-a guide to experiments in quantum optics, second edition (2004)

A Guide to Experiments in Quantum Optics
Second, Revised and Enlarged Edition

A Guide to Experiments

Timothy C. Ralph
Cover Picture
Real experimental data for squeezed states, generated at the ANU by Jin Wei Wu; U. L.Andersen, B. C.Buchler, P. K. Lam, J. W. Wu, J. R. Gao, H.-A. Bachor Eur.Phys.J. D. 27, No. 2 (2003)
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1.3 How to use this guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1 Classical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The Gaussian beam . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Quadrature amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.5 A classical mode of light . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.6 Classical modulations . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 The origin of fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Correlation functions . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.4 Noise spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.5 An idealized classical case: Light from a chaotic source. . . . . . . . 31
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Photons – the motivation to go beyond classical optics 37
3.1 Detecting light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Light from a thermal source . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Interference experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5.1 Polarization of a single photon . . . . . . . . . . . . . . . . . . . . . 47
3.5.2 Some mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.3 Polarization states . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Quantization of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.1.1 Some general comments on quantum mechanics . . . . . . . . . . . 60
4.1.2 Quantization of cavity modes . . . . . . . . . . . . . . . . . . . . . 61
4.1.3 Quantized energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Quantum states of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.1 Number or Fock states . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.2 Coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2.3 Mixed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.2 Probability and quasi-probability distributions . . . . . . . . . . . . 70
4.3.3 Photon number distributions, Fano factor . . . . . . . . . . . . . . . 75
4.4 Propagation and detection of quantum optical fields . . . . . . . . . . . . . . 77
4.4.1 Propagation in quantum optics . . . . . . . . . . . . . . . . . . . . . 77
4.4.2 Detection in quantum optics . . . . . . . . . . . . . . . . . . . . . . 81
4.4.3 An example: The beamsplitter . . . . . . . . . . . . . . . . . . . . . 82
4.5 Quantum transfer functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.5.1 A linearized quantum noise description . . . . . . . . . . . . . . . . 84
4.5.2 An example: The propagating coherent state . . . . . . . . . . . . . 86
4.5.3 Real laser beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5.4 The transfer of operators, signals and noise . . . . . . . . . . . . . . 88
4.5.5 Sideband modes as quantum states . . . . . . . . . . . . . . . . . . . 90
4.6 Quantum correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6.1 Photon correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.6.2 Quadrature correlations . . . . . . . . . . . . . . . . . . . . . . . . . 93
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.1 Beamsplitters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.1.2 The beamsplitter in the quantum operator model . . . . . . . . . . . 102
5.1.3 The beamsplitter with single photons . . . . . . . . . . . . . . . . . 104
5.1.4 The beamsplitter and the photon statistics . . . . . . . . . . . . . . . 106
5.1.5 The beamsplitter with coherent states . . . . . . . . . . . . . . . . . 108
5.1.6 The beamsplitter in the noise sideband model . . . . . . . . . . . . . 110
5.1.7 Comparison between a beamsplitter and a classical current junction . 111
5.2 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.4 Transfer of intensity noise through the interferometer . . . . . . . . . 116
5.2.5 Sensitivity limit of an interferometer . . . . . . . . . . . . . . . . . . 117
5.3 Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3.3 The phase response . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3.4 Spatial properties of cavities . . . . . . . . . . . . . . . . . . . . . . 128
5.3.5 Equations of motion for the cavity mode . . . . . . . . . . . . . . . . 132
5.3.6 The quantum equations of motion for a cavity . . . . . . . . . . . . . 133
5.3.7 The propagation of fluctuations through the cavity . . . . . . . . . . 133
5.3.8 Single photons through a cavity . . . . . . . . . . . . . . . . . . . . 137
5.4 Other optical components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.1 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.3 Modulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4.6 Nonlinear processes . . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.1.1 Technical specifications of a laser . . . . . . . . . . . . . . . . . . . 148
6.1.2 Rate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.1.4 Examples of lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.1.5 Laser phase noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2 Amplification of optical signals . . . . . . . . . . . . . . . . . . . . . . . . . 162
6.3 Parametric amplifiers and oscillators . . . . . . . . . . . . . . . . . . . . . . 164
6.3.1 The second-order non-linearity . . . . . . . . . . . . . . . . . . . . . 165
6.3.2 Parametric amplification . . . . . . . . . . . . . . . . . . . . . . . . 167
6.3.4 Pair production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.3 Photon sources and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 Detecting photocurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.5 Spectral analysis of photocurrents . . . . . . . . . . . . . . . . . . . . . . . 187
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.1.1 Direct detection and calibration . . . . . . . . . . . . . . . . . . . . 200
8.1.2 Balanced detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.1.4 Homodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.1.5 Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . 210
8.2 Intensity noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
8.3.1 Classical intensity control . . . . . . . . . . . . . . . . . . . . . . . 213
8.3.2 Quantum noise control . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.4 Frequency stabilization, locking of cavities . . . . . . . . . . . . . . . . . . 221
8.4.1 How to mount a mirror . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.5 Injection locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
9.1.1 Tools for squeezing, two simple examples . . . . . . . . . . . . . . . 232
9.1.2 Properties of squeezed states . . . . . . . . . . . . . . . . . . . . . . 238
9.2 Quantum model of squeezed states . . . . . . . . . . . . . . . . . . . . . . . 242
9.2.1 The formal definition of a squeezed state . . . . . . . . . . . . . . . 242
9.2.2 The generation of squeezed states . . . . . . . . . . . . . . . . . . . 245
9.2.3 Squeezing as correlations between noise sidebands . . . . . . . . . . 247
9.3 Detecting squeezed light . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
9.3.1 Reconstructing the squeezing ellipse . . . . . . . . . . . . . . . . . . 253
9.3.2 Summary of different representations of squeezed states . . . . . . . 254
9.3.3 Propagation of squeezed light . . . . . . . . . . . . . . . . . . . . . 254
9.4 Four wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
9.5 Optical parametric processes . . . . . . . . . . . . . . . . . . . . . . . . . . 263
9.6 Second harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.7 Kerr effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.7.2 Fibre Kerr Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9.7.3 Atomic Kerr squeezing . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.8 Atom-cavity coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.9 Pulsed squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
9.9.2 Pulsed squeezing experiments with Kerr media . . . . . . . . . . . . 285
9.9.3 Pulsed SHG and OPO experiments . . . . . . . . . . . . . . . . . . . 287
9.9.4 Soliton squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
9.9.5 Spectral filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
9.9.6 Nonlinear interferometers . . . . . . . . . . . . . . . . . . . . . . . 290
9.11 Twin photon beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

9.14 Summary of squeezing results . . . . . . . . . . . . . . . . . . . . . . . . . 300
9.14.1 Loopholes in the quantum description . . . . . . . . . . . . . . . . . 303
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
10.1 Optical communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
10.3 Optical sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.4.1 The origin and properties of GW . . . . . . . . . . . . . . . . . . . . 321
10.4.2 Quantum properties of the ideal interferometer . . . . . . . . . . . . 323
10.4.3 The sensitivity of real instruments . . . . . . . . . . . . . . . . . . . 328
10.4.4 Interferometry with squeezed light . . . . . . . . . . . . . . . . . . . 333
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
11.2 Classification of QND measurements . . . . . . . . . . . . . . . . . . . . . . 346
11.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
12.1 Wave-Particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.3.3 Bell inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
13.2 Postselection and coincidence counting . . . . . . . . . . . . . . . . . . . . 376
13.3 True single photon sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
13.3.1 Heralded single photons . . . . . . . . . . . . . . . . . . . . . . . . 377
13.3.2 Single photons on demand . . . . . . . . . . . . . . . . . . . . . . . 379
13.4 Characterizing photonic qubits . . . . . . . . . . . . . . . . . . . . . . . . . 381
13.5 Quantum key distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
13.5.1 QKD using single photons . . . . . . . . . . . . . . . . . . . . . . . 383
13.5.2 QKD using continuous variables . . . . . . . . . . . . . . . . . . . . 385
13.5.3 No cloning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
13.6.2 Continuous variable teleportation . . . . . . . . . . . . . . . . . . . 390
13.7 Quantum computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
15 Appendices 407
Appendix B: List of quantum operators, states and functions . . . . . . . . . . . . 408
Appendix C: The full quantum derivation of quantum states . . . . . . . . . . . . . 410
Appendix D: Calculation of of the quantum properties of a feedback loop . . . . . 412
Appendix E: Symbols and abbreviations . . . . . . . . . . . . . . . . . . . . . . . 414
Index 416
The idea behind this guide
Some of the most interesting and sometimes puzzling phenomena in optics are those where
the quantum mechanical nature of light is apparent. Recent years have seen a rapid expansion
of experimental optics into this area known as Quantum Optics. Beautiful demonstrations and
applications of the quantum nature of light are now possible and optics has been shown to be
one of the best areas of physics to actually make use of quantum mechanical ideas. This book
is intended to guide you through the many experiments published, to present and interpret
them in one common style. It also provides a practical background in opto-electronics.
Several excellent textbooks have already been written on this topic. However, in the same
way as this field of research has been initiated by theoretical ideas, most of these books have
been written from a theoretical point of view. While this results in a very solid and reliable
description of the field, we found that it frequently leaves out some of the important simpler
pictures and intuitive interpretations of the experiments. Here we are using a different and
complementary approach: This guide focuses on the actual experiments and what we can
learn from them. It explains the underlying physics in the most intuitive way we could find. It
addresses questions such as: what are the limitations of the equipment; what can be measured
and what remains a goal for the future. The answers prepare the reader to make independent
predictions of the outcome of their own experiments.
We assume that most of the readers will already have a fairly good understanding of optical
phenomena, a reliable idea of how light propagates and how it interacts with components
such as lenses, mirrors, detectors etc. The reader can picture these processes in action in
an optical instrument.. It is useful in the everyday work of a scientist to have such pictures
of what is going on inside the experiments. They have been shaped over many years by
many lectures and through ones own experimentation. In optics three different pictures are
used simultaneously: Light as waves, light as photons and light as the solution to operator
equations. To these is added the picture of noise propagating through an optical system, a
description using noise transfer functions. All of these pictures are useful; all of them are
correct within their limits. They are based on specific interpretations of the one formalism,
they are mathematically equivalent.
In this guide all these four descriptions are introduced rigorously and are used to discuss
a series of experiments. These start with simple demonstrations, gradually getting more com-
plex and include most of the experiments on quantum noise detection, squeezed states and
quantum non demolition measurements published to date. All four pictures – waves, pho-

and intuitive interpretation is highlighted, the limitations of all three interpretations are stated.
Through this approach we hope to maintain and present a lot of the fascination of quantum
optics that has captured us and many of our colleagues.
Preface to the 2nd edition
The development of the field has been rapid in recent years and the interest in quantum optics
is growing. Experiments such as the demonstration of teleportation created headlines. The
quest for communicating and storing quantum information, for quantum logic and quantum
computing by optical means is a major driving force behind this popularity. Optics is an area
that makes the concept of entanglement accessible. In the last few years several important
demonstration experiments have been published and real applications seem to be very close.
The ideas are drawn equally from the techniques of single photon detection and single
event logic as well as the ideas of continuous variable logic and the use of continuous coherent
and squeezed beams. In the experiments on entanglement so far more work has been done
with single photons. On the other hand the continuous variable approach overlaps well with
the existing technology in optical communication and optical sensing systems. We can expect
both areas to grow and to complement each other.
Thus, we felt that we should show more clearly the link between the single photon and
the photo-current case. We have expanded and reorganized the theoretical section to represent
both cases in equal detail. We have included a technical chapter on techniques for detecting
single photons and the evaluation of correlations through coincidence counting. In regard
to applications we included an introductory chapter on quantum information communication
and processing. All technical chapters were updated to take into account the recent scientific
developments and progress in technology. The book follows the well established idea of
providing independent building blocks in theory and experimental techniques which are used
in later chapters to discuss complete experiments and applications. Working as a team has
allowed us to provide more rigour and to cover a wider range of topics to make this an even
more useful book.
September 2003
Acknowledgments
This book would not have been possible without the teaching we received from many of the
creative and motivating pioneers of the field of quantum optics. We have named them in
their context throughout the book. HAB is particularly in debt to Marc Levenson who chal-
lenged him first with ideas about quantum noise. From him he learned many of the tricks
of the trade and the art of checking one’s sanity when the experiment produces results that
seem to be impossible. Over the years we learned from and received help for this book
from many colleagues and students, including W. Bowen, B.C. Buchler, J. Close, G. Dennis,
C. Fabre, P. Fisk, D. Gordon, M. Gray, C.C. Harb, D. Hope, J. Hope, E. Huntington, M. Hsu,
P.K. Lam, G. Leuchs, P. Manson, D.E. McClelland, G.J. Milburn, G. Moy, W. Munro, C. Sav-
age, R. Schnabel, R. Scholten, D. Shaddock, A. Stevenson, T. Symul, M. Taubman, N. Treps,
A.G. White, H.M. Wiseman, J.W. Wu, and from the many others who are always keen to
discuss at length the mysteries of quantum optics. We like to thank M. Colla for his expert
technical support and A. Dolinska for the creative graphics work. TCR wishes to thank Bron
Vincent and Bruce Harper for all their support and love and Isaac for putting up with a dis-
tracted Daddy. Most of all HAB wants to thank his wife Cornelia for the patience she showed
with an absentminded and many times frustrated writer.
Hans Bachor and Timothy Ralph

1.1 Historical perspective
Ever since the quantum interpretation of the black body radiation by M. Planck [Pla00], the discovery of the photoelectric effect by H. Hellwarth [Hel1898] and P. Lennard [Len02] and its interpretation by A. Einstein [Ein05] has the idea of photons been used to describe the origin of light. For the description of the generation of light a quantum model is essential. The emission of light by atoms, and equally the absorption of light, requires the assumption that light of a certain wavelength λ, or frequency ν , is made up of discrete units of energy each with the same energy hc/λ = hν .
This concept is closely linked to the quantum description of the atoms themselves. Several models have been developed to describe atoms as quantum systems, their properties can be described by the wave functions of the atom. The energy eigenstates of atoms lead directly to the spectra of light which they emit or absorb. The quantum theory of atoms has been developed extensively, it led to many practical applications [Tho00] and has played a crucial role in the formulation of quantum mechanics itself. Spectroscopy relies almost completely on the quantum nature of atoms. However, in this guide we will not be concerned with the quantum theory of atoms, but concentrate on the properties of the light, its propagation and its applications in optical measurements.
The description of optical phenomena, or physical optics, has developed largely independently from quantum theory. Almost all physical optics experiments can be explained on the basis of classical electromagnetic theory [Bow59]. An interpretation of light as a classical wave is perfectly adequate for the understanding of effects such as diffraction, interference or image formation. Even nonlinear optics, such as frequency doubling or wave mixing, are well described by classical theory. Even many properties of the laser, the key component of most modern optical instruments, can be described by classical means. Most of the present photonics and optical communication technologies are essentially applications of classical optics.
However, some experiments with extremely low intensities and those which are based on detecting individual photons raised new questions. For instance, the famous interference experiments by G.I. Taylor [Tay09] where the energy flux corresponds to the transit of individual photons. He repeated T. Young’s double slit experiment [You1807] with typically less than one photon in the experiment at any one time. In this case the classical explanation of interference based on electromagnetic waves and the quantum explanation based on the interference of probability amplitudes can be made to coincide by the simple expedient of making the probability of counting a photon proportional to the intensity of the classical field. The experiment cannot distinguish between the two explanations. Similarly, the modern versions
A Guide to Experiments in Quantum Optics, 2nd Edition. Hans-A. Bachor and Timothy C. Ralph
Copyright c 2004 Wiley-VCH Verlag GmbH & Co. KGaA
ISBN: 3-527-40393-0
2 1 Introduction
of these experiments, using the technologies of low noise current detection or, alternatively, photon counting, show the identity of the two interpretations.
The search for uniquely quantum optical effects continued through experiments concerned with intensity, rather than amplitude interferometry. The focus shifted to the measurement of fluctuations and the statistical analysis of light. Correlations between the arrival of photons were considered. It started with the famous experiment by R. Hanbury-Brown and R. Twiss [Bro56] who studied the correlation between the fluctuations of the two photo- currents from two different detectors illuminated by the same light source. They observed with a thermal light source an enhancement in the two-time intensity correlation function for short time delays. This was a consequence of the large intensity fluctuations of the thermal light and was called photon bunching. This phenomenon can be adequately explained using a classical theory which includes a fluctuating electro-magnetic field. Once laser light was available it was found that a strong laser beam, well above threshold shows no photon bunching, instead the light has a Poissonian counting statistic. One consequence is that laser beams cannot be perfectly quiet, they show noise in the intensity which is called shot noise, a name that reminds us of the arrival of many particles of light. This result can be derived from both classical and quantum models.
Next it was shown by R.J. Glauber in 1963 [Gla63] that additional, unique predictions could be made from his quantum formulation of optical coherence. One such prediction is
photon anti-bunching, where the initial slope of the two-time correlation function is positive. This corresponds to greater than average separations between the arrival times of photons, the photon counting statistics may be sub-Poissonian and the fluctuations of the resulting photo- current would be smaller than the shot noise. It was shown that a classical theory based on fluctuating field amplitudes would require negative probabilities in order to predict anti- bunching and this is clearly not a classical concept but a key feature of a quantum model. It was not until 1976, when H.J. Carmichael and D.F. Walls [Car76] predicted that light generated by resonance fluorescence from a two level atom would exhibit anti-bunching, that a physically accessible system was identified which exhibits non classical behaviour. This phenomenon was observed in experiments by H.J. Kimble, M. Dagenais and L. Mandel [Kim77], opening the era of quantum optics. More recently, the ability to trap and study individual ions allows the observation of both photon anti-bunching and sub-Poissonian statistics.

1.2 Motivation: Practical effects of quantum noise 3
quantum properties do not allow the copying of the information. Any eavesdropping will introduce noise and can therefore be detected. Quantum cryptography [Mil96] is now a viable technology, completely based on the weirdness of the concept of photons.
There is speculation of other future technological applications of photons. The superposition of different states of photons can be used to send more complex information, allowing the transfer not only of classical bits of information, such as 0 and 1, but of q-bits based on the superposition of states 0 and 1. The coherent evolution of several such quantum systems could lead to the development of quantum logic connections, quantum gates and eventually whole quantum computers which promise to solve certain classes of mathematical problems much more efficiently and rapidly [Mil98]. Even more astounding is the concept of teleportation, the perfect communication of the complete quantum information from one place to another. This is presently being tested in experiments, see Chapter 13. Optics is now a testing ground for future quantum technologies, for data communication and processing.
1.2 Motivation: Practical effects of quantum noise
Apart from their fundamental importance, quantum effects now increasingly play a role in the design and operation of modern optical devices. The 1960s saw a rapid development of new laser light sources and improvements in light detection techniques. This allowed the distinction between incoherent (thermal) and coherent (laser) light on the basis of photon statistics. The groups of A. Arecchi [Arr66], L. Mandel and R.E. Pike all demonstrated in their experiments that the photon counting statistic goes from super-Poissonian at threshold to Poissonian far above threshold. The corresponding theoretical work by R. Glauber [Gla63] was based on the concept that both the atomic variables and the light are quantised and showed that light can be described by a coherent state, the quantum analogy counterpart to a classical field. The results are essentially equivalent to a classical treatment of an oscillator. However, it is an important consequence of the quantum model, that any measurement of the properties of this state, intensity, amplitude or phase of the light, will be limited by quantum noise.
Quantum noise can impose a limit to the performance of lasers, sensors and communication systems and near the quantum limit the performance can be quite different. To illustrate this consider the result of a very simple and practical experiment: Use a laser, such as a laser printer with a few mW of power, detect all of the light with a photo-diode and measure the fluctuations of the photo-current with an electronic spectrum analyser, as shown in Fig. 1.1(a) This produces an electronic signal which represents the intensity noise at one single detection frequency. The fluctuations vary with the frequency, a plot of the noise power is called an intensity noise spectrum of the laser light. Such spectra will be discussed in detail in this book.

MHz
(i)
(ii)
(a)
(b)
Figure 1.1: The intensity noise of a laser. (a) Schematical layout of the experiment. (b) The
noise spectrum, for detection frequencies 4–12 [MHz]. The noise power is shown on a logarith-
mic scale. Two traces are shown: (i) laser on, (ii) laser off. Details of these measurements are
discussed in Chapter 9.
individual components at 9 [MHz] are due to modulations of the light. This is the way information can be sent via a laser beam. The diagram shows that the quality of the information, given by the ratio between the size of the signal and the size of the noise, is limited.
The flat noise background in trace (i), which is frequency independent and thus classed as “white noise”, is the feature of greatest interest in terms of quantum optics. This is quantum
noise and represents a phenomenon which is not included in the classical electro-magnetic model of light. It is a direct consequence of the quantum theory of light and will be discussed in great detail in this book. This quantum noise represents some unavoidable fluctuations in the intensity, it forms the quantum noise limit , or QNL, for the intensity noise. It is an intrinsic property of the light and it appears for both laser and thermal light. The magnitude of the noise is directly linked to the intensity of the light detected, or the average of the photo-current, and is expressed by the shot noise formula.

1.2 Motivation: Practical effects of quantum noise 5
These particles do not interact with each other and a certain degree of randomness, represented by a Poissonian distribution, is the natural state of this system. We will find that light tends to approach such a Poissonian distribution during the arrival times of the photons at the detector. As a consequence the photo-current, which is the quantity actually measured in the experiments, will display fluctuations of a magnitude dictated by the Poissonian distribution. Such a statistical model is very useful for the interpretation of intensity, but we will find that it has severe shortcomings whenever other properties of light, such as phase and interference, are investigated.
Alternatively, quantum noise can be regarded as a consequence of the photo-detection process, as a randomness of the stream of electrons produced in the photo-detector. This view prevailed for a long time, particularly in engineering text books. But this view is misleading as we find in the recent squeezing experiments, described in this book. It cannot account for situations where nonlinear optical processes modify the quantum noise, while the detector remains unaffected. In the squeezing experiments the quantum noise is changed optically and consequently we have to assume it is a property of the light, not of the detector.
Finally, it is possible to expand the concept of classical waves and to expand the concepts of beat notes and modulation, which we already used in the description of the origin of the discrete components contained in the noise spectrum, see Fig. 1.1. Consider any noise detected as the outcome of a beat experiment. A signal at frequency corresponds to the beat, or product, of at least two waves with optical frequencies ν and ν ± . Since there is only one dominant frequency component in the spectral distribution of a laser beam, which is at the centre frequency ν L of the laser, the noise at can be regarded as the consequence of randomly fluctuating fields at the laser sidebands ν L + and ν L − beating with the centre component at ν L. The randomness of the field in the sidebands is not included in the classical wave model, where the amplitude is strictly zero away from the centre component. However, the quantum effect can be incorporated by adding fluctuations of a fixed size to the otherwise noiseless classical sidebands. This idea is very successful in interpreting experiments concerned with quantum noise, for example the description of the properties of beamsplitters and interferometers. It uses all of the components familiar from physical optics and electronics (monochromatic waves, sidebands, phase difference, interference and beat signals) and adds only one extra component, namely a randomly varying field. This model, which is rarely used in most of the research literature, has a prominent position in this book due to its simplicity and practicality.
Laser

6 1 Introduction
It is worthwhile to explore the properties of quantum noise somewhat further. It was found that, in clear distinction to classical noise, no technical trick can eliminate quantum noise. To illustrate the differences consider the following schemes for the suppression of the noise. The first scheme, see Fig. 1.2, involves difference detection, as it is frequently employed in absorption experiments. A beamsplitter after the laser is used to generate a second beam, which is detected on a second detector. Any intensity modulation of the intensity of the laser beam will appear on both beams and results in changes of the two photo-currents. The modulations of the currents will be strongly correlated. Using a difference amplifier the common fluctuations will be subtracted. When the gain of the two amplifiers has been chosen appropriately the resulting difference current will contain no modulation. This idea works not only for modulations but also for technical noise, which can be regarded as a random version of the modulations. Such technical noise can be subtracted. However, this scheme fails for the suppression of quantum noise. Both light beams contain quantum noise, which leads to white noise in the two photo-currents. Experiments show that these fluctuations are not subtracted by the difference amplifier. The noise of the difference is actually larger than the noise of the individual beams, it is the quadrature sum of the noise from the two individual currents. This result remains unchanged if the difference is replaced by a sum or if the two currents are added with an arbitrary phase difference. The resulting noise is always the quadrature sum independent of the sign, or phase, of the summation. This is equivalent to the statement that the noise in the two photo-currents is not correlated.
At this stage it is not easy to identify the point in the experiment where this uncorrelated noise is generated. One interpretation assumes that the noise in the currents is generated in the photo-detectors. Another interpretation assumes that the beamsplitter is a random selector for photons and consequently the intensities of the two beams are random and thus uncorrelated. A distinction can only be made by further experiments with squeezed light, which are discussed later in this book.
An alternative scheme for noise suppression is the use of feedback control, as shown in Fig. 1.3. It achieves equivalent results to difference detection. The intensity of the light can be controlled with a modulator, such as an acousto-optic modulator or an electro-optic modulator. Using a feedback amplifier with appropriately chosen gain and phase lag the intensity noise can be reduced, all the technical noise can be eliminated [Rob86]. It is possible to get very close to the quantum noise limit, but the quantum noise itself cannot be suppressed [Mer93], [Tau95]. This phenomenon can be understood by considering the properties of photons. As mentioned before, the quantum noise measured by the two detectors is not correlated, thus the feedback control, when operating only on quantum noise on one detector, will not be able to control the noise in the beam that reaches the other detector. This can be explained using a full quantum theory. Alternatively, it can be interpreted as a consequence of the properties of the beamsplitter. It will randomly select the photons going to the control detector and consequently the control system has no information about the quantum fluctuations of the light leaving the experiment.

Laser
Modulator
Figure 1.3: The second attempt to eliminate laser noise: An improved apparatus using a feed-
back controller, or ‘noise eater’.
Laser
.
Figure 1.4: Feedback control of the current below the Poissonian limit. A laser with high
quantum efficiency can convert the stabilised current into light with intensity noise suppressed
below the quantum noise limit.
concerned with, and the fluctuations can be controlled with ease to levels well below the shot noise level. For sources with high quantum efficiencies the sub-Poissonian statistics of the drive current is transferred directly to the statistics of the light emitted. This is illustrated in the extension of our little experiment shown in Fig. 1.4. Such experiments were pioneered for the case of diode lasers by the group of Y. Yamamoto [Mac89]. They showed that intensity fluctuations can be suppressed in a high impedance semiconductor laser driven by a constant current and similar work was carried out with LEDs by several groups.
A two dimensional description of the light, with properties we will call quadratures will be necessary to explain this quantum effect. Noise can be characterized by the variance in both the amplitude and phase quadrature. This work shows one of the limits of the quantum models discussed above: they were restricted to one individual laser beam but should have included the entire apparatus. We will derive a simple formalism that allows us to predict the laser noise at any location within the instrument.

8 1 Introduction
may have reduced quantum fluctuations at the expense of increased fluctuations in the other quadrature. Such squeezed light could be used to beat the standard quantum limit. After the initial theoretical predictions, the race was on to find such a process. A number of nonlinear processes were tried simultaneously by several competing groups. The first observation of a squeezed state of light was achieved by the group of R.E. Slusher in 1985 in four-wave mixing in sodium atomic beam [Slu85]. This was soon followed by a demonstration of squeezing by four wave mixing in optical fibres by the group of M.D. Levenson and R. Shelby [She86] and in an optical parametric oscillator by the group of H.J. Kimble [Wu87]. In recent years a number of other nonlinear processes have been used to demonstrate the quantum noise suppression based on squeezing [Sp.Issues]. An generic layout is shown in Fig. 1.5. The experiments are now reliable and practical applications are feasible.
Laser
+ _
Figure 1.5: A typical squeezing experiment. The nonlinear medium generates the squeezed
light which is detected by a homodyne detection scheme.
In the last few years the concept of quantum information has brought even more life to quantum optics. Plans for using the complexity of quantum states to code information, to transmit it without out losses, also known as teleportation, to use it for secure communication and cryptography, to store quantum information and possibly use it for complex logical processes and quantum computing have all been widely discussed. The concept of entanglement
has emerged as one of the key qualities of quantum optics. As it will be shown in this guide, it is now possible to create entangled beams of light either from pairs of individual photons or from the combination of two squeezed beams and the demand and interest in non-classical states of light has sharply risen. We can see quantum optics playing a large role in future communication and computing technologies [Mil96,Mil98].
For this reason the guide covers both single photon and CW beam experiments parallel to each other. It provides a unified description and compares the achievements as well as tries to predict the future potential of these experiments.
1.3 How to use this guide

1.3 How to use this guide 9
challenges of these experiments and gives an interpretation of their results. One of the current difficulties in understanding the field of quantum optics is the diversity of the models used. On the one hand, the theory, and most of the publications in quantum optics, are based on a rigorous quantum model which is rather abstract. On the other hand, the teaching of physical optics and the experimental training in using devices such as modulators, detectors and spectrum analysers are based on classical wave ideas. This training is extremely useful, but frequently does not include the quantum processes. Actually, the language used by these two approaches can be very different and it is not always obvious how to relate a result from a theoretical model to a technical device designed and vice versa. As an example compare the schematical representation of a squeezing experiment, given in Fig. 1.6, both in terms of the theoretical treatments for photons and laser beams and in an experimental description. The purpose of this guide is to bridge the gap between theory and experiment. This is done by describing the different building blocks in separate chapters and combining them into complete experiments as described in recent literature.
Experiment vs. Theory
Experiment
1 211 2 3
Figure 1.6: Comparison between an experiment (middle) and the theoretical description for few
photon states (top) and laser beams (bottom)

10 1 Introduction
niques such as cavity locking and feedback controller, given in Chapter 8. The concept of squeezing is central to all attempts to improve optical devices beyond the standard quantum limit and is introduced and discussed in Chapter 9. It also describes the various squeezing experiments and their results are discussed, the different interpretations are compared. In a similar way Chapter 11 discusses quantum non-demolition experiments. Finally, the potential applications of squeezed light are described in Chapter 10. In Chapter 12 experiments which test the fundamental concepts of quantum mechanics are discussed. Finally, the concepts of quantum information and the present state of art of experiments using either single photon or CW beams are presented in Chapter 13.
This guide can be used in different ways. A reader who is primarily interested in learning about the ideas and concepts of quantum optics would best concentrate on Chapters 2, 3, 4, 9, 12, and 13 but may leave out many of the technical details. For these readers Chapter 5 would provide a useful exercise in applying the concepts introduced in Chapter 4. In contrast, a reader who wishes to find out the limitations of optical engineering or wants to learn about the intricacies of experimentation would concentrate more on Chapters 2, 6, 5, 8 and for an extension into experiments involving squeezed light Chapters 9, 10 can be added. A quick overview of the possibilities opened by quantum optics can be gained by reading Chapters 3, 4, 6, 9, 11, and 12. We hope that in this way our book provides a useful guide to the fascinating world of quantum optics.
This book is accompanied by a Web page http://photonics.anu.edu.au/qoptics/people/bachor/book.html
which provides updates, exercises, links and, if required, errata.
Bibliography
[Arr66] Measurement of the time evolution of the radiation field by joint photocurrent distributions, F.T. Arecchi, A. Berne, A. Sona, Phys. Rev. Lett. 17, 260 (1966)
[Asp82] Experimental realisation of Einstein-Podolsky-Rosen-Bohm Gedankenexperi- ment: A new violation of Bell’s inequalities, A. Aspect, P. Grangier, G. Roger, Phys. Rev. Lett. 49, 91 (1982)
[Bel64] On the Einstein-Podolsky-Rosen paradox, J.S. Bell, Physics 1, 195 (1964)
[Ben93] Teleporting an unknown quantum state via dual classical and Einstein-Podolsky- Rosen channels, C.H. Bennett, G. Brassard, C. Crepau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70, 1895 (1993)
[Bow59] The principles of Optics, M. Born and E. Wolf (1959)
[Bro56] Correlation between photons in two coherent beams of light, R. Hanbury-Brown, R.Q. Twiss, Nature 177, 27-29 (1956)

[Ein35] Can a quantum-mechanical description of physical reality be considered complete?, A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. A 47, 777 (1935)
[Gla63] The quantum theory of optical coherence, R.J. Glauber, Phys. Rev. Lett. 130, 2529 (1963)
[Hel1898] Elektrometrische Untersuchungen, W. Hallwachs, Annalen der Physik und Chemie, neue Folge Band XXIX (1886)
[Kim77] Photon antibunching in resonance fluorescence, H.J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977)
[Len02] Ueber die lichtelektrische Wirkung, P. Lenard, Ann. Physik 8, 149 (1902) [Mac89] Observation of amplitude squeezing from semiconductor lasers by balanced direct
detectors with a delay line, S. Machida, Y. Yamamoto, Opt. Lett. 14, 1045 (1989) [Mer93] Photon noise reduction by controlled deletion techniques, J. Mertz, A. Heidmann,
J. Opt. Soc. Am. B 10, 745 (1993) [Mil96] Quantum technology, G. Milburn, Allen & Unwin (1996) [Mil98] The Feynman processor , G. Milburn, Allen & Unwin (1998) [Pla00] Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum, M. Planck,
Verhandlungen der Deutschen Physikalischen Gesellschaft 2, 237 (1900) [Rob86] Intensity stabilisation of an Argon laser using an electro-optic modulator: perfor-
mance and limitation, N.A. Robertson, S Hoggan, J.B. Mangan, J. Hough, Appl. Phys. B 39, 149 (1986)
[She86] Generation of squeezed states of light with a fiber-optics ring interferometer, R.M. Shelby, M.D. Levenson, D.F. Walls, A. Aspect, G.J. Milburn, Phys. Rev. A33, 4008 (1986)
[Slu85] Observation of squeezed states generated by four-wave mixing in an optical cavity, R.E. Slusher, L.W. Hollberg, B. Yurke, J.C. Mertz, J.F. Valley, Phys. Rev. Lett. 55, 2409 (1985)
[Sp.Issues] Special issues on squeezed states of light , J. Opt. Soc. Am. B4 (1987) and J. Mod. Opt. 34 (1987)
[Tap91] Sub-shot-noise measurement of modulated absorption using parametric down- conversion, P.R. Tapster, S.F. Seward, J.G. Rarity, Phys. Rev. A44, 3266 (1991)
[Tau95] Quantum effects of intensity feedback, M.S. Taubman, H. Wiseman, D.E. Mc- Clelland, H-A. Bachor, J. Opt. Soc. Am. B 12, 1792 (1995)
[Tay09] Interference fringes with feeble light, G.I. Taylor, Proc. Cambridge Phil. Soc.15, 114 (1909)
[Tho00] Spectrophysics, principles and applications, 3rd edition, A. Thorne, U. Litzen, S. Johansson, Springer Verlag (1999)
[Wal83] Squeezed states of light, D. Walls, Nature 306, 141 (1983) [Wu87] Squeezed states of light from an optical parametric oscillator, Ling-An Wu, Min
Xiao, H.J. Kimble, J. Opt. Soc. Am. B 4, 1465 (1987) [You1807] Course of lectures on natural philosophy and mechanical arts , Th. Young, Lon-
don (1807) [Zei00] Quantum cryptography with entangled photons, A. Zeilinger, Phys. Rev. Lett. 84,
4729 (2000)
2 Classical models of light
The development of optics was driven by the desire to understand and interpret the many op-
tical phenomena which were observed. Over the centuries more and more refined instruments
were invented to make observations, revealing ever more fascinating details. All the optical
phenomena investigated during the 19th century were observed either directly or with pho-
tographic plates. The spatial distribution of light was the main attraction. The experiments
measured long term averages of the intensity, no time resolved information was available.
The formation of images and the interference patterns were the main topics of interest. In
this context the classical optical models, namely geometrical optics, dating back to the Greeks
and developed ever since, and wave optics, introduced through Huygens’ ideas and Fresnel’s
mathematical models, were sufficient.
A revolution came with the invention of the laser. This changed optics in several ways:
It allowed the use of coherent light, bringing the possibilities of wave optics to their full
potential, for example through holography. Lasers allowed spectroscopic investigations with
unprecedented resolution, testing the quantum mechanical description of the atom in even
more detail. The fast light pulses generated by lasers allowed the study of the dynamics
of chemical and biological processes. In addition, a range of unforeseen effects emerged:
The high intensities resulted in nonlinear optical effects. One fundamental assumption of the
wave model, namely that the frequency of the light remained unchanged, was found to be
seriously violated in this new regime. The sum-, difference- and higher harmonic frequencies
were generated. Finally, optics was found to be one of the best testing grounds of quantum
mechanics. The combination of sensitive photoelectric detection with reliable, stable laser
sources meant not only that interesting quantum mechanical states could be produced but also
that quantum mechanical effects had to be accounted for in technical applications.
Light is now almost exclusively observed using photo-electric detectors which can register
individual events or produce photocurrents. A description of these effects requires the field of
statistical optics.
We need a description of the beam of light which emerges from a laser. A laser beam
is a well confined, coherent electro-magnetic field oscillating at a frequency ν , or angular
frequency ω = 2πν . This frequency is constant for all times and locations and can only be
changed by the interaction of light with nonlinear materials. Furthermore, the beam prop-
agates and maintains its shape unperturbed by diffraction. In this chapter we will develop
classical models suited to describing this type of light. In following chapters we will see how
these must be modified to account for quantum mechanical effects.
A Guide to Experiments in Quantum Optics, 2nd Edition. Hans-A. Bachor and Timothy C. Ralph
Copyright c 2004 Wiley-VCH Verlag GmbH & Co. KGaA
ISBN: 3-527-40393-0
2.1.1 Mathematical description of waves
An electromagnetic wave in an isotropic, insulating medium is described by the wave equation
∇2E(r, t)− 1 c
∂t2 E(r, t) = 0 (2.1.1)
with a solution for the electric field vector E(r, t), given by
E(r, t) = E 0
p(r, t) (2.1.2)
The wave oscillates at a frequency ν in [Hz]. For now we consider only a single frequency
component but in general the solution can be an arbitrary superposition of such components.
The electric field vector is always perpendicular to the local propagation direction. The ori-
entation of the electric field on the plane tangential to the local wavefront is known as the
polarisation and is described by the vector p(r, t). The dimensionless complex amplitude
α(r, t) of the wave can be written as a magnitude α0(r, t) and a phase φ(r, t).
α(r, t) = α0(r, t)exp{iφ(r, t)} (2.1.3)
The magnitude α0(r, t) will change in time when the light is modulated or pulsed. It will also
vary in space, describing the extent of the wave. The wave will typically be constant in space
for dimensions of the order of several wavelengths. The direction and shape of the wave front
is determined by the phase term φ(r, t) and the spatial distribution of the φ(r, t) describes the
curvature of the waves.
Simple examples of specific solutions are monochromatic plane waves and spherical waves.
These represent the extreme cases in terms of angular and spatial confinement. The monochro-
matic plane wave is the ideal case in which the complex amplitude magnitude α0 is a constant
and the phase is given by
φ(r) = −k · r (2.1.4)
The wave propagates in a direction and with a wavelength λ represented by the wave vector k,
where k/k is a unit vector that provides the direction of the local wave front and k = 2π/λ.
The wavelength is related to the frequency via λ = c/nν , where n is the refractive index of
the medium in which the wave propagates and c is the speed of light in the vacuum. Thus we
can write
α(z) = α0 exp{−ikz} (2.1.5)
where we have assumed that the wave propagates in the z direction. The wave has a constant
amplitude in any plane perpendicular to this optical axis. Substituting Equation (2.1.5) into
Equation (2.1.2) and assuming α0 as real we obtain the familiar solution

14 2 Classical models of light
Linear polarisation means that p(r, t) has a fixed direction perpendicular to k. Circular po-
larisation means that in any one location p(r, t) is rotating in the plane perpendicular to k.
Equation (2.1.6) represents a wave which has no angular spread. In contrast the spherical
wave originates from one singular point. All its properties are described by one parameter r,
the distance to its point of origin. The amplitude magnitude reduces with 1/r, and the phase
is constant on spheres around the origin,
α(r, t) = α(r) = α0
r exp{ikr} (2.1.7)
2.1.2 The Gaussian beam
Both of these models are not physical, they are only crude approximations to a beam of light
emitted by a real laser. Such a beam is confined to a well defined region and should be as close
as possible to the ideal of a spatially localized but non divergent wave. It has to be immune
to spreading by diffraction and therefore has to be a stable solution of the wave equation.
An example of such solutions are the Gaussian modes. The simplest of these is circularly
symmetric around the optical axis, thus depending only on the radial distance ρ from the
beam axis
ρ2 = x2 + y2
This special solution is the Gaussian beam or fundamental mode, with a complex amplitude
distribution of
exp
−ikρ2
, q (z) = z + iz0 (2.1.8)
where z0 is a constant. The Gaussian beam is a good spatial representation of a laser beam.
Laser light is generated inside a cavity, constructed of mirrors, by multiple interference of
waves and simultaneous amplification. These processes result in a well defined, stable field
distribution inside the cavity, the so called cavity modes [Kog66, Sie86, Tei96]. The freely
propagating beam is the extension of this internal field. It has the remarkable property that
it does not change its shape during propagation. Both the internal and the external fields are
described by Equation (2.1.8). By taking the square of the complex amplitude α(r, t) the
intensity is found, resulting in a Gaussian transverse intensity distribution.
I (ρ, z) = I 0

z
z0
2
(2.1.9)
The properties of this Gaussian beam are the following: The shape of the intensity distribution
remains unchanged by diffraction. The size of the beam is given by W (z). The beam has its
narrowest width at z = 0, the so-called waist of the beam. It corresponds to the classical focus,
but rather than point-like, the Gaussian beam has a size of W 0. This value gives the radial

Figure 2.1: Properties of a Gaussian beam.
At the waist the phase front is a plane, φ(ρ) is a constant. Moving away from the waist the
beam radius W (z) expands, the intensity at the axis reduces and, at the same time, the wave
front curves. Along the optical axis the Gaussian beam scales with the parameter z0 which is
known as the Rayleigh length
z0 = πW 20
λ
It describes the point where the intensity on axis has dropped to half the value at the waist,
I (0, z0) = 1 2 I 0, and the beam radius has expanded by a factor of
√ 2. This can be regarded
as the spatial limit to the focus region, the value 2z0 is frequently described as the depth of
focus or as the confocal parameter. Far away from the waist, for z z0, the Gaussian beam
has essentially the wave front of a spherical wave originating at the waist. Obviously, the
Gaussian beam is still centered on the beam axis and has only a finite extent W (z). At these
large distances the size of the beam W (z) depends linearly on z, the beam has a constant
divergence angle of
2πW 0 (2.1.10)
In addition to the wave front curvature a gradual phase shift occurs on the beam axis between
a uniform plane wave with identical phase at z = 0 and the actual Gaussian beam. The phase
difference increases to π/4 at z = z0 and asymptotically approaches π/2.
Table 2.1: Properties of a Gaussian beam
Location z = 0 z = z0 z z0
Beam Waist W = W 0 W = √
2W 0 W (z) = z/z0W 0 Intensity I 0 I 0/2 I 0(z0/z)2
Wave front plane wave curved wave front spherical wave
Phase retardation 0 π/4 π/2

As known from experiments with laser beams, the Gaussian beam is very robust in regard
to propagation through optical components. Any component which is axially symmetric and
the effect of which can be described by a paraboloidal wave front will keep the Gaussian char-
acteristic of the beam. This means the emerging beam will be Gaussian again, however, now
with a new position z = z of the waist. All spherical lenses, spherical and parabolic mirrors
transform a Gaussian beam into another Gaussian beam. Another important consequence of
this transformation is the fact that Gaussian beams are self reproducing under reflection with
spherical mirrors.
However, the Gaussian beam, Equation (2.1.8), is not the only stable solution of the parax-
ial wave equation. There are many other spatial solutions with non-Gaussian intensity dis-
tributions which can be generated by a laser and propagate without change in shape. One
particular set of solutions are the Hermite-Gaussian beams. These beams have the same un-
derlying phase, except from an excess term ζ l,m(z) which varies slowly with z . They have
the same paraboloidal wave front curvature as the Gaussian beam, which matches the curva-
ture of spherical optical components aligned on the axis. Their intensity distribution is fixed
and scales with W (z). The distribution is two dimensional, with symmetry in the x and y directions and usually not with a maximum on the beam axis. The intensity on the beam axis
decreases with the same factor W (z) as the Gaussian beam along the z axis. The various
solutions can be represented by a product of two Hermite-Gaussian functions, one in the x the
other in the y direction, which directly leads to the following notation
αl,m(x,y,z) = α0Al,m
(2.1.11)
Here Al,m is a constant and Gl(u) and Gm(u) are standard mathematical functions, the Her-
mite polynomials of order l, m respectively, which can be found in tables [Sie86, Abr72]. In
particular, G0(u) = exp{−u2/2} and thus the solution α0,0(x,y,z) is identical to the Gauss-
ian beam described above. This set of solutions forms a complete basis for all functions with
paraboloidal wave fronts. Any such beam can be described by a superposition of Hermite-
Gaussian beams. All these components propagate together, maintaining the amplitude dis-
tribution. The only component with circular symmetry is the Gaussian beam α0,0(x,y,z).
However, it is possible to describe other circularly symmetric intensity distributions. Consider
for example the superposition of two independent beams with α1,0(x,y,z) and α0,1(x,y,z) and equal intensity; if these beams have random phase their interference can be neglected and
the sum of the intensities results in a donut shaped circularly symmetric function, occasionally
observed in lasers with high internal gain.
2.1.3 Quadrature amplitudes

2.1 Classical waves 17
Figure 2.2: Phasor diagram (a) for a constant single wave, and (b) the representation of inter-
ference of several coherent waves.
As shown above, the phase distribution φ(r, t) describes the shape of the wavefront as well
as the absolute phase, in respect to a reference wave. However, it is useful to separate the
two, to consider the absolute phase independently from the phase distribution, or wave front
curvature. An equivalent form of Equation (2.1.2) can be given in terms of the quadrature
amplitudes, X 1 and X 2, the amplitudes of the associated sine and cosine waves
E (r, t) = E 0 [X 1(r, t) cos(2πνt) + X 2(r, t) sin(2πνt)] p(r, t) (2.1.12)
The quadrature amplitudes are proportional to the real and imaginary parts of the complex
amplitude.
X 2(r, t) = i
(2.1.13)
In this notation the absolute phase φ0 of the wave is associated with the distribution of the
amplitude between the quadratures X 1 and X 2.
φ0 = tan−1
(2.1.14)
For X 2 = 0 the wave is simply a cosine wave, Equation (2.1.6). We can regard this wave
arbitrarily as the phase reference. A common description of classical waves whenever inter-
ference or diffraction is described is phasor diagram, a two dimensional diagram of the values
X 1 and X 2 as shown in Fig. 2.2(a). In such a diagram, which represents the field at one point
in space and time, each wave corresponds to one particular vector of length α from the origin
to the point (X 1, X 2), and this vector could be measured as the complex amplitude averaged
over a few optical cycles of length 1/ν . The magnitude of the wave is given by the distance
α of the point from the origin, the relative phase, φ0 is given by the angle with the X 1 axis,
since the absolute phase is arbitrary. It is useful to choose α as real (φ = 0), that means the
vector parallel to the X 1 axis.
If several waves are present at one point at the same time, they will interfere. Each indi-
vidual wave is represented by one vector, the total field is given by the sum of the vectors, as
shown in Fig. 2.2(b). These types of diagrams are a useful visualisation of the interference
effects. After the total amplitude αtotal has been determined we can again select a phase for
this total wave and choose αtotal as being real. All forms of interference can be treated this

E
X1
X2
φ
X1
E
Figure 2.3: Fluctuations shown in phasespace: the field changes in time and moves from point
to point in the phasor diagram. This corresponds to waves with different amplitude and phase.
In addition, the wave can have fluctuations or modulations in both amplitude and phase.
Such variations correspond to fluctuations in the quadrature values. We can describe the time
dependent wave with the complex amplitude
α(t) = α + δX 1(t) + i δX 2(t) (2.1.15)
In the limit α δX 1(t), that means for waves that change or fluctuate only very little around
a fixed value α, we see that δX 1(t), δX 2(t) describe the changes in the amplitude and phase
respectively. This is the regime which we use to describe beams of light detected by detectors
which produce a photocurrent. We will call them the amplitude and phase quadratures. The
direct link between variations in the quadrature values and those of the field magnitude and
phase is shown diagrammatically in Fig. 2.3.
2.1.4 Field energy, intensity, power
The energy passing through a small surface area element dx, dy around the point x, y in a short
time interval dt around the time t is obtained by integrating the square of Equation (2.1.2).
Note that because the field is propagating in the z-direction this is equivalent to the energy in
the volume element dx, dy,c dt. We get
E (x,y,t) = E 20 2 α ∗α dt dx dy (2.1.16)
where we have assumed that dx dy dt is sufficiently small that α does not vary over the
interval, but that dt 1/ is satisfied and the time average will be over a large number of
optical cycles. As a result only the phase independent terms survive. The intensity of the beam
is
E 20 2 α ∗α dt (2.1.17)
What is normally measured by a photo-detector is the power P of the light, the total flux of
energy reaching the detector in a second, integrated over the area of the detector
P =

The energy in a small volume element, Equation (2.1.16), can also be expressed in terms of
the quadratures as
E (x,y,t) = E 20 (X 12 + X 22) dt dx dy (2.1.19)
where we have used Equation (2.1.13). The parameter E 20 is proportional to the frequency of
the light, ν , specifically:
4 (2.1.20)
where is Planck’s constant. It is the power in Equation (2.1.18) which is measured in most
experiments and from this we infer the amplitude, the phase and the fluctuations of the beam.
2.1.5 A classical mode of light
The idea of a phasor diagram can now be combined with the concept of spatial modes as
described in Equation (2.1.3). Let us concentrate on the simplest type of beam, a Gaussian
beam. Equation (2.1.8) relates the relative amplitude and phase at any one point back to a point
at z = 0 at the centre of the waist. Once the size W 0 is given the entire spatial distribution
is well defined. We can assign one overall optical power, or peak intensity I 0, one overall
phase to the entire spatial distribution. Thus the entire laser beam can be described by only
six parameters.
Parameters required to describe one mode of light
Intensity of the mode I 0 Frequency of optical field ν
Absolute phase of this mode φ0 Direction of propagation z
Location of the waist (z = 0) r0
Waist diameter of the beam W 0 Direction of polarisation P
Such a mode of light is strictly speaking a constant in time. It corresponds to a laser
which has a stable, time independent output, it produces an intensity distribution which is
not changing at all in time. More realistically the light mode will not be perfectly constant.
Experiments show that there are always fluctuations. Ultimately quantum mechanics limits
how small the fluctuations can be.
In addition, we wish to communicate some sort of information with the light and that
means we must modulate it either in time or in space. Any such changes in the intensity
have to be considered separately by introducing time variable parameters such as I (t), ν (t) etc. . We will encounter two different types of experiments. In the case of continuous wave

W o P
x
y
Figure 2.4: Graphical representation of the parameters of a mode of light.
dependence of the parameters is slow compared to the time constant of the detection process.
That means we can actually resolve and process the fluctuations and modulations of the light.
Pulses of light are the other extreme case. Laser pulses are frequently shorter than the
response time of the detector. Consequently only the integrated energy of many pulses is
measured and the details of the time development of an individual pulse are lost. However, it
is still possible to determine the fluctuations of the beam and this becomes a measurement of
the statistics of the energy of the pulses. This special case will be discussed in Section 9.9.
2.1.6 Classical modulations
Information carried by a beam of light can be described in the form of modulation sidebands.
Note that the equations used here are identical to those describing radio waves where AM
and FM (or PM) modulation are the familiar, standard techniques. Many of the technical
expressions, such as carrier for the fundamental frequency components and sidebands, are
taken directly from this much older field of technology.
A beam of light which is amplitude modulated by a fraction M at one frequency mod can
be described by
(2.1.21)
We see that the effect of the modulation is to create new frequency components at ν L + mod
and ν L − mod. These are known as the upper and lower sidebands respectively whilst the
component remaining at frequency ν L is known as the carrier Fig. 2.5(a). For completely
modulated light, that means M = 1, the sidebands have exactly 1 2 the amplitude of the fun-
damental component. The sidebands are perfectly correlated, they have the same amplitude
and phase, that means the two beat signals, generated by the sidebands at ν L + mod and
ν L − mod beating with the carrier, are in phase. A second representation is in the time do-

and δα− rotate in opposite directions with angular frequency 2π. A third representation is
in frequency space in the phasor diagram Fig. 2.5(c). Notice that for a given modulation depth
M the sideband amplitudes scale with the amplitude of the carrier, α0. Thus even very small
modulation depths, M , can result in significant sideband amplitude if the carrier amplitude is
large. The detected intensity is given by
I (t) = I 0 |α(t)|2 = I 0

2
(2.1.22)
and oscillates at the frequency mod around the long term average I 0. The resulting photo-
current has two components, a long term average iDC and a constant component at frequency
mod, labelled i(mod). A common (and useful) situation is when the modulation depth is
very small, M 1. Then the resulting intensity modulation is linear in the modulation depth
I (t) ≈ I 0
(2.1.23)
A light beam with modulated phase can be described in a similar way. The amplitude can
be represented by
α(t) = α0 exp
exp(i2πν Lt) (2.1.24)
Clearly the intensity will not be modulated as |α(t)|2 = |α0|2. Expanding αt we obtain
α(t) = α0
2 cos2(2πmodt) + · · ·
−
+ · · ·
(2.1.25)
Suppose we measure the quadrature amplitudes (X 1 and X 2) of this field relative to a cosine
reference wave (i.e. we take α to be real). The X 2 quadrature will have pairs of sidebands
separated from the carrier by ±mod, ±3mod, . . .. The X 1 quadrature on the other hand will
have sidebands separated from the carrier by ±2mod,± 4mod, . . .. Again a simple, and
useful, situation is for the modulation depth to be very small, M 1. The amplitude is
approximately

(2.1.26)
and the sidebands will only appear on measurements of the X 2 quadrature. Exact expressions

to this double sideband approximation is shown in Fig. 2.5(b) and Fig. 2.5(d). The sidebands
have the same magnitude. The relative phase of the amplitudes is such that the two resulting
beat signals have the opposite sign and the beat signals cancel. Thus, as we have seen, phase
modulation cannot be detected with a simple intensity detector. Another way of saying this
is that the sidebands for phase modulation are perfectly anti-correlated. The treatment of
frequency modulation proceeds in a similar way. Any information carried by the light can be
expressed as a combination of amplitude and phase modulation. Any modulation, in turn, can
be expressed in terms of the corresponding sidebands.
AM= L
Figure 2.5: Three representations of amplitude (AM) and frequency (FM) modulation with the
frequencies sidebands 1 and 2 respectively. (a) shows the sidebands ν L ±1 and ν L ±2,
(b) shows the same modulations in the time domain and in a phasor diagram, and (c) shows the
phasor diagram in frequency space.

2.2 Statistical properties of classical light
2.2.1 The origin of fluctuations
The classical model so far assumes that light consists of a continuous train of electro-magnetic
waves. This is unrealistic: atoms emitting light have a finite lifetime, they are emitting bursts
of light. Since these lifetimes are generally short compared to the detection interval td, these
bursts will not be detected individually. The result will be a well defined average intensity,
which will fluctuate on time scales > td. Similarly in most classical, or thermal, light sources
there will be a large number of atoms, each emitting light independently. In addition, the
atoms will move, which leads to frequency shifts, and they can collide, which disrupts the
emission process and leads to phase shifts. All these effects have two major consequences:
1. Any light source has a spectral line shape. The frequency of the real light is not perfectly
constant, it is changing rapidly in time. The description of any light requires a spectral
distribution which is an extended spectrum, not a delta-function.
2. Any light source will have some intensity noise. The intensity will not be constant.
The amplitude of the light will change in time. There will be changes in the amplitude
due to the fluctuations of the number of emitting atoms but also due to the jumps, or
discontinuities, in the phase of the light emitted by the individual atoms.
The first point leads to the study of spectral line shapes, an area of study which can reveal
in great detail the condition of the atoms. The study of spectra is a well established and well
documented area of research. However, in the context of this guide we will not expand on this
subject. The relevant results are cited, where required. For the analysis of our quantum optics
experiments we are not concerned as much with the spectral distribution of the light. We are
more interested in the noise, and thus the statistical properties, of intensity, amplitude and
phase. Only the intensity fluctuations can be directly measured in the experiments. However,
with the homodyne detector, which will be explained in Section 8.1.4, we have a tool to
investigate both the phase and amplitude of the light.
The measurement of the statistical properties of light can be carried out in two alternative
ways. Either through a measurement of the fluctuations themselves in the form of temporal
noise traces or noise spectra. Or, alternatively, through the measurement of coherence or cor-
relation functions. While requiring entirely different equipment, the two concepts are related
and in this chapter predictions for both types of measurement will be derived. The emphasis
here is on the results and the underlying assumptions and not on the details of the deriva-
tion. Many of the details can be found in the excellent monographs by Loudon, Louisell and
Gardiner [Lou73, Loi73, Gar91]. In the experimental Chapter 8 it will be shown how these
statistical properties can actually be measured and how they have been used to distinguish
between classical and quantum properties of the light.
2.2.2 Coherence
Light emitted by a point-like source has properties closely resembling one single wave. The
amplitudes at two closely spaced points are linked with each other, as part of the same wave.

be identical. The question how far the detectors can be separated in space, or in time, before
the amplitudes will differ depends on an additional property of the light: its coherence. Any
source of light has a coherence length and a coherence time. The amplitude at two points
within the coherence length and coherence time is in phase and has the same fluctuations.
Interference can be observed. Light at two points separated by more than the coherence length,
or the coherence time, has independent amplitudes which will not interfere.
One clear example of this concept is the technique of measuring the size of a star, which
has a small angular size but is not a point source, using a stellar interferometer. The size of a
distant star can usually not be resolved with a normal optical telescope. The starlight forms
an image in the focal plane of the telescope which is given by the diffraction of the light.
The image depends on the geometrical shape and the size of the aperture of the telescope and
has little to do with the actual shape and size of the object, the star. Diffraction limits the
resolution. Only a bigger aperture of the telescope, larger than technically feasible, would
increase the resolution. To overcome this limitation, A.H. Fizeau proposed the idea of a
stellar interferometer which was first used by A.A. Michelson and his colleagues [Mic21].
The concept of the interferometer is shown in Fig. 2.6.
M3 M4
M2M1
Detector
Lens
δθ
Figure 2.6: Stellar interferometer as proposed and developed by A. Michelson [Mic21]
In this apparatus the starlight is collected by two mirrors, M 1 and M 2, which direct the
light via the mirrors M 3 and M 4 to a telescope. There the two rays of light are combined
in the focal plane. In addition we have optical filters in the instrument which select only a
limited range of optical wavelengths. The mirrors M 1 and M 2 are separated by a distance d and the apparatus is carefully aligned such that the distances from the focal plane to M 1 and
to M 2 are identical. As a consequence we have interference of the two partial waves in the
focal plane. An interference system will appear, due to the fact that the mirrors M 1 and M 2 are not exactly at 45 degrees to the axis of the telescope. This arrangement is equivalent to
a Michelson interferometer with an optical wedge in one arm and it measures the first order
coherence of the light.
2.2 Statistical properties of classical light 25
For an extended source we get light from different parts of the source reaching the tele-
scope under slightly different directions, let us say with a very small angular difference of δθ .
Each wave will produce an interference system, but at a slightly different location in the focal
plane. This is due to the additional optical path l = d sin(δθ) ≈ d δθ inside the interferome-
ter. We assume that the light from the different parts of the star is independent. Consequently,
it will interfere with light from the same part of the star, but not with light from other parts and
the intensity of the interference patterns from the different directions are added. The interfer-
ence pattern in the focal plane will be visible as long as the path difference is small compared
to an optical wavelength. If the separation d between M 1 and M 2 is increased the total in-
terference pattern will eventually disappear, since the individual patterns are now shifted in
respect to each other.
An estimate for the separation at which fringes are no longer visible is d δθ = λ. In this
situation the two individual patterns from the edges of the star are shifted by one fringe period
and all other patterns lie in between, washing out the total pattern. A more detailed analysis
predicts the change in the visibility of the interference patterns as a function of the separation
d. The interferometer measures the coherence of the light. In this case the coherence is
limited due to the independence of the different parts of the star and the coherence length is
the path difference for the light travelling from different edges of the star to the earth. The
corresponding coherence time for the entire starlight is the difference in travel time. We find
that interferometry can be used to measure the coherence length and coherence time.
Michelson built an instrument based on this idea and was able to measure star diameters
down to about 0.02 arc seconds [Mic21, Pea31]. There are improved versions of this instru-
ment in operation, in particular an instrument called SUSI (Sydney University Stellar Inter-
ferometer) which have achieved resolutions as good as 0.004 arc seconds. However, there are
technical limitations. It is very easy to lose the visibility of the fringes, already the wavelength
is restricted to an interval of about 5 nm and even then the path difference inside the interfer-
ometer has to be kept to less than 0.01 mm. A wider spectrum, containing more light, would
require an even better tolerance. It is very difficult to maintain such a small path difference for
the entire time of the observation, while the mirrors track the star and the separation between
the mirrors is changed by up to several meters.
To avoid these complications, R. Hanbury Brown & R.Q. Twiss proposed an entirely dif-
ferent scheme to measure coherence. Theirs is a very elegant approach which uses intensity
fluctuations rather than interference, an idea that had shortly before been used in radio astron-
omy. In this way they founded a new discipline based on the statistical properties of light.
They set out to measure the correlation between the intensities measured at different locations
rather than the amplitude and thereby avoided all the complications associated with build-
ing an optical interferometer. In their experiment they replaced the two mirrors M 1 and M 2 (Fig. 2.6) with curved mirrors which collected the light onto two different photo-multipliers,
as shown in Fig. 2.7 [Bro56]. The photo-currents from these two detectors are amplified inde-
pendently and then multiplied with each other and the time average of the product is recorded.
This signal reflects the correlation between the intensities. This apparatus does not require
the mechanical stability of an interferometer. However, some questions arise: In which way
is this experiment related to an interferometer? How can it provide similar information if we
have disbanded the crucial phase measurement?
To answer these questions, we remind ourselves that stellar interferometers measure the

Multiplier
x
d
Figure 2.7: Intensity interferometer proposed and developed by Hanbury Brown & Twiss
[Bro56]
phase of the light fluctuates wildly and so does the amplitude of light from a thermal source.
Generally the amplitude and the phase are not directly linked and both fluctuate independently.
However, for two points within a coherence

-a guide to experiments in quantum optics, second edition (2004)

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