each arm of the cavity and that

its amplitude is constant within each interaction zone, the transition probability is

given by the expression [2]

P(δ) =

sin

(Ωτ/2)



cos(Ωτ/2) cos[(δT +φ)/2] −

sin(Ωτ/2) sin[(δT +φ)/2]



(13.8)

Here τ and T are the transit times of an atom through one interaction region and

through the space between the two regions, respectively. The detuning δ is positive

when the frequency of the microwave is higher than the resonance frequency of the

atom, and Ω =

√

+δ

.TheRabi frequency b = μB

/ is a measure of the

strength of the interaction, containing both the magnetic dipole moment μ of the

transition and the amplitude B

of the microwave magnetic ﬁeld.

Figure 13.7 shows an example of the resulting line shape as a function of δ.The

graph of Ramsey transition probability as a function of frequency detuning looks

just like the graph of the light intensity as a function of position in the far ﬁeld

of a double-slit diffraction pattern in optics. Of course, this is not a coincidence

because in both cases there are two pathways along which the system can reach

the same ﬁnal state from the same initial state. While in the optical diffraction case

the photon can have gone through one slit or the other, so to speak, here the atom

can undergo the transition in the ﬁrst or in the second Ramsey zone. Since the end

results are indistinguishable, the rules of quantum mechanics state that one has to

add the probability amplitudes for the two pathways and square the result to obtain

the outcome seen on the detector, leading to interference fringes. The atomic clock

with Ramsey excitation therefore constitutes a quantum-mechanical device, an atom

interferometer.

One recognizes the dotted line as the line shape when only one zone of twice

the length is present, i.e. T = 0 in Eq. (13.8) with properly adjusted b. Under

this envelope the transition probability oscillates rapidly with detuning. This is

13 Atomic Clocks 377

–200 –100 0 100 200

0.0

0.5

1.0

Frequency detuning (Hz)

Transition probability

Fig. 13.7 Examples of a calculated Ramsey interference pattern as a function of frequency detun-

ing δ. The parameters used in Eq. (13.8) are τ = 11.12 ms, T = 40 ms, b = 141.21/s. The dotted

line is the envelope of the fringe pattern, obtained by averaging over rapidly oscillating terms in

(13.8)

analogous to the optical case where the double-slit diffraction pattern consists of

high spatial-frequency fringes under an envelope given by the diffraction pattern for

a single slit of the same width.

The Ramsey technique has a number of advantages even over a single interaction

zone of the same total length 2l + L, some of which are listed here without further

comments:

• The central Ramsey fringe is actually narrower almost by a factor of 2 than if the

interaction time would have been 2τ + T in a single long zone.

• For a laterally extended beam of atoms susceptible to Zeeman shift of the res-

onance frequency, each atom only has to see the same average magnetic ﬁeld

along its trajectory. As long as this condition is fulﬁlled one need not fulﬁl the

much harder condition that the ﬁeld is constant everywhere on and off axis.

• When the phase difference of the microwave ﬁelds in the two zones is constant

there is no ﬁrst-order Doppler shift or broadening.

There are other useful features of the Ramsey technique that are important in

other ﬁelds of physics. They have been omitted here because they have less rele-

vance for the application of the method in atomic clocks.

Although there is no ﬁrst-order Doppler broadening of the central Ramsey fringe

there is nonetheless a large inﬂuence of the velocity distribution of the atoms

(Fig. 13.8). In a thermal-beam clock, fast atoms experience a shorter time T between

the two parts of the Ramsey interaction, resulting in a larger fringe width than for

slower atoms with larger T . As a result, the fringe contrast washes out with increas-

ing detuning from the central fringe. This is once again completely analogous to the

optical case, with the role of the atomic velocity distribution played by a limited

coherence length of the light illuminating the double slit. The background that the

remaining fringes sit on is called the Rabi pedestal; it corresponds to the resonance

Muga G. Time in Quantum Mechanics - Vol. 2

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