The quantum physics of atomic frequency standarda

The Quantum
Physics of
Atomic
Frequency
Standards
Recent Developments


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The Quantum
Physics of
Atomic
Frequency
Standards
Recent Developments

Jacques Vanier

Université de Montréal, Montréal, Canada

Cipriana Tomescu

Université de Montréal, Montréal, Canada


CRC Press
Taylor & Francis Group
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Version Date: 20150617
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Contents
Preface.................................................................................................................... xiii
Introduction ............................................................................................................xvii
Authors ....................................................................................................................xix

Chapter 1

Microwave Atomic Frequency Standards: Review and Recent
Developments .......................................................................................1
1.1

Classical Atomic Frequency Standards .....................................2
1.1.1 Cs Beam Frequency Standard ......................................2
1.1.1.1 Description of the Approach Using
Magnetic State Selection ..............................3
1.1.1.2 Review of Frequency Shifts and Accuracy .... 7
1.1.1.3 Frequency Stability of the Cs Beam
Standard ...................................................... 15
1.1.1.4 Recent Accomplishments ........................... 16
1.1.2 Hydrogen Maser ......................................................... 33
1.1.2.1 Active Hydrogen Maser .............................. 33
1.1.2.2 Passive Hydrogen Maser ............................. 48
1.1.2.3 Frequency Stability of the Hydrogen
Maser .......................................................... 53
1.1.2.4 State of the Art of Recent
Developments and Realizations .................. 57
1.1.3 Optically Pumped Rb Frequency Standards .............. 69
1.1.3.1 General Description .................................... 69
1.1.3.2 State-of-the-Art Development .................... 71
1.2 Other Atomic Microwave Frequency Standards ..................... 82
1.2.1 199Hg+ Ion Frequency Standard .................................. 83
1.2.1.1 General Description .................................... 83
1.2.1.2 Frequency Shifts ......................................... 85
1.2.1.3 Linear Trap ................................................. 88
1.2.2 Other Ions in a Paul Trap ...........................................90

1.2.2.1 171Yb+ and 173Yb+ Ion Microwave
Frequency Standards .................................. 91
1.2.2.2 201Hg+ Ion Microwave Frequency
Standard ...................................................92
1.3 On the Limits of Classical Microwave Atomic Frequency
Standards ................................................................................. 93
Appendix 1.A: Formula for Second-Order Doppler Shift..................94
Appendix 1.B: Phase Shift between the Arms of Ramsey Cavity ......95

v


vi

Contents

Appendix 1.C: Square Wave Frequency Modulation
and Frequency Shifts .......................................................................... 95
Appendix 1.D: Ring Cavity Phase Shift ............................................97
Appendix 1.E: Magnetron Cavity ...................................................... 98
Chapter 2

Recent Advances in Atomic Physics That Have Impact
on Atomic Frequency Standards Development ................................ 101
2.1

2.2

2.3
2.4

2.5

Solid-State Diode Laser......................................................... 102
2.1.1 Basic Principle of Operation of a Laser Diode ..........102
2.1.2 Basic Characteristics of the Semiconductor
Laser Diode .............................................................. 105
2.1.3 Types of Laser Diodes .............................................. 106
2.1.4 Other Types of Lasers Used in Special Situations.....108
Control of Wavelength and Spectral Width
of Laser Diodes .................................................................. 109
2.2.1 Line Width Reduction .............................................. 109
2.2.1.1 Simple Optical Feedback .......................... 109
2.2.1.2 Extended Cavity Approach ....................... 109
2.2.1.3 Feedback from High-Q Optical Cavities ...112
2.2.1.4 Electrical Feedback .................................. 112
2.2.1.5 Other Approaches ..................................... 112
2.2.1.6 Locking the Laser to an Ultra-Stable
Cavity ........................................................ 113
2.2.2 Laser Frequency Stabilization Using an Atomic
Resonance Line ........................................................ 116
2.2.2.1 Locking the Laser Frequency to Linear
Optical Absorption ................................... 116
2.2.2.2 Locking the Laser Frequency
to Saturated Absorption ............................ 117
Laser Optical Pumping.......................................................... 119
2.3.1 Rate Equations ......................................................... 120
2.3.2 Field Equation and Coherence ................................. 122
Coherent Population Trapping ............................................... 127
2.4.1 Physics of the CPT Phenomenon ............................. 129
2.4.2 Basic Equations ........................................................ 131

Laser Cooling of Atoms ........................................................ 136
2.5.1 Atom–Radiation Interaction ..................................... 138
2.5.1.1 Effect of a Photon on Atom External
Properties: Semi-Classical Approach ....... 138
2.5.1.2 Quantum Mechanical Approach............... 143
2.5.2 Effect of Fluctuations in Laser Cooling
and Its Limit ............................................................. 158
2.5.3 Cooling below Doppler Limit: Sisyphus Cooling .... 160
2.5.3.1 Physics of Sisyphus Cooling ..................... 160
2.5.3.2 Capture Velocity ....................................... 164


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Contents

2.5.3.3 Friction Coefficient ................................... 165
2.5.3.4 Cooling Limit Temperature ...................... 166
2.5.3.5 Recoil Limit .............................................. 166
2.5.3.6 Sub-Recoil Cooling .................................. 167
2.5.4 Magneto-Optical Trap .............................................. 167
2.5.5 Other Experimental Techniques in Laser
Cooling and Trapping............................................... 170
2.5.5.1 Laser Atom-Slowing Using a
Frequency Swept Laser System:
Chirp Laser Slowing ................................. 171
2.5.5.2 Laser Atom-Slowing Using Zeeman
Effect: Zeeman Slower ............................. 173
2.5.5.3 2D Magneto-Optical Trap ........................ 177
2.5.5.4 Isotropic Cooling ......................................180

2.5.5.5 Optical Lattice Approach ......................... 183
Appendix 2.A: Laser Cooling—Energy Considerations.................. 189
Chapter 3

Microwave Frequency Standards Using New Physics ..................... 191
3.1

3.2

Cs Beam Frequency Standard ............................................... 192
3.1.1 Optically Pumped Cs Beam Frequency Standard.... 192
3.1.1.1 General Description .................................. 192
3.1.1.2 Frequency Shifts and Accuracy ................ 194
3.1.1.3 Experimental Determination
of Those Shifts .......................................... 197
3.1.1.4 Frequency Stability ................................... 198
3.1.1.5 Field Application ......................................200
3.1.2 CPT Approach in a Beam ......................................200
3.1.2.1 General Description ..................................200
3.1.2.2 Analysis .................................................... 201
3.1.2.3 Experimental Results ................................206
3.1.3 Classical Cs Beam Standard Using Beam
Cooling .............................................................. 208
Atomic Fountain Approach ................................................... 210
3.2.1 In Search of a Solution ............................................. 210
3.2.2 General Description of the Cs Fountain................... 211
3.2.3 Functioning of the Cs Fountain ................................ 213
3.2.3.1 Formation of the Cooled Atomic
Cloud: Zone A .......................................... 213
3.2.3.2 Preparation of the Atoms: Zone B ............ 217

3.2.3.3 Interrogation Region: Zone C ................... 218
3.2.3.4 Free Motion: Zone D ................................ 218
3.2.3.5 Detection Region: Zone E......................... 218
3.2.4 Physical Construction of the Cs Fountain ................ 219
3.2.4.1 Vacuum Chamber ..................................... 219
3.2.4.2 Microwave Cavity ..................................... 220


viii

Contents

3.2.4.3
3.2.4.4
3.2.4.5
3.2.4.6
3.2.4.7
3.2.4.8

3.3

3.4

Magnetic Field .......................................... 221
Temperature Control ................................. 221
Capture and Selection Zone...................... 221
Detection Zone ......................................... 221
Supporting Systems .................................. 221
Advantages and Disadvantages
of a Pulsed Fountain ................................. 222

3.2.5 Frequency Stability of the Cs Fountain.................... 223
3.2.5.1 Photon Shot Noise.....................................224
3.2.5.2 Quantum Projection Noise........................ 225
3.2.5.3 Electronic Noise........................................ 225
3.2.5.4 Reference Oscillator Noise: Dicke Effect ... 225
3.2.6 Rubidium and Dual Species Fountain Clock ........... 226
3.2.7 Frequency Shifts and Biases Present in the
Fountain.................................................................... 229
3.2.7.1 Second-Order Zeeman Shift ..................... 230
3.2.7.2 Black Body Radiation Shift ...................... 232
3.2.7.3 Collision Shift ........................................... 237
3.2.7.4 Cavity Phase Shift ....................................240
3.2.7.5 Cavity Pulling ........................................... 242
3.2.7.6 Microwave Spectral Purity ....................... 247
3.2.7.7 Microwave Leakage .................................. 247
3.2.7.8 Relativistic Effects ....................................248
3.2.7.9 Other Shifts............................................... 249
3.2.7.10 Conclusion on Frequency Shifts
and Accuracy ............................................ 250
3.2.8 An Alternative Cold Caesium Frequency
Standard: The Continuous Fountain ........................ 251
3.2.8.1 Light Trap ................................................. 252
3.2.8.2 Interrogation Zone, Microwave Cavity..... 253
3.2.8.3 Preliminary Results .................................. 255
3.2.9 Cold Atom PHARAO Cs Space Clock .................... 257
Isotropic Cooling Approach .................................................. 258
3.3.1 External Cavity Approach: CHARLI ...................... 258
3.3.2 Approach Integrating Reflecting Sphere
and Microwave Cavity: HORACE ...........................260
3.3.3 Different HORACE Approach ................................. 261

Room Temperature Rb Standard Approach Using Laser
Optical Pumping.................................................................... 262
3.4.1 Contrast, Line Width, and Light Shift ..................... 263
3.4.2 Effect of Laser Radiation Beam Shape .................... 272
3.4.3 Expectations Relative to Short-Term Frequency
Stability .................................................................... 273
3.4.4 Review of Experimental Results on Signal Size,
Line Width, and Frequency Stability ....................... 273


ix

Contents

3.4.5

Frequency Shifts....................................................... 278
3.4.5.1 Buffer Gas Shift ........................................ 278
3.4.5.2 Magnetic Field Shift ................................. 279
3.4.5.3 Light Shift ................................................. 279
3.4.5.4 Spin-Exchange Frequency Shift ...............284
3.4.5.5 Microwave Power Shift............................. 285
3.4.5.6 Cavity Pulling ........................................... 286
3.4.6 Impact of Laser Noise and Instability on Clock
Frequency Stability .................................................. 287
3.4.6.1 Spectral Width, Phase Noise, and
Intensity Noise of Laser Diodes ............... 288
3.4.6.2 Impact of Laser Noise on Clock
Short-Term Frequency Stability ................290
3.4.6.3 Medium- and Long-Term Frequency

Stability ..................................................... 295
3.4.7 Other Approaches Using Laser Optical Pumping
with a Sealed Cell..................................................... 297
3.4.7.1 Maser Approach........................................ 297
3.4.7.2 Laser Pulsing Approach............................ 297
3.4.7.3 Wall-Coated Cell Approach ..................... 299
3.5 CPT Approach .......................................................................300
3.5.1 Sealed Cell with a Buffer Gas in Continuous
Mode: Passive Frequency Standard..........................300
3.5.1.1 Signal Amplitude and Line Width ............302
3.5.1.2 Practical Implementation and Its
Characteristics ..........................................307
3.5.2 Active Approach in a Cell: The CPT Maser ............ 315
3.5.2.1 Basic CPT Maser Theory ......................... 315
3.5.2.2 Frequency Stability ................................... 318
3.5.2.3 Frequency Shifts ....................................... 320
3.5.3 Techniques for Improving S/N Ratio in the
Passive IOP and CPT Clock Approach ....................322
3.5.4 CPT in Laser-Cooled Ensemble for Realizing a
Frequency Standard.................................................. 323
3.6 Laser-Cooled Microwave Ion Clocks .................................... 324
3.6.1 9Be+ 303 MHz Radio-Frequency Standard .............. 325
3.6.2 113Cd+ and 111Cd+ Ion Trap ........................................ 327
3.6.3 171Yb+ Laser-Cooled Microwave Frequency
Standard ................................................................... 328
Appendix 3.A: Frequency Stability of an Atomic Fountain ............ 329
3.A.1 Shot Noise ................................................................ 333
3.A.2 Quantum Projection Noise ....................................... 334
Appendix 3.B: Cold Collisions and Scattering Length .................... 337
Appendix 3.C: Optical Absorption of Polarized Laser Radiation

Including Optical Pumping .............................................................. 338
Appendix 3.D: Basic CPT Maser Theory ........................................ 341


x

Chapter 4

Contents

Optical Frequency Standards ........................................................... 345
4.1
4.2
4.3
4.4

4.5

4.6

Early Approach Using Absorption Cells ............................... 347
Some Basic Ideas ................................................................... 349
MOT Approach...................................................................... 351
Single Ion Optical Clocks ...................................................... 352
4.4.1 The Concept ............................................................. 352
4.4.2 Outline of Particular Implementations with
Individual Ions.......................................................... 357
4.4.2.1 27Al+ (I = 5/2) ............................................ 357
4.4.2.2 40Ca+ (I = 0) and 43Ca+ (I = 7/2) ............... 359
4.4.2.3 87Sr + (I = 9/2) and 88Sr+ (I = 0) ................. 361

4.4.2.4 115In+ (I = 9/2) ........................................... 362
4.4.2.5 137Ba+ (I = 3/2) and 138Ba+ (I = 0) ............. 363
4.4.2.6 171Yb+ (I = 1/2), 172Yb+ (I = 0),
and 173Yb+ (I = 5/2) ...................................364
198
4.4.2.7
Hg+ (I = 0) and 199Hg+ (I = 1/2) ............. 366
4.4.3 Systematic Frequency Shifts in Single Ion Clocks ....366
4.4.3.1 Doppler Effect........................................... 366
4.4.3.2 Zeeman Effect .......................................... 368
4.4.3.3 Biases due to the Presence of Electric
Fields ......................................................... 371
Optical Lattice Neutral Atoms Clock .................................... 377
4.5.1 The Concept ............................................................. 377
4.5.1.1 Trapping Characteristics ........................... 382
4.5.1.2 Atom Recoil .............................................. 383
4.5.1.3 Atom Localization .................................... 383
4.5.1.4 Magic Wavelength .................................... 384
4.5.1.5 Clock Transition........................................ 385
4.5.2 Type of Atoms Used in Optical Lattice Clocks ....... 386
4.5.2.1 Strontium Atom ........................................ 386
4.5.2.2 Mercury Atom .......................................... 387
4.5.2.3 Ytterbium Atom ........................................ 389
4.5.2.4 Magnesium Atom ..................................... 390
4.5.2.5 Calcium Atom ........................................... 391
4.5.3 Important Frequency Biases..................................... 391
4.5.3.1 Zeeman Effect .......................................... 391
4.5.3.2 BBR Shift.................................................. 392
4.5.3.3 Lattice Light Shift..................................... 393
4.5.3.4 Other Shifts............................................... 394

4.5.4 Frequency Stability of an Optical Lattice Clock...... 395
4.5.5 Practical Realizations ............................................... 395
Frequency Measurement of Optical Clocks .......................... 397
4.6.1 Optical Comb ........................................................... 398
4.6.2 Clock Frequencies and Frequency Stabilities
Realized .................................................................... 399


xi

Contents

Chapter 5

Summary, Conclusion, and Reflections............................................ 401
5.1
5.2

5.3

Accuracy and Frequency Stability ........................................402
Selected Applications of Atomic Frequency Standards ........404
5.2.1 The SI: Towards a Redefinition of the Second .........405
5.2.2 Tests of Fundamental Physical Laws .......................407
5.2.2.1 Fundamental Constants ............................407
5.2.2.2 Time Dilation and Gravitational
Red Shift ...................................................408
5.2.2.3 Fundamental Physics in Space..................409
5.2.3 Clocks for Astronomy and Earth Science ................ 410
5.2.3.1 VLBI and Geodesy ................................... 410

5.2.3.2 Deep Space Network ................................ 410
5.2.3.3 Earth Clocks Network .............................. 410
5.2.3.4 Navigation Systems................................... 411
Last Reflections ..................................................................... 412

References ............................................................................................................. 415
Index ...................................................................................................................... 457


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Preface
Volumes 1 and 2 of The Quantum Physics of Atomic Frequency Standards, henceforth
referred to as QPAFS (1989), were written in the 1980s and were published in 1989.
They covered, in some detail, work done up to 1987 on the development of atomic
frequency standards. The text included a description of their development at that
time, as well as a description of the research on the physics supporting that development. Since that time, the field has remained a very active part of the research
program of many national laboratories and institutes. Work has remained intensive
in many sectors connected to the refinement of classical frequency standards based
on atoms such as rubidium (Rb), caesium (Cs), hydrogen (H), and selected ions in the
microwave range, while new projects were started in connection to the realization of
stable and accurate frequency standards in the optical range.
For example, intensive studies were made on the use of lasers in the optical pumping and cooling of Rb and Cs as well as on the development of a new type of standard
based on the quantum-mechanical phenomenon called coherent population trapping
(CPT). Regarding Cs and Rb, laser cooling of atoms has made possible the realization of an old dream in which a small blob of atoms, cooled in the microkelvin range,
is projected upward at a slow speed in the gravitational field of the earth and the
atoms fall back like water droplets in a fountain. In their path, the atoms are made
to pass through a microwave cavity, and upon falling back after having spent their
kinetic energy, pass through the same cavity, mimicking, with a single cavity, the

classical double-arm Ramsey cavity approach. The system is called atomic fountain.
Its advantage over the classical approach resides in the reduction of the width of the
resonance hyperfine line by a factor of the order of 100 relative to that observed in
the room temperature approach. The resulting line width is of the order of 1  Hz.
Work has also continued on the development of smaller H masers, in particular in
the development of passive devices and in the use of a new smaller so-called magnetron cavity. The advent of the solid-state laser in the form of the conventional edgeemitting type (GaAs) and vertical structure (VCSEL) has opened the door to a new
approach in optical pumping for implementing smaller and more performing Rb and
Cs cell frequency standards.
Since the 1990s, laser cooling has been studied extensively and aside from providing a means for realizing the fountain clock mentioned above, it has allowed the
realization of clocks based on microwave transitions in ions such as mercury (Hg+),
barium (Ba+), strontium (Sr+), and ytterbium (Yb+) confined within an electromagnetic trap.
On the other hand, intense work has been carried out in several laboratories in
extending the work done at microwave frequencies to the optical frequency range. The
gain in that approach relies mainly on the increase in the frequency of the atomic transitions involved, which provides for a line width similar to that obtained in the microwave
range a resonance quality factor millions of times larger. Laser cooling has been applied
successfully to such atoms as mercury (Hg), ytterbium (Yb), and strontium (Sr)
xiii


xiv

Preface

stored in optical lattice traps in order to reduce their thermal motion. Laser cooling
has also been used in the mono-ion trap to implement optical frequency standards. In
that case, a single ion, say Sr+ or Yb+, is maintained in a Paul or Penning trap and its
motion within the trap is damped by laser cooling. Clocks at optical frequencies have
been implemented as laboratory units with unsurpassed accuracy and frequency stability reaching the 10−16 to 10−18 range. In both cases, the clock frequency is derived
from a transition between the ground S state of the atom and an excited metastable
state with a lifetime of the order of 1  second or more leading to a very narrow

resonance line. The clock transition is detected by means of monitoring changes in
the fluorescence level created by the cooling radiation when the clock transition is
excited.
The large gap in frequency between the microwave and the optical range has
always been an roadblock in the use of optical frequencies in various applications
such as frequency standards or still high precision spectroscopy and fundamental
research. The reason is mainly due to the fact that gaps between available optical
frequencies for the realization of clocks are very large. It is extremely difficult to
connect those frequencies to the microwave range. This connection is required
because most of the applications are in the low frequency range of the spectrum and,
furthermore, because the SI (International System of Units) definition of the second
is based on a microwave hyperfine transition in Cs, in the X band. We have given in
Volume 2 of QPAFS examples of the conventional method used to make that connection. That method comprises frequency- and phase-locking together banks of lasers
with appropriate heterodyning in several steps in order to interconnect various optical frequencies to reach finally the microwave range. The connection has to be done
over a large number of steps and involves tremendous investment of space and time
to finally measure what very often happens to be just a single frequency. Such a task
has been reduced considerably by the invention of the so-called optical comb, which
comprises locking the repetition rate of a femtosecond laser to a stable atomic frequency standard of high spectral purity, such as an H maser referenced in frequency
to a primary Cs atomic clock. When observed by means of a nonlinear optical fibre,
the resulting laser spectrum consists of a spectrum of sharp lines, themselves called
the teeth of the comb, which covers a frequency range of the order of 1  octave.
Frequencies over a broad range are then measured essentially in a single step on an
optical table, resulting in a considerable reduction in work and size as compared to
the previous heterodyning technique, which required entire rooms filled with lasers.
This volume covers those subjects in some detail. It is divided into five chapters.
Chapter 1 is an introduction, presenting a review of recent developments made on
the improvement of conventional atomic frequency standards described in the two
volumes of QPAFS. It highlights the main limitations of those frequency standards
and the physical basis of those limitations and outlines the progress made during the
last 25 years. Chapter 2 is a description of recent advances in atomic physics, theory

and applications, that opened new avenues. Chapter 3 is concerned with  research
and development done in the development of new microwave frequency standards.
Chapter 4 describes research and development done in the optical range to implement
optical frequency standards based on new results in atomic physics as described in
Chapter 2. Chapter 5 summarizes the results in frequency stability and accuracy


Preface

xv

achieved with those new frequency standards and outlines selected applications.
A short reflection is included giving some insight into future work.
Such a text cannot be written without significant help from experts in the field.
We wish to recognize the contribution and collaboration of many scientists. In particular, we wish to recognize the invaluable help of André Clairon, who has read the whole
manuscript and helped in improving its exactness and completeness. We also show our
gratitude to the following scientists who helped us through their encouragement, supplied original figures or material, and contributed by means of comments on various sections of the text: C. Affolderbach, A. Bauch, S. Bize, J. Camparo, C. Cohen-Tannoudji,
E. De Clercq, A. Godone, D. Goujon, S. Guérandel, P. Laurent, T. Lee, S. Micalizio,
G. Mileti, J. Morel, W.D. Phillips, P. Rochat, P. Thomann, R.F.C. Vessot, and S. Weyers.
Jacques Vanier
and
Cipriana Tomescu
University of Montreal


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Introduction
This book is about recent developments in the field of atomic frequency standards,

developments that took place after the publication in 1989 of the first two volumes
with the same title. Atomic frequency standards are systems providing an electrical signal at a cardinal frequency of, say, 10 MHz, a signal generated usually by
a quartz crystal oscillator locked in phase or in frequency to a quantum transition
inside an atom. The atom is selected for its properties such as easy detection of the
particular quantum transition chosen and relative independence of its frequency of
the environment. In early work, those conditions limited development around hydrogen and alkali atoms, which have transitions in the microwave range and could be
manipulated easily as beams or atomic vapour with the techniques available at that
time. Progress in the development of lasers and their stabilization extended that
work to the optical range. A major task encountered in the early development of
microwave standards has always been the elimination of Doppler effect. Atoms at
room temperature travel at speeds of several hundred metres per second and, consequently, Doppler effect causes frequency shifts and line broadening of the resonance
signal. This effect is generally eliminated by means of various storage techniques
based on Dicke effect, or still beam techniques using the Ramsey double-arm cavity
approach. These techniques are not well adapted to optical frequencies because of
the shorter wavelengths involved. However, progress in the understanding of interactions between atoms and electromagnetic interactions has provided new means
of reducing the velocity of atoms and reducing, if not eliminating, the constraints
introduced by Doppler effect.
An atomic frequency standard that is operated continuously becomes an atomic
clock. The operation is essentially a process of integration and the date set as the
constant of integration provides the basis for implementing a timescale. This is the
origin of atomic timescales, in particular the one maintained by the International
Bureau of Weights and Measures. Various systems in operation have their own
timescale, for example, the global positioning system (GPS) of the United States,
the Russian Glonass system, the Chinese Beidou system, and the European Galileo
systems under development, all playing an important role in navigation on or near
the surface of the earth.
Although time is central to physics and is used in our day-to-day life, it is a concept that is difficult to grasp, let alone to define. We use it without questioning its
origin and its exact nature. It is basic in physics for describing the dynamics of systems and ensembles of systems by means of equations that model the evolution of
objects forming our universe. The concept is used as such without questioning much
its exact nature and origin. In Newtonian mechanics, objects evolve in space and

their behaviour is described by means of differential equations and functions of time
and space. Both space and time are independent and in common language they are
said to be absolute. In that context, time is not a function of space and space is not a
function of time. However, in attempts to relate mechanics and electromagnetism by
xvii


xviii

Introduction

space and time transformations, a difficulty arose. This is due to the finiteness and
invariability of the speed of light, made explicit in Maxwell’s equations, whatever
the motion of the frame of reference in which it is generated and measured. In this
context, with Einstein, Poincaré, Lorentz, Minkowski, and others, time and space
become entangled and functions of each other. There is no such thing as an absolute
space in which objects evolve in an absolute time framework, both independent of
each other. Time and space form a single four-dimensional framework and cannot be
treated independently. This concept forms the basis of the theory of relativity. This
theory has been shown to be valid through multiple experiments and verifications to
a level that raises its validity to a high degree. It should be pointed out that the most
accurate verifications were done with atomic clocks, the instruments that are the
content of this book. There is another question also often raised regarding the nature
of time: Could it be discrete? If so what would be the size of its smallest quantity,
the time quantum? Could it be that Planck’s time is the smallest time entity? This
is a totally unknown subject and appears to be a roadblock to in the development of
a sustainable quantum theory that includes the concepts elaborated in the theory of
general relativity.
Although we may feel somewhat uncomfortable in the context of such questions,
time remains the most basic concept in physics, is fundamental, and is the quantity

that is measured with the greatest precision. Current atomic clocks can commonly
keep time to an accuracy of 1 s in a million years, or in other words are stable to
better than 1 ms in a year. For example, the timescale generated by the GPS satellites
for navigation, based on atomic frequency standards on satellites and on ground,
is stable after appropriate processing and filtering to about 1 ns/day. On the other
hand, on the basis of our inability to measure time by astronomical means with such
accuracy, it was decided in 1967 to replace the astronomical definition of the second
by one in terms of a particular atomic hyperfine transition in the Cs atom. The frequency of that transition is set at 9,192,631,770 Hz. Furthermore, since now the speed
of light is defined exactly as 299,792,458 m/s, providing at the same time a definition
of the metre, the mechanical units of the SI become essentially determined by the
basic time unit, the second. The concept of unifying all SI units in terms of a single
quantity goes further due to the Josephson effect phenomenon, which relates voltage
to frequency in a most fundamental expression, 2e/ℏ, involving only fundamental
constants. This is the subject that will be described in Chapter 5.
From this discussion, it is evident that time plays a most important role in physics
and technology and the realization of the highest accuracy and precision of the SI
unit, the second, has remained one of the most active preoccupations of several laboratories and institutes over the past 50 years. Starting with tremendous improvements
in the realization of the second within the microwave range, work has extended to
the optical range with proven increase in frequency stability and accuracy by several
orders of magnitude. These achievements were possible mainly through a better
understanding of the interactions between electromagnetic radiation and atoms, providing a means of altering the properties of atoms. This book is about those improvements that have taken place mainly during the past 25 years, on the realization of
stable and accurate frequency standards.


Authors
Jacques Vanier completed his undergraduate studies in
physics at the University of Montreal, Québec, Canada,
before moving to McGill University to undertake his graduate studies. During his career he has worked in various
industries (Varian, Hewlett-Packard); taught physics; and
carried out research at Laval University, Montreal, Québec,

Canada, and has also been an active member of the National
Research Council of Canada, in Ottawa, Ontario, Canada.
His research work is oriented towards the understanding
and the application of the quantum electronics phenomena
and he has been a consultant for several companies engaged in the development of
atomic clocks. Jacques has also been very active on the academic circuit, giving lectures and presenting at numerous conferences in universities, national institutes, and
summer schools around the world. He has written more than 120 journal articles and
proceedings papers and is the author of review articles and books on masers, lasers,
and atomic clocks. His book The Quantum Physics of Atomic Frequency Standards,
written with C. Audoin, is recognized as a main reference in the field. He is the author
of The Universe: A Challenge to the Mind published by Imperial College Press/
World Scientific. Jacques is a fellow of the Royal Society of Canada, the American
Physical Society, and the Institute of Electrical and Electronic Engineers. He has
received several awards for his contributions to the field of measurement science. He
is currently an adjunct professor in the Physics Department, University of Montreal,
Québec, Canada.
Cipriana Tomescu completed her studies in physics at
the University of Bucharest, Romania, where she obtained
her PhD degree.
From 1982 to 2004, she was a researcher at the National
Institute of Laser Physics, Plasma and Radiation, Bucharest,
Romania. In the early years of her employment, she participated in the construction of H masers used by the
Bucharest Observatory, the Institute of Metrology, and
the Faculty of Physics. During the period 1996–2004, she
was laboratory director. During the period 1992–2006, she
also worked in various national laboratories, in particular,
Paris Observatory, LNE-SYRTE, France; Neuchâtel Observatory, Switzerland; and
Communication Research Laboratory, Japan. At those locations, she contributed to
the development of advanced state-of-the-art atomic frequency standards, such as Rb
and Cs fountains using atom trapping techniques and laser atom cooling. From 2008

to 2012, she worked at the University of Liege, IPNAS, and at Gillam-Fei. She was
responsible for the implementation of the first H maser realized in Belgium under
xix


xx

Authors

Plan Marshall: SKYWIN-TELECOM. She is the author of numerous publications
in scientific journals and conference proceedings and she has been invited to make
presentations at numerous symposia, universities, and national institutes. In 1985, she
received the D. Hurmuzescu Prize of the Romanian Academy for work on the physics
of the H maser. She is currently an invited researcher in the Physics Department of the
University of Montreal, Québec, Canada.


1

Microwave Atomic
Frequency Standards
Review and Recent
Developments

At the end of the 1980s, atomic frequency standards reached a level of refinement
that made it the envy of many other fields of physics. The accuracy of primary caesium (Cs) standards maintained in operation at national institutes reached a level
better that 10−13 and the frequency stability of the hydrogen (H) maser in the medium
term was better than 10−14. These characteristics made possible the verification to
great accuracy of basic physics predictions such as those resulting from the theory of
relativity and made possible the maintenance of a timescale to an unsurpassed stability. It also opened the use of such devices in many applications. The time unit, the

second, became the most accurate unit of the International System of Units (SI), with
consequences for the implementation of other units such as the metre, the volt, and
the ohm. On the other hand, Rb standards had reached a level of development that
made them an excellent support of digital communication systems with improved
reliability and also made them appropriate for navigation systems using satellites
requiring medium frequency stability and small size.
There has been extensive research on the possibility of using other atoms as the basis
for new types of frequency standards. However, those systems are still under study in laboratories; Cs, H, and Rb therefore remain the atoms at the heart of atomic frequency standards used at large either as references in basic research or in practical systems requiring
precise and accurate timing. Although the Cs standard in its original beam implementation using magnetic state selection has been dethroned as the most accurate primary
standard with the introduction of optical pumping and laser cooling, it still remains in
many laboratories the work horse for implementing a local timescale, for confirming the
accuracy of other standards, and, to a limited extent, for reliable reporting to the BIPM
(Bureau International des Poids et Mesures) in the maintenance of the second.
In this chapter, we recall the physical construction and the characteristics of
those frequency standards based on Cs, H, and Rb, as well as of some selected other
types of microwave frequency standards, which still show promise regarding possible specific applications. We examine the physics at the heart of the operation of
those standards and behind their limitations relative to size, accuracy, and frequency
stability. We also see that those limitations were overcome to some extent, showing
that, with some imagination, improvements could still be made on instruments that
had already attained a very high level of maturity.
1


2

1.1

The Quantum Physics of Atomic Frequency Standards

CLASSICAL ATOMIC FREQUENCY STANDARDS

We usually group Cs beam frequency standards, H masers, and optically pumped
Rb standards under the terminology “classical atomic frequency standards.” In the
following paragraphs, we review their physical construction and recall the essential theoretical results that were developed in parallel with their implementation.
Theoretical investigations were required for an understanding of the various phenomena causing biases observed when evaluated relative to accuracy and frequency
stability. The reader will find in Volumes 1 and 2 of The Quantum Physics of Atomic
Frequency Standards (QPAFS) a detailed description of the operation of such standards and a description of the basic physics involved. In the following sections, we
recall the main concepts behind their operation in order to simplify reading of subsequent sections, in which we discuss recent progress in understanding the physics
involved. We then present new analysis and realizations that have resulted in better
understanding of their operation, greater accuracy, better frequency stability, and in
some instances reduction in size and weight.

1.1.1

Cs Beam FrequenCy standard

A frequency standard using Cs and the separate oscillatory field approach proposed
by Ramsey (Ramsey 1950) was implemented as early as 1955 (Essen and Parry
1955). Intense laboratory and industrial development followed (see, e.g., McCoubrey
1996). Development showed great success and soon after the construction of Cs
primary standards in several laboratories, the frequency of the Cs ground state
hyperfine transition was adopted for the definition of the second by the Conférence
Générale des Poids et Mesures (CGPM, 1967–1968). The frequency adopted was
9,192,631,770 Hz. It was the best number obtained by means of precise astronomical measurements by which the Cs hyperfine frequency was determined relative to
the second, whose formal definition at the time of measurement was the ephemeris
second based on astronomical observation (Markowitz et al. 1958). That choice has
remained till date (2015).
Why was Cs selected for providing the basis of the definition of the second? First,
the choice of the Cs atom in the implementation of a frequency standard has resulted
from the considerable accumulation of knowledge on that atom over the years and

from the several advantages that it provides over other candidates. In particular, Cs
has a single stable isotope, 133Cs, and is relatively abundant in nature. Its melting
point is 28.4°C. Its vapour pressure is such that it is possible to implement a rather
intense atomic beam from an oven at a relatively low temperature of the order of 425
to 500 K. Its ionization energy is low, 3.9 eV, making it easy to detect by conventional
procedures such as ionization with a hot wire detector and ion counting. Finally, Cs
has a ground state hyperfine frequency falling in the X band, a microwave region
that has known extensive development, which makes possible atom–microwave
interaction by means of small structures such as cavities whose dimensions are in
the centimetre range.
The Cs atom has a nuclear spin I = 7/2 and has a single s electron outside closed
electronic shells. Its ground state consists of two hyperfine levels F = 3 and F = 4


3

Microwave Atomic Frequency Standards

E/hνhf

2.0

4
3
2
1
0
−1
−2
−3

1.5
1.0

F=4

0.5
0.0

F = 3 −0.5
−1.0
−1.5
−2.0
0.0

0.2

0.4

B (T )

0.6

0.8

−4
−3
−2
−1
0

1
2
3

1.0

FIGURE 1.1 Ground state energy level manifold of the caesium atom as a function of the
magnetic induction B in tesla.

and in a low magnetic field the structure consists of two manifolds of 7 and 9 energy
levels, respectively. This ground state is shown in Figure  1.1 as a function of the
magnetic induction B.
1.1.1.1 Description of the Approach Using Magnetic State Selection
A conceptual diagram of the classical Cs beam frequency standard using magnetic
state selection is shown in Figure 1.2 (Vanier and Audoin 2005).
A beam of Cs atoms is formed by proper collimation from an oven heated at a
temperature of the order of 50–100°C depending on the intensity required. This
beam is directed as to pass through a so-called Ramsey cavity that provides a region
of electromagnetic interaction and excite transitions between the two ground state
hyperfine levels mF = 0 of the atoms. Magnets A and B are generally dipole magnets
and create an intense inhomogeneous field in which atomic trajectories are deflected.
They are called Stern–Gerlach selector magnets or filters. The deflection is caused
by the interaction of the atom magnetic moment with the magnetic field gradient
and by the tendency of atoms to seek states of low potential energy. Consequently,
according to Figure  1.1, atoms having higher energy in high magnetic fields are
deflected to regions of low magnetic field in order to lower their potential energy.
Similarly, those atoms having lower energies at high magnetic fields seek regions
of high magnetic field for the same reason. Selection is accomplished by means of
magnet A whose orientation is such as to force atoms in level F = 4, mF = 0 to pass
through the Ramsey microwave cavity, and reach the second deflecting magnet B.

Atoms in the other F = 3, mF = 0 level, being deflected away from the F = 4, mF = 0
atoms, are eliminated from the beam by appropriate collimation. The analysis of the


4

The Quantum Physics of Atomic Frequency Standards
Modulation,
synchronous detector,
and frequency lock
system

Microwave generator

Hot wire
detector

Electron
multiplier

Ramsey cavity

Cs oven

Signal out
Bo
Magnet A
state selector

Magnetic shields

Signal out

Magnet B
analyzer
ν

FIGURE 1.2 Simplified conceptual diagram of the Cs beam frequency standard using magnetic
state selection. The inset shows the shape of the resonant signal observed when the frequencylock loop is open and the microwave frequency is scanned slowly over the atomic hyperfine
resonance. Although in the figure the magnetic induction is shown parallel to the beam direction, in practice it is very often made perpendicular to the beam. (Data from Vanier, J. and
Audoin, C., Metrologia, 42, S31, 2005. Copyright Bureau International des Poids et Mesures.
Reproduced by kind permission of IOP Publishing. All rights reserved.)

beam composition is done by the combination of magnet B, called the analyzer, and
a hot wire ionizer followed by a counter usually assisted by an electron multiplier. In
their transit through the Ramsey cavity, the atoms are submitted to an electromagnetic field of angular frequency ω in the two arms of the cavity called the interaction
regions. In the first arm of the cavity, atoms are excited into a Rabi oscillation that
puts them into a quantum superposition of the two hyperfine levels F = 4, mF = 0
and F = 3, mF = 0 of the ground state. We define τ, the time of transit of an atom at
speed v inside that first arm of length l. The power fed into the cavity is adjusted to
such a value as to make the electromagnetic radiation appear as a π/2 pulse, that is to
say a microwave pulse that puts the atoms in an exact superposition state of the two
hyperfine levels when they exit that first arm at the most probable speed. The atoms
are subsequently left to drift unperturbed in the space within the double arm cavity.
A uniform magnetic induction Bo provides an axis of quantization and the atoms
remain in the same state. They then penetrate inside the second arm at distance L
from the first arm. We call T the time of transit between the two arms of the cavity.
If v is the speed of a given atom, then T is simply L/v and is affected the spread in v.
In that second arm, the atoms are again submitted to a field of the same intensity and
same frequency as in the first arm. Atoms having the same speed as in the first arm

are, thus, submitted again to a π/2 pulse. The atoms at the exit of the second arm find
themselves in the lower state F = 3, mF = 0 and the transition is complete as if they
had been submitted to a π pulse. If the frequency applied to the cavity is not exactly
the resonance frequency of the atoms, the phase of the field in the second arm is not