Multi-Photon Quantum Interference

Zhe-Yu Jeff Ou

Multi-Photon QuantumInterference

Zhe-Yu Jeff OuDepartment of PhysicsIndiana University-Purdue University IndianapolisIndianapolis, IN 46202

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ISBN 978-0-387-25532-3 e-ISBN 978-0-387-25554-5

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For my parents,who did so much for me.

Preface

Quantum interference, as a fundamental phenomenon of quantum mechanics,is what makes quantum physics different from classical Newtonian physics.Optical interference has played some essential roles in the understanding oflight. It has fascinated Dirac, the pioneer of the quantum theory of light,since the very beginning, as seen in his famous statement on photon interfer-ence: “Each photon ... interferes only with itself. Interference between differ-ent photons never occurs.” Feynman, who was one of the founders of quantumelectrodynamics, wrote, in his well-known lecture series on physics, that theinterference phenomenon “has in it the heart of quantum mechanics..., it con-tains the only mystery.” As we explore into the quantum regime in the 21stcentury, we will find even more presence of quantum interference in our life.

Most commonly-occurring interference phenomena are optical interference,in the form of some beautiful interference fringe patterns. These phenomenahave been well-studied by the classical theory of coherence, and documented inBorn and Wolf’s classic book Principle of Optics. In terms of the language ofphoton, these phenomena can be categorized as the single-photon interferenceeffect and described by the first part of Dirac’s statement given above. Onthe other hand, in the situation when more than one photon is involved,the second part of Dirac’s statement is not correct. In this book, we try tounderstand the phenomena of quantum interference through the multi-photoneffects of photon correlation. Our major concern is the temporal correlationamong photons and how it influences the interference effect. Because of this,we resort to the multi-frequency description of an optical field.

The multi-photon interference effects discussed in this book will find theirapplications in many of the quantum information protocols, such as, quantumcryptography and quantum state teleportation. However, the emphasis of thisbook is on the fundamental physical principle in those protocols. Therefore,we will not cover the broad topics of quantum information processing. Never-theless, readers may find the multi-frequency description of optical fields to bea good complement to the single-mode treatment found in most discussionson quantum information, and closer to a real experimental environment.

VIII Preface

This book is organized into two parts. The first part deals mainly withthe two-photon interference effect. The second part studies the effects of morethan two photons. In addition to the interference effects, Chapter 2 is devotedto the generation and the spectral properties of a two-photon state in theprocess of parametric down-conversion, which is the main photon source forthe effects studied in this book. We also investigate the coherence of themulti-photon source in Chapter 7, which is the preparation for Chapters 8-10. The complementary principle of quantum mechanics is demonstrated ina quantitative fashion in Chapter 9, when we discuss the relation betweenphoton distinguishability and multi-photon interference effects.

This book is based on a tutorial lecture series held during the Yellow Moun-tain Workshop on Quantum Information in 2001. I would like to thank Pro-fessor Guang-can Guo of the University of Science and Technology of Chinafor inviting me to the workshop and for his generous support.

Indianapolis, September, 2006 Zhe-Yu Jeff Ou

Contents

Part I Two-Photon Interference

1 Historical Background and General Remarks . . . . . . . . . . . . . . 31.1 Interference between Independent Lasers: Magyar-Mandel

and Pfleegor-Mandel Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Two-Photon Interpretation of Pfleegor-Mandel Experiment . . . 61.3 Two-Photon Interference with Quantum Sources . . . . . . . . . . . . 8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Quantum State from Parametric Down-Conversion . . . . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Spontaneous Parametric Down-Conversion Process . . . . . . . . . . 212.3 Phase Matching Condition and Spectral Bandwidth . . . . . . . . . . 27

2.3.1 Type-I Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Type-II Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4 Quantum State with a Narrow Band Pump Field . . . . . . . . . . . . 342.5 Quantum State with a Wide Band Pump Field . . . . . . . . . . . . . . 39References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Hong-Ou-Mandel Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.1 Single-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip . . . . . . . . . . . 45

3.2.1 Narrow Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.2 Wide Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2.3 Dispersion Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 A Nonlocal Two-Photon Interference Effect . . . . . . . . . . . . . . . . . 563.4 Photon Bunching in Hong-Ou-Mandel Interferometer . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

X Contents

4 Phase-Independent Two-Photon Interference . . . . . . . . . . . . . . 634.1 Two-Photon Polarization Entanglement . . . . . . . . . . . . . . . . . . . . 63

4.1.1 Polarization Hong-Ou-Mandel Interferometerand Violation of Bell’s Inequalities . . . . . . . . . . . . . . . . . . . 63

4.1.2 Photon Anti-Bunching Effect and Bell StateMeasurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Two-Photon Frequency Entanglement and Spatial Beating . . . . 704.3 Two-Photon Interference Fringes and Ghost Fringes . . . . . . . . . . 73

4.3.1 Two-Photon Interference Fringes . . . . . . . . . . . . . . . . . . . . 744.3.2 Spatial Correlation and Ghost Fringes . . . . . . . . . . . . . . . 78

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Phase-Dependent Two-Photon Interference:Two-Photon Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1 Two-Photon Interferometer with Two Down-Converters . . . . . . 83

5.1.1 Phase Memory by Entanglement with Vacuum . . . . . . . . 835.1.2 Multi-Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2 Franson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.1 Entanglement in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.2 Two-Photon Coherence versus One-Photon Coherence:

Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Two-Photon De Broglie Interferometer . . . . . . . . . . . . . . . . . . . . . 95

5.3.1 Maximally Entangled Photon State – the NOON State . 955.3.2 Detailed Analysis of Two-Photon De Broglie

Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Interference between a Two-Photon State and a CoherentState . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.1 Anti-Bunching by Two-Photon Interference . . . . . . . . . . . . . . . . . 1016.2 Multi-Mode Analysis I: CW Case . . . . . . . . . . . . . . . . . . . . . . . . . . 1046.3 Multi-Mode Analysis II: Pulsed Case . . . . . . . . . . . . . . . . . . . . . . . 106References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Part II Quantum Interference of More Than Two Photons

7 Coherence and Multiple Pair Production in ParametricDown-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.1 Coherence Properties of Spontaneous Parametric

Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1137.1.1 Field Correlation Functions and Coherence of

Parametric Down-Converted Fields . . . . . . . . . . . . . . . . . . 1147.1.2 Generation of Transform-Limited Single-Photon Wave

Packet by Gated Photon Detection . . . . . . . . . . . . . . . . . . 118

Contents XI

7.2 Multi-Pair Production and Stimulated Pair Emission . . . . . . . . . 1217.2.1 Pair Statistics and Photon Bunching . . . . . . . . . . . . . . . . . 1227.2.2 Stimulated Pair Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.2.3 Induced Coherence without Induced Emission . . . . . . . . . 130

7.3 Distinguishable or Indistinguishable Pairs of Photons . . . . . . . . 132References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8 Quantum Interference with Two Pairs of Down-ConvertedPhotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1 Hong-Ou-Mandel Interferometer for Independent Photons . . . . 137

8.1.1 Two-Photon Interference without Gating . . . . . . . . . . . . . 1388.1.2 Two-Photon Interference with Gating: Hong-

Ou-Mandel Interferometer for Two IndependentPhotons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

8.2 Quantum State Teleportation and Swapping . . . . . . . . . . . . . . . . 1428.2.1 Quantum State Teleportation: Single-Mode Case . . . . . . 1438.2.2 Quantum State Teleportation: Multi-Mode Case . . . . . . . 1458.2.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

8.3 Distinguishing a Genuine Polarization Entangled Four-PhotonState from Two Independent EPR Pairs . . . . . . . . . . . . . . . . . . . . 1498.3.1 Single-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.3.2 Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons . . . . 1558.4.1 Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 1558.4.2 Asymmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . 159

8.5 Generation of a NOON State by Superposition . . . . . . . . . . . . . . 1638.5.1 Interference between a Coherent State and Parametric

Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1638.5.2 A Special N-Photon Interference Scheme . . . . . . . . . . . . . 166

8.6 Multi-Photon De Broglie Wavelength by ProjectionMeasurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1698.6.1 Projection by Asymmetric Beam Splitters . . . . . . . . . . . . 1698.6.2 NOON State Projection Measurement . . . . . . . . . . . . . . . 172

8.7 Stimulated Emission and Multi-Photon Interference . . . . . . . . . 1778.8 Remarks on E and A and General Discussion . . . . . . . . . . . . . . . 181References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9 Temporal Distinguishability of a Multi-Photon State . . . . . . 1859.1 Hong-Ou-Mandel Interferometer for Characterizing

Two-Photon Temporal Distinguishability . . . . . . . . . . . . . . . . . . . 1869.2 Characterizing an N-Photon State in Time . . . . . . . . . . . . . . . . . 187

9.2.1 General Description of a Multi-Mode N-Photon State . . 1889.2.2 Direct Photon Detection from Glauber’s Coherence

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

XII Contents

9.2.3 A NOON-State Projection Measurement andGeneralized Hong-Ou-Mandel Interferometer . . . . . . . . . . 195

9.3 The First Example of |1H , 2V 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1979.4 The General Case of |1H , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2009.5 The General Case of |kH , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.5.1 General Formula for the Visibility . . . . . . . . . . . . . . . . . . . 2069.5.2 The Special Cases of |2H , 2V 〉, |2H , 3V 〉, and |2H , 4V 〉 . . 2069.5.3 The Special Case of |3H , 3V 〉 . . . . . . . . . . . . . . . . . . . . . . . . 208

9.6 The Scheme for Characterizing the TemporalDistinguishability by an Asymmetric Beam Splitter . . . . . . . . . . 2089.6.1 The Temporal Distinguishability of |1H , NV 〉 . . . . . . . . . . 2109.6.2 The Case of |2H , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

9.7 Experimental Realization of the Cases of |2H , 1V 〉, |2H , 2V 〉with Two Pairs of Down-Converted Photons . . . . . . . . . . . . . . . . 2149.7.1 Generation of the State of |2H , 1V 〉 with Tunable

Temporal Distinguishability . . . . . . . . . . . . . . . . . . . . . . . . 2149.7.2 Distinguishing 4 × 1 Case from 2 × 2 Case for the

State of |2H , 2V 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

10 Homodyne of a Single-Photon State: A SpecialMulti-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22510.1 Interference with a Single-Photon State and an N-Photon

State at a Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . 22510.2 Interference of a Single-Photon State and an Arbitrary State . . 22810.3 Multi-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

A Lossless Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

B Derivation of the Visibility for |kH , NV 〉 . . . . . . . . . . . . . . . . . . . 245B.1 The Case of |2H , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

B.1.1 The Scenario of 2HmV + (N −m)V . . . . . . . . . . . . . . . . . 245B.1.2 The Scenario of 1HmV + 1HnV + (N − n−m)V . . . . . 249

B.2 The Case of |3H , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251B.2.1 The Scenario of 3HmV + (N −m)V . . . . . . . . . . . . . . . . . 251B.2.2 The Scenario of 2HmV + 1HnV + (N −m− n)V . . . . . 255B.2.3 The Scenario of 1HmV + 1HnV + 1HpV

+ (N−m−n−p)V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258B.3 The General Case of |kH , NV 〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

B.3.1 The Scenario of k1V mH + k2V nH . . . . . . . . . . . . . . . . . . 262B.3.2 The Most General Scenario . . . . . . . . . . . . . . . . . . . . . . . . . 265

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267

Part I

Two-Photon Interference

1

Historical Background and General Remarks

For many years, the phenomena of the violations of local realism by photons[1.1, 1.2] and those of photon interference [1.3, 1.4, 1.5] were not intertwined.In the first few experimental demonstrations [1.2] of quantum nonlocality inthe sense of the violations of Bell’s inequalities, the EPR-Bohm type [1.6] ofpolarization singlet state of two photons in the form

|Ψ〉12 = (|x〉1|y〉2 − |y〉1|x〉2)/√

2 (1.1)

is naturally produced from atomic cascade due to angular momentum conser-vation, where x, y represent the two orthogonal polarizations. In demonstrat-ing the violation of Bell’s inequalities, a two-photon polarization correlationmeasurement is made. However, even though two-photon polarization inter-ferometry is involved in the process, no one has attempted to interpret thepolarization correlation of Eq.(1.1) in terms of two-photon interference untilthe experiments of Shih-Alley [1.7, 1.8] and Ou-Mandel [1.9], who created atwo-photon polarization entangled state in Eq.(1.1) by superposing two fieldsfrom parametric down-conversion.

Investigation of multi-photon entanglement truly began with the introduc-tion of the two-photon correlation technique in parametric down-conversionby Burnham and Weinberg in 1970 [1.10], where entanglement in various de-grees of freedom is demonstrated. These include polarization [1.7, 1.8, 1.9],frequency [1.11], space [1.4, 1.12], phase and momentum [1.13], etc. Manyquantum information protocols are based on multi-photon interference effects.To better understand photon entanglement, we start with the simplest case:two-photon interference.

Historically, the investigation of two-photon interference began as early as1967 with the classic Pfleegor-Mandel experiment [1.14, 1.15]. Early studies byMandel [1.3] emphasized the nonclassical effects in two-photon interference.This emphasis follows naturally the demonstrations of a series of nonclassicaleffects of light such as photon anti-bunching [1.16], sub-Poissonian photonstatistics [1.17], and violation of Cauchy-Schwartz inequality [1.18].

4 1 Historical Background and General Remarks

In this chapter, we will recall some of the experiments, and their inter-pretation in the early development of two-photon interference. Then, we willmake some general remarks before proceeding to the subject of parametricdown conversion.

1.1 Interference between Independent Lasers:Magyar-Mandel and Pfleegor-Mandel Experiments

In the early development of coherence theory in quantum optics, one im-portant milestone was the Magyar-Mandel [1.19] experiment that demon-strated the interference effect between two independent lasers. This experi-ment showed that besides the high brightness, a laser also has a long coherencetime. This property is quite different from a thermal source. This interferenceeffect can be well understood with the classical optical coherence theory [1.20]:the second-order amplitude correlation function between the two lasers,

Γ12(τ) = 〈V ∗1 (t)V2(t+ τ)〉, (1.2)

is non-zero when the observation time τ is within the long coherence time ofthe two lasers.

However, in terms of the language of photon, this immediately poses aserious challenge to the second part of Dirac’s famous statement on photoninterference: each photon interferes only with itself; different photons neverinterfere with each other [1.21]. The early experiment by Taylor [1.22] wherehe introduced an extremely weak source of light in Young’s double slit in-terference experiment, supported the first part of Dirac’s statement. But thelight level in the Magyar and Mandel experiment was too high to rule outthe possibility of photon interaction in interference. Under this circumstance,Pfleegor and Mandel [1.14, 1.15, 1.23] performed an experiment similar to theone by Taylor but with two independent lasers.

Laser 1

Laser 2

κ1^

κ2^

x1

x2..Δk = k2−k1

θ

Fig. 1.1. The schemefor interference betweentwo independent lasersin Magyar-Mandel andPfleegor-Mandel experi-ments.

In the Pfleegor-Mandel experiment, however, due to low light levels, longexposure time well over the coherence time of the lasers is needed and as aresult, the fringe pattern constantly moves, due to the random phase diffusionof each laser within the coherence time. It would, therefore, be difficult to

1.1 Interference between Independent Lasers 5

directly observe the fringe pattern at low light levels in the traditional waywith just one detector, i.e., the intensity of light varies as a function of thedetector position. In order to reveal the interference fringe pattern, Pfleegorand Mandel invented an ingenious method based on intensity correlation, or,more precisely, anti-correlation, as they first described in Ref.[1.14].

Pfleegor and Mandel noticed that with a moving fringe pattern, intensi-ties at locations separated by half the fringe spacing tend to go in oppositedirections. At locations with one full fringe spacing, on the other hand, theintensities tend to go in the same direction. These two correlated effects areillustrated in Fig.1.2. So when two detectors are placed half a fringe spacingapart, they should exhibit negative intensity correlation, while at one fringespacing separation, the two detectors will have positive correlation.

Direction of fringe motion

L/2

L/2 L/2L/2L/2 L/2

A AB B

C C

D D

Fig. 1.2. Intensity correla-tion at full fringe separation(between points of same let-ters A,or B, or C, or D) andintensity anti-correlation athalf fringe spacing (betweenA and B or C and D) for amoving fringe pattern.

More quantitatively, it is straightforward to explain the above using theclassical wave theory. Let us describe the fields from the two lasers by twoplane waves denoted as V1(r, t) and V2(r, t) with

Vn(r, t) = Aei(kn·r−ωt+ϕn(t)) (n = 1, 2). (1.3)

Here, ϕn(t) (n = 1, 2) is the phase of the lasers. It diffuses with a time scalein the order of the coherence time of the lasers. The amplitudes of the lasersare represented by A and, for simplicity, we assume they are the same. Whenwe superpose the two fields, we have the intensity at a certain location r as:

I(r, t) =∣∣∣Aei(k1·r−ωt+ϕ1(t)) +Aei(k2·r−ωt+ϕ2(t))

∣∣∣

2

= 2A2[1 + cos((k1 − k2) · r +Δϕ(t))]= 2A2[1 + cos(2πx/L+Δϕ(t))], (1.4)

where L = λ/θ is the fringe spacing with θ as the small angle between k1 andk2. x is the distance along the direction of k1 − k2. Δϕ(t) = ϕ1(t) − ϕ2(t)is the phase difference between the two independent lasers. It is constant ina short time scale, as in the experiment of Magyar and Mandel [1.19]. Butin a long time scale ( >> coherence time), it is randomly fluctuating due tothe independent phase diffusions of the two lasers, as in the experiment ofPfleegor and Mandel [1.14, 1.15, 1.23].

6 1 Historical Background and General Remarks

Therefore, for long time observation, there is no fringe: 〈I(r, t)〉Δϕ = 2A2.But for the intensity correlation at two locations, we have:

G(2)12 = 〈I(x1)I(x2)〉Δϕ = 4A4[1 + 0.5 cos 2π(x1 − x2)/L], (1.5)

which shows a modulation despite the fluctuations of Δϕ. The normalizedintensity correlation function is given by:

g(2)12 =

〈I(x1)I(x2)〉〈I(x1)〉〈I(x2)〉

= 1 + 0.5 cos 2π(x1 − x2)/L. (1.6)

More specifically, for x1 − x2 = L/2, we have g(2)12 = 0.5 < 1, which gives rise

to the anti-correlation effect.Notice that in Eq.(1.6), the relative depth of modulation of the fringe pat-

tern, namely the visibility, is only 50%. This is actually a general conclusion,which we will prove for classical fields later in Sect.1.4.

Pfleegor and Mandel observed the above predicted intensity correlationeffects with a light level so low that on average, a photon is detected wellbefore the next one is emitted from the lasers. This seems to support thesecond part of the Dirac statement, i.e., different photons never interfere.However, Dirac’s statement on photon interference is too crude to accountfor details such as the 50% visibility in Eq.(1.5). In the following section, wewill see how Dirac’s statement can be modified to suit the Pfleegor-Mandelexperiment and provide a quantitative explanation in the language of two-photon correlation.

1.2 Two-Photon Interpretation of Pfleegor-MandelExperiment

The major difference between the Magyar-Mandel and Pfleegor-Mandel ex-periments is that the former records interference fringe in intensity, whereasthe observed interference effect in the latter case is in intensity correlationwith two detectors. To measure intensity of a field, we need only one detectorwhich responds to single photons. On the other hand, in intensity correlation,the measurement device will produce a signal only when there are two pho-tons, one at each detector. So intensity measurement registers single-photonevents, whereas intensity correlation measures two-photon events (Fig. 1.3).

With this picture in mind, we find that Dirac’s statement on single-photoninterference still applies to the Magyar-Mandel experiment. But for two-photon detection in the Pfleegor-Mandel experiment, it is not appropriate.We need to modify Dirac’s statement to suit two-photon detection as follows:

In interference involving two−photon detection, a pairof photons only interferes with the pair itself .

1.2 Two-Photon Interpretation of Pfleegor-Mandel Experiment 7

CoincidenceUnit

(b)

(a) Fig. 1.3. The differencebetween intensity mea-surement and intensitycorrelation measurement:(a) single-photon will pro-duce a signal out (b) Onlytwo photons registered ateach of the two detectorswill produce a coincidencesignal.

Here, the pair and the pair itself correspond to two parts of a wave that isassociated only with two photons. The two parts are the two indistinguishablepaths for the two photons together. Thus, the interference is between the twopossibilities. The special wave that is related only to two photons is differentfrom the traditional waves that we usually encounter in a Michelson inter-ferometer, which is associated only with one photon, since only one detectoris involved. As we will see in Chapt.5, the two kinds of waves have differentcoherence times. Thus, in order to separate these two different waves, we referto them as “two-photon wave” and “single-photon wave”, respectively. Theabove picture for two-photon interference was discussed briefly by Glauber[1.24], with an equivalent view of interference of two-photon amplitude. Thegeneralization to an N -photon case is straightforward using the N -photonwave packet concept. Next, we will attempt to interpret the Pfleegor-Mandelexperiment with the modified Dirac statement, or the two-photon wave con-cept.

Consider the situation depicted in Fig.1.4, where two photons are detectedwith one at each detector. There are four possibilities for the two photons thatproduce the two-photon coincidence signal: the two photons are both fromone of the two lasers in cases (A) and (B) or one from each laser in cases (C)and (D). The first two cases do not produce interference because of the phasediffusion (fluctuation of Δϕ in a time interval much longer than the coherencetime of the lasers). Later in Chapter 5, we will see a situation when cases (A)and (B) do produce interference. But here, cases (A) and (B) will simply adda constant and raise the baseline. Cases (C) and (D) are indistinguishable andcorrespond to the cases of a pair of photons and the pair itself in the modifiedDirac statement. So they will produce interference and give the fringe patternin two-photon coincidence.

Because of the randomness of photon statistics of a laser, all four caseshave equal chances if the two lasers have the same intensity. So, if we denoteA4 as the two-photon probability from one laser, we can write the intensitycorrelation function from the above discussion as

G(2)12 = A4 +A4 + 2A2A2[1 + cos 2π(x1 − x2)/L], (1.7)

8 1 Historical Background and General Remarks

(B)(A)

(C) (D)

Fig. 1.4. Four possibleorigins for the two photonsdetected by two detectors:no interference between (A)and (B) but 100% visibilityinterference between (C)and (D).

where the first two terms are from cases (A) and (B) and the last term is fromthe two-photon interference of cases (C) and (D). Eq.(1.7) is exactly same asEq.(1.5), derived from classical wave theory.

From the above argument leading to Eq.(1.7), we find that the reason for50% visibility in Eq.(1.7) is due to the existence of cases (A) and (B), i.e., somenonzero two-photon probability from one source only. As a matter of fact, forany classical source, the two-photon probability P2 is always greater than orequal to the square of one-photon probability or the so-called accidental two-photon probability P 2

1 . For coherent state from a laser, we have P2 = P 21 as

described above in the Pfleegor-Mandel experiment. On the other hand, fora thermal source, we have P2 = 2P 2

1 , which leads to the so-called photonbunching effect. With this source, cases (A) and (B) are twice as probable ascases (C) and (D). This will give rise to a visibility of 1/3, from the argumentabove [1.3, 1.25].

To obtain a visibility over 50%, we must consider quantum sources withsub-Poissonian photon statistics or the anti-bunching effect. For example, ifboth fields are in single-photon state, the probabilities for cases (A) and (B)are simply zero. This leads to

G(2)12 = 2A4[1 + cos 2π(x1 − x2)/L], (1.8)

which shows a visibility of 100% in two-photon interference.Next we will consider two-photon interference with general sources and

treat optical fields and optical detection in a quantum mechanical fashion.We will show rigorously that the upper bound of the visibility for classicalstates is 50% and certain quantum sources with photon anti-bunching effectcan produce a visibility exceeding this classical limit.

1.3 Two-Photon Interference with Quantum Sources

Two-photon interference with quantum light was first studied by Fano [1.26],who used a complicated QED treatment for the problem of detection by twoatom-detectors of the light emitted from two other simultaneously excited

1.3 Two-Photon Interference with Quantum Sources 9

atoms. Fano showed for the first time that it is possible to achieve 100% vis-ibility for the interference effect in intensity correlation. This was also men-tioned briefly by Richter [1.27] with regard to a similar problem. But it wasMandel [1.3] who first pointed out the difference between the predictions ofquantum and classical theories on the visibility of two-photon interference.Although the early theoretical quantum treatments of this problem were ap-plied to light emitted from two simultaneously excited atoms, the paramet-ric down-conversion process proved to be more suitable for an experimentaldemonstration of the quantum nature of two-photon interference. Two-photoninterference in the parametric down-conversion process was first analyzed byGhosh et al. [1.28], and the subsequent experiments by Ghosh and Mandel[1.4] and by Hong, Ou, and Mandel [1.5] demonstrated two-photon interfer-ence with a visibility of more than 50%.

In this section, we will show that the general classical limit is 50% for thevisibility of two-photon interference and derive the necessary condition forthe fields to have a nonclassical two-photon interference effect with visibilitylarger than 50%. Then we will examine a couple of special cases. We start witha quantum description of a free optical field [1.29] by the positive frequencypart of an electric field operator:

E(+)(r, t) =1

(2π)3/2

∫

d3k akei(k·r−ωt). (1.9)

Here, we treat the field as quasi-monochromatic and assume that all the modesin the field have the same polarization so that we can treat them as scalarfields. In the interference of two optical fields, we further assume that mostof the modes in the integral are in the vacuum state and only modes in twodirections with unit vectors κ1 and κ2 are excited (see Fig. 1.1). For simplicity,we omit all the unoccupied modes and re-write the field operator in Eq.(1.9)in one-dimensional form:

E(+)(r, t) = E(+)1 (r, t) + E

(+)2 (r, t), (1.10)

with

E(+)n (r, t) =

1√2π

∫

dωan(ω)ei(kn·r−ωt) (n = 1, 2), (1.11)

where kn = κnω/c. In order to produce macroscopic fringe pattern, we assumethe angle θ between the two directions κ1 and κ2 is small, i.e., cos θ ≡ κ1 ·κ2 ≈1. Let the bandwidths of the two interfering fields (characterized by directionsκ1 and κ2) be Δω1 and Δω2, respectively. We assume that the two fields arequasi-monochromatic and have the same center frequency, that is, ω10 = ω20

>> Δω1 and Δω2.We now evaluate the joint probability of detecting one photon at position

r and at time t and the other at r′ at t+ τ . This probability P2(r, t; r′, t+ τ)is given by

10 1 Historical Background and General Remarks

P2(r, t; r′, t+ τ) ∝ 〈E(−)(r, t)E(−)(r′, t+ τ)E(+)(r′, t+ τ)E(+)(r, t)〉= 〈T : I(r, t)I(r′, t+ τ) :〉, (1.12)

where T is time ordering and :: is normal ordering. E(−) = [E(+)]† is thenegative frequency part of the field operator.

Before we proceed, let us now introduce the assumption made earlier thatthere is no interference effect in the field intensity (single-photon detection)so that we concentrate only on a genuine two-photon effect. The easiest wayto achieve this is to assume that the two fields have independent phase fluctu-ations. We then find, after substituting Eq.(1.10) into Eq.(1.12) and makingexpansion of the product, that all unpaired terms like E(−)

1 E(−)1 E

(+)2 E

(+)2 ,

E(−)1 E

(−)2 E

(+)2 E

(+)2 ,etc. vanish. Only six terms survive and Eq.(1.12) becomes:

P2(r, t; r′, t+ τ)∝ 〈T : I1(r, t)I1(r′, t+ τ) :〉 + 〈T : I2(r′, t+ τ)I2(r, t) :〉+

+〈T : I1(r, t)I2(r′, t+ τ) :〉 + 〈T : I1(r′, t+ τ)I2(r, t) :〉++〈E(−)

1 (r, t)E(−)2 (r′, t+ τ)E(+)

1 (r′, t+ τ)E(+)2 (r, t)〉+

+〈E(−)2 (r, t)E(−)

1 (r′, t+ τ)E(+)1 (r′, t+ τ)E(+)

2 (r, t)〉. (1.13)

It is obvious from the above expression that the first two terms are the con-tribution from cases (A) and (B) in Fig.1.4, i.e., two photons are all from thefield of one direction, and the rest correspond to one photon from each field,as in cases (C) and (D).

For the nearly parallel plane waves described by Eq.(1.12) and the nearlyperpendicular observation plane where the detectors are located (see Fig.1.1for detailed geometry), the first four terms in Eq.(1.13) are independent ofr and r′ and are always positive. Actually, they are the auto-correlation andcross correlation of the corresponding fields, that is,

⎧

⎪⎪⎨

⎪⎪⎩

〈T : I1(r, t)I1(r′, t+ τ) :〉 = 〈I1(t)〉〈I1(t+ τ)〉[1 + λ1(t, τ)],〈T : I2(r′, t+ τ)I2(r, t) :〉 = 〈I2(t)〉〈I2(t+ τ)〉[1 + λ2(t, τ)],〈T : I1(r, t)I2(r′, t+ τ) :〉 = 〈I1(t)〉〈I2(t+ τ)〉[1 + λ12(t, τ)],〈T : I1(r′, t+ τ)I2(r, t) :〉 = 〈I2(t)〉〈I1(t+ τ)〉[1 + λ21(t, τ)],

(1.14)

where λ-functions are independent of t for stationary fields. λn(0)(n = 1, 2) isnon-negative for classical fields but can be −1 for some quantum fields [1.30].

The last two terms in Eq.(1.13) involve mixed amplitudes of the two fieldsat positions r and r′ and change rapidly with r and r′. These are the inter-ference terms which give rise to modulation. Hence, the relative magnitudeof these two terms, compared with the first four terms in Eq.(1.13), will de-termine the relative depth of modulation or the visibility of the interferencepattern.

To evaluate their relative magnitude, we consider the operator

O ≡ E(+)1 (r′, t+ τ)E(+)

2 (r, t) − E(+)2 (r′, t+ τ)E(+)

1 (r, t)eiϕ, (1.15)

1.3 Two-Photon Interference with Quantum Sources 11

where ϕ is an arbitrary phase. We then construct the non-negative quantity

〈O†O〉 ≥ 0, (1.16)

where the average is on an arbitrary quantum state of the fields. SubstitutingEq.(1.15) into Eq.(1.16) and expanding the product, we obtain:

〈T : I1(r, t)I1(r′, t+ τ) :〉 + 〈T : I2(r′, t+ τ)I2(r, t) :〉≥ 〈E(−)

1 (r, t)E(−)2 (r′, t+ τ)E(+)(r′, t+ τ)E(+)

2 (r, t)〉e−iϕ + c.c. (1.17)

Since ϕ is arbitrary, we can choose it to be 0 or π. Then Eq.(1.17) becomes:

〈T : I1(r, t)I2(r′, t+ τ) :〉 + 〈T : I1(r′, t+ τ)I2(r, t) :〉≥

∣∣∣〈E(−)

1 (r, t)E(−)2 (r′, t+ τ)E(+)

1 (r′, t+ τ)E(+)2 (r, t)〉 + c.c.

∣∣∣. (1.18)

This inequality ensures that the joint probability P2 is non-negative for anystate of light.

Next we will prove, for classical fields only, that the first two terms inEq.(1.13) are larger than or equal to the last two interference terms, that is,

〈T : I1(r, t)I1(r′, t+ τ) :〉c + 〈T : I2(r′, t+ τ)I2(r, t) :〉c≥

∣∣∣〈E(−)

1 (r, t)E(−)2 (r′, t+ τ)E(+)

1 (r′, t+ τ)E(+)2 (r, t)〉 + c.c.

∣∣∣, (1.19)

where the subscript c indicates that the quantum average is only over classicalstates. To prove Eq.(1.19), we write an arbitrary quantum state described bya density operator in the Glauber-Sudarshan P-representation [1.31, 1.32]:

ρ =∫

d{α}P ({α})|{α}〉〈{α}|, (1.20)

where {α} is a set of variables covering all the excited modes of the fields andP ({α}) satisfies the normalization relation:

∫

d{α}P ({α}) = 1. (1.21)

In general, P ({α}) may be negative. But classical fields are those with well-behaved and non-negative P ({α}) so that P ({α}) can be treated as a trueprobability distribution.

We now use Eq.(1.20) to express the first two terms in Eq.(1.13) as

〈T : I1(r, t)I1(r′, t+ τ) :〉 =∫

d{α}P ({α})|E1(r, t)|2|E1(r′, t+ τ)|2 (1.22)

〈T : I2(r′, t+ τ)I2(r, t) :〉 =∫

d{α}P ({α})|E2(r, t)|2|E2(r′, t+ τ)|2 (1.23)

and the last interference terms as

12 1 Historical Background and General Remarks

〈E(−)1 (r, t)E(−)

2 (r′, t+ τ)E(+)1 (r′, t+ τ)E(+)

2 (r, t)〉=

∫

d{α}P ({α})E∗1 (r, t)E∗

2 (r′, t+ τ)E2(r, t)E1(r′, t+ τ), (1.24)

whereEn(r, t) =

1√2π

∫

dωαn(ω)ei(kn·r−ωt). (1.25)

Consider the following quantity with an arbitrary phase ϕ:

O ≡ E1(r, t)E∗1 (r′, t+ τ) − E2(r, t)E∗

2 (r′, t+ τ)eiϕ. (1.26)

Obviously,

|O|2 = O∗O ≥ 0. (1.27)

So, we have

|E1(r, t)|2|E∗1 (r′, t+ τ)|2 + |E2(r, t)|2|E∗

2 (r′, t+ τ)|2≥ E∗

1 (r, t)E∗2 (r′, t+ τ)E2(r, t)E1(r′, t+ τ)eiϕ + c.c. (1.28)

Since P ({α}) ≥ 0 for classical fields, we can multiply it to the above inequalityand integrate over {α}. We then obtain an inequality exactly the same asEq.(1.19), if we choose ϕ = 0 or π.

Therefore, we may conclude generally from Eqs.(1.18, 1.19) that classicalfields can only give rise to two-photon interference with a maximum visibilityof 50%, whereas quantum fields may achieve 100% visibility. We have alreadyseen one example of this in Sect.1.2 [Eq.(1.8)].

From inequalities (1.18) and (1.19), we may obtain a necessary conditionfor the nonclassical effect to occur in two-photon interference as

〈T : I1(r, t)I1(r′, t+ τ) :〉 + 〈T : I2(r′, t+ τ)I2(r, t) :〉<

∣∣∣〈E(−)

1 (r, t)E(−)2 (r′, t+ τ)E(+)

1 (r′, t+ τ)E(+)2 (r, t)〉 + c.c.

∣∣∣M, (1.29)

where the subscript M stands for the maximum value. The reason for usingthe maximum value in the inequality (1.29) is that the interference terms onthe right side of Eq.(1.19) modulate as r and r′ change. With the inequalityin Eq.(1.18), we can write the necessary condition in Eq.(1.29) in a differentform as

〈T : I1(r, t)I1(r′, t+ τ) :〉 + 〈T : I2(r′, t+ τ)I2(r, t) :〉< 〈T : I1(r, t)I2(r′, t+ τ) :〉 + 〈T : I1(r′, t+ τ)I2(r.t) :〉. (1.30)

By using Eqs.(1.14) for stationary fields in the expression above, we can fur-ther simplify the necessary condition to

〈I1〉2[1 + λ1(τ)] + 〈I2〉2[1 + λ2(τ)] < 〈I1〉〈I2〉[2 + λ12(τ) + λ21(τ)]. (1.31)

1.3 Two-Photon Interference with Quantum Sources 13

If λn(τ) = −1(n = 1, 2), the left hand side of the above expression has aminimum value of

〈I1〉〈I2〉√

[1 + λ1(τ)][1 + λ2(τ)], (1.32)

when the intensities 〈I1〉, 〈I2〉 satisfy

〈I1〉2[1 + λ1(τ)] = 〈I2〉2[1 + λ2(τ)].

Using this minimum value in Eq.(1.31), we arrive at the necessary conditionin its simplest form:

√

[1 + λ1(τ)][1 + λ2(τ)] < 1 +λ12(τ) + λ21(τ)

2. (1.33)

For the special case of λn(τ) = −1 for any of the two fields (n = 1 or 2), thecondition in Eq.(1.31) is always satisfied if we allow the intensity to be largeenough for the field with λn(τ) = −1(n =1 or 2). In this case, the necessarycondition (1.33) is certainly satisfied. So, the condition in Eq.(1.33) is a moregeneral necessary condition for any two fields to exhibit a nonclassical two-photon interference effect of larger than 50% visibility.

The above-mentioned special case with, say, λ1(τ) = −1 presents an in-teresting phenomenon in two-photon interference. If we set 〈I1〉 >> 〈I2〉, i.e.,one field is much stronger than the other field, the condition in Eq.(1.31)is well satisfied and we will have a visibility larger than 50% in two-photoninterference. This is in sharp contrast to the same situation in one-photoninterference (fringe pattern is exhibited in intensity), where if one field isdominant, the other field can hardly disturb the final intensity distribution,resulting in nearly zero visibility.

As a matter of fact, the visibility of two-photon interference is nearly100% in the above case. To see the physical picture of this, let us go backto Fig.1.4. When λ1(τ) = −1 or 〈I1I1〉 = 0, there is no contribution fromcase (A). Since the contribution from case (B) is proportional to 〈I2〉2 andthose from case C and D are proportional to 〈I1〉〈I2〉, the contributions totwo-photon coincidence from (C) and (D) will be much larger than (A) and(B) if 〈I1〉 >> 〈I2〉. Therefore, we obtain nearly 100% visibility, according tothe discussion in Sect.1.2. Two-photon interference of nearly 100% visibilitywith 〈I1〉 >> 〈I2〉 was demonstrated by Ou and Mandel [1.12].

In a real experiment, we measure the joint two-photon detection proba-bility with some finite time resolution. So, the coincidence count registered inthe lab will be

Nc ∝∫

T

dt

∫

TR

dτ P2(r, t; r′, t+ τ), (1.34)

where T is the total time of data-taking for the stationary cw case or theduration of the field for the non-stationary pulsed case, and TR is the coin-cidence window (time) for the two photoelectric pulses after the detection ofthe two photons. TR is normally limited by the detectors’ resolving time.

14 1 Historical Background and General Remarks

In the case where the detectors are fast enough so that we may choose asmall TR within which P2(r, t; r′, t+ τ) ≈ P2(r, t; r′, t), i.e., we can set τ = 0in all the expressions from Eq.(1.13) to Eq.(1.33), in particular, Eq.(1.33)becomes

√

[1 + λ1(0)][1 + λ2(0)] < 1 + λ12(0). (1.35)

Note that if we use Eq.(1.14), Eq.(1.35) is just the opposite of the followingSchwartz inequality for classical fields:

〈I21 〉〈I2

2 〉 ≥ 〈I1I2〉2. (1.36)

So, the necessary condition for having over 50% visibility in two-photon in-terference is that the two interfering fields must be nonclassical fields thatviolate the Schwartz inequality in Eq.(1.36).

In order to show that some quantum sources can give rise to 100% visibilityin two-photon interference, let us consider a two-photon quantum state of theform

|Ψ〉 = |1〉k1 |1〉k2 , (1.37)

that is, one photon from each direction. The contributions from cases (A) and(B) in Fig.1.4 are zero because there is simply no two-photon event from field 1or 2 alone in the quantum state depicted in Eq.(1.37). From the simple pictureof Sect.1.3, we should expect 100% visibility in two-photon interference. Wewill confirm this by calculation.

Because only one mode is excited (occupied) in field 1 or 2, we can simplifyEq.(1.10) as

E(+) = ak1ei(k1·r−ωt) + ak2e

i(k2·r−ωt). (1.38)

The change from continuous mode integral in Eq.(1.10) to discrete sum herewill take away the 1/

√2π coefficient in Eq.(1.10). We can easily find the

intensity I(r) ≡ 〈E(−)E(+)〉 as

I(r) = 2. (1.39)

So there is no interference effect in intensity. This is because the photon num-ber Fock state in Eq.(1.37) does not have definite phases for the two fields.For intensity correlation, however, we have:

〈E(−)(r1, t)E(−)(r2, t+ τ)E(+)(r2, t+ τ)E(+)(r1, t)〉=

∣∣∣ei(k1·r1−ωt)ei[k2·r2−ω(t+τ)] + ei[k1·r2−ω(t+τ)]ei(k2·r1−ωt)

∣∣∣

2

= 2[

1 + cos(k1 − k2) · (r1 − r2)]

= 2[1 + cos 2π(x1 − x2)/L], (1.40)

where x1, x2 are the coordinates along the direction of k1 − k2 and L ≡1/2π|k1−k2| = λ/θ (θ is the small angle between k1 and k2). Indeed, Eq.(1.40)shows two-photon interference with 100% visibility.

References 15

The interference phenomena that we have discussed above only involveparameters of the position of the detectors. They are the most commonly seeninterference phenomena. The other kind of interference phenomena, knownas beating, are associated with time. As is well-known, the observation ofbeating effects requires a fast detection system. However, recent experimentson spatial beating [1.11, 1.33] have shown that this is not necessarily so. Thephenomenon of spatial beating, which exhibits beat note in space domain, onlyexists in two-photon interference and will be studied in detail in Sect.4.2.

Notice that the interference fringe exhibited in Eq.(1.40) does not dependon the phases of each field. This can be understood again with the picture inFig.1.4 where the two interfering paths of cases (C) and (D) are overlapping,so that any phase change in one field will influence both paths. When the twointerfering paths can be separated, two-photon interference phenomena willdepend on the phases of the interfering fields [1.34, 1.35, 1.36, 1.37, 1.38]. Thefirst experimental demonstration of phase-dependent two-photon interferencewas carried out by Ou et al. [1.39]. This will be the subject of Chapt.5 andChapt.6.

The quantum state in Eq.(1.37) can be produced in many ways. Earlyproposals are based on two atoms simultaneously excited [1.3, 1.26, 1.27].Atomic cascade can also produce two photons of different frequencies [1.18].Others are based on single photon on-demand [1.40]. However, none of theseis as easy as the parametric down-conversion process. It can produce a two-photon state with entanglement in many degrees of freedom. This book isdevoted to the understanding and applications of multi-photon interferencewith the parametric down-conversion process.

References

1.1 J. S. Bell, Physics (N. Y.) 1, 195 (1965).1.2 J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978).1.3 L. Mandel, Phys. Rev. A28, 929 (1983).1.4 R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987).1.5 C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).1.6 D. Bohm, Quantum Theory (Prentice Hall, Englewood Cliffs, N. J., 1951).1.7 C. O. Alley and Y. H. Shih, Proceedings of the Second International Symposium

on Foundations of Quantum Mechanics in the Light of New Technology, editedby M. Namiki et al. (Physical Society of Japan, Tokyo, 1987).

1.8 Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921 (1988).1.9 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988).

1.10 D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970).1.11 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 54 (1988).1.12 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989).1.13 J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).1.14 R. L. Pfleegor and L. Mandel, Phys. Lett. 24A, 766 (1967).1.15 R. L. Pfleegor and L. Mandel, Phys. Rev. 159, 1084 (1967).

16 1 Historical Background and General Remarks

1.16 H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977).1.17 R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983).1.18 J. F. Clauser, Phys. Rev. D 9, 853 (1974).1.19 G. Magyar and L. Mandel, Nature (London) 198, 255 (1963).1.20 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959;

7th ed., 1999).1.21 P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford,

1st ed., 1930; 5th ed., 1958).1.22 G. I. Taylor, Proc. Camb. Phil. Soc. 15, 114 (1909).1.23 R. L. Pfleegor and L. Mandel, J. Opt. Soc. Am. 58, 946 (1968).1.24 R. J. Glauber, Quantum Optics and Electronics (Les Houches Lectures), p.63,

edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon andBreach, New York, 1965).

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419, 594 (2002).

2

Quantum State fromParametric Down-Conversion

As we discussed in Sect.1.3, the situation when λn(0) = −1(n = 1, 2) givesthe largest quantum effect in two-photon interference. We showed briefly inSect.1.3 that the quantum state |1k1, 1k2〉 gives a visibility of 100% in two-photon interference. This is not a surprise because λn(0) = −1(n = 1, 2) forthis state. In this chapter, we will see how to generate a quantum state ofthe form |1a, 1b〉 with a, b representing various kinds of modes of light fieldssuch as polarization and frequency. This will also give rise to two-photonentanglement of various degrees of freedom.

2.1 Introduction

The parametric process was initially developed in radio wave and microwaveas low noise amplifiers, well before the invention of the laser [2.1]. With theemergence of nonlinear optics, most of the concepts and even the terminologieshave been adapted in the optical range of electromagnetic wave. Parametricamplification was first demonstrated in the optical region by Giordmaine andMiller [2.2]. It soon became an important technique for generating intense,coherent, and tunable radiation. Usually, the amplified input field (labelled as“signal”) is accompanied by another field (called “idler”), which is requiredby the principle of quantum mechanics to preserve the commutation relation.The energy is from another field or fields (labelled as “pump”). In the opticalrange, there are two basic processes that can give rise to the parametric gain.They are three-wave mixing and four-wave mixing. Both of them are wellstudied in nonlinear optics with classical wave theory [2.3].

Quantum mechanically, the parametric process is described simply by theinteraction Hamiltonian with the signal and idler fields in single-mode:

HI = ihχa†sa†i +H.c., (2.1)

18 2 Quantum State from Parametric Down-Conversion

where as(i) is the annihilation operator for the signal (idler) field and χ is aparameter related to the pump fields (which are usually treated as classicalfields described by numbers).

In the Heisenberg picture, the evolution of the annihilation operator canbe solved, and follows the Bogoliubov transformation:

{

as(t) = (μas + νa†i )e−iωt,

ai(t) = (μai + νa†s)e−iωt,

(2.2)

with

μ = cosh |χ|t, ν = (χ/|χ|) sinh |χ|t. (2.3)

The above expressions describe the parametric amplification process with anamplitude gain of μ. The νa†i term [or νa†s in the second line of Eq.(2.2)] givesrise to the accompanying “idler” field. In the interaction picture, the quantumstate evolves as

|Ψ(t)〉 = U(t)|Ψ(0)〉 (2.4)

with

U = exp(−iHIt/h) = exp(ηa†sa†i −H.c.), (2.5)

where η ≡ χt. This is the two-mode squeezed state [2.4] and for the generationof large squeezing, it is normally operated in the regime of |η| >> 1. This isthe high gain regime of parametric amplifier. However, to generate the two-photon state, we work in the regime of low gain, with μ ≈ 1 or |η| << 1. ThenEq.(2.4) can be approximately rewritten as

|Ψ(t)〉 ≈ (1 − |η|2/2)|0〉 + η|1s, 1i〉 + η2|2s, 2i〉, (2.6)

where we drop terms higher than second order in η and take the initialstate |Ψ(0)〉 as vacuum. Because |η| << 1, the first nontrivial contributionin Eq.(2.6) to photo-detection is the two-photon state |1s, 1i〉, which is ex-actly in the form of Eq.(1.37) for 100% visibility in two-photon interference.Although the vacuum state in Eq.(2.6) does not directly contribute to photo-detection, it plays an important role in preserving the two-photon phase andproducing entanglement in photons, as we will see in later chapters. FromEq.(2.6), we see that the physical meaning of |η|2 is simply the probability ofpump photon conversion to down-converted photons. The pair generation israndom, so the probability for two pairs is proportional to |η|4 [the last termin Eq.(2.6)].

However, what makes parametric process so interesting in quantum inter-ference is not only its ability to generate a two-photon state of the form inEq.(1.37), but also the possibility for it to produce an entangled two-photonstate. When we operate at the low gain regime of |η| << 1, it is a spontaneous

2.1 Introduction 19

process that can create the signal and idler fields without an input. Usuallythere is not just one possibility for spontaneous process. Any field (or moreprecisely, any mode of the fields) that is coupled in the process may radiate.The ability to have multiple paths in parametric process creates two-photonentanglement among the different paths. For example, for two such possiblepaths, we may rewrite the Hamiltonian in two-mode form for each of the signaland idler fields as

HI = hχ1as1ai1 + hχ2as2ai2 +H.c., (2.7)

and it produces an entangled two-photon state of the form:

|Ψ〉 ≈ ...+ η1|1s1, 1i1〉 + η2|1s2, 1i2〉 + ..., (2.8)

where we write down only the most dominating terms in two-photon detection.Here, s1, i1, s2, i2 describe different modes of the signal and idler fields andcan be any degree of freedom for an optical field. For polarization degree offreedom, for example, if we choose 1 → x and 2 → y, then Eq.(2.8) is similarto the EPR-Bohm singlet state in Eq.(1.1), introduced in the beginning ofthe book. However, most common is the frequency degree of freedom, for, thespontaneous parametric process has a wide spectrum.

To fully study the spectral structure and other degrees of freedom, weneed to concentrate on some specific processes. As we mentioned earlier, boththree-wave mixing and four-wave mixing can give rise to the Hamiltonianin Eq.(2.1). Most of the investigations on two-photon entanglement are onthe three-wave mixing process through χ(2) nonlinear optical coupling. Inthe nonlinear three-wave mixing process, a light beam of higher frequencyinteracts with a nonlinear medium and pumps it to a virtual state (Fig. 2.1a).This beam of light is known as the “pump” beam. When a second beam, calledthe “signal”, with lower frequency is introduced on the interaction region in acertain direction, its strength is amplified, while the pump beam is depletedand partly converted into the signal beam. Energy conservation requires athird beam, named the “idler”, be generated at the same time (Fig.2.1b). Thefrequency of the idler beam is the difference frequency of the other two beams.Since the energy transfer is from a higher frequency to lower frequencies, thisprocess is also known as the parametric down-conversion (PDC) process.

Depending on the polarization of the signal and idler photons, we have twotypes of parametric down-conversion: type-I process produces two photons ofthe same polarization and type-II process generates two orthogonally polarizedphotons. Because almost all nonlinear media are birefringent, there is a largedifference in the properties of down-converted photons between the two typesof processes.

Entangled photon pairs can also be produced in a four-wave mixing processvia χ(3) nonlinear coupling. There are two completely different approaches inthis direction. One approach utilizes optical fibers to increase the interactionlength to compensate for the relatively small χ(3) [2.5]. But the newly discov-ered micro-structured fiber has a very large nonlinear χ(3) coefficient and can