Multi-Photon Quantum Interference

Multi-Photon Quantum Interference Zhe-Yu Jeff Ou Multi-Photon Quantum Interference Zhe-Yu Jeff Ou Department of Ph...

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Multi-Photon Quantum Interference

Zhe-Yu Jeff Ou

Multi-Photon Quantum Interference

Zhe-Yu Jeff Ou Department of Physics Indiana University-Purdue University Indianapolis Indianapolis, IN 46202

Library of Congress Control Number: 2007922125 ISBN 978-0-387-25532-3

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

c 2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com

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 of light. It has fascinated Dirac, the pioneer of the quantum theory of light, since the very beginning, as seen in his famous statement on photon interference: “Each photon ... interferes only with itself. Interference between diﬀerent photons never occurs.” Feynman, who was one of the founders of quantum electrodynamics, wrote, in his well-known lecture series on physics, that the interference phenomenon “has in it the heart of quantum mechanics..., it contains the only mystery.” As we explore into the quantum regime in the 21st century, 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 phenomena have been well-studied by the classical theory of coherence, and documented in Born and Wolf’s classic book Principle of Optics. In terms of the language of photon, these phenomena can be categorized as the single-photon interference eﬀect and described by the ﬁrst part of Dirac’s statement given above. On the 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 to understand the phenomena of quantum interference through the multi-photon eﬀects of photon correlation. Our major concern is the temporal correlation among 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 their applications in many of the quantum information protocols, such as, quantum cryptography and quantum state teleportation. However, the emphasis of this book is on the fundamental physical principle in those protocols. Therefore, we will not cover the broad topics of quantum information processing. Nevertheless, readers may ﬁnd the multi-frequency description of optical ﬁelds to be a good complement to the single-mode treatment found in most discussions on quantum information, and closer to a real experimental environment.

VIII

Preface

This book is organized into two parts. The ﬁrst part deals mainly with the two-photon interference eﬀect. The second part studies the eﬀects of more than two photons. In addition to the interference eﬀects, Chapter 2 is devoted to the generation and the spectral properties of a two-photon state in the process of parametric down-conversion, which is the main photon source for the eﬀects studied in this book. We also investigate the coherence of the multi-photon source in Chapter 7, which is the preparation for Chapters 810. The complementary principle of quantum mechanics is demonstrated in a quantitative fashion in Chapter 9, when we discuss the relation between photon distinguishability and multi-photon interference eﬀects. This book is based on a tutorial lecture series held during the Yellow Mountain Workshop on Quantum Information in 2001. I would like to thank Professor Guang-can Guo of the University of Science and Technology of China for 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 . . . . . . . . . . . . . . 3 1.1 Interference between Independent Lasers: Magyar-Mandel and Pﬂeegor-Mandel Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Two-Photon Interpretation of Pﬂeegor-Mandel Experiment . . . 6 1.3 Two-Photon Interference with Quantum Sources . . . . . . . . . . . . 8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2

Quantum State from Parametric Down-Conversion . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spontaneous Parametric Down-Conversion Process . . . . . . . . . . 2.3 Phase Matching Condition and Spectral Bandwidth . . . . . . . . . . 2.3.1 Type-I Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Type-II Phase Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Quantum State with a Narrow Band Pump Field . . . . . . . . . . . . 2.5 Quantum State with a Wide Band Pump Field . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 21 27 27 30 34 39 41

3

Hong-Ou-Mandel Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Single-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip . . . . . . . . . . . 3.2.1 Narrow Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wide Band Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Dispersion Cancellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 A Nonlocal Two-Photon Interference Eﬀect . . . . . . . . . . . . . . . . . 3.4 Photon Bunching in Hong-Ou-Mandel Interferometer . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 45 49 52 55 56 59 61

X

4

5

6

Contents

Phase-Independent Two-Photon Interference . . . . . . . . . . . . . . 4.1 Two-Photon Polarization Entanglement . . . . . . . . . . . . . . . . . . . . 4.1.1 Polarization Hong-Ou-Mandel Interferometer and Violation of Bell’s Inequalities . . . . . . . . . . . . . . . . . . . 4.1.2 Photon Anti-Bunching Eﬀect and Bell State Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two-Photon Frequency Entanglement and Spatial Beating . . . . 4.3 Two-Photon Interference Fringes and Ghost Fringes . . . . . . . . . . 4.3.1 Two-Photon Interference Fringes . . . . . . . . . . . . . . . . . . . . 4.3.2 Spatial Correlation and Ghost Fringes . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase-Dependent Two-Photon Interference: Two-Photon Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Two-Photon Interferometer with Two Down-Converters . . . . . . 5.1.1 Phase Memory by Entanglement with Vacuum . . . . . . . . 5.1.2 Multi-Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Franson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Entanglement in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Two-Photon Coherence versus One-Photon Coherence: Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Two-Photon De Broglie Interferometer . . . . . . . . . . . . . . . . . . . . . 5.3.1 Maximally Entangled Photon State – the NOON State . 5.3.2 Detailed Analysis of Two-Photon De Broglie Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 63 67 70 73 74 78 82 83 83 83 85 89 89 92 95 95 96 99

Interference between a Two-Photon State and a Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1 Anti-Bunching by Two-Photon Interference . . . . . . . . . . . . . . . . . 101 6.2 Multi-Mode Analysis I: CW Case . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Multi-Mode Analysis II: Pulsed Case . . . . . . . . . . . . . . . . . . . . . . . 106 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Part II Quantum Interference of More Than Two Photons 7

Coherence and Multiple Pair Production in Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1 Coherence Properties of Spontaneous Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.1.1 Field Correlation Functions and Coherence of Parametric Down-Converted Fields . . . . . . . . . . . . . . . . . . 114 7.1.2 Generation of Transform-Limited Single-Photon Wave Packet by Gated Photon Detection . . . . . . . . . . . . . . . . . . 118

Contents

XI

7.2 Multi-Pair Production and Stimulated Pair Emission . . . . . . . . . 121 7.2.1 Pair Statistics and Photon Bunching . . . . . . . . . . . . . . . . . 122 7.2.2 Stimulated Pair Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2.3 Induced Coherence without Induced Emission . . . . . . . . . 130 7.3 Distinguishable or Indistinguishable Pairs of Photons . . . . . . . . 132 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 8

Quantum Interference with Two Pairs of Down-Converted Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 8.1 Hong-Ou-Mandel Interferometer for Independent Photons . . . . 137 8.1.1 Two-Photon Interference without Gating . . . . . . . . . . . . . 138 8.1.2 Two-Photon Interference with Gating: HongOu-Mandel Interferometer for Two Independent Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 Quantum State Teleportation and Swapping . . . . . . . . . . . . . . . . 142 8.2.1 Quantum State Teleportation: Single-Mode Case . . . . . . 143 8.2.2 Quantum State Teleportation: Multi-Mode Case . . . . . . . 145 8.2.3 Entanglement Swapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 8.3 Distinguishing a Genuine Polarization Entangled Four-Photon State from Two Independent EPR Pairs . . . . . . . . . . . . . . . . . . . . 149 8.3.1 Single-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.3.2 Multi-Mode Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons . . . . 155 8.4.1 Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.4.2 Asymmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . 159 8.5 Generation of a NOON State by Superposition . . . . . . . . . . . . . . 163 8.5.1 Interference between a Coherent State and Parametric Down-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 8.5.2 A Special N-Photon Interference Scheme . . . . . . . . . . . . . 166 8.6 Multi-Photon De Broglie Wavelength by Projection Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.6.1 Projection by Asymmetric Beam Splitters . . . . . . . . . . . . 169 8.6.2 NOON State Projection Measurement . . . . . . . . . . . . . . . 172 8.7 Stimulated Emission and Multi-Photon Interference . . . . . . . . . 177 8.8 Remarks on E and A and General Discussion . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9

Temporal Distinguishability of a Multi-Photon State . . . . . . 185 9.1 Hong-Ou-Mandel Interferometer for Characterizing Two-Photon Temporal Distinguishability . . . . . . . . . . . . . . . . . . . 186 9.2 Characterizing an N-Photon State in Time . . . . . . . . . . . . . . . . . 187 9.2.1 General Description of a Multi-Mode N-Photon State . . 188 9.2.2 Direct Photon Detection from Glauber’s Coherence Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

XII

Contents

9.2.3 A NOON-State Projection Measurement and Generalized Hong-Ou-Mandel Interferometer . . . . . . . . . . 195 9.3 The First Example of |1H , 2V . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9.4 The General Case of |1H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.5 The General Case of |kH , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 9.5.1 General Formula for the Visibility . . . . . . . . . . . . . . . . . . . 206 9.5.2 The Special Cases of |2H , 2V , |2H , 3V , and |2H , 4V . . 206 9.5.3 The Special Case of |3H , 3V . . . . . . . . . . . . . . . . . . . . . . . . 208 9.6 The Scheme for Characterizing the Temporal Distinguishability by an Asymmetric Beam Splitter . . . . . . . . . . 208 9.6.1 The Temporal Distinguishability of |1H , NV . . . . . . . . . . 210 9.6.2 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 . . . . . . . . . . . . . . . . 214 9.7.1 Generation of the State of |2H , 1V with Tunable Temporal Distinguishability . . . . . . . . . . . . . . . . . . . . . . . . 214 9.7.2 Distinguishing 4 × 1 Case from 2 × 2 Case for the State of |2H , 2V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 10.1 Interference with a Single-Photon State and an N-Photon State at a Symmetric Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . 225 10.2 Interference of a Single-Photon State and an Arbitrary State . . 228 10.3 Multi-Mode Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 A

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

B

Derivation of the Visibility for |kH , NV . . . . . . . . . . . . . . . . . . . 245 B.1 The Case of |2H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 B.1.1 The Scenario of 2HmV + (N − m)V . . . . . . . . . . . . . . . . . 245 B.1.2 The Scenario of 1HmV + 1HnV + (N − n − m)V . . . . . 249 B.2 The Case of |3H , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 B.2.1 The Scenario of 3HmV + (N − m)V . . . . . . . . . . . . . . . . . 251 B.2.2 The Scenario of 2HmV + 1HnV + (N − m − n)V . . . . . 255 B.2.3 The Scenario of 1HmV + 1HnV + 1HpV + (N −m−n−p)V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 B.3 The General Case of |kH , NV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 B.3.1 The Scenario of k1 V mH + k2 V nH . . . . . . . . . . . . . . . . . . 262 B.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 in the sense of the violations of Bell’s inequalities, the EPR-Bohm type [1.6] of polarization singlet state of two photons in the form √ (1.1) |Ψ 12 = (|x1 |y2 − |y1 |x2 )/ 2 is naturally produced from atomic cascade due to angular momentum conservation, where x, y represent the two orthogonal polarizations. In demonstrating the violation of Bell’s inequalities, a two-photon polarization correlation measurement is made. However, even though two-photon polarization interferometry is involved in the process, no one has attempted to interpret the polarization correlation of Eq.(1.1) in terms of two-photon interference until the experiments of Shih-Alley [1.7, 1.8] and Ou-Mandel [1.9], who created a two-photon polarization entangled state in Eq.(1.1) by superposing two ﬁelds from parametric down-conversion. Investigation of multi-photon entanglement truly began with the introduction of the two-photon correlation technique in parametric down-conversion by Burnham and Weinberg in 1970 [1.10], where entanglement in various degrees 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. Many quantum 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 as 1967 with the classic Pﬂeegor-Mandel experiment [1.14, 1.15]. Early studies by Mandel [1.3] emphasized the nonclassical eﬀects in two-photon interference. This emphasis follows naturally the demonstrations of a series of nonclassical eﬀects of light such as photon anti-bunching [1.16], sub-Poissonian photon statistics [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 interpretation in the early development of two-photon interference. Then, we will make some general remarks before proceeding to the subject of parametric down conversion.

1.1 Interference between Independent Lasers: Magyar-Mandel and Pﬂeegor-Mandel Experiments In the early development of coherence theory in quantum optics, one important milestone was the Magyar-Mandel [1.19] experiment that demonstrated the interference eﬀect between two independent lasers. This experiment showed that besides the high brightness, a laser also has a long coherence time. This property is quite diﬀerent from a thermal source. This interference eﬀect can be well understood with the classical optical coherence theory [1.20]: the second-order amplitude correlation function between the two lasers, Γ12 (τ ) = V1∗ (t)V2 (t + τ ),

(1.2)

is non-zero when the observation time τ is within the long coherence time of the two lasers. However, in terms of the language of photon, this immediately poses a serious challenge to the second part of Dirac’s famous statement on photon interference: each photon interferes only with itself; diﬀerent photons never interfere with each other [1.21]. The early experiment by Taylor [1.22] where he introduced an extremely weak source of light in Young’s double slit interference experiment, supported the ﬁrst part of Dirac’s statement. But the light level in the Magyar and Mandel experiment was too high to rule out the possibility of photon interaction in interference. Under this circumstance, Pﬂeegor and Mandel [1.14, 1.15, 1.23] performed an experiment similar to the one by Taylor but with two independent lasers.

Laser 2

θ ^ κ 1

Laser 1

∆k = k2−k1

^ κ 2

..

x2 x1

Fig. 1.1. The scheme for interference between two independent lasers in Magyar-Mandel and Pﬂeegor-Mandel experiments.

In the Pﬂeegor-Mandel experiment, however, due to low light levels, long exposure time well over the coherence time of the lasers is needed and as a result, the fringe pattern constantly moves, due to the random phase diﬀusion of 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 way with just one detector, i.e., the intensity of light varies as a function of the detector position. In order to reveal the interference fringe pattern, Pﬂeegor and 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, intensities at locations separated by half the fringe spacing tend to go in opposite directions. At locations with one full fringe spacing, on the other hand, the intensities tend to go in the same direction. These two correlated eﬀects are illustrated in Fig.1.2. So when two detectors are placed half a fringe spacing apart, they should exhibit negative intensity correlation, while at one fringe spacing separation, the two detectors will have positive correlation.

L/2 A

L/2 B

L/2 C

L/2 A

B

D

C L/2

Direction of fringe motion

L/2 D

Fig. 1.2. Intensity correlation at full fringe separation (between points of same letters A,or B, or C, or D) and intensity anti-correlation at half fringe spacing (between A and B or C and D) for a moving fringe pattern.

More quantitatively, it is straightforward to explain the above using the classical wave theory. Let us describe the ﬁelds from the two lasers by two plane 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 scale in the order of the coherence time of the lasers. The amplitudes of the lasers are represented by A and, for simplicity, we assume they are the same. When we superpose the two ﬁelds, we have the intensity at a certain location r as: 2 I(r, t) = Aei(k1 ·r−ωt+ϕ1 (t)) + Aei(k2 ·r−ωt+ϕ2 (t)) = 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 and k2 . 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 in a short time scale, as in the experiment of Magyar and Mandel [1.19]. But in a long time scale ( >> coherence time), it is randomly ﬂuctuating due to the independent phase diﬀusions of the two lasers, as in the experiment of Pﬂeegor 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: (2)

G12 = 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 normalized intensity correlation function is given by: (2)

g12 =

I(x1 )I(x2 ) = 1 + 0.5 cos 2π(x1 − x2 )/L. I(x1 )I(x2 )

(1.6)

(2)

More speciﬁcally, for x1 − x2 = L/2, we have g12 = 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 pattern, 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 correlation eﬀects with a light level so low that on average, a photon is detected well before the next one is emitted from the lasers. This seems to support the second part of the Dirac statement, i.e., diﬀerent photons never interfere. However, Dirac’s statement on photon interference is too crude to account for details such as the 50% visibility in Eq.(1.5). In the following section, we will see how Dirac’s statement can be modiﬁed to suit the Pﬂeegor-Mandel experiment and provide a quantitative explanation in the language of twophoton correlation.

1.2 Two-Photon Interpretation of Pﬂeegor-Mandel Experiment The major diﬀerence between the Magyar-Mandel and Pﬂeegor-Mandel experiments is that the former records interference fringe in intensity, whereas the observed interference eﬀect in the latter case is in intensity correlation with two detectors. To measure intensity of a ﬁeld, we need only one detector which responds to single photons. On the other hand, in intensity correlation, the measurement device will produce a signal only when there are two photons, one at each detector. So intensity measurement registers single-photon events, whereas intensity correlation measures two-photon events (Fig. 1.3). With this picture in mind, we ﬁnd that Dirac’s statement on single-photon interference still applies to the Magyar-Mandel experiment. But for twophoton 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 pair of photons only interferes with the pair itself .

1.2 Two-Photon Interpretation of Pﬂeegor-Mandel Experiment

(a)

Coincidence Unit

(b)

7

Fig. 1.3. The diﬀerence between intensity measurement and intensity correlation measurement: (a) single-photon will produce a signal out (b) Only two photons registered at each of the two detectors will produce a coincidence signal.

Here, the pair and the pair itself correspond to two parts of a wave that is associated only with two photons. The two parts are the two indistinguishable paths for the two photons together. Thus, the interference is between the two possibilities. The special wave that is related only to two photons is diﬀerent from the traditional waves that we usually encounter in a Michelson interferometer, which is associated only with one photon, since only one detector is involved. As we will see in Chapt.5, the two kinds of waves have diﬀerent coherence times. Thus, in order to separate these two diﬀerent waves, we refer to them as “two-photon wave” and “single-photon wave”, respectively. The above picture for two-photon interference was discussed brieﬂy by Glauber [1.24], with an equivalent view of interference of two-photon amplitude. The generalization to an N -photon case is straightforward using the N -photon wave packet concept. Next, we will attempt to interpret the Pﬂeegor-Mandel experiment with the modiﬁed Dirac statement, or the two-photon wave concept. Consider the situation depicted in Fig.1.4, where two photons are detected with one at each detector. There are four possibilities for the two photons that produce the two-photon coincidence signal: the two photons are both from one 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 phase diﬀusion (ﬂuctuation of Δϕ in a time interval much longer than the coherence time 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 add a constant and raise the baseline. Cases (C) and (D) are indistinguishable and correspond to the cases of a pair of photons and the pair itself in the modiﬁed Dirac statement. So they will produce interference and give the fringe pattern in two-photon coincidence. Because of the randomness of photon statistics of a laser, all four cases have equal chances if the two lasers have the same intensity. So, if we denote A4 as the two-photon probability from one laser, we can write the intensity correlation function from the above discussion as (2)

G12 = A4 + A4 + 2A2 A2 [1 + cos 2π(x1 − x2 )/L],

(1.7)

8

1 Historical Background and General Remarks

(A)

(B)

(C)

(D)

Fig. 1.4. Four possible origins for the two photons detected by two detectors: no interference between (A) and (B) but 100% visibility interference between (C) and (D).

where the ﬁrst two terms are from cases (A) and (B) and the last term is from the two-photon interference of cases (C) and (D). Eq.(1.7) is exactly same as Eq.(1.5), derived from classical wave theory. From the above argument leading to Eq.(1.7), we ﬁnd that the reason for 50% visibility in Eq.(1.7) is due to the existence of cases (A) and (B), i.e., some nonzero two-photon probability from one source only. As a matter of fact, for any classical source, the two-photon probability P2 is always greater than or equal to the square of one-photon probability or the so-called accidental twophoton probability P12 . For coherent state from a laser, we have P2 = P12 as described above in the Pﬂeegor-Mandel experiment. On the other hand, for a thermal source, we have P2 = 2P12 , which leads to the so-called photon bunching eﬀect. With this source, cases (A) and (B) are twice as probable as cases (C) and (D). This will give rise to a visibility of 1/3, from the argument above [1.3, 1.25]. To obtain a visibility over 50%, we must consider quantum sources with sub-Poissonian photon statistics or the anti-bunching eﬀect. For example, if both ﬁelds are in single-photon state, the probabilities for cases (A) and (B) are simply zero. This leads to (2)

G12 = 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 classical states is 50% and certain quantum sources with photon anti-bunching eﬀect can 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 two atom-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% visibility for the interference eﬀect in intensity correlation. This was also mentioned brieﬂy by Richter [1.27] with regard to a similar problem. But it was Mandel [1.3] who ﬁrst pointed out the diﬀerence between the predictions of quantum and classical theories on the visibility of two-photon interference. Although the early theoretical quantum treatments of this problem were applied to light emitted from two simultaneously excited atoms, the parametric down-conversion process proved to be more suitable for an experimental demonstration of the quantum nature of two-photon interference. Two-photon interference in the parametric down-conversion process was ﬁrst analyzed by Ghosh 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 interference with a visibility of more than 50%. In this section, we will show that the general classical limit is 50% for the visibility of two-photon interference and derive the necessary condition for the ﬁelds to have a nonclassical two-photon interference eﬀect with visibility larger than 50%. Then we will examine a couple of special cases. We start with a quantum description of a free optical ﬁeld [1.29] by the positive frequency part of an electric ﬁeld operator: 1 ˆ (+) (r, t) = E ˆk ei(k·r−ωt) . (1.9) d3 k a (2π)3/2 Here, we treat the ﬁeld as quasi-monochromatic and assume that all the modes in the ﬁeld have the same polarization so that we can treat them as scalar ﬁelds. In the interference of two optical ﬁelds, we further assume that most of the modes in the integral are in the vacuum state and only modes in two directions with unit vectors κ ˆ 1 and κ ˆ 2 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: ˆ (+) (r, t) + Eˆ (+) (r, t), ˆ (+) (r, t) = E E 1 2 with ˆ (+) (r, t) = √1 E n 2π

dωˆ an (ω)ei(kn ·r−ωt) (n = 1, 2),

(1.10)

(1.11)

where kn = κ ˆ n ω/c. In order to produce macroscopic fringe pattern, we assume the angle θ between the two directions κ ˆ 1 and κ ˆ 2 is small, i.e., cos θ ≡ κ ˆ1 · κ ˆ2 ≈ 1. Let the bandwidths of the two interfering ﬁelds (characterized by directions κ ˆ 1 and κ ˆ 2 ) be Δω1 and Δω2 , respectively. We assume that the two ﬁelds are quasi-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 ) ∝ Eˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t) ˆ t)I(r ˆ , t + τ ) :, = T : I(r, (1.12) ˆ (+) ]† is the where T is time ordering and :: is normal ordering. Eˆ (−) = [E negative frequency part of the ﬁeld operator. Before we proceed, let us now introduce the assumption made earlier that there is no interference eﬀect in the ﬁeld intensity (single-photon detection) so that we concentrate only on a genuine two-photon eﬀect. The easiest way to achieve this is to assume that the two ﬁelds have independent phase ﬂuctuations. We then ﬁnd, after substituting Eq.(1.10) into Eq.(1.12) and making (−) (−) (+) (+) expansion of the product, that all unpaired terms like Eˆ1 Eˆ1 Eˆ2 Eˆ2 , (−) (−) (+) (+) ˆ E ˆ E ˆ ,etc. vanish. Only six terms survive and Eq.(1.12) becomes: ˆ E E 1 2 2 2 P2 (r, t; r , t + τ ) ∝ T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) :+ +T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : + T : Iˆ1 (r , t + τ )Iˆ2 (r, t) :+ ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t)+ +E 1 2 1 2 ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t). +E 2

1

1

2

(1.13)

It is obvious from the above expression that the ﬁrst two terms are the contribution from cases (A) and (B) in Fig.1.4, i.e., two photons are all from the ﬁeld 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 nearly perpendicular observation plane where the detectors are located (see Fig.1.1 for detailed geometry), the ﬁrst four terms in Eq.(1.13) are independent of r and r and are always positive. Actually, they are the auto-correlation and cross correlation of the corresponding ﬁelds, that is, ⎧ ⎪ T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : = Iˆ1 (t)Iˆ1 (t + τ )[1 + λ1 (t, τ )], ⎪ ⎨ T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : = Iˆ2 (t)Iˆ2 (t + τ )[1 + λ2 (t, τ )], (1.14) ⎪ T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : = Iˆ1 (t)Iˆ2 (t + τ )[1 + λ12 (t, τ )], ⎪ ⎩ T : Iˆ1 (r , t + τ )Iˆ2 (r, t) : = Iˆ2 (t)Iˆ1 (t + τ )[1 + λ21 (t, τ )], where λ-functions are independent of t for stationary ﬁelds. λn (0)(n = 1, 2) is non-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 ﬁelds at positions r and r and change rapidly with r and r . These are the interference terms which give rise to modulation. Hence, the relative magnitude of these two terms, compared with the ﬁrst four terms in Eq.(1.13), will determine the relative depth of modulation or the visibility of the interference pattern. To evaluate their relative magnitude, we consider the operator ˆ (+) (r , t + τ )Eˆ (+) (r, t)eiϕ , ˆ≡E ˆ (+) (r , t + τ )Eˆ (+) (r, t) − E O 1 2 2 1

(1.15)

1.3 Two-Photon Interference with Quantum Sources

11

where ϕ is an arbitrary phase. We then construct the non-negative quantity ˆ † O ˆ ≥ 0, O

(1.16)

where the average is on an arbitrary quantum state of the ﬁelds. Substituting Eq.(1.15) into Eq.(1.16) and expanding the product, we obtain: T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : ˆ (−) (r, t)Eˆ (−) (r , t + τ )Eˆ (+) (r , t + τ )Eˆ (+) (r, t)e−iϕ + c.c. ≥ E 1 2 2

(1.17)

Since ϕ is arbitrary, we can choose it to be 0 or π. Then Eq.(1.17) becomes: ˆ ˆ ˆ T : Iˆ1 (r, t)I2 (r , t + τ ) : + T : I1 (r , t + τ )I2 (r, t) : ˆ (−) (−) (+) (+) ≥ E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c..

(1.18)

This inequality ensures that the joint probability P2 is non-negative for any state of light. Next we will prove, for classical ﬁelds only, that the ﬁrst two terms in Eq.(1.13) are larger than or equal to the last two interference terms, that is, ˆ ˆ ˆ T : Iˆ1 (r, t)I1 (r , t + τ ) :c + T : I2 (r , t + τ )I2 (r, t) :c ˆ (−) (−) (+) (+) ≥ E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c.,

(1.19)

where the subscript c indicates that the quantum average is only over classical states. To prove Eq.(1.19), we write an arbitrary quantum state described by a density operator in the Glauber-Sudarshan P-representation [1.31, 1.32]: ρˆ = d{α}P ({α})|{α}{α}|, (1.20) where {α} is a set of variables covering all the excited modes of the ﬁelds and P ({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 wellbehaved and non-negative P ({α}) so that P ({α}) can be treated as a true probability distribution. We now use Eq.(1.20) to express the ﬁrst two terms in Eq.(1.13) as ˆ ˆ T : I1 (r, t)I1 (r , t + τ ) : = d{α}P ({α})|E1 (r, t)|2 |E1 (r , t + τ )|2 (1.22) T : Iˆ2 (r , t + τ )Iˆ2 (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 (−) ˆ (−) ˆ (+) ˆ (+) Eˆ1 (r, t)E2 (r , t + τ )E1 (r , t + τ )E2 (r, t) = d{α}P ({α})E1∗ (r, t)E2∗ (r , t + τ )E2 (r, t)E1 (r , t + τ ),

where

1 En (r, t) = √ 2π

(1.24)

dωαn (ω)ei(kn ·r−ωt) .

(1.25)

Consider the following quantity with an arbitrary phase ϕ: O ≡ E1 (r, t)E1∗ (r , t + τ ) − E2 (r, t)E2∗ (r , t + τ )eiϕ .

(1.26)

|O|2 = O∗ O ≥ 0.

(1.27)

Obviously,

So, we have |E1 (r, t)|2 |E1∗ (r , t + τ )|2 + |E2 (r, t)|2 |E2∗ (r , t + τ )|2 ≥ E1∗ (r, t)E2∗ (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 inequality and integrate over {α}. We then obtain an inequality exactly the same as Eq.(1.19), if we choose ϕ = 0 or π. Therefore, we may conclude generally from Eqs.(1.18, 1.19) that classical ﬁelds can only give rise to two-photon interference with a maximum visibility of 50%, whereas quantum ﬁelds may achieve 100% visibility. We have already seen one example of this in Sect.1.2 [Eq.(1.8)]. From inequalities (1.18) and (1.19), we may obtain a necessary condition for the nonclassical eﬀect to occur in two-photon interference as ˆ ˆ ˆ T : Iˆ1 (r, t)I1 (r , t + τ ) : + T : I2 (r , t + τ )I2 (r, t) : ˆ (−) (−) (+) (+) < E1 (r, t)Eˆ2 (r , t + τ )Eˆ1 (r , t + τ )Eˆ2 (r, t) + c.c. , (1.29) M

where the subscript M stands for the maximum value. The reason for using the maximum value in the inequality (1.29) is that the interference terms on the right side of Eq.(1.19) modulate as r and r change. With the inequality in Eq.(1.18), we can write the necessary condition in Eq.(1.29) in a diﬀerent form as T : Iˆ1 (r, t)Iˆ1 (r , t + τ ) : + T : Iˆ2 (r , t + τ )Iˆ2 (r, t) : < T : Iˆ1 (r, t)Iˆ2 (r , t + τ ) : + T : Iˆ1 (r , t + τ )Iˆ2 (r.t) :.

(1.30)

By using Eqs.(1.14) for stationary ﬁelds in the expression above, we can further simplify the necessary condition to Iˆ1 2 [1 + λ1 (τ )] + Iˆ2 2 [1 + λ2 (τ )] < Iˆ1 Iˆ2 [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 a minimum value of (1.32) Iˆ1 Iˆ2 [1 + λ1 (τ )][1 + λ2 (τ )], when the intensities Iˆ1 , Iˆ2 satisfy Iˆ1 2 [1 + λ1 (τ )] = Iˆ2 2 [1 + λ2 (τ )]. Using this minimum value in Eq.(1.31), we arrive at the necessary condition in 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), the condition in Eq.(1.31) is always satisﬁed if we allow the intensity to be large enough for the ﬁeld with λn (τ ) = −1(n =1 or 2). In this case, the necessary condition (1.33) is certainly satisﬁed. So, the condition in Eq.(1.33) is a more general necessary condition for any two ﬁelds to exhibit a nonclassical twophoton interference eﬀect of larger than 50% visibility. The above-mentioned special case with, say, λ1 (τ ) = −1 presents an interesting phenomenon in two-photon interference. If we set Iˆ1 >> Iˆ2 , 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-photon interference. This is in sharp contrast to the same situation in one-photon interference (fringe pattern is exhibited in intensity), where if one ﬁeld is dominant, 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 nearly 100% in the above case. To see the physical picture of this, let us go back to Fig.1.4. When λ1 (τ ) = −1 or Iˆ1 Iˆ1 = 0, there is no contribution from case (A). Since the contribution from case (B) is proportional to Iˆ2 2 and those from case C and D are proportional to Iˆ1 Iˆ2 , the contributions to two-photon coincidence from (C) and (D) will be much larger than (A) and (B) if Iˆ1 >> Iˆ2 . Therefore, we obtain nearly 100% visibility, according to the discussion in Sect.1.2. Two-photon interference of nearly 100% visibility with Iˆ1 >> Iˆ2 was demonstrated by Ou and Mandel [1.12]. In a real experiment, we measure the joint two-photon detection probability with some ﬁnite time resolution. So, the coincidence count registered in the lab will be dt dτ P2 (r, t; r , t + τ ), (1.34) Nc ∝ T

TR

where T is the total time of data-taking for the stationary cw case or the duration of the ﬁeld for the non-stationary pulsed case, and TR is the coincidence window (time) for the two photoelectric pulses after the detection of the 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 a small TR within which P2 (r, t; r , t + τ ) ≈ P2 (r, t; r , t), i.e., we can set τ = 0 in 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 following Schwartz inequality for classical ﬁelds: I12 I22 ≥ I1 I2 2 .

(1.36)

So, the necessary condition for having over 50% visibility in two-photon interference is that the two interfering ﬁelds must be nonclassical ﬁelds that violate the Schwartz inequality in Eq.(1.36). In order to show that some quantum sources can give rise to 100% visibility in two-photon interference, let us consider a two-photon quantum state of the form |Ψ = |1k1 |1k2 ,

(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 1 or 2 alone in the quantum state depicted in Eq.(1.37). From the simple picture of Sect.1.3, we should expect 100% visibility in two-photon interference. We will conﬁrm this by calculation. Because only one mode is excited (occupied) in ﬁeld 1 or 2, we can simplify Eq.(1.10) as ˆ (+) = a E ˆk1 ei(k1 ·r−ωt) + a ˆk2 ei(k2 ·r−ωt) .

(1.38)

The change from continuous mode integral in Eq.(1.10) to discrete sum here √ will take away the 1/ 2π coeﬃcient in Eq.(1.10). We can easily ﬁnd the intensity I(r) ≡ Eˆ (−) Eˆ (+) as I(r) = 2.

(1.39)

So there is no interference eﬀect in intensity. This is because the photon number Fock state in Eq.(1.37) does not have deﬁnite phases for the two ﬁelds. For intensity correlation, however, we have: Eˆ (−) (r1 , t)Eˆ (−) (r2 , t + τ )Eˆ (+) (r2 , t + τ )Eˆ (+) (r1 , t) 2 = ei(k1 ·r1 −ωt) ei[k2 ·r2 −ω(t+τ )] + ei[k1 ·r2 −ω(t+τ )] ei(k2 ·r1 −ωt)

= 2 1 + cos(k1 − k2 ) · (r1 − r2 ) (1.40) = 2[1 + cos 2π(x1 − x2 )/L], 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 involve parameters of the position of the detectors. They are the most commonly seen interference phenomena. The other kind of interference phenomena, known as beating, are associated with time. As is well-known, the observation of beating eﬀects requires a fast detection system. However, recent experiments on spatial beating [1.11, 1.33] have shown that this is not necessarily so. The phenomenon of spatial beating, which exhibits beat note in space domain, only exists 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 depend on the phases of each ﬁeld. This can be understood again with the picture in Fig.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 two interfering paths can be separated, two-photon interference phenomena will depend on the phases of the interfering ﬁelds [1.34, 1.35, 1.36, 1.37, 1.38]. The ﬁrst experimental demonstration of phase-dependent two-photon interference was carried out by Ou et al. [1.39]. This will be the subject of Chapt.5 and Chapt.6. The quantum state in Eq.(1.37) can be produced in many ways. Early proposals 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 these is as easy as the parametric down-conversion process. It can produce a twophoton state with entanglement in many degrees of freedom. This book is devoted to the understanding and applications of multi-photon interference with the parametric down-conversion process.

References 1.1 1.2 1.3 1.4 1.5 1.6 1.7

1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15

J. S. Bell, Physics (N. Y.) 1, 195 (1965). J. F. Clauser and A. Shimony, Rep. Prog. Phys. 41, 1881 (1978). L. Mandel, Phys. Rev. A28, 929 (1983). R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987). C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). D. Bohm, Quantum Theory (Prentice Hall, Englewood Cliﬀs, N. J., 1951). C. O. Alley and Y. H. Shih, Proceedings of the Second International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, edited by M. Namiki et al. (Physical Society of Japan, Tokyo, 1987). Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921 (1988). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988). D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 54 (1988). Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989). J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990). R. L. Pﬂeegor and L. Mandel, Phys. Lett. 24A, 766 (1967). R. L. Pﬂeegor and L. Mandel, Phys. Rev. 159, 1084 (1967).

16 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24

1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40

1 Historical Background and General Remarks H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977). R. Short and L. Mandel, Phys. Rev. Lett. 51, 384 (1983). J. F. Clauser, Phys. Rev. D 9, 853 (1974). G. Magyar and L. Mandel, Nature (London) 198, 255 (1963). M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). P. A. M. Dirac, The Principles of Quantum Mechanics (Clarendon, Oxford, 1st ed., 1930; 5th ed., 1958). G. I. Taylor, Proc. Camb. Phil. Soc. 15, 114 (1909). R. L. Pﬂeegor and L. Mandel, J. Opt. Soc. Am. 58, 946 (1968). R. J. Glauber, Quantum Optics and Electronics (Les Houches Lectures), p.63, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965). S. J. Kuo, D. T. Smithey, and M. G. Raymer, Phys. Rev. A43, 4083 (1991). U. Fano, Am. J. Phys. 29, 539 (1961). G. Richter, Abh. Acad. Wiss. DDR, 7N, 245 (1977). R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. A34, 3962 (1986). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995). R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1st ed., 1973; 3rd ed., 2000). R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, Opt. Comm. 69, 1 (1988). P. Grangier, M. J. Potasek, and B. Yurke, Phys. Rev. A 38, 3132 (1988). B. J. Oliver and C. R. Stroud, Jr., Phys. Lett. 135A, 407 (1989). Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989). J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989). M. A. Horne, A. Shimony, and A. Zeilinger, Phys. Rev. Lett. 62, 2209 (1989). Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. A 41, 566 (1990). C. Santori, D. Fattal, J. Vukovi, G. S. Solomon, and Y. Yamamoto, Nature 419, 594 (2002).

2 Quantum State from Parametric Down-Conversion

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

2.1 Introduction The parametric process was initially developed in radio wave and microwave as low noise ampliﬁers, well before the invention of the laser [2.1]. With the emergence of nonlinear optics, most of the concepts and even the terminologies have been adapted in the optical range of electromagnetic wave. Parametric ampliﬁcation was ﬁrst demonstrated in the optical region by Giordmaine and Miller [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 required by the principle of quantum mechanics to preserve the commutation relation. The energy is from another ﬁeld or ﬁelds (labelled as “pump”). In the optical range, 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 well studied in nonlinear optics with classical wave theory [2.3]. Quantum mechanically, the parametric process is described simply by the interaction Hamiltonian with the signal and idler ﬁelds in single-mode: ˆ I = i¯hχˆ H a†s a ˆ†i + H.c.,

(2.1)

18

2 Quantum State from Parametric Down-Conversion

where a ˆs(i) is the annihilation operator for the signal (idler) ﬁeld and χ is a parameter related to the pump ﬁelds (which are usually treated as classical ﬁelds described by numbers). In the Heisenberg picture, the evolution of the annihilation operator can be solved, and follows the Bogoliubov transformation: a ˆs (t) = (μˆ as + νˆ a†i )e−iωt , (2.2) a ˆi (t) = (μˆ ai + νˆ a†s )e−iωt , with μ = cosh |χ|t,

ν = (χ/|χ|) sinh |χ|t.

(2.3)

The above expressions describe the parametric ampliﬁcation process with an a†s in the second line of Eq.(2.2)] gives amplitude gain of μ. The νˆ a†i term [or νˆ rise to the accompanying “idler” ﬁeld. In the interaction picture, the quantum state evolves as ˆ (t)|Ψ (0) |Ψ (t) = U

(2.4)

ˆ = exp(−iH ˆ I t/¯h) = exp(ηˆ U a†s a ˆ†i − H.c.),

(2.5)

with

where η ≡ χt. This is the two-mode squeezed state [2.4] and for the generation of large squeezing, it is normally operated in the regime of |η| >> 1. This is the high gain regime of parametric ampliﬁer. However, to generate the twophoton state, we work in the regime of low gain, with μ ≈ 1 or |η| << 1. Then Eq.(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 initial state |Ψ (0) as vacuum. Because |η| << 1, the ﬁrst nontrivial contribution in Eq.(2.6) to photo-detection is the two-photon state |1s , 1i , which is exactly 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 photodetection, it plays an important role in preserving the two-photon phase and producing entanglement in photons, as we will see in later chapters. From Eq.(2.6), we see that the physical meaning of |η|2 is simply the probability of pump photon conversion to down-converted photons. The pair generation is random, so the probability for two pairs is proportional to |η|4 [the last term in Eq.(2.6)]. However, what makes parametric process so interesting in quantum interference is not only its ability to generate a two-photon state of the form in Eq.(1.37), but also the possibility for it to produce an entangled two-photon state. 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. Usually there is not just one possibility for spontaneous process. Any ﬁeld (or more precisely, any mode of the ﬁelds) that is coupled in the process may radiate. The ability to have multiple paths in parametric process creates two-photon entanglement among the diﬀerent paths. For example, for two such possible paths, we may rewrite the Hamiltonian in two-mode form for each of the signal and idler ﬁelds as ˆI = h ¯ χ1 a ˆs1 a ˆi1 + ¯hχ2 a ˆs2 a ˆi2 + H.c., H

(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 and can be any degree of freedom for an optical ﬁeld. For polarization degree of freedom, for example, if we choose 1 → x and 2 → y, then Eq.(2.8) is similar to the EPR-Bohm singlet state in Eq.(1.1), introduced in the beginning of the book. However, most common is the frequency degree of freedom, for, the spontaneous parametric process has a wide spectrum. To fully study the spectral structure and other degrees of freedom, we need to concentrate on some speciﬁc processes. As we mentioned earlier, both three-wave mixing and four-wave mixing can give rise to the Hamiltonian in Eq.(2.1). Most of the investigations on two-photon entanglement are on the three-wave mixing process through χ(2) nonlinear optical coupling. In the nonlinear three-wave mixing process, a light beam of higher frequency interacts 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, called the “signal”, with lower frequency is introduced on the interaction region in a certain direction, its strength is ampliﬁed, while the pump beam is depleted and partly converted into the signal beam. Energy conservation requires a third beam, named the “idler”, be generated at the same time (Fig.2.1b). The frequency 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, this process is also known as the parametric down-conversion (PDC) process. Depending on the polarization of the signal and idler photons, we have two types of parametric down-conversion: type-I process produces two photons of the same polarization and type-II process generates two orthogonally polarized photons. Because almost all nonlinear media are birefringent, there is a large diﬀerence in the properties of down-converted photons between the two types of processes. Entangled photon pairs can also be produced in a four-wave mixing process via χ(3) nonlinear coupling. There are two completely diﬀerent approaches in this direction. One approach utilizes optical ﬁbers to increase the interaction length to compensate for the relatively small χ(3) [2.5]. But the newly discovered micro-structured ﬁber has a very large nonlinear χ(3) coeﬃcient and can

20

2 Quantum State from Parametric Down-Conversion

(a)

(b) Signal ωs

ωp

Pump

PDC Idler

ωi

ki

ks kp

Fig. 2.1. The process of parametric down-conversion: (a) energy diagram; (b) geometry and phase matching.

give rise to eﬃcient conversion [2.6]. To avoid some notorious noise in the ﬁber system, two entangled photons are produced at two well-separated frequencies with a bandwidth as wide as a few tens of nanometers. This is intended for quantum information applications in the conventional optical communication environment. The other approach relies on some resonant atomic Raman transitions to enhance the χ(3) coeﬃcient [2.7, 2.8]. The entangled photon pairs are the correlated Stokes and anti-Stokes photons in the process. This produces a rather narrow spectrum of an entangled two-photon state due to narrow atomic transitions. The advantage is the utilization of atomic media as the intermediate storage for quantum memory so that there may exist a controllable delay between the generation of the signal and idler photons. Fig.2.2a shows the energy diagram for the Raman processes, but the fourwave mixing process, in general, has the geometry and phase matching shown in Fig.2.2b, where two pumping ﬁelds of “write” and “read” interact with the nonlinear medium and produce two correlated signal and idler photons. The two pumping ﬁelds may also travel in opposite directions.

(b)

(a) ωr |c〉

Signal

Write Atomic Medium

ωi Read

ωw |b〉 |a〉

|d〉

ωs

Idler ks kw

ki kr

Fig. 2.2. The four-wave mixing process for the generation of entangled photon pairs: (a) energy diagram; (b) geometry and phase matching.

Generation of entangled two-photon states in four-wave mixing is a newly emerging ﬁeld and requires more research to fully understand its impact.

2.2 Spontaneous Parametric Down-Conversion Process

21

Therefore, it is not the goal of this book to cover this topic, although some of the multi-photon interference schemes may well be applied to the states produced in the process. We will concentrate on the more mature process of three-wave mixing. In the following, we will work in the low gain limit without injection of signal beam. This is a spontaneous process and has to be treated quantum mechanically.

2.2 Spontaneous Parametric Down-Conversion Process The process of spontaneous parametric down-conversion can be understood as the splitting of the pump photon into the signal and idler photons after interacting with a nonlinear medium. Energy conservation law of the form ωp = ωs + ωi

(2.9)

is satisﬁed in this process, but the momentum conservation law of the form kp = ks + ki

(2.10)

is only met when the phase matching condition is satisﬁed (Fig.2.1b) and the maximum down-conversion is generated (see below for details). The downconverted beams usually do not propagate in the same direction; instead, they emerge from the nonlinear medium at some small angle determined by the conditions in Eqs.(2.9, 2.10). The ﬁrst demonstration of spontaneous parametric down-conversion was carried out by Magde and Mahr [2.9] and by Akhmanov et al. [2.10] and more thoroughly by Byer and Harris [2.11]. Later, Burnham and Weinberg [2.12] performed an experiment proposed by Zel’dovich and Klyshko [2.13], and found that signal and idler photons are produced simultaneously within the 4 nsec resolution of the detection process. This limit was later lowered to 100 psec by Friberg et al. [2.14], with an improved detection system. But later experiments [2.15, 2.16] showed that the signal and idler photons are generated within an extremely short period of time, of the order of 100 fsec. The quantum theory of these experiments was developed by Giallorenzi and Tang [2.17], Mollow and Glauber [2.18], Zel’dovich and Klyshko [2.13], Mollow [2.19] and by Hong and Mandel [2.20]. These treatments mostly concentrated on either the problem of production and phase matching of the down-converted photons, or on the problem of time correlation between the two photons. The formalism is not suited for the discussion of two-photon interference. Although the conclusion that the two down-converted photons are highly correlated in time was reached in some of these treatments, none of the treatments were based on the two-photon Fock-state in the form of Eq.(1.37). The ﬁrst treatment to use a model of a multi-mode two-photon Fock-state for the spontaneous parametric down-conversion process was given by Ghosh

22

2 Quantum State from Parametric Down-Conversion

et al. [2.21]. Another similar but more rigorous treatment was given by Ou et al. [2.22]. This treatment is simple and straightforward and is well-suited to the discussion of two-photon interference. In the following, we will follow the treatment of Ou et al. [2.22] to derive the output state of the spontaneous parametric down-conversion process. The treatment is based on perturbative expansion in the interaction picture.

z=0

z= -L Pump

a Fig. 2.3. The detailed geometry of the parametric down-conversion process.

L

Consider a classical pump wave travelling in z-direction and coupled to quantum ﬁelds through a nonlinear medium (Fig.2.3). For simplicity, we assume that the interaction region has a length of L in z-direction and a cross section of a so that the interaction volume v ≡ aL. We also choose the origin at the center of the cross section and at the end of the interaction medium. In doing so, the locations of subsequent elements (such as mirrors, wave plates, and detectors) are all referenced to the end of the nonlinear medium. The three-wave mixing process involves the second-order nonlinear susceptibility (2) tensor χijk , which relates the medium polarization vector P(N L) to the electric ﬁeld vector E [2.23] in a quadratic form: (N L) (2) Pi (r, t) = dt1 dt2 χijk (t − t1 , t − t2 )Ej (r, t1 )Ek (r, t2 ), (2.11) (2)

where χijk is the second-order nonlinear susceptibility tensor. Here, we assume that the nonlinear response of the medium is local and we use the Einstein notation of repeated indices for vector (tensor) products. The Hamiltonian of the electromagnetic system is given by

1 1 3 H= d r D·E+ B·B . (2.12) 8π v μ0 If we use the deﬁnition D = E + 4πP, we immediately obtain the nonlinear interaction Hamiltonian as 1 1 (2) d3 rP(N L) · E = d3 rEi dt1 dt2 χijk Ej Ek . HI = 2 v 2 v

(2.13)

(2.14)

2.2 Spontaneous Parametric Down-Conversion Process

23

After quantization of the electromagnetic ﬁelds, the interaction Hamiltonian describing the nonlinear coupling is in the form of the corresponding Hilbert space operators ˆi (r, t) dt1 dt2 χ(2) Eˆj (r, t1 )Eˆk (r, t2 ), ˆ I = 1 d3 rE (2.15) H ijk 2 v where the electric ﬁeld operators take the free ﬁeld plane wave form of ˆ (+) (r, t) ˆ t) = E ˆ (−) (r, t) + E E(r,

(2.16)

† ˆ (+) (r, t) ˆ (−) (r, t) = E E

(2.17)

with

and ˆ (+) (r, t) = √1 E 2π

d3 k

k,ν l(ω)ˆ ak,ν exp[i(k · r − ωt)],

(2.18)

ν=1,2

where l(ω) = i(¯hω/2c)1/2 with |k| = nω/c and k,ν are the unit vectors for two independent polarizations perpendicular to k. a ˆk,ν is the annihilation operator for the mode given by k, ν and satisﬁes the commutation relation: ˆ†k ,ν ] = δνν δ(k − k ). [ˆ ak,ν , a

(2.19)

Next, we consider the actual situation of parametric down-conversion and for the purpose of interference, we assume the down-converted ﬁelds are nearly degenerate: |ωs − ωi | << ωs , ωi . This enables us to divide the spectrum of interacting optical ﬁelds into two separate bands: one centered at the angular frequency ωp of the pump ﬁeld and the other at the angular frequency ω0 = ωp /2 of the down-converted ﬁelds. By dividing the electric ﬁelds into positive and negative frequency parts [Eq.(2.16)], we may expand Eq.(2.15). After omitting two terms that do not satisfy energy conservation and, therefore, are of no importance in the steady state, we obtain: (2) ˆI = d3 k3 d3 k1 d3 k2 χijk (ω1 , ω2 , ω3 ) H [ωp ]

[ω0 ]

ν1 ,ν2 ,ν3

×(k3 ,ν3 )i (k1 ,ν1 )∗j (k2 ,ν2 )∗k a ˆk3 ,ν3 a ˆ†k1 ,ν1 a ˆ†k2 ,ν2 ×ei(ω1 +ω2 −ω3 )t d3 r exp(iΔk · r) + H.c.,

(2.20)

v (2)

where Δk ≡ k3 − k1 − k2 is the so-called phase mismatch. χijk (ω1 , ω2 , ω3 ) is related to the Fourier transformation of the nonlinear response func(2) tion χijk (t1 , t2 ) and it also includes the term of l∗ (ω1 )l∗ (ω2 )l(ω3 ). Normally, (2)

χijk (ω1 , ω2 , ω3 ) is a slowly varying function.

24

2 Quantum State from Parametric Down-Conversion

The evolution of the state of the system is described in the interaction picture by the unitary operator t ˆ I (τ )dτ . ˆ (t, t ) = exp 1 H U (2.21) i¯h t Through this unitary operator, the state of the system at time t is related to the state at time t by ˆ (t, t )|Ψ (t ). |Ψ (t) = U

(2.22)

When the steady state is reached and the interaction is over, the system does not have a memory of the initial and ﬁnal time, so we may set t = −∞ and t = ∞. Then, we have: ∞ dτ ei(ω1 +ω2 −ω3 )τ = 2πδ(ω3 − ω1 − ω2 ), (2.23) −∞

which gives the energy conservation relation in Eq.(2.9). When the interaction region is much larger than the wavelength, the spatial integration can be approximated to d3 r exp(iΔk · r) ≈ (2π)3 δ (3) (Δk), (2.24) v

which corresponds to the momentum conservation relation in Eq.(2.10). However, we do not use this approximation when we want to ﬁnd the bandwidth of down-conversion, which is determined by the range of allowed phase mismatching (Δk = 0) and is usually related to the length of the nonlinear medium. See more on this in Sect.2.3. The energy and momentum conservation relations are quite strict, especially in birefringent crystals. This means that only certain polarization arrangements are allowed. In practice, depending on the polarization of the signal and idler photons, we have two types of parametric down-conversion: type-I process produces two photons of same polarization and type-II process generates two orthogonally polarized photons. Because almost all nonlinear media are birefringent, there is a large diﬀerence in the properties of downconverted photons between the two types of processes. In any case, the polarizations of the ﬁelds are ﬁxed, so that we can drop the summation over ν’s in Eq.(2.20) and the time integral of the Hamiltonian in Eq.(2.20) becomes: 1 ∞ ˆ d3 k3 d3 k1 d3 k2 χ(ω1 , ω2 , ω3 )ˆ ak3 ,p HI (τ )dτ = i¯ h −∞ [ωp ] [ω ] 0 ×ˆ a†k1 ,s a ˆ†k2 ,i δ(ω1 + ω2 − ω3 ) d3 r exp(iΔk · r) + H.c., (2.25) v

where we put all the extra constants into the χ-function, which is proportional (2) to the original tensor product χijk (ω1 , ω2 , ω3 )(k3 ,p )i (k1 ,s )∗j (k2 ,i )∗k . Furthermore, in most of the applications, the directions of the down-converted ﬁelds

2.2 Spontaneous Parametric Down-Conversion Process

25

are ﬁxed due to the strict energy and momentum conservation relations in Eqs.(2.9, 2.10), so that we can treat the problem as a one-dimensional case of nearly collinear propagation. Then Eq.(2.25) becomes ∞ 1 ˆ I (τ )dτ = χ dω3 dω1 dω2 δ(ω1 + ω2 − ω3 ) H i¯ h −∞ [ωp ] [ω 0] ×ˆ ap (ω3 )ˆ a†s (ω1 )ˆ a†i (ω2 ) d3 r exp(iΔk · r) + H.c., (2.26) v

where we take the χ-function out of the integral because it is a slowly varying function of ω’s, as compared to the spatial integral term. We assign a new constant χ to take care of all the new constants emerging when we go from 3-dimension to 1-dimension. The annihilation operators now have the commutation relations in the one-dimensional form: [ˆ a(ω), a ˆ† (ω )] = δ(ω − ω ).

(2.27)

In the collinear case when the two down-converted ﬁelds travel in the same direction as the pump ﬁeld, the spatial integral in Eq.(2.26) can be carried out (please refer to Fig.2.3 for the geometry of the spatial integral) and Eq.(2.26) becomes 1 ∞ ˆ dω3 dω1 dω2 δ(ω1 + ω2 − ω3 ) HI (τ )dτ = ξ i¯ h −∞ [ωp ] [ω0 ] ×ˆ ap (ω3 )ˆ a†s (ω1 )ˆ a†i (ω2 )h(LΔk) + H.c., (2.28) where

h(x) ≡

0

dzeixz =

−1

1 − e−ix = e−ix/2 sinc(x/2), ix

(2.29)

with Δk = [n(ωp )ωp − n(ωs )ωs − n(ωi )ωi ]/c and L is the length of the nonlinear medium. ξ = χv is another new constant characterizing the nonlinear interaction strength. The origin of the system is chosen to be at the end of the nonlinear medium. For a non-collinear case, the result is similar but with a slight modiﬁcation of Δk to Δk¯ ≡ [n(ωp )ωp −n(ωs )ωs cos θs −n(ωi )ωi cos θi ]/c. (For one-dimensional approximation, we take Δkx ≈ 0, Δky ≈ 0.) Next, we ﬁrst make an expansion of the exponential function to the ﬁrst order for the evolution operator in Eq.(2.21) as ∞ ˆ I (τ )dτ. ˆ (∞, −∞) ≈ 1 + 1 H U (2.30) i¯h −∞ Then, we apply it to the initial state of |Ψ0 = |{αp (ω)} ⊗ |0s , 0i ,

(2.31)

which is a multi-mode coherent state with only the modes around the pump frequency (ω ∼ ωp ) occupied and all the other modes in vacuum, that is,

26

2 Quantum State from Parametric Down-Conversion

a ˆp (ω)|Ψ0 = αp (ω)|Ψ0 ,

a ˆs,i (ω)|Ψ0 = 0.

(2.32)

αp (ω) is the spectral amplitude of the pump ﬁeld. With the above expression, we obtain: ˆ |Ψ = U(∞, −∞)|Ψ0 = |{αp (ω)} ⊗ |0s , 0i + ξ

dω1 dω2 h(LΔk) ×δ(ω1 + ω2 − ω3 )αp (ω3 )|ω1s , ω2i , [ωp ]

dω3

[ω0 ]

(2.33)

ˆ†s (ω1 )ˆ a†i (ω2 )|0s , 0i is a two-photon state. If we only care where |ω1s , ω2i ≡ a about the down-converted photons, we can trace out the state of the pump ﬁeld and have the ﬁnal state for the down-converted ﬁelds as |Ψ P DC = |0s , 0i + ξ dω1 dω2 Φ(ω1 , ω2 )|ω1s , ω2i , (2.34) with Φ(ω1 , ω2 ) ≡ αp (ω1 + ω2 )h(ω1 , ω2 ),

(2.35)

h(ω1 , ω2 ) = h(LΔk)|ω3 =ω1 +ω2 .

(2.36)

where

The h-function is given in Eq.(2.29). The quantum state in Eq.(2.34) is for one pair of photons and it is suitable for the discussion of two-photon interference. For four-photon interference, we need two pairs of down-converted photons. To ﬁnd the four-photon state from ˆ parametric down-conversion, we may expand U(∞, −∞) to the next higher order than that in Eq.(2.30): ˆ (∞, −∞) ≈ 1 + 1 U i¯h

∞ 2 ˆ I (τ )dτ + 1 1 ˆ I (τ )dτ . H H 2! i¯h −∞ −∞

∞

(2.37)

Substituting Eq.(2.28) into Eq.(2.37) and expanding the square term, we obtain, after tracing oﬀ the pump ﬁeld state:

η2 |0s , 0i + ξ dω1 dω2 Φ(ω1 , ω2 )|ω1s , ω2i + |Ψ P DC = 1 − 2 ξ2 dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )× + 2 , ω2i , ω2i , (2.38) ×Φ(ω1 , ω2 )|ω1s , ω1s where a normalization constant η 2 ≡ |ξ|2

dω1 dω2 |Φ(ω1 , ω2 )|2

(2.39)

2.3 Phase Matching Condition and Spectral Bandwidth

27

is introduced. Note the state in Eq.(2.38) is properly normalized up to ξ 2 . Normally, the state in Eq.(2.34) is the quantum state for two-photon interference, while the state in Eq.(2.38) is the starting point for the four-photon case. From Eqs.(2.34, 2.38), we ﬁnd that the properties of parametric downconversion are determined from the pump amplitude function αp (ωp ) and the phase matching function h(LΔk). In the following section, we will ﬁrst concentrate on the phase matching function h(LΔk). We will discuss the eﬀect of pump bandwidth later in Sects.2.4 and 2.5.

2.3 Phase Matching Condition and Spectral Bandwidth The phase matching function h(LΔk) is given in a simple form in Eq.(2.29). But the phase mismatch quantity Δk ≡ kp (ωp ) − ks (ωs ) − ki (ωi ) is a complicated function of frequencies and indices of refraction dependent on some speciﬁc processes of parametric down-conversion. Between the type-I and the type-II down-conversion processes, the major diﬀerence is the polarization of the two photons. In the type-I case, two photons of the same polarization are produced. In the near degenerate frequency and collinear case, the two photons are nearly identical (indistinguishable). For type-II process, on the other hand, the two photons have orthogonal polarizations. Because of the birefringence of the nonlinear medium, the two photons in the type-II process have diﬀerent indices of refraction and diﬀerent group velocities, which, in turn, give rise to a totally diﬀerent function for Δk. Let us now consider Δk for these two types of processes separately. 2.3.1 Type-I Phase Matching Spectrum in Collinear Propagation In the type-I case, the nonlinear crystal is usually cut so that for one speciﬁc frequency ω0 , the harmonic generation is phase matched: Δk = kh (2ω0 ) − 2kf (ω0 ) = 0

or nh (2ω0 ) = nf (ω0 ),

(2.40)

where h, f denote harmonic and fundamental waves, respectively. Because of dispersion, the above condition can only be satisﬁed if the harmonic and fundamental ﬁelds have diﬀerent polarization. This is achieved by using the birefringence of the nonlinear crystal and assigning, say, the harmonic ﬁeld as the extraordinary ray and the fundamental as the ordinary ray [2.24]. The vice versa situation is a little more complicated but the result is similar. For simplicity we will only consider the former case. Then, Eq.(2.40) becomes: Δk = ke (2ω0 , ϕ0 ) − 2ko (ω0 ) = 0

or ne (2ω0 , ϕ0 ) = no (ω0 ),

(2.41)

28

2 Quantum State from Parametric Down-Conversion

and [2.24] cos2 ϕ0 sin2 ϕ0 1 = + , n2e (ϕ0 ) n2e n2o

(2.42)

where ϕ0 is the angle between the optic axis and the direction of propagation of the e-ray. Eq.(2.40) indicates that the harmonic wave is collinear with the fundamental wave. Parametric down-conversion is the reverse process of the second harmonic generation. So, Eq.(2.41) is for the degenerate and collinear case of ωs = ωi = ω0 under the strict phase match condition, which gives h(LΔk) = 1. But as we know, the phase matching condition in Eq.(2.40) is not strictly satisﬁed because of the ﬁnite size L of the crystal. The allowed range of ωs , ωi for which |h(LΔk)| ∼ 1 will give rise to the spectrum of down-conversion. We ﬁrst consider Δk in the collinear case for a ﬁxed pump frequency at ωp0 = 2ω0 . We expand ko (ω) around ω0 by setting ωs = ω0 + Ω with ωi = ω0 − Ω [from Eq.(2.9)] to the ﬁrst non-zero order of Ω: Δk = ke (2ω0 , ϕ0 ) − ko (ωs ) − ko (ωi ) ≈ −ko Ω 2 ,

(2.43)

where ko = d2 ko /dω 2 |ω=ω0 . Since the sinc-function in Eq.(2.29) has a range of π, the bandwidth of the type-I down-converted ﬁelds is thus determined by (I) L|Δk| = 2π or ΔωP DC = 2π/L|ko |. (2.44) The bandwidth of the down-converted ﬁelds is typically 1012∼13 Hz. For broad band pumping, ωp is not ﬁxed but equals ωs + ωi by energy conservation. Now, ωs and ωi are independent variables. Let us set ωs = ω0 + Ω1 , ωi = ω0 + Ω2 and expand ke (ωp ) around 2ω0 , and ko (ωs,i ) around ω0 . The phase mismatch then becomes: Δk|ωp =ωs +ωi = ke (ωp , ϕ) − ko (ωs ) − ko (ωi ) ≈ (ke − ko )(Ω1 + Ω2 )+ +ke (Ω1 + Ω2 )2 /2 − ko (Ω12 + Ω22 )/2.

(2.45)

So, the phase match function h(Ω1 , Ω2 ) in Eq.(2.29) is symmetric with Ω1 , Ω2 : h(Ω1 , Ω2 ) = h(Ω2 , Ω1 ).

(2.46)

In a normal situation, we have ke = ko , i.e., the group velocities (vg = 1/k ) of the two waves are not matched. Then the shape of h(Ω1 , Ω2 ) is highly irregular. Fig.2.4a shows a 3-D plot of |h(Ω1 , Ω2 )| with scaled axes. On the other hand, when the condition of group velocity match is met, i.e., ke = ko , the ﬁrst linear term in Eq.(2.45) disappears. Fig.2.4b shows a better behaved h-function.

2.3 Phase Matching Condition and Spectral Bandwidth (a)

29

(b)

Ω2

Ω1 Ω2

Ω1

Fig. 2.4. 3-D plot of the function |h(Ω1 , Ω2 )| when (a) the group velocities are mismatched (ke = ko ) and (b) the group velocities are matched ke = ko .

ks Pump

Nonlinear Crystal

al Sign θs

ki

ks kp

θi

ki

Idler

θs

kp

θi

Fig. 2.5. The geometry for non-collinear phase matching.

Exact Phase Match in Non-collinear Propagation For the non-degenerate case of ωs = ωi , phase matching can only occur in a non-collinear fashion (Fig.2.5). The angles oﬀ the pump direction for the down-converted ﬁelds can be calculated from ke (2ω0 , ϕ0 ) = ko (ωs ) cos θs + ko (ωi ) cos θi , (2.47) ko (ωs ) sin θs = ko (ωi ) sin θi . In the near degenerate case, we have both θs , θi << 1 and |ωs − ωi | ≡ 2Ω << ω0 . Therefore, Eq.(2.47) can be approximately solved when we use Eq.(2.41): θs ≈ θi ≡ θ,

and ko Ω 2 − ko θ2 = 0.

(2.48)

If we tune the angle ϕ between the optic axis and the direction of the pump ﬁeld away from ϕ0 in Eq.(2.47), then Eq.(2.48) changes to ko Ω 2 − ko θ 2 = (∂ke /∂ϕ)Δϕ

(Δϕ ≡ ϕ − ϕ0 )

(2.49)

Δn Δϕ (Δn ≡ no − ne ). n2o

(2.50)

or cko Δλ2 /λ20 − ko θ2 = ko sin 2ϕ0

In Fig.2.6, we plot the angle θ as a function of Δλ for a number of values of ΔnΔϕ. When ΔnΔϕ > 0, no degenerate case is allowed. On the other hand,

30

2 Quantum State from Parametric Down-Conversion

∆n∆ϕ < 0 ∆n∆ϕ > 0

θ

∆λ

Fig. 2.6. Angular dependence on wavelength for non-collinear phase matching at around degenerate wavelength. The dotted line corresponds to the degenerate case of Δλ = 0.

if ΔnΔϕ < 0, the angle of the degenerate wavelength can be adjusted by changing ϕ (dotted line in Fig.2.6). For a large deviation of wavelength from the degenerate one, we need to solve numerically Eq.(2.47) together with energy conservation in Eq.(2.9) and the expression in Eq.(2.42). This will show asymmetry for the signal and idler photons with θs = θi because of the diﬀerence in wavelength [2.25]. Because no is more or less independent of the direction of propagation for small angles of θs , θi , there is a cylindrical symmetry around the direction of the pump ﬁeld. For a ﬁxed wavelength, the down-converted ﬁeld is, thus, in the shape of a cone, originated from the crystal (Fig.2.5). Diﬀerent wavelengths correspond to cones with diﬀerent angles of θ. However, the conjugate pair of photons with ωs + ωi = ωp always lie on the opposite sides of the pump direction. The spectrum of the non-collinear case is similar to the collinear case. 2.3.2 Type-II Phase Matching Spectrum in Collinear Propagation For the type-II phase matching, the polarizations of the signal and idler ﬁelds are orthogonal and the crystal is cut, so that it is phase matched for type-II second harmonic generation at a speciﬁc frequency of ω0 : kp (2ω0 ) = ks (ω0 ) + ki (ω0 , ϕ0 ),

(2.51)

2np (2ω0 ) = ns (ω0 ) + ni (ω0 , ϕ0 ),

(2.52)

or

where we choose the idler ﬁeld as the extra-ordinary ray (i = e) and ni = ne (ϕ0 ) is given in Eq.(2.42). The polarization of the pump ﬁeld must be same as either the signal or the idler. For simplicity, we consider only the case when the pump is o-ray (i.e., p = o). Because of diﬀerent polarizations, the signal and the idler ﬁelds will not have the kind of symmetry seen in the type-I case. Let us look at the spectrum in the collinear case. We start with singlefrequency pumping at 2ω0 . Expanding around the degenerate frequency of ω0 with ωs = ω0 + Ω and ωi = ω0 − Ω, we ﬁnd the phase mismatch as

2.3 Phase Matching Condition and Spectral Bandwidth

Δk = kp (2ω0 ) − ks (ωs ) − ki (ωi , ϕ0 ) 1 ≈ −(ks − ki )Ω − (ks + ki )Ω 2 . 2

31

(2.53)

Normally, ks = ki , so that the ﬁrst term dominates and the type-II down(II) conversion bandwidth is then ΔωP DC = 2π/L|D| with D ≡ ks − ki . (b)

(a)

Ω1

Ω2 Ω1

Ω2

Fig. 2.7. 3-D plot of the function h(Ω1 , Ω2 ) for the type-II phase matching case. (a) tan γ = −3, (b) tan γ = 1.

For broad band pumping, we have, similar to Eq.(2.45): Δk|ωp =ωs +ωi = kp (ωp ) − ks (ωs ) − ki (ωi , ϕ0 ) ≈ (kp − ks )Ω1 + (kp − ki )Ω2 + 1 1 + kp (Ω1 + Ω2 )2 − (ks Ω12 + ki Ω22 ), 2 2

(2.54)

where kp ≡ [dkp (ωp )/dωp ]|ωp =2ω0 . In general, the quadratic terms are much smaller than the linear terms, so that only the linear terms contribute to h(Ω1 , Ω2 ). Normally, ks = ki , so the phase matching function h(Ω1 , Ω2 ) in Eq.(2.36) is highly asymmetric with respect to Ω1 , Ω2 , as seen in Fig.2.7a. The orientation of the shape of h(Ω1 , Ω2 ) is determined by the ratio (kp − ki )/(ks − kp ) ≡ tan γ. However, when γ = 45◦ , the shape is oriented along the Ω1 = Ω2 line and h(Ω1 , Ω2 ) becomes symmetric again with respect to Ω1 , Ω2 , as seen in Fig.2.7b. The condition for this is: tan γ = 1

or

2kp = ks + ki .

(2.55)

This is the so-called extended phase matching (EPM) condition, ﬁrst discussed by Kuzucu et al. [2.26]. It is so named in contrast to the original phase matching condition in Eq.(2.51). The symmetry property of the phase matching function h(Ω1 , Ω2 ) is very important for temporal mode match in two-photon interference, as we will see later in Sect.3.2.2.

32

2 Quantum State from Parametric Down-Conversion

Exact Phase Matching in Non-collinear Propagation Quite diﬀerent from the type-I case, it is still possible to have non-collinear down-conversion at degenerate frequency when the crystal is set for a collinear case, as required by Eq.(2.52). To conﬁrm this, let us calculate the angles θs , θi for the down-converted ﬁelds. Because of the involvement of e-ray, there will not be a cylindrical symmetry here. So, we need to consider a three-dimensional case and use two parameters to determine the direction of km (m = o, e or s, i). Let the angles between km and the x-axis and y-axis be 90◦ − θmx and 90◦ − θmy , respectively. Then, we can write km as km =

nm ω (sin θmx , sin θmy , cos θm ), (m = o, e), c

(2.56)

2 2 2 ≈ θmx + θmy for |θmx |, |θmy | << 1 with sin2 θm ≡ sin2 θmx + sin2 θmy or θm (θm is the angle between km and kp or the z-axis). Therefore, the threedimensional phase matching condition of Δk = 0 becomes ⎧ ⎨ ko (2ω0 ) − ko (ω0 ) cos θo − ke (ω0 , ϕe ) cos θe = 0, ko (ω0 ) sin θox + ke (ω0 , ϕe ) sin θex = 0, (2.57) ⎩ ko (ω0 ) sin θoy + ke (ω0 , ϕe ) sin θey = 0,

or

⎧ ⎨ 2no (2ω0 ) = no (ω0 ) cos θo + ne (ω0 , ϕe ) cos θe , no (ω0 ) sin θox = −ne (ω0 , ϕe ) sin θex , ⎩ no (ω0 ) sin θoy = −ne (ω0 , ϕe ) sin θey .

(2.58)

Note that because of the non-collinear situation, the angle between the optic axis of the crystal and ke is no longer ϕ0 but ϕe . We can calculate ϕe from the direction of the optic axis (0, sin ϕ0 , cos ϕ0 ) and that of ke in Eq.(2.56) as cos ϕe = (sin θex , sin θey , cos θe ) · (0, sin ϕ0 , cos ϕ0 ) ≈ θey sin ϕ0 + cos ϕ0 ,

(2.59)

from which we obtain Δϕe ≡ ϕe − ϕ0 ≈ −θey . Since no ≈ ne , we have from Eq.(2.58) that θo ≈ −θe ≡ θ. For small angles, Eq.(2.58) becomes 2no (2ω0 ) = no (ω0 )(1 − θ2 /2) + [ne (ω0 , ϕ0 ) + Δne ](1 − θ2 /2),

(2.60)

where Δne ≡ ne (ω0 , ϕe ) − ne (ω0 , ϕ0 ) = Δϕe [dne (ϕ)/dϕ]ϕ=ϕ0 and from Eq.(2.42), we have Δne ≈ θey (no − ne ) sin 2ϕ0 . Using Eq.(2.52) and θ2 = θx2 + θy2 , we simplify Eq.(2.60) to 2 2 + θey − 2aθey = 0 θex

with a≡

no − ne sin 2ϕ0 . no (2ω0 )

(2.61)

2.3 Phase Matching Condition and Spectral Bandwidth

33

θy e-ray

θe θx θo

A

e-ray

B o-ray

o-ray (a)

(b)

(c)

(d)

Fig. 2.8. Various angular distributions for degenerate non-collinear type-II phase matching: (a) aΔϕ = 0; (b) aΔϕ < 0; (c) aΔϕ > 0; (d) Δϕ = a/2.

So, the tip of the ke vector draws a circle with an angular radius of |a| [2.27]. Since θo = −θe , the tip of the ko vector draws a circle on the opposite side of kp , with the same angular radius (Fig.2.8a). Note that both circles pass through the origin, which corresponds to θx = θy = 0, or the collinear case. If we allow the orientation of the crystal to be adjusted away from ϕ0 , the direction of the optic axis is then (0, sin ϕ, cos ϕ). Replacing (0, sin ϕ0 , cos ϕ0 ) with (0, sin ϕ, cos ϕ) in Eq.(2.59), we have Δϕe = Δϕ − θey with Δϕ ≡ ϕ − ϕ0 and Eq.(2.61) is modiﬁed to 2 2 + θey − 2a(θey − Δϕ) = 0. θex

(2.62)

Now, the trajectories are still circles at the same centers but the radius is changed to (2.63) r = a2 − 2aΔϕ. Depending on the sign of aΔϕ, we have enlarged circles as in Fig.2.8b or shrunk circles as in Fig.2.8c. Particularly worth noting are two features: (i) When Δϕ = a/2, the radii of the circles are shrunk to zero. Only one direction is possible for ko and ke , respectively, and the down-converted ﬁelds become beam-like instead of cone-like (Fig.2.8d). Under this condition, because the down-converted ﬁelds from all directions in the cones collapse into two directions, respectively, the beam-like ﬁelds become much stronger than the cone-like ﬁelds in certain directions. This scheme of beam-like down-converted ﬁelds was ﬁrst proposed and demonstrated by Takeuchi [2.28] and by Kurtsiefer et al. [2.29]. (ii) For the situation in Fig.2.8b, the two circles are intercepted at two locations (A and B in Fig.2.8). At these two directions, both o-ray and e-ray are allowed. Since photons in o-ray and e-ray are correlated as the signal and idler ﬁelds in Eq.(2.34), these two locations, together, give rise to a quantum state of |Ψ = ... + η1 |1Ao , 1Be + η2 |1Ae , 1Bo + ...,

(2.64)

34

2 Quantum State from Parametric Down-Conversion

where A, B denote the two directions of propagation. If we arrange the right parameters so that η1 = η2 and we only detect photons in the directions of A, B, the contribution of |Ψ comes only from a state of 1 |EP R = √ (|1AH , 1BV + |1AV , 1BH ), 2

(2.65)

where H, V denote the horizontal and vertical polarizations for the o-ray and e-ray, respectively. This is exactly the EPR-Bohm two-photon entangled polarization state in Eq.(1.1) for the demonstration of locality violation of quantum mechanics. This scheme for the direct generation of the EPR-Bohm type two-photon entangled polarization state from parametric down-conversion was ﬁrst proposed and experimentally realized by Kwiat et al. [2.30]. Next, we will examine the role of the bandwidth of the pump ﬁeld on the properties of the quantum state in Eq.(2.34) from parametric downconversion.

2.4 Quantum State with a Narrow Band Pump Field We start with a special ideal case when the pump ﬁeld has only one frequency component ωp = ωp0 and αp (ωp ) = Vp δ(ωp −ωp0 ), with Vp being the amplitude of the pump ﬁeld. Eq.(2.34) becomes (cw) |Ψ P DC = |0s , 0i + ζ dω1 dω2 Φ(ω1 , ω2 )|ω1s , ω2i , (2.66) with Φ(ω1 , ω2 ) = Vp δ(ω1 + ω2 − ωp )ψ(ω1 ). Here, the new constant ζ is now chosen so that dω1 |ψ(ω1 )|2 = 1

(2.67)

(2.68)

and ψ(ω1 ) ∝ h(ω1 , ωp0 − ω1 ) = h(LΔk)|ω3 =ωp0 ,

ω2 =ωp0 −ω1 .

(2.69)

As expected, the frequencies of the down-converted photons are perfectly correlated with ω1 + ω2 = ωp0 . This case corresponds to single-frequency operation of the pump laser in an actual experiment. Note that the state in Eq.(2.66) is not normalizable because we use the δ-function for the spectral amplitude αp (ωp ). This is not a problem in most of the cases. Using the state in Eq.(2.66) signiﬁcantly simpliﬁes the derivation without loss of generality in most cases when the coherence of the pump ﬁeld is not an inﬂuencing factor.

2.4 Quantum State with a Narrow Band Pump Field

35

However, in a more general case when the pump ﬁeld is a stationary continuous wave (cw) but with some ﬁnite bandwidth, we may treat the pump ﬁeld spectral amplitude αp (ωp ) as a random variable satisfying the correlation relation [2.31] α∗p (ω)αp (ω )p = 2πnp (ω)δ(ω − ω ),

(2.70)

where np (ω) is the power spectrum of the pump ﬁeld. From the discussion in Sect.2.3, we know that the spectra of the downconverted ﬁelds are very broad for a ﬁxed pump frequency. Therefore, for a stationary cw pump ﬁeld, its spectrum is much narrower than those of the down-converted ﬁelds. So, to a good approximation, we have: (2.71) dω1 |h(ω1 , ωp − ω1 )|2 = f (ωp ) ≈ f (ωp0 ) for |ωp − ωp0 | << ΔωP DC . We can rewrite the state in Eq.(2.34) as |Ψ P DC = |0s , 0i + ζ dω1 dω2 αp (ω1 + ω2 )ψ(ω1 , ω2 )|ω1s , ω2i , (2.72) where we introduce a new constant ζ ≡ ξ

and

f (ωp0 ) such that ψ(ω1 , ω2 ) = h(ω1 , ω2 )/ f (ωp0 )

(2.73)

dω1 |ψ(ω1 , ωp − ω1 )|2 = f (ωp )/f (ωp0 ) ≈ 1.

(2.74)

This normalization is similar to the one in Eq.(2.68), so the situation with a ﬁnite bandwidth pump ﬁeld is not much diﬀerent from the ideal case of a single-frequency pump ﬁeld. However, this approximation is only true for a narrow band pump ﬁeld. For an ultra-fast pump ﬁeld, its bandwidth is as wide as those of the down-converted ﬁelds, so we cannot utilize the normalization in Eq.(2.74) but use a diﬀerent normalization (see Sect.2.5). Note that although they are deﬁned diﬀerently, the two ζ’s in Eqs.(2.66, 2.72) are actually the same, if we set αp (ωp ) = Vp δ(ωp − ωp0 ) for single-frequency pumping. The subscript p in Eq.(2.70) stands for the classical ensemble average over the pump wave. So, in calculating the expectation value (measurement outcome) of any operator, we need to take not only the quantum average over the state in Eq.(2.72) but also the classical average over the pump ﬁeld. With this notation, the pump ﬁeld function Vp (r, t), which is the eigenvalue of the one-dimensional ﬁeld operator in Eq.(1.11) on the pump state in Eq.(2.31): ˆp(+) (r, t)|{αp (ω)} = Vp (r, t)|{αp (ω)}, E has the form of

(2.75)

36

2 Quantum State from Parametric Down-Conversion

1 Vp (r, t) = √ 2π

dωp αp (ωp )ei(kp ·r−ωp t)

(2.76)

and is also a classical random variable. Its ﬁeld correlation function can be calculated from the correlation relation in Eq.(2.70) as ∗ (2.77) Γp (τ ) ≡ Ep (r, t)Ep (r, t + τ )p = dωp np (ωp )e−iωp τ , (p)

and its range gives the coherence time Tc of the pump ﬁeld. Note that because of the commutation relation in Eq.(2.27) and the correlation function in Eq.(2.70), the power spectrum np (ω) is a dimensionless quantity and Ip ≡ Γp (0) = |Vp (r, t)|2 p = dωp np (ωp ) (2.78) has a unit of Hz or s−1 . So, Ip = |Vp (r, t)|2 p is the photon ﬂux of the pump ﬁeld. Next we will investigate the properties of the down-converted ﬁelds, such as coherence and intensity correlation. From now on, we will be dealing with the photo-detection process, which is long after the nonlinear interaction. So for this process, we can use the Heisenberg picture and choose |Ψ P DC given in Eq.(2.72) as the initial quantum state. We consider the one-dimensional ﬁeld operators as in Eq.(1.11) for signal and idler ﬁelds 1 (+) ˆm (r, t). (m = s, i). (2.79) Eˆm dωm a (r, t) = √ ˆm (ωm )ei(km ·r−ωm t) ≡ E 2π These operators are slightly diﬀerent from the electric ﬁeld operators given in Eq.(2.18) but they only diﬀer by a constant for quasi-monochromatic ﬁelds ˆ E ˆ † to replace E ˆ (+) , E ˆ (−) . [2.32]. From now on, for brevity we will use E, The ﬁrst quantity we will calculate is the ﬁeld correlation function: † ˆm (r, t)Eˆm (r, t + τ ). Γm (τ ) ≡ E

(m = s, i).

(2.80)

This function gives rise to the coherence property and the spectra of the downconverted ﬁelds. With the state in Eq.(2.72) and ﬁeld operators in Eq.(2.79), we have: 2 dω1 dω2 np (ω1 + ω2 )|ψ(ω1 , ω2 )|2 e−iω1 τ Γs (τ ) = |ζ| (2.81) and Γi (τ ) = |ζ|

2

dω1 dω2 np (ω1 + ω2 )|ψ(ω1 , ω2 )|2 e−iω2 τ ,

(2.82)

where we used the correlation relation in Eq.(2.70) for the classical average over the pump ﬁeld and the commutator in Eq.(2.27) for the quantum average.

2.4 Quantum State with a Narrow Band Pump Field

37

For a narrow band pump ﬁeld, we have ω1 + ω2 ≈ ωp0 , so that Eq.(2.81) becomes: 2 dωp np (ωp ) dω1 |ψ(ω1 , ωp − ω1 )|2 e−iω1 τ Γs (τ ) = |ζ| ≈ |ζ|2 Ip dω1 |ψ(ω1 , ωp0 − ω1 )|2 e−iω1 τ (2.83) and, similarly, Γi (τ ) ≈ |ζ|2 Ip

dω2 |ψ(ω1 , ωp0 − ω1 )|2 e−i(ωp0 −ω1 )τ = e−iωp0 τ Γs∗ (τ ). (2.84)

It is obvious that the two down-converted ﬁelds have similar coherence properties but are exactly complex conjugates of each other. Their spectra are given by |ψ(ω1 , ωp0 − ω1 )|2 and are completely independent of the coherence property of the pump ﬁeld. The bandwidth of the down-converted ﬁelds is determined by the range of function h(LΔk). Of course, this conclusion is drawn under the condition of narrow band pumping. For wide band pumping, the situation is completely diﬀerent, as will be seen later in Sects.2.5 and 7.1. In order to interpret the physical meaning of the coeﬃcient ζ, let us express the rate of signal or idler photons in terms of the rate of pump photons. Note † (r, t)Eˆm (r, t) has the unit of s−1 , so it represents the that the quantity Eˆm photon ﬂux of the ﬁeld at time t and position r. It is related, for a quasimonochromatic light ﬁeld, to the rate Rm (r, t) of registering a photon by a detector placed at position r and time t by the relation [2.32] † Rm (r, t) = βm Eˆm (r, t)Eˆm (r, t) = βm Γm (0),

(2.85)

where βm is the quantum eﬃciency of the detector labeled by m. From Eqs.(2.83, 2.84) and the normalization relation in Eq.(2.74), we have: 2 (2.86) Rm (r, t) ≈ βm |ζ| dωp np (ωp ) = βm |ζ|2 Ip . For the single-mode pump ﬁeld with a quantum state in Eq.(2.66), we obtain a similar result as above, but with Ip ≡ |Vp |2 /2π. For ideal detectors with unit quantum eﬃciency, we obtain: Rs (r, t) = Ri (r, t) = |ζ|2 Ip .

(2.87)

Therefore, the physical meaning of |ζ|2 is the probability of the conversion of a pump photon into two down-converted photons in certain directions via the nonlinear interaction. However, this interpretation is based on the onedimensional treatment for the down-converted ﬁelds. A more precise threedimensional treatment will relate |ζ|2 to a diﬀerential cross section, as in any scattering process. The coincidence rate Rc of registering a signal photon with detector A and an idler photon with detector B within the resolving time of the detectors TR

38

2 Quantum State from Parametric Down-Conversion

ˆ t), for a quasi-monochromatic light ﬁeld, can also be expressed in term of E(r, in the form [2.32] dτ G(2) (τ ) (2.88) Rc = βs βi TR

with the intensity correlation function ˆ † (rs , t + τ )Eˆs (rs , t + τ )Eˆi (ri , t), G(2) (τ ) ≡ Eˆi† (ri , t)E s

(2.89)

where βs , βi are the quantum eﬃciencies of the detectors A and B. In calculating the quantity in Eq.(2.89), we ﬁnd it easier to ﬁrst evaluate ˆs (rs , t + τ )Eˆi (ri , t)|Ψ P DC . With the state in Eq.(2.66) and the ﬁeld opE erators in Eq.(2.79), we have: ˆs (rs , t + τ )Eˆi (ri , t)|Ψ P DC E ζ dω1 dω2 Vp δ(ω1 + ω2 )ψ(ω1 ) = 2π ×e−iω1 (t+τ −zs/c) e−iω2 (t−zi /c) |0s , 0i ,

(2.90)

where we used the commutation relation in Eq.(2.27) and denote zs(i) = κ ˆ s(i) · rs(i) as the distance from detector A(B) to the end of the nonlinear medium (origin of the system) and Δz ≡ zs −zi . Then, the intensity correlation function in Eq.(2.89) can be calculated as ζV 2 p G(2) (τ ) = ||Eˆs (rs , t + τ )Eˆi (ri , t)|Ψ P DC ||2 = g(τ − Δz/c) , (2.91) 2π with

g(τ ) ≡

dω1 ψ(ω1 )e−iω1 τ .

(2.92)

Here, we use the state in Eq.(2.66) for ease of calculation. The function g(τ ) has a range of Tc ∼ 1/ΔωP DC where ΔωP DC is the range of ψ(ω1 ) in Eq.(2.69) or the bandwidth of down-conversion. Normally the bandwidth of down-conversion is as wide as 1012∼13 Hz so that Tc ∼ 1 ps. Earlier attempts by Burnham and Weinberg [2.12] and later by Friberg et al. [2.14] to directly measure the correlation function G(2) (τ ) failed to ﬁnd the dependence on the function g(τ ) because optical detectors have a typical resolving time of the order of TR ∼ 100 ps. So, the measurement by Burnham and Weinberg and by Friberg et al. merely gave the electronic response function. Nevertheless, these pioneering experiments proved that the photons produced in the parametric down-conversion process are highly correlated in time with far shorter correlation time than with any other source known at that time. Indirect measurement of G(2) (τ ) via nonlinear interaction [2.15] and quantum interference [2.16] conﬁrmed the relation in Eq.(2.91). Further measurement on narrow band parametric down-conversion [2.33] gives a direct conﬁrmation of Eq.(2.91) by photo-detection.

2.5 Quantum State with a Wide Band Pump Field

39

Since TR >> Tc in most of the cases, we can take the range of the integral in Eq.(2.88) as (−∞, ∞). Under this condition, the coincidence measurement catches every pair of signal and idler photons with any delay. Then, we have the coincidence rate as 1 2 ∞ dτ |g(τ )|2 . (2.93) Rc = βA βB |ζVp |2 2π −∞ But from Eq.(2.68), we have: ∞ 1 dτ |g(τ )|2 = dω|ψ(ω)|2 = 1. 2π −∞

(2.94)

So, the coincidence rate becomes: Rc = βA βB |ζVp |2 /2π = βA βB |ζ|2 Ip = βB Rs = βA Ri .

(2.95)

Note that for the state in Eq.(2.66), we have Ip ≡ |Vp |2 /2π. In the ideal case when βA = βB = 1, we have: Rc = Rs = Ri .

(2.96)

In this ideal case, the detectors catch all the incoming photons and Eq.(2.96) means that the two-photon detection rate is exactly equal to the one photon detection rate, indicating that the two down-converted photons are exactly correlated within Tc . This conclusion was ﬁrst reached via a diﬀerent approach by Hong and Mandel [2.20].

2.5 Quantum State with a Wide Band Pump Field When the pump ﬁeld is an ultra short pulse (∼ 100 fs), its bandwidth is as wide as the spectra of the down-converted ﬁelds. Together with the phase matching function h(ω1 , ω2 ), the pump spectral amplitude αp (ωp ) will determine the two-photon wave function Φ(ω1 , ω2 ) in the state in Eq.(2.34). The main diﬀerence between this section and the previous one is that for pulsed pumping, the ﬁelds become non-stationary, and, normally, the time frame is much shorter than the response time of any photo-detector. Therefore, the output from any detection process is a time integral from −∞ to ∞. For example, the single-photon detection rate in Eq.(2.85) is changed to singlephoton detection probability by detector “m” (m = s, i): (m) † P1 (r) = βm dtEˆm (r, t)Eˆm (r, t). (2.97) With a quantum state in Eq.(2.34) and the ﬁeld operators in Eq.(2.79), we (s) may calculate P1 (r) with

40

2 Quantum State from Parametric Down-Conversion

ˆ † (r, t)Eˆs (r, t) E s |ξ|2 dω1 dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω1 , ω2 )e−i(ω1 −ω1 )t . = 2π

(2.98)

Then Eq.(2.97) becomes (s)

P1

= βs |ξ|2

dω1 dω2 |Φ(ω1 , ω2 )|2 = βs |η|2 ,

(2.99)

where the time integral becomes a δ-function, and we used the normalization (i) relation in Eq.(2.39). For the idler ﬁeld, we have similarly that P1 = βi |η|2 . Two-photon joint detection probability from detector “s” and detector “i” is, likewise, related to the two-photon detection rate in Eq.(2.88) as (2.100) P2 = βs βi dt1 dt2 G(2) (t1 , t2 ) with G(2) (t1 , t2 ) given in Eq(2.89), but t = t2 and τ = t1 − t2 or, more explicitly, ˆ † (ri , t2 )Eˆ † (rs , t1 )Eˆs (rs , t1 )E ˆi (ri , t2 ). G(2) (t1 , t2 ) = E s i

(2.101)

For the quantum state in Eq.(2.34), we ﬁnd, similar to Eq.(2.90), that ˆs (rs , t1 )Eˆi (ri , t2 )|Ψ P DC E ξ dω1 dω2 Φ(ω1 , ω2 )e−iω1 (t1 −zs /c) e−iω2 (t2 −zi /c) |0s , 0i , = 2π

(2.102)

so that G(2) (t1 , t2 ) = |ξg(t1 − zs /c, t2 − zi /c)|2 with g(t1 , t2 ) ≡

1 2π

dω1 dω2 Φ(ω1 , ω2 )e−iω1 t1 e−iω2 t2 .

(2.103)

(2.104)

Then, the two-photon detection probability P2 = βs βi dt1 dt2 |ξg(t1 − zs /c, t2 − zi /c)|2 = βs βi |ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 = βs βi |η|2 .

(2.105)

For ideal detectors, we have the quantum eﬃciencies βs = βi = 1, so that (s)

P2 = P1

(i)

= P1 = |η|2 .

(2.106)

References

41

Eq.(2.106) for pulsed pumping is equivalent to Eq.(2.96) for cw pumping. So, the physical meaning of |η|2 is the probability of two-photon pair generation in one pump pulse. Before we leave this chapter, let us look at the spectra of the downconverted ﬁelds in a pulsed pumping case. For a non-stationary ﬁeld, the spectrum S(ω) is the Fourier transformation of the time-averaged amplitude correlation function in Eq.(2.80): † (r, t)Eˆm (r, t + τ ). (m = s, i). (2.107) Γm (τ )T = dtEˆm So, similar to Eq.(2.99), we have: Γs (τ )T = |ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 e−iω1 τ and Ss (ω1 ) =

1 2π

dtΓs (t)T eiω1 t = |ξ|2

dω2 |Φ(ω1 , ω2 )|2 .

(2.108)

(2.109)

Likewise, Si (ω2 ) = |ξ|2

dω1 |Φ(ω1 , ω2 )|2 .

(2.110)

For the type-I phase matching case, we have the symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ). This leads to the same spectra Ss (ω) = Si (ω) for both downconverted ﬁelds. For the type-II phase matching case, however, we have Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ) in general, so that Ss (ω) = Si (ω). Such an asymmetry in spectra from the two down-converted ﬁelds will have a detrimental eﬀect on two-photon interference.

References 2.1 W. H. Louisell, Coupled Mode and Parametric Electronics (Wiley, New York, 1960). 2.2 J. A. Giordmaine and R. C. Miller, Phys. Rev. Lett. 14, 973 (1965). 2.3 R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992). 2.4 C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985). 2.5 X. Li, J. Chen, P. L. Voss, J. E. Sharping, and P. Kumar, Opt. Express, 12, 3737 (2004); X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, Phys. Rev. Lett. 94, 053601 (2005). 2.6 J. Fan, A. Dogariu, and L. J. Wang, Opt. Lett. 30, 1530 (2005); J. Fan, A. Migdall, and L. J. Wang, Opt. Lett. 30, 3368 (2005). 2.7 A. Kuzmich, W. P. Bowen, A. D. Boozer, A. Boca, C. W. Chou, L. M. Duan, and H. J. Kimble, Nature 423, 731 (2003).

42

2 Quantum State from Parametric Down-Conversion

2.8 W. Jiang, C. Han, P. Xue, L.-M. Duan, and G.-C. Guo, Phys. Rev. A 69, 043819 (2004). 2.9 D. Magde and H. Mahr, Phys. Rev. Lett. 18, 905 (1967). 2.10 S. A. Akhmanov, V. V. Fadeev, R. V. Khoklov, and O. N. Chunaev, Sov. Phys. JETP Lett. 6, 85 (1967). 2.11 R. L. Byer and S. E. Harris, Phys. Rev. 168, 1064 (1968). 2.12 D. C. Burnham and D. L. Weinberg, Phys. Rev. Lett. 25, 84 (1970). 2.13 B. Ya. Zel’dovich and D. N. Klyshko, JETP Lett. 9, 40 (1969). 2.14 S. R. Friberg, C. K. Hong, and L. Mandel, Phys. Rev. Lett. 54, 2011 (1984). 2.15 I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, Phys. Rev. Lett. 57, 2516 (1986). 2.16 C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 2.17 T. G. Giallorenzi and C. L. Tang, Phys. Rev. 166, 225 (1968). 2.18 B. R. Mollow and R. J. Glauber, Phys. Rev. 160, 1067 (1967); Phys. Rev. 160, 1097 (1967). 2.19 B. R. Mollow, Phys. Rev. A 8, 2684 (1973). 2.20 C. K. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985). 2.21 R. Ghosh, C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. A34, 3962 (1986). 2.22 Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989). 2.23 N. Bloembergen, Nonlinear Optics; a lecture note and reprinted volume (Benjamin, New York, 1965). 2.24 A. Yariv, Quantum Electronics (Wiley, New York, 1st ed., 1967; 3rd ed., 1989) 2.25 S. R. Friberg, Ph.D. Thesis, University of Rochester (1986). 2.26 O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. K¨ artner, Phys. Rev. Lett. 94, 083601 (2005). 2.27 P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao, Phys. Rev. A 49, 3209 (1994). 2.28 S. Takeuchi, Opt. Lett. 26, 843 (2001). 2.29 C. Kurtsiefer, M. Oberparleiter, and H. Weinfurter, J. Mod. Opt. 48, 1997 (2001). 2.30 P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995). 2.31 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). 2.32 L. Mandel, Phys. Rev. 144, 1071 (1966). 2.33 Z. Y. Ou and Y. J. Lu, Phys. Rev. Lett. 83, 2556 (1999).

3 Hong-Ou-Mandel Interferometer

In this chapter, we will study the most popular arrangement in two-photon interference: the Hong-Ou-Mandel interferometer [3.1]. This unique interference eﬀect, due to its simplicity in geometry and clarity in physics, has been widely used as a criterion for testing the degree of entanglement of two photons. We will start with a single-mode situation to demonstrate the simple physical principle, and then look into the more realistic multi-mode case, where the results are quite diﬀerent for various experimental situations. This will allow us to analyze a special feature, i.e., the dispersion-free property of the Hong-Ou-Mandel interferometer. We conclude the chapter with a discussion on photon bunching in the Hong-Ou-Mandel interferometer.

3.1 Single-Mode Consideration From the look of its geometrical shape (Fig.3.1), the Hong-Ou-Mandel interferometer is hardly an interferometer in the traditional sense, i.e., waves are ﬁrst split and then recombined. Rather, it involves a pair of photons entering on a lossless beam splitter from two separate input ports of the beam splitter. Nevertheless, there is, indeed, an interference eﬀect in this seemingly simple geometry.

BS

a^2 ^

a^1

b1 ^

b2

T, R

Fig. 3.1. Layout of HongOu-Mandel interferometer: two photons enter a lossless beam splitter from two sides.

44

3 Hong-Ou-Mandel Interferometer

Because of the involvement of two photons, there are four possibilities for the output of the beam splitter: both are transmitted (Fig.3.2a), both are reﬂected (Fig.3.2b), and one is transmitted while the other is reﬂected (Fig.3.2c and Fig.3.2d). The ﬁrst two possibilities are indistinguishable, resulting in amplitude addition. However, because of energy conservation in the process, there is an overall π-phase diﬀerence for the two photons together, between the two cases. (This phase shift is universal and independent of the speciﬁcs of the beam splitter. See discussion in Sect.3.3 and Appendix A.) This leads to destructive interference, and the two cases completely cancel each other when their amplitudes are the same. This eﬀect was ﬁrst discovered by Hong, Ou, and Mandel in 1987 [3.1] and independently by Fearn and Loudon [3.2].

(a)

(b)

(c)

(d)

Fig. 3.2. Four possibilities in the output for a Hong-Ou-Mandel interferometer.

More rigorously, if the two photons entering the beam splitter have the same frequency, same polarization, and same spatial mode, we can treat each of them in a single mode denoted as a ˆ1 , a ˆ2 , respectively. These correspond to the two input modes of the beam splitter (Fig.3.1). The input state is then: |Ψ in = |1a1 , 1a2 .

(3.1)

ˆ1 , a ˆ2 by If the output modes are labeled by ˆb1 , ˆb2 , they are related to a √ √ ˆb1 = T a a2 , √ ˆ1 + √Rˆ (3.2) ˆb2 = T a ˆ2 − Rˆ a1 , where T and R (≥ 0) are the transmissivity and reﬂectivity of the beam splitter, respectively, and T + R = 1. (In general, coeﬃcients in Eq.(3.2) are complex numbers. For simplicity, we let them be real. The following results can be generalized easily to complex numbers. See Appendix A for more.) The above relationship, together with the input state in Eq.(3.1), is enough to determine the properties at the output ports. For example, the probability P12 for detecting one photon at each output port is given by P12 = in Ψ |ˆb†1ˆb†2ˆb2ˆb1 |Ψ in . With the state in Eq.(3.1) and operators in Eq.(3.2), we easily arrive at

(3.3)

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

P12 = (T − R)2 .

45

(3.4)

When T = R = 1/2, P12 = 0. This corresponds to the complete destructive two-photon interference. However, to better interpret the picture in Fig.3.2, we need to ﬁnd the output state of the beam splitter for the input state in Eq.(3.1). We will leave the task of deriving the output state to Appendix A and write only the state as follows: √ |Ψ out = (T − R)|1b1 , 1b2 + 2T R |2b1 , 0b2 − |0b1 , 2b2 , (3.5) where the ﬁrst term corresponds to the situations in Fig.3.2a and Fig.3.2b, while the other two terms correspond to Fig.3.2c and Fig.3.2d. For a 50:50 beam splitter, we obtain a two-photon entangled state: √ |Ψ out = (|2, 0 − |0, 2)/ 2. (3.6) From the above state, we ﬁnd the |1, 1 term is noticeably missing due to complete destructive two-photon interference. The above photon bunching eﬀect, i.e., the tendency that photons go together to either side of the beam splitter, is the Bosonic property of photons. This property requires the wave function be symmetric about the two photons. However, the symmetry is for the global wave function. If we can manipulate some other degrees of freedom of the photons, say, the polarization, so that the polarization state of photons is anti-symmetric, the two photons will behave like fermions at the beam splitter and go in separate directions, to maintain global symmetry. This eﬀect will be seen in Chapter 4 when we discuss polarization entangled states. It is interesting to see what classical Newtonian particle theory would predict for the outcome. In this case, the two input photons are identiﬁable, so that the two situations in Figs.3.2a and 3.2b are distinguishable and no interference occurs. For a 50:50 beam splitter, all four situations in Fig.3.2 are equally probable, so the probability for |1, 1 would be a nonzero value of 1/2. This classical situation corresponds to the situation when the two photons do not overlap at the beam splitter and, thus, are identiﬁable by their order of arrival. But, this situation may never occur in the single-mode treatment above, where the ﬁelds are inﬁnitely long wave trains and photons are equally probable at any place in the wave trains. Therefore, we will never be able to distinguish the arrival of the two photons at the beam splitter. A more realistic model would contain multi-modes for the photons so that they are in ﬁnite wave packets.

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip We now consider the multi-frequency case when there are more than one frequency modes being excited. As we derived earlier, the down-converted

46

3 Hong-Ou-Mandel Interferometer

ﬁelds are in a multi-mode two-photon state, given generally in Eq.(2.34). We rewrite it as a†s (ω1 )ˆ a†i (ω2 )|vac. (3.7) |Ψ P DC = |0s , 0i + ξ dω1 dω2 Φ(ω1 , ω2 )ˆ For single-frequency pumping at 2ω0 , we have, from Eq.(2.67), that Φ(ω1 , ω2 ) is in the form of Φ(ω1 , ω2 ) = Vp δ(ω1 + ω2 − 2ω0 )ψ(ω1 ), with

ψ(ω) ∝ h(LΔk),

dω|ψ(ω)|2 = 1.

(3.8)

(3.9)

From Eqs.(2.29, 2.43, 2.53), the h-function has the form of h(LΔk) = e−iLΔk/2 sinc(LΔk/2)

(3.10)

with ΔkI = −ko Ω 2 ΔkII = −DΩ

for type−I phase matching and for type−II phase matching,

(3.11) (3.12)

where Ω = ω − ω0 and D ≡ ks − ki . So, ψ(Ω) = ψ(−Ω) in the type-I case and ψ(Ω) = ψ ∗ (−Ω) in the type-II case. For wide band pumping, we have, from Eqs.(2.35, 2.36, 2.45, 2.54), Φ(ω1 , ω2 ) ∝ αp (ω1 + ω2 )h(LΔk)

(3.13)

1 1 ΔkI = (ke − ko )(Ω1 + Ω2 ) + ke (Ω1 + Ω2 )2 − ko (Ω12 + Ω22 ) 2 2

(3.14)

with

for type-I phase matching and ΔkII = (kp − ks )Ω1 + (kp − ki )Ω2 + 1 1 + kp (Ω1 + Ω2 )2 − (ks Ω12 + ki Ω22 ) 2 2

(3.15)

for type-II phase matching (Ω1 = ω1 − ω0 , Ω2 = ω2 − ω0 ). To separate the two down-converted photons, we choose the geometry of non-collinear propagation in Fig.2.5. Combining this geometry of a two-photon state with the Hong-Ou-Mandel interferometer in Fig.3.1, we have the scheme in Fig.3.3. Recall from Sect.2.2 that we set the origin of the system at the end of the nonlinear medium. For unidirectional propagation, the input ﬁeld operators at the beam splitter are similar to Eq.(2.79):

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

signal PDC Pump

idler

1 (in) ˆm (t) = √ E 2π

^ (o)

z1 E^ (in) 1

E2

^ (in)

E1

z2

^ (o)

E2

dωˆ am (ω)e−iω(t−zm /c)

47

Fig. 3.3. Sketch of the Hong-Ou-Mandel interferometer with two photons from parametric downconversion in non-collinear fashion.

(m = 1, 2),

(3.16)

where zm is the optical path for the down-converted photons (m = 1, 2 or s, i) from the origin (the end of the nonlinear medium) to the beam splitter. The diﬀerence Δz = z1 − z2 reﬂects the diﬀerence in their paths from the nonlinear crystal to the beam splitter (Fig.3.3). Assuming that the spatial modes of the two input ﬁelds are perfectly aligned at the beam splitter, we have the output ﬁeld operators of the beam splitter as √ √ ˆ (in) (t) + RE ˆ (in) (t), ˆ (o) (t) = T E E 1 1 2 √ √ (3.17) ˆ (o) (t) = T E ˆ (in) (t) − RE ˆ (in) (t). E 2 2 1 The probability of detecting one photon at each side of the beam splitter is related to the two-photon correlation function (o)†

ˆ G(2) (t1 , t2 ) = E 1

(o)†

ˆ (t1 )E 2

(o)

(o)

ˆ (t2 )E ˆ (t1 ). (t2 )E 2 1

(3.18)

With the quantum state in Eq.(3.7) and the ﬁeld operators in Eq.(3.17), we may derive G(2) (t1 , t2 ) in Eq.(3.18) with some lengthy calculation similar to Eq.(2.90) as G(2) (t1 , t2 ) = |ξ|2 T g(t1 − z1 /c, t2 − z2 /c)− 2 (3.19) −Rg(t2 − z1 /c, t1 − z2 /c) , where g(t1 , t2 ) is given in Eq.(2.104) or, more explicitly, 1 dω1 dω2 Φ(ω1 , ω2 )e−iω1 t1 −iω2 t2 . g(t1 , t2 ) ≡ 2π

(3.20)

The two terms in Eq.(3.19) correspond to the two cases of (a) and (b) in Fig.3.2. In the special case when Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ),

(3.21)

g(t2 , t1 ) = g(t1 , t2 )

(3.22)

we have

48

3 Hong-Ou-Mandel Interferometer

and if the two paths are balanced with z1 = z2 and a 50:50 beam splitter is used, we have G(2) (t1 , t2 ) = 0, i.e., there is absolutely no coincidence between the two detectors at the two sides of the beam splitter. This is exactly the Hong-Ou-Mandel interference eﬀect, but with multi-frequency mode analysis. Before we discuss further the details in Φ(ω1 , ω2 ), let us consider a more practical situation when optical ﬁlters are placed before the detectors. This is usually the case in experimental implementation where there is a huge background scattering from the pump beam, so optical ﬁlters are used to reduce the background noise. In quantum mechanics, optical spectral ﬁlters can be modeled as frequencydependent beam splitters as a ˆ (ω) = f (ω)ˆ a(ω) + 1 − f 2 (ω) a ˆ0 (ω), (3.23) where a ˆ0 (ω) is in vacuum. Normally, f (ω) is centered at some speciﬁc frequency ω0 with a Gaussian shape of standard deviation σ: f (ω) = exp[−(ω − ω0 )2 /2σ 2 ].

(3.24)

For simplicity, let us assume that the ﬁlters in front of each detector are identical, with a center frequency ω0 = ωp0 /2. The ﬁeld operators in Eq.(3.17) ˆm (ω) in Eq.(3.23) then need to be modiﬁed by replacing a ˆm (ω) in Eˆm (t) with a for the corresponding ﬁelds. It is easy to show that Eq.(3.19) still stands, but with Φ(ω1 , ω2 ) modiﬁed to Φ (ω1 , ω2 ) = f (ω1 )f (ω2 )Φ(ω1 , ω2 ).

(3.25)

For a narrow band ﬁlter with σ << ΔωP DC , Φ (ω1 , ω2 ) is dominated by f (ω1 )f (ω2 ). Φ(ω1 , ω2 ) simply varies slowly within σ and can be treated as a constant. So, the condition in Eq.(3.21) is always satisﬁed and we always have G(2) (t1 , t2 ) = 0, or complete destructive two-photon interference. For other values of z1 , z2 , we need to evaluate G(2) (t1 , t2 ) for Φ in Eq.(3.25). Then Eq.(3.20) becomes g(t1 , t2 ) = F (t1 )F (t2 ) with

F (t) ∝

(3.26)

dω exp[−(ω − ω0 )2 /2σ 2 ]e−iωt

∝ e−iω0 t e−(σt)

2

/2

.

(3.27)

Under most experimental conditions, the detectors are slow, so their resolution time TR is much larger than the correlation time Tc of the detected ﬁeld (Tc ∼ 1/σ in the current case). Then, the two-photon coincidence count Nc is proportional to a time average of the two-photon correlation function in Eq.(3.19) and with g(t1 , t2 ) in Eq.(3.26), it has the form of

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

Nc ∝

∞

−∞

∝ 1−

49

dt1 dt2 G(2) (t1 , t2 ) 2T R −(σΔz)2 /2c2 e , + R2

T2

(3.28)

where Δz ≡ z1 − z2 is the path diﬀerence for the two photons to arrive at the beam splitter. In Fig.3.4, we plot Nc as a function of Δz (solid line). The ﬁgure shows a dip with a width of c/σ. This is the Hong-Ou-Mandel interference dip, as ﬁrst demonstrated by Hong et al. [3.1].

Fig. 3.4. Two-photon coincidence as a function of the position of the beam splitter shows the Hong-Ou-Mandel interference dip. Reprinted with permission from C. K. c Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 1987 by the American Physical Society.

When the ﬁlters have a band pass much wider than the spectrum of the down-conversion, the ﬁlters will lose their eﬀect and the Hong-Ou-Mandel interference dip depends on Φ(ω1 , ω2 ). We learned from Sect.2.3, that Φ(ω1 , ω2 ) is not always symmetric with respect to ω1 , ω2 . Depending on the bandwidth of the pump ﬁeld, we have the following two diﬀerent cases. 3.2.1 Narrow Band Pumping For narrow band pumping, Φ(ω1 , ω2 ) is given in Eq.(3.8) and we have: g(t1 , t2 ) = Vp e−iω0 (t1 +t2 ) f (τ ), where τ = t1 − t2 and

(3.29)

50

3 Hong-Ou-Mandel Interferometer

1 f (τ ) = 2π

dΩψ(ω0 + Ω)e−iΩτ .

(3.30)

In this case, the coincidence rate is obtained by integrating G(2) in Eq.(3.19) over τ in the range of resolving time TR of detectors. The result is Rc (Δz) ∝ TR

2 dτ T f (τ − Δz/c) − Rf (−τ − Δz/c) .

(3.31)

Usually the resolving time of detectors is much larger than the range of f (τ ), so we can extend the range of the integral in Eq.(3.31) to {−∞, +∞}. Then, Eq.(3.31) changes to Rc (Δz) ∝ 1 −

2T R V(Δz), + R2

where V(Δz) =

(3.32)

T2

∞

−∞

dτ [f ∗ (τ )f (−2Δz/c − τ )]

∞

−∞

dτ |f (τ )|2 ,

(3.33)

or V(Δz) =

dΩ[ψ ∗ (ω0 + Ω)ψ(ω0 − Ω)e−i2ΩΔz/c ] , dΩ|ψ(ω0 + Ω)|2

(3.34)

if we use Eq.(3.30) for the f -function. For type-I phase matching, ψ(ω) is given in Eqs.(3.9, 3.10, 3.11). Therefore, the function V(Δz) in Eq.(3.34) becomes: VI (Δz) =

3v(Δz/wI ) √ , 4 π

(3.35)

where v(z) ≡

∞

dy −∞

sin2 y 2 −iyz e , y4

(3.36)

√ (I) (I) and wI ≡ c L|ko |/8 = πc/2ΔωP DC , with ΔωP DC = 2π/L|ko | as the bandwidth of degenerate type-I PDC in Eq.(2.44). Note that VI (0) = 1 because of the identity: √ sin2 y 2 4 π dy = . y4 3 −∞

(I)

∞

(3.37)

So, Rc in Eq.(3.32) is zero at Δz = 0 for a 50:50 beam splitter. Fig.3.5 plots (I) (I) the normalized coincidence rate Rc (Δz)/Rc (∞) as a function of Δz/wI

Normalized Coincidence rate

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

51

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -10

-5

0

5

∆z /wI

10

Fig. 3.5. Normalized coincidence rate as a function of relative delay shows the Hong-Ou-Mandel interference dip without ﬁlters.

for a 50:50 beam splitter. This ﬁgure shows the typical Hong-Ou-Mandel in(I) terference dip. The size of the dip is of the order of c/ΔωP DC , which is the coherence length, or the size of the wave packet of the down-converted ﬁelds. Experimentally, interference ﬁlters are often placed before the detectors and the ψ-function is modiﬁed by the approximately Gaussian proﬁle of the ﬁlters, as we have seen before. However, an observation without ﬁlters was made by Rarity and Tapster [3.3] and shows the oscillatory behavior depicted in Fig.3.5. For type-II phase matching, we have, from Eqs.(3.9, 3.10, 3.12), f (τ ) ∝ dΩe−iΩτ +iLDΩ/2 sinc(LDΩ/2) ∝ qII (bτ ) where

qII (ξ) ≡

(3.38)

dxe

ixξ−ix

sinc(x) =

π 0

for |ξ − 1| ≤ 1, for |ξ − 1| > 1,

(3.39)

and b ≡ 2/LD. So, the function V(Δz) in Eq.(3.33) becomes: ⎧ ⎨ 1 − |1 + bΔz/c| for |1 + bΔz/c| ≤ 1, VII (Δz) = ⎩ 0 for |1 + bΔz/c| > 1.

(3.40) (II)

Note that VII = 1 for Δz = −wII with wII ≡ LDc/2, so that Rc = 0 in Eq.(3.32) at a nonzero delay of Δz = −LDc/2. This is because of the group velocity diﬀerence for the o-ray (signal) and e-ray (idler) in a birefringent crystal, which results in an extra delay between the o-ray and e-ray from inside the (II) (II) crystal. Fig.3.6 shows the normalized coincidence rate Rc (Δz)/Rc (∞) as a function of Δz/wII for a 50:50 beam splitter. The V-shaped interference dip is typical for a type-II Hong-Ou-Mandel interferometer, as was ﬁrst discussed by Rubin et al. [3.4] and observed by Sergienko et al. [3.5].

3 Hong-Ou-Mandel Interferometer

Normalized Coincidence rate

52

1.2 1.0 0.8 0.6 0.4 0.2 0.0

-2

-3

-1

0

1

∆z /wII

Fig. 3.6. The Hong-OuMandel interference eﬀect with a two-photon state from Type-II parametric down-conversion.

3.2.2 Wide Band Pumping In this case, the two-photon detection probability is a double integration of G(2) (t1 , t2 ) with respect to t1 , t2 , as discussed in Sect.2.5, and from Eq.(3.19) we have: P2 ∝ |ξ|2 dt1 dt2 T g(t1 − z1 /c, t2 − z2 /c)− 2 −Rg(t2 − z1 /c, t1 − z2 /c) . (3.41) With g(t1 , t2 ) in Eq.(3.20), we then carry out the time integral and obtain 2T R V(Δz), T 2 + R2

(3.42)

dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω2 , ω1 )e−i(ω1 −ω2 )Δz/c . dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω1 , ω2 )

(3.43)

P2 ∝ 1 − with V(Δz) =

If the Φ-function is symmetric: Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ),

(3.44)

dω1 dω2 |Φ(ω1 , ω2 )|2 e−i(ω1 −ω2 )Δz/c . dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω1 , ω2 )

(3.45)

then Eq.(3.43) becomes: V(Δz) =

Note that V(0) = 1. This is the case for the two photons from parametric down-conversion with type-I phase matching. From Eqs.(3.13, 3.14, 3.10), we ﬁnd Φ(ω1 , ω2 ) as

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

|Φ(ω1 , ω2 )|2 ∝ e−x

2

/2σp2

sin2 [L(Δk x − ko y 2 /4)/2] , [L(Δk x − ko y 2 /4)/2]2

53

(3.46)

where x = ω1 + ω2 − 2ω0 , y = ω1 − ω2 , Δk = ke − ko and we take the pump spectrum as a Gaussian with a width of σp . We also drop the x2 term in ΔkI . Substituting Eq.(3.46) into Eq.(3.45), we have: 2 2 2 2 sin [L(Δk x − k y /4)/2] o V(Δz) = C dxdye−x /2σp e−iyΔz/c , (3.47) [L(Δk x − ko y 2 /4)/2]2 where C is such that V(0) = 1. For wide band pumping with σp >> 1/L|Δk |, the sinc-function dominates in the integral in Eq.(3.47) and can be approximated by a δ-function. Eq.(3.47) then becomes: 4 −1 V(Δz) = 2Γ (1/4) dye−y e−iy(σΔz/c) , (3.48)

1.0 0.8 0.6 0.4 0.2 0.0 −10

−5

0

σ ∆ z/c (a)

5

10

Normalized Coincidence

Normalized Coincidence

with σ = 2 |Δk σp /ko |. Γ (1/4) is the Euler Gamma function. Fig.3.7a shows the Hong-Ou-Mandel dip for this case. On the other hand, if σp << 1/L|Δk|, we can approximate the Gaussian function in Eq.(3.47) by a δ-function. Then, Eq.(3.47) becomes Eq.(3.35) for the case of type-I narrow band pumping.

1.0 (iv)

0.8

(iii)

0.6 0.4

(ii)

0.2 (i) 0.0 −1.0

−0.5

0

0.5

1.0

ζ (b)

Fig. 3.7. The Hong-Ou-Mandel dip for broad band pumped parametric downconversion. (a) Type-I, σp >> 1/L|Δk |; (b) Type-II, (i) σp /2Ω+ << 1, (ii) σp /2Ω+ = 1, (iii) σp /2Ω+ = 2, (iv) σp /2Ω+ = 4. Reprinted ﬁgure with permission c from W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1632 (1997). 1997 by the American Physical Society.

For type-II phase matching, however, we normally do not have the symmetry condition in Eq.(3.44). After dropping out the higher order terms in Eq.(3.15), we have the Φ-function as Φ(ω1 , ω2 ) ∝

sin(LΔk/2) −iLΔk/2 −(Ω1 +Ω2 )2 /4σp2 e , e LΔk/2

(3.49)

54

3 Hong-Ou-Mandel Interferometer

with Δk = Δks Ω1 + Δki Ω2 and Δks = kp − ks , Δki = kp − ki . After deﬁning new constants 1/Ω± ≡ L(Δks ± Δki ) and carrying out the integral in Eq.(3.43) with the Φ-function in Eq.(3.49), we obtain: ⎧√ ⎨ π|Ω+ | erf σp (1 − |ζ|) , for |ζ| < 1, σp 2Ω+ (3.50) V(Δz) = ⎩ 0, otherwise, with ζ = 1 − ΔzΩ− /c, where 2 erf(x) ≡ √ π

x

e−t dt 2

(3.51)

(3.52)

0

is the error function. In Fig.3.7b, we plot the Hong-Ou-Mandel dip for various values of σp /2Ω+ . Obviously, the visibility drops as the bandwidth σp of the pump increases. Note, that when σp /|Ω+ | << 1, we have: ⎧ for |ζ| ≤ 1, ⎨ 1 − |ζ|, (3.53) V(Δz) = ⎩ 0, otherwise, which is exactly same as Eq.(3.40) for the case of type-II narrow band pumping. That is why case (i) in Fig.3.7b is exactly same as Fig.3.6. So, now we may set the condition for narrow band pumping as − ko − ke |. σp << |Ω+ | ≡ 1/L|2k2o

(3.54)

|Ω+ | is roughly the bandwidth of type-II parametric down-conversion. The reduced interference eﬀect for an asymmetric Φ(ω1 , ω2 ) is the result of temporal /spectral distinguishability of the two photons. See more in Chapt.9. Grice and Walmsley [3.6] were the ﬁrst to derive the expressions in Eq.(3.50) and discuss the eﬀect of the asymmetric Φ(ω1 , ω2 ) on the visibility of the Hong-Ou-Mandel dip. Another interesting feature is that when we have 2kp − ks − ki = 0, or Δks = −Δki ≡ Δk ,

(3.55)

the condition in Eq.(3.54) is always satisﬁed for arbitrary pumping bandwidth. This is the so-called extended phase matching (EPM) condition, previously discussed in Sect.2.3 [Eq.(2.55)]. When the EPM condition in Eq.(3.55) is satisﬁed, the Φ-function in Eq.(3.49) becomes Φ(ω1 , ω2 ) ∝

sin[LΔk (Ω1 − Ω2 )/2] −iLΔk (Ω1 −Ω2 )/2 −(Ω1 +Ω2 )2 /4σp2 e , (3.56) e LΔk (Ω1 − Ω2 )/2

3.2 Multi-Mode Treatment and Hong-Ou-Mandel Dip

55

which, except for the Gaussian, is similar to that in Eqs.(3.9, 3.10, 3.12) for the case of type-II narrow band pumping. Therefore, it is not a surprise that under the condition in Eq.(3.55), we obtain the visibility in Eq.(3.53), which is exactly same as Eq.(3.40). Experimentally, Atat¨ ure et al. [3.7] observed the degradation of the HongOu-Mandel dip with increasing pump bandwidth in type-II parametric downconversion, as predicted in Fig.3.7b. But, Kuzucu et al. [3.8] later repeated the same experiment with a nonlinear crystal that satisﬁes the extended phase matching condition in Eq.(3.55), and recovered the Hong-Ou-Mandel dip to the full depth in case (i) of Fig.3.7b. 3.2.3 Dispersion Cancellation One of the interesting features of the Hong-Ou-Mandel interferometer is the property of dispersion independence, ﬁrst discovered by Franson [3.9] and by Steinberg et al. [3.10], and demonstrated by Steinberg et al. [3.11]. Consider the situation when one of the photons, say, the signal photon, passes through a dispersive medium of length l before hitting the beam splitter. The ﬁeld operator after the medium is then: ˆ (in) (t) = √1 E dω1 a ˆ1 (ω1 )eik(ω1 )l e−ω1 (t−z1 /c) , (3.57) 1 2π where the extra phase k(ω1 )l is due to the dispersive medium. After substituting Eq.(3.57) into Eqs.(3.17, 3.18), we obtain G(2) (t1 , t2 ) in the same form as in Eq.(3.19), but with g(t1 , t2 ) function modiﬁed as 1 dω1 dω2 Φ(ω1 , ω2 )eik(ω1 )l e−iω1 t1 −iω2 t2 . (3.58) g˜(t1 , t2 ) ≡ 2π For the narrow band case, we still have the g˜(t1 , t2 )-function in the form of Eq.(3.29), but the f -function is changed to 1 f˜(τ ) = (3.59) dΩψ(ω0 + Ω)eik(ω0 +Ω)l e−iΩτ 2π so that the V(Δz) function in Eq.(3.34) becomes dΩψ ∗ (ω0 + Ω)ψ(ω0 − Ω)eiδk(Ω)l e−i2ΩΔz/c ˜ V(Δz) = , dΩ|ψ(ω0 + Ω)|2

(3.60)

where we have an extra phase of eiδk(Ω)l with δk(Ω) = k(ω0 − Ω) − k(ω0 + Ω) ≈ −2k Ω + o(Ω 2 ).

(3.61)

The above expression is expanded up to the Ω 2 term, but with the Ω 2 term clearly missing. Hence,

56

3 Hong-Ou-Mandel Interferometer

˜ V(Δz) = V(z1 − z2 + ck l),

(3.62)

with the function V(z) given in Eq.(3.34). So, the eﬀect of the dispersion of the medium is cancelled out. The dispersion, though, only shifts the HongOu-Mandel dip by the amount of the group velocity delay of ck l, but does not aﬀect the shape of the dip. As a matter of fact, Eq.(3.61) shows that all the even orders of dispersion are cancelled, leaving only the odd orders. For the case of wide band pumping, on the other hand, the dispersion eﬀect cannot be cancelled. As seen from Eq.(3.58), The new Φ˜ = Φ(ω1 , ω2 )eik(ω1 )l is not symmetric with respect to ω1 , ω2 . In fact, the dispersion will destroy the original symmetry and have a detrimental eﬀect on two-photon interference. However, if another identical medium is put in the other ﬁeld (the idler ﬁeld), then we have Φ˜ = Φ(ω1 , ω2 )eik(ω1 )l+ik(ω2 )l . When we apply that to Eq.(3.43), we ﬁnd the dispersion eﬀect is completely cancelled.

3.3 A Nonlocal Two-Photon Interference Eﬀect The Hong-Ou-Mandel dip is a signature of two-photon destructive interference. But why can’t it be a bump due to constructive interference? After all, an arbitrary beam splitter will have some arbitrary phase shifts at both transmission and reﬂection for an incoming wave. The answer lies in the fact that the Hong-Ou-Mandel eﬀect involves two-photon amplitude: when counting phase shifts, we need to consider the phase sum experienced by two photons together. Therefore, the phase diﬀerence Δϕ that determines whether it is destructive or constructive interference will be Δϕ = (ϕt + ϕt ) − (ϕr + ϕr ). Although all the phases involved may be arbitrary for any given beam splitter, Δϕ is always π because of energy conservation (See Appendix A for details). This means that the Hong-Ou-Mandel eﬀect always appears as a destructive interference dip. This two-photon phase sum also highlights the underlying physics in the dispersion cancellation eﬀect discussed in the previous section: the dispersive phase shift experienced by one photon is exactly the opposite of that for its conjugate counterpart, and the two-photon phase will experience no dispersive phase shift, at least in the lowest order.

z1 PDC

z2

Pump d z'2

signal idler

Fig. 3.8. The modiﬁed Hong-Ou-Mandel interferometer for a demonstration of a nonlocal eﬀect.

3.3 A Nonlocal Two-Photon Interference Eﬀect

57

The two-photon amplitude concept discussed here sometimes may lead to some startling nonlocal eﬀect if we use the ”two-photon” part of the concept, that is, the two photons may be separated. Consider an extra delay introduced at one arm, say, ﬁeld 2, of the interferometer, as depicted in Fig.3.8. Because of the extra delayed path, one may expect two dips, separated by the delay, when the path of the other arm is scanned. However, as will be seen from the following calculation, there is a third dip in the middle, which corresponds to half of the delay. Because of the extra delay line, the input ﬁeld operators in Eq.(3.16) for the beam splitter need to be modiﬁed for ﬁeld 2 as (in) ˆ (in) (t) + c2 Eˆ (in) (t − d), Eˆ2 (t) = c1 E 2 2

(3.63)

(in)

ˆ where E (t) is given in Eq.(3.16) and d ≡ z2 − z2 is the delayed distance for 2 ﬁeld 2. c1 , c2 are the probability amplitudes for ﬁeld 2 in the short path or the long path, respectively. To simplify the situation, let us assume d/c >> Tc , the coherence time of ﬁeld 2, so that there is no interference from ﬁeld 2 itself. Then, the intensity correlation function at two outputs deﬁned in Eq.(3.18) becomes, for a 50:50 beam splitter, z1 z2 z1 z G(2) (t1 , t2 ) ∝ c1 g t1 − , t2 − + c2 g t1 − , t2 − 2 − c c c z c z z2 z 2 1 1 − c2 g t2 − , t1 − 2 , (3.64) −c1 g t2 − , t1 − c c c c where g(t1 , t2 ) is given in Eq.(3.20). For simplicity, let us consider only the narrow band pumping case. From Eq.(3.29) for g(t1 , t2 ), we have: Δz Δz + c2 eiω0 z2 /c f − τ − − G(2) (t1 , t2 ) ∝ c1 eiω0 z2 /c f − τ − c c 2 Δz Δz − c2 eiω0 z2 /c f τ − −c1 eiω0 z2 /c f τ − , (3.65) c c where τ = t1 − t2 , Δz = z1 − z2 , Δz = z1 − z2 and f (τ ) is deﬁned in Eq.(3.30). Because d/c ≡ (z2 − z2 )/c >> Tc or the width of f (τ ), some of the cross terms in the expansion of the expression in Eq.(3.65) are zero. Usually the resolving time of the detectors is quite large, so that TR >> Δz/c, Δz /c, Tc . The measurable coincidence rate is an integral of G(t1 , t2 ) over τ in the range from −∞ to +∞. Then, similar to Eq.(3.32), the coincidence rate Rc is given by Rc ∝ 1 − V(Δz, Δz ),

(3.66)

with

|c1 |2 V(Δz) + |c2 |2 V(Δz ) + [2c1 c∗2 V( Δz+Δz )e−id/c ] 2 . (3.67) V(Δz, Δz ) ≡ |c1 |2 + |c2 |2

58

3 Hong-Ou-Mandel Interferometer

As expected, the ﬁrst two terms give rise to the traditional Hong-Ou-Mandel dips with the sizes of the dips proportional to the partition probabilities |c1 |2 , |c2 |2 . This eﬀect has been used to implement a scheme of quantum optical coherent tomography (QOCT) [3.12]. The advantage of QOCT over traditional optical coherent tomography (OCT) is its dispersion-independent property, as we found in the previous section. So QOCT will maintain its resolution even in the presence of dispersion.

time D1

time D1

D2

space z1 z2,z2 transmission

time

time D2

d/2c

D2 D1

space z1 z2,z2 reflection

(a) z1= z2 or z2'

d/2c

D2

D1

space z1

space

z2

z1 z2 reflection transmission (b) z1 - z2 = d/2 and z1 - z2' = −d/2

Fig. 3.9. Time line for the modiﬁed Hong-Ou-Mandel interferometer: (a) a regular situation (b) a nonlocal situation.

Surprisingly, however, there is a third term in Eq.(3.67) that is nonzero around Δz + Δz = 0 or z1 = (z2 + z2 )/2. It is right at the midway between the two expected dips. The unexpected term can be understood if we take a detailed look at the time line of photo-detection in Fig.3.9 of the modiﬁed Hong-Ou-Mandel interferometer in Fig.3.8. The ﬁgure shows the interference of two possibilities: both photons are transmitted or reﬂected. Fig.3.9a shows the traditional Hong-Ou-Mandel interference eﬀect when both photons arrive in the beam splitter at the same time. In Fig.3.9b, z2 − z2 ≡ d is the path diﬀerence between the two arms of the interferometer and photo-detections of the two photons are not simultaneous but have a time diﬀerence of Δτ = d/2c. The two overlapping possibilities are from two diﬀerent cases: two photons are separated by a delay of z1 − z2 or z2 − z1 . The two possibilities are coherent to each other and will produce interference. It can be seen that the twophoton wave packets overlap at the two detectors when z1 − z2 = z2 − z1 or Δz = −Δz = d/2. Since the interference is between the shorter path and the longer path, the path diﬀerence d appears as a phase of eiω0 d/c in the third term of Eq.(3.67). Depending on the delay, we may have a dip (d < λ0 /2) or a bump (d > λ0 /2), as shown in Fig.3.10. We will have more discussion on the phase dependent two-photon interference eﬀects in Chapter 5. Note that in Fig.3.9b the two photons never meet at the beam splitter and yet there still exists the seemingly “nonlocal” interference eﬀect. The two-photon amplitudes, on the other hand, do overlap at the detectors so the detectors cannot distinguish which path the two photons take, and thus,

59

Coincidence Rate

3.4 Photon Bunching in Hong-Ou-Mandel Interferometer

z1

(z1+z'1)/2

z'1

z2

Fig. 3.10. Coincidence rate as a function of z1 for QOCT.

interference occurs. This further demonstrates the statement that the twophoton interference eﬀect is not the interference between the two photons, but the interference of the two photons with themselves. A similar eﬀect was observed with a mode-locked two-photon state [3.13]. This type of nonlocal two-photon interference was ﬁrst discovered by Pittman et al. [3.14] in a polarization Hong-Ou-Mandel interferometer that will be discussed in the next chapter.

3.4 Photon Bunching in Hong-Ou-Mandel Interferometer In all the previous discussions of the Hong-Ou-Mandel interferometer, we place the two detectors at the opposite sides of the beam splitter. Destructive twophoton interference leads to the disappearance of the |1, 1 state in the output state in Eq.(3.6), and a dip in the coincidence measurement between the two detectors occurs as the relative delay between the two input photons is scanned. Let us now concentrate on the |2, 0 and |0, 2 terms in Eq.(3.6), from which we ﬁnd the probability of discovering the two photons in one side, say, b1 side, is P2 (b1 ) = 1/2. On the other hand, if the two photons were independent (cl) classical particles, the probability would be P2 (b1 ) = (1/2) × (1/2) = 1/4. The enhancement factor of two in P2 (b1 ) is thus due to the wave nature of photons or the interference eﬀect. In this case, it is always constructive interference. Although it is not as straightforward to visualize the constructive interference in Fig.3.2 as it is the destructive interference, the reason for the constructive interference can be revealed in Eq.(1.8), together with Fig.1.4C and Fig.1.4D. When both detectors are on the same side of the beam splitter, the situation is the same as in cases (C) and (D) in Fig.1.4, but with x1 = x2 (2) in Eq.(1.8). We then obtain G12 = 4A4 . However, there is no interference for two classical particles, i.e., there is no cosine-term in Eq.(1.8) for the

60

3 Hong-Ou-Mandel Interferometer (2)

classical particles. So, we have G12 (cl.) = 2A4 and an enhancement factor of (2) (2) G12 /G12 (cl.) = 2. Perhaps a better way to visualize the situation is depicted in Fig.3.11, where there are two indistinguishable possibilities to arrange the two photons with two detectors, leading to constructive interference.

(a)

(b)

Fig. 3.11. Two possibilities in two-photon constructive interference for photon bunching eﬀect.

Of course, our discussion so far is of a single-mode. Multi-mode analysis of the situation is similar to Sect.3.2. Here, we present only the case of narrow band pumping. For the detection scheme in Fig.3.11, we have the two-photon correlation function as: (2) ˆ (o)† (t1 )E ˆ (o)† (t2 )E ˆ (o) (t2 )E ˆ (o) (t1 ), (3.68) G (t1 , t2 ) = E 11

1

1

1

1

instead of Eq.(3.18), and with the quantum state in Eq.(3.7) and ﬁeld operators in Eq.(3.17), we obtain: (2) G11 (t1 , t2 ) = |ξ|2 T g(t1 − z1 /c, t2 − z2 /c)+ 2 +Rg(t2 − z1 /c, t1 − z2 /c) . (3.69) Note that the only diﬀerence between Eq.(3.19) and Eq.(3.69) is the sign in the second term. This is the main the diﬀerence between destructive and constructive interference. For narrow band pumping, we have g(t1 , t2 ) in Eq.(3.29) and the two-photon coincidence rate becomes 2T R V(Δz), (3.70) + R2 which gives a bump instead of a dip, as Δz is scanned. This bump in the HongOu-Mandel interferometer was ﬁrst observed by Rarity and Tapster [3.3]. It is interesting to note that the excess coincidence between the two detectors resembles the Hanbury-Brown and Twiss photon bunching eﬀect from a thermal source [3.15]. As a matter of fact, Glauber [3.16] explained the photon bunching eﬀect using exactly the two-photon interference picture in Fig.3.11. We will discuss more of the connection between the two eﬀects in Sect.7.2. A similar bunching eﬀect in a four-photon case will be studied in Sect.8.4.1 and the more general N -photon case will be addressed in Sect.9.2.2. R11 (Δz) ∝ 1 +

T2

References

61

References 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16

C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). H. Fearn and R. Loudon, Opt. Commun. 64, 485 (1987). J. G. Rarity and P. R. Tapster, J. Opt. Soc. Am. B 6, 1221 (1989). M. H. Rubin, D. N. Klyshko, Y. H. Shih, and A. V. Sergienko, Phys. Rev. A 50, 5122 (1994). A. V. Sergienko, Y. H. Shih, and M. H. Rubin, J. Opt. Soc. Am. B 12, 859 (1995). W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997). M. Atat¨ ure, A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, Phys. Rev. Lett. 83, 1323 (1999). O. Kuzucu, M. Fiorentino, M. A. Albota, F. N. C. Wong, and F. X. K¨ artner, Phys. Rev. Lett. 94, 083601 (2005). J. D. Franson, Phys. Rev. A 45, 3126 (1992). A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. A 45, 6659 (1992). A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Phys. Rev. Lett. 68, 2421 (1992). M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 91, 083601 (2003) Y. J. Lu, R. L. Campbell, and Z. Y. Ou, Phys. Rev. Lett. 91, 163602 (2003). T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 77, 1917 (1996). R. Hanbury-Brown and R. W. Twiss, Nature 177, 27 (1956). R. J. Glauber, Quantum Optics and Electronics (Les Houches Lectures), p.63, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965).

4 Phase-Independent Two-Photon Interference

One striking feature in the Hong-Ou-Mandel interferometer is its phase independent nature. The reason for this is the overlap of the two interference paths for the two photons, as seen in Fig.3.2a,b. But, because of the involvement of two photons, even though their paths overlap, there may still be some minute diﬀerences between the two photons, resulting in adiﬀerence in the two overlapping paths. In this chapter, we will discuss some additional two-photon interference phenomena that are independent of phase changes. We start with the polarization Hong-Ou-Mandel interferometer.

4.1 Two-Photon Polarization Entanglement Two-photon polarization entangled states have played an important role in the demonstration of locality violation by quantum mechanics via the violations of Bell’s inequalities. As we noted at the beginning of Chapt.1, polarization entangled states emerge naturally from atomic cascade and many demonstrations have been performed in that system. However, as was ﬁrst pointed out by Chubarov and Nikolayev [4.1], photon sources with sub-Poissonian statistics can be utilized for a demonstration of locality violation via Bell’s inequalities. Later, it was independently proposed by Alley and Shih [4.2] and by Ou, Hong, and Mandel [4.3] that the two photons from parametric down-conversion are, indeed, such sources for creating a polarization entangled state for the demonstration of locality violation. Experiments were ﬁrst performed by Shih and Alley [4.2, 4.4] and by Ou and Mandel [4.5] to implement these ideas. 4.1.1 Polarization Hong-Ou-Mandel Interferometer and Violation of Bell’s Inequalities In order to generate the polarization entangled two-photon state of the form

64

4 Phase-Independent Two-Photon Interference

1 |Ψ = √ |x1 |y2 + |y1 |x2 , 2

(4.1)

we consider a polarization Hong-Ou-Mandel interferometer (Fig.4.1), where the two photons entering the beam splitter have orthogonal polarizations, that is, |Ψin = |xa1 |ya2 . Because of the involvement of the polarization degree, diﬀerent polarization modes have diﬀerent phase shifts. If we continue to use Eq.(3.2) for x-polarization, that is, ⎧ √ √ ⎨ ˆb1x = Tx a ˆ1x + Rx a ˆ2x , (4.2) √ √ ⎩ˆ b2x = Tx a ˆ2x − Rx a ˆ1x , then, for the y-polarization, we have ⎧ ⎨ ˆb1y = Ty a ˆ1y − Ry a ˆ2y , ⎩ˆ ˆ2y + Ry a ˆ1y , b2y = Ty a

(4.3)

where the extra π phase shift for the reﬂected y-polarization stems from the Fresnel coeﬃcients for transmission and reﬂection (See Ref.[4.6] and Appendix A). From Appendix A, we ﬁnd the output state to be: Rx Ry |y |Ψout = Tx Ty |x b1 |yb1 + b1 |xb2 − (4.4) − Tx Ry |xb1 |yb1 − Rx Ty |xb2 |yb2 . For a 50:50 beam splitter, Eq.(4.4) becomes: 1 |Ψout = |xb1 |yb2 + |yb1 |xb2 − |xb1 |yb1 − |xb2 |yb2 . 2

(4.5)

If coincidence measurement is made at two outputs of the beam splitter, only the ﬁrst two terms will contribute to the coincidence count, and the state in Eq.(4.5) will be projected into the entangled state in Eq.(4.1).

| y a2 a^2x,y

BS | x a1

a^1x,y

^

b1x,y

^

b2x,y

Tx,y , Rx,y

Fig. 4.1. Polarization Hong-Ou-Mandel interferometer.

In order to conﬁrm this, let us evaluate the polarization correlation function of one photon polarized in the θ1 -direction and the other in θ2 :

4.1 Two-Photon Polarization Entanglement

P (θ1 , θ2 ) = ˆb†2 (θ2 )ˆb†1 (θ1 )ˆb1 (θ1 )ˆb2 (θ2 ),

65

(4.6)

where ˆb1(2) (θ1 or θ2 ) is the annihilation operator after the polarizer of θ1 (θ2 ) orientation in the output side of 1 (or 2) and is related to ˆb1x , ˆb1y or ˆb2x , ˆb2y by ˆb1 (θ1 ) = ˆb1x cos θ1 + ˆb1y sin θ1 , ˆb2 (θ2 ) = ˆb2x cos θ2 + ˆb2y sin θ2 .

(4.7) (4.8)

For the output state in Eq.(4.5), we ﬁnd easily that P (θ1 , θ2 ) =

1 sin2 (θ1 + θ2 ). 2

(4.9)

The invariance of P (θ1 , θ2 ) as a function of θ1 + θ2 is typical with the entanglement in Eq.(4.1). This is so because Eq.(4.1) is invariant if we rotate photon 1’s base of polarization by an angle of θ and photon 2’s base by an angle of −θ with x ˆ1 = xˆ1 cos θ + yˆ1 sin θ (4.10) yˆ1 = −ˆ x sin θ + yˆ cos θ, and

x ˆ2 = x ˆ2 cos θ − yˆ1 sin θ yˆ2 = x ˆ2 sin θ + yˆ2 cos θ.

(4.11)

We can check this easily by solving {x, y} in terms of {x , y } from Eqs.(4.10, 4.11) and substituting them into Eq.(4.1). Then, Eq.(4.1) becomes: 1 |Ψ = √ |x 1 |y 2 + |y 1 |x 2 . 2

(4.12)

The above situation is the ideal single-frequency mode case for the two orthogonally polarized photons. In practice, if we use the two photons from parametric down-conversion, they have very wide bandwidths and each photon may be viewed as a ﬁnite size wave packet. In order to obtain the polarization entangled state in Eq.(4.5), the two wave packets need to overlap at the beam splitter. This is very similar to the Hong-Ou-Mandel interferometer. As a matter of fact, the case when θ1 = −θ2 = 45◦ corresponds exactly to the HongOu-Mandel interferometer since we have P (θ1 , −θ1 ) = 0. To understand this in a more clear physical picture, we just need to notice the fact that because of the mirror imaging eﬀect of the beam splitter, θ1 = −θ2 corresponds to the same direction of polarization for the two opposite sides of the beam splitter. Thus, the polarization correlation measurement at θ1 = −θ2 is simply a projection measurement of the x- and y-polarized photon, along the same polarization direction given by θ1 = −θ2 . This is equivalent to the same polarization input at the beam splitter, which is exactly the situation in a

66

4 Phase-Independent Two-Photon Interference

Hong-Ou-Mandel interferometer. Of course, the maximum projection occurs at θ1 = −θ2 = 45◦ for both x- and y-polarized photons. We can conﬁrm the above simple argument by calculating the multi-mode correlation function Γ (2) (θ1 , t1 ; θ2 , t2 ) ˆ (o)† (θ1 , t1 )Eˆ (o) (θ1 , t1 )E ˆ (o) (θ2 , t2 ), ˆ (o)† (θ2 , t2 )E = E 2 1 1 2

(4.13)

where from Eqs.(4.2, 4.3, 4.7, 4.8), we have with Tx = Rx = Ty = Ry = 1/2 √ ˆ1x (t) cos θ1 − Eˆ2y (t) sin θ1 ]/ 2, ˆ (o) (θ1 , t) = [E E 1 (4.14) √ , ˆ (o) (θ2 , t) = [−E ˆ1x (t) cos θ2 + E ˆ2y (t) sin θ2 ]/ 2 E 2 where

ˆ1x (t) = √1 dωs a ˆs (ωs )e−iωs (t−z1 /c) , E 2π 1 ˆ E2y (t) = √ dωi a ˆi (ωi )e−iωi (t−z2 /c) 2π

(4.15) (4.16)

are the ﬁeld operators for the signal and idler photons, respectively. Here we dropped the E2x , E1y terms in Eq.(4.14) because they are in vacuum. Then, similar to Eq.(3.19), we have: Γ (2) (θ1 , t1 ; θ2 , t2 ) =

|ξ|2 g(t1 − z1 /c, t2 − z2 /c) sin θ1 cos θ2 2 2 +g(t2 − z1 /c, t1 − z2 /c) sin θ2 cos θ1 ,

(4.17)

where g(t1 , t2 ) is given in Eq.(3.20). Similar to Eq.(3.41) in Sect.3.2.2, after carrying out the time integral, we obtain: P2 (θ1 ; θ2 ) ∝ sin2 θ1 cos2 θ2 + sin2 θ2 cos2 θ1 +2V(Δz) sin θ1 cos θ2 sin θ2 cos θ1 ,

(4.18)

where V(Δz) is given in Eq.(3.43). When θ1 = −θ2 , Eq.(4.18) is same as Eq.(3.42) with T = R, which is the Hong-Ou-Mandel interference eﬀect. On the other hand, when Δz = 0 and Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ), we have V(0) = 1 and Eq.(4.18) becomes P2 (θ1 ; θ2 ) ∝ sin2 (θ1 + θ2 ),

(4.19)

which is same as the single-mode result in Eq.(4.6). The polarization correlation in the form of Eqs.(4.6, 4.19) is exactly what is needed in the violation of Bell’s inequalities and it was demonstrated by Shih and Alley [4.4] and by Ou and Mandel [4.5].

4.1 Two-Photon Polarization Entanglement

67

4.1.2 Photon Anti-Bunching Eﬀect and Bell State Measurements The photon bunching eﬀect, i.e., the tendency that both photons go together to either side of the beam splitter, in the Hong-Ou-Mandel interferometer is often thought to be a consequence of the Bosonic nature of photons, which requires that the global wave function be symmetric, with respect to the two photons. When input photons have the same polarizations and, thus, have symmetric internal wave function, their spatial wave function must also be symmetric in order to maintain a symmetric overall wave function. Therefore, when a beam splitter is used to combine the photons together, they will naturally fall into the same spatial modes: either to the one side or the other of the beam splitter. On the other hand, if we let an anti-symmetric polarization state of the form 1 (4.20) |Ψin = √ |xa1 |ya2 − |ya1 |xa2 2 enter a 50:50 beam splitter, the two photons’ spatial wave function must also be anti-symmetric in order to have a symmetric global wave function. This will lead to a photon anti-bunching eﬀect at the beam splitter, i.e., the two photons will go their separate ways to two sides of the beam splitter because the Fermionic behavior of an anti-symmetric wave function does not allow two particles to occupy the same mode. The above symmetry argument is valid only if the beam splitter is a polarization-preserving beam splitter that has the same action for all polarizations of input ﬁeld, that is, the input-output relation for a 50:50 beam splitter must have the following form: √ √ ˆb1x = (ˆ a1x + a ˆ2x )/ 2, ˆb2x = (ˆ a2x − a ˆ1x )/ 2, (4.21) for the x-polarization and √ ˆb1y = (ˆ a1y + a ˆ2y )/ 2,

√ ˆb2y = (ˆ a2y − a ˆ1y )/ 2,

(4.22)

for the y-polarization. Note that there is no extra π-phase shift in the reﬂected y-polarization, as compared to Eq.(4.3). A metallic beam splitter usually preserves the polarization. To conﬁrm the above symmetry argument, let us consider the antisymmetric input state in Eq.(4.20) for a 50:50 polarization-preserving beam splitter. From Appendix A, we may derive the following state changes for the operator relationship in Eqs.(4.21, 4.22): |xa1 |ya2 → and

1 |xb1 |yb2 − |yb1 |xb2 + |xb1 |yb1 − |xb2 |yb2 , 2

(4.23)

68

4 Phase-Independent Two-Photon Interference

|ya1 |xa2 →

1 |yb1 |xb2 − |xb1 |yb2 + |xb1 |yb1 − |xb2 |yb2 . 2

Combining Eqs.(4.23, 4.24), we obtain the output state as 1 |Ψout = √ |xb1 |yb2 − |yb1 |xb2 . 2

(4.24)

(4.25)

Notice that the output photons go their separate ways at two sides of the beam splitter. This is typical Fermionic behavior, consistent with the antisymmetric input state in Eq.(4.20). The input state in Eq.(4.20) belongs to a class of polarization entangled states, called the Bell states [4.7], which can violate the Bell’s inequalities by the maximum amount. The complete set of Bell states are denoted as 1 |Ψ (±) = √ |xa1 |ya2 ± |ya1 |xa2 , (4.26) 2 1 (4.27) |Φ(±) = √ |xa1 |xa2 ± |ya1 |ya2 . 2 So |Ψin in Eq.(4.20) is simply |Ψ (−) with an output state in Eq.(4.25). Notice that the other Bell states are all symmetric states with respect to the two photons. Therefore, if they are input to a polarization-preserving beam splitter, the output state will show the Bosonic photon bunching eﬀect. This can be conﬁrmed from their output states: 1 (+) |Ψout = √ |xb1 |yb1 − |xb2 |yb2 , (4.28) 2 1 (±) |2xb1 − |2xb2 ± |2yb1 ∓ |2yb2 . (4.29) |Φout = 2 In the above states, both photons go to either side b1 or b2 together, i.e., they bunch together. The Bell states form an orthonormal base set for the polarization state of two photons in two separate spatial modes (a1 , a2 ). An arbitrary state of the form |Θ(x, y)a1a2 = c1 |xa1 |xa2 + c2 |xa1 |ya2 +c3 |ya1 |xa2 + c4 |ya1 |ya2

(4.30)

can be decomposed into the superposition of the Bell states: |Θ(x, y)a1a2 = c+ |Ψ (+) + c− |Ψ (−) + d+ |Φ(+) + d− |Φ(−) .

(4.31)

A Bell measurement then measures the input state |Θ(x, y) in the Bell state base and projects the input state into one of the four Bell states. The scheme depicted in Fig.4.2 will realize a partial Bell measurement [4.8, 4.9]. Using the traditional photon coincidence measurement technique, we can distinguish |Ψ (−) from other Bell states by the photon anti-bunching eﬀect:

4.1 Two-Photon Polarization Entanglement

69

referring to Fig.4.2, if there is a coincidence between either A and C or B and D, we have a projection to the |Ψ (−) state. We can further distinguish |Ψ (+) from |Φ(±) by polarization beam splitters after the ﬁrst beam splitter: if there is a coincidence between either A and B or C and D, we have a projection to the |Ψ (−) state. However, it seems that we cannot distinguish between the |Φ(+) and |Φ(−) states by coincidence measurement in this scheme.

|Θ (x, y)

a2

a1a2

x

a1

y

C y x

A

B

D

Fig. 4.2. A partial Bell state measurement with a beam splitter and two polarization beam splitters. Coincidence measurement is performed on all pairs of the four detectors.

A dielectric beam splitter, however, does not preserve polarization because of the extra π-phase shift in the reﬂected y-polarization. The above simple symmetry argument fails. Nevertheless, it is straightforward to directly derive the output states for the Bell state input, with the help of the operator relationship in Eqs.(4.2, 4.3). The results are: 1 (+) (4.32) |Ψout = √ |xb1 |yb2 + |yb1 |xb2 , 2 1 (−) (4.33) |Ψout = − √ |xb1 |yb1 + |xb2 |yb2 , 2 1 (±) (4.34) |Φout = |2xb1 − |2xb2 ± |2yb2 ∓ |2yb1 . 2 (±)

(±)

The output states of |Ψout are just switched, as compared to |Ψout . So, we (±) (±) are still able to distinguish |Ψout , but |Φout is now similar to Eq.(4.29) and (±) we are unable to distinguish |Φout as before. Since the Bell states form a base set for the two-photon polarization space, according to the quantum measurement theory, Bell state measurement will deﬁne a physical observable that has four possible eigenvalues corresponding to the projections to the four Bell states in Eq.(4.26) and Eq.(4.27). Bell state measurement is the key ingredient in quantum state teleportation [4.8, 4.10, 4.11] (see more in Sect.8.2) and in dense coding [4.12]. Since we cannot distinguish between |Φ(+) and |Φ(−) states, the measurement scheme in Fig.4.2 gives only an incomplete Bell measurement with a 50% success rate. Some other schemes are proposed and realized by Kwiat and Weinfurter [4.13], by Walborn et al. [4.14], and by Schuck et al. [4.15], who utilized hyper-entanglement in polarization and space-time to achieve a complete Bell measurement.

70

4 Phase-Independent Two-Photon Interference

4.2 Two-Photon Frequency Entanglement and Spatial Beating In the Hong-Ou-Mandel interferometer, the two photons entering the beam splitter have the same frequency, which is ensured experimentally by placing identical interference ﬁlters centered at ωp /2, the half frequency of the pump ﬁeld. Indistinguishability leads to quantum interference. When the two photons have a diﬀerent center frequency, distinguishability in color usually washes out the interference eﬀect, unless the response band widths of the detectors are much wider than the frequency diﬀerence and the interference fringe manifests as a beat note in time. On the other hand, if the two photons are in a frequency entangled state of the form 1 |Ψ (ω1 , ω2 ) = √ |ω1 a1 |ω2 a2 − |ω2 a1 |ω1 a2 , 2

(4.35)

the output from a Hong-Ou-Mandel interferometer will show a spatial beating eﬀect, as seen in the following.

signal Pump

PDC

z1

f2(ω2) D2

|Ψ (ω1 ,ω2)

idler z 2

cδτ f1(ω1)

D1

Fig. 4.3. Hong-Ou-Mandel interferometer with frequency entangled input state for the spatial beating eﬀect.

Consider the situation in Fig.4.3 where the setup is basically a HongOu-Mandel interferometer but with ﬁlters of diﬀerent frequencies in front of the detectors. From experience based on second-order interference, one might intuitively expect a reduced interference eﬀect because of distinguishability in the frequency of the photons. However, with the quantum state in Eq.(4.35), the detectors that detect the (ω1 , ω2 ) coincidence cannot tell which terms in Eq.(4.35) contribute to the coincidence (Fig.4.3). This indistinguishability leads to two-photon interference. Although Fig.4.3 is similar to Fig.3.3, there is an additional two-photon phase diﬀerence between the transmitted and reﬂected two-photon waves, besides the π-phase shift from the beam splitter (Fig.3.2). This extra twophoton phase diﬀerence is caused by the frequency diﬀerences of the two photons when the beam splitter is displaced from the symmetric position by an amount Δz = cδτ and is equal to ω1 z1 /c + ω2 z2 /c − ω2 z1 /c − ω1 z2 /c = (ω1 − ω2 )δτ . As the beam splitter moves away from the symmetric position of δτ = 0, this extra phase increases from 0 to π and eventually compensates for the π phase diﬀerence caused by the beam splitter. Then, destructive interference changes to constructive interference for the two possibilities. As

4.2 Two-Photon Frequency Entanglement and Spatial Beating

71

the position of the beam splitter is moved, cδτ increases continuously and destructive and constructive interference occur alternatively with a period related to the beat frequency |ω1 − ω2 |. The result is a beat in the space domain — spatial beating. Quantitatively, we may calculate the joint probability for detecting photons at frequency ω1 by D1 and at frequency ω2 by D2, for the frequency entangled state in Eq.(4.35). Since there are two frequency components, the input ﬁeld operators for the beam splitter have the form ˆ1 (t) = a E ˆ1 (ω1 )e−iω1 t + a ˆ1 (ω2 )e−iω2 t , (4.36) −iω1 t ˆ ˆ2 (ω1 )e +a ˆ2 (ω2 )e−iω2 t , E2 (t) = a and then the output ﬁeld operators after the beam splitter become

√ ˆ1 (t) + E ˆ2 (t + δτ ) / 2, Eˆ1out (t) = E

√ ˆ2 (t) − E ˆ1 (t − δτ ) / 2, Eˆ out (t) = E

(4.37)

2

where cδτ gives the distance away from the balanced symmetric position for the beam splitter. With the ﬁlters at each detector, the ﬁeld operators at detectors D1 and D2 have the form:

√ ˆD1 (t) = a ˆ1 (ω1 )e−iω1 t + a ˆ2 (ω1 )e−iω1 (t+δτ ) / 2, E

√ (4.38) ˆD2 (t) = a ˆ2 (ω2 )e−iω2 t − a ˆ1 (ω2 )e−iω2 (t−δτ ) / 2. E So, for the state in Eq.(4.35), the joint two-photon detection probability can be calculated as † ˆ † (t)EˆD1 (t)EˆD2 (t ) P (δτ ) ∝ EˆD2 (t )E D1 2 1 −iω1 t −iω2 t e − e−iω1 (t+δτ ) e−iω2 (t −δτ ) = e 8 1 = [1 − cos(ω1 − ω2 )δτ ]. 4

(4.39)

Indeed, this shows a beat note in the variable δτ , the spatial delay. This result is consistent with the qualitative argument above. In practice, however, one never has a state as simple as that given by Eq.(4.35). For the multi-mode case, let us consider the general two-photon state given in Eq.(3.7). This state has frequency entanglement in a wide spectrum, determined by the function Ψ (ω1 , ω2 ). If we place two ﬁlters in front of D1 and D2 of diﬀerent central frequency ω10 , ω20 , only two frequency bands around Ψ (ω10 , ω20 ) and Ψ (ω20 , ω10 ) will contribute to coincidence measurement leading to a frequency entangled state similar to that in Eq.(4.35). To calculate the coincidence rate in this case, our treatment is similar to that in the Hong-Ou-Mandel interferometer in Sect.3.2. We can still use Eq.(3.17) for the ﬁeld operators after the beam splitter. But, the ﬁeld operators after the ﬁlters need to be modiﬁed according to Eq.(3.23) for ﬁlter transmission. Now, the two ﬁlters are diﬀerent in central frequencies:

72

4 Phase-Independent Two-Photon Interference

fm (ω) = e−(ω−ωm0 )

2

/2σ 2

(m = 1, 2).

(4.40)

Here, we assume a Gaussian shape of the same width for the two ﬁlters. The ﬁeld operators at the detectors then have the following form: √ √ ˆ (f 1) (r1 , t) + REˆ (f 1) (r2 , t), ˆD1 (t) = T E E 1 √ √ 2 (4.41) ˆD2 (t) = T E ˆ (f 2) (r2 , t) − REˆ (f 2) (r1 , t), E 2

1

where

ˆ (f m) (r1 , t) = E 1 (f m) ˆ E (r2 , t) = 2

√1 2π √1 2π

dωfm (ω)ˆ a1 (ω)e−iω(t−z1 /c) dωfm (ω)ˆ a2 (ω)e−iω(t−z2 /c)

(m = 1, 2), (4.42)

and where we dropped the vacuum operators, which give no contribution to photon counting. The two-photon correlation function Γ (2) can be calculated and has the following form: Γ (2) (t1 , t2 ) = |ξ|2 T g1 (t1 − z1 /c, t2 − z2 /c)− 2 (4.43) −Rg2 (t2 − z1 /c, t1 − z2 /c) , where

g1 (t1 , t2 ) = g2 (t1 , t2 ) =

dω1 dω2 f1 (ω1 )f2 (ω2 )Ψ (ω1 , ω2 )e−iω1 t1 −iω2 t2 , dω1 dω2 f1 (ω1 )f2 (ω2 )Ψ (ω2 , ω1 )e−iω2 t1 −iω1 t2 .

(4.44)

If σ << ΔωP DC , Ψ (ω1 , ω2 ) can be approximated by a constant and Eq.(4.44) becomes g1 (t1 , t2 ) = Ψ (ω10 , ω20 )F (t1 )F (t2 )e−iω10 t1 −iω20 t2 , g2 (t1 , t2 ) = Ψ (ω20 , ω10 )F (t1 )F (t2 )e−iω20 t1 −iω10 t2 ,

(4.45)

with F (t) ∝ e−(σt)

2

/2

.

(4.46)

Substituting the above into Eq.(4.43) and integrating over t1 , t2 , we have, similar to Eq.(3.28): ∞ Nc ∝ dt1 dt2 Γ (2) (t1 , t2 ) −∞

∝ 1−

2T R −(σδτ )2 /2 e cos[(ω10 − ω20 )δτ + Δψ], T 2 + R2

(4.47)

where T = T |Ψ (ω10 , ω20 )|, R = R|Ψ (ω20 , ω10 )|, δτ = (z1 − z2 )/c, and Δψ = arg[Ψ (ω10 , ω20 )] − arg[Ψ (ω20 , ω10 )]. Fig.4.4 plots Nc as a function of

4.3 Two-Photon Interference Fringes and Ghost Fringes

73

Normalized Coincidence

Δz = z1 −z2 and shows clearly the spatial beating eﬀect, as concluded from the simple two-mode calculation. The asymmetry in Fig.4.4 is a result of Δψ = 0. The exponential contour gives the size of the wave packet for each photon. This eﬀect of spatial beating was ﬁrst demonstrated by Ou and Mandel [4.16] with type-I parametric down-conversion, and later by Ou et al. [4.17] with classical source of laser. A similar eﬀect with type-II parametric down-conversion was demonstrated by Shih and Sergienko [4.18]. More recently, Legero et al. [4.19] observed a similar two-photon beating eﬀect in time with two photons from independent single-photon sources.

2.0 1.5 1.0 0.5 -30

-20

-10

0

∆z

10

20

30

Fig. 4.4. Two-photon coincidence rate as a function of the relative delay Δz = cδτ for the spatial beating effect.

The beating eﬀect discussed here is quite diﬀerent from traditional beating eﬀects, which are in the time domain, i.e., the measured quantity oscillates in time. Those eﬀects are non-stationary and the observation requires a fast detection system, because the beat is usually very fast. The beating eﬀect discussed here, however, does not depend on time, although it does depend on the time delay δτ = (z1 − z2 )/c between the signal and idler photons. The time delay is actually related to the optical path diﬀerence, which is a spatial quantity. Therefore, the beating is displayed in space and the beat pattern is stationary. Although the beat corresponds to two diﬀerent frequencies, the observed interference eﬀect is not an interference between diﬀerent photons with different frequencies; instead, the interference is between two two-photon waves, i.e., between the pair of (ω1 , ω2 ) and the pair of (ω2 , ω1 ). In fact, the counting rate of each detector does not exhibit any beat.

4.3 Two-Photon Interference Fringes and Ghost Fringes Interference eﬀects are most familiar as fringe patterns, probably because the ﬁrst interference experiment by Thomas Young involved a fringe pattern and most interference phenomena appear as various kinds of beautiful fringe patterns. Thus far, we have only brieﬂy discussed the two-photon interference fringe pattern in the Pﬂeegor-Mandel experiment in Chapt.1. But we have

74

4 Phase-Independent Two-Photon Interference

not dealt with the spatial or directional correlation between the two conjugate photons from parametric down-conversion. In this section, we will take this correlation into account and reveal two-photon interference fringe patterns and some related eﬀects. 4.3.1 Two-Photon Interference Fringes To produce two-photon interference fringes, let us go back to the Hong-OuMandel interferometer. Only now we realign the two incoming ﬁelds so that they form a small angle Δθ, after the beam splitter as shown in Fig.4.5. So, the superposed ﬁelds do not overlap exactly with regard to direction. Therefore, interference fringe patterns may form in the detector planes A and B with a fringe spacing of L = c/2πω0 Δθ. For two-photon interference, the situation can again be understood in terms of the two-photon wave picture introduced in Sect.1.2. There are two possible ways for the two photons to be detected by the two detectors at planes A and B, as shown in Fig.4.5 by the solid and dashed lines, respectively. These are similar to Figs.3.2a,b and there is a π two-photon phase diﬀerence for the two possibilities. In addition to this π phase diﬀerence, an extra space dependent two-photon phase diﬀerence is introduced because the two superposed ﬁelds propagate in slightly diﬀerent directions. A careful examination of the two-photon phases in the two possible paths, illustrated in Fig.4.5, shows that this extra phase diﬀerence is (k1 · r1 + k2 · r2 ) − (k2 · r1 + k1 · r2 ) = (k1 − k2 ) · r1 − (k1 − k2 ) · r2 = 2π(x1 − x2 )/L, where k1 , k2 are the mirror images of k1 , k2 , respectively and x1 , x2 are the coordinates along k1 − k2 and k1 − k2 , respectively. L = λ0 /Δθ is the fringe spacing. Therefore, the interference of the two possibilities (two two-photon waves) produces a space dependent coincidence rate for the two detectors D1 and D2, depending on the extra phase diﬀerence. This is a two-photon interference fringe pattern.

k2 r20 r10

k1

x1 ∆θ

x2

r1

k1

∆θ

k2

k2' A

k 1' r2 B

Fig. 4.5. Hong-Ou-Mandel interferometer with a slight misalignment for twophoton interference fringe and nonlocal spatial correlation eﬀect.

To conﬁrm the above argument, let us consider a two-photon quantum state of the simple form

4.3 Two-Photon Interference Fringes and Ghost Fringes

|Ψ = |1k1 |1k2 ,

75

(4.48)

that is, one photon from each direction entering the beam splitter. The ﬁeld operators at the output can be written as √ √ ik1 ·r1 ˆ1 (r1 ) = T a ak2 eik2 ·r1 , E √ ˆk1 eik ·r + √Rˆ (4.49) ˆ2 (r2 ) = T a ˆk2 e 2 2 − Rˆ ak1 eik1 ·r2 , E where we drop the time dependent part and k1 , k2 are mirror reﬂections of k1 , k2 . † ˆ We can easily ﬁnd the intensities Im (rm ) ≡ Eˆm Em (m = 1, 2) at the two locations as I1 (r1 ) = I2 (r2 ) = T + R = 1.

(4.50)

Therefore, there is no interference eﬀect in intensity because the photon number Fock state in Eq.(4.48) does not have deﬁnite phases for the two ﬁelds. For two-photon correlation, however, we have: ˆ † (r2 )Eˆ2 (r2 )E ˆ1 (r1 ) ˆ † (r1 ) E E 1 2 2 = T eik1 ·r1 eik2 ·r2 − Reik2 ·r1 eik1 ·r2

= T 2 + R2 − 2T R cos (k1 − k2 ) · r1 − (k1 − k2 )r2 2T R ∝ 1− 2 cos 2π(x1 − x2 )/L, T + R2

(4.51)

where x1 , x2 are the coordinates along the directions of k1 − k2 and k1 − k2 , respectively. Because of the mirror reﬂection, we have L ≡ 1/2π|k1 − k2 | = 1/2π|k1 − k2 | = λ0 /Δθ (Δθ is the small angle between k1 and k2 ). When T = R, Eq.(4.51) shows a two-photon interference fringe with 100% visibility as we change the position of either D1 or D2, and conﬁrms our qualitative discussion above. If we consider a more practical multi-mode situation, then the single-mode two-photon state in Eq.(4.48) is replaced by the multi-mode state in Eq.(3.7) and the ﬁeld operators at the output of the beam splitter in Eq.(4.49) are replaced by √ √ ˆ (o) (r1 , t) = T E ˆ (in) (r1 , t) + RE ˆ (in) (r1 , t), E 1 1 2 √ √ (4.52) ˆ (in) (r2 , t) − RE ˆ (in) (r2 , t), ˆ (o) (r2 , t) = T E E 2 2 1 with

1 (in) ˆ E1 (r, t) = √ dωˆ as (ω)ei[ks ·(r−r10 )−ω(t−z1 /c)] , 2π ˆ (in) (r, t) = √1 dωˆ ai (ω)ei[ki ·(r−r20 )−ω(t−z2 /c)] , E 2 2π

(4.53)

where r10 , r20 are the crossing points at which the central rays of the downconverted ﬁelds meet at the beam splitter (see Fig.4.5). z1 , z2 are the same as

76

4 Phase-Independent Two-Photon Interference

in Eq.(3.16) and are the optical path from the end of the nonlinear medium to the beam splitter, i.e., from the origin to r10 , r20 for the signal and idler ﬁelds, respectively. Here, it is equivalent to think that the origin of the coordinates is moved to r10 , r20 and z1 , z2 give the extra phases, due to this change. (in) (in) ˆm E (r, t) (m = 1, 2) is same as Eˆm (r, t) (m = 1, 2), but with km replaced by the mirror reﬂection km (m = s, i). If the signal and idler ﬁelds are collimated in certain directions, we may have ks = κ ˆ s ω/c and ki = κ ˆ i ω/c in Eq.(4.53). Note that the only diﬀerence between Eq.(4.53) and Eq.(3.16) is the spatial dependent phases ikm · (r − rn0 ) (m = s, i; n = 1, 2). Therefore, the intensity correlation between detectors D1 and D2 has the similar form as Eq.(3.19) but with extra terms in g(t1 , t2 ), resulting from the extra phases. A detailed calculation gives Γ (2) (t1 , t2 ) = |ξ|2 T g(t1 − xs1 − z1 /c, t2 − xi2 − z2 /c)− 2 (4.54) −Rg(t2 − xs2 − z1 /c, t1 − xi1 − z2 /c) , where xs1 = κ ˆ s · (r1 − r10 ), xi2 = κ ˆ i · (r2 − r20 ), xs2 = κ ˆ s · (r2 − r10 ), and xi1 = κ ˆ i · (r1 − r20 ). Carrying out the time integral, we obtain the coincidence count rate as P2 (r1 , r2 , Δz) ∝ 1 − where W (r1 , r2 , Δz) ≡ C −1

2T R [W (r1 , r2 , Δz)], T 2 + R2

(4.55)

dω1 dω2 Ψ ∗ (ω1 , ω2 )Ψ (ω2 , ω1 )×

×e−iω1 (xs1 −xi1 +Δz/c) e−iω2 (xi2 −xs2 −Δz/c) , with

C≡

dω1 dω2 |Ψ (ω1 , ω2 )|2 .

(4.56)

(4.57)

When the angle Δθ is small between κ ˆ s and κ ˆi or between κ ˆ i and κ ˆ s , we can deﬁne the unit vectors n ˆ1, n ˆ 2 by κ ˆs − κ ˆ i ≈ Δθˆ n1 and κ ˆs − κ ˆ i ≈ Δθˆ n2 . Now let us move the fast oscillating terms in the exponential function outside of the integral in Eq.(4.56): W (r1 , r2 , Δz) = e−iω0 (x1 −x2 )Δθ V (Δz), with V (Δz) ≡ C −1

(4.58)

dω1 dω2 Ψ ∗ (ω1 , ω2 )Ψ (ω2 , ω1 )e−i(ω1 −ω2 )Δz/c × ×e−i[(ω1 −ω0 )x1 −(ω2 −ω0 )x2 ]Δθ ,

(4.59)

4.3 Two-Photon Interference Fringes and Ghost Fringes

77

where xm ≡ n ˆ m · (rm − r0 ), (m = 1, 2). Here, for Δθ << 1, we have r10 ≈ r20 ≡ r0 , and n ˆ1 ⊥ κ ˆ s , κi and n ˆ2 ⊥ κ ˆ i , κs . So, x1 , x2 are the coordinates along the fringe directions in the observation planes at D1 and D2. Substituting Eq.(4.58) into Eq.(4.55), we have the two-photon interference fringes: P2 (r1 , r2 , Δz) ∝ 1 −

2T R V (Δz) cos 2π(x1 − x2 )/L, T 2 + R2

(4.60)

where L ≡ λ0 /Δθ is the fringe spacing and V (Δz) is the fringe visibility when T = R. If D1 and D2 are not far from the central rays of the signal and idler ﬁelds, we can drop the e−i[(ω1 −ω0 )x1 +(ω2 −ω0 )x2 ]θ term in Eq.(4.59) and the visibility becomes dω1 dω2 Ψ ∗ (ω1 , ω2 )Ψ (ω2 , ω1 )e−i(ω1 −ω2 )Δz/c , (4.61) V (Δz) ≈ dω1 dω2 |Ψ (ω1 , ω2 )|2 which is exactly same as Eq.(3.43) for the Hong-Ou-Mandel dip. When we plot the coincidence counts as a function of the position of, say, D1, the functionality will depend on where the other detector D2 is located. Notice that D1 and D2 are placed at two sides of the beam splitter and, therefore, are well-separated. As demonstrated by polarization entanglement and as will be seen in the future, this type of nonlocal eﬀect is intrinsic with the two-photon interference eﬀect. So far, the two detectors are located at two separate sides of the beam splitter. But what happens if the two detectors are placed on the same side? ˆ2 is replaced by E ˆ1 in Eq.(4.52). The two-photon correlation function Then, E in Eq.(4.54) is then modiﬁed as Γ (2) (t1 , t2 ) = T R|ξ|2 g(t1 − xs1 − z1 /c, t2 − xi2 − z2 /c)+ 2 (4.62) +g(t2 − xs2 − z1 /c, t1 − xi1 − z2 /c) . This is the same situation encountered in Sect.3.4 regarding the photon bunching eﬀect in the Hong-Ou-Mandel interferometer. Similar to Eq.(4.60), we have: P2 (r1 , r2 , Δz) ∝ 1 + V (Δz) cos 2π(x1 − x2 )/L,

(4.63)

where the visibility V (Δz) is given in Eq.(4.61). Now, D1 and D2 are in the same detection plane and x1 , x2 are their coordinates along n ˆ 2 direction. An interesting situation occurs when x1 = x2 ≡ x. Eq.(4.63) becomes P2 (r, Δz) ∝ 1 + V (Δz).

(4.64)

This is the same expression as in Eq.(3.70) in Sect.3.4 for the photon bunching eﬀect in the same output side of the Hong-Ou-Mandel interferometer. Experimentally, the two-photon interference fringe pattern exhibited in Eq.(4.63) was ﬁrst observed by Ghosh and Mandel [4.20] and was later conﬁrmed by Ou and Mandel using the nonlocal pattern in Eq.(4.60) [4.21].

78

4 Phase-Independent Two-Photon Interference

4.3.2 Spatial Correlation and Ghost Fringes Two-photon fringes can also be formed with a double slit as in Young’s double slit experiment. However, diﬀerent from Young’s double slit experiment, we make a two-photon detection and obtain two-photon coincidence as our outcome. Because we can separate the two photons in the measurement process, some intriguing nonlocal eﬀects will appear. PBS pump Type-II PDC

o-ray

e-ray B A Double xi Slit

xs

Fig. 4.6. The arrangement of type-II parametric down-conversion for the demonstration of twophoton ghost interference fringe.

Consider a situation depicted in Fig.4.6, where we use a collinear type-II parametric down-conversion process from a relatively thin crystal (see below for the reason). The signal and idler ﬁelds are separated by a polarization beam splitter (PBS). Let us put a double slit in the center of the signal ﬁeld (e-ray). The direction of the signal ﬁeld is perpendicular to the plane of the slit. The orientation of the slit is parallel to the polarization of the e-ray. Note that the orientation of the slit is crucial here, as we will see in the following. A detector is placed at some distance behind the slit and another one is located in the idler ﬁeld. Both the single rate from each detector and the coincidence count rate between the two detectors are measured as the transverse positions of the detectors are changed. It has been observed [4.22] that single rates are constant at both detectors, but the coincidence rate between the two detectors shows interference fringes when either one of the detectors is moved while the other is ﬁxed. It is not surprising that single rates are constant, for the down-converted ﬁelds are not point sources, but are extended across the cross section of the pump ﬁeld. The extended ﬁelds are incoherent of each other. The lack of spatial coherence at the two slits leads to the absence of interference at the single detector. What’s surprising is the interference eﬀect in the two-photon coincidence count as the detector in the idler ﬁeld is moved. Note, that there is no slit in the idler ﬁeld. This type of “ghost” interference eﬀect is a result of nonlocal spatial correlation between the signal and idler photons. Because the slit is placed across the signal ﬁeld, the ﬁelds entering the two slits will have diﬀerent directions of propagation. Since the separation of the slit is very small, we only need to consider the direction that is slightly oﬀ the collinear direction (paraxial rays). We will consider the directional or spatial correlation between the two down-converted photons. From the angular distribution (Fig.2.8a) in the phase matching of typeII parametric down-conversion, we ﬁnd that in the region near the collinear

4.3 Two-Photon Interference Fringes and Ghost Fringes

79

direction, only the horizontal direction (x-direction) away from the collinear direction is possible for phase matching, but the vertical direction (y-direction) oﬀ the collinear direction is not allowed. This is why we must put the slit orientation in the vertical direction (parallel to the e-ray). No down-converted photons will reach the two slits if placed otherwise. If the angle oﬀ the collinear direction is small, the two arches in the spatial distribution become two overlapping horizontal lines. So, the down-converted ﬁelds have ky ≈ 0 and ksx = −kix , due to momentum conservation for xdirection, or θs = −θi

(4.65)

at outside of the nonlinear medium. Note that Eq.(4.65) is exactly same as the law of reﬂection in geometric optics, that is, the idler ray is the mirror reﬂection of the signal ray or vice versa, with the end surface of the nonlinear medium as the mirror surface. Thin Crystal

o-ray e-ray

k sA k sB

o-ray

kiB ksA ksB kiA

Fig. 4.7. Geometry of spatial correlation in type-II parametric down-conversion near the collinear region.

The quantum state derived in Sect.2.1 needs to be modiﬁed to take the direction into consideration. We recall the interaction Hamiltonian in Eq.(2.25) before the one-dimensional approximation. For simplicity, we consider only the single-mode case here. So, we rewrite Eq.(2.25) in discrete mode form as ˆI = H ηq(ks , ki )ˆ a†ks a ˆ†ki + h.c., (4.66) ks ,ki

where we treat the pump ﬁeld as a classical wave and put it in the parameter η, which also include other parameters such as the nonlinear coeﬃcient. The function q(ks , ki ) is from the phase matching function and gives rise to the spatial correlation in Eq.(4.65), as shown in Fig.4.7. With the angular relation in Eq.(4.65), we may think the signal and idler beams are mirror images of each other at the crystal surface (Fig.4.7), if the crystal is short compared to the characteristic distance of the system (such as the distance between the slits and observation plane). So, the three-dimensional single-mode quantum state has the form of |Ψ = |0 + η q(ks , ki )ˆ a†ks a ˆ†ki |0. (4.67) ks ,ki

80

4 Phase-Independent Two-Photon Interference

But, according to the phase matching diagram for type-II parametric downconversion and the discussion about the spatial correlation, we may write the q-function as q(ks , ki ) ∝ δ(ksx + kix )δ(ksy )δ(kiy )δ(Δkz ),

(4.68)

for paraxial rays with |θs |, |θi | << 1 and ksx , kix change continuously, i.e., θs , θi can take any value as long as |θs |, |θi | << 1. Here, we assume the crystal length is large enough so that we can approximate the sinc-function with a δ-function. For the two-photon double slit experiment, consider a detailed geometry, as shown in Fig.4.8. The ﬁeld operator at the observation plane after the slit can be written as ˆs (rs ) = E ˆsA eikrsA + E ˆsB eikrsB , E

(4.69)

where rsA , rsB are the distance from the observation point S to the two slits ˆsA , E ˆsB are the two ﬁeld operators at the two slits, A and B, respectively. E respectively, and have the form of ⎧ iks1 ·rA ˆsA = a , ⎨E ks1 ˆ ks1 e (4.70) ⎩ˆ EsB = ks2 a ˆks2 eiks2 ·rB , where the sum is over all possible ks allowed in Eq.(4.68). Take P as the image of P with respect to the PBS and ri the image of ri , then the ﬁeld operator at the idler observation point P has the form of ˆi (ri ) = a ˆki eiki ·ri , (4.71) E ki

where ki is correlated to ks , according to the q-function in Eq.(4.68).

r

crystal surface

k sB

GB

O

r

k sA

rB B

ksB

GA

Di

kiB kiA ksA

PBS

P ri )

S(rs)

rA A P(ri)

Ds

Fig. 4.8. Detailed geometry for two-photon ghost interference fringe. GA , GB are where photon pairs are generated.

The two-photon coincidence count is thus proportional to

4.3 Two-Photon Interference Fringes and Ghost Fringes

ˆ † (r )E ˆ † (r )E ˆ (r )Eˆ (r ) Γ (2) = E i i s s s s i i 2 q(ks , ki )eiki ·ri ei(ks ·rA +krsA ) + ei(ks ·rB +krsB ) . ∝

81

(4.72)

ks ,ki

Let us consider the quantity q(ks , ki )ei(ki ·ri +ks ·rA,B ) in more detail. By Eq.(4.65), the signal and idler rays are mirror reﬂections of each other from the end surface of the crystal. So, from Fig.4.8, if A and ksA are the images of A and ksA with respect to the end surface of the crystal, then by mirror symmetry, we have ksA = −kiA and A GA is the same line as GA P , i.e., A GA P is a straight line, so that ksA · rA = ksA · rA = krA O with O as the projection of the origin O on the A P line. With kiA · ri = krO P , we then have ksA · rA + kiA · ri = krA O + krO P = krA P

(4.73)

ksB · rB + kiB · ri = krB P .

(4.74)

and, similarly,

Hence, Eq.(4.72) becomes

2 Γ (2) ∝ eik(rsA +rA P ) + eik(rsB +rB P ) 2 = 1 + eik(Δri +Δrs ) ,

(4.75)

where Δrs = rsA − rsB , Δri = rA P − rB P . From any optics textbook about Young’s double slit experiment, we ﬁnd that kΔrs = 2πxs a/λDs , kΔri = 2πxi a/λDi ,

(4.76)

where xs,i are the x-coordinates in the observation plane {S, P }, a is the slit separation, and Ds(i) is the distance from the slit (image of the slit) to the observation plane. So, ﬁnally, we have: (4.77) Γ (2) ∝ 1 + cos 2π xs /Ls + xi /Li . Therefore, the coincidence count shows a two-photon fringe pattern in both the signal observation plane and the idler plane. But there is no slit in the idler ﬁeld. So, the observer will think a ghost fringe is recorded in coincidence as the idler detector changes its position. Note that because of the spatial correlation in Eq.(4.68), the coincidence measurement will select the idler photons, satisfying Eq.(4.68). So, even though there is no slit in the idler ﬁeld, the slits in the signal ﬁeld will force the selected idler photons to pass through some virtual slits (ghost slits). The eﬀect of ghost fringe is a manifestation of the nonlocal spatial entanglement exhibited in the state in Eq.(4.67). It was ﬁrst reported by Strekalov et al. [4.22]. Further application of the spatial correlation gives rise to the ghost image eﬀect [4.23] and other interesting spatial eﬀects [4.24, 4.25, 4.26]. But these topics are beyond the scope of this book.

82

4 Phase-Independent Two-Photon Interference

References 4.1 M. S. Chubarov and E. P. Nikolayev, Phys. Lett. 110A, 199 (1985). 4.2 C. O. Alley and Y. H. Shih, Proceedings of the Second International Symposium on Foundations of Quantum Mechanics in the Light of New Technology, edited by M. Namiki et al. (Physical Society of Japan, Tokyo, 1987). 4.3 Z. Y. Ou, C. K. Hong, and L. Mandel, Phys. Lett. 122A, 11 (1987). 4.4 Y. H. Shih and C. O. Alley, Phys. Rev. Lett. 61, 2921 (1988). 4.5 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988). 4.6 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). 4.7 S. L. Braunstein, A. Mann, M. Revzen, Phys. Rev. Lett. 68, 3259 (1992). 4.8 S. L. Braunstein and A. Mann, Phys. Rev. A 51, R1727 (1995). 4.9 M. Michler, K. Mattle, H. Weinfurter, and A. Zeilinger, Phys. Rev. A 53, R1209 (1996). 4.10 C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wooters, Phys. Rev. Lett. 70, 1895 (1993). 4.11 D. Bouwmeester, J. W. Pan, K. Mattel, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature (London) 390, 575 (1997). 4.12 K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, Phys. Rev. Lett. 76, 4656 (1996). 4.13 P. G. Kwiat and H. Weinfurter, Phys. Rev. A 58, R2623 (1998). 4.14 S. P. Walborn, S. P´ adua, and C. H. Monken, Phys. Rev. A 68, 042313 (2003). 4.15 C. Schuck, G. Huber, C. Kurtsiefer, and H. Weinfurter, Phys.l Rev. Lett. 96, 190501 (2006). 4.16 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 54 (1988). 4.17 Z. Y. Ou, E. C. Gage, B. E. Magill, and L. Mandel, Opt. Comm. 69, 1 (1988). 4.18 Y. H. Shih and A. V. Sergienko, Phys. Rev. A 50, 2564 (1994). 4.19 T. Legero, T. Wilk, M. Hennrich, G. Rempe, and A. Kuhn, Phys. Rev. Lett. 93, 070503 (2004). 4.20 R. Ghosh and L. Mandel, Phys. Rev. Lett. 59, 1903 (1987). 4.21 Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 62, 2941 (1989). 4.22 D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, Phys. Rev. Lett. 74, 3600 (1995). 4.23 T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, Phys. Rev. A 52, R3429 (1995); T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Phys. Rev. A 53, 2804 (1996). 4.24 H. H. Arnaut and G. A. Barbosa, Phys. Rev. Lett. 85, 286 (2000); A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001). 4.25 A. R. Altman, K. G. K¨ opr¨ ul¨ u, E. Corndorf, P. Kumar, and G. A. Barbosa, Phys. Rev. Lett. 94, 123601 (2005). 4.26 S. P. Walborn, A. N. de Oliveira, S. P´ adua, and C. H. Monken, Phys. Rev. Lett. 90, 143601 (2003).

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

The phase independent two-photon interference fringe patterns discussed in the previous two chapters are the consequence of nearly overlapping paths for the two interfering possibilities. When we change one optical path, the phases of the photon pair change by the same amount; the net result is that the phase changes cancel and the interference is phase-insensitive. However, we can certainly separate the two paths, as in the Mach-Zehnder and Michelson interferometers in single-photon interference. Then, we can change the phases independently by diﬀerent amounts. This leads to phase-sensitive two-photon interference. Furthermore, and diﬀerent from single-photon interference, twophoton interference involves two photons that can be separated and go through diﬀerent paths. This leads to some intriguing nonlocal eﬀects. In this chapter, we will discuss various schemes of phase dependent twophoton interference eﬀects. Some of the eﬀects rely on the coherent process of parametric down-conversion and others simply take advantage of parametric down-conversion as a process that naturally creates photon pairs with various kinds of entanglement.

5.1 Two-Photon Interferometer with Two Down-Converters The best way to separate the two paths of a pair of photons is by creating them in two separate ways. Of course, to have interference, the two processes need to be coherent. We start with a simple single-mode analysis. 5.1.1 Phase Memory by Entanglement with Vacuum From the general expression in Eq.(2.34) for the quantum state from parametric down-conversion, we can simplify the quantum state in a single-mode case as

84

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

|η = |0 + ηVp |1s , 1i .

(5.1)

Note that this is a two-photon state entangled with the vacuum. In all previous chapters, the vacuum state does not play any role and only two-photon terms contribute to the two-photon detection. However, the state in Eq.(5.1) contains the amplitude of the pump ﬁeld, or the phase of the pump ﬁeld. Without the vacuum state, this phase information is lost. Thus, the vacuum state plays an essential role in keeping the phase memory of the pump ﬁeld in the process of parametric down-conversion. The process can be thought of as a two-photon coherent process. Grangier, Potasek and Yurke [5.1] ﬁrst proposed a homodyne scheme to show this kind of phase memory by mixing the two down-converted photons with two coherent local oscillators, respectively. In the following, we will consider a scheme proposed by Ou et al. [5.2] using two parametric down-conversion processes, and superpose the down-converted ﬁelds to look for two-photon interference, as depicted in Fig.5.1.

Vp1 t1

s1 PDC1 i1 s2

t2

ts1 ti1

PDC2 Vp2

i2

Fig. 5.1. Scheme for demonstrating two-photon phase memory with two parametric downconverters.

A

ts2 ti2

B

With two such processes, the quantum state of the system is then: |SY S = |η1 ⊗ |η2 ,

(5.2)

with |ηj = |0 + ηj Vpj |1sj , 1ij

(j = 1, 2).

(5.3)

The quantum state can be further written as |SY S = |01 , 02 + η1 Vp1 |1s1 , 1i1 ⊗ |02 + η2 Vp2 |01 ⊗ |1s2 , 1i2 ,

(5.4)

after dropping out the higher order four-photon term. Note that this state is an entangled state when projection is made in two-photon detection. The entanglement can be understood as the two possible ways of generating the pair of photons. Since the process is coherent, the two possibilities are entangled. The two-photon coincidence probability at detectors A and B is given as P2 ∝ SY S|ˆb†i ˆb†sˆbsˆbi |SY S, with

(5.5)

5.1 Two-Photon Interferometer with Two Down-Converters

√ ˆbs = (ˆ as1 + a ˆs2 )/√ 2, ˆbi = (ˆ ai1 + a ˆi2 )/ 2.

85

(5.6)

For the quantum state in Eq.(5.4), P2 can be calculated easily as P2 ∝ |η1 Vp1 + η2 Vp2 |2 ∝ 1 + V2 cos(ϕp1 − ϕp2 ),

(5.7)

with a visibility of V2 =

2|η1 Vp1 η2 Vp2 | |η1 Vp1 |2 + |η2 Vp2 |2 .

(5.8)

When the two-photon rates in the two processes are equal, i.e., |η1 Vp1 | = |η2 Vp2 |, the visibility V2 is 100%. It is obvious from Eq.(5.4) that the vacuum contribution to the state in Eq.(5.3) plays an essential role in this eﬀect. From Eq.(5.8), we can see that the visibility, which is usually a measure of optical coherence of a ﬁeld, is related to the pump ﬁelds Vp1 , Vp2 . Since what we measured is the visibility in twophoton interference, which is a measure of two-photon coherence, this suggests that the two-photon coherence is directly related to the pump ﬁeld coherence. In other words, the two down-converted ﬁelds together have the memory of the pump ﬁeld coherence. However, the single-mode model in Eq.(5.3) does not show clearly how optical coherence of the pump ﬁeld is related to the visibility. To see this point more directly, we resort to the multi-mode model of parametric down-conversion, discussed thoroughly in Chapt.2. 5.1.2 Multi-Mode Theory For the demonstration of pump coherence preservation in the process, we consider two parametric down-conversion processes pumped by two diﬀerent cw ﬁelds Vp1 and Vp2 , described by classical random ﬁelds of the form 1 Vpj (t) = √ dωp αpj (ωp )e−iωp t (j = 1, 2), (5.9) 2π with α∗pj (ωp )αpj (ωp )p = 2πnpj (ωp )δ(ωp − ωp ). npj (ωp ) describes the power spectrum of the narrow band stationary pump ﬁelds Vp1 and Vp2 . The quantum state for this case is given by Eq.(2.72) for narrow band pumping, and we rewrite it here with diﬀerent notations for the two processes: (cw) |Ψ P DCj = |0s , 0i + ηj dωdω αpj (ω + ω )× ×ψ(ω, ω )ˆ a†sj (ω)ˆ a†ij (ω )|0,

(5.10)

with j = 1, 2. Here, we assume the spectra for both processes are the same and are deﬁned by ψ(ω, ω ). The quantum state of the system is simply |Ψ sys = (cw) (cw) |Ψ P DC1 ⊗ |Ψ P DC2 .

86

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

Consider the arrangement shown in Fig.5.1 where the signal beams from the two nonlinear crystals are combined by a 50:50 beam splitter BSA and the idler beams by BSB . The mixed beams are directed to detectors A and B, respectively. Let us introduce some delays for the beams arriving at BSA and BSB and denote them as ts1 , ts2 , ti1 , ti2 . Then, the ﬁeld operators at the two detectors are of the form √ √ ˆA = (Eˆs1 + E ˆs2 )/ 2, E ˆB = (Eˆi1 + E ˆi2 )/ 2, E (5.11) with ˆj (t) = √1 E 2π

dωˆ aj (ω)e−iω(t−tj ) ,

(5.12)

where j = s1, s2, i1, i2 and a ˆj is the photon annihilation operators for the four signal and idler modes, respectively. The coincidence rate RAB for detecting two photons at A and B simultaneously is proportional to (2) RAB ∝ dτ ΓAB (t, t + τ ), (5.13) TR

(2)

with ΓAB (t, t + τ ) = |C2 (t, t + τ )|2 p and C2 (t, t + τ ) ≡ vac|EˆB (t + τ )EˆA (t)|Ψ sys .

(5.14)

C2 (t, t + τ ) is the two-photon wave function. Here, the average p is on the pump ﬁelds. TR is the resolution time window of the detectors. With the help of Eqs.(5.10) and (5.11), the calculation of C2 (t, t + τ ) is straightforward and the result is obtained as follows: 1 ˆi2 (t + τ )Eˆs2 (t)|Ψ sys vac|Eˆi1 (t + τ )Eˆs1 (t) + E 2 = η1 G1 (t, t + τ ) + η2 G2 (t, t + τ ), (5.15)

C2 (t, t + τ ) =

with Gj (t, t + τ ) ≡

1 4π

1 = 8π

dω1 dω2 αpj (ω1 + ω2 )ψ(ω1 , ω2 )×

×e−iω1 (t−tsj ) e−iω2 (t+τ −tij ) dωp dΩ αpj (ωp )e−iωp (t+τ /2−tsj /2−tij /2) × ω + Ω ω − Ω p p , e−iΩ(τ −Δj )/2 , ×ψ 2 2

(5.16)

where we made the variable change: ωp = ω1 + ω2 , Ω = ω1 − ω2 and Δj ≡ tsj − tij (j = 1, 2). For narrow band pump ﬁelds, as deﬁned in Sect.2.4, we have: ω + Ω ω − Ω ω + Ω ω − Ω p p p0 p0 ψ , ≈ψ , ≡ φ(Ω). (5.17) 2 2 2 2

5.1 Two-Photon Interferometer with Two Down-Converters

87

So, we obtain: tsj tij τ Δj τ 1 − g − , Gj (t, t + τ ) ≈ √ Vpj t + − 2 2 2 2 2 4 2π where Vpj is given in Eq.(5.9) and g(τ ) = dΩφ(Ω)e−iΩτ .

(5.18)

(5.19)

After averaging over the pump ﬁelds, we obtain: (2)

|η2 |2 Ip2 |g2 |2 + ΓAB (t, t + τ ) ∝ |η1 |2 Ip1 |g1 |2 + ∗ (5.20) +(η1 η2 Ip1 Ip2 γp12 g1 g2∗ + c.c.), ∗ where γp12 = Vp1 (tp1 )Vp2 (tp2 )/ Ip1 Ip2 with tpj = t + τ /2 − tsj /2 − tij /2 and gj = g(τ /2 − Δj /2)(j = 1, 2). If the two pump ﬁelds are from a common source Vp , i.e., Vpj (t) = Vp (t − tj ), but with delays of t1 and t2 , respectively, as in many optical interferometers, we can rewrite γp12 as γp12 = Vp (0)Vp∗ (Δ)/Ip = γp (Δ) = |γp (Δ)|eiωp0 Δ eiϕγ ,

(5.21)

with Δ ≡ t1 + ts1 /2 + ti1 /2 − t2 − ts2 /2 − ti2 /2. ϕγ is a slowly varying phase function of Δ and is zero for the symmetric spectrum of the pump ﬁeld. Let us set η1 = η2 = η and, as before, assume that TR >> Tc so that we can take the range of the time integral in Eq.(5.13) as (−∞, +∞). Then, Eq.(5.13) becomes

RAB ∝ |η|Ip 1 + |γp (Δ)| eiωp0 Δ F (Δ1 − Δ2 ) + c.c. /2

(5.22) = |η|Ip 1 + V2 cos(ωp0 Δ + ϕF + ϕγ ) , with V2 ≡ |γp (Δ)F (Δ1 − Δ2 )|, and F (τ ) ≡

∞ −∞

dt g(t)g ∗ (t + τ )

∞

−∞

dt|g(t)|2 = |F (τ )|eiϕF .

(5.23)

(5.24)

Eq.(5.22) shows an interference fringe in two-photon correlation with a visibility of V2 . The fringe is phase sensitive to both the phases of the pump ﬁelds ωp0 (t1 − t2 ) and of the down-converted ﬁelds (ωp0 /2)(ts1 + ti1 − ts2 − ti2 ). With a narrow band pump ﬁeld, the spectrum of the pump is much narrower than the spectrum of the down-conversion so γp (Δ) is a slowly varying function, whereas F (τ ) has a much smaller range of Tc .

88

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

The dependence of the visibility on the F -function, whose maximum is achieved when Δ1 = Δ2 or Δs ≡ ts1 − ts2 = ti1 − ti2 ≡ Δi , means that the relative positions of the single-photon wave packets from each down-conversion must be adjusted to form the same shape so as to obtain identical two-photon waves for both down-conversion processes (Fig.5.2). This shape match of the two-photon waves is the same as the mode match in any interference scheme for the maximum interference eﬀect (visibility). Only after the shapes of the two-photon waves are matched do the two-photon waves acquire the coherence of the pump ﬁelds, which leads to γp in Eq.(5.23). s2

s1

A

B i2

i1

Fig. 5.2. Match of the shapes (areas inside the dashed lines) of the twophoton waves from the two crystals: the formation of the single-photon waves must be identical for both crystals in order to obtain the maximum two-photon interference eﬀect.

Let Tj ≡ (tsj + tij )/2 be the average time delay of the two down-converted photons. The visibility then becomes V2 = γp (Δ) with Δ = t1 − t2 + T1 − T2 , which is directly related to the coherence of the pump ﬁelds, as if we were allowing the two pump ﬁelds to interfere directly. Although we deal with signal and idler beams, we end up with a result that depends on the coherence of the pump ﬁelds. We may, therefore, think of the parametric down-conversion process as a process that transfers the amplitude (or phase) of the pump ﬁeld to the pair of down-converted photons. Notice that we are assigning an amplitude (or a phase) to the photon pair. This is just the amplitude of the two-photon waves discussed in Sect.1.2, and the two twophoton waves from the two nonlinear media interfere here. In some sense, the two-photon waves are duplications of the pump waves and they should have the coherence of the pump waves rather than of single-photon waves. So, it is quite diﬀerent from the coherence of single-photon waves. In terms of photons, the observed interference eﬀect is based on pairs of photons, and it can be easily understood with the help of the modiﬁed Dirac’s statement on photon interference (see Sect.1.2). The two photons detected in coincidence in Fig.5.1 must have originated either from one nonlinear medium or from another, and they are in the superposition state of Eq.(5.4). Therefore, the detected photon pair interferes with itself via these two possibilities, just as in the Pﬂeegor-Mandel experiment described in Sect.1.2. What is diﬀerent here, from those experiments, is that the two possible paths of the two photons are separated here, and this results in phase-sensitive two-photon interference.

5.2 Franson Interferometer

89

If we consider that the two-photon waves are the continuation of the singlephoton waves of the pump ﬁelds, the scheme in Fig.5.1 can be thought of as a gigantic Mach-Zehnder interferometer. The interferometric scheme in Fig.5.1, using two crystals, was ﬁrst demonstrated by Ou et al. [5.3] and later was conﬁrmed with χ(3) ﬁber optical systems [5.4].

5.2 Franson Interferometer For the interferometric scheme described in the previous section, the pair of photons exists only after interacting with the two crystals. Before the nonlinear interaction, light is in the form of single (pump) photons. Thus, the waves in the equivalent Mach-Zehnder interferometer are in both two-photon form and single-photon form. Therefore, it is not a true “two-photon” interferometer. In 1989, Franson [5.5] proposed a new and simple form of a two-photon interference scheme, with pairs of photons throughout the entire interferometer. 5.2.1 Entanglement in Time The principle of the Franson interferometer is based on the superposition of two-photon waves generated at diﬀerent times. As we demonstrated in the previous section, the two-photon wave has the same coherence as the pump ﬁeld. Thus, the two-photon waves generated at diﬀerent times will be coherent as long as they are within the coherence time of the pump ﬁeld. Quantum mechanically, this kind of two-photon state entangled in time can be demonstrated best in the quantum state for parametric down-conversion pumped by a monochromatic ﬁeld (inﬁnite coherence time) in Eqs.(2.66, 2.67). For the sake of argument, we rewrite it as (cw) |Ψ P DC = |0 + ηVp dΩψ(Ω)ˆ a†s (ωp0 /2 + Ω)ˆ a†i (ωp0 /2 − Ω)|0, (5.25) with ψ(Ω) ∝ Ψ (ωp0 /2 + Ω, ωp0 /2 − Ω) = h(LΔk)|ω3 =ωp0 , ω2 =ωp0 /2−Ω,

ω1 =ωp0 /2+Ω ,

(5.26)

and η is such that:

dΩ|ψ(Ω)|2 = 1.

(5.27)

Obviously, Eq.(5.25) shows frequency entanglement. However, the two-photon part in Eq.(5.25) can be rearranged as follows:

90

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

|Ψ2 = ηVp = η

dΩ1 dΩ2 δ(Ω1 + Ω2 ) ψ(Ω1 ) ψ(−Ω2 )× ×ˆ a†s (ωp0 /2 + Ω1 )ˆ a†i (ωp0 /2 + Ω2 )|0

dT Vp e−iωp0 T |T s |T i ,

(5.28)

where η ≡ C −2 η and |T k ≡ Aˆ†k (T )|0 (k = s, i) with Aˆ†k (T ) ≡ C dΩ ψ(Ω)ei(ωp0 /2+Ω)T a ˆ†k (ωp0 /2 + Ω).

(5.29)

Here, C is such that |C|2 dΩ|ψ(Ω)| = 1, which leads to [Aˆk (T ), Aˆ†k (T )] = 1 (k = s, i). |T k (k = s, i) is a single-photon state of single temporal mode deﬁned by the temporal function w(τ − T ): w(τ ) ≡ C dΩ ψ(Ω)e−iΩτ . (5.30) Hence, the state |T k (k = s, i) describes a single-photon wave packet peaked at t = T . See Chapter 9 for more on the characterization of temporal distribution of multiple photons.

Vp(T)

Signal

Idler T1

T2

Tk

Fig. 5.3. Production of time-entangled photon pair of V (T )|T s |Tk i in p k k k parametric down-conversion with a long two-photon coherence time.

Therefore, the last expression in Eq.(5.28) presents a two-photon state entangled in time. Pictorially, the process of parametric down-conversion pumped by a monochromatic ﬁeld produces a two-photon wave packet at any time with equal probability (Fig.5.3). The two-photon wave packets at two diﬀerent times are coherent and, thus, should give rise to two-photon interference if they are within the coherence time of the pump ﬁeld. This is the underlying principle behind the proposal by Franson [5.5]. The original proposal is on atomic system. The realization of the Franson interferometer with parametric downconversion is illustrated in Fig.5.4, where a pair of down-converted photons is sent directly into two unbalanced interferometers. Since there are always two photons in the interferometer, this is a genuine two-photon interferometer. The path diﬀerences cTA and cTB of the two interferometers are much larger than the coherence length of each photon or the length of each single-photon

5.2 Franson Interferometer

pump

TA

A

TB

B

91

PDC

Fig. 5.4. Schematics for realizing Franson interference eﬀect by time-entangled photon pairs with unbalanced interferometers.

wave packet [described by w(τ ) in Eq.(5.30)], so there is no single-photon interference eﬀect, i.e., each detector does not exhibit interference fringe. However, if the pair of photons carries some phase coherence with coherence time longer than TA and TB , as does the pair of photons generated in the downconversion process, then the photon pair may interfere with itself via the two shorter paths and the two longer paths in the two interferometers. In other words, because the two-photon waves associated with the photon pair from the down-conversion have a coherence time much longer than the single-photon waves, if this coherence time is longer than the time diﬀerence between the longer and shorter paths, the two-photon waves for the longer and the shorter paths will interfere. This interference eﬀect is expected to appear in coincidence measurements, with the two detectors placed at the outputs of the two interferometers, respectively. The two-photon interference eﬀect described above is a manifestation of photon entanglement in time, as discussed in the beginning of this section. Consider two photons in the Fock state |1A |1B injected into the two unbalanced interferometers. The state after the ﬁrst two beam splitters will have the following form, if the beam splitters are 50:50: |SY S = (|1AS + |1AL ) ⊗ (|1BS + |1BL )/2 = (|1AS , 1BS + |1AL , 1BL + + |1AS , 1BL + |1AL , 1BS )/2,

(5.31)

where S and L represent the shorter and the longer paths, respectively. If the detection system can resolve the time diﬀerence between the shorter and longer paths, the last two terms in the second expression of Eq.(5.31) will not contribute to the measured coincidence rate. The other two terms that do contribute to the coincidence rate are in the form of the entangled state, and they lead to an interference eﬀect with 100% visibility. On the other hand, if the detection system is unable to resolve the time diﬀerence between the shorter and the longer paths, then all terms in Eq.(5.31) contribute to the coincidence rate. The last two terms in Eq.(5.31), however, do not contribute to the interference, because the length of each single-photon wave packet is much shorter than the path diﬀerence between the longer and the shorter paths. Therefore, the two single-photon wave packets cannot combine to form

92

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

a two-photon wave packet with a longer coherence time. Rather, they simply contribute background to the interference and result in reduced visibility of 50%. In connection with the two terms that give rise to interference, for which the corresponding two photons both follow either the shorter or the longer path, the two single-photon wave packets in each case need to form a twophoton wave packet of the same shape as shown in Fig.5.2. This requires that the path diﬀerences TA and TB of the two unbalanced interferometers be close to within the length of each single-photon wave packet, i.e., |TA − TB | << Tc = 1/Δω, where Δω is the bandwidth of the single-photon wave packets. The above discussion of the Franson-type experiment is based on a qualitative picture of two-photon interference and the idea of an entangled state. Next, we examine the multi-mode theory of the Franson interferometer and provide a more quantitative description. We shall ﬁnd that the qualitative physical picture does, indeed, lead to the correct answer. 5.2.2 Two-Photon Coherence versus One-Photon Coherence: Multi-Mode Analysis The two photons from the parametric down-conversion process have all the properties required for a Franson-type experiment. Since the emission time is involved, a full analysis of the Franson interferometer requires a multi-mode description of the light source. In order to have long two-photon coherence, we consider the parametric down-conversion with a narrow band pump ﬁeld. The quantum state of the ﬁelds has the form given in Eq.(5.10) and is rewritten here, for only one crystal, as follows: (cw) |Ψ P DC = |0s , 0i + η dω1 dω2 αp (ω1 + ω2 )× ×ψ(ω1 , ω2 )|ω1s , ω2i .

(5.32)

αp (ωp ) is the spectral amplitude for the pump ﬁeld and has a narrow range around ωp0 . Consider the arrangement shown in Fig.5.4. The ﬁelds at the two detectors can be expressed in terms of the ﬁeld operators before the interferometers as ˆs (t − TA )]/2, ˆA = [Eˆs (t) + E E with ˆk (t) = √1 E 2π

ˆB = [E ˆi (t) + E ˆi (t − TB )]/2, E

dωˆ ak (ω)e−iωt (k = s, i),

(5.33)

(5.34)

where we dropped the operators from the vacuum ports of the beam splitters. In writing Eq.(5.33), we assumed that the two down-converted photons arrive simultaneously at the ﬁrst beam splitters of the two interferometers.

5.2 Franson Interferometer

93

ˆA or E ˆB . But, this can be Otherwise, we may introduce a common delay in E adjusted in the two-photon correlation function. The measured two-photon coincidence is related to the intensity correlation: (2) † ˆ † (t + τ )EˆB (t + τ )EˆA (t) (t)E ΓAB (t, t + τ ) = EˆA B ˆB (t + τ )EˆA (t)|Ψ (cw) 2 . = 0|E P DC p

(5.35)

As before, the subscript p is for the average over the pump ﬁeld. With the quantum state in Eq.(5.10), it is straightforward to derive (cw)

ˆB (t + τ )EˆA (t)|Ψ 0|E P DC = G(t, t) + G(t − TA , t − TB )+ +G(t − TA , t) + G(t, t − TB ),

(5.36)

with

η G(t1 , t2 ) = dω1 dω2 αp (ω1 + ω2 )ψ(ω1 , ω2 )e−iω1 t1 −iω2 (t2 +τ ) 8π η dωp αp (ωp )e−iωp (t1 +t2 +τ )/2 × = 16π

ωp + Ω ωp − Ω iΩ(t2 +τ −t1 )/2 e , . × dΩψ 2 2

(5.37)

The four terms in Eq.(5.36) correspond to SS, LL, LS, and SL paths for the two photons, as discussed in the previous section. For a narrow band pump ﬁeld around ωp0 , we have ψ(ωp /2+Ω/2, ωp/2−Ω/2) ≈ ψ(ωp0 /2+Ω/2, ωp0/2− Ω/2) ≡ φ(Ω) and Eq.(5.37) becomes:

η t1 + t2 + τ t2 + τ − t1 G(t1 , t2 ) = √ Vp g , (5.38) 2 2 8 2π where Vp (t) is given in Eq.(5.9) and g(t) in Eq.(5.19). In the arrangement for the Franson interferometer, we have TA , TB >> Tc = 1/ΔωP DC , the reciprocal bandwidth of the down-converted light or the range of the function g(t). Then, when we substitute Eq.(5.36) into Eq.(5.35), there is only one non-zero cross term and Eq.(5.35) becomes: (2)

ΓAB (τ ) =

|η|2 Ip |g(τ )|2 + |g(τ + ΔT )|2 + 128π T + T A B g ∗ (τ )g(τ + ΔT ) + c.c. + + γp 2 +|g(τ + TA )|2 + |g(τ − TB )|2 ,

(5.39)

where ΔT ≡ TA − TB . The measured photon coincidence rate RAB is an inte(2) gral of ΓAB (τ ), with respect to τ over the coincidence window with resolving time TR . Normally, TR >> Tc = 1/Δω, so we can eﬀectively set the integral range from −∞ to +∞. There are two diﬀerent situations:

94

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

(i) When the delays in the unbalanced interferometers are long enough so that we have the condition TR << TA , TB , corresponding to the situation when the detection system is able to resolve the shorter and longer paths, there is no contribution from the two terms in the last line of Eq.(5.39). So we have, after the time integration, T + T A B (2) RAB ∝ 1 + F (ΔT )γp cos[ωp0 (TA + TB )/2 + ϕ0 ], (5.40) 2 where F (ΔT ) is given in Eq.(5.24). The visibility of the two-photon interference fringe is, thus: V2 = |γp |F (ΔT ).

(5.41)

The second-order coherence function γp of the pump ﬁeld appears in the visibility, implying once again that information about the amplitude of the pump ﬁeld transfers to the two down-converted photons, as we discussed earlier in Eq.(5.23). When |γp | = 1, we ﬁnd that the visibility of the interference is 100% only when |ΔT | ≡ |TA − TB | << Tc = 1/Δω, which leads to F (ΔT ) = 1. This is exactly the result of the qualitative discussion given earlier: the condition of |ΔT | << Tc is to assure again that the two two-photon waves (short-short and long-long) have the same shape. (ii) On the other hand, if TR >> TA , TB , which corresponds to the situation when the detection system is unable to resolve the shorter and the longer paths, then the last two terms in Eq.(5.39) contribute and we have, after the time integration: T + T 1 A B (2) RAB ∝ 1 + F (ΔT )γp cos[ωp0 (TA + TB )/2 + ϕ0 ]. (5.42) 2 2 This time, the maximum visibility is only 50% in the ideal condition. It is easy to check that the counting rate for each detector is constant with changes of path diﬀerence, indicating that the interference eﬀect described above is a purely two-photon interference eﬀect. Experimental realization of the Franson interferometer with two unbalanced interferometers was achieved by Ou et al. [5.6], who demonstrated the relation in Eq.(5.42) for visibility reduction due to ﬁnite pump coherence time with a multi-mode pump laser. A similar experiment was performed by Kwiat et al. [5.7], but with only one unbalanced interferometer for both the signal and idler ﬁelds. Furthermore, they made TA , TB much smaller than the coherence time of the pump ﬁeld, and observed two-photon interference with visibility close to 50%, with no reduction due to pump incoherence. Note that between the two demonstrations in Refs.[5.6, 5.7], only the former exhibits the nonlocal feature and potentially can be used to demonstrate Bell’s inequality violation in phase variables. However, the limited size of visibility (only a maximum of 50%) is not enough for a violation. Later, Brendel

5.3 Two-Photon De Broglie Interferometer

95

et al. [5.8] achieved the condition in case (i), i.e., TR << TA , TB and observed a visibility greater than 50%, as predicted in Eq.(5.40). Indeed, subsequent experiments [5.9] demonstrated violations of Bell’s inequalities for phase variables.

5.3 Two-Photon De Broglie Interferometer The phenomena described in the previous sections have clearly demonstrated the phase sensitive two-photon interference. Yet, as each photon of the pair goes through a separate interferometer, the two-photon waves formed in this way are nonlocally separated and not the usual “two-photon waves” in which both photons are in one wave packet. Because of this separation, the visibility in Eqs.(5.23, 5.41) depends on a ﬁne structure function of F (τ ) in Eq.(5.24), which relies on the details of the match between the shapes of the two-photon waves formed by the signal and idler wave packets. In this section, we describe a scheme in which the two photons completely overlap and stick together as they travel through either one or the other arm of an interferometer. The physical picture of a two-photon wave packet is then much clearer.

|Ψ in = |1a1, 1a2

a^2 ^

b1

a^1 ^

b2

50:50

|Ψ out = (|2, 0 − |0, 2 )/21/2

Fig. 5.5. Hong-Ou-Mandel interferometer for the generation of a two-photon NOON state.

5.3.1 Maximally Entangled Photon State – the NOON State In Sect.3.1, we showed that when two identical photons enter a 50:50 beam splitter from diﬀerent sides, as shown in Fig.5.5, the two photons tend to “stick together” and both emerge from the same side of the beam splitter; the situation in which one photon emerges from one side and the other one from the other side never occurs. This is clearly indicated by Eq.(3.5): when T = R = 50%, the output state from the beam splitter becomes √ (5.43) |Ψ out = (|2, 0 − |0, 2)/ 2. The two photons in this state are inseparable and become completely indistinguishable. The two states in the superposition are genuine two-photon Fock states with the two photons in one single-mode.

96

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

signal Pump

D2

∆T

cδΤ

PDC idler

BS2

BS1

D1

Fig. 5.6. Two-photon de Broglie wave interferometer.

Next, we combine the two outputs again with a second beam splitter (BS2), as shown in Fig.5.6. Since the two photons, acting as one entity, travel in either one or the other path, this will lead to an interference eﬀect with a wavelength equal to half of the single-photon wavelength. This wavelength is the so called de Broglie wavelength of a two-photon wave. To observe the two-photon de Broglie wavelength, we must take a two-photon coincidence measurement either at two outputs (Fig.5.6) or at one output. A simple calculation of the two-photon coincidence gives: P2 ∝ 1 ± cos 4πΔL/λ,

(5.44)

where λ is the wavelength of a single photon and ΔL = cΔT is the optical path diﬀerence between the two arms of the interferometer. The plus sign corresponds to the case of a one-side two-photon coincidence measurement, whereas the minus sign is for the case of a two-side coincidence measurement, as shown in Fig.5.6. The state in Eq.(5.43) is known as the maximally entangled photon number state or the so-called NOON state [5.10]. Later in Part II, we will generalize this state to the cases with N > 2. Although the argument above is based on a simple single-mode model for the two-photon state, the conclusion holds even for a multi-mode twophoton wave packet, as long as the two wave packets of the two individual photons overlap exactly at the beam splitter to form one single wave packet for the two photons (bottom of the Hong-Ou-Mandel dip). However, if the two photons do not overlap in the ﬁrst beam splitter (wings of the Hong-OuMandel dip), the two photons in the interferometer may separate and each will act independently. This is similar to the Franson interferometer discussed in the previous section, except that the two interferometers in the original Franson interferometer become one here. This situation, of course, cannot be reﬂected in the single-mode argument given above. So, in the following, we will analyze this scheme by using the multi-mode model for the parametric down-conversion and consider the two diﬀerent situations mentioned above. 5.3.2 Detailed Analysis of Two-Photon De Broglie Interferometer Referring to Fig.5.6, where we introduce a delay δT between the signal and idler before they arrive at the ﬁrst beam splitter (BS1), we obtain the ﬁeld operators right after BS1 in the form of

5.3 Two-Photon De Broglie Interferometer

√ ˆA = [E ˆs (t) + E ˆi (t − δT )]/ 2, E

97

√ ˆB = [Eˆs (t) − E ˆi (t − δT )]/ 2, (5.45) E

ˆs,i (t) given in Eq.(5.34). After considering an optical path diﬀerence of with E cΔT , we ﬁnd the ﬁeld operators at the outputs (BS2) of the interferometer, in terms of the input operators in Eq.(5.45), as ⎧ ˆs (t − ΔT ) + E ˆi (t − ΔT − δT ) + E ˆs (t) − E ˆi (t − δT )]/2, ⎨ Eˆ1 (t) = [E (5.46) ⎩ ˆ ˆ ˆ ˆ ˆ E2 (t) = [Es (t − ΔT ) + Ei (t − ΔT − δT ) − Es (t) + Ei (t − δT )]/2. For the two-photon coincidence rates between the two outputs, or in one output, say output 1, we have, respectively: (2) (2) R12 ∝ dτ Γ12 (t, t + τ ), R11 ∝ dτ Γ11 (t, t + τ ), (5.47) with (2)

Γ12 (t, t + τ ) = Eˆ2† (t)Eˆ1† (t + τ )Eˆ1 (t + τ )Eˆ2 (t), (2)

Γ11 (t, t + τ ) = Eˆ1† (t)Eˆ1† (t + τ )Eˆ1 (t + τ )Eˆ1 (t),

(5.48) (5.49)

where the average is over the two-photon state from parametric downconversion given in Eq.(5.32), and over the pump ﬁeld. In the following, we will calculate R12 but only present results for R11 . To evaluate the expression in Eq.(5.48), we ﬁrst calculate the two-photon ˆ1 (t + τ )Eˆ2 (t)|ΨP DC . Due to the two-photon nature in the amplitude 0|E ˆ1 (t + τ )Eˆ2 (t) give state in Eq.(5.32), only eight terms in the expansion of E contributions, and they are:

ˆs (t − ΔT + τ ) E ˆ (t − ΔT − δT ) + E ˆi (t − δT ) + E i

ˆi (t − ΔT − δT ) + E ˆi (t − δT ) + ˆs (t + τ ) E +E

ˆ ˆs (t) − ˆ + Ei (t − ΔT − δT + τ ) Es (t − ΔT ) − E

ˆs (t − ΔT ) − E ˆs (t) . ˆi (t − δT + τ ) E (5.50) −E ˆi (t2 ). So, let us denote G(t1 , t2 ) ≡ All of these terms are in the form of Eˆs (t1 )E ˆ ˆ 0|Es (t1 )Ei (t2 )|ΨP DC . Then, it is straightforward to obtain G(t1 , t2 ) ∝ η dω1 dω2 αp (ω1 + ω2 )ψ(ω1 , ω2 )e−i(ω1 t1 +ω2 t2 ) ∝ ηVp (t1 /2 + t2 /2)g(t1 − t2 ),

(5.51)

where Vp is the pump amplitude in Eq.(5.9) and g(t) is the single-photon temporal shape in Eq.(5.19). As before, we assume a narrow band pump ﬁeld and make the approximation: ψ(ωp /2 + Ω/2, ωp /2 − Ω/2) ≈ ψ(ωp0 /2 + Ω/2, ωp0 /2 − Ω/2) ≡ φ(Ω). Then, the two-photon detection amplitude has the form of

98

5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

ˆ1 (t + τ )Eˆ2 (t)|ΨP DC 0|E

∝ η Vp (t − ΔT ) + Vp (t ) g(δT − τ ) + g(δT + τ ) + +Vp (t − ΔT /2) g(δT + τ + ΔT ) + g(δT + τ − ΔT )− −g(δT − τ + ΔT ) − g(δT − τ − ΔT ) , (5.52)

where t = t + (τ − δT )/2. Now let us consider the situation when δT = 0, which corresponds to a perfect overlap of the two wave packets for the signal and idler photons at BS1, or the location of the bottom of the Hong-Ou-Mandel dip. If ψ(ω1 , ω2 ) is symmetric, which gives rise to a perfect match between the signal and idler wave packets and leads to 100% visibility in the Hong-Ou-Mandel dip, g(τ ) will be an even function of τ : g(−τ ) = g(τ ). Then, Eq.(5.52) becomes

ˆ1 (t + τ )Eˆ2 (t)|ΨP DC = 2g(τ ) Vp (t − ΔT ) + Vp (t ) . 0|E (5.53) The complete disappearance in Eq.(5.53) of the last term in Eq.(5.52) is a result of the photon bunching eﬀect at BS1, where the two incoming photons stick together to travel to either arm of the interferometer. Inside the interferometer, the two photons are indistinguishable and are in one single temporal mode deﬁned by g(τ ). They can be represented by the single-mode expression in Eq.(5.43). This is a genuine two-photon interferometer in the sense that the two photons become one entity and should exhibit interference fringes with a two-photon de Broglie wavelength of λ0 /2 = λp0 . This is reﬂected exactly in the two-photon coincidence rate from Eq.(5.47): R12 ∝ dτ |0|Eˆ1 (t + τ )Eˆ2 (t)|ΨP DC |2 p ∝ 1 + γp (ΔT ) cos 2πcΔT /λp0 + ϕγ . (5.54) It is interesting to note that even if there is no pump ﬁeld inside the interferometer, its amplitude appears in Eq.(5.53) as if the interferometer is for the pump ﬁeld. The reason for this can be traced back to the discussion of phase memory in the parametric down-conversion process, given in the previous sections (especially Sect. 5.1). We recall that the two down-converted photons actually carry information about the amplitude of the pump ﬁeld. Therefore, the two photons, as one entity, behave just like one pump photon. The fact that the second-order coherence function |γp | of the pump ﬁeld determines the visibility of the interference further conﬁrms what we just stated. Next, we consider the situation when |δT | >> 1/Δω = Tc , the width of the function g(τ ). This corresponds to the case when the signal and idler photon do not overlap at the ﬁrst beam splitter of BS1 for interference. Instead, they each go through the interferometer separately. If the path diﬀerence satisﬁes ΔT >> 1/ΔωP DC = Tc , this becomes the Franson interferometer, discussed in the previous section. Indeed, after carrying out the time integral

References

99

in Eq.(5.47) and using the fact of δT, ΔT >> 1/ΔωP DC = Tc , we ﬁnd the two-photon coincidence rate to be: R12 ∝ 1 + V2 cos 2πcΔT /λp0 + ϕγ , (5.55) with

V2 =

|γp (ΔT )|, |γp (ΔT )|/2,

if TR << ΔT, if TR >> ΔT.

(5.56)

This result is exactly same as that given by Eqs.(5.40) and (5.42), with TA = TB = ΔT for the Franson-type interferometer in the previous section. However, there is an exception to the second line of Eq.(5.56), that is, when δT = ±ΔT , we have V2 = 2/3 from Eq.(5.52). This exception to the Franson interferometer in the previous section is because here we have the overlap of the two un-balanced interferometers. In this case, although the signal and the idler photons do not meet at the ﬁrst beam splitter (BS1) due to δT = 0, they may meet at the second beam splitter (BS2) when δT = ±ΔT . The HongOu-Mandel eﬀect then cancels the part of the background due to either the L-S or S-L case in the Franson interferometer, thus raising the visibility from 1/2 to 2/3. But this will not inﬂuence the result in the ﬁrst line of Eq.(5.56) because the detectors can now resolve the L-S and S-L cases so that they won’t contribute to the coincidence count anyway. Note that when δT = 0, the signal and idler photon wave packets coincide to form one two-photon wave. Therefore, there is no need to adjust and match the shapes of the two-photon waves. That is why the visibility in Eq.(5.54) does not depend on the function F (τ ), which is in Eqs.(5.23, 5.40, 5.42). Here, F is always one. Even for δT = 0, the visibility in Eq.(5.56) still does not rely on F (τ ). This is because the overlap of the two unbalanced interferometers here ensures TA = TB , for a perfect match between the shapes of the twophoton waves (long-long and short-short). Experimentally, the two-photon interferometer of the de Broglie wave was ﬁrst realized by Ou et al. [5.11] and by Rarity et al. [5.12]. Boto et al. [5.10] proposed using this type of interferometer for better resolutions in optical lithography and this was tested by D’Angelo et al. [5.13] for the two-photon case. There are many other forms of two-photon interferometers that exhibit phase dependent two-photon coincidence for various applications. For example, Rarity and Tapster [5.14] used a scheme ﬁrst proposed by Oliver and Stroud [5.15] to demonstrate a violation of Bell’s inequalities with the phase variables and Herzog et al. [5.16] used a frustrated two-photon interferometer to demonstrate a quantum eraser.

References 5.1 P. Grangier, M. J. Potasek and B. Yurke, Phys. Rev. A 38, 3132 (1988).

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5 Phase-Dependent Two-Photon Interference: Two-Photon Interferometry

5.2 Z. Y. Ou, L. J. Wang, and L. Mandel, Phys. Rev. A 40, 1428 (1989). 5.3 Z. Y. Ou, L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. A 41, 566 (1990). 5.4 X. Li, P. L. Voss, J. E. Sharping, and P. Kumar, Phys. Rev. Lett. 94, 053601 (2005). 5.5 J. D. Franson, Phys. Rev. Lett. 62, 2205 (1989). 5.6 Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 65, 321 (1990) 5.7 P. G. Kwiat, W. A. Vareka, C. K. Hong, H. Nathel, and R. Y. Chiao, Phys. Rev. A 41, 2910 (1990). 5.8 J. Brendel, E. Mohler, and W. Martienssen, Phys. Rev. Lett. 66, 1142 (1991). 5.9 T. B. Pittman, Y. H. Shih, A. V. Sergienko, and M. H. Rubin, Phys. Rev. A 51, 3495 (1995); W. Tittel, J. Brendel, H. Zbinden, and N. Gisin, Phys. Rev. Lett. 81, 3563 (1998). 5.10 A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). 5.11 Z. Y. Ou, X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. A 42, 2957 (1990). 5.12 J. G. Rarity, P. R. Tapster, E. Jakeman, T. Larchuk, R. A. Campos, M. C. Teich, and B. E. A. Saleh, Phys. Rev. Lett. 65, 1348 (1990). 5.13 M. D’Angelo, M. V. Chekhova, and Y. Shih, Phys. Rev. Lett. 87, 013602 (2001). 5.14 J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990). 5.15 B. J. Oliver and C. R. Stroud, Phys. Lett. 135A, 407 (1989). 5.16 T. J. Herzog, P. G. Kwiat, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 75, 3034 (1995).

6 Interference between a Two-Photon State and a Coherent State

In the two-photon interference phenomena discussed earlier, the two-photon ﬁelds are either from the same source (Sect.5.2 and Sect.5.3) or from two similar sources, pumped by same ﬁeld (Sect.5.1). Similar to one-photon interference, two-photon interference between independent sources may occur under certain circumstances. In this chapter, we will consider the two-photon interference between a coherent source and parametric down-conversion. The phenomenon is also phase sensitive, similar to the previous chapter. However, because of the involvement of the special sources and its potential application, we discuss this phenomenon separately in this chapter.

6.1 Anti-Bunching by Two-Photon Interference Interference between a two-photon state from parametric down-conversion and a weak coherent state was initially studied by Stoler [6.1] as an alternative way to generate photon anti-bunching, although Stoler never mentioned anything about two-photon interference in his original paper. This phenomenon was ﬁrst demonstrated experimentally by Koashi et al. [6.2] using a pico-second pulse laser. Anti-bunched light was ﬁrst demonstrated by Kimble, Dagenais, and Mandel [6.3] as the ﬁrst example of nonclassical behavior of light. With the advent of quantum information, it was found that such a light source is extremely important for quantum cryptography. Traditionally, an anti-bunched photon ﬁeld was produced by a two-level single quantum system, which can be a single atom, ion, molecule, and a quantum dot. The physics for these systems is such that when the single system emits one photon, it is in the lower energy level and, therefore, unable to emit a second photon immediately. Only after waiting some time for the system to be re-pumped to the higher energy level, can the system emit the second photon. The crucial part in this approach is a single quantum system; but, it is a technical challenge to isolate a single quantum system.

102

6 Interference between a Two-Photon State and a Coherent State

Another approach for anti-bunching is to use selective absorption to take out the unwanted two-photon events while leaving the single-photon events unchanged. This leads naturally to some two-photon processes such as twophoton absorption [6.4] and harmonic generation [6.5]. Indeed, it has been shown that these light sources exhibit more or less some kind of sub-Poissonian photon statistics [6.4, 6.5]. However, because of the relatively low absorption cross-section in these processes, the eﬀect is not pronounced for practical application. |α

|Ψ

Parametric Amplifier

Fig. 6.1. Parametric ampliﬁer with a coherent state as input.

out

On the other hand, since harmonic generation and its reverse process, parametric down-conversion, are coherent processes involving two photons, we may use two-photon interference to suppress the two-photon events. This is the physics behind Stoler’s proposal, in which a coherent state is injected into a parametric ampliﬁer, as depicted in Fig.6.1. In a single-mode treatment, a parametric ampliﬁer can be described by the unitary operator: [6.6] ˆ2 − ζˆ a†2 )/2], Uˆ = exp[(ζ ∗ a

(6.1)

where ζ is the gain parameter. The output state is then related to the input state by ˆ in . |Ψ out = U|Ψ

(6.2)

For a coherent state input, as shown in Fig.6.1, the output state becomes Yuen’s two-photon coherent state [6.6]: ˆ |Ψ out = U|α.

(6.3)

This state has been studied extensively in Ref.[6.6] and has the following expansion in photon number state basis: |Ψ out =

∞

Cn |n,

(6.4)

n=0

with Cn = √

n ν∗ 1 ν 2 |α|2 α α2 − Hn √ exp , 2μ 2 2μν n!μ 2μ

where Hn is the nth-order Hermite polynomial and μ ≡ cosh(|ζ|), ν ≡

ζ sinh(|ζ|), or μ2 = 1 + |ν|2 . |ζ|

(6.5)

6.1 Anti-Bunching by Two-Photon Interference

103

More speciﬁcally, we list the ﬁrst few coeﬃcients as ν∗ 1 1 C0 = √ exp( α2 − |α|2 ), μ 2μ 2 α C1 = C0 , μ C0 ν α 2 C2 = √ − + , μ μ 2 C0 3να α 3 C3 = √ − 2 + , μ μ 6 C0 ν 2 να2 α 4 3 −6 3 + C4 = √ . μ μ μ 24

(6.6) (6.7) (6.8) (6.9) (6.10)

It can be seen that when the relative strength between the coherent state and the ampliﬁer is such that α2 = μν, the two-photon probability P2 = |C2 |2 is zero [6.8, 6.9], leading to an anti-bunching eﬀect in the output of the parametric ampliﬁer. From the expression for C2 in Eq.(6.8), we ﬁnd that it is the superposition of two terms: the ﬁrst one is from two-photon amplitude of the coherent state, while the second is from the parametric down-conversion process. Destructive interference between these two amplitudes leads to the cancellation of twophoton probability and the photon anti-bunching eﬀect. If we don’t have two-photon events in the output, then where do the two photons from the injected coherent state go in the anti-bunching eﬀect? It turns out that in a parametric ampliﬁer, although down-conversion process from pump ﬁeld to the signal and idler ﬁelds is the dominating process, the reverse process of up-conversion still exists if there is also an injection of ﬁeld at the fundamental frequency. These up and down conversion processes are coherent and when the strengths of the two processes are just right, the two photons from the input coherent state are converted to a harmonic pump photon with 100% eﬃciency [6.7]. However, this is only for a two-photon state and will not work for states with other photon numbers. So, it does not have any practical use in harmonic generation. The essence of this eﬀect is twophoton interference between the two-photon wave from the coherent state and the two-photon wave from parametric down-conversion. If the two waves are from the same origin, they are coherent to each other. Two-photon interference between the two waves will occur. When it is complete destructive interference, the two-photon waves cancel each other, leading to zero probability for the two-photon event or the anti-bunching eﬀect. Two-photon interference between a coherent state and the state from parametric down-conversion can also be implemented with the help of a beam splitter. Consider the scheme in Fig.6.2, where the coherent state is superimposed with parametric down-conversion by a 50:50 beam splitter. In single-mode format, the input states have the form of |BSin = |PDC1 ⊗ |α2 ,

(6.11)

104

6 Interference between a Two-Photon State and a Coherent State

|α 50:50

| BS

out

| PDC = | 0 − (ζ / 2 ) | 2

Fig. 6.2. Interference between a weak coherent state and a two-photon state with a 50:50 beam splitter.

or α2 ζ |BSin ≈ |0, 0 + α|0, 1 + √ |0, 2 − √ |2, 0, (6.12) 2 2 √ √ if we use |PDC ≈ |0 − (ζ/ 2)|2 and |α ≈ |0 + α|1 + (α2 / 2)|2, up to two-photon terms. From Appendix A, we may ﬁnd the output state for an input state of |m, n. We list only the ones to our interest here: √ |0, 1out = |1, 0 + |0, 1 / 2 √ |0, 2out = |2, 0 + |0, 2 + 2|1, 1 /2 √ |2, 0out = |2, 0 + |0, 2 − 2|1, 1 /2. Up to the two-photon state, we then have the output state as √ |BSout = |0, 0 + α |1, 0 + |0, 1 / 2+ α2 − ζ α2 + ζ + √ |1, 1. |2, 0 + |2, 0 + 2 2 2

(6.13)

As can be seen, if we set α2 = ζ, the coeﬃcient in front of |2, 0 + |0, 2 is zero and there will be no two-photon state in either output port. In the discussion above, although we can cancel the two-photon terms and, thus, lead to the photon anti-bunching eﬀect, there still exist three- or morephoton states in the higher order terms. We will consider the higher photon number state later in Sect.8.5.1. In the next two sections, we will treat the more practical multi-mode situations.

6.2 Multi-Mode Analysis I: CW Case In the cw case, the coherent state is from a single-mode laser with only one frequency component, and the parametric down-conversion is pumped by a cw ﬁeld that is from harmonic generation of the single-mode laser. We will consider only the scheme of interference by a 50:50 beam splitter (Fig.6.2). Now, let us consider the input for ﬁeld 1 from parametric down-conversion with single-frequency pumping. The state is in the form of Eq.(2.66) or, more

6.2 Multi-Mode Analysis I: CW Case

105

explicitly, Eq.(5.25). However, since we are dealing with degenerate photons, the two photons belong to the same ﬁeld. So, the state is rewritten as a†1 (ωp0 /2 − Ω)|0. (6.14) a†1 (ωp0 /2 + Ω)ˆ |BS1in = |0 + ηVp dΩΦ(Ω)ˆ The input for ﬁeld 2 is from the single-mode laser, which is in a single-mode coherent state of |BS2in = |α(ωp0 /2),

(6.15)

a ˆ2 (ω)|α(ωp0 /2) = αδ(ω − ωp0 /2)|α(ωp0 /2).

(6.16)

where

When the detectors are fast enough to resolve the arrivals of diﬀerent photons, the time-delayed two-photon coincidence rate is proportional to the intensity correlation function: † † Γ (2) (t, t + τ ) = Eˆ1out (t)Eˆ2out (t + τ )Eˆ2out (t + τ )Eˆ1out (t),

with

√ ˆ2in (t)]/ 2, Eˆ1out (t) = [Eˆ1in (t) + E √ ˆ2in (t)]/ 2. Eˆ2out (t) = [Eˆ1in (t) − E

(6.17)

(6.18)

a1,2 (ω)e−iωt . Substituting the input states in Here, Eˆ1,2in (t) = (2π)−1/2 dωˆ Eqs.(6.14, 6.15) into Eq.(6.17) and after some straightforward calculation, we have: Γ (2) (t, t + τ ) ∝ |α2 − f (τ )|2 , with

(6.19)

f (τ ) = ηVp

dΩΦ(Ω)eiΩτ .

(6.20)

Eq.(6.19) is in the form of two-photon interference. From our previous discussion on two-photon detection in parametric down-conversion (Sect 2.4), we know that the shape of |f (τ )|2 provides the time-delay distribution between the two down-converted photons. So f (τ ) is the amplitude of the twophoton detection, with a time delay of τ between the two photons. In the meantime, α2 is the two-photon detection amplitude from a coherent state. The time independent nature of this amplitude is a reﬂection of the randomness of the Poisson statistics for the photons from a coherent state (laser). Fig.6.3 shows the shape of the normalized intensity correlation function of g (2) (τ ) ≡ Γ (2) (τ )/Γ (2) (∞) for the three diﬀerent relative strengths (shown in the insets) between the coherent state and the parametric down-conversion. The phase diﬀerence between α and f (τ ) is chosen to be zero so that we

106

6 Interference between a Two-Photon State and a Coherent State

have destructive interference. Fig.6.3a shows the perfect anti-bunching eﬀect, which is exactly the same as the situation in the resonance ﬂuorescence [6.3]. In this case, we have α2 = f (0), so that a complete destructive interference occurs at zero delay of τ = 0. The rise of the function at non-zero delay is due to an amplitude mismatch that leads to incomplete destructive interference. Although the physics is quite diﬀerent here, we can mimic the behavior in resonance ﬂuorescence for the anti-bunching eﬀect. In Fig.6.3b, |α|2 > |f (0)|. We never obtain complete cancellation. An interesting double dip feature occurs in Fig.6.3c when |α|2 < |f (0)|. As can be seen in the inset of Fig.6.3c, we have α2 = f (±τ0 ) for some nonzero τ = τ0 , leading to complete two-photon cancellation at those two locations, whereas at other locations the two amplitudes do not match.

Normalized Intensity Correlation Function g(2)(τ)

2.0

1.5

(c)

(b)

(a)

1.0

0.5

0.0 -10

-5

0

5

Time Delay τ (scaled)

10 -10

-5

0

5

-10 10

Time Delay τ (scaled)

-5

0

5

10

Time Delay τ (scaled)

Fig. 6.3. Normalized two-photon correlation function as a function of the relative delay between the two detectors for the interference between a coherent state and a two-photon state. (a) |α|2 = f (0); (b) |α|2 > f (0); |α|2 < f (0). Reprinted ﬁgure with permission from Y. J. Lu and Z. Y. Ou, Phys. Rev. Lett. 88, 023601 (2001). c 2001 by the American Physical Society.

While Fig.6.3a,b show the typical anti-bunching eﬀects of g (2) (0) < 1 and g (0) < g (2) (τ ), Fig.6.3c demonstrates another type of nonclassical eﬀect by violating the classical inequality of |g (2) (0) − 1| ≥ |g (2) (τ ) − 1| [6.10]. (Obviously from Fig.6.3c, |g (2) (0) − 1| = 0 and |g (2) (τ ) − 1| = 1.) The antibunching behavior in Fig.6.3 was observed by Lu and Ou [6.11], with a narrow band two-photon state. (2)

6.3 Multi-Mode Analysis II: Pulsed Case In the cw case discussed in the previous section, we have complete two-photon cancellation only at some speciﬁc delay locations (τ = 0 in Fig.6.3a and τ = ±τ0 in Fig.6.3c). In the situation when the photo-detectors are slow, we need to average over a range of τ and we cannot have complete cancellation of

6.3 Multi-Mode Analysis II: Pulsed Case

107

the two-photon events. This arises from the mismatch between the temporal modes of the two ﬁelds: the coherent state has a ﬂat time dependence, whereas the parametric down-conversion has a well-localized temporal shape. Since the eﬀect is due to interference between the two ﬁelds, complete destructive interference requires all the modes be matched, no matter whether the modes are temporal or spatial. Spatial mode match can be achieved easily by careful alignment or by coupling into a single-mode ﬁber. Temporal mode match requires spectral ﬁltering. This is better handled with pulsed operation, where both the coherent ﬁeld and the two-photon ﬁeld have some temporal proﬁles that need to be matched. The quantum state from the type-I parametric down-conversion pumped by a pulse is derived in Eq.(2.34) in Chapt.2 and is rewritten here for the degenerate case as |Ψ P DC = |0 + ξ dω1 dω2 Φ(ω1 , ω2 )ˆ a†1 (ω1 )ˆ a†1 (ω2 )|0. (6.21) The input state to the other port of the beam splitter is a multi-mode coherent state |α with a ˆ2 (ω)|α = α(ω)eiωtp |α, (6.22) where α(ω) is a well-deﬁned real function of ω with a ﬁnite bandwidth. It forms a transform-limited coherent pulse centered at t = tp : (6.23) A(t) = dωα(ω)e−iω(t−tp ) . Therefore, it is synchronized with the pump pulse. The electric ﬁeld operators at the output of the beam splitter are same as Eq.(6.18). The two-photon detection probability rate at one of the output ports, say port 1, is then: ˆ1out (t2 )|Ψ P DC |α||2 . p2 (t1 , t2 ) = ||Eˆ1out (t1 )E

(6.24)

After some lengthy calculation, to the lowest nonzero order of |η| or α(ω), Eq.(6.24) becomes p2 (t1 , t2 ) = |A(t1 )A(t2 ) + F (t1 , t2 )|2 , where F (t1 , t2 ) is deﬁned as F (t1 , t2 ) = ξ dω1 dω2 e−iω1 t1 −iω2 t2 [Φ(ω1 , ω2 ) + Φ(ω2 , ω1 )].

(6.25)

(6.26)

Since the pulse is usually faster than the detectors, the overall probability of two-photon detection is an integration over all times (6.27) P2 = dt1 dt2 p2 (t1 , t2 ) = A4 + 4|η|2 − 2B,

108

6 Interference between a Two-Photon State and a Coherent State

where η is deﬁned in Eq.(2.39) [we used the symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ) for the degenerate two-photon state] and 2 (6.28) A ≡ dt|A(t)|2 , B ≡ dt1 dt2 A(t1 )A(t2 )F ∗ (t1 , t2 ). (6.29) Here, we choose the relative phase so that it gives rise to destructive interference in Eq.(6.27). It can be shown easily from Schwartz inequality that 2B ≤ |A|4 + 4|η|2 . Let us now deﬁne a parameter (6.30) M ≡ B/2A2 η . Then, Eq.(6.27) becomes P2 = dt1 dt2 p2 (t1 , t2 ) = A4 + 4|η|2 − 2M A2 |η|.

(6.31)

The quantity M characterizes the degree of temporal mode match between the coherent ﬁeld and the down-converted ﬁeld. This can be seen from the time integral in Eq.(6.29), where the modes are now the two-photon waves of F (t1 , t2 ) from down-conversion and A(t1 )A(t2 ) from the coherent state. Therefore, if the two ﬁelds have a perfect mode match, then we have M = 1, which leads to the perfect two-photon cancellation of P2 in Eq.(6.31) when A4 = 4|η|2 (amplitude match). The temporal mode match of M = 1 is equivalent to the spectral match: Φ(ω1 , ω2 ) ∝ α(ω1 )α(ω2 ).

(6.32)

But, the spectral of the parametric down-conversion is complicated, depending on the intrinsic characteristic of the nonlinear crystal and the type of the phase match [see Eqs.(2.35, 2.45)]. A passive ﬁltering of the parametric downconversion is then needed to reshape its spectrum, but at the sacriﬁce of the signal level. Demonstration of two-photon reduction by pulsed ﬁelds was performed by Koashi et al. [6.2].

References 6.1 D. Stoler, Phys. Rev. Lett. 33, 1397 (1974). 6.2 M. Koashi, K. Kono, T. Hirano, and M. Matsuoka, Phys. Rev. Lett. 71, 1164 (1993). 6.3 H. J. Kimble, M. Dagenais, and L. Mandel, Phys. Rev. Lett. 39, 691 (1977). 6.4 M. S. Zubairy and J. J. Yeh, Phys. Rev. A 21, 1624 (1980). 6.5 J. Mostowski and K. Rz¸az˙ ewski, Phys. Lett. 66A, 275 (1978).

References

109

6.6 H. P. Yuen, Phys. Rev. A 13, 2226 (1976). 6.7 K. J. Resch, J. S. Lundeen, and A. M. Steinberg, Phys. Rev. Lett. 87, 123603 (2001). 6.8 M. Matsuoka and T. Hirano, Phys. Rev. A 67, 042307 (2003). 6.9 Y. J. Lu, L. Zhu, and Z. Y. Ou, Phys. Rev. A 71, 032315 (2005). 6.10 P. R. Rice and H. J. Carmichael, IEEE J. Quantum Electron. QE24, 1351 (1988); G. T. Foster, S. L. Mielke, and L. A. Orozco, Phys. Rev. A61, 053821 (2000). 6.11 Y. J. Lu and Z. Y. Ou, Phys. Rev. Lett. 88, 023601 (2002).

Part II

Quantum Interference of More Than Two Photons

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

For the generation of a photon state with a photon number greater than two, we will need to consider multiple pair production in parametric downconversion. However, as we will see, the coherence property of each downconverted ﬁeld is closely related to the temporal distribution of photon pairs from the two ﬁelds. So, we will ﬁrst investigate the optical coherence of one of the down-converted ﬁelds. The case of the cw pump is trivial and is fully covered in Sect.2.4. Therefore, we will only consider the case when the pump ﬁeld is a pulse.

7.1 Coherence Properties of Spontaneous Parametric Down-Conversion The process of spontaneous parametric down-conversion is an intrinsically incoherent process. This can be seen from the fact that even with a singlefrequency continuous wave (cw) pumping, the down-converted ﬁelds have a ﬁnite coherence time Tc , which is the reciprocal bandwidth (1/ΔωP DC ) of the ﬁelds. Since this is the coherence time from the cw-pumped down-conversion, (cw) we denote it as cw coherence time Tc . Normally, the bandwidth is more (cw) of about sub-picosecond. than THz and leads to a cw coherence time Tc (cw) is also the correlation time between the two The cw coherence time Tc down-converted photons (see Chapt.2). Therefore, we usually describe the down-converted ﬁelds pictorially as in Fig.7.1, where the source emits a pair of wave packets with a single photon in each of them (a two-photon wave packet), but the emission of the pair is completely random. This source is by no means a coherent source. Because of the randomness in pair production, when considering two pairs, we can usually treat them as independent. This, however, will create a problem if we want to combine two pairs of photons from parametric down-conversion

114

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

Signal

Idler

Fig. 7.1. Temporal distribution of down-converted photon pairs: there is a perfect temporal correlation between the signal and the idler photons, but the pair production is completely random.

to form a four-photon entangled state, that is, when will the two pairs overlap to become an indistinguishable four-photon state? One solution is to use an ultra short pulse to pump the parametric downconversion. This will force the pair production to occur only during the pump pulse duration. The same can be said for multi-pair production [7.1, 7.2]. With a coherent pulse as the pump ﬁeld, Fig.7.2 shows the likely scenario, where the down-converted photons are generated within the pump pulse duration. With this picture, it seems that the down-converted ﬁelds will become transform-limited, when the pump duration is shorter than the cw coherence (cw) time Tc . However, as we will see in the following, this picture is only approximately correct. It is the goal of this section to study in detail the second order (single-photon) coherence of the down-converted ﬁelds. Multi-photon coherence will be related to the single-photon coherence of the signal or idler ﬁeld.

Pump Profile (cw)

Tc

Tp

Fig. 7.2. The proﬁle of pump pulse (Tp ) and that of the down-converted wave (cw) packet (Tc ).

7.1.1 Field Correlation Functions and Coherence of Parametric Down-Converted Fields To study the coherence property of parametric down-conversion, we start by investigating the second-order ﬁeld correlation function of either the signal or the idler ﬁeld [7.3]: ˆ † (t1 )Eˆk (t2 ), Γk (t1 , t2 ) = E k

(k = s, i)

(7.1)

where the average is over the quantum state of parametric down-conversion. When t1 = t2 = t, we have Γk (t, t) = Ik (t) as the instantaneous intensity of the ﬁeld k(= s, i). The degree of coherence is related to the visibility of secondorder (or single-photon) interference fringes and is the correlation function in Eq.(7.1) normalized to the intensity Ik (t) [7.4]:

7.1 Coherence Properties of Spontaneous Parametric Down-Conversion

γk (t1 , t2 ) =

Γk (t1 , t2 ) Γk (t1 , t2 ) = . Γk (t1 , t1 )Γk (t2 , t2 ) Ik (t1 )Ik (t2 )

(k = s, i)

115

(7.2)

The perfect coherence occurs when the visibility or |γk (t1 , t2 )| = 1. But from Schwartz inequality, we have |γk (t1 , t2 )| ≤ 1 in general and |γk (t1 , t2 )| = 1 when t1 = t2 . If a ﬁeld has |γk (t1 , t2 )| = 1 for all t1 , t2 , this ﬁeld is transformlimited, in the sense that the temporal proﬁle of the ﬁeld is the Fourier transform of its spectrum. From the process of proving the Schwartz inequality, we ﬁnd that |γk (t1 , t2 )| = 1 for all t1 , t2 if and only if Γk (t1 , t2 ) is factorized as Γk (t1 , t2 ) = u∗ (t1 )u(t2 ).

(7.3)

Obviously, Ik (t) = |u(t)|2 . Because |γk (t1 , t2 )| ≤ 1 for all t1 , t2 , we have another inequality dt1 dt2 |Γk (t1 , t2 )|2 ≤ dτ dtIk (t)Ik (t + τ ) , (7.4) where the equality stands if and only if Γk (t1 , t2 ) is factorized as in Eq.(7.3) or the ﬁelds are transform-limited. Note that the quantity inside the square bracket on the right hand side of Eq.(7.4) is the intensity auto-correlation of the ﬁeld, whose range provides a measure of pulse duration. On the other hand, the left hand side of Eq.(7.4) is related to the ﬁeld correlation function, whose range is a measure of the coherence time of the ﬁeld. Let us now apply the above to the ﬁelds from parametric down-conversion. The quantum state for the non-degenerate case has a general form of a†s (ω1 )ˆ a†i (ω2 )|vac, (7.5) |Ψ P DC = |vac + ξ dω1 dω2 Φ(ω1 , ω2 )ˆ where Φ(ω1 , ω2 ) has the form of Eq.(2.36). The intensities of the ﬁelds are deﬁned as † Iˆs,i (t) = Eˆs,i (t)Eˆs,i (t),

(7.6)

√ ˆs,i (t) = dωˆ where E as,i (ω)e−jωt / 2π are the ﬁeld operators for the downconverted ﬁelds. It is straightforward to show that Iˆs,i (t) = dω2,1 |As,i (t, ω2,1 )|2 , (7.7) with ξ As,i (t, ω2,1 ) ≡ √ 2π Obviously, we have:

dω1,2 Φ(ω1 , ω2 )e−jω1,2 t .

(7.8)

116

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

dtIˆs (t) =

dtIˆi (t) =

dω1 dω2 |ξΦ(ω1 , ω2 )|2 .

(7.9)

Next, we examine the ﬁeld correlation function Γs,i (t1 , t2 ): † ˆs,i (t2 ) (t1 )E Γs,i (t1 , t2 ) ≡ Eˆs,i = dω2,1 A∗s,i (t1 , ω2,1 )As,i (t2 , ω2,1),

(7.10)

where A∗s,i (t, ω2,1 ) is given in Eq.(7.8). By using Schwartz inequality, we have: 2 dω2,1 A∗ (t1 , ω2,1)As,i (t2 , ω2,1 ) s,i ≤ dω2,1 |As,i (t1 , ω2,1 )|2 dω2,1 |As,i (t2 , ω2,1 )|2 ,

(7.11)

or by Eq.(7.2): |γ(t1 , t2 )| ≤ 1.

(7.12)

The equality sign in the above relation stands if and only if As,i (t1 , ω2,1 ) = CAs,i (t2 , ω2,1 )

with C = C(t1 , t2 ).

(7.13)

C is independent of ω2,1 . An examination of Eq.(7.8) shows that Eq.(7.13) is satisﬁed if and only if Φ(ω1 , ω2 ) factorizes as Φ(ω1 , ω2 ) = ψ1 (ω1 )ψ2 (ω2 ).

(7.14)

However, from the complicated form of Φ(ω1 , ω2 ) in Eq.(2.35) (see also Figs.2.4,2.7), this is impossible for the ﬁelds directly from parametric downconversion. Therefore, |γ(t1 , t2 )| < 1 for t1 = t2 and the signal and idler ﬁelds can never be transform-limited. But Eq.(7.12) does not tell us how far the ﬁelds are non-transform limited. To quantitatively characterize the coherence of the ﬁelds, we need to compare the coherence time Tc and the pulse duration Td of the ﬁelds. There are a number of methods to deﬁne Tc and Td . One straightforward way is to use inequality in Eq.(7.4) by deﬁning the ratio (7.15) 1 ≡ dt1 dt2 |Γk (t1 , t2 )|2 dτ dtIk (t)Ik (t + τ ) . For the state from parametric down-conversion in Eq.(7.5), we obtain: 1 = E/A, where

(7.16)

7.1 Coherence Properties of Spontaneous Parametric Down-Conversion

dω1 dω1 dω2 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2 ,

(7.17)

dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 ).

(7.18)

A≡ and

E≡

117

Obviously, 1 = E/A = 1 corresponds to the transform-limitedness of the ﬁeld. Note that because of the Schwartz inequality, we have E ≤ A and the equality stands if and only if Φ(ω1 , ω2 ) factorizes, as in Eq.(7.14). Sect.7.3 has a detailed discussion on the value of E/A. The factorization of the Φ-function can be achieved with the help of spectral ﬁltering. If we place ﬁlters with transmission amplitude of f (ω) in front of the detectors, we need to replace the operator a ˆ(ω) in the ﬁeld operator ˆ E(t) with a ˆ (ω) = f (ω)ˆ a(ω) + 1 − |f (ω)|2 a ˆ0 (ω), (7.19) where a ˆ0 (ω) is in vacuum. Then, it is straightforward to show that we may ¯ 1 , ω2 ) ≡ Φ(ω1 , ω2 )f (ω1 )f (ω2 ) and everything simply replace Φ(ω1 , ω2 ) by Φ(ω else remains the same. When the pass band of f (ω) is much narrower than that ¯ 1 , ω2 ) is approximately Cf (ω1 )f (ω2 ), which is factorized, and of Φ(ω1 , ω2 ), Φ(ω this will lead to E/A = 1. Another way to characterize the transform-limitedness of an optical ﬁeld is by the method of characterizing optical chirping for a classical ultra fast optical ﬁeld [7.5]. For this purpose, consider the ﬁeld correlation function in the fourth order Γ (2,2) (τ ) = dtE ∗2 (t)E 2 (t + τ ). (7.20) This quantity will be compared to the intensity correlation function G(2) (τ ) = dtI(t)I(t + τ ).

(7.21)

Note that Γ (2,2) (0) = G(2) (0) and |Γ (2,2) (τ )| ≤ G(2) (τ ). So, a quantitative measure for transform-limitedness is the quantity: (2,2) 2 ≡ dτ |Γ (τ )| dτ G(2) (τ ). (7.22) For a quantum ﬁeld, Eqs.(7.20, 7.21) are modiﬁed as ˆ + τ )]2 , Γ (2,2) (τ ) = dt[Eˆ † (t)]2 [E(t and

(7.23)

118

7 Coherence and Multiple Pair Production in Parametric Down-Conversion (2)

G

(τ ) =

ˆ † (t)E ˆ † (t + τ )E(t ˆ + τ )E(t), ˆ dtE

(7.24)

where the angle brackets correspond to the average over the quantum state of the ﬁeld. With the state in Eq.(7.5), we may ﬁrst evaluate the quantity ˆs (t2 )]2 , which, after some manipulation, has the form of [Eˆs† (t1 )]2 [E [Eˆs† (t1 )]2 [Eˆs (t2 )]2 = 2[Γs (t1 , t2 )]2 , where

Γs (t1 , t2 ) ≡

dω2 A∗s (ω2 , t1 )As (ω2 , t2 ),

with As (ω2 , t) ≡

1 2π

dω1 Φ(ω1 , ω2 )e−iω1 t .

(7.25)

(7.26)

(7.27)

The intensity correlation function can be calculated as ˆ † (t1 )Eˆ † (t2 )E ˆs (t2 )Eˆs (t1 ) Γs(2) (t1 , t2 ) = E s s 1 = dω2 dω2 |A(ω2 , t1 )A(ω2 , t2 ) + A(ω2 , t2 )A(ω2 , t1 )|2 . (7.28) 2 It is straightforward to see that ˆs† (t1 )]2 [E ˆs (t2 )]2 |. dτ |Γs(2,2) (τ )| ≤ dt1 dt2 |[E

(7.29)

But,

and

ˆs (t2 )]2 | = 2 dt1 dt2 |[Eˆs† (t1 )]2 [E

dτ G(2) s (τ ) =

dt1 dt2 |Γs (t1 , t2 )|2 = 2E,

(7.30)

ˆs† (t1 )Eˆs† (t2 )E ˆs (t2 )Eˆs (t1 ) = A + E. dt1 dt2 E

(7.31)

So, the parameter to characterize the transform-limitedness of the ﬁeld is 2 ≤ 2E/(A + E).

(7.32)

Once again, the coherence of the ﬁeld is related to the quantity E/A. 7.1.2 Generation of Transform-Limited Single-Photon Wave Packet by Gated Photon Detection Generation of a single-photon state in a single temporal mode is important in characterizing a single-photon state by quantum state tomography [7.6, 7.7],

7.1 Coherence Properties of Spontaneous Parametric Down-Conversion

119

where a strong coherent pulse is used as a local oscillator for the homodyne of the single-photon state (see Chapt.10 for more). Since homodyne is basically an interference eﬀect between the local oscillator and the single-photon state, it requires that both ﬁelds be in the same temporal mode, which means that the single-photon state must be transform-limited. Consider, ﬁrst, a non-stationary single-photon state described by ˆ† (ω)|vac, (7.33) |Ψ1 = dωψ(ω)ejωt0 a with normalization condition

dω|ψ(ω)|2 = 1.

(7.34)

The intensity of the ﬁeld in this state has the form of ˆ = |g(t − t0 )|2 , I(t) = Eˆ † (t)E(t) with 1 g(t) = √ 2π

dωψ(ω)e−jωt .

(7.35)

(7.36)

It can be conﬁrmed easily that the quantity |γ(t1 , t2 )| = 1 for the state in Eq.(7.33) for arbitrary t1 , t2 . Thus, it is a transform-limited single-photon pulse centered around t = t0 . Furthermore, from Eq.(7.34), we have the total photon number in the pulse as ∞ ∞ dτ I(τ ) = dτ |g(τ − t0 )|2 = 1. (7.37) −∞

−∞

Next, let us examine the possibility of using down-converted ﬁelds to produce a single-photon state |Ψ1 in Eq.(7.33). It is known that by gating the detection of the signal ﬁeld upon the detection of a photon in the idler ﬁeld, one can realize a single-photon state for the signal ﬁeld [7.8]. However, because of a ﬁnite resolution time of the detectors, the detection of the idler photon is uncertain within the resolution time. Although there is a good time correlation between the signal and the idler photons (< 100 fsec), the time uncertainty in the idler photon detection will make the temporal proﬁle of the signal photon undeﬁned and lead to incoherence in the signal photon. On the other hand, with an ultra fast pulse pumping the parametric down-conversion process, one may expect that everything will be synchronized to the ultra fast pump pulse and the down-converted photon may have a well-deﬁned temporal mode. Thus, the detection of the idler photon merely indicates the existence of the signal photon, but does not determine its timing. However, as we have seen in the previous section, this is not always the case because the down-converted ﬁelds are not transform-limited even with

120

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

ultra fast pumping. With the aid of narrow band ﬁlters, the down-converted ﬁelds may become transform-limited. But, since a ﬁlter is equivalent to a beam splitter, ﬁltering the signal ﬁeld will introduce vacuum noise and the signal ﬁeld will not be in a pure single-photon state. In the following, we will demonstrate that by only detecting a narrow band of the idler ﬁeld to generate the gating current, we can project the gated signal ﬁeld into a transform-limited single-photon ﬁeld synchronized to the pumping pulse. This is because of the frequency correlation due to energy conservation in Eq.(2.9) in parametric down-conversion. Thus, merely ﬁltering the idler ﬁeld will project the signal ﬁeld to a narrow band ﬁeld and there is no need to ﬁlter the signal ﬁeld. Consider, now, the two-photon state in Eq.(7.5) from parametric downconversion process. Suppose that detector Di registers an idler photon at time t = ti . Then, at the moment of detection, the state of the signal ﬁeld collapses to a state that is obtained by projecting the two-photon state |ΦP DC in ˆ † (ti )|vaci [7.9] with Eq.(7.5) onto the state |ψ(ti )i = K E i ˆ † (ti ) = √1 dωˆ a†i (ω)fi (ω)ejωti , (7.38) E i 2π where we used Eq.(7.19) for a ﬁlter characterized by amplitude transmission fi (ω) and placed in front of the Di detector and K is a normalization constant. We dropped the vacuum operator a ˆ0 because it has no contribution in our calculation. Next, let us take the pump pulse centered at t = tp by assuming the pump spectral amplitude has the form αp (ωp ) = α ˜ p (ωp )ejωp tp with α ˜p = real. The state after the projection is then given by |Ψ (ti )s = i ψ(t i )|ΦP DC ˜ 1 , ω2 )fi (ω2 )e−jω2 ti a dω1 dω2 ej(ω1 +ω2 )tp Φ(ω =K ˆ†s (ω1 )|vac = dω1 Ψ (ω1 , ti )ˆ a†s (ω1 )|vac, (7.39) with Ψ (ω1 , ti ) = K ejω1 tp

dω2 α ˜ p (ω1 + ω2 )h(ω1 , ω2 )fi (ω2 )e−jω2 (ti −tp ) ,

(7.40)

√ and K ≡ K ∗ / 2π. Here, we used Eq.(2.35) for Φ(ω1 , ω2 ). Eq.(7.39) has the form of Eq.(7.33), indicating that the gated ﬁeld is in a transform-limited single-photon state. But, the shape of the transform-limited pulse is determined by Ψ (ω1 , ti ), which seems to depend on the detection time ti . Let us consider the following two situations: (i) If the bandwidth σi of the ﬁlter fi and the bandwidth ΔωP DC of the downconverted ﬁelds are much larger than σp of the pump, i.e., {σi , ΔωP DC } >> σp , then α ˜ p (ω1 + ω2 ) behaves like a δ-function centered at ωp0 . This is close to the case of cw pumping. We then can make a change of variable ω = ω1 + ω2 in the integral in Eq.(7.40) and obtain:

7.2 Multi-Pair Production and Stimulated Pair Emission

Ψ (ω1 , ti ) = K ejω1 ti

121

dω α ˜ p (ω)h(ω1 , ω − ω1 )fi (ω − ω1 )ejω(tp −ti )

≈ ejω1 ti K α ˜p (ωp0 )fi (ωp0 − ω1 )h(ω1 , ωp0 − ω1 )ejωp0 (tp −ti ) . (7.41) Therefore, Ψ (ω1 , ti ) has a frequency dependent phase term of ejω1 ti , which indicates that the transform-limited pulse is peaked at ti and has a pulse width determined by Max(1/σi, 1/ΔωP DC ). Since the detection of the idler photon is uncertain to within the resolving time TR of the detector, which is usually much larger than 1/σi , 1/ΔωP DC , the location of the single-photon state at the signal ﬁeld is uncertain, which makes it impossible to match its shape to that of the coherent pulse for homodyne. In this situation, the detection of the idler photon not only indicates the existence of the signal photon but also determines the location of the signal photon. The uncertainty in idler photon detection transfers to the uncertainty in the signal photon. (ii) On the other hand, when σi is much smaller than σp and ΔωP DC , α ˜p and h-function are slowly varying functions of ω2 and we can take them outside of the integral and obtain: ˜ p (ω1 + ωi0 )h(ω1 , ωi0 )Fi (ti ), (7.42) Ψ (ω1 , ti ) ≈ ejω1 tp K α where ωi0 is the center frequency of fi (ω2 ) and Fi (τ ) ≡ dω2 fi (ω2 )e−jω2 (τ −tp ) . So, Ψ (ω1 , ti ) has both a bandwidth of Min(σp , ΔωP DC ) and a phase of ejωtp that determine the shape and location of the signal photon and are independent of the detection time ti of the idler photon. Normally, the bandwidth of down-conversion (∼ 10 THz) is larger than that of the pump. Then, the transform-limited single-photon state at the signal ﬁeld has exactly the shape of the pump ﬁeld and is centered at tp with a width of 1/σp . Both the peak and the shape of the single-photon pulse are, thus, synchronized with the pump pulse. The key in obtaining a pump-synchronized transform-limited single-photon pulse is in the fact that the frequencies of the two down-converted photons are correlated in the way that their sum is equal to the pump frequency, so that the bandwidth-limited detection of the idler photon lengthens the correlation time between the idler and signal photons. This ensures that the gated signal ﬁeld has a bandwidth determined by the pump bandwidth σp (if it is narrower than the down-converted bandwidth).

7.2 Multi-Pair Production and Stimulated Pair Emission The consideration of the coherence properties of the down-converted photons is virtually the preparation for the case of multi-pair production. We have seen in Part I that we can achieve high visibility in two-photon interference, indicating that the temporal modes of the signal and idler photons are perfectly matched (with the exception of pulse-pumped type-II parametric

122

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

down-conversion). For a photon number higher than two, multi-pair production is needed. The temporal mode match between diﬀerent pairs is, thus, important in order to obtain a good multi-photon interference eﬀect (high visibility), which will be the main topic of this part of the book. We will see that this is related to the coherence properties of the down-converted ﬁelds. For multi-pair production, we ﬁrst investigate the statistics of the pairs from parametric down-conversion. 7.2.1 Pair Statistics and Photon Bunching We start by considering the single-mode model of parametric down-conversion. As described in Sect.2.1, the single-mode quantum state of spontaneous nondegenerate parametric down-conversion is given by ˆ |vac |Φ = U

ˆ = exp(ηˆ with U a†s a ˆ†i − H.c).

(7.43)

The normal ordering of the unitary operator can be found with a technique by Yuen [7.10] as

∗ ˆ = 1 N exp ν α∗ α∗ − ν αs αi + 1 − μ α∗ αs + α∗ αi , (7.44) U s i μ μ s i μ μ ˆ† a ˆ is with μ = cosh |η|, ν = (η/|η|) sinh |η|. Here, N{α∗ α} = N{αα∗ } = a the normal ordering operation. Immediately, we have the quantum state of Eq.(7.43) in the Fock state expansion as:

k 1 ν ν∗ 1−μ ∗ |vac N α∗s α∗i − αs αi + αs αs + α∗i αi μk! μ μ μ k=0 νk νk a ˆ†k ˆ†k |ks |ki . (7.45) = s a i |vac = k+1 k!μ μk+1

|Φ =

k=0

k=0

If we consider only one ﬁeld, either the signal or the idler, we need to trace out the other ﬁeld and then obtain the density operator as ρˆs,i = Pk |ks,i k|s,i , (7.46) k=0

with

Pk = |ν|2k /μ2k+2 = nk /(n + 1)k+1 , 2

(7.47)

where n = |ν| is the average photon number. The photon statistics in Eq.(7.47) are exactly that of a thermal state. From Eq.(7.45), we see that Pk is also the probability of k pairs of photons. So, the pair statistics of nondegenerate parametric down-conversion are of a thermal nature. This observation was ﬁrst made by Yurke and Potasek in 1987 [7.11]. Actually, if we consider that parametric down-conversion is an ampliﬁcation process, as described by

7.2 Multi-Pair Production and Stimulated Pair Emission

123

Eq.(2.2), it is not surprising at all that the spontaneously down-converted ﬁelds are respectively thermal. As early as 1975, Carusotto [7.12] showed that any linear ampliﬁer produces an output that is the superposition of a thermal ﬁeld and a ﬁeld statistically similar to the input. The thermal noise in a parametric ampliﬁer was ﬁrst observed by Ou, Pereira, and Kimble in 1993 [7.13] and directly measured by Vasilyev et al. [7.14]. But the best way to characterize a thermal ﬁeld is the so called photon bunching eﬀect, ﬁrst observed by Hanbury-Brown and Twiss in 1956 [7.15]. With the state in Eq.(7.45), we can easily conﬁrm the bunching eﬀect, that is, the normalized intensity correlation function is: g (2) ≡ n2 /n2 = 2.

(7.48)

The above discussion is based on a single-mode description. For a multimode case with weak pumping, the process of parametric down-conversion pumped by a coherent pulse is given in Eq.(2.38) and is rewritten as |Ψ P DC = (1 −

|η|2 ξ2 )|vac + ξ|Φ2 + |Φ4 2 2

(7.49)

up to the second order of η (see Sect.2.5 for the physical meaning of η), where |η|2 = |ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 , with |Φ2 = dω1 dω2 Φ(ω1 , ω2 )ˆ a†s (ω1 )ˆ a†i (ω2 )|vac, and (7.50) |Φ4 = dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) ×ˆ a†s (ω1 )ˆ a†i (ω2 )ˆ a†s (ω1 )ˆ a†i (ω2 )|vac.

(7.51)

The information about the pump ﬁeld and the parametric interaction is contained in the function Φ(ω1 , ω2 ), which has the explicit form in Eq.(2.35) and is rewritten as Φ(ω1 , ω2 ) = αp (ω1 + ω2 )h(ω1 , ω2 ), (7.52) where αp (ω) describes the pump ﬁeld spectral amplitude and h(ω1 , ω2 ) is the sinc-function part from parametric down-conversion [Eq.(2.36)]. For a near degenerate type-I phase matching case, we have h(ω1 , ω2 ) = h(ω2 , ω1 ), so that Φ(ω1 , ω2 ) has the exchange symmetry: Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ). For a coherent pulse (transform-limited), αp (ωp ) is a well-behaved deterministic function of ωp . Here, the pump ﬁeld is treated as a classical ﬁeld. The ﬁeld operators for the down-converted ﬁelds are given by 1 ˆ Es (t) = √ (7.53) dωˆ as (ω)e−iωt , 2π ˆi (t) = √1 dωˆ ai (ω)e−iωt . E (7.54) 2π

124

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

To observe photon bunching in either the signal or the idler ﬁeld, let us calculate the two-photon detection probability rate at one ﬁeld, say, the signal ﬁeld as ˆs (t1 )Eˆs (t2 )|Ψ P DC ||2 = p2s (t1 , t2 ) = ||E

|ξ|4 ˆ ||Es (t1 )Eˆs (t2 )|Φ4 ||2 , 4

(7.55)

where we used Eq.(7.49) for the state |Ψ P DC . The states |vac and |Φ2 do not contribute in Eq.(7.55). By making use of the commutation relation ak (ω ), a ˆ†k (ω1 )ˆ a†k (ω1 )] (k = s, i) [ˆ ak (ω)ˆ = δ(ω − ω1 )δ(ω − ω1 ) + δ(ω − ω1 )δ(ω − ω1 ),

(7.56)

we ﬁnd the quantity in Eq.(7.55) becomes |ξ|4 dω2 dω2 [|F (ω2 , ω2 )|2 + F (ω2 , ω2 )F ∗ (ω2 , ω2 )], (7.57) p2s (t1 , t2 ) = 4(2π)2 where F (ω2 , ω2 ) ≡

dω1 dω1 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) × e−i(ω1 t1 +ω1 t2 ) + e−i(ω1 t1 +ω1 t2 ) .

(7.58)

Next, we calculate the overall probability for detecting two photons in one pump pulse by integrating the time variables t1 , t2 over the pulse period and we obtain: |ξ|4 P2s = dt1 dt2 p2s (t1 , t2 ) = dω2 dω2 dt1 dt2 |F (ω2 , ω2 )|2 . (7.59) 2(2π)2 Here, we used the fact that F (ω2 , ω2 ) = F (ω2 , ω2 ) because of the symmetry among t1 , t2 , ω1 , ω1 , ω2 , ω2 in Eq.(7.58). After some lengthy manipulation, Eq.(7.59) becomes P2s = |ξ|4 dω1 dω1 dω2 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2 +Φ(ω1 , ω2 )Φ(ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 ) ≡ |ξ|4 (A + E),

(7.60)

where A and E are given in Eqs.(7.17, 7.18), respectively. Note that |ξ|4 A = 2 |η|4 = P1s corresponds to the accidental two-photon probability and |ξ|4 E is the excess two-photon probability due to photon bunching. Since E ≤ A, we have the normalized intensity correlation function as 2 = 1 + E/A ≤ 2. g (2) ≡ P2s /P1s

(7.61)

7.2 Multi-Pair Production and Stimulated Pair Emission

125

Therefore, the best we can achieve is g (2) = 2, or the perfect photon bunching. But the equality holds if and only if the function Φ(ω1 , ω2 ) is factorized into ψ1 (ω1 )ψ2 (ω2 ), which is impossible because of the complicated dependence of Φ on ω1 , ω2 in Eq.(2.35). However, with the help of narrow band ﬁlters before detection, we can modify the function of Φ(ω1 , ω2 ) to achieve E → A, as discussed in Sect.7.1.1. The fact that the photon bunching eﬀect depends on the quantity of E/A seems to suggest that the bunching eﬀect is a result of two-photon interference, for, as we have seen in Sect.7.1.1, E/A characterizes how close the downconverted ﬁelds are to the transform-limited pulses and, thus, determines how well the temporal mode match is between the two photons detected in the signal ﬁeld. These two photons belong to one of the correlated pairs of photons and are, thus, from diﬀerent pairs. This temporal mode match, as we will see in Chapt.9, is crucial for indistinguishability between the two detected photons. As a matter of fact, Glauber was the ﬁrst to explain the photon bunching eﬀect by two-photon interference [7.16]. Another interpretation for the photon bunching eﬀect is the stimulated emission. We will understand more about this interpretation in the following section. It turns out that these two interpretations are equivalent. 7.2.2 Stimulated Pair Emission From the mechanism of pair production, we learned that the pair emission is from the splitting of the pump photon. Therefore, it is natural to believe the pair statistics should be the same as that of the pump ﬁeld, which is Poissonian for a coherent ﬁeld. As we have seen in the previous section, there is a bunching eﬀect in pair production – the pair statistics are not Poissonian but thermal. Where does the extra pair come from? The answer is the stimulated emission. The stimulated emission was ﬁrst proposed by Einstein, [7.17] to study energy balance in blackbody radiation. It was ﬁrst demonstrated by Purcell and Pound [7.18] in a nuclear spin system and later by Gordon, Zeiger, and Townes [7.19] with radio wave for the ampliﬁcation in the operation of a maser. To further study the stimulated pair emission, we consider the scheme shown in Fig.7.3, where two parametric down-conversion processes are used, with the signal ﬁeld from the ﬁrst one injected into the second one for ampliﬁcation. The input state to the second crystal is simply the quantum state from parametric down-conversion and has the form given by Eq.(7.49). To ﬁnd the output state from the second crystal, we need the evolution operator for parametric down-conversion. This was derived in Eq.(2.37) in Chapt.2 when we ﬁrst studied the parametric down-conversion process. We write it explicitly as ˆ = 1 + ξ dω1 dω2 Φ (ω1 , ω2 )ˆ U a†s (ω1 )ˆ a†i (ω2 )+

126

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

A pump

B

∆T PDC

Φ

s

Φ T C

i D

+

ξ 2 2

Fig. 7.3. A simple scheme for studying the stimulated emission in parametric down-conversion.

dω1 dω2 dω1 dω2 Φ (ω1 , ω2 )Φ (ω1 , ω2 ) ×ˆ a†s (ω1 )ˆ a†i (ω2 )ˆ a†s (ω1 )ˆ a†i (ω2 ).

(7.62)

Here we use “ ” to denote the second crystal. To characterize the relative timing between the two crystals, we may introduce a time delay ΔT for the second crystal by adding an extra phase ei(ω1 +ω2 )ΔT in Φ -function. Since the signal ﬁeld of the ﬁrst crystal is injected into the second one, we use the same creation operator a ˆ†s for the signal mode in the second crystal. The output state then becomes ˆ |Ψ P DC . |Ψ out = U (7.63) Up to the second order of ξ and ξ , we have: 1 2 ξ |Φ4 + ξ 2 |Φ4 + 2ξξ |Φ¯4 , |Ψ out = |0 + ξ|Φ2 + ξ |Φ2 + 2 where |Φ2 , |Φ4 are given in Eqs.(7.50, 7.51) and ˆ†s (ω1 )ˆ a†i (ω2 )|vac, |Φ2 = dω1 dω2 Φ (ω1 , ω2 )ei(ω1 +ω2 )ΔT a |Φ4 = dω1 dω2 dω1 dω2 Φ (ω1 , ω2 )ei(ω1 +ω2 )ΔT Φ (ω1 , ω2 )

|Φ¯4 =

ˆ†s (ω1 )ˆ a†i (ω2 )ˆ a†s (ω1 )ˆ a†i (ω2 )|vac, ei(ω1 +ω2 )ΔT a

(7.64)

(7.65)

(7.66)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ (ω1 , ω2 )

ei(ω1 +ω2 )ΔT a ˆ†s (ω1 )ˆ a†i (ω2 )ˆ a†s (ω1 )ˆ a†i (ω2 )|vac.

(7.67)

Let us ﬁrst evaluate the coincidence between detectors A and B, which is simply the auto-correlation of the signal ﬁeld: ˆs (t2 )|Ψ out ||2 pAB (t1 , t2 ) = ||Eˆs (t1 )E 4 ˆs (t1 )E ˆs (t2 )|Φ4 ||2 + (|ξ |4 /4)||E ˆs (t1 )Eˆs (t2 )|Φ ||2 = (|ξ| /4)||E 4 2 ˆ ˆ +|ξξ | ||Es (t1 )Es (t2 )|Φ¯4 ||2 , (7.68) where the cross terms between |Φ4 , |Φ4 , |Φ¯4 are zero because i = i . The ﬁrst two terms are same as Eq.(7.57) except that Φ(ω1 , ω2 ) is replaced by

7.2 Multi-Pair Production and Stimulated Pair Emission

127

Φ (ω1 , ω2 )ei(ω1 +ω2 )ΔT for the second term. The last term is not quite same as Eq.(7.57) in that there is no second term due to i = i and |ξ|4 is replaced by 4|ξξ |2 and F by F (ω2 , ω2 , ΔT ) ≡ dω1 dω1 Φ(ω1 , ω2 )Φ (ω1 , ω2 )ei(ω1 +ω2 )ΔT

× e−iω1 t1 −ω1 t2 + e−iω1 t1 −iω1 t2 . (7.69) After integrating the time variables t1 , t2 over the pulse period for the overall probability for detecting two photons, we obtain: PAB (ΔT ) = |ξ|4 (A + E) + |ξ |4 (A + E √ )+ ¯ )], +2|ξξ |2 [ AA + E(ΔT

(7.70)

where A, E, A , E are given in Eqs.(7.17, 7.18) with corresponding Φ-function and ¯ E(ΔT ) ≡ dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ (ω1 , ω2 )× × Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 −ω2 )ΔT . (7.71) Obviously, the ﬁrst two terms in Eq.(7.70) are the photon bunching eﬀects from two crystals, respectively, while the last term is the contribution from both crystals, with one photon from the ﬁrst and another from the second. ¯ What is interesting is the ΔT -dependent term E(ΔT ). It is maximum when ΔT = 0 — the extra coincidence depends on the overlap between the photons from the ﬁrst and the second crystals. This is the result of the stimulated emission, i.e., the second photon is stimulated by the ﬁrst one. In the case when the two crystals are identical, we have ξ = ξ and Φ(ω1 , ω2 ) = Φ (ω1 , ω2 ) and Eq.(7.70) becomes PAB (ΔT ) = 2|ξ|4 [2A + E + E(ΔT )], with

E(ΔT ) ≡

(7.72)

dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )×

× Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 −ω2 )ΔT .

(7.73)

If we deﬁne the visibility as V2 ≡

PAB (0) − PAB (∞) , PAB (∞)

then we have V2 (AB) = E/(2A + E) ≤ 1/3.

(7.74)

128

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

We may concentrate on only the last term of Eq.(7.70) by gating the coincidence measurement on the detection of the two idler photons with a fourphoton coincidence measurement among ABCD detectors (Fig.7.3). In this way, contribution from one crystal alone is zero. The four-photon coincidence rate is proportional to ˆs (t2 )Eˆi (t3 )E ˆi (t4 )|Ψ out ||2 pABCD (t1 , t2 , t3 , t4 ) = ||Eˆs (t1 )E 2 ˆ ˆs (t2 )Eˆi (t3 )E ˆi (t4 )|Φ¯4 ||2 . = |ξξ | |Es (t1 )E

(7.75)

As expected, the ﬁrst and the second terms in Eq.(7.70) are absent in Eq.(7.75). A straightforward calculation leads to pABCD (t1 , t2 , t3 ,t4 ) = |ξξ |2 dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ (ω1 , ω2 ) 2 −i(ω1 t1 +ω1 t2 ) −i(ω1 t2 +ω1 t1 ) −i(ω2 t3 +ω2 t4 ) e × e +e .

(7.76)

A time integral over all the time variables gives the overall four-photon coincidence probability: √ ¯ PABCD (ΔT ) = 2|ξξ |2 [ AA + E(ΔT )] 2¯ (7.77) = 2P2 P2 + 2|ξξ | E(ΔT ). As expected, we have only the last term of Eq.(7.70) in Eq.(7.77). From Eq.(7.77), we ﬁnd that there are two contributions to PABCD : the ﬁrst one is the accidental coincidence, whereas the second term gives some extra coincidence that is due to the stimulated emission. This term depends on the relative delay ΔT between the two signal ﬁelds. It is zero when ΔT = ±∞ and becomes maximum for ΔT = 0. Thus, this term can be interpreted as the stimulated emission of the second crystal by the ﬁrst signal photon. However, since the two signal ﬁelds are superposed, the enhanced coincidence can also be considered a result of constructive interference between the two signal ﬁelds (somewhat similar to the Hong-Ou-Mandel bunching eﬀect discussed in Sect.3.4). See Sect.8.1.2 for more details concerning this kind of interference. The visibility V4 (ABCD) of this interference eﬀect is,√thus, deﬁned in a ¯ AA ≤ 1 where similar way as in Eq.(7.74) and is simply V4 (ABCD) = E/ E¯ ≡ dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ (ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 ). (7.78) The factor 2 in the ﬁrst term of Eq.(7.77) arises from constructive interference between the two possibilities when detectors A and B detect two photons: photon 1 goes to A and photon 2 goes to B and vice versa. This is somewhat similar to Glauber’s explanation of the photon bunching eﬀect by two-photon interference [7.16].

7.2 Multi-Pair Production and Stimulated Pair Emission

129

Perhaps a more convincing and more direct evidence of stimulated emission is from the two-photon coincidence between the two idler ﬁelds: ˆi (t1 )E ˆi (t2 )|Ψ out ||2 pCD (t1 , t2 ) = ||E 2 ˆ ˆi (t2 )|Φ¯4 ||2 . = |ξξ | ||Ei (t1 )E

(7.79)

A simple calculation gives: 2

pCD (t1 , t2 ) = |ξξ |

with H(ω1 , ω1 ) ≡

dω1 dω1 |H(ω1 , ω1 )|2 +

+H(ω1 , ω1 )H ∗ (ω1 , ω1 ) ,

(7.80)

dω2 dω2 Φ(ω1 , ω2 )Φ (ω1 , ω2 )ei(ω1 +ω2 )ΔT −iω2 t1 −iω2 t2 . (7.81)

This leads to the overall two-photon coincidence after the time integral: √ ¯ PCD (ΔT ) = |ξξ |2 [ AA + E(ΔT )] 2¯ = P2 P2 + |ξξ | E(ΔT ). (7.82) Once again, the ﬁrst term is from the accidental coincidence between the two idler ﬁelds and the second term is some access coincidence. Since there is no superposition of the two idler ﬁelds, the only explanation for the extra coincidence at ΔT = 0 is stimulated emission. This is the most direct √ evidence ¯ AA = for stimulated emission. The visibility in this case is V2 (CD) = E/ V4 (ABCD). There is no factor 2 in the ﬁrst term of Eq(7.82) because the two photons arriving at detectors C and D are well distinguishable. √ The necessary and suﬃcient condition for E¯ = AA or V4 (ABCD) = 1 = V2 (CD) is Φ(ω1 , ω2 ) = Φ (ω1 , ω2 ) = ψ1 (ω1 )ψ2 (ω2 ), that is, the spatial modes of the two crystals are perfectly matched and the spectrum is factorized. The ﬁrst condition can be easily achieved with a single-mode ﬁber coupling for the spatial mode ﬁltering of the signal ﬁeld and the latter by spectral ﬁltering. Although the visibilities for PABCD and PCD are the same, the two measurements are quite diﬀerent in practice. Since the signal and idler photons are correlated, it is not necessary to mode match the two idler ﬁelds when observing photon bunching in PABCD . On √ the other hand, a spatial ¯ AA cannot be made with mode match between C and D in V2 (CD) = E/ a single-mode ﬁber in the signal ﬁelds because unlike PABCD , no detection of the signal ﬁelds is made in PCD . This makes the observation of the bunching eﬀect in PCD much harder than in PABCD and PAB . Another diﬀerence between PABCD and PCD is in their dependence on loss. Let us introduce some loss in between the two crystals by adding a beam splitter of transmission T in the signal ﬁeld, before √ right √ the second crystal (Fig.7.3). We need to replace a ˆ†s (ω1 ) by T a ˆ†s (ω1 ) + 1 − T a ˆ†s0 (ω1 )

130

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

in |Φ4 , |Φ¯4 in Eqs.(7.51, 7.67) with a ˆ†s0 (ω1 ) representing the vacuum mode from the loss. Then, it is straightforward to show that PAB is changed to

and

PAB (ΔT ) = T 2 |ξ|4 (A + E) + |ξ |4 (A√ + E )+ ¯ )] +2T |ξξ |2 [ AA + E(ΔT

(7.83)

√ ¯ PCD (ΔT ) = |ξξ |2 [ AA + T E(ΔT )],

(7.84)

but PABCD is only reduced by a factor of T from Eq.(7.77). Therefore, V4 (ABCD) is the same, but V2 (CD) is reduced by a factor of T and V2 (AB) to V2 (AB) =

2T E . (1 + T 2 )(A + E) + 2T A

(7.85)

The dependence of V2 (AB) on the transmission T is typical of two-photon interference of thermal sources. Thus, while the photon bunching eﬀects in PABCD and PAB may be interpreted both by stimulated emission and by two-photon interference, photon bunching in PCD can only be explained by stimulated emission. 7.2.3 Induced Coherence without Induced Emission In the scheme depicted in Fig.7.3, there is one more interesting phenomenon that is not related to multi-photon correlation but concerns coherence between the two idler ﬁelds. In our treatment of parametric down-conversion, we always assume weak pumping so that we can use perturbative expansion for the state. This is also the regime where the gain is close to one. Thus, the stimulated emission is much weaker than the spontaneous emission. However, even with the negligible induced emission, the signal ﬁeld injected into the second crystal may still be able to create coherence between the two idler ﬁelds. Consider the output state in Eq.(7.64), up to only two-photon terms. Let us calculate the second-order ﬁeld correlation function between the two idler ﬁelds: ˆ † (t1 )Eˆi (t2 ). ΓCD (t1 , t2 ) ≡ E i A straightforward calculation with ΔT = 0 leads to ΓCD (t1 , t2 ) = ξξ dω1 q ∗ (ω1 , t1 )q (ω1 , t2 ), with

q(ω1 , t1 ) ≡ q (ω1 , t2 ) ≡

(7.86)

(7.87)

dω2 Φ(ω1 , ω2 )e−iω2 t1 ,

(7.88)

dω2 Φ (ω1 , ω2 )e−iω2 t2 .

(7.89)

7.2 Multi-Pair Production and Stimulated Pair Emission

131

Experimentally observed correlation function is a time integral of ΓCD (t1 , t2 ): ΓCD (τ ) = dt1 ΓCD (t1 , t1 + τ ) (7.90) = ξξ dω1 dω2 Φ∗ (ω1 , ω2 )Φ (ω1 , ω2 )e−iω2 τ . To ﬁnd the degree of coherence, we evaluate the intensities of the two ﬁelds: √ (7.91) IC = dtΓCC (t, t) = |ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 = |ξ|2 A, √ ID = dtΓDD (t, t) = |ξ |2 dω1 dω2 |Φ (ω1 , ω2 )|2 = |ξ |2 A . (7.92) Then, the degree of coherence or the visibility of interference is γCD (τ ) ≡ ΓCD (τ )/ IC ID = (AA )−1/4

dω1 dω2 Φ∗ (ω1 , ω2 )Φ (ω1 , ω2 )e−iω2 τ .

(7.93)

When τ = 0 and the modes are matched so that Φ(ω1 , ω2 ) = Φ (ω1 , ω2 ), we have γCD = 1 or 100% coherence between the two idler ﬁelds. On the other hand, γCD degrades with imperfect mode match. As a matter of fact, similar to V2 (CD), γCD is sensitive to the loss of the signal ﬁeld before it is coupled into the second crystal. a beam splitter inserted, a ˆ†s (ω1 ) in |Φ2 √ †With √ † in Eq.(7.65) is replaced by T a ˆs (ω1 ) + 1 − T a ˆs0 (ω1 ). Then, γCD is changed to T dω1 dω2 Φ∗ (ω1 , ω2 )Φ (ω1 , ω2 )e−iω2 τ . γCD (τ ) = (7.94) (AA )1/4 When T = 0, that is, the coupling to the second crystal is blocked, γCD = 0 and we lose completely the coherence between the two idler ﬁelds. To explain the induced coherence in a simple physical picture, we look at the single-mode description: |Ψ out = ... + η|1s , 1i + η |1s , 1i + ...,

(7.95)

where we dropped the irrelevant terms. The degree of coherence between i and i is simply: ˆ a†i a ˆi γii = = 1s |1s . ˆ a†i a ˆi ˆ a†i a ˆi

(7.96)

When 1s = 1s , which is equivalent to the mode match between the two signal ﬁelds, γii = 1. In this case, the output state becomes |Ψ out = ... + |1s (η|1i + η |1i ) + ....

(7.97)

132

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

This is a pure state for the two idler ﬁelds and exhibits complete coherence. But when 1s |1s = 0, which corresponds to the complete mode mismatch, γii = 0. The state of the two idler ﬁelds can be described by a density operator after tracing out the signal ﬁelds: ρˆout = ... + |η|2 |1i 1i | + |η |2 |1i 1i | + ...,

(7.98)

which is a mixed state with no coherence. From the above, we see that the existence of coherence between the two idler ﬁelds is not due to the stimulated emission but because of overlap of the two signal ﬁelds, or the indistinguishability of the signal photons. The state in Eq.(7.95) shows that the idler photon can be identiﬁed by its companion signal photon. If we cannot distinguish the signal photons, as in Eq.(7.97), by the complementary principle of quantum mechanics, the indistinguishability leads to coherence between the two idler ﬁelds. However, if the two signal photons can be completely distinguished, we lose the coherence, as in Eq.(7.98). This phenomenon of induced coherence without stimulated emission was ﬁrst observed by Zou et al. in 1991 [7.20].

7.3 Distinguishable or Indistinguishable Pairs of Photons From the discussion in previous sections, we ﬁnd that the quantity E/A characterizes the temporal distinguishability between two photons in one of the signal and the idler ﬁelds. (For more discussion on temporal distinguishability of a multi-photon state, see Chapt.9.) When E/A = 0, the two photons are independent of each other and are completely distinguishable in time, whereas when E/A = 1, the two photons are in the same temporal mode and become indistinguishable in time, leading to the perfect photon bunching eﬀect. Since the photons between the signal and the idler ﬁelds are correlated in time, we may as well consider the quantity E/A to be a characterization of the temporal distinguishability between two pairs of photons in parametric down-conversion. With this picture in mind, we ﬁnd the situation of E/A = 1 more interesting, for it corresponds to the case when the two pairs of photons merge into four temporally indistinguishable photons, i.e., the four photons are in one temporal mode. We then have a genuine four-photon state. We refer to this case as the 4 × 1 case, where the number 4 represents the four photons and the number 1 represents the single temporal mode. On the other hand, the other extreme case of E/A = 0 corresponds to the situation when the two pairs of photons are independent from each other. We refer to this case as the 2 × 2 case, meaning two pairs are each in two well-separated temporal modes. In reality, however, E/A lies between those two extreme values, indicating partial overlap of the two pairs. The multi-mode model of parametric downconversion covers this arbitrary situation. However, can this quantity serve to describe partial distinguishability in the intermediate case? To answer this,

7.3 Distinguishable or Indistinguishable Pairs of Photons

133

we resort to quantum interference of four photons. From the complementary principle of quantum mechanics, we know that the eﬀect of quantum interference, which is usually described by the visibility of interference, is related to the degree of indistinguishability. If the quantity E/A is to serve as the measure of pair (in)distinguishability in time, it should be related to the visibility of four-photon interference. This is the topic of the next chapter. Before we proceed to next chapter, let us evaluate the value of E/A for various situations in parametric down-conversion and ﬁnd out how to increase its value and make it approach the ideal value of one. From Eqs.(7.17, 7.18, 2.35), we have: A ≡ dω1 dω1 dω2 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2 (7.99) and

E≡

dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 ), (7.100)

with Φ(ω1 , ω2 ) = αp (ω1 + ω2 )h(LΔk|ω3 =ω1 +ω2 ),

(7.101)

where αp is the spectral amplitude of the pump ﬁeld and Δk is given in Eqs.(2.45, 2.54) for the two types of parametric down-conversion. Without loss of generality, we assume αp take a Gaussian form of αp (ωp ) = e−(ωp −ωp0 )

2

/2σp2

.

(7.102)

The Schwartz inequality leads to E ≤ A, with the equality holding if and only if Φ(ω, ω2 ) is factorized as in Eq.(7.14). With the complicated form of the h-function (Figs.2.4,2.7), it is impossible to have E/A = 1 without any help. 0.30 0.25

ε/A

0.20 0.15 0.10 0.05 0 0

2

4

6

8

10 12 14 16 18 20

σp/Ωcw

Fig. 7.4. Typical E/A as a function of the pump bandwidth σp with a Gaussian shape normalized to the bandwidth Ωcw of cw downconversion for type-II parametric down-conversion.

For type-II parametric down-conversion without extended phase matching (EPM), Fig.7.4 shows a typical dependence of 1 = E/A on the pump bandwidth σp normalized to the bandwidth Ωcw = 1/L|kp − ks | with tan γ = 2.

134

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

It can be seen that E/A = 0 for both cw pumping with σp → 0 and for an ultra-short pump pulse with σp → ∞. A maximum of 0.3 is achieved when σp ∼ 2Ωcw . The zero value of E/A for an ultra short pump pulse is in contradiction with the intuitive picture discussed at the beginning of Sect.7.1. The reason is that when the pump bandwidth becomes very large (for ultra-short pulses), the spectrum of the down-converted ﬁelds is mainly determined by the h-function. But from Fig.2.7, there exists some kind of frequency correlation as in Eq.(2.54) for the signal and idler ﬁelds for type-II parametric down-conversion. So, when we look only at one ﬁeld, it becomes incoherent. In this case, one ﬁeld serves as the label for the other ﬁeld, which will lose its coherence because of distinguishability, similar to the eﬀect discussed in Sect.7.2.3. On the other hand, the frequency correlation disappears for type-I downconversion when the group velocities are matched and it has a well-behaved h-function, as shown in Fig.2.6b. We should expect a better E/A value. This is, indeed, the case, as demonstrated in Fig.7.5. Notice that E/A does not go to zero as the pump bandwidth goes to inﬁnity. 1.00

ε/A

0.80 0.60

Fig. 7.5. E/A as a function of the pump bandwidth σp with a Gaussian shape normalized to the bandwidth Ωcw of cw down-conversion for group velocity-matched type-I parametric downconversion.

0.40 0.20 0 0

1

2

3

4

5

σp/Ωcw

As mentioned in Sect.7.1.1, the value of E/A may be increased by extra spectral ﬁltering. When the bandwidth of the ﬁlters are much narrower than the pump bandwidth and down-conversion bandwidth, Φ is mainly determined by the ﬁlter function and becomes factorized, as in Eq.(7.14). In this condition, E/A → 1. The other extreme case is when the ﬁlter bandwidth is much larger than the bandwidth of both the pump and the down-conversion. Then, the ﬁlters have no eﬀect and E/A is same as in Figs.7.4 and 7.5. The intermediate case is when the bandwidths of the ﬁlters are much narrower than the downconversion bandwidth but comparable to the pump bandwidth. To account for this intermediate situation of a ﬁnite pass band of f (ω), we take a Gaussian form for f (ω) as f (ω) = e−(ω−ωp0 /2)

2

/2σf2

,

(7.103)

7.3 Distinguishable or Indistinguishable Pairs of Photons

135

and assume the bandwidth Ωcw of the cw parametric down-conversion is much wider than the bandwidths of the ﬁlters, i.e., Ωcw >> σf . Then, the sincfunction in the h-function can be approximated by one and we have: ¯ 1 , ω2 ) ≈ e−(ω1 +ω2 −ωp0 )2 /2σp2 e−[(ω1 −ωp0 /2)2 +(ω2 −ωp0 /2)2 ]/2σf2 . Φ(ω

(7.104)

With this Φ-function, we can easily evaluate E and A from Eqs.(7.99, 7.100) and obtain: 2σf2 + σp2 σ p E . (7.105) = A σp2 + σf2 It can be seen that E/A → 1 when σp >> σf . If we consider the coherence time of down-converted photons to be Tc = 1/σf and the pulse duration of the pump to be Tp = 1/σp , then Eq.(7.105) is consistent with the intuitive picture at the beginning of Sect.7.1. de Riedmatten et al. [7.21] were the ﬁrst to investigate the eﬀect of the ﬁltering on the coherent property of the downconverted ﬁelds. On the other hand, because of the ﬁnite bandwidth Ωcw of down-converted ﬁelds, we must consider it, even with the insertion of spectral ﬁlters. Fig.7.6 shows the value of E/A as a function of ﬁlter bandwidth σf normalized to Ωcw . The values are optimized for the pump bandwidth for maximum E/A. It can be seen that E/A → 0.27 for wide ﬁlter bandwidth, which is the maximum value in Fig.7.4 for the case without ﬁlters. 1.2 1.0 0.8

ε/A

0.6 0.4 0.2 0 0

5

10

σf /Ωcw

15

20

Fig. 7.6. Typical E/A as a function of the ﬁlter bandwidth σf normalized to Ωcw at optimum pump bandwidth.

From the expressions for A and E in Eqs.(7.99, 7.100), we ﬁnd the condition for A = E is that the wave function Φ(ω1 , ω2 ) is factorized: Φ(ω1 , ω2 ) = ψ(ω1 )φ(ω2 ).

(7.106)

Although it is impossible for the parametric down-conversion from a bulk crystal to produce such a function, eﬀorts [7.22, 7.23, 7.24, 7.25, 7.26, 7.27]

136

7 Coherence and Multiple Pair Production in Parametric Down-Conversion

are underway to engineer the structure of the nonlinear media for achieving the factorization condition in Eq.(7.106).

References 7.1 M. Zukowski, A. Zeilinger, and H. Weinfurter, in Fundamental Problems in Quantum Theory, edited by D. M. Greenberger and A. Zeilinger, Ann. NY Acad. Sci. 755, 91 (1995). 7.2 J. G. Rarity, in Fundamental Problems in Quantum Theory, edited by D. M. Greenberger and A. Zeilinger, Ann. NY Acad. Sci. 755, 624 (1995). 7.3 R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). 7.4 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). 7.5 K. Naganuma, K. Mogi, and H. Yamada, IEEE J. of Quan. Eletr. 25, 1225 (1989). 7.6 D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244(1993). 7.7 A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller Phys. Rev. Lett. 87, 050402 (2001). 7.8 C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986). 7.9 Z. Y. Ou, L. J. Wang, and L. Mandel, J. Opt. Soc. Am. B7 211 (1990). 7.10 H. P. Yuen, Phys. Rev. A 13, 2226 (1976). 7.11 B. Yurke and M. Potasek, Phys. Rev. A 36, 3464 (1987). 7.12 S. Carusotto, Phys. Rev. A 11, 1629 (1975). 7.13 Z. Y. Ou, S. F. Pereira, and H. J. Kimble, Phys. Rev. Lett. 70, 3239 (1993). 7.14 M. Vasilyev, S.-K. Choi, P. Kumar, and G. M. D’Ariano, Phys. Rev. Lett. 84, 2354 (2000). 7.15 R. Hanbury-Brown and R. W. Twiss, Nature 177, 27 (1956). 7.16 R. J. Glauber, Quantum Optics and Electronics (Les Houches Lectures), p.63, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965). 7.17 A. Einstein, Phys. Z. 18, 121 (1917). 7.18 E. M. Purcell and R. V. Pound, Phys. Rev. 81, 279 (1951). 7.19 J. P. Gordon, H. J. Zeiger, and C. H. Townes, Phys. Rev. 99, 1264 (1955). 7.20 X. Y. Zou, L. J. Wang, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991). 7.21 H. de Riedmatten, V. Scarani, I. Marcikic, A. Acin, W. Tittel, H. Zbinden, and N. Gisin, J. Mod. Opt. 51, 1637 (2004). 7.22 V. Giovannetti, L. Maccone, J. H. Shapiro, and F. N. C. Wong, Phys. Rev. Lett. 88, 183602 (2002). 7.23 A. B. U Ren, K. Banaszek, and I. A. Walmsley, Quant. Inf. and Comp. 3, 480 (2003). 7.24 Z. D. Walton, M. C. Booth, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 67, 053810 (2003). 7.25 Z. D. Walton, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 70 052317 (2004). 7.26 M. G. Raymer, Jaewoo Noh, K. Banaszek, and I. A. Walmsley, Phys. Rev. A 72, 023825 (2005). 7.27 J. P. Torres, F. Maci´ a, S. Carrasco, and Torner L. Opt. Lett., 30 314 (2005).

8 Quantum Interference with Two Pairs of Down-Converted Photons

As more qubits are involved in quantum information processing [8.1], we inevitably start to deal with the quantum interference of more than two photons. However, multi-photon states are diﬃcult to produce. So, it is natural to consider four-photon interference by using two pairs of photons from parametric down-conversion. Since the two photons within one pair are already correlated in time, we just need to consider the relationship between diﬀerent pairs. This was discussed extensively in the previous chapter. The quantity E/A is used to characterize this relationship. In this chapter, we will see how the quantity E/A aﬀects the visibility of the interference eﬀect involving two pairs of down-converted photons.

8.1 Hong-Ou-Mandel Interferometer for Independent Photons As we will see in later sections, many of the quantum information protocols such as quantum state teleportation [8.2, 8.3] and swapping [8.4, 8.5, 8.6] involve interference between two independent photons. It is the simplest interference scheme, with two pairs of down-converted photons. The Hong-OuMandel interferometer discussed in Chapt.3 deals with two correlated photons. Here we will consider two independent photons, one from one pair and the other from another pair of down-converted photons. The scheme is depicted in Fig.8.1 where two parametric down-conversion processes are considered, with each providing one pair of photons. When gated on the detection of the idler photon, the signal ﬁeld is in a single-photon state [8.7] (Fig.1a). Therefore, a four-photon coincidence measurement will be equivalent to the two-photon coincidence measurement in the Hong-Ou-Mandel interferometer. Since the production of the two pairs is independent, the gated single photons entering the Hong-Ou-Mandel interferometer will be independent. But before investigating this situation, we consider the case without gating, by looking at the two-photon coincidence from A and B detectors only, as in Fig.8.1b.

138

8 Quantum Interference with Two Pairs of Down-Converted Photons

D

trigger i1

i1 PDC1 s1

B

∆T

Pump

s2

PDC2

A

PDC1

s1 ∆T

Pump PDC2

i2 C (a)

s2

B A

i2

trigger (b)

Fig. 8.1. Two-photon interference with photons from two independent processes of parametric downconversion: (a) with and (b) without the gating triggers from the detection of the idler photons.

8.1.1 Two-Photon Interference without Gating For the situation in Fig.8.1b, there will be contributions from two pairs generated entirely from one crystal alone, in addition to the contribution from two crystals with one pair from each. The former contributions are similar to the photon bunching eﬀect discussed in Sect.7.2 and do not produce an interference eﬀect, whereas the latter give rise to the Hong-Ou-Mandel interference eﬀect. To take account of all these contributions, we start by writing down the quantum state that describes the overall system: |Ψ = |Ψ (1) ⊗ |Ψ (2) ,

(8.1)

where |Ψ (1,2) describes the quantum state from each parametric process and has the form of Eq.(7.49) with a ˆs changed to a ˆs1 or a ˆs2 and a ˆi changed to a ˆi1 or a ˆi2 . We assume the two processes are identical with the same Φ. From Fig.8.1b, we ﬁnd that the s1 and s2 ﬁelds are superposed with a beam splitter but the i1 and i2 ﬁelds are left unattended. Two-photon coincidence between the two outputs of the beam splitter is measured. The output ﬁelds of the beam splitter are connected to s1 and s2 ﬁelds by

√ ˆA (t) = E ˆ (t) + E ˆs2 (t + ΔT ) / 2, E s1

√ (8.2) ˆs2 (t) − E ˆs1 (t − ΔT ) / 2 , ˆB (t) = E E where we introduced a time delay ΔT on the reﬂected ﬁelds so that the transmitted and the reﬂected ﬁelds arrive at the detectors at diﬀerent times. This time delay is important because when ΔT is zero, we should have complete overlap of the two ﬁelds and maximum interference eﬀect; but when ΔT is larger than the coherent time of the two ﬁelds, no interference should occur. This provides a base line for calculating the visibility of the interference. So, by comparing the coincidence at these two values of ΔT , we may deduce the visibility of the interference. ˆB ﬁelds is proportional to Two-photon coincidence between EˆA and E ˆ † (t2 )E ˆ † (t1 )EˆA (t1 )EˆB (t2 )|Ψ pAB (t1 , t2 ) = Ψ |E B A ˆA (t1 )E ˆB (t2 )|Ψ ||2 . = ||E

(8.3)

8.1 Hong-Ou-Mandel Interferometer for Independent Photons

139

Using Eq.(8.2), we have: ˆB (t2 ) = 1 Eˆs2 (t1 + ΔT )E ˆs2 (t2 ) − E ˆs1 (t1 )Eˆs1 (t2 − ΔT ) ˆA (t1 )E E 2 ˆs2 (t2 ) − E ˆs2 (t1 + ΔT )E ˆs1 (t2 − ΔT ) . ˆs1 (t1 )E +E

(8.4)

Because of the existence of idler ﬁelds in |Ψ , it can be shown easily that ˆs1 |Ψ , E ˆs2 |Ψ , and (E ˆs2 − Eˆs2 E ˆs1 )|Ψ are mutually ˆs1 E ˆs2 E ˆs1 E the states E orthogonal in Eq.(8.3). So we have: pAB (t1 , t2 ) =

1 ˆ ˆs1 (t2 − ΔT )|Ψ ||2 + ||Eˆs2 (t1 + ΔT )E ˆs2 (t2 )|Ψ ||2 ||Es1 (t1 )E 4

ˆs1 (t1 )E ˆs2 (t2 ) − E ˆs1 (t2 − ΔT )E ˆs2 (t1 + ΔT ) |Ψ ||2 . (8.5) +|| E

The observed coincidence is a time integral of pAB (t1 , t2 ): ∞ PAB (ΔT ) = dt1 dt2 pAB (t1 , t2 ).

(8.6)

−∞

The contributions from the ﬁrst two terms in Eq.(8.5) have been calculated in Eq.(7.60). We simply rewrite the result as follows: ˆs1 (t1 )E ˆs1 (t2 − ΔT )|Ψ 2 dt1 dt2 E ˆs2 (t2 )|Ψ 2 ˆs2 (t1 + ΔT )E = dt1 dt2 E = |ξ|4 (A + E).

(8.7)

Notice that these two terms are independent of the time delay ΔT and simply add to the base line. The contribution from the last term in Eq.(8.5) gives ˆs1 (t1 )Eˆs2 (t2 )|Ψ : rise to interference. To calculate it, we start with E ˆs2 (t2 )|Ψ = Eˆs1 (t1 )|Ψ (1) ⊗ E ˆs2 (t2 )|Ψ (2) Eˆs1 (t1 )E 2 ξ = dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) 2π ˆ†i1 (ω2 )ˆ a†i2 (ω2 )|vac. ×e−iω1 t1 −iω1 t2 a

(8.8)

For the other term, we can obtain it simply by replacing t1 with t2 − ΔT and t2 with t1 + ΔT in Eq.(8.8). So we have: ˆs1 (t1 )Eˆs2 (t2 ) − E ˆs1 (t2 − ΔT )E ˆs2 (t1 + ΔT )]|Ψ 2 [E |ξ|4 dω dω1 dω1 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) = dω 2 2 (2π)2 2 −iω1 t1 −iω1 t2 −iω1 (t2 −ΔT )−iω1 (t1 +ΔT ) × e −e

140

8 Quantum Interference with Two Pairs of Down-Converted Photons

=

|ξ|4 (2π)2

dω2 dω2 dω1 dω1 e−iω1 t1 −iω1 t2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) 2 i(ω1 −ω1 )ΔT −Φ(ω1 , ω2 )Φ(ω1 , ω2 )e (8.9) ,

a†i2 (ω2 )|vac} is a set of orthogonal states where we used the fact that {ˆ a†i1 (ω2 )ˆ and we made the switch ω1 ↔ ω1 in the second term in the integrand. Now we can carry out the time average. The result is ˆs2 (t2 ) − Eˆs1 (t2 − ΔT )E ˆs2 (t1 + ΔT )]|Ψ 2 dt1 dt2 [Eˆs1 (t1 )E 4 dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) = |ξ| 2 i(ω1 −ω1 )ΔT −Φ(ω1 , ω2 )Φ(ω1 , ω2 )e = 2|ξ|4 [A − E(ΔT )]

(8.10)

with E(ΔT ) given in Eq.(7.73). Notice that E(0) = E, and E(∞) = 0 if Φ(ω1 , ω2 ) has a ﬁnite bandwidth. Combining Eqs.(8.5-8.7, 8.10), we ﬁnd the overall coincidence equal to PAB (ΔT ) = (|ξ|4 /2)[2A + E − E(ΔT )].

(8.11)

Therefore, as the time delay ΔT scans through 0 from −∞ to +∞, the fourphoton detection probability will show a dip at ΔT = 0 with a minimum value of |ξ|4 A. The baseline for the coincidence is at ΔT = ±∞ with a value of |ξ|4 (2A + E)/2. So the visibility of the interference pattern is V2 =

E |PAB (∞) − PAB (0)| = . PAB (∞) 2A + E

(8.12)

Since E ≤ A, the maximum value of V2 is 1/3. As we found in Sect.7.2.1, parametric down-conversion, when only one ﬁeld, either the signal or the idler, is concerned, is a source of thermal nature. The visibility of 1/3 derived above is consistent with the theory of two-photon interference between two thermal ﬁelds [8.8, 8.9]. The visibility in Eq.(8.12) is exactly same as that derived from Eq.(7.74), indicating that, indeed, the bunching eﬀect discussed in Sect.7.2.2 can be explained in terms of constructive interference, although it is destructive interference here. Constructive interference occurs if detectors A and B are placed on the same side of the beam splitter in Fig.8.1b (see Sect.3.4) and Eq.(8.11) is changed to PAB (ΔT ) = (|ξ|4 /2)[2A + E + E(ΔT )],

which gives a bump instead of a dip as ΔT is scanned.

(8.13)

8.1 Hong-Ou-Mandel Interferometer for Independent Photons

141

8.1.2 Two-Photon Interference with Gating: Hong-Ou-Mandel Interferometer for Two Independent Photons In the interference scheme discussed in the previous section, the two signal ﬁelds from two independent parametric down-conversion are used without any participation of the conjugate idler ﬁelds. So, there are two-photon contributions from one crystal that raise the baseline leading to reduced visibility. However, if we can gate the coincidence measurement on the detection of the two idler photons in the two parametric processes, we may guarantee that the two photons arriving at detectors A and B are, respectively, from two crystals, thus eliminating the unwanted baseline. The gated signal ﬁelds in this case will be in a single-photon Fock state [8.7] and the situation is no diﬀerent from the correlated two-photon case [8.10]. We should expect the visibility of the interference will reach 100% in an ideal condition and the coincidence will dip down to zero at zero delay (ΔT = 0). A similar technique was used in Sect.7.2.2 in the study of the stimulated emission. In the following, we will calculate the visibility in this interference scheme and relate it to the quantities A and E. If we gate the coincidence measurement on the detection of the two idler photons, this corresponds to quadruple coincidence measurement of two output ﬁelds of the beam splitter and two idler ﬁelds (Fig.8.1a). The coincidence rate is proportional to ˆi1 (t3 )Eˆi2 (t4 )|Ψ ||2 . p4 (t1 , t2 , t3 , t4 ) = ||EˆA (t1 )EˆB (t2 )E

(8.14)

To calculate p4 , we make use of Eq.(8.4) for EˆA (t1 )EˆB (t2 ). But, it can be shown easily that there is no contribution from the ﬁrst two terms, that is, ˆs1 (t2 − ΔT )E ˆi1 (t3 )Eˆi2 (t4 )|Ψ = 0, ˆs1 (t1 )E E ˆs2 (t1 + ΔT )E ˆs2 (t2 )Eˆi1 (t3 )Eˆi2 (t4 )|Ψ = 0, E (1,2)

(8.15)

up to the order of ξ 2 because of the two-photon nature of |Φ2 in Eq.(7.50). For the contribution from the last two terms in Eq.(8.4), we have:

ˆs2 (t2 ) − Eˆs1 (t2 − ΔT )E ˆs2 (t1 + ΔT ) E ˆi1 (t3 )Eˆi2 (t4 )|Ψ ˆs1 (t1 )E E 2 ξ = dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) e−iω1 t1 −iω1 t2 −iω2 t3 −iω2 t4 (2π)2

−e−iω1 (t1 +ΔT )−iω1 (t2 −ΔT )−iω2 t3 −iω2 t4 |vac ξ2 = dω1 dω1 dω2 dω2 e−iω1 t1 −iω1 t2 −iω2 t3 −iω2 t4 (2π)2

× Φ(ω1 , ω2 )Φ(ω1 , ω2 ) − Φ(ω1 , ω2 )Φ(ω1 , ω2 )ei(ω1 −ω1 )ΔT |vac. (8.16) So, the coincidence rate is |ξ|4 dω1 dω1 dω2 dω2 e−iω1 t1 −iω1 t2 −iω2 t3 −iω2 t4 p4 (t1 , t2 , t3 , t4 ) = 4 4(2π)

142

8 Quantum Interference with Two Pairs of Down-Converted Photons

2 i(ω1 −ω1 )ΔT × Φ(ω1 , ω2 )Φ(ω1 , ω2 ) − Φ(ω1 , ω2 )Φ(ω1 , ω2 )e , (8.17) and the overall quadruple detection probability in one pulse is the time average over the pulse duration and is calculated to be: |ξ|4 dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) P4 (ΔT ) = 4 2 −Φ(ω1 , ω2 )Φ(ω1 , ω2 )ei(ω1 −ω1 )ΔT |ξ|4 [A − E(ΔT )], (8.18) = 2 where A and E(ΔT ) are the quantities deﬁned in Eq.(7.17) and Eq.(7.73), respectively. As expected, the gated detection takes out the extra ΔT independent contribution given in Eq.(8.7) to the overall coincidence. Therefore, the visibility of the interference fringe is V4 = E/A.

(8.19)

In the ideal case when E = A, we will have V4 = 1, i.e., 100% visibility, which is exactly what we expect for the interference of two Fock states. Notice that the visibility is directly connected to the photon bunching quantity E/A. Such a connection was ﬁrst demonstrated by Wang and Rhee [8.11] and later conﬁrmed by de Riedmatten et al. [8.12]. When the A and B detectors are placed on the same side of the beam splitter in Fig.8.1a, it is straightforward to show that similar to Eq.(8.13), P4 (ΔT ) =

|ξ|4 [A + E(ΔT )], 2

(8.20)

which leads to a bump of the same visibility as in Eq.(8.19). This is similar to the photon bunching eﬀect discussed in Sect.3.4. The expression in Eq.(8.20) is the same as that in Eq.(7.77), conﬁrming the interpretation by two-photon interference of the stimulated emission.

8.2 Quantum State Teleportation and Swapping Another type of interference eﬀect between independent ﬁelds from parametric down-conversion involves polarization entanglement. Interference occurs by projecting ﬁelds with correlated polarization into a certain common direction. Quantum superposition stems from projection of ﬁelds of diﬀerent polarizations. We already discussed the violation of Bell’s inequality by a Bohm-type EPR state in two-photon interference of correlated photons with diﬀerent polarizations in Sect.4.1. The application of the two-photon interference eﬀect

8.2 Quantum State Teleportation and Swapping

143

between independent ﬁelds to parametric down-conversion with polarization entanglement leads to quantum information protocols of quantum state teleportation and entanglement swapping [8.2, 8.3, 8.4, 8.5, 8.6]. The underlying principle of these phenomena is the same as that discussed in the previous section. We will address them in detail in this section.

|1θ

Bell Measurement

Classical Channel |1θ R EPR (|Ψ(−) )

Fig. 8.2. Quantum state teleportation scheme by Bennett et al. [8.2]

8.2.1 Quantum State Teleportation: Single-Mode Case In 1990, Bennett et al. [8.2] proposed a scheme that can teleport an arbitrary polarization state of a single photon: |1θ 1 = cos θ|H1 + sin θ|V 1 , with H, V denoting horizontal and vertical polarizations, respectively. It is achieved (Fig.8.2) by mixing it with a pair of photons in a Bohm-type EPR singlet state of √ (8.21) |Ψ (−) 23 = (|H2 , V3 − |V2 , H3 )/ 2. The quantum state of the three-photon system is then: (−) |SY S123 = |1 θ 1 ⊗ |Ψ 23 = cos θ|H1 |H2 , V3 − cos θ|H1 |V2 , H3

√ + sin θ|V 1 |H2 , V3 − sin θ|V 1 |V2 , H3 / 2. (8.22)

Now by rewriting the part of the above state involving photons 1 and 2, in terms of the Bell states of (Sect.4.1.2), namely: 1 (±) (8.23) |Ψ12 = √ |H1 |V2 ± |V1 |H2 , 2 1 (±) (8.24) |Φ12 = √ |H1 |H2 ± |V1 |V2 , 2 we obtain: |SY S123 =

1 (−) (+) |Ψ (cos θ|H3 + sin θ|V 3 ) + |Ψ12 (cos θ|H3 − sin θ|V 3 ) 2 12

144

8 Quantum Interference with Two Pairs of Down-Converted Photons

(−) (+) +|Φ12 (sin θ|H3 + cos θ|V 3 ) + |Φ12 (cos θ|V 3 − sin θ|H3 ) . (8.25) Since the Bell states in Eqs.(8.23, 8.24) are orthogonal and form a complete base set for the two-photon polarization state, we can make projection measurement of them on photon 1 and 2 in the state of |SY S123 . The third photon will be projected to the corresponding states in Eq.(8.25), with four diﬀerent outcomes in the measurement. The ﬁrst projected state of photon 3 is exactly the same as the original state |1θ of photon 1. The remaining three states can be rotated by a local operation R to the original state |1θ of photon 1, upon knowledge of the outcomes of the Bell measurement (Fig.8.2). Thus, the original polarization state |1θ of photon 1 is teleported to photon 3 in another location. Bell State Measurement

trigger D

i1 PDC1 s1 Pump rotator θ1 s2 PDC2 i2 θ2

B A

C

Four-Photon Coincidence

Fig. 8.3. The detailed geometry of quantum state teleportation with parametric down-conversion processes.

Realization of teleportation with parametric down-conversion was ﬁrst proposed by Braunstein and Mann [8.13] and was demonstrated experimentally by Bouwmeester et al. [8.3]. The schematic is shown in Fig.8.3. Two parametric down-converters are used in the demonstration. The ﬁrst one prepares a single-photon polarization state |θ1 s1 = cos θ1 |Hs1 + sin θ1 |V s1 in s1 by gating all the measurement on the detection of i1. The second √ one produces a Bohm-EPR singlet state: |Ψ (−) = (|Hs2 , Vi2 − |Vs2 , Hi2 )/ 2. By gating on the coincidence measurement between the outputs of the beam splitter with s1 and s2 as input (Bell measurement of |Ψ (−) , see Sect.4.1.2), the state of i2 is expected to be projected into |θ1 i2 = cos θ1 |Hi2 + sin θ1 |V i2 , thus achieving teleportation. The output polarization state in i2 can be checked by taking a polarization measurement in the direction of θ2 . By Malus’ law, we should ﬁnd the detection rate after the polarizer to be proportional to cos2 (θ1 − θ2 ), if the teleportation is successful. Notice the similarity in the arrangement between Fig.8.2 and Fig.8.3. The diﬀerence lies only in the polarization measurement on i2 and the state of the second parametric down-conversion (PDC2).

8.2 Quantum State Teleportation and Swapping

145

8.2.2 Quantum State Teleportation: Multi-Mode Case The multi-mode description of the down-conversion states is similar to Eq.(8.1). To cope with the polarization state of the ﬁelds, we introduce an extra degree of freedom for polarization entanglement. Keeping only the four-photon term, we may write the quantum state of the system as |Ψ = |Ψ (1) (θ1 ) ⊗ |Ψ (2) (EP R), with |Ψ (1) (θ1 ) = ξ

dω1 dω2 Φ(ω1 , ω2 ) a ˆ†s1H (ω1 ) cos θ1 +ˆ a†s1V (ω1 ) sin θ1 a ˆ†i1 (ω2 )|vac,

(8.26)

(8.27)

where the polarization of the signal photon is rotated by an angle of θ1 and 1 † ˆ (ω1 )ˆ a†i2V (ω2 ) |Ψ (2) (EP R) = ξ dω1 dω2 Φ(ω1 , ω2 ) √ a 2 s2H a†i2H (ω2 ) |vac. (8.28) −ˆ a†s2V (ω1 )ˆ Here, we consider a multi-mode polarization entangled two-photon state sim(−) ilar to the single-mode state of |Ψ12 in Eq.(8.23) for the second parametric down-conversion process. So, after the expansion of the product, Eq.(8.26) becomes ξ2 dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) |Ψ = √ 2 a†i1 (ω2 )ˆ a†s2H (ω1 )ˆ a†i2V (ω2 ) cos θ1 × a ˆ†s1H (ω1 )ˆ +ˆ a†s1V (ω1 )ˆ a†i1 (ω2 )ˆ a†s2H (ω1 )ˆ a†i2V (ω2 ) sin θ1 −ˆ a†s1H (ω1 )ˆ a†i1 (ω2 )ˆ a†s2V (ω1 )ˆ a†i2H (ω2 ) cos θ1 † † † −ˆ as1V (ω1 )ˆ ai1 (ω2 )ˆ as2V (ω1 )ˆ a†i2H (ω2 ) sin θ1 |vac. (8.29) Similar to Eq.(8.5), we have, for the ﬁelds of two outputs of the beam splitter: ⎧

√ ˆAH (t) = E ˆs1H (t) + E ˆs2H (t + ΔT ) / 2, ⎪ E ⎪

√ ⎪ ⎨E ˆs2H (t) − E ˆs1H (t − ΔT ) / 2, ˆBH (t) = E

√ (8.30) ˆs1V (t) + E ˆs2V (t + ΔT ) / 2, ˆAV (t) = E ⎪ E ⎪ ⎪ √

⎩ ˆ ˆs2V (t) − E ˆs1V (t − ΔT ) / 2, EBV (t) = E with 1 Eˆj (t) = √ 2π

dωˆ aj (ω)e−iωt (j = s1H, s1V, s2H, s2V, i1).

(8.31)

146

8 Quantum Interference with Two Pairs of Down-Converted Photons

Here, because two polarizations are involved, we consider both of them in the ﬁeld operators. The polarization measurement at detector C is related to the ﬁeld operator of 1 dω[ˆ ai2H (ω) cos θ2 + a ˆi2V (ω) sin θ2 ]e−iωt , (8.32) EˆC (t) = √ 2π and the detection rate at detector C, gated on the other three detectors, is proportional to the four-photon coincidence rate: p4 (t1 , t2 , t3 , t4 ) = : IˆA (t1 )IˆB (t2 )IˆC (t3 )IˆD (t4 ) :,

(8.33)

with ˆ† E ˆ† ˆ ˆ IˆA = E AH AH + EAV EAV , ˆ† E ˆ† ˆ ˆ IˆB = E BH BH + EBV EBV , ˆC , ˆ† E IˆC = E C

ˆ† E ˆ IˆD = E i1 i1 . The expansion of the product IˆA (t1 )IˆB (t2 ) gives rise to four terms: ˆ† ˆ ˆ ˆ ˆ† E ˆ† ˆ† ˆ IˆA (t1 )IˆB (t2 ) = E AH BH EAH EBH + EAV EBV EAV EBV † † † † ˆ E ˆ ˆ ˆ ˆ ˆ ˆ ˆ +E AH BV EAH EBV + EAV EBH EAV EBH . (8.34) Substituting Eq.(8.34) into Eq.(8.33), we have: ˆAH (t1 )E ˆBH (t2 )E ˆC (t3 )E ˆD (t4 )|Ψ ||2 p4 (t1 , t2 , t3 , t4 ) = ||E ˆ ˆ ˆ ˆD (t4 )|Ψ ||2 +||EAV (t1 )EBV (t2 )EC (t3 )E ˆ ˆ ˆ +||EAH (t1 )EBV (t2 )EC (t3 )EˆD (t4 )|Ψ ||2 ˆAV (t1 )EˆBH (t2 )EˆC (t3 )EˆD (t4 )|Ψ ||2 . (8.35) +||E The overall probability is a time integral over all times and is given by ∞ dt1 dt2 dt3 dt4 p4 (t1 , t2 , t3 , t4 ). (8.36) P4 (ΔT, θ1 , θ2 ) = −∞

Besides the subscripts {H, V }, each term in Eq.(8.35) is similar to those in Eq.(8.14). We can follow the same procedure to calculate their contributions to the overall probability. Due to its complexity, we omit the detail of the calculation here and present the results as follows: P4 (1st term) = α[A − E(ΔT )] cos2 θ1 sin2 θ2 , P4 (2nd term) = α[A − E(ΔT )] sin2 θ1 cos2 θ2 , P4 (3rd term) = P4 (4th term) α = A(cos2 θ1 cos2 θ2 + sin2 θ1 sin2 θ2 ) 2

+2E(ΔT ) cos θ1 cos θ2 sin θ1 sin θ2 .

(8.37)

8.2 Quantum State Teleportation and Swapping

147

Here,α is a proportional constant. So the overall probability becomes P4 (ΔT, θ1 , θ2 ) = α[A − E(ΔT ) sin2 (θ1 − θ2 )].

(8.38)

In the ideal case when ΔT = 0 and E = A, Eq.(8.38) becomes P4 (θ1 , θ2 ) = αA cos2 (θ1 − θ2 ),

(8.39)

which is exactly what we expect for the output of the polarizer in ﬁeld i2 if the input state to the polarizer is |θ1 i2 = cos θ1 |Hi2 + sin θ1 |V i2 . Therefore, we achieved the teleportation of the state |θ1 from i1 to i2. When θ1 is ﬁxed at π/4 and θ2 is varied, we have from Eq.(8.38) with ΔT = 0 and θ1 = π/4 P4 (θ2 ) =

α (2A − E + E sin 2θ2 ). 2

(8.40)

When we scan θ2 , P4 exhibits sinusoidal oscillation with visibility V=

E . 2A − E

(8.41)

In the experimental demonstration by Bouwmeester et al. [8.3], the angle θ1 is set at +45◦ and θ2 at -45◦ . This leads Eq.(8.38) to P4 (ΔT, +45◦ , −45◦ ) = α[A − E(ΔT )],

(8.42)

which shows a dip as ΔT is scanned, as observed in Ref.[8.3]. The visibility of the dip is V4 = E/A. Note that this visibility is exactly the same as that in Eq.(8.19), indicating that what happens here is basically a two-photon polarization interference eﬀect with independent photons, similar to Sect.8.1.2. 8.2.3 Entanglement Swapping Entanglement swapping [8.4, 8.5, 8.6] is, in essence, the same as teleportation. The only diﬀerence is that entanglement swapping teleports a part of an entangled state while a whole polarization state is teleported in quantum state teleportation. Speciﬁcally, the state to be teleported in entanglement swapping is an EPR entangled state (Fig.8.4). By taking a Bell measurement on s1 and s2 and gating the polarization measurement of i2 on the result of the Bell measurement, we can transfer (swap) the EPR correlation between i1 and s1 to the EPR correlation between i1 and i2, even though i1 and i2 are independent of each other. This scheme can also be used for an “event-ready” Bell experiment, where the trigger signals from the Bell measurement indicate that the EPR pair in i1 , i2 is ready for the test [8.4, 8.5]. To conﬁrm the EPR correlation between i1 and i2, we need to perform polarization correlation measurement on both i1 and i2 and gate the measurement on the Bell measurement of s1 and s2 (quadruple coincidence measurement). The multi-mode description of the ﬁelds is similar to Eq.(8.26), except that |Ψ (1) (θ1 ) is replaced by an EPR state |Ψ (1) (EP R) similar to Eq.(8.28):

148

8 Quantum Interference with Two Pairs of Down-Converted Photons

D

i1

Polarizer (θ1) |EPR〉1

PDC

A s1 s2

C B

PDC

|EPR〉2 Polarizer (θ2) i2 C

|Ψ

(1)

(EP R) = ξ

P4

Fig. 8.4. Entanglement swapping with parametric down-conversion processes. Reprinted ﬁgure with permission from Z. Y. Ou, J. -K. Rhee, and L. J. Wang, Phys. Rev. A 60, c 593 (1999). 1999 by the American Physical Society.

1 † dω1 dω2 Φ(ω1 , ω2 ) √ [ˆ a†i1V (ω2 ) as1H (ω1 )ˆ 2 −ˆ a†s1V (ω1 )ˆ a†i1H (ω2 )|vac.

(8.43)

The ﬁelds for A, B, and C are the same as in Eq.(8.30) and Eq.(8.32), but the ﬁeld for D is changed to ˆD (t) = dω[ˆ ai1H (ω) cos θ1 + a ˆi1V (ω) sin θ1 ]e−iωt (8.44) E for polarization measurement of i1. Following the same calculation leading to Eq.(8.38), we have, for the quadruple coincidence measurement: P4 (ΔT, θ1 , θ2 ) = α[A − E(ΔT ) cos2 (θ1 − θ2 )].

(8.45)

In the ideal case when ΔT = 0 and E = A, we have P4 (θ1 , θ2 ) = αA sin2 (θ1 − θ2 ),

(8.46)

which is exactly the polarization correlation for the EPR singlet state |EPR = √ (|Hi1 , Vi2 − |Vi1 , Hi2 )/ 2. For less than the ideal case, we set θ1 = π/4 and look at P4 as a function of θ2 , as in the experimental demonstration [8.6]. P4 (θ2 ) will be a sinusoidal function of θ2 : P4 (π/4, θ2 ) =

α (2A − E − E sin 2θ2 ). 2

(8.47)

So, the visibility of the modulation is then V = E/(2A − E), which is same as in Eq.(8.41). As can be seen, the visibility in both quantum state teleportation and entanglement swapping is related to the quantity E/A, as in the interference schemes discussed in the previous section. This is not surprising, in the sense that two-photon interference of s1 and s2 requires indistinguishability in their

8.3 Distinguishing a Four-Photon State from Two Independent Pairs

149

temporal modes, but as we learned in Sect.7.1, E/A quantiﬁes the closeness of the down-converted ﬁelds to transform-limitedness. Since s1 and s2 are independent of each other, their temporal mode match will depend on how close they are to the transform-limitedness. Although two pairs are involved here, the photons participating in interference are only s1 and s2. So we do not deal directly with two pairs together and we still cannot discuss the temporal relationship between diﬀerent pairs in parametric down-conversion. But, next, we will look at a scheme that can truly tell the diﬀerence in various temporal relations between diﬀerent pairs.

8.3 Distinguishing a Genuine Polarization Entangled Four-Photon State from Two Independent EPR Pairs To study the temporal structure of photon pairs from parametric downconversion, Tsujino et al. [8.14] proposed and demonstrated a novel method to characterize the entanglement of the four photons. This method can distinguish a genuine polarization entangled four-photon state from two independent EPR pairs of photons. We start with a simple single-mode picture.

B1

Pump

Type-II PDC

HWP or QWP

B2

PBS

A1

Four-Photon Coincidence

A2

Fig. 8.5. The scheme by Tsujino et al. [8.14] to distinguish between a genuine four-photon state and two independent pairs of photons. HWP: half wave plate; QWP: quarter wave plate; PBS: polarization beam splitter. Reprinted ﬁgure with c permission from Z. Y. Ou, Phys. Rev. A 72, 053814 (2005). 2005 by the American Physical Society.

8.3.1 Single-Mode Analysis As we discussed in Sect.7.3, when two pairs of photons from parametric downconversion overlap in time, the four photons become indistinguishable and form a genuine four-photon state of |4. In contrast, when the two pairs are

150

8 Quantum Interference with Two Pairs of Down-Converted Photons

well-separated, the four-photon state is a product state of two two-photon states of form |21 ⊗ |22 . These are totally diﬀerent states, especially when other degrees of freedom are involved for entanglement. For example, in TypeII parametric down-conversion (PDC) (Fig.8.5), a pair of photons are generated in two modes A and B in an EPR polarization-entangled state: √ (8.48) |2HV = (|HA |V B − |V A |HB )/ 2, where H, V represent horizontal and vertical polarizations of photon, respectively. When two such pairs are generated randomly, the quantum state describing them has the form of 1 |2HV 1 ⊗ |2HV 2 = √ (|HA1 |V B1 − |V A1 |HB1 ) 2 1 ⊗ √ (|HA2 |V B2 − |V A2 |HB2 ), 2

(8.49)

where 1, 2 denote the two separate pairs, respectively. But when the two pairs are produced simultaneously, the quantum state becomes a spin-one polarization state [8.15]: √ |4HV = (|2HA |2V B + |2V A |2HB − |HV A |HV B )/ 3. (8.50) Note that |2H, |2V and |HV form an ortho-normal base set for the polarization states of two indistinguishable photons. In this spirit, the state in Eq.(8.50) is also called an “entangled two-photon polarization state (ETP)” of two modes by Tsujino et al. [8.14]. They proposed a method to experimentally distinguish between the two states in Eq.(8.49) and Eq.(8.50). Their method is based on the fact that the state in Eq.(8.50) can be re-written in a diﬀerent form of √ |4HV = −(|P M A |P M B − |RLA |RLB + |HV A |HV B )/ 3, (8.51) where

√ |P M = (|2H − |2V )/√ 2, |RL = (|2H + |2V )/ 2,

(8.52)

with √ √ |P = (|H + |V )/ √2, |M = (|H − |V )/ 2, √ |R = (|H + i|V )/ 2, and |L = (|H − i|V )/ 2 denoting the 45 degree, 135 degree, right-circular, and left-circular polarization states, respectively. Note that |P M , |RL and |HV together also form an ortho-normal base set for the two-photon polarization states. So, when measurement is made at a diﬀerent base for the two modes A and B, say, two-photon probability in mode A is measured in the base of |HV while

8.3 Distinguishing a Four-Photon State from Two Independent Pairs

151

that in B in the base of |P M , the outcome, denoted by C⊥ , will be zero because Eq.(8.51) does not contain a term of |HV A |P M B and all the terms in Eq.(8.51) are orthogonal to |HV A |P M B . But, when measurement is made in the same base, say, |P M A |P M B , the outcome, denoted by C , is a non-zero value of 1/3 so that we have: r ≡ C⊥ /C = 0.

(8.53)

The situation is quite diﬀerent for the distinguishable four-photon state in Eq.(8.49). For the projection to the same base, say, two photons in mode A are detected in |HA1 |V A2 or |HA2 |V A1 and two photons in mode B are detected in |HB1 |V B2 or |HB2 |V B1 , we can rewrite Eq.(8.49) so that both mode A and mode B are in HV base: |2HV 1 ⊗ |2HV 2 1 |HA1 |HA2 |V B1 |V B2 + |V A1 |V A2 |HB1 |HB2 − = 2 −|HA1 |V A2 |V B1 |HB2 − |V A1 |HA2 |HB1 |V B2 . (8.54) Therefore, only the last two terms contribute to the measurement in the base of HV and we have C = 1/2. For the projection to the diﬀerent bases, say, mode A in HV and mode B in PM, we rewrite Eq.(8.54) so that mode B is in the PM base. Because the ﬁrst two terms in Eq.(8.54) are orthogonal to |HV A , we only need to work on the last two terms: |2HV 1 ⊗ |2HV 2 1 = ... − |HA1 |V A2 (|P B1 |M B2 − |M B1 |P B2 ) 4 −|V A1 |HA2 (|M B1 |P B2 − |P B1 |M B2 ) .

(8.55)

We did not write out those terms that do not contribute to C⊥ . This gives rise to C⊥ = 4 × (1/16) = 1/4, so that r = C⊥ /C = 1/2. Therefore, by measuring the quantity r in the three diﬀerent bases of HV, P M , and RL, we should be able to distinguish between the genuine fourphoton state of Eq.(8.50) and the two separate pairs of photons in Eq.(8.49). In practice, however, the two pairs may neither overlap completely nor separate completely. So, what comes out of the parametric down-conversion will be something with r in between 0 and 1/2. To fully examine the temporal relation between photons, we need to consider all the frequency components from parametric down-conversion. We will calculate the quantity r in the next section, using a multi-mode model. 8.3.2 Multi-Mode Analysis The multi-mode description of the quantum state from parametric downconversion is given in Eq.(7.49) in Chapt.7, without polarization entanglement. A simple modiﬁcation will lead to the state for polarization-entangled photons:

152

8 Quantum Interference with Two Pairs of Down-Converted Photons (e)

|Ψ P DC = (1 − η 2 /2)|0 + ξ|Φ2 + with (e)

|Φ2 = and (e) |Φ4

dω1 dω2

ξ 2 (e) |Φ , 2 4

(8.56)

Φ(ω1 , ω2 ) † √ a ˆAH (ω1 )ˆ a†BV (ω2 ) − a ˆ†AV (ω1 )ˆ a†BH (ω2 ) |0, (8.57) 2

† 1 = ˆAH (ω1 )ˆ a†BV (ω2 ) dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) a 2

†

−ˆ a†AV (ω1 )ˆ a†BH (ω2 ) a ˆAH (ω1 )ˆ a†BV (ω2 ) − a ˆ†AV (ω1 )ˆ a†BH (ω2 ) |0, (8.58)

with Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ) describing the spectrum of down-conversion. The (e) superscript “(e) ” denotes polarization entanglement. |Φ4 is associated with two pairs of down-converted photons. Let us calculate the four-photon correlation function equivalent to the quantity C . We start with the arrangement that both modes A and B are measured in HV base and calculate the four-photon correlation function (4)

ΓHV HV (t1 , t2 , t3 , t4 ) = : IˆAH (t1 )IˆAV (t2 )IˆBH (t3 )IˆBV (t4 ) :,

(8.59)

† Iˆkl (t) ≡ Eˆkl (t)Eˆkl (t),

(8.60)

where

with k = A, B; l = H, V , and ˆkl (t) = √1 E 2π

dωˆ akl (ω)e−iωt .

(8.61)

(e)

Only |Φ4 contributes to the four-photon correlation function and it is easier ˆBH (t3 )E ˆBV (t4 )|Φ(e) . After some calculation, to start with EˆAH (t1 )EˆAV (t2 )E 2 we ﬁnd it in the form of ˆBH (t3 )E ˆBV (t4 )|Φ(e) ≡ G4 (t1 , t2 , t3 , t4 )|0, EˆAH (t1 )EˆAV (t2 )E 4

(8.62)

with G4 (t1 , t2 , t3 , t4 )

1 =− dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )× 2 2(2π) × e−i(ω1 t1 +ω1 t2 +ω2 t3 +ω2 t4 ) + e−i(ω1 t1 +ω1 t2 +ω2 t3 +ω2 t4 ) . (8.63)

The overall probability of detecting four photons during one pulse duration is then proportional to the integration over all times: (4) PHV HV = dt1 dt2 dt3 dt4 ΓHV HV (t1 , t2 , t3 , t4 )

8.3 Distinguishing a Four-Photon State from Two Independent Pairs

=

dt1 dt2 dt3 dt4 |ξ 2 G4 (t1 , t2 , t3 , t4 )/2|2 .

153

(8.64)

After carrying out the time integration, we arrive at PHV HV = with

A≡

|ξ|4 A, 4

dω1 dω2 dω1 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2 .

(8.65)

(8.66)

Similarly, we can calculate PP MP M and PRLRL . This is done by replacing ˆkH (k = A, B) in Eq.(8.60) with E √ ˆkP = (EˆkH + E ˆkH )/ 2, E (8.67) or √ ˆkH + iE ˆkH )/ 2, EˆkR = (E

(8.68)

√ ˆkM = (EˆkH − E ˆkH )/ 2, E

(8.69)

√ ˆkH )/ 2. ˆkL = (EˆkH − iE E

(8.70)

and EˆkV with

or

We obtain, after some calculation: PP MP M = PRLRL =

|ξ|4 A. 4

(8.71)

This is consistent with the single-mode theory in Sect.8.3.1 that PP MP M = PRLRL = PHV HV = C . Next, let us calculate C⊥ . We will look only at PHV P M . The others will ˆBP (t3 )EˆBM (t4 )|Φ(e) be similar. It is easier to ﬁrst derive EˆAH (t1 )EˆAV (t2 )E 4 and it has the form of 1ˆ ˆBV (t3 )EˆBV (t4 )+ EAH (t1 )EˆAV (t2 ) EˆBH (t3 )EˆBH (t4 ) − E 2 ˆBH (t3 )E ˆBV (t4 ) |Φ(e) ˆBV (t3 )EˆBH (t4 ) − E +E 4 1 = − [G4 (t1 , t2 , t3 , t4 ) − G4 (t1 , t2 , t4 , t3 )]|0. (8.72) 2 Taking the integration over all times, we obtain: PHV P M =

|ξ|4 (A − E), 8

(8.73)

154

8 Quantum Interference with Two Pairs of Down-Converted Photons

with

E≡

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 ).

(8.74)

Both E and A are the same as those in Eqs.(7.17, 7.18). Similarly, we can prove that PHV RL = PRLP M = PHV P M ≡ C⊥ =

|ξ|4 (A − E). 8

(8.75)

Therefore, we obtain the quantity r as r ≡ C⊥ /C =

E 1 A−E 1− . = 2A 2 A

(8.76)

When E = A, r = 0 and when E = 0, r = 1/2. So, E = 0 corresponds to the case of two distinguishable pairs, while E = A corresponds to an indistinguishable four-photon state. This correspondence is a direct result of the fact that the quantity E/A is a measure of the photon bunching eﬀect in mode A or B alone, that is, how well we can characterize the two photons in mode A or B as two indistinguishable photons rather than two distinct photons. If we examine Eq.(8.76) more closely, we ﬁnd it is very similar to Eq.(8.18), which gives rise to the Hong-Ou-Mandel interference eﬀect from two independent photons. As a matter of fact, the above scheme is exactly equivalent to the scheme of Hong-Ou-Mandel interference of two signal photons from two parametric down-converters gated on the idler photons, as discussed in Sect.8.1.2. To see why, let us examine the state of Eq.(8.49) or (8.50). When we detect two photons in mode A, one in H and other in V, the state in mode B is projected into |1H , 1V . When we measure mode B with coincidence between P and M or between R and L, it is equivalent to a polarization Hong-Ou-Mandel interferometer (Sect.4.1). The projection into |1AH , 1AV is same as gating on two idler photons (if we label mode A as “idler”). Therefore, it is not surprising that Eq.(8.76) is similar to Eq.(8.18). To further conﬁrm PP this, we may calculate the quantity PHV P P ≡ C⊥ , which corresponds to two-photon detection at the same side of a Hong-Ou-Mandel interferometer (photon bunching eﬀect in Sect.3.4). In the distinguishable case, the quantity PP s ≡ C⊥ /C should be the same as r and is 1/2, but for the ideally indistinguishable case, the bunching eﬀect gives s = 2 × 1/2 = 1. It can be shown easily, in the same manner that leads to Eq.(8.76), that PP /C = s ≡ C⊥

1 E 1+ , 2 A

(8.77)

which gives precisely the above values for the appropriate cases. It is easy to understand the physical situation for the two extreme cases of r = 0 and 1/2 (or s = 2 and 1/2). In reality r (or s) lies between those two extreme values, indicating partial overlapping (indistinguishability) of the two

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons

155

pairs. The fact that the quantity r is directly related to the quantity E/A, as in Eq.(8.76), indicates that E/A is a measure of the degree of temporal distinguishability of the two pairs of photons produced in parametric downconversion.

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons Although quadruple coincidence is measured in Sects.8.1-8.3, it is still twophoton amplitudes that are involved in interference. The other two photons merely serve as gating signals to guarantee that the gated ﬁelds are in singlephoton states. So, thus far, our discussion has been limited to two-photon interference. Next, we consider two interference schemes where it is truly the four-photon amplitudes that are superposed, so that the interference phenomenon is a genuine four-photon eﬀect.

|2a, 2b

b

D1 D2

A a B

BS

D3 D4

Fig. 8.6. The geometry of a beam splitter with |2a , 2b input.

8.4.1 Symmetric Beam Splitter Consider now the situation when two pairs of photons enter a 50:50 beam splitter (BS) with each pair in one input port (Fig.8.6). We can treat the input state as a photon number Fock state and in the simple single-mode description, it has the form of |2a , 2b . Here, a, b denote the two input modes of the beam splitter. It can be shown easily (see Appendix A) that the output state is given by !

1 3 |4A , 0B + |0A , 4B − |2A , 2B , |Ψ4 out = (8.78) 8 2 where the subscripts {A,B} denote the two output modes. Notice that the states |3A , 1B and |1A , 3B are missing in Eq.(8.78). This can be easily understood in terms of two-photon interference. Recall from Sect.3.1 that for an input state of |1a , 1b to the 50:50 BS, the output state is given by

1 |Ψ2 out = √ |2A , 0B − |0A , 2B . (8.79) 2

156

8 Quantum Interference with Two Pairs of Down-Converted Photons

If we consider the state |2a , 2b as two pairs of photons with each pair in the state |1a , 1b , then according to Eq.(8.79), only |4A , 0B , |0A , 4B , and |2A , 2B are possible. The disappearance of |3A , 1B and |1A , 3B terms in Eq.(8.78) is, thus, a direct result of the absence of the |1A , 1B term in Eq.(8.79) due to two-photon interference. This picture of two pairs of photons is usually referred to as the 2 × 2 situation. However, such a picture is inappropriate in explaining the probability of |4A , 0B or |0A , 4B , for the probability of |2A , 0B is 1/2 for one pair from Eq.(8.79) so the probability of |4A , 0B is simply (1/2)2 = 1/4 for two pairs. But, from Eq.(8.78), the probability for |4A , 0B is 3/8. The diﬀerence comes from the fact that the four photons in the 2 × 2 case correspond to two independent (uncorrelated) pairs while the four photons in |2a , 2b are indistinguishable. To understand the partition probability for the |4A , 0B output state, we ﬁrst treat the four photons as classical particles and consider the classical partition probability. As classical particles, the four photons can be thought of as independent particles and their partition at the BS simply follows the Bernoulli distribution and P4 = (1/2)4 = 1/16. The ratio between quantum and classical predictions is then P4q /P4c = 6.

(8.80)

We can understand the six-fold increase for quantum prediction in terms of four-photon interference: if we consider the two photons entering each side of the BS as indistinguishable, then there are 6 possible ways to arrange the four photons (Fig.8.7). The four numbered slots for the four photons can be viewed as four photo-detectors. Since the four photons in the state |4A , 0B are indistinguishable, the amplitudes for the 6 possibilities are added to give an overall amplitude of 6A due to constructive four-photon interference. Here, A is the probability amplitude for each possibility. So, the overall probability for |4A , 0B is then P4q = (6A)2 = 36A2 for quantum prediction. However, for classical particles, there is no interference and we simply add probability A2 for each possibility to obtain an overall probability P4c = 6A2 . So, we have the ratio P4q /P4c = 36A2 /6A2 = 6, or a six-fold increase from classical prediction to quantum prediction. The four-photon interference picture discussed above can be applied also to the three other situations of |3A , 1B , |1A , 3B and |2A , 2B in Eq.(8.78). However, because the three situations involve diﬀerent combinations of reﬂected photons, which experience an extra π phase shift from the a-side reﬂection for a lossless beam splitter, the probability amplitudes for the six possibilities in Fig.8.7 will have diﬀerent phases. For example, in the case of |3A , 1B , the total phases (experienced by all four photons) for each probability amplitude of the six possibilities in Fig.8.7 (D4 is now placed in side B) are π, 0, π, 0, 0, π, respectively. This results in probability amplitudes of −A, A, −A, A, A, −A and a total probability amplitude of −A + A − A + A + A − A = 0, which explains the disappearance of the |3A , 1B and |1A , 3B terms in Eq.(8.78). We can likewise explain the coeﬃcient for the |2A , 2B term.

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons

D1 D2 D3 D4

157

D1 D2 D3 D4

D1 D2 D3 D4

D1 D2 D3 D4

D1 D2 D3 D4

D1 D2 D3 D4

Fig. 8.7. Six possible ways to arrange two pairs of photons with four detectors. Reprinted ﬁgure with permission from Z. Y. Ou, J. -K. Rhee, and L. J. Wang, Phys. Rev. A 60, c 593 (1999). 1999 by the American Physical Society.

To produce the input state of |2a, 2b , we can use parametric downconversion process with two pairs of photons. The last term |Φ4 of the quantum state from parametric down-conversion in Eq.(7.49) is a state of the form of |2s , 2i . However, the state in Eq.(7.49) is a multi-mode state, so we need to treat the problem with a multi-mode theory. Since there is a more profound diﬀerence among P4q (four-photon interference), P4c (no interference), (2×2) and P4 (the prediction from two-photon interference) for the |4A , 0B or |0A , 4B case, we will only study this case and omit the |2A , 2B and |3A , 1B cases. Consider the quadruple detection probability at one output port (say, port A) of the beam splitter: ˆA (t3 )E ˆA (t4 )|Ψ ||2 , p4 (t1 , t2 , t3 , t4 ) = ||EˆA (t1 )EˆA (t2 )E where

(8.81)

ˆi (t + ΔT )], ˆA (t) = √1 [Eˆs (t) + E (8.82) E 2 with a delay ΔT between the arrivals of signal and idler ﬁelds at the BS. Obviously, there is no contribution in Eq.(8.81) from the ﬁrst two terms of |Ψ of Eq.(7.49). There are 16 terms in the expansion of Eq.(8.81) when we substitute Eq.(8.82) into Eq.(8.81). Among them, 10 terms of the form Es Es Es Es , Ei Ei Ei Ei , Es Es Es Ei , and Es Ei Ei Ei give zero result when applied to the state |Φ4 . The six nonzero terms correspond to the six possibilities in the simple picture of four-photon interference in Fig.8.7. They are listed as follows: ˆs (t1 )E ˆs (t2 )Eˆi (t3 + ΔT )E ˆi (t4 + ΔT )|Ψ , E ˆi (t1 + ΔT )E ˆi (t2 + ΔT )E ˆs (t3 )E ˆs (t4 )|Ψ , E ˆi (t2 + ΔT )E ˆs (t3 )E ˆi (t4 + ΔT )|Ψ , ˆs (t1 )E E ˆ ˆ ˆ ˆs (t4 )|Ψ , Es (t1 )Ei (t2 + ΔT )Ei (t3 + ΔT )E

158

8 Quantum Interference with Two Pairs of Down-Converted Photons

ˆi (t1 + ΔT )E ˆs (t2 )Eˆi (t3 + ΔT )E ˆs (t4 )|Ψ , E ˆs (t2 )Eˆs (t3 )E ˆi (t4 + ΔT )|Ψ . ˆi (t1 + ΔT )E E The evaluation of these terms is similar to those carried out in Sect.8.1. We will leave the derivation to readers who are interested in it. But readers may always refer Ref.[8.16] for details. We present the ﬁnal result for the overall four-photon detection probability as follows:

2 (1) (1) P4 (ΔT ) = α A 1 + q(ΔT ) + E + 2E1 (ΔT ) + 2E2 (ΔT )+ (1) +2E3 (ΔT ) + E (2) (ΔT ) , (8.83) where α is some proportional constant and 2 i(ω1 −ω2 )τ q(τ ) = dω1 dω2 |Φ(ω1 , ω2 )| e dω1 dω2 |Φ(ω1 , ω2 )|2 , (8.84) and

(1)

E1 (τ ) =

(1)

E2 (τ ) = (1) E3 (τ )

=

E (2) (τ ) =

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) ×Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 −ω1 )τ ;

(8.85)

dω1 dω2 dω1 dω2 Φ(ω1 , ω1 )Φ(ω2 , ω2 ) ×Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 −ω1 )τ ;

(8.86)

dω1 dω2 dω1 dω2 Φ(ω1 , ω1 )Φ(ω2 , ω2 ) ×Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 −ω1 )τ ;

(8.87)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )

×Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω2 +ω2 −ω1 −ω1 )τ ,

(8.88)

where we assumed the symmetry relation Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ). It is easy to (1)∗ (1) (1)∗ (1) check that E1 = E2 , E3 = E3 , and E (2)∗ = E (2) . Notice that q(0) = 1, (1) (2) and E1,2,3 (0) = E (0) = E. So, we have: P4 (0) = α(4A + 8E).

(8.89)

When the delay is zero, maximum interference occurs. This is the multimode equivalent of the quantum mechanical case described in Eq.(8.78). On the other hand, if Φ(ω1 , ω2 ) has a nonzero bandwidth, then q(∞) = 0, and E1,2,3,4 (∞) = 0. Therefore, P4 (∞) = α(A + E).

(8.90)

When the delay is large, no interference occurs, which corresponds to the classical case. So, the ratio

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons

4E P4 (0) =4+ , P4 (∞) A+E

159

(8.91)

is the multi-mode result for the quantity P4q /P4c in Eq.(8.80). In the ideal condition when E/A = 1, we have P4 (0)/P4 (∞) = 6, which is exactly the same as in Eq.(8.80) for the single-mode case. On the other hand, when E = 0, we have P4 (0)/P4 (∞) = 4, which is the result for the 2 × 2 case, i.e., the case when the two pairs are totally independent. So, the quantity E/A is again the characterization of the degree of the temporal (in)distinguishability of the pairs in parametric down-conversion. Although the six-fold increase in the four-photon coincidence is a result of a four-photon interference eﬀect, it may be explained as well in terms of a generalized photon bunching eﬀect similar to the two-photon case in Sect.7.2, but here for two pairs of photons. In fact, later in Sect.9.2, we will generalize it to an N -photon case with an enhancement factor of N !. This bunching eﬀect is an N -photon interference eﬀect due to N -photon indistinguishability. The factor of six for two pairs of photons comes from the four-photon enhancement factor 4! divided by 2!2!. The 2!2! factor is for over-counting in two-photon indistinguishability within each pair. Experimentally, the enhancement in the four-photon coincidence rate shown in Eq.(8.91) was observed by Ou, Rhee and Wang [8.17]. 8.4.2 Asymmetric Beam Splitter The four-photon interference scheme discussed in the previous section is similar to a Hong-Ou-Mandel interferometer except that the term |2A , 2B is still in the output state of Eq.(8.78), which is the diﬀerence between the twophoton and the four-photon cases. The disappearance of |3A , 1B and |1A , 3B is the result of two-photon interference. In this section, we will not use a symmetric beam splitter with T = R = 50%. Instead, we employ an asymmetric beam splitter with T = R. We will choose T so that the |2A , 2B term disappears from the output state. It turns out that this disappearance is a result of four-photon interference and the visibility of this interference phenomenon will depend on the E/A-quantity. Consider a beam splitter of transmissivity T and reﬂectivity R with a input state of |2a , 2b . From Eq.(A.18) in Appendix A, we ﬁnd the output state with M = N = 2 as √ |Ψ4 out = 6T R |4A , 0B + |0A , 4B + (T − R)2 − 2T R |2A , 2B √ + 6T R(T − R) |3A , 1B − |1A , 3B , (8.92) where T = t2 , R = r2 . Note that Eq.(8.92) becomes Eq.(8.78) when T = R = 1/2. On the other hand, when (T − R)2 − 2T R = 0 or √ T = (3 ± √3)/6, (8.93) R = (3 ∓ 3)/6,

160

8 Quantum Interference with Two Pairs of Down-Converted Photons

a

|2a,2b b

T,R

B Fig. 8.8. The geometry of an asymmetric beam splitter with |2a , 2b input.

A

the |2A , 2B term will disappear in Eq.(8.92). But since T = R, the |3A , 1B and |1A , 3B terms stay in Eq.(8.92). The input state |2a , 2b can be formed from two pairs of photons generated from parametric down-conversion. The disappearance of the |2A , 2B term in the output state is a result of four-photon interference in which the two pairs are indistinguishable in time. This is the so called 4 × 1 case. If the two pairs are independent, as the 2×2 case discussed in Sects.8.3.1 and 8.4.1, the output state becomes |Ψ4 out (2 × 2) = |Ψ2 1 ⊗ |Ψ2 2 , with |Ψ2 j =

√ 2T R |2A , 0B j − |0A , 2B j + (T − R)|1A , 1B j ,

(8.94)

(8.95)

where j = 1, 2 is the label for the two separate pairs. Expanding Eq.(8.94) with |Ψ2 j in Eq.(8.95), we obtain: 2 |Ψ4 out (2 × 2) = ... + (T − R) |1A , 1B 1 |1A , 1B 2 −2T R |2A 1 |2B 2 + |2B 1 |2A 2 + ...,

(8.96) (2,2)

where we omit the terms that have no contribution to the probability P4 of detecting two photons at A and B, respectively. From Eq.(8.96), we obtain (2,2)

P4

(2 × 2) = (T − R)4 + (2T R)2 + (2T R)2 = (T − R)4 + 8T 2 R2 ,

(8.97)

which is never zero. Now consider the situation when all four photons are distinguishable and become classical particles. This occurs when there is a large delay between the ﬁelds at two sides, so that the two photons from side a arrive at the beam splitter at a diﬀerent time from those from side b. They follow the Bernoulli binomial distribution and we have: (2,2)

P4

(cl.) = T 4 + R4 + 4T 2 R2 .

(8.98)

8.4 Hong-Ou-Mandel Interferometer for Two Pairs of Photons

161

With T, R in Eq.(8.93), we have: (2,2)

P4

(cl.) = 1/2;

(2,2)

P4

(2 × 2) = 1/3.

(8.99)

So, as we scan the relative delay between the two sides, we should observe a (2,2) drop in P4 . For the state in Eq.(8.92), the probability drops all the way to zero, whereas for the state in Eq.(8.94), it drops from 1/2 to 1/3. If we use visibility to describe the relative depth of the dip, we have: V4 (4 × 1) = 1;

V4 (2 × 2) = 1/3.

(8.100)

For the situation in between the 4 × 1 and 2 × 2 cases, we resort to a multimode model for parametric down-conversion. The quantum state is |Ψ P DC , given in Eq.(7.49) with only |Φ4 in a four-photon state that contributes to what we are interested in. The four-photon detection probability at A and B (Fig.8.8) is proportional to P4 (2A, 2B) = dt1 dt2 dt3 dt4 p4 (t1 , t2 , t3 , t4 ), (8.101) with ˆA (t1 )E ˆA (t2 )EˆB (t3 )EˆB (t4 )|Ψ P DC ||2 , p4 (t1 , t2 , t3 , t4 ) = ||E where

√ √ ˆ ˆ ˆA (t) = T E E √ s (t) + √ REi (t + ΔT ), ˆB (t) = T E ˆi (t) − RE ˆs (t − ΔT ). E

(8.102)

(8.103)

After substituting Eq.(8.103) into Eq.(8.102), we ﬁnd only the following terms are non-zero: ˆA (t2 )E ˆB (t3 )EˆB (t4 )|Ψ P DC ˆA (t1 )E E ˆs E ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆE ˆ ˆ = − TR E i s Ei + Ei Es Ei Es + Es Ei Ei Es + Ei Es Es Ei ˆs Eˆs E ˆi E ˆi + R2 Eˆi E ˆi E ˆs E ˆs |Ψ P DC , +T 2 E (8.104) ˆi (t + ΔT ), E ˆs ≡ E ˆs (t − ΔT ) and we keep the order of time for where Eˆi ≡ E each term. It is straightforward, then, to obtain: ˆi (t3 )E ˆi (t4 )|Ψ P DC = ˆs (t1 )Eˆs (t2 )E E with G4 (t1 , t2 , t3 , t4 ) =

ξ2 G4 (t1 , t2 , t3 , t4 )|vac, (8.105) 2(2π)2

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) × e−i(ω1 t1 +ω1 t2 ) + e−i(ω1 t2 +ω1 t1 )

162

8 Quantum Interference with Two Pairs of Down-Converted Photons

× e−i(ω2 t3 +ω2 t4 ) + e−i(ω2 t4 +ω2 t3 )

=2

dω1 dω2 dω1 dω2 e−i(ω1 t1 +ω1 t2 +ω2 t3 +ω2 t4 ) × Φ(ω1 , ω2 )Φ(ω1 , ω2 ) + Φ(ω1 , ω2 )Φ(ω1 , ω2 ) . (8.106)

Substituting Eq.(8.105) into Eq.(8.104), we obtain: ˆA (t1 )E ˆA (t2 )E ˆB (t3 )E ˆB (t4 )|Ψ P DC E ξ2 2 T G4 (t1 , t2 , t3 , t4 ) + R2 G4 (t3 − ΔT, t4 − ΔT, t1 + ΔT, t2 + ΔT ) = 2(2π)2

−T R G4 (t1 , t3 − ΔT, t2 + ΔT, t4 ) + G4 (t2 , t4 − ΔT, t1 + ΔT, t3 ) +G4 (t1 , t4 − ΔT, t2 + ΔT, t3 ) + G4 (t2 , t3 − ΔT, t1 + ΔT, t4 ) |vac. (8.107)

Substituting Eq.(8.107) into Eq.(8.102) and carrying out the time integral in Eq.(8.101), we ﬁnally arrive at:

P4 (ΔT ) = 2|ξ|4 (T 4 + R4 + 4T 2R2 )(A + E) + 6T 2R2 Aq 2 (ΔT ) + E (2) (ΔT )

(1) (1) (1) −4T R(1 − 3T R) Aq(ΔT ) + E1 (ΔT ) + E2 (ΔT ) + E3 (ΔT )

2 |ξ|4 = 2A + 3E + A 1 − q(ΔT ) + E (2) (ΔT ) 3 (1)

(1) (1) −2 E1 (ΔT ) + E2 (ΔT ) + E3 (ΔT ) , (8.108) (1)

where we used T, R in Eq.(8.93) and q(ΔT ), E1,2,3 (ΔT ), E (2) (ΔT ) are given in Eqs.(8.84-8.88). Similar to Eq.(8.83), we have: P4 (0) = 2|ξ|4 (A − E)/3, P4 (∞) = |ξ|4 (A + E),

(8.109)

from which we obtain the visibility V4 ≡

A + 5E P4 (∞) − P4 (0) = . P4 (∞) 3(A + E)

(8.110)

In the extreme case of E = 0 and E = A, we have: V4 (2 × 2) = 1/3,

V4 (4 × 1) = 1.

(8.111)

These are exactly what are predicted in the simple single-mode picture at the beginning of this section. But Eq.(8.110) also gives the visibility for the intermediate case of E/A.

8.5 Generation of a NOON State by Superposition

163

8.5 Generation of a NOON State by Superposition A NOON state is a maximally entangled N-photon state of two modes [8.18]. For polarization modes of the ﬁeld, it has the form of √ (8.112) |N OON = |N H |0V + ejN ϕ |0V |N H / 2, where ϕ is a phase experienced by single photons. For example, the Hong-OuMandel interferometer discussed in Chapt.3 has an output in a two-photon NOON state (see also Sect.5.3). A signature of this state is in the N-photon coincidence measurement in the superposition of H and V. The coincidence rate has the form of Rc ∝ 1 + cos(N ϕ).

(8.113)

This type of interference fringe is typical of N-photon interference when all the N photons are stuck together as one entity, resulting in an N-photon de Broglie wavelength of λ/N with λ as the single-photon wavelength. The fringe in Eq.(8.113) has a super-resolved sensitivity to the phase change [8.19, 8.20] and may lead to the so-called Heisenberg limit [8.21, 8.22] in the precision of phase measurement. Notice that the middle terms of |N − kH |kV (k = 0, N ) do not appear in Eq.(8.112). Those middle terms usually give a cos[(N −k)ϕ]-dependence on ϕ, reducing the phase resolution. Normally, if we superpose two ﬁelds in N-photon states, those middle terms inevitably appear in the resultant ﬁelds, making it diﬃcult to generate a NOON state. However, as we have seen in the previous section, the coeﬃcients of some of the terms may be set to zero by quantum interference. In fact, the reason that a Hong-Ou-Mandel interferometer gives a two-photon NOON state as its output is precisely the cancellation of the |1, 1 term by two-photon interference. In this section, we will discuss a couple of schemes that utilize the multi-photon interference eﬀect to generate NOON states. 8.5.1 Interference between a Coherent State and Parametric Down-Conversion We have discussed the interference between a weak coherent state and parametric down-conversion in Chapt.6. But in that chapter, we considered only the two-photon case where a two-photon interference eﬀect leads to the cancellation of two-photon events and the photon anti-bunching eﬀect. We can likewise consider the three-photon case and hope a three-photon interference eﬀect may cancel some unwanted terms and lead to a three-photon NOON state. Consider the scheme shown in Fig.8.9, where a coherent state is injected into one input port of a 50:50 beam splitter while light from spontaneous

164

8 Quantum Interference with Two Pairs of Down-Converted Photons

parametric down-conversion (PDC) is in the other input port. This scheme was investigated in Sect.6.1, but only up to two-photon terms. Let us now concentrate on the three-photon terms.

|PDC

a

ϕ

B 50:50

|α

b

A

Fig. 8.9. Generation of a three-photon NOON state by interference between a coherent state and parametric down-conversion (PDC).

For weak ﬁelds, the input state from spontaneous down-conversion is given by √ ˆ ≈ |0 − (ζ/ 2)|2 + ..., |PDC = U|0

(8.114)

where Uˆ is from Eq.(6.1) and |0 is the vacuum state. We dropped terms with four or more photons in weak ﬁeld approximation. Under this approximation we can also write the coherent state as α2 α3 |α ≈ |0 + α|1 + √ |2 + √ |3 + ... . 2 6

(8.115)

So, the overall input state for the beam splitter is: |BSin = |PDCa ⊗ |αb α3 ζα ≈ ... + √ |0a , 3b − √ |2a , 1b + ..., 6 2

(8.116)

where we omit the terms with a photon number less than and higher than three. To obtain the output state of the beam splitter, we will ﬁrst ﬁnd the output state for each term. We write |m, nout as the output state for the input state of |m, n. From Appendix A, we have, for a 50:50 beam splitter: √ √ √ |0, 3out = |3, 0 + |0, 3 + 3|2, 1 + 3|1, 2 / 8 (8.117) √ √ √ (8.118) |2, 1out = − |2, 1 − |1, 2 + 3|3, 0 + 3|0, 3 / 8. For the input state in Eq.(8.116), we then have the output state as 2 α − 3ζ)α (α2 + ζ)α √ |3, 0 + |0, 3 . (8.119) |BSout = |2, 1 + |1, 2 + 4 4 3 When α2 = −ζ, the coeﬃcients of |2, 1 and |1, 2 are zero and the output state becomes:

8.5 Generation of a NOON State by Superposition

|BSout

√ = α3 |3, 0 + |0, 3 / 3,

165

(8.120)

which is a three-photon NOON state. To conﬁrm the entanglement, we can recombine the outputs of the beam splitter with another beam splitter and measure the three-photon coincidence of the outputs, as in Fig.8.9. A result similar to Eq.(8.113) with N = 3 is the three-photon coincidence rate. From the process of constructing the output state in Eq.(8.119) we ﬁnd that the cancellation of the coeﬃcients of the |2, 1 and |1, 2 terms occurs because of the complete destructive three-photon interference between the |0a , 3b and |2a , 1b input terms when α2 = −ζ. But this method cannot be generalized to a larger photon number because we only have one degree of freedom of adjustment, i.e., the relative amplitude α2 and ζ. There are more than one terms we need to cancel when N ≥ 4.

|PDC

a

B NPA

|α

b

A

|NPA

out

Fig. 8.10. Generation of a four-photon NOON state with a non-degenerate parametric ampliﬁer (NPA) for extra adjustment parameter.

More free parameters may be introduced by adding a non-degenerate parametric ampliﬁcation (NPA) process after the beam splitter, as depicted in Fig.8.10. The evolution operator for the NPA is given by ˆN P A = exp(β ∗ a U ˆ1 a ˆ2 − βˆ a†1 a ˆ†2 ) ˆ†2 + β 2 a ˆ†2 ˆ†2 ≈ 1 − βˆ a†1 a 1 a 2 /2,

(8.121)

up to the four-photon terms. The output state of the NPA is then: ˆN P A |BSout , |N P Aout = U

(8.122)

where |BSout is given in Eq.(8.119). However, since we are interested in the four-photon case, we need to write |BSout up to the four-photon number terms. Following the same procedure that leads to Eq.(8.119), we have: |BSout = |0, 0 + C10 (|1, 0 + |0, 1) + C11 |1, 1 + C20 (|2, 0 + |0, 2) +C30 (|3, 0 + |0, 3) + C21 (|2, 1 + |1, 2) + C22 |2, 2 +C40 (|4, 0 + |0, 4) + C31 (|3, 1 + |1, 3), (8.123) where C10 C20 C30 C40

√ = α/ 2, √ = (α2 − ζ)/2 2,√ C11 = (α2 + ζ)/2, 3 = (α3 − 3αζ)/4 3, √ C21 = (α + αζ)/4, 4 2 2 = (α − 6α ζ + 3ζ )/8 6,

166

8 Quantum Interference with Two Pairs of Down-Converted Photons

√ C31 = (α4 − 3ζ 2 )/4 6,

C22 = (α4 + 2α2 ζ + 3ζ 2 )/8.

The reason that we list all the coeﬃcients is that they will contribute to higher photon number terms. So, as we mentioned before, we can only manipulate two-photon and three-photon terms with just α2 and ζ. But with the NPA, we have the output state as (|4, 0 + |0, 4) + C22 |2, 2 |N P Aout = ... + C40 +C31 (|3, 1 + |1, 3),

(8.124)

where we write down only the four-photon terms and

√ = α4 − 3ζ 2 − 6(α2 − ζ)β /4 6, C31

= α4 + 2α2 ζ + 3ζ 2 − 8(α2 + ζ)β + 8β 2 /8, C22 √ C40 = (α4 − 6α2 ζ + 3ζ 2 )/8 6. With two relative parameters to √ √ adjust, we may set C22 = 0 = C31 by choosing 2 α = (−3 ± 12)ζ and β = ± 3ζ/2. Then Eq.(8.124) becomes

1 |N P Aout = ... + √ α4 (|4, 0 + |0, 4), 4 6

(8.125)

which is a four-photon NOON state. Likewise, we can go to the ﬁve-photon case and ﬁnd that the α, ζ, β parameters are enough because we only have two coeﬃcients C32 and C41 to cancel (C23 and C14 are same as C32 and C41 , respectively). We leave it for interested readers to work out the details. Even higher photon number cases are possible if we can involve more processes. Next, however, we will discuss a scheme that does not rely on adding more processes for cancellation of all the unwanted terms. To discriminate against lower photon number states, we usually perform an N-photon coincidence measurement. However, higher photon number states still contribute. If we make the ﬁelds weak enough, these higher order contributions become negligible. Even if they do contribute, we can always estimate how much they add to the background and subtract them out in the data analysis. 8.5.2 A Special N-Photon Interference Scheme This scheme utilizes an ingenious arrangement of beam splitters to cancel all the unwanted terms at once by N-photon interference. This scheme was proposed by Hofmann [8.23], who considered N photons in N diﬀerent spatial modes. Each photon is prepared in a slightly diﬀerent polarization state. By bringing them into one spatial mode, an N-photon NOON state of two polarization modes may be generated. To see how this works, consider the scheme in Fig.8.11, where N photons are fused into one spatial mode by N − 1 beam splitters. The N photons are

8.5 Generation of a NOON State by Superposition

b2

b1

bN−1

1/2

|δ1

a1

a2 |δ2

167

1/3

1/N

|NOON

aN

a3

bN

|δN

|δ3

Fig. 8.11. Interferometric scheme for canceling all unwanted terms of |kH , N − kV (k = 0, N ).

each in a diﬀerent polarization state, which is twisted a bit, from each other. To be more speciﬁc, the k-th photon is in a polarization state of √ |δk k = (|Hk − ejδk |V k )/ 2 1 † = √ (ˆ aHk − ejδk a ˆ†V k )|vac, (8.126) 2 with δk = 2π(k − 1)/N, (k = 1, 2, ..., N ). The input state is then: " |ΨN in = ⊗|δk k k

= 2−N/2

" † (ˆ aHk − ejδk a ˆ†V k )|vac.

(8.127)

k

For the output state, the N photons will be distributed around all N output ports ˆb1 , ..., ˆbN . Now, we search for the projected state |ΨN (bN )out ≡ PN (bN )|ΨN out , when all N photons are in the output port ˆbN . Here, PN (bN ) is the projection operator and ˆN |ΨN in , |ΨN out = U

(8.128)

ˆN as the evolution operator, from input to output, of the beam splitter with U system. Generalizing the formalism of a four-port beam splitter in Appendix A to an N -port beam splitter, we obtain: 1 " † jδk † ˆN ˆ† ˆ |ΨN out = U (ˆ a − e a ˆ ) Hk V k UN UN |vac 2N/2 k 1 " ˆ † ˆ† ˆN a ˆ † )|vac, (UN a ˆHk UN − ejδk U ˆ†V k U (8.129) = N/2 N 2 k

ˆ † is the operator evolution ˆN a ˆk U According to Eq.(A.14) in Appendix A, U N in reverse direction, that is, a ˆk expressed in terms of ˆbk . It is straightforward to ﬁnd that in the reverse direction, 1 a ˆHk = √ (ˆbHN + ...), N so that

1 a ˆV k = √ (ˆbV N + ...), N

(8.130)

168

8 Quantum Interference with Two Pairs of Down-Converted Photons

ˆN a ˆ † = √1 (ˆ ˆk U U aN + ...). N N

(8.131)

Now, change a ˆ to ˆb for the output operator, as in Appendix A, and make use of the projection operation: PN (bN )ˆb†Hk = 0

(k = N ).

(8.132)

Then, we have: √ ˆN a ˆ † = ˆb† / N . ˆ†k U PN (bN ) U N N

(8.133)

Finally, the projected state is ˆN |ΨN in |ΨN (bN )out = PN (bN )U 1 " ˆN a ˆ† ) = N/2 ˆ†Hk U PN (bN )(U N 2 k

ˆN a ˆ † ) |vac ˆ†V k U −ejδk PN (bN )(U N

=

" † 1 (ˆb − ejδk ˆb†V N )|vac. (2N )N/2 k HN

Next, let us apply the algebraic identity " (x − yejδk ) xN − y N =

(8.134)

(8.135)

k

to Eq.(8.134) and we obtain: 1 (ˆb†N − ˆb†N V N )|vac N/2 HN (2N ) # N! , 0 − |0 , N . = |N H V H V (2N )N

|ΨN (bN )out =

(8.136)

This is a NOON state of two polarization modes. The state is not normalized since it is a projected state. The projection probability is given by PN =

out ΨN (bN )|ΨN (bN )out

= 2N !/(2N )N ,

(8.137)

which goes to zero very quickly as N increases. For example, P3 = 1/18. Experimentally, the easiest way to project to an N -photon state in bN is to make an N-photon coincidence measurement. This is achieved by dividing the ﬁeld into N equal parts and sending each part to a detector. The division further deduces the probability by another factor of N −N , making the N photon coincidence incredibly small. Demonstration of an N = 3 case was performed by Mitchell et al. [8.20].

8.6 Multi-Photon De Broglie Wavelength by Projection Measurement

169

8.6 Multi-Photon De Broglie Wavelength by Projection Measurement Since N-photon coincidence measurement has to be performed in the schemes discussed in previous sections in order to project out the correct states, we may incorporate it in the interference scheme altogether. In this case, although the output state of the system is usually not a NOON state, the projection measurement responds only to the NOON state part of the state and cancellation of the unwanted terms occurs at measurement. The N -photon coincidence measurement will have an interference fringe pattern with an N-photon de Broglie wavelength, as in Eq.(8.113). 8.6.1 Projection by Asymmetric Beam Splitters In Sect.8.4.2 and Sect.8.5.1, we ﬁnd that some terms may be canceled by using a beam splitter. However, to cancel all the unwanted terms, we need more free parameters to adjust. With two beam splitters (minimum to form a phasedependent interferometer), we have two parameters. Therefore, we may use those parameters to cancel two terms (three-photon), or three (four-photon) if symmetry is used. We start with the three-photon case, which was ﬁrst considered by Wang and Kobayashi [8.24]. Three-photon case For a three-photon state of |2a , 1b entering a beam splitter of transmissivity T and reﬂectivity R, we ﬁnd, from Appendix A, the output state as √ √ R |3A , 0B + 3T R2 |0√ |BS(3)out = 3T 2√ A , 3B + T (T − 2R)|2A , 1B + R(R − 2T )|1A , 2B . (8.138) When R = 2T = 2/3, the |1A , 2B term will disappear from Eq.(8.138) due to three-photon interference, and Eq.(8.138) becomes √ √ 2 2 3 (8.139) |BS(3)out = |3A , 0B + |0A , 3B − |2A , 1B . 3 3 3 But, the |2A , 1B term is still in Eq.(8.139). Now, we can arrange a projection measurement to take out the |2A , 1B term in Eq.(8.139). Let us combine A and B with another beam splitter (Fig.8.12) that has same transmissivity and reﬂectivity (R = 2T = 2/3) as the ﬁrst BS. According to Eq.(8.139), |2A , 1B will not contribute to the probability P3 (1C , 2D ). So, only |3A , 0B and |0A , 3B in Eq.(8.139) will contribute to P3 (1C , 2D ). Although the coeﬃcients of |3A , 0B and |0A , 3B in Eq.(8.139) are not equal, their contributions to P3 (1C , 2D ) are the same after considering the unequal T and R. Therefore, the projection measurement of P3 (1C , 2D ) is responsive only to the three-photon NOON state.

170

8 Quantum Interference with Two Pairs of Down-Converted Photons

a

B

|2a ,1b b

ϕ C

Fig. 8.12. Interferometric scheme for three-photon de Broglie wave length by projective measurement via asymmetric beam splitters.

T, R D

A

The above argument can be conﬁrmed by calculating P3 (1C , 2D ) directly for the scheme in Fig.8.12 [8.25]: ˆ †2 D ˆ 2 C|2 ˆ a , 1b /1!2!, P3 (1C , 2D ) = 2a , 1b |Cˆ † D with

√ √ ˆ Cˆ = (Aˆ + ejϕ √2B)/ √3, ˆ = (ejϕ B ˆ ˆ − 2A)/ D 3,

(8.140)

(8.141)

where we introduce a phase ϕ between A and B. But for the ﬁrst BS, we have √ √ Aˆ = (ˆ a + √ 2ˆb)/ √3, (8.142) ˆ = (ˆb − 2ˆ a)/ 3. B Substituting Eq.(8.141) into Eq.(8.140) with Eq.(8.142), we obtain 16 (1 + cos 3ϕ), (8.143) 81 which is same as Eq.(8.113) with N = 3. Note that P3 (1C , 2D ) has a maximum value of 32/81 when ϕ = 0. This value is of the order of one, indicating that we do not lose many photons in this projection measurement. This is the advantage of this projection scheme over the one in Sect.8.5.2. Experimentally, a photon state of |2a , 1b can be produced from two pairs of parametric down-conversion photons in a scheme similar to that discussed in Sect.7.2.2. More detailed multi-mode analysis is performed later in Sect.9.7.1. As an exercise, readers who are interested in the multi-mode result can work out the dependence of the visibility on the E/A quantity. Here, we present only the result: P3 (1C , 2D ) =

P3 ∝ 1 + V3 cos 3ϕ + V1 cos ϕ,

(8.144)

with V3 =

8(A + 2E) , 17A + 7E

V1 =

9(A − E) . 17A + 7E

(8.145)

The existence of the cos ϕ term in Eq.(8.144) is due to the less-than-perfect cancellation of the |1A , 2B term in the multi-mode case. For small A − E, this term is small compared to the cos 3ϕ term. But, it does give imbalance in the peaks of the interference fringe, as observed by Liu et al. [8.26].

8.6 Multi-Photon De Broglie Wavelength by Projection Measurement

a

B

|2a, 2b b

171

ϕ C

50:50

T, R

Fig. 8.13. Interferometric scheme for four-photon de Broglie wave length by projective measurement.

D

A

Four-photon case For an input of |2a , 2b into a 50:50 beam splitter, we already have both |3A , 1B and |1A , 3B canceled due to symmetry, as presented in Eq.(8.78). The only thing we need to do is to choose the second beam splitter so that the |2A , 2B term won’t contribute to P4 (2C , 2D ) (Fig.8.13). This situation was√already treated in √ Sect.8.4.2 and the beam splitter must have T = (3 ± 3)/6, R = (3 ∓ 3)/6 [Eq.(8.92)]. In this case, the only contributions to P4 (2C , 2D ) are from the |4A , 0B and |0A , 4B terms in Eq.(8.78), which should result in P4 (2C , 2D ) as in Eq.(8.113) with N = 4. To conﬁrm this, again, we carry out the calculation of ˆ †2 D ˆ 2 Cˆ 2 |2a , 2b /2!2!, P4 (2C , 2D ) = 2a , 2b |Cˆ †2 D with

√ √ jϕ ˆ Cˆ = √T Aˆ + Re √ B, ˆ ˆ − RA, ˆ = T ejϕ B D

[T = (3 ±

√ 3)/6, R = 1 − T ],

(8.146)

(8.147)

and

√ Aˆ = (ˆ a + ˆb)/ √2, ˆ = (ˆb − a B ˆ)/ 2.

(8.148)

After substituting Eq.(8.147) into Eq.(8.146) with Eq.(8.148), we obtain: P4 (2C , 2D ) =

1 (1 + cos 4ϕ). 8

(8.149)

Note that the maximum value of 1/4 for P4 (2C , 2D ) is much better than P4 = 3/256 in Sect.8.5.2. For multi-mode treatment, we use a state in Eq.(7.49) as the input to the interferometer in Fig.8.13. Similar to the calculation in Sect.8.4.2, we obtain P4 ∝ 1 + V4 cos 4ϕ + V2 cos 2ϕ.

(8.150)

Due to the non-trial values of T, R in Eq.(8.93), V4 and V2 have a complicated dependence on the E/A quantity. We will not present that here. The

172

8 Quantum Interference with Two Pairs of Down-Converted Photons

existence of the cos 2ϕ term in Eq.(8.150) again is due to the less-than-perfect cancellation of the |2A , 2B term in the multi-mode case, which is superposed with the |4A , 0B and |0A , 4B terms to produce a cos 2ϕ-dependence. A reverse arrangement of the interferometer in Fig.8.13 should also work to demonstrate the four-photon de Broglie wavelength. In the reverse arrangement, the ﬁrst beam splitter has T, R as given in Eq.(8.93), but the second one is a symmetric beam splitter. The following is the simple physical picture. For a |2a , 2b state input to the ﬁrst beam splitter, the output for T, R in Eq.(8.93) has the form of 1 1 |Ψ4 out = √ |4A , 0B + |0A , 4B + √ |3A , 1B − |1A , 3B . (8.151) 6 3 With a symmetric beam splitter, |3A , 1B and |1A , 3B will not contribute to P4 (2C , 2D ) because, from Appendix A, we ﬁnd the output states for |3A , 1B and |1A , 3B are, respectively: ⎧ ⎨ |3A , 1B out = |4C , 0D − |0C , 4D + |1C , 3D − |3C , 1D /2, (8.152) ⎩ |1A , 3B out = |4C , 0D − |0C , 4D + |3C , 1D − |1C , 3D /2, which do not contain the |2C , 2D term. A direct calculation of P4 (2C , 2D ) from Eq.(8.146) with √ ˆ Cˆ = (Aˆ + ejϕ B)/ √2 (8.153) ˆ = (ejϕ B ˆ − A)/ ˆ 2 D and

√ √ Aˆ = √T a ˆ + √ Rˆb, ˆ = T ˆb − Rˆ a B

(8.154)

can conﬁrm the above argument: P4 (2C , 2D ) =

1 (1 + cos 4ϕ), 8

(8.155)

which is same as Eq.(8.149). 8.6.2 NOON State Projection Measurement The two projection schemes discussed in previous sections cannot be generalized to an arbitrary number of photons because the projection cancellation is only on some speciﬁc terms. A general cancellation of all the unwanted terms is required for the generalization. Such a scheme is just a time-reversal of the scheme in Sect.8.5.2, where a photon state is sent at the bN port and is output at all the a-ports for detection, as in Fig.8.14. It is straightforward to show that the output is related to the input by

8.6 Multi-Photon De Broglie Wavelength by Projection Measurement 1 N−1

1 N

V

|ΨΝ

1 2

δΝ−1

δΝ

Η

δ2

a^Ν−1

a^Ν

173

δ1

a^1

a^2

Fig. 8.14. NOON state projection measurement. Reprinted ﬁgure with permission c from F. W. Sun, Z. Y. Ou, and G. C. Guo, Phys. Rev. A 73, 023808 (2006). 2006 by the American Physical Society.

1 ˆ bH − ejδk ˆbV + ..., a ˆk = √ N

(8.156)

where we omit the other input ports which are in vacuum and have no contribution to the N-photon coincidence measurement. N-photon coincidence measurement probability at the N detectors is proportional to " †" PN ∝ a ˆk a ˆk k k 2 1 " ˆ bH − ejδk ˆbV |Ψ = N N k 2 1 ˆN ˆN = N bH − bV |Ψ , N

(8.157)

where we used the algebraic identity in Eq.(8.135). If the input state |Ψ is an N-photon state of two polarization modes in the form of |Ψ = |ΨN = ck |N − kH |kV , (8.158) k

then Eq.(8.157) becomes: 2 1 ˆbN − ˆbN |ΨN vac| H V NN 2 2N ! = N N OON |ΨN N 2N ! = N PN OON (ΨN ), N

PN ∝

(8.159)

where |N OON is the NOON state in Eq.(8.112) with ϕ = π/N and PN OON (ΨN ) is the projection probability of |ΨN to the NOON state. That is why we call this scheme the NOON state projection measurement. Substituting the state in Eq.(8.158) into Eq.(8.159), we obtain:

174

8 Quantum Interference with Two Pairs of Down-Converted Photons

PN ∝

N! |c0 − cN |2 . NN

(8.160)

If there is a relative phase change of ϕ between the H- and V-polarizations in the input ﬁeld b and |c0 | = |cN | ≡ c, then Eq.(8.160) becomes: PN ∝

2|c|2 N ! 1 − cos(N ϕ) , NN

(8.161)

which shows the N-photon de Broglie wavelength. Just like the scheme in Sect.8.5.2, all the unwanted middle terms of |N − k, k with k = 0, N have no contribution to PN , due to destructive N-photon interference in the NOON state projection measurement. Later in Chapt.9, we will see how we can use this destructive interference to characterize quantitatively the degree of temporal distinguishability of photons. Note that as long as c0 and cN are not zero, PN always shows the N-photon de Broglie wavelength, regardless of the detail of the input state |Ψ . In fact, if |Ψ = |αH |αV or the coherent state in 45◦ -polarization, we will have: PN

2 2e−2|α| |α|2N 1 − cos(N ϕ) . ∝ NN

(8.162)

Of course, the projection probability is very small for large N , just like the scheme in Sect.8.5.2. In the following, we will apply the projection scheme to the state in Eq.(8.78), which can be obtained with the help of a beam splitter from two indistinguishable pairs (4 × 1) of photons in parametric down-conversion. In the case of two polarization modes, we inject a state of |2+ , 2− (“+ = 45◦ polarized and “− = 135◦ polarized) into a polarization beam splitter. The output state is then: ! 1 3 (8.163) |Ψ4 = |4H , 0V + |0H , 4V + |2H , 2V . 8 2 Note that the above state can also be produced from two orthogonally polarized type-I parametric down-conversion processes. Substituting Eq.(8.163) into Eq.(8.159), we obtain: 9 P4 ∝ 1 − cos(4ϕ) , (8.164) 128 which has a visibility of 100%. On the other hand, as we discussed in Sect.8.4.1, the two pairs from parametric down-conversion may not be completely indistinguishable, and in the extreme case may be totally independent of each other (2 × 2 case). In the latter case, the state is not in the form of Eq.(8.163), but in a product state of |Ψ4 = |Ψ2 (τ1 ) ⊗ |Ψ2 (τ2 ),

(8.165)

8.6 Multi-Photon De Broglie Wavelength by Projection Measurement

175

with 1 |Ψ2 = √ |2H + e2iϕ |2V . 2

(8.166)

Here, we label the times at which the two pairs are generated as τ1 , τ2 , respectively. If we let Tc be the size of the wave packet for each photon, then for the 2 × 2 case, we have |τ1 − τ2 | >> Tc , but for the 4 × 1 case, |τ1 − τ2 | << Tc .

V Η

50:50

R

A QWP R

B C

D

Fig. 8.15. NOON state projection measurement for N = 4 case. QWP=quarter wave plate; R=45◦ -rotator. Reprinted ﬁgure with permission from F. W. Sun, Z. Y. Ou, and G. C. Guo, Phys. Rev. A 73, 023808 c (2006). 2006 by the American Physical Society.

The NOON state projection measurement scheme is redrawn in Fig.8.15 for the N = 4 case. To calculate the four-photon detection probability, we need to rewrite Eq.(8.156) to include the detection time: ⎧

ˆ ˆ ˆ ⎪ ⎪ E1 (t) = EH (t) + EV (t) /2 + ..., ⎪ ⎨ ˆ ˆ (t) − EˆV (t) /2 + ..., E2 (t) = E H

(8.167) ˆ ˆH (t) + iE ˆV (t) /2 + ..., ⎪ E3 (t) = E ⎪ ⎪

⎩ ˆ ˆH (t) − iE ˆV (t) /2 + ..., E4 (t) = E and the four-photon coincidence is proportional to the four-photon correlation function of ˆ1 (t1 )E ˆ2 (t2 )Eˆ3 (t3 )Eˆ4 (t4 )|Ψ4 ||2 . Γ (4) (t1 , t2 , t3 , t4 ) = ||E

(8.168)

When applying the above to the state in Eq.(8.165), we ﬁnd six nonzero terms in Eq.(8.168), that is, when (i) t1 = t2 = τ1 , t3 = t4 = τ2 or t1 = t2 = τ2 , t3 = t4 = τ1 ; (ii) t1 = t3 = τ1 , t2 = t4 = τ2 or t1 = t3 = τ2 , t2 = t4 = τ1 ; (iii) t1 = t4 = τ1 , t2 = t3 = τ2 or t1 = t4 = τ2 , t2 = t3 = τ1 . The case (i) can be calculated as ˆ2 (τ1 )Eˆ3 (τ2 )E ˆ4 (τ2 )|Ψ4 Eˆ1 (τ1 )E ˆ ˆ ˆ ˆ4 (τ1 )|Ψ4 = E1 (τ2 )E2 (τ2 )E3 (τ1 )E ˆ ˆ ˆ3 (τ2 )E ˆ4 (τ2 )|Ψ2 (τ2 ) = E1 (τ1 )E2 (τ1 )|Ψ2 (τ1 ) ⊗ E i2ϕ i2ϕ ∝ (1 − e )(1 + e )|0; Similarly, the case (ii) gives:

(8.169)

176

8 Quantum Interference with Two Pairs of Down-Converted Photons

ˆ1 (τ1 )Eˆ2 (τ2 )E ˆ3 (τ1 )E ˆ4 (τ2 )|Ψ4 E ˆ ˆ ˆ3 (τ2 )Eˆ4 (τ1 )|Ψ4 = E1 (τ2 )E2 (τ1 )E i2ϕ ∝ (1 + ie )(1 + iei2ϕ )|0;

(8.170)

and the case (iii) gives: ˆ1 (τ1 )Eˆ2 (τ2 )E ˆ3 (τ2 )E ˆ4 (τ1 )|Ψ4 E ˆ ˆ ˆ3 (τ1 )Eˆ4 (τ2 )|Ψ4 = E1 (τ2 )E2 (τ1 )E i2ϕ ∝ (1 − ie )(1 − iei2ϕ )|0.

(8.171)

When the two pairs overlap and become indistinguishable, i.e., |τ1 −τ2 | << Tc (the 4 × 1 case), we add the six contributions in amplitudes before taking the absolute value for the overall four-photon detection probability: 2 P4 ∝ 2(1 − ei4ϕ ) + 2(1 + iei2ϕ )2 + 2(1 − iei2ϕ )2 ∝ 1 − cos 4ϕ, (8.172) which is same as Eq.(8.164) in the single-mode model. On the other hand, when the two pairs are well-separated, i.e, |τ1 − τ2 | >> Tc (the 2 × 2 case), all six contributions are distinguishable and we add their absolute values to give P4 : 2 4 4 P4 ∝ 2(1 − ei4ϕ ) + 2(1 + iei2ϕ ) + 2(1 − iei2ϕ ) 3 (8.173) ∝ 1 − cos 4ϕ. 7 In both cases, the four-photon coincidence measurement has a sinusoidal modulation with 4ϕ — typical of a four-photon de Broglie wave. The ﬁrst case has a 100% visibility, a result from a true four-photon NOON state projection but the second case only produces 3/7= 42% visibility. This reduced visibility is a result of temporal distinguishability between the two pairs. However, there is still indistinguishability for the two photons within one pair. That is why the visibility is not zero in the 2 × 2 case. For the case in between the two extreme cases of 4 × 1 and 2 × 2, we need to use the more rigorous multi-mode theory. Consider two collinear degenerate type-I parametric down-conversion processes in series, but with their orientations orthogonal to each other. One of the processes gives the |2H state while the other produces |2V . If the two processes are pumped from a common source, the ﬁnal state will be in the form of Eq.(8.166). In the multi-mode theory, the quantum state for the system up to the four-photon state can be derived along the same line leading to Eq.(2.38) and it has the form of 1 2 |Ψ4 = ... + ξ1 |ΦH4 + ξ22 |ΦV 4 + 2ξ1 ξ2 |ΦH2 |ΦV 2 , (8.174) 2 where

8.7 Stimulated Emission and Multi-Photon Interference

|ΦM2 = and

|ΦM4 =

dω1 dω2 Φ(ω1 , ω2 )ˆ a†M (ω1 )ˆ a†M (ω2 )|0,

177

(8.175)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) ×ˆ a†M (ω1 )ˆ a†M (ω2 )ˆ a†M (ω1 )ˆ a†M (ω2 )|0,

(8.176)

where M = H, V . |ΦM2 is a two-photon state and |ΦM4 is a four-photon state. Now we calculate the four-photon correlation function Γ (4) in Eq.(8.168) with the state in Eq.(8.174). After integrating over all time variables and with some lengthy calculation, we obtain the four-photon detection probability with |ξ1 | = |ξ2 | ≡ |ξ| and ξ1 /ξ2 = e2iϕ (ξ is the pump amplitude so ϕ is the phase of the down-converted ﬁelds): P4 ∝ 64|ξ|4 E + 7A/2 1 − V4 cos 4ϕ , (8.177) with V4 ≡

3(A + 2E) . 7A + 2E

(8.178)

Here, the quantities A, E are given in Eqs.(7.17, 7.18). When E = A, we have: P4 ∝ 1 − cos 4ϕ,

(8.179)

which is exactly the same as Eq.(8.172) and corresponds to the situation when the two pairs overlap to form an indistinguishable four-photon entangled state. However, when E = 0, Eq.(8.177) becomes: P4 ∝ 1 −

3 cos 4ϕ, 7

(8.180)

which is same as Eq.(8.173) and in this situation the two pairs of downconverted photons are well-separated and independent of each other. Experimentally, this scheme of NOON state projection for the demonstration of multi-photon de Broglie wavelength was demonstrated by Resch et al. [8.27] with a coherent state and by Sun et al. [8.28] with a state in Eq.(8.163) from parametric down-conversion.

8.7 Stimulated Emission and Multi-Photon Interference When Einstein [8.29] ﬁrst proposed the stimulated emission in 1917 to explain the blackbody radiation spectrum, he only presented the process phenomenologically, that is, he assumed that an incoming photon will give rise to another

178

8 Quantum Interference with Two Pairs of Down-Converted Photons

chance for an excited atom to radiate in addition to the spontaneous emission. Although the stimulated emission forms the basis for the operation of laser, no one has attempted to understand the process from fundamental principles. The theory for laser relied totally on Einstein’s original idea [8.30]. Of course, the modern quantum electrodynamics theory can calculate the Einstein’s B-coeﬃcient and relates it to the A-coeﬃcient that is responsible for the spontaneous emission. But is there a simpler fundamental physical principle that governs the phenomenon of stimulated emission? It turns out that the underlying physics in stimulated emission is the multi-photon interference eﬀect that we discussed throughout this book. It stems from indistinguishablity between the incoming photons and the photon emitted by the excited atom. At the fundamental single-photon level, stimulated emission is seen as the emission by the excited atom of an identical photon to the incoming photon. However, the same photon can also be produced even without the incoming photon, due to spontaneous emission. Thus, the existence of the input photon will enhance the production rate, as compared to the case without the input photon. Indeed, in recent study of stimulated emission of single photons, a doubled rate is observed in photon production that is correlated to the input photon [8.31, 8.32, 8.33]. In our discussion of stimulated emission in Sect.7.2.2, we already associated the stimulated emission of one photon to the two-photon interference eﬀect, as in the Hong-Ou-Mandel photon bunching eﬀect. This suggests that the underlying physics in stimulated emission is simply multi-photon quantum interference. Indeed, Ali-Khan and Howell [8.34] and Irvine et al. [8.35] recently utilized a beam splitter and the two-photon Hong-Ou-Mandel interference eﬀect [8.10, 8.36] to duplicate the result in the photon cloning process observed in stimulated emission [8.33]. This further strengthens the connection between stimulated emission and the multi-photon interference.

{

{

...

N

...

(a)

N

excited atom

(b)

{

...

{

...

N

BS

N

Fig. 8.16. Similarity between (a) the stimulated emission in an excited atom and (b) the multiphoton interference with a beam splitter (BS). The empty circle in (a) represents the photon emitted by the excited atom.

To understand further the connection between the stimulated emission and multi-photon interference, we consider the two situations in Fig.8.16. The process of stimulated emission of an N -photon state is shown in Fig.8.16(a), where N photons interact with an atom in an excited state. The atom will

8.7 Stimulated Emission and Multi-Photon Interference

179

emit one photon regardless of the input (the empty circle in Fig.8.16a). Total photon number is N + 1. Assume that the spontaneous emission rate is R into the same mode of the input photons. It is known that the emission rate stimulated by one photon is the same. Then because each input photon may stimulate the excited atom, the total rate is then (N + 1)R. The extra R is from spontaneous emission. There is an enhancement of N times in the photon production rate. The case of N = 1 was observed in Ref.[8.33]. Mathematically, any phase insensitive linear ampliﬁer can be modelled as a parametric ampliﬁer, which is described quantum mechanically by [8.37] a ˆ(out) = Gˆ as + gˆ a†i , s

(8.181)

where a ˆi corresponds to the internal modes of the ampliﬁer and is the idler mode for the parametric ampliﬁer. It is usually independent of a ˆs and is in vacuum. To preserve the commutation relation, we need |G|2 − |g|2 = 1. At microscopic level of atoms, we have a small value of |g| << 1 or |G| ∼ 1. The unitary evolution operator for Eq.(8.181) then has the form of ˆ ≈ 1 + (gˆ ˆ†i + h.c.) U a†s a

(8.182)

With a vacuum input, we have the output state as (0)

|Φout ≈ |0 + g|1s ⊗ |1i .

(8.183)

The second term gives the spontaneous emission with a probability of |g|2 . When the input is a single-photon state |1s ⊗ |0i , we have (1)

† |Φout ≈ |1s |0i + g(ˆ √ as |1s ) ⊗ |1i = |1s |0i + 2g|2s ⊗ |1i .

(8.184)

The probability for the emission from the ampliﬁer is then 2|g|2 . The extra emission probability of |g|2 is usually attributed to the stimulated emission, which is similar to the photon bunching eﬀect. In general, when the input state is an N -photon state of |Ns |0i , we have the output state as (N )

† |Φout ≈ |Ns |0i + g(ˆ √ as |Ns ) ⊗ |1i = |Ns |0i + N + 1g|(N + 1)s ⊗ |1i .

(8.185)

The photon emission rate from the ampliﬁer is now N + 1 times the spontaneous rate. As the single-photon input case, each fold of increase in the rate can be attributed to the stimulated emission from one individual photon in the input N -photon state. Notice that when the input photons are not in the same mode as the ampliﬁer and thus are distinguishable from the photon emitted by the ampliﬁer, the output state becomes

180

8 Quantum Interference with Two Pairs of Down-Converted Photons (N )

|Φout ≈ |0s |Ns |0i + g(ˆ a†s |0s ) ⊗ |Ns ⊗ |1i = |0s |Ns |0i + g|1s |Ns |1i .

(8.186)

So the photon production rate is exactly the same as the spontaneous emission. The above analysis with the parametric ampliﬁer conﬁrms the previous results obtained from the pictorial argument based on Fig.8.16(a). ...

...

...

... N

(1)

(2)

...

...

{

{

{

N

...

N

...

(N+1)

Fig. 8.17. The N + 1 possibilities to arrange the photons in Fig.8.16(b), i.e., the input N photons (solid circles) and the single photon from the other input port (empty circle) in the N + 1 detectors for coincidence measurement.

In multi-photon interference scheme in Fig.8.16(b), on the other hand, a single photon and N photons are combined by a 50:50 beam splitter. The probability of detecting all N + 1 photons in one side is easily calculated to be (N + 1)/2N +1 (see below). However, when the single photon is distinguishable from the N photons and no multi-photon interference occurs, we ﬁnd the detection probability is simply 1/2N +1 . The N + 1 factor is a result of constructive interference of N + 1 possibilities in detecting the N + 1 photons (Fig.8.17). Each possibility corresponds to the situation when the single input photon (empty circle in Fig.8.17) is detected by a speciﬁc detector. We add the amplitudes of the N + 1 possibilities before taking the absolute value for the indistinguishable case but we add the absolute values of the amplitude of each possibility for the distinguishable case. The ratio between the two cases is then N + 1. The case of N = 1 is the Hong-Ou-Mandel photon bunching eﬀect [8.36] discussed in Sect.3.4. A similar four-photon bunching eﬀect was observed by Ou et al. [8.17] with a |2, 2 input state, which is also a result of four-photon interference, as discussed in Sect.8.4.1. To conﬁrm the intuitive argument analytically, we consider the output state of the situation depicted in Fig.8.16(b) where the input N photons are superposed with a single photon by a 50:50 beam splitter. It has the form of (see Appendix A) ! N +1 (BS) |Φout = |N + 11 |02 + ... , (8.187) 2N +1 where we only write down the state for which all the N + 1 photons exit at one port (port 1) of the beam splitter. On the other hand, if the input N

8.8 Remarks on E and A and General Discussion

181

photons are distinguishable from the single photon from the other input port, they behave like classical particles and follow the Bernoulli distribution and the output state becomes ! 1 (BS) |N 1 |11 |02 + ... . (8.188) |Φout = 2N +1 Therefore, the ratio of the probabilities of detecting N + 1 photons in port 1 is N + 1 between the case when the photons are all indistinguishable and the case when the N photons are distinguishable from the one photon. The enhancement factor here is exactly the same as the stimulated emission in Fig.8.16(a). Therefore, the spontaneous emission rate R corresponds to the situation when the input N photons to the atom are distinguishable from the emitted photon by the atom, so that the atom is not inﬂuenced by the input photons and only emits spontaneously. This case is exactly the same as the case in Fig.8.16(b) but when the single photon is distinguishable from the N photons. But when the input N photons are indistinguishable from the emitted photon by the atom, constructive multi-photon interference leads to a factor of N enhancement in photon detection rate. From the above analysis, we thus claim that the fundamental principle at play in stimulated emission is nothing but multi-photon constructive interference due to photon indistinguishability between the input photons and the photon emitted by the atom.

8.8 Remarks on E and A and General Discussion Throughout this chapter, we ﬁnd that the quantity E/A describes the temporal distinguishability between the two pairs of photons generated in parametric down-conversion. From the formulae of E, A in Eqs.(7.17, 8.18), if we deﬁne Φ4 (ω1 , ω2 ; ω1 , ω2 ) ≡ Φ(ω1 , ω2 )Φ(ω1 , ω2 ), then E, A have the form of E = dω1 dω2 dω1 dω2 Φ∗4 (ω1 , ω2 ; ω1 , ω2 )Φ4 (ω1 , ω2 ; ω1 , ω2 ), and

A=

dω1 dω2 dω1 dω2 Φ∗4 (ω1 , ω2 ; ω1 , ω2 )Φ4 (ω1 , ω2 ; ω1 , ω2 ).

(8.189)

(8.190)

(8.191)

So, E is diﬀerent from A in the exchange between ω1 and ω1 in only one Φ4 (ω1 , ω2 ; ω1 , ω2 ) function in the integral. Note that the exchange is on the variables for two photons from diﬀerent pairs (between unprimed and primed variables). If we have the exchange symmetry of

182

8 Quantum Interference with Two Pairs of Down-Converted Photons

Φ4 (ω1 , ω2 ; ω1 , ω2 ) = Φ4 (ω1 , ω2 ; ω1 , ω2 ),

(8.192)

which indicates the symmetry between the exchange of the two pairs, then we have E = A, which, from previous sections, corresponds to completely indistinguishable pairs. On the other hand, the orthogonal relation for the exchange

dω1 dω2 dω1 dω2 Φ∗4 (ω1 , ω2 ; ω1 , ω2 )Φ4 (ω1 , ω2 ; ω1 , ω2 ) = 0

(8.193)

leads to E = 0, which corresponds to totally separate pairs. Thus the exchange relations in Eq.(8.192) and Eq.(8.193) deﬁne the two scenarios in temporal distinguishability between pairs of photons in parametric down-conversion. In the next chapter, we will discuss temporal distinguishability in a more general sense. There are a number of interference schemes involving two pairs of downconverted photons that are omitted from our discussion. They are nevertheless quite important in quantum information science and the foundation of quantum mechanics. They include a realization of the GHZ state [8.38] for three [8.39] and four photons [8.40], a violation of generalized Bell’s inequality for four photons [8.41], and a Bell state bunching eﬀect [8.42]. This list is by no means complete and, undoubtedly, more interesting schemes will emerge as more quantum information protocols are discovered.

References 8.1 D. Bouwmeester, A. Ekert, and A. Zeilinger, eds. The Physics of Quantum Information, (Springer, New York, 1999). 8.2 C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wooters, Phys. Rev. Lett 70, 1895 (1993). 8.3 D. Bouwmeester, J. W. Pan, K. Mattel, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature (London) 390, 575 (1997). 8.4 M. Zukowski, A. Zeilinger, M. A. Horne, and A. Ekert, Phys. Rev. Lett. 71, 4287 (1993). 8.5 M. Pavi˘ci´c and J. Summhammer, Phys. Rev. Lett. 73, 3191 (1994). 8.6 J. W. Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998). 8.7 C. K. Hong and L. Mandel, Phys. Rev. Lett. 56, 58 (1986). 8.8 L. Mandel, Phys. Rev. A28, 929 (1983). 8.9 S. J Kuo, D. T Smithey, and M.G. Raymer, Phys. Rev. A 43, 4083 (1991). 8.10 C.K. Hong, Z.Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 8.11 L. J. Wang and J.-K. Rhee, in Quantum Electronics and Laser Science Conference, OSA Technical Digest, pp.143-144 (Optical Society of America, Washington DC, 1999). 8.12 H. de Riedmatten, I. Marcikic, W. Tittel, H. Zbinden, and N. Gisin, Phys. Rev. A 67, 022301 (2003). 8.13 S. L. Braunstein and A. Mann, Phys. Rev. A 51, R1727 (1995).

References

183

8.14 K. Tsujino, H. F. Hofmann, S. Takeuchi, and K. Sasaki, Phys. Rev. Lett. 92, 153602 (2004). 8.15 A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, Nature 412, 887 (2001). 8.16 Z. Y. Ou, J. -K. Rhee, and L. J. Wang, Phys. Rev. A 60, 593 (1999). 8.17 Z. Y. Ou, J. -K. Rhee, and L. J. Wang, Phys. Rev. Lett. 83, 959 (1999). 8.18 A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). 8.19 P. Walther, J.-W. Pan, M. Aspelmeyer, R. Ursin, S. Gasparoni, and A. Zeilinger, Nature (London) 429, 158 (2004). 8.20 M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, Nature (London) 429, 161 (2004). 8.21 W. Heisenberg, Z. Phys. 43, 172 (1927). 8.22 Z. Y. Ou, Phys. Rev. A 55, 2598 (1997). 8.23 H. F. Hofmann, Phys. Rev. A 70, 023812 (2004). 8.24 H. Wang and T. Kobayashi, Phys. Rev. A 71, 021802(R) (2005). 8.25 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995). 8.26 B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, quant-ph/0610266. 8.27 K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, quant-ph/0511214. 8.28 F. W. Sun, B. H. Liu, Y. F. Huang, Z. Y. Ou, and G. C. Guo, Phys. Rev. A 74, 033812 (2006). 8.29 A. Einstein, Phys. Z. 18, 121 (1917). 8.30 P. W. Milonni and J. H. Eberly, Lasers (Wiley, New York, N.Y., 1988). 8.31 Z. Y. Ou, L. J. Wang, and L. Mandel, J. Opt. Soc. Am. B 7, 211 (1990). 8.32 A. Lamas-Linares, J. C. Howell, and D. Bouwmeester, Nature 412, 6850 (2001). 8.33 A. Lamas-Linares, C. Simon, J. C. Howell, and D. Bouwmeester, Science 296, 712 (2002). 8.34 I. Ali-Khan and J. C. Howell, Phys. Rev. A 70, 010303(R) (2004). 8.35 W. T. M. Irvine, A. Lamas-Linares, M. J. A. de Dood, and D. Bouwmeester, Phys. Rev. Lett. 92, 047902 (2004). 8.36 J. G. Rarity and P. R. Tapster, J. Opt. Soc. Am. B 6, 1221 (1989). 8.37 C. M. Caves, Phys. Rev. D 26, 1817 (1982). 8.38 D. M. Greenberger, M. A. Horne, and A, Zeilinger, in Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, edited by M. Katafos, (Kluwer Academic, Dordrecht, The Netherlands, 1989). 8.39 D. Bouwmeester, J. -W. Pan, M. Daniell, H.d Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 82, 1345 (1999). 8.40 J. -W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 86, 4435 (2001). 8.41 M. Eibl, S. Gaertner, M. Bourennane, C. Kurtsiefer, M. Zukowski, and H. Weinfurter, Phys. Rev. Lett. 90, 200403 (2003). 8.42 H. S. Eisenberg, J. F. Hodelin, G. Khoury, and D. Bouwmeester, Phys. Rev. Lett. 96, 160404 (2006).

9 Temporal Distinguishability of a Multi-Photon State

Optical coherence theory was developed in late 1940 [9.1] to describe how close an optical ﬁeld is to a single train of wave. The coherence time, or length of an optical ﬁeld, is roughly the size of the wave packet within which the ﬁeld can be regarded as an uninterrupted wave train with a ﬁxed initial phase. The quantum counterpart of the theory was later constructed by Glauber [9.2], and explained a vast realm of phenomena in quantum optics [9.3]. Its primary success has been in the discovery of a new class of optical ﬁelds, the so-called nonclassical ﬁelds, which do not have counterparts in classical wave theory [9.4]. Later, it was found that this class of nonclassical optical ﬁelds have some extraordinary properties and are sometimes superior to the classical ﬁelds [9.5]. For example, the squeezed state of light is a kind of nonclassical ﬁeld that has less noise in quadrature-phase amplitudes than classical ﬁelds, including vacuum [9.6, 9.7, 9.8]. On the other hand, the coherence theory is based on the concept of waves via interference observed in intensity of the ﬁeld or the single-photon interference. This description becomes short-handed when we begin to deal with cases involving more than one photon in quantum information. For example, in the previous chapters, the multi-photon interference eﬀects cannot be described simply by the optical coherence theory with the correlation functions. For four-photon interference with two pairs of parametric down-converted photons, we found a quantity of E/A that is best to describe the interference eﬀect. This quantity is related to how the two pairs are distributed in time. In this chapter, we will develop a general theory to best describe the multiphoton interference eﬀect. We ﬁnd that the interference eﬀect is directly related to the distinguishability of the photons, providing a direct proof of the fundamental quantum principle, that is, interference is a consequence of indistinguishability. We consider some interferometric techniques that can provide a quantitative measure of the degree of distinguishability. We will concentrate only on the temporal distinguishability. Other degrees of freedom can be discussed in a similar formalism. In the following, let us ﬁrst

186

9 Temporal Distinguishability of a Multi-Photon State

introduce the concept with the simplest multi-photon interference scheme, i.e., the Hong-Ou-Mandel two-photon interferometer.

9.1 Hong-Ou-Mandel Interferometer for Characterizing Two-Photon Temporal Distinguishability Let us revisit the Hong-Ou-Mandel interferometer. Consider a photon pair from a type-II non-degenerate parametric down-conversion process. From Eq.(3.43), we ﬁnd that for an input state of a†s (ω1 )ˆ a†i (ω2 )|0, (9.1) |Φ2 = dω1 dω2 Φ(ω1 , ω2 )ˆ the maximum visibility in a two-photon Hong-Ou-Mandel interferometer occurs at Δz = 0 with dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω2 , ω1 ) V(0) = M2 ≡ . (9.2) dω1 dω2 |Φ(ω1 , ω2 )|2 M2 is deﬁned as the degree of permutation symmetry. Note that M2 = M2∗ and 0 ≤ |M2 | ≤ 1. The visibility, or the degree of permutation symmetry, is one if and only if Φ(ω1 , ω2 ) satisﬁes the permutation symmetry relation: Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ).

(9.3)

This permutation relation is a signature of spectral indistinguishability of the two photons, that is, we cannot tell the diﬀerence between the two photons through their spectra. This, in turn, yields temporal indistinguishability if we consider the Fourier transformation: 1 (9.4) G(t1 , t2 ) = dω1 dω2 Φ(ω1 , ω2 )e−i(ω1 t1 +ω2 t2 ) . 2π A combination of Eqs.(9.3, 9.4) gives directly the symmetric relation: G(t1 , t2 ) = G(t2 , t1 ),

(9.5)

for all times of t1 , t2 . From Sect.2.5, we ﬁnd that |G(t1 , t2 )|2 is proportional to the probability density of detecting one photon at t1 and another one at t2 . So, the symmetric relation in Eq.(9.5) tells us that we cannot distinguish the two photons in time as well. On the other hand, the visibility is zero if Φ(ω1 , ω2 ) does not have any overlap with Φ(ω2 , ω1 ), which is characterized by the orthogonal relation: (9.6) dω1 dω2 Φ∗ (ω1 , ω2 )Φ(ω2 , ω1 ) = 0, or, in time,

9.2 Characterizing an N-Photon State in Time

dt1 dt2 G∗ (t1 , t2 )G(t2 , t1 ) = 0.

187

(9.7)

This orthogonal relation indicates that the two functions G(t1 , t2 ), G(t2 , t1 ) have no overlap. At this point, it is not easy to see the physical meaning of Eq.(9.7). However, if we go back to Eq.(3.43) for nonzero delay (Δz = 0), we ﬁnd that the equivalent Φ(ω1 , ω2 ) in Eq.(9.2) in this case will be Φ (ω1 , ω2 ) ≡ Φ(ω1 , ω2 )eiω1 Δz/c , which is not symmetric with respect to ω1 , ω2 , even if Φ(ω1 , ω2 ) is. In particular, if Δz/c is large enough [>> Tc ∼ 1/ΔωP DC with ΔωP DC as the range of Φ(ω1 , ω2 )], we may have V2 (Δz) = 0 or Φ (ω1 , ω2 ) satisﬁes Eq.(9.6). Since Δz is the relative delay between the two photons before they meet at the beam splitter of the Hong-Ou-Mandel interferometer in Fig.3.3, we may believe that there is a large enough delay between the two photons so that the two photons become distinguishable in time when they arrive at the beam splitter. So, the orthogonal relation in Eq.(9.6) or Eq.(9.7) corresponds to the situation when the two photons are well-separated in time and form two non-overlapping and distinguishable wave packets. Therefore, the visibility in the Hong-Ou-Mandel interferometer is a direct measure of the temporal distinguishability of the two photons. This is very similar to the role of the ﬁeld correlation function Γ12 of Eq.(1.2) in deﬁning optical coherence of a ﬁeld. Furthermore, we have learned from Chapt.8 that the visibility in fourphoton interference is directly related to the quantity E/A, which is a measure of the temporal distinguishability of photon pairs from parametric downconversion. This quantity is again dependent on the permutation symmetry as shown in Eqs.(8.190, 8.191). The symmetry relation in Eq.(8.192) and the orthogonal relation in Eq.(8.193) again are related to the pair indistinguishability and distinguishability in time, respectively. Next, we will generalize Eqs.(9.3, 9.6) and Eqs.(8.192, 8.193) to an arbitrary N -photon case and relate them to the visibility of some N -photon interference experiments.

9.2 Characterizing an N-Photon State in Time In this section, we will present a general formalism for describing photons in diﬀerent temporal distributions. The general question is how to characterize an N -photon state with diﬀerent temporal distributions, as shown in Fig.9.1. We will mainly concentrate on some extreme cases, i.e., some photons are either completely indistinguishable or completely distinguishable in time. Our discussion is only on the temporal/spectral modes. Other degrees of freedom, such as polarization, are only auxiliary in the discussion. We also consider only pure states here.

188

9 Temporal Distinguishability of a Multi-Photon State

(a)

Fig. 9.1. Diﬀerent temporal distributions for an N -photon state: (a) All N photons are in one temporal mode (the N × 1 case); (b) all N photons are in diﬀerent and orthogonal temporal modes (the 1 × N case); (c) the intermediate case.

(b)

(c)

9.2.1 General Description of a Multi-Mode N-Photon State An arbitrary N-photon state of wide spectral range can be described by |ΦN = N −1/2 dω1 dω2 ...dωN Φ(ω1 , ..., ωN )× ×ˆ a† (ω1 )ˆ a† (ω2 )...ˆ a† (ωN )|0, where the normalization factor N is given by Φ(P {ω1 , ..., ωN }), N = dω1 dω2 ...dωN Φ∗ (ω1 , ..., ωN )

(9.8)

(9.9)

P

where P is the permutation operator on the indices of 1, 2, ..., N and the sum is over all possible permutation. There are totally N ! terms in the sum. So, the value of N ranges from AN to N !AN with AN ≡ dω1 dω2 ...dωN |Φ(ω1 , ..., ωN )|2 . (9.10) The maximum value of N !AN is reached when Φ(ω1 , ..., ωN ) = Φ(P {ω1 , ..., ωN }),

(9.11)

for all P . As we have noted before for a two-photon case, this corresponds to a case when the N photons are indistinguishable. We will refer to this case as the N × 1 case, meaning that all the N photons are in one indistinguishable temporal mode. In the special case when Φ(ω1 , ..., ωN ) is factorized as φ(ω1 )φ(ω2 )...φ(ωN ), the N-photon state simply becomes: |ΦN = with

1 ˆ †N A(φ) |0 ≡ |N φ , N!

ˆ A(φ) =

dωφ(ω)ˆ a(ω)

( dω|φ(ω)|2 = 1).

(9.12)

(9.13)

9.2 Characterizing an N-Photon State in Time

189

ˆ ˆ Aˆ† ] = 1 and represents the annihilation operator Note that A(φ) satisﬁes [A, for a single temporal mode characterized by φ(ω). The single-photon state |1φ has a single-photon detection probability of |g(τ )|2 with a temporal shape of 1 √ (9.14) dωφ(ω)e−iωt . g(τ ) = 2π The other extreme case of N = AN is achieved when dω1 ...dωN Φ∗ (ω1 , ..., ωN )Φ(P {ω1 , ..., ωN }) = 0,

(9.15)

for all P except the identity operation. From the discussion of the two-photon case in Sect.9.1, we ﬁnd this corresponds to the situation when all photons are well-separated in time. For the situations in between the two extreme cases, the value of N is between AN and N !AN . For example, if the spectral amplitude Φ({ω}) have partial permutation symmetry, that is, Φ(ω1 , ..., ωN ) = Φ(P{ni } {ω1 , ..., ωN }),

(9.16)

where the permutation P{ni } with i = 1, ..., k only applies to a subgroup of ni variables in {ω1 , ω2 , ..., ωN } and the permutation between diﬀerent groups is orthogonal, that is, dω1 ...dωN Φ∗ (ω1 , ..., ωN )Φ(P{ni ,nj } {ω1 , ..., ωN }) = 0, (i = j), (9.17) then it can be easily shown that N = n1 !n2 !...nk !AN . In the simple case when Φ(ω) can be factorized as Φ(ω1 , ..., ωN ) = φ1 (ω1 )...φ1 (ωn1 )φ2 (ωn1 +1 )...φ2 (ωn1 +n2 )...φk (ωN ), (9.18) the N-photon state in Eq.(9.8) becomes: |ΦN =

1 1 1 |n1 φ1 |n2 φ2 ... |nk φk . n1 ! n2 ! nk !

(9.19)

This is the situation when the N photons are divided into k subgroups with ni (i = 1, 2, ..., k) photons in each group in a single temporal mode characterized by φi . This situation is denoted as the n1 + ... + nk case. For the special case of n1 = ... = nN = 1, we have |ΦN = |11 ⊗ |12 ⊗ ... ⊗ |1N and N = AN . So, the extreme case of N = AN corresponds to the situation when all the photons are in a mode of their own and are wellseparated from the others. We refer to this case as the 1 × N case. For simplicity of our later argument, let us consider another special kind of N-photon state with Φ(ω1 , ..., ωN ) = φ(ω1 )eiω1 T1 ...φ(ωN )eiωN TN .

(9.20)

190

9 Temporal Distinguishability of a Multi-Photon State

With this Φ, the N-photon state can be viewed as a direct product of N identical single-photon wave packets: |N T = |T1 ⊗ |T2 ⊗ ... ⊗ |TN , with

|Tj =

(9.21)

dωφ(ω)eiωTj a ˆ† (ω)|0.

(9.22)

However, the quantum state in Eq.(9.21) is not normalized. Substituting Eq.(9.20) into Eq.(9.9), we have the normalization factor as " P [exp{ iωk Tk }], (9.23) |φ(ωk )|2 e−iωk Tk N = dω1 dω2 ...dωN P

k

k

where P operates on {ω1 , ..., ωN }. When T1 = T2 = ... = TN , we recover the case when all N photons are in one single temporal mode with N = N !AN (the N × 1 case). On the other hand, if |Ti −Tj | >> 1/Δω(i = j) with Δωas the bandwidth of φ(ω), diﬀerent |Ti states are orthogonal with Tj |Ti = dtg ∗ (t − Tj )g(t − Ti ) = 0 and we have N = AN . This is the case when all the photons are well-separated from each other (the 1 × N case). The N-photon state in Eq.(9.8) describes a state when all photons are in one spatial and polarization mode. They only diﬀer in spectral mode. To include spatial and polarization modes, a more general N-photon state then has the following shape: −1/2 (1) (k) |ΦN = Nk dω1 ...dωn(1) ...dω1 ...dωn(k) Φ({ω (1) }, ..., {ω (k) }) 1 k (1)

a†1 (ωn(1) )...ˆ a†k (ωn(k) )|0, ×ˆ a†1 (ω1 )...ˆ 1 k (1)

(9.24)

(1)

where {ω (1) } = ω1 , ..., ωn1 , etc, and the normalization factor Nk is then: Nk = d{ω (1) }...d{ω (k) }Φ∗ ({ω (1) }, ..., {ω (k) })× × Φ(P1 {ω (1) }, ..., Pk {ω (k) }). (9.25) P1 ,...,Pk

Nk now ranges from AN to n1 !...nk !AN . The special case when Φ({ω (1) }, ..., {ω (k) }) factorizes is similar as before. 9.2.2 Direct Photon Detection from Glauber’s Coherence Theory Next, we consider an N-photon joint measurement with the joint probability density given by the correlation function from the quantum coherent theory as [9.2]

9.2 Characterizing an N-Photon State in Time

ˆ † (t1 )...Eˆ † (tN )E(t ˆ N )...E(t ˆ 1 ), Γ (N ) (t1 , t2 , ..., tN ) = E where ˆ = √1 E(t) 2π

dωˆ a(ω)e−iωt ,

191

(9.26)

(9.27)

and the average is over the quantum state of the system given in Eq.(9.8) for an arbitrary N-photon state. For simplicity, we ﬁrst apply it to the state in Eq.(9.21). To carry out the quantum average, it is easier to ﬁrst ﬁnd the N-photon detection probability amplitude: ˆ N )...E(t ˆ 1 )|ΦN . C (N ) (t1 , t2 , ..., tN ) = 0|E(t

(9.28)

Then, Γ (N ) (t1 , t2 , ..., tN ) = |C (N ) (t1 , t2 , ..., tN )|2 . From Eq.(9.27) for the ﬁeld operator and Eq.(9.20) for the Φ-function in |ΦN , it is straightforward to obtain: C (N ) (t1 , t2 , ..., tN ) = P [g(t1 − T1 )...g(tN − TN )], (9.29) P

where the permutation operation P is on t1 t2 ...tN and there are N ! terms in the sum. The overall probability of detecting N photons together (N-photon coincidence) is proportional to an integral of Γ (N ) (t1 , t2 , ..., tN ) over all times t1 , ..., tN : PN = dt1 ...dtN Γ (N ) (t1 , t2 , ..., tN ) 2 = dt1 ...dtN P [g(t1 − T1 )...g(tN − TN )] . (9.30) P

In the extreme case when T1 = T2 = ... = TN , we obtain PN (N × 1) = (N !)2 AN , while in the other extreme case when |Ti − Tj | >> 1/ΔΩ, we have PN (1 × N ) = N !AN . Therefore, we seem to have: PN (N × 1) = N !PN (1 × N ),

(9.31)

that is, the N-photon detection probability is N ! larger in the case of N identical photons than in the case of N separated photons. One may claim that this is the Bosonic photon bunching eﬀect. However, as we know, the N-photon state in Eq.(9.21) is not normalized. With the normalization factor considered, we have, instead: PN (N × 1) = PN (1 × N ) = N !.

(9.32)

For the case in between the two extreme cases, the expression for PN is complicated. For the un-normalized general state in Eq.(9.8) and Eq.(9.21), we may write:

192

9 Temporal Distinguishability of a Multi-Photon State

C (N ) (t1 , t2 , ..., tN ) =

G(P {t1 , ..., tN }),

(9.33)

P

with

G(t1 , ..., tN ) =

dω1 ...dωN Φ(ω1 ..., ωN )e−i(ω1 t1 +...+ωN tN ) . (2π)N/2

For the N-photon detection probability, we have: PN = dt1 ...dtN |C(t1 , t2 , ..., tN )|2 2 = dt1 ...dtN G(P {t1 , ..., tN }) .

(9.34)

(9.35)

P

Because the sum is over all permutations, we then have: G(P {t1 , ..., tN }) dt1 ...dtN G∗ (P {t1 , ..., tN }) P = P {dt1 ...dtN }G∗ (t1 , ..., tN ) G(P {t1 , ..., tN }) P = dt1 ...dtN G∗ (t1 , ..., tN ) G(P {t1 , ..., tN }),

(9.36)

P

where P is an arbitrary permutation operation and where we made the change of variables from t1 , ..., tN to P {t1 , ..., tN } in the integral. So, Eq.(9.35) becomes: G(P {t1 , ..., tN }). (9.37) PN = N ! dt1 ...dtN G∗ (t1 , ..., tN ) P

It is straightforward to show that dt1 ...dtN G∗ (t1 , ..., tN ) G(P {t1 , ..., tN }) = N ,

(9.38)

P

where N is given in Eq.(9.23). Thus, we have: PN = N !N .

(9.39)

For a normalized N-photon state, we then have PN = N ! in any case, as in Eq.(9.32). For the multi-spatial and polarization state in Eq.(9.24), we may ﬁnd PN , after some lengthy manipulation similar to that leading to Eq.(9.39), to be: PN = n1 !...nk !Nk ,

(9.40)

for the un-normalized state and P4 = n1 !...nk ! for the normalized state.

9.2 Characterizing an N-Photon State in Time

193

Thus, it is impossible to characterize diﬀerent cases of temporal entanglement with just simple direct multi-photon detection for the normalized state. Furthermore, even for the un-normalized state, we cannot explore the temporal indistinguishability among diﬀerent spatial and polarization modes with multi-photon detection, for Nk in Eq.(9.25) depends only on the permutation symmetry within photons in one spatial and polarization mode. Before we proceed further, it is interesting to evaluate the single-photon detection rate P1 which is proportional to: ˆ † (t)E(t)|Φ ˆ P1 = dtΦN |E (9.41) N . With some manipulation, it can be shown that P1 = N N for an un-normalized N-photon state and P1 = N for a normalized N-photon state. The reason that we still discuss the un-normalized case of an N-photon state is because we still encounter this kind of state in practice when a projection measurement is involved. Consider, for example, a multi-photon state from degenerate parametric down-conversion, which, for small ξ, has the form of (see Sect.2.2) ξ2 |ΨDP DC = α |0 + ξ|Φ2D + |Φ4D + ... , 2 with

|Φ2D =

and

|Φ4D =

dω1 dω2 Φ(ω1 , ω2 )ˆ a† (ω1 )ˆ a† (ω2 )|0,

(9.42)

(9.43)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )× ×ˆ a† (ω1 )ˆ a† (ω2 )ˆ a† (ω1 )ˆ a† (ω2 )|0.

(9.44)

Here, α in Eq.(9.42) is a normalization factor but because |ξ| << 1, |α| ≈ 1 no matter what the function Φ(ω1 , ω2 ) is. Two-photon and four-photon detection project the state to ξ|Φ2D and ξ 2 |Φ4D /2, respectively, which are not normalized. So, from Eqs.(9.9, 9.39), we have: (9.45) P2 = 2|ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 + Φ∗ (ω1 , ω2 )Φ(ω2 , ω1 ) . For a state from parametric down-conversion in the degenerate case, as in Eq.(9.42), we usually have the symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ) so that (9.46) P2D = 4|ξ|2 dω1 dω2 |Φ(ω1 , ω2 )|2 . Similarly,

194

9 Temporal Distinguishability of a Multi-Photon State

P4D = 48|ξ|4 (A + 2E) = 3P22 (1 + 2E/A), where

dω1 dω2 dω1 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2

(9.48)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )Φ(ω1 , ω2 )Φ(ω1 , ω2 )

(9.49)

A= and

E=

(9.47)

are the same quantities deﬁned in Sects.7.1 and 8.7 to characterize pair temporal distribution. Another example is from non-degenerate parametric down-conversion in a type-II χ(2) medium. The quantum state is similar to that in Eq.(9.42): ξ2 |ΨN P DC = α |0 + ξ|Φ2N + |Φ4N + ... , 2 with

|Φ2N =

and

|Φ4N =

dω1 dω2 Φ(ω1 , ω2 )ˆ a†H (ω1 )ˆ a†V (ω2 )|0

(9.50)

(9.51)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )× ×ˆ a†H (ω1 )ˆ a†V (ω2 )ˆ a†H (ω1 )ˆ a†V (ω2 )|0.

From Eq.(9.40), it is straightforward to show that 2 P2N = |ξ| dω1 dω2 |Φ(ω1 , ω2 )|2

(9.52)

(9.53)

and P4N = 2|ξ|4 (A + E) = 2P22 (1 + E/A).

(9.54)

The diﬀerence between degenerate and non-degenerate cases lies in the normalization factor N , which is not included in the un-normalized states in Eqs.(9.42, 9.50). N takes the form of Eq.(9.9) for the degenerate case but Eq.(9.25) for the non-degenerate case. The smaller values in Eqs.(9.53, 9.54) are due to distinguishability in polarization. The degenerate case gives an indistinguishable two-photon or four-photon state in Eq.(9.42), whereas photons are in two distinguishable polarization modes for the non-degenerate case, as in Eq.(9.24). The dependence of the four-photon coincidence in Eqs.(9.47, 9.54) on the quantity E/A is similar to the photon bunching eﬀect discussed in Sect.7.2.1.

9.2 Characterizing an N-Photon State in Time

195

In fact, the N-photon coincidence in Eq.(9.39) for the un-normalized N-photon state in Eq.(9.21) is proportional to the normalization factor of N , which has a maximum value of N !AN for the N × 1 case. This is the generalized photon bunching for the N-photon case, similar to the bunching eﬀect in Sect.8.4. So, the extra term in P4D that is proportional to E/A is the fourphoton bunching eﬀect, while that in P4N is for the two-pair bunching eﬀect, which is discussed in detail in Sect.7.2.1. In P4N , the two photons from the same polarization mode become indistinguishable when E = A, but photons of diﬀerent polarizations are still distinguishable. 9.2.3 A NOON-State Projection Measurement and Generalized Hong-Ou-Mandel Interferometer As seen in the previous section, a direct N-photon detection scheme cannot characterize the temporal indistinguishability in the general case. Therefore, we need to seek another method. Since the direct result of photon indistinguishability is the interference eﬀect, our scheme will be an N-photon interference scheme. As we discussed in Sect.9.1, a Hong-Ou-Mandel interferometer can be used to characterize the temporal distinguishability of two photons. However, this method cannot be applied simply to a higher photon number. For example, for a four-photon input state of |in = |21 , 22 , the output state of the HongOu-Mandel interferometer is (see Appendix A): !

1 3 (9.55) |41 , 02 + |01 , 42 + |21 , 22 , |out = 8 2 where we have a nonzero probability for |21 , 22 , which is equivalent to |11 , 12 in the Hong-Ou-Mandel interferometer. Therefore, the Hong-Ou-Mandel interferometer needs to be modiﬁed in order to generalize from a two-photon case to an arbitrary N-photon case. It turns out that the generalized Hong-Ou-Mandel interferometer is the NOONstate projection measurement scheme discussed in Sect.8.6.2. The following is the process for constructing the generalized Hong-Ou-Mandel interferometer. To account for the contribution of all the photons in an N -photon input state, we would like to have some kind of N -photon coincidence measurement, which requires N separate detectors. Let us denote the ﬁeld operator at each detector as ˆbj (j = 1, ..., N ). If we limit the input ports to two that are labeled as a ˆ1,2 , similar to the Hong-Ou-Mandel interferometer, the outputs are likely connected to the inputs by a linear relation for the linear optical system ˆbj = cj1 a ˆ1 + cj2 a ˆ2 + ...,

(9.56)

where due to commutation relation, there will be more than two inputs, but we put in vacuum the irrelevant ones, as denoted by the “...” symbol in Eq.(9.56) and they have no contribution to N -photon detection.

196

9 Temporal Distinguishability of a Multi-Photon State

For the Hong-Ou-Mandel interferometer with two outputs, the product of the output operators is related to the input operators as ˆb1ˆb2 = (ˆ a21 − a ˆ22 )/2.

(9.57)

The cross term of a ˆ1 a ˆ2 is notably missing in Eq.(9.57), due to two-photon interference. For generalization to the N -photon case, we would likewise want $ ˆto have −k ˆN (k = 0, N ) missing in the product of the middle terms like a ˆk1 a 2 j bj . This can be achieved with the algebraic factorial identity [9.9] N "

(x − yeiδj ) = xN − y N

(9.58)

j=1

with δj = 2π(j − 1)/N (j = 1, ..., N ). So, if we assign ˆbj ∝ a ˆ1 − a ˆ2 eiδj + ...,

(9.59)

then we have: N " j=1

ˆbj ∝

N "

(ˆ a1 − a ˆ2 eiδj ) = a ˆN ˆN 1 −a 2 ,

(9.60)

j=1

−k ˆN (k = 0, N ) are all missing. where as required, the middle terms of a ˆk1 a 2 This is exactly the NOON-state projection measurement scheme that was discussed in Sect.8.6.2 for the demonstration of an N-photon de Broglie wavelength without the need for a NOON-state. The actual implementation of the scheme for two polarization modes is depicted in Fig.9.2, where the input is an arbitrary N-photon state of two polarization modes in the form of

|ΨN =

N

ck |kH , (N − k)V ,

(9.61)

k=0

ˆV − a ˆH eiδj + .... Then, from Eq.(9.60), we ﬁnd the N-photon and ˆbj ∝ a coincidence probability from the N detectors to be proportional to PN ∝ |N OON |ΨN |2 , (9.62) √ with |N OON = (|NH , 0V − |0H , NV )/ 2. If the input state is of the form |k, N − k(k = 0, N ), the outcome of this projection measurement is zero, due to the orthogonality of N OON |k, N − k = 0 for (k = 0, N ). Notice that when N = 2, the NOON-state projection measurement scheme becomes a Hong-Ou-Mandel interferometer. From the construction of the NOON-state, we ﬁnd that this orthogonal projection is a result of N-photon interference. So, just like the Hong-Ou-Mandel interferometer, the NOON-state projection measurement scheme can be used to characterize the temporal distinguishability of an N -photon state by the visibility in the interference. We will demonstrate this in the following sections.

9.3 The First Example of |1H , 2V

δΝ

Η

1 N−1

1 N

V

|ΨΝ

1 2

δΝ−1

^ bΝ

197

δ2

^ bΝ−1

δ1

^ b1

^ b2

Fig. 9.2. The schematics for the NOON-state projection measurement in polarization representation. Reprinted ﬁgure with permission from Z. Y. Ou, Phys. Rev. A c 74, 063808 (2006). 2006 by the American Physical Society.

9.3 The First Example of |1H , 2V Besides the Hong-Ou-Mandel interferometer with two photons, the next simplest case is the three-photon case where the input state is |1H , 2V . Let us ﬁnd out how the photon distinguishability inﬂuence the visibility in the NOON state projection measurement. For the projection measurement in Fig.9.2 with N = 3, we have the electric ﬁeld operators at three detectors as ⎧ √ ˆV (t) − E ˆH (t)]/ 6 + ..., ˆ1 (t) = [E ⎨E √ ˆV (t) − ei2π/3 E ˆH (t)]/ 6 + ..., (9.63) E (t) = [E √ ⎩ ˆ2 i4π/3 ˆ ˆ E3 (t) = [EV (t) − e EH (t)]/ 6 + ..., ˆ where E(t) is given in Eq.(9.27). To ﬁnd the three-photon coincidence probability, we ﬁrst calculate the time correlation function ˆ † (t2 )Eˆ † (t1 )E ˆ1 (t1 )E ˆ2 (t2 )Eˆ3 (t3 ). Γ (3) (t1 , t2 , t3 ) = Eˆ3† (t3 )E 2 1 Here, the average is over the three-photon state of a†H (ω1 )ˆ a†V (ω2 )ˆ a†V (ω3 )|vac, |Φ3 = dω1 dω2 dω3 Φ(ω1 , ω2 , ω3 )ˆ

(9.64)

(9.65)

where, for simplicity, we take Φ(ω1 , ω2 , ω3 ) in the form of Eq.(9.20). It is easy ˆ1 (t1 )E ˆ2 (t2 )Eˆ3 (t3 )|Φ3 : to ﬁrst calculate E ˆ1 (t1 )Eˆ2 (t2 )Eˆ3 (t3 )|Φ3 E

−1 ˆ ˆ ˆ i2π/3 i4π/3 ˆ ˆ ˆ ˆ ˆ ˆ |Φ3 . (9.66) + EV EV EH e = √ EH EV EV + EV EH EV e 6 6 Here, we dropped the terms that have no contribution. The order of the operators is kept for the time variables t1 t2 t3 . With the state in Eq.(9.65) and Φ3 in Eq.(9.20), it is straightforward to ﬁnd: ˆ1 (t1 )Eˆ2 (t2 )E ˆ3 (t3 )|Φ3 E

198

9 Temporal Distinguishability of a Multi-Photon State

−1 G(t3 , t1 , t2 ) + G(t3 , t2 , t1 ) ei4π/3 = √ 6 6 + G(t2 , t1 , t3 ) + G(t2 , t3 , t1 ) ei2π/3 % + G(t1 , t2 , t3 ) + G(t1 , t3 , t2 ) |0,

(9.67)

where

1 G(t1 , t2 , t3 ) = dω1 dω2 dω3 Φ(ω1 , ω2 , ω3 )e−i(ω1 t1 +ω2 t2 +ω3 t3 ) (2π)3 = g(t1 − T1 )g(t2 − T2 )g(t3 − T3 ) (9.68)

with 1 g(τ ) = √ 2π

dωφ(ω)e−iωτ .

(9.69)

The three-photon joint detection probability is an integral of the correlation function in Eq.(9.64) over all time variables: (9.70) P3 = dt1 dt2 dt3 Γ (3) (t1 , t2 , t3 ). To evaluate P3 , let us consider some extreme cases in the following: (i) The two V-photons are indistinguishable with T2 = T3 ≡ T : There is permutation symmetry between t2 and t3 in the function in Eq.(9.68) and Eq.(9.67) becomes: ˆ2 (t2 )Eˆ3 (t3 )|Φ3 ˆ1 (t1 )E E −1 = √ G(t3 , t1 , t2 )ei4π/3 + G(t2 , t1 , t3 )ei2π/3 + G(t1 , t2 , t3 ) |0.(9.71) 3 6 Then, we evaluate the integral in Eq.(9.70) and obtain: P3 = where

A3 − E3 (T1 − T ) , 18

(9.72)

dω1 dω2 dω3 |Φ(ω1 , ω2 , ω3 )|2

3 2 = dω|φ(ω)| ,

A3 ≡

(9.73)

and E3 (τ ) ≡

2 2 −iωτ dω|φ(ω)| dω|φ(ω)| e . 2

(9.74)

9.3 The First Example of |1H , 2V

199

Note that E3 (0) = A3 and P3 = 0. This is the 1H2V case when all three photons are indistinguishable in time. But, when |T1 − T | >> 1/Δω, P3 = A3 /18. So, as we scan the relative delay T1 − T between the H-photon and the V-photons, we will observe a dip all the way down to zero, similar to the Hong-Ou-Mandel dip in the two-photon case. The visibility of the dip is then 100%. (ii) The two V-photons are distinguishable in time with |T2 − T3 | >> 1/Δω: The interchange between t2 and t3 in G(t1 , t2 , t3 ) in Eq.(9.68) gives the orthogonal relation: dt1 dt2 d3 G∗ (t1 , t2 , t3 )G(t1 , t3 , t2 ) = 0. (9.75) Therefore, when we evaluate the integral in Eq.(9.70), some of the cross terms will vanish and we obtain: A3 − E3 (ΔT1 )/2 − E3 (ΔT2 )/2 , 36

P3 =

(9.76)

with ΔT1 = T1 − T2 and ΔT2 = T1 − T3 . We will then have two dips that drop to half of the initial value when T1 scans through T2 and T3 . The visibility of the dips is 50%. Furthermore, let us take the general form of Φ3 (ω1 , ω2 , ω3 ) and carry out the integral in Eq.(9.70). After some lengthy manipulation, we obtain: P3 =

1 (2A3 + 2E23 − E12 − E13 − E231 − E312 ), 72

where

A3 ≡

and

E23 ≡ E12 ≡

E13 ≡ E231 ≡ E312 ≡

dω1 dω2 dω3 |Φ(ω1 , ω2 , ω3 )|2 ,

(9.77)

(9.78)

dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω1 , ω3 , ω2 ) dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω2 , ω1 , ω3 ) dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω3 , ω2 , ω1 ) dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω2 , ω3 , ω1 ) dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω3 , ω1 , ω2 ).

(9.79)

∗ ∗ and E231 = E312 . When the two V-photons are comNote that Eij = Eij pletely indistinguishable (the 1H2V case), this corresponds to Φ(ω1 , ω2 , ω3 ) = Φ(ω1 , ω3 , ω2 ) or E23 = A3 and E12 = E13 = E231 = E312 . So, Eq.(9.77) becomes:

200

9 Temporal Distinguishability of a Multi-Photon State

P3 =

1 (A3 − E12 ), 18

and the visibility is simply V3 = M3 with dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω2 , ω1 , ω3 ) E12 = M3 ≡ A3 dω1 dω2 dω3 |Φ(ω1 , ω2 , ω3 )|2

(9.80)

(9.81)

deﬁned as the degree of permutation symmetry for the three-photon case. Similar to the Hong-Ou-Mandel two-photon case, the visibility is directly related to the degree of permutation symmetry, and, it is equal to one when we have the symmetry relation of Φ(ω1 , ω2 , ω3 ) = Φ(ω2 , ω1 , ω3 ) = Φ(ω1 , ω3 , ω2 ).

(9.82)

The other extreme scenario of 1H1V + 1V gives E23 = 0 and either E12 = A3 or E13 = A3 . This leads to the symmetry relation of Φ(ω1 , ω2 , ω3 ) = Φ(ω2 , ω1 , ω3 )

(9.83)

Φ(ω1 , ω2 , ω3 ) = Φ(ω3 , ω2 , ω1 ),

(9.84)

or

and the orthogonal relation of E23 = dω1 dω2 dω3 Φ∗ (ω1 , ω2 , ω3 )Φ(ω1 , ω3 , ω2 ) = 0.

(9.85)

In either case, we have E231 = E312 = 0 and P3 drops to half of the value when both E12 = 0 and E13 = 0, or the visibility is 50%. This is consistent with the result in Eq.(9.76). For the case other than the two extreme scenarios, the closeness of P3 to zero depends on the degree of permutation symmetry among the three variables of ω1 , ω2 , ω3 in function Φ(ω1 , ω2 , ω3 ) or, equivalently, the degree of temporal distinguishability in G(t1 , t2 , t3 ). To summarize, we can distinguish diﬀerent scenarios in the three-photon case by measuring the visibility of the interference dips in the NOON-state projection measurement. Later in Sect.9.7.1, we will consider a realistic situation when the three photons are produced from two pairs of down-converted photons.

9.4 The General Case of |1H , NV Let us now generalize the conclusion in the previous section to the general case of |1H , NV with an arbitrary integer N . The most general scenario in this case is when the single horizontal photon (H) is indistinguishable from

9.4 The General Case of |1H , NV

201

m vertical photons (V) while other N − m V-photons are well-separated in time from the m + 1 photons (1H + mV ). From the discussion in Sect.9.1, the N-photon state for this case is labeled as |1HmV and has the form of |1HmV = dω1 dω2 ...dωN +1 Φ(ω1 ; ω2 , ..., ωN +1 ) ×ˆ a†H (ω1 )ˆ a†V (ω2 )...ˆ a†V (ωN +1 )|vac,

(9.86)

with Φ(ω1 ; ω2 , ..., ωN +1 ) = Φ(P {ω1 ; ω2 , ..., ωm+1 }, ωm+2 , ..., ωN +1 ),

(9.87)

where P is an arbitrary permutation operation. This expression is from the indistinguishability among the one H-photon and m V-photons. But, because the other N − m V-photons are well-separated in time from the 1HmV photons, we have the orthogonal relations: dωk dωj Φ∗ (ω1 ; ω2 , ..., ωN +1 )Φ(Pkj {ω1 ; ω2 , ..., ωN +1 }) = 0, (9.88) where Pkj interchanges ωk with ωj and 1 ≤ k ≤ m + 1, m + 2 ≤ j ≤ N + 1. Or, we can write similar relations in time domain: G(t1 ; t2 , ..., tN +1 ) = G(P {t1 ; t2 , ..., tm+1 }, tm+2 , ..., tN +1 ),

dtk dtj G∗ (t1 ; t2 , ..., tN +1 )G(Pkj {t1 ; t2 , ..., tN +1 }) = 0,

(9.89)

(9.90)

where G(t1 ; t2 , ..., tN +1 ) is given in Eq.(9.34). With the expressions in Eqs.(9.89, 9.90), we are ready to evaluate the joint N + 1-photon probability PN +1 in the NOON-state projection measurement scheme with an input state of |1HmV in Eq.(9.86). PN +1 is a time integral of the correlation function from (N + 1) detectors: † ˆ† Γ (N ) (t1 , t2 , ..., tN ) = 1HmV |EˆN +1 (tN +1 )...E1 (t1 ) ˆ1 (t1 )...E ˆN +1 (tN +1 )|1HmV , ×E

(9.91)

with ˆj (t) ∝ E ˆV (t) − E ˆH (t)eiδj + ..., E where ˆH,V (t) = √1 E 2π

dωˆ aH,V (ω)e−iωt .

(9.92)

(9.93)

ˆ1 (t1 )...EˆN +1 (tN +1 )|1HmV . After expanding It is easy to ﬁrst evaluate E the product, we ﬁnd only N + 1 nonzero terms of the form:

202

9 Temporal Distinguishability of a Multi-Photon State

−

N +1

ˆV (tN +1 )|1HmV . eiδk EˆV (t1 )...EˆH (tk )...E

(9.94)

k=1

For the state |1HmV in Eq.(9.86), we have: ˆV (t1 )...EˆH (tk )...EˆV (tN +1 )|1HmV = G(P1k {t1 , ..., tN +1 })|vac, (9.95) E with G(t1 , ..., tN +1 ) =

G(t1 ; P {t2 , ..., tN +1 }),

(9.96)

P

where P is an arbitrary permutation of t2 , ..., tN +1 . The overall (N + 1)-photon coincidence probability is then given by 2 N +1 iδ k ∝ dt1 ...dtN +1 e G(P1k {t1 , ..., tN +1 }) k=1 = dt1 ...dtN +1 ei(δk −δj ) G(P1k {t1 , ..., tN +1 })

PN +1

k,j

×G ∗ (P1j {t1 , ..., tN +1 }).

(9.97)

Diagonal terms of k = j in the double sum are all the same because the integration is over all time variables: 2 2 dt1 ...dtN +1 G(P1k {t1 , ..., tN +1 }) = dt1 ...dtN +1 G(t1 , ..., tN +1 ) . (9.98) Furthermore, 2 dt1 ...dtN +1 G(t1 , ..., tN +1 ) = dt1 ...dtN +1 G(t1 , P {t2 ..., tN +1 }) G∗ (t1 , P {t2 ..., tN +1 }) P P = N ! dt1 ...dtN +1 G(t1 , P {t2 ..., tN +1 })G∗ (t1 , t2 ..., tN +1 ) P

= N !NN +1 , with

(9.99)

NN +1 ≡

dω1 ...dωN +1 Φ∗ (ω1 ; ω2 , ..., ωN +1 ) × Φ(ω1 ; P {ω2 , ..., ωN +1 }),

(9.100)

P

where we used a similar trick that leads to Eq.(9.39). So, the diagonal terms are simply (N + 1)N !NN +1 .

9.4 The General Case of |1H , NV

203

The cross terms are given by dt1 ...dtN +1

ei(δk −δj ) G(P1k {t1 , ..., tN +1 })G ∗ (P1j {t1 , ..., tN +1 }).

(9.101)

k =j

Let us consider one arbitrary term in the sum. The time integral part can be rewritten as G(tk , P {t2 , ..., t1 , ..., tN +1 }) dt1 ...dtN +1 P × G∗ (tj , P {t2 , ..., t1 , ..., tN +1 }). (9.102) P

Because of the permutation properties in Eqs.(9.89, 9.90) and tk = tj , the only way to obtain a non-zero integral is for tk in {t2 , ..., t1 , ..., tN +1 } to be permuted to the ﬁrst m positions by P . Then we can use the permutation relation in Eq.(9.89) to interchange it with tj so that for these P s, we have: G(tj , P {t2 , ..., t1 , ..., tN +1 }) = G(tk , P {t2 , ..., t1 , ..., tN +1 }).

(9.103)

Permutation by P to other positions for tk cannot be interchanged with tj and by the orthogonal relation in Eq.(9.90), the integral is zero. Since this is only about tk in P , other N − 1 time variables in P are free to move. So, there will be m(N − 1)! permutation terms in the sum over P that are nonzero and, as before in Eq.(9.38), they will all have the same time integral result of NN +1 , with G(t1 , P {t2 , ..., tN +1 }). (9.104) NN +1 ≡ dt1 ...dtN +1 G∗ (t1 , ..., tN +1 ) P

Therefore, the cross terms are equal to ei(δk −δj ) G(P1k {t1 , ..., tN +1 })G ∗ (P1j {t1 , ..., tN +1 }) dt1 ...dtN +1 k =j

= m(N − 1)!NN +1

ei(δk −δj ) .

(9.105)

k =j

But, because

k

eiδk = 0, we have: i(δk −δj ) e = − ei(δk −δj ) k =j

=

k=j k,j eiδk e−iδj − (N + 1) k

j

= −(N + 1). So, the ﬁnal result is:

(9.106)

204

9 Temporal Distinguishability of a Multi-Photon State

PN +1 (1HmV ) ∝ NN +1 (N + 1)(N − m) − 1)!(N m . = (N + 1)!NN +1 1 − N

(9.107)

For the generalized Hong-Ou-Mandel interferometer, we scan the delay of the H-photon relative to the V-photons. When it does not overlap with any of the V-photons, no interference occurs and PN +1 is a straight line which corresponds to m = 0 in Eq.(9.107) with PN +1 (∞) = (N + 1)!NN +1 . The value in Eq.(9.107) corresponds to the case when the delay is zero and a local maximum interference is achieved. So, the visibility is: VN +1 (1HmV ) ≡

m PN +1 (∞) − PN +1 (1HmV ) = . PN +1 (∞) N

(9.108)

Note that this visibility depends only on N and m, i.e., the total number of Vphotons and the number of V-photons that overlap with the single H-photon. It is independent of the normalization factor NN +1 or how the other N − m photons distribute in time.

(a)

V

m

H

PN+1 (b) 0

1

m N

Fig. 9.3. (a) A temporal distribution with well-separated groups of V-photons and (b) the corresponding normalized PN+1 as the position of the H-photon is scanned. Reprinted ﬁgure with permission from Z. Y. Ou, Phys. Rev. A 74, 063808 c (2006). 2006 by the American Physical Society.

So, for a temporal distribution of well-separated groups of V-photons, shown in Fig.9.3a, as we scan the location of the single H-photon, we will have more dips of various visibility (Fig.9.3b) and the visibility is m/N when the single H-photon overlaps with the group of m V-photons that are in one temporal mode and are well-separated from other V-photons. In general, for a temporal distribution with m partially overlapping Vphotons, the visibility will be a value less than m/N . Therefore, the experimentally measurable visibility of the dips can be used to characterize the degree of temporal indistinguishability of an N-photon state. Before we examine a more general case, let us consider the case when the H-photon may not be completely indistinguishable in time from the N V-photons. We treat only the scenario when the N V-photons are in one temporal mode with the permutation symmetry relation:

9.5 The General Case of |kH , NV

Φ(ω1 ; ω2 , ..., ωN +1 ) = Φ(ω1 ; P {ω2 , ..., ωN +1 }),

205

(9.109)

for all permutation P . Then, from Eq.(9.96), we have: G(t1 , ..., tN +1 ) = N !G(t1 ; t2 , ..., tN +1 ),

(9.110)

and Eq.(9.97) becomes: 2 N +1 2 iδk e PN +1 ∝ dt1 ...dtN +1 (N !) G(t1 , t2 , ..., tN +1 ) + G(t2 , t1 , ..., tN +1 ) k=2 2 2 = dt1 ...dtN +1 (N !) G(t1 , t2 , ..., tN +1 ) − G(t2 , t1 , ..., tN +1 ) ∝ 1 − VN +1 , with VN +1 = MN +1 , where dω1 ...dωN +1 Φ∗ (ω1 ; ω2 , ..., ωN +1 )Φ(ω2 ; ω1 , ..., ωN +1 ) MN +1 ≡ dω1 ...dωN +1 |Φ(ω1 ; ω2 , ..., ωN +1 )|2

(9.111)

(9.112)

is the degree of permutation symmetry in the (N + 1)-photon case. Note that ∗ MN +1 = MN +1 and 0 ≤ |MN +1 | ≤ 1. So similar to the two-photon case in Eq.(9.2) and the three-photon case in Eq.(9.81), the visibility for the (N + 1)photon case is directly related to the degree of permutation symmetry. Since Φ(ω1 ; ω2 , ..., ωN +1 ) is related to G(t1 , ..., tN +1 ) by Eq.(9.34), we also have: dt1 ...dtN +1 G∗ (t1 ; t2 , ..., tN +1 )G(t2 ; t1 , ..., tN +1 ) . (9.113) MN +1 = dt1 ...dtN +1 |G(t1 ; t2 , ..., tN +1 )|2 So, MN +1 is also a degree of temporal indistinguishability. Although they are derived for the case of all N V-photons in one indistinguishable temporal mode, the relations in Eq.(9.112, 9.113) also apply to the less ideal case of only m(≤ N ) V-photons in one temporal mode and the other (N − m) V-photons well-separated from the H-photon and the m single-mode V-photons. But, the visibility VN +1 is related to MN +1 by m VN +1 = MN +1 . (9.114) N This time the H-photon does not quite overlap with the m single-mode Vphotons. So, instead of the symmetry relations in Eq.(9.87) and in Eq.(9.109), we have only: Φ(ω1 ; ω2 , ..., ωN +1 ) = Φ(ω1 ; P {ω2 , ..., ωm+1 }, ωm+2 , ..., ωN +1 ). (9.115)

9.5 The General Case of |kH , NV This general case does not have a simple formula, as in Eq.(9.107), and is very complicated. We refer the readers who are interested in the details of this general case to Appendix B. Here, we will present a general formula and a few special cases.

206

9 Temporal Distinguishability of a Multi-Photon State

9.5.1 General Formula for the Visibility The most general case is when the k H-photons do not overlap in time, but rather, are split into r temporally well-separated subgroups with kj indistinguishable photons in the jth group and k1 + ... + kr = k. We also divide the N V-photons into r + 1 subgroups with mj V-photons overlapping in time with the jth H-photon group. The rest N − m1 − ... − mr V-photons are in a separate group by themselves. The visibility in the (N + k)-photon NOON-state projection measurement has a complicated form of VN +k =

k l=1

(−1)

l−1

l i1 ...ir i1 +...+ir =l

(i ) (i ) Cki11 ...Ckirr m1 1 ...mr r l! , (9.116) i1 !...ir ! (N + k − 1)...(N + k − l)

where Cki = k!/(k−i)!i!, m(0) = 1, m(m+1) = 0, and m(i) ≡ m(m−1)...(m−i+ 1) for 0 < i ≤ m. A ﬁrst special case is when there is only one subgroup, i.e., r = 1. In this case, all k H-photons are indistinguishable in time. We denote this as the kHmV + (N − m)V case (or kHmV for short) with a visibility of VN +k (kHmV ) =

k

(−1)l−1 Ckl

l=1

m(m − 1)...(m − l + 1) . (N + k − 1)...(N + k − l)

(9.117)

For k = 2 but the two H-photons are separated, we have the 1HmV +1HnV + (N − m − n)V case (or 1HmV + 1HnV for short) with a visibility of VN +2 (1HmV + 1HnV ) =

m+n 2mn − . N + 1 N (N + 1)

(9.118)

For k = 3, we have two cases with separated H-photons: the 2H + 1H and 1H + 1H + 1H cases. The visibilities are, respectively: VN +3 (2HmV 1HnV ) 3nm(m − 1) 2m + n 4mn + m(m − 1) − + , = N +2 (N + 1)(N + 2) N (N + 1)(N + 2)

(9.119)

and VN +3 (1HmV 1HnV 1HpV ) m + n + p 2(mn + mp + np) 6mnp = − + . (9.120) N +2 (N + 1)(N + 2) N (N + 1)(N + 2) In the following, we consider a few speciﬁc examples with various scenarios. 9.5.2 The Special Cases of |2H , 2V , |2H , 3V , and |2H , 4V To gain a sense of how the visibility is aﬀected by the overlapping of diﬀerent photons, we ﬁrst consider a few simple examples with k = 2, which are covered

9.5 The General Case of |kH , NV

207

by the visibility formulae in Eq.(9.117) for k = 2 and by Eq.(9.118). We start with the case of |2H , 2V . There are four diﬀerent scenarios for the case of |2H , 2V , i.e., |2H2V , |2H1V + 1V , |1H1V + 1H1V , and |1H1V + 1H + 1V . Their corresponding visibilities are listed in Table 9.1. The scenarios of |2H2V and |1H1V + 1H1V were experimentally realized with parametric down-conversion (see more details on this in Sect.9.7) with exactly the visibility in Table 9.1. Table 9.1. Visibility for 2 H-photons and 2 V-photons input 2H2V 2H1V+1V 1HV+1HV 1HV+H+V 1

2/3

1/3

1/3

The case of |2H , 3V has eight diﬀerent scenarios. They are: (i) |2H3V , |2H2V + 1V , |2H1V + 2V , and (ii) |1H3V + 1H, |1H2V + 1H1V , |1H2V + 1H + 1V , |1H1V + 1H1V + 1V , |1H1V + 1H + 2V . Their visibilities are listed in Table 9.2. Table 9.2. Visibility for 2 H-photons and 3 V-photons input 2H3V 2H2V 2H1V 1H3V 1H2V 1H2V HV+V HV+V +V +2V +H +HV +H+V +HV +H+V 1

5/6

1/2

3/4

5/12

1/2

1/3

1/4

The case of |2H , 4V has 12 diﬀerent scenarios. They are: (i) |2H4V , |2H3V + 1V , |2H2V + 2V , |2H1V + 3V , and (ii) |1H4V + 1H, |1H3V + 1H1V , |1H3V + 1H + 1V , |1H2V + 1H2V , |1H2V + 1H1V + 1V , |1H2V + 1H + 2V , |1H1V + 1H1V + 2V , |1H1V + 1H + 3V . The scenarios with diﬀerent visibilities are listed in Table 9.3. |1H2V +1H1V + 1V and |1H2V +1H +2V have the same visibility of 2/5 as |1H2V +1H2V . In general, they follow the trend that a smaller visibility corresponds to less photon overlapping. However, there are exceptions: 1H2V + HV has less visibility than 1H2V + 1H + V in Table 9.2 and 1H3V + HV has less visibility than 1H3V + 1H + 1V in Table 9.3. So, the runaway HV does not help when H and V overlap in these cases.

208

9 Temporal Distinguishability of a Multi-Photon State Table 9.3. Visibility for 2 H-photons and 4 V-photons input 2H4V 2H3V 2H2V 2H1V 1H4V 1H3V 1H3V 1H2V 2×HV 1H1V +V +2V +3V +H +HV +H+V +1H2V +2V +1H+3V 1

9/10 7/10

2/5

4/5

1/2

3/5

2/5

3/10

1/5

9.5.3 The Special Case of |3H , 3V This case is covered by Eq.(9.117) for k = 3 and by Eqs.(9.119, 9.120). There is a total of 11 diﬀerent scenarios in this case: (i) |3H3V , |3H2V + V , |3H1V + 2V ; (ii) |2H2V + 1H1V , |2H2V + 1H + 1V , |2H1V + 1H2V , |2H1V + 1H1V + 1V , |2H1V + 1H + 2V ; (iii) |1H1V + 1H1V + 1H1V , |1H1V + 1H1V + 1H + 1V , |1H1V + 1H + 1V + 1H + 1V . In Table 9.4, we list the visibilities for most of the scenarios. |2H1V + 1H1V + 1V and |2H1V + 1H + 2V have the same visibility of 2/5, as |2H1V + 1H2V , and are not listed. As can be seen, anomaly occurs for |2H2V + 1H1V and |2H2V + 1H + 1V where the visibility is greater for the case with less photon overlap. The scenarios of |3H3V , |2H2V + 1H1V , and |3 × HV were observed experimentally by Xiang et al. [9.10], with the corresponding visibilities in Table 9.4. Table 9.4. Visibility for 3 H-photons and 3 V-photons input 3H3V 3H2V 3H1V 2H2V 2H2V 2H1V HV×3 HV×2 HV+V +V +2V +HV +H+V +1H2V +H+V +H+H+V 1

9/10

3/5

3/5

7/10

2/5

2/5

3/10

1/5

9.6 The Scheme for Characterizing the Temporal Distinguishability by an Asymmetric Beam Splitter This scheme is another generalization of the Hong-Ou-Mandel interferometer. Consider a k-photon state and an N -photon state entering a lossless beam splitter from port 1 and port 2, respectively, as shown in Fig.9.4. Diﬀerent from the two-photon case, we do not use a 50:50 symmetric beam splitter, but leave the amplitude transmissivity t and reﬂectivity r to be determined later. The operator relation is given by

9.6 Another Scheme by an Asymmetric Beam Splitter

(9.121)

a^2

|k 1|N 2

N N

k

a^1

ˆb1 = tˆ a1 + rˆ a2 , ˆb2 = tˆ a2 − rˆ a1 .

209

^

Fig. 9.4. The scheme for an asymmetric beam splitter with adjustable amplitude transmissivity t and reﬂectivity r.

b1 t, r ^

b2

|N 1|k 2

k

Now, we search for the probability of detecting only N photons in the output port 1 and k photons in output port 2. From Eq.(A.18) in Appendix A, we obtain the probability amplitude as AN,k =

k

k−m m 2(k−m) N −k+2m (−1)m CN Ck t r ,

(k ≤ N )

(9.122)

m=0

and specially for k = 1, we have the amplitude as AN,1 = N t2 rN −1 − rN +1 .

(9.123)

When r2 = N/(N + 1) ≡ ρ21 , t2 = 1/(N + 1) ≡ τ12 , we have AN,1 = 0. N = 1 gives the Hong-Ou-Mandel interference scheme. The cases of k = 1, N = 2 and k = N = 2 were discussed in Sect.8.4.2 and Sect.8.6.1.

{

N−1

...

...

{

N

(b)

{

...

{

...

(a)

N

N−1

Fig. 9.5. The possibilities for obtaining |N 1 |12 in the output: (a) all input photons are reﬂected; (b) the single input photon and one of the input N photons transmit while the rest are reﬂected. There are N equal possibilities in (b).

There is a simple physical picture of (N+1)-photon interference for AN,1 = 0, as depicted in Fig.9.5. There are N + 1 possibilities to obtain an output state of |N1 , 12 . In the ﬁrst one (Fig.9.5a), the single photon from input port 1 is reﬂected to port 2 and the are all reﬂected to port 1. This other N photons case has an amplitude of − N/(N + 1) × ( N/(N + 1))N . The minus sign

210

9 Temporal Distinguishability of a Multi-Photon State

comes from the reﬂection of the single photon from input port 1. The other N possibilities (Fig.9.5b) correspond to the input a single photon transmits to port 1 while one of the input N photons from port 2 transmits to port 2 are reﬂected. Each of these cases has an equal amplitude of and the rest 1/(N + 1) × 1/(N + 1) × ( N/(N + 1))N −1 . If the N + 1 photons are all in one mode, we add the amplitudes of all the possibilities and obtain: ! ! !

N −1 1 1 N PN,1 = N × N +1 N +1 N +1 ! !

N 2 N N × − N +1 N +1 = 0. (9.124) Therefore, this probability cancellation is a result of (N + 1)-photon interference when all the photons are in one temporal mode. On the other hand, if not all the photons are in the same temporal mode, there will be no perfect cancellation of the probability amplitude. Like the NOON-state projection measurement, this simple interferometer can be used to characterize the temporal distinguishability of an N-photon state. This interference scheme was ﬁrst implemented by Sanaka et al. [9.11] for demonstrating a nonlinear phase gate, and later by the same group [9.12] for ﬁltering out the N-photon state. From Eq.(9.122), we may obtain for k = 2 the transmission and reﬂectance for which A2,N = 0. They must satisfy the following equation: AN,2 = rN −2 [r4 − 2N t2 r2 + N (N − 1)t4 /2] = 0.

(9.125)

Solving the quadratic equation, we have: 1 ≡ τ22 , N +1± N (N + 1)/2 N (N + 1)/2 N ± ≡ ρ22 . r2 = N + 1 ± N (N + 1)/2 t2 =

(9.126)

Unfortunately, for k > 2, we are unable to give an analytical expression of t, r for which AN,k = 0. This would require solving an algebraic equation of the order ≥ 3. Next, we will consider the scenarios when the N photons may not belong to a single temporal mode. 9.6.1 The Temporal Distinguishability of |1H , NV For the consistency of notations, let us consider the equivalent polarization beam splitter. The two input modes are the H- and the V-polarizations. The operator relation becomes:

9.6 Another Scheme by an Asymmetric Beam Splitter

ˆbH = a ˆH cos θ + a ˆV sin θ, ˆbV = a ˆV cos θ − a ˆH sin θ,

211

(9.127)

where cos θ ≡ t, sin θ ≡ r. The angle θ is the angle of rotation of the polarizations (see Fig.9.6). The input state will be |1H , NV . V

Rotator

|Ψin

θ

Η

1 N

PBS

1 2

Η V

^ (o)

EV

^

^ (o)

EH

^ (o)

EH(o)

EH

Fig. 9.6. The scheme of polarization beam splitter for characterizing N -photon temporal distinguishability.

Let us consider the situation when m V-photons overlap with the sole H-photon among the N input V-photons and the rest of the (N − m) Vphotons are completely distinguishable in time from the (m + 1) photons. This is the 1HmV + (N − m)V case. The state is given in Eq.(9.86) with the same symmetry and orthogonal relations in Eqs.(9.89, 9.90). To obtain the probability for |NH , 1V in the output, we place N detectors in the H-port and one detector in the V-port of the polarization beam splitter (PBS) and measure the (N + 1)-photon coincidence (Fig.9.6). Similar to Sect.9.4, the coincidence rate is proportional to a time integral of the correlation function of ˆ (o)† (tN +1 )...Eˆ (o)† (t2 )E ˆ (o)† (t1 ) Γ (N +1) (t1 , ...tN +1 ) = 1HmV |E H H V (o) (o) (o) ˆ (t1 )Eˆ (t2 )...E ˆ (tN +1 )|1HmV . (9.128) ×E V H H ˆ (o) ˆ (o) (t1 )Eˆ (o) (t2 )...Eˆ (o) (tN +1 )|1HmV , where E It is easy to ﬁrst evaluate E V H H V (o) follows the second line of Eq.(9.127) while EˆH follow the ﬁrst line. For the input state of |Ψin = |1HmV in Eq.(9.86) and t = cos θ = τ1 , r = sin θ = ρ1 in Eq.(9.127), the non-zero contribution in the expansion is from the following terms: N −1 EˆV (t1 )...EˆH (tj+1 )...EˆV (tN +1 ) τ12 ρN 1 j=1

+1 ˆ ˆV (tN +1 ) |1HmV , EH (t1 )EV (t2 )...E −ρN 1

(9.129)

which is equal to +1 ρN 1

+1 1 N G(P1j {t1 , ..., tN +1 }) − G(t1 ; t2 , ..., tN +1 ) |vac, N j=2

(9.130)

212

9 Temporal Distinguishability of a Multi-Photon State

where we used τ12 = 1/(N + 1), ρ21 = N/(N + 1). Substituting the above into Eq.(9.128) and carrying out the time integral, we obtain the (N + 1)-photon coincidence probability as PN +1 = (I1 − 2I2 /N + I3 /N 2 )[N/(N + 1)]N +1 , with

I1 = I2 =

2 dt1 ...dtN +1 G(t1 , ..., tN +1 ) = N !NN +1 ,

dt1 ...dtN +1 G ∗ (t1 , ..., tN +1 )

= N m(N − 1)!NN +1 , I3 =

dt1 ...dtN +1

N +1

N +1

(9.132)

G(P1j {t1 , ..., tN +1 })

j=2

G ∗ (P1k {t1 , ..., tN +1 })

k=2

(9.131)

(9.133) N +1

G(P1j {t1 , ..., tN +1 })

j=2

= N N !NN +1 + m(N − 1)!NN +1 N (N − 1).

(9.134)

Hence, we have:

N +1 N 2m N + m(N − 1) + = 1− N !NN +1 N N2 N +1

N N = N !NN +1 1 − VN +1 (1HmV ) , (9.135) N +1

PN +1

with VN +1 (1HmV ) = m/N . This visibility is exactly the same as that in Sect.9.4, under the same input condition. Thus, similar to the NOON-state projection scheme, we can use this simple scheme for characterizing the temporal distinguishability of an N-photon state. 9.6.2 The Case of |2H , NV Unfortunately, because of the nontrivial form of τ22 , ρ22 in Eq.(9.126) for this case, the visibility has a very complicated form. The derivation process becomes cumbersome, although it is straightforward if one follows the same line of argument as in Appendix B for the NOON-state projection scheme. Here we present only the ﬁnal results for the visibility and list in table form a couple of special cases. Like the NOON-state projection scheme, there are two scenarios here. The ﬁrst is when the two H-photons are indistinguishable and there are m Vphotons overlapping with the H-photons in time. This is the case of 2HmV + (N − m)V . The visibility in this case is

9.6 Another Scheme by an Asymmetric Beam Splitter

VN +2 (2HmV ) =

mD1 m(m − 1)D2 − , D0 D0

213

(9.136)

where D0 ≡ 6N 2 ρ22 − N (N − 1)(7N − 2)τ22 /4 D1 ≡ (10N − 4)ρ22 − (N − 1)(3N − 2)τ22 D2 ≡ 4ρ22 − (5N − 6)τ22 /4,

(9.137)

with τ22 , ρ22 given in Eq.(9.126). Note that when m = N , we have VN +2 (2HN V ) = 1, which is consistent with the result in Eq.(9.125). The second scenario is when the two H-photons are well-separated in time and become distinguishable. Assume m and n V-photons overlap with them, respectively. This is the case of 1HmV + 1HnV + (N − m − n)V , and the corresponding visibility is VN +2 (1HmV + 1HnV ) =

(m + n)D1 2mnD2 − . 2D0 D0

(9.138)

Using Eqs.(9.136, 9.138), we may evaluate the special cases of N √ = 2, 3. The scenarios of N = 2 are listed in Table 9.5, where τ22 , ρ22 = (3 ± 3)/6. This case is exactly the same as the one in Table 9.1. However, the similarity stops here. The cases of N > 2 do not give rational numbers, as in Tables 9.2 and 9.3. For √ example, for N =√3, the scenarios are listed in Table 9.6, where τ22 = (4 ± 6)/10, ρ22 = (6 ∓ 6)/10 from Eq.(9.126). Table 9.6 still follows the general trend of Table 9.2. Table 9.5. Visibility for 2 H-photons and 2 V-photons input in an asymmetric BS 2H2V 2H1V+1V 1HV+1HV 1HV+H+V 1

2/3

1/3

1/3

Table 9.6. Visibility for 2 H-photons and 3 V-photons input in an asymmetric BS 2H3V 2H2V 2H1V 1H3V 1H2V 1H2V HV+V HV+V +V +2V +H +HV +H+V +HV +H+V 1

0.81

0.48

0.72

0.43

0.48

1/3

0.24

214

9 Temporal Distinguishability of a Multi-Photon State

9.7 Experimental Realization of the Cases of |2H , 1V , |2H , 2V with Two Pairs of Down-Converted Photons The previous discussions were all concerned with extreme cases when the photons are either well-separated and completely distinguishable or overlap and completely indistinguishable in time. In this section, we will consider the situation of partial distinguishability. Because of the complexity of the problem, we will present only two examples, which can be realized experimentally. 9.7.1 Generation of the State of |2H , 1V with Tunable Temporal Distinguishability The situation with |2H , 1V was discussed in Sect.9.3 for the case when the three photons are independent, and it applies to the situation when the photons are from single-photon sources such as quantum dots. The case of a general three-photon state was also presented in Eq.(9.77). Here, we consider how to realize such a state with two pairs of photons from parametric downconversion. So, the starting point is not a three-photon state, as in Eq.(9.65), but a four-photon state from parametric down-conversion. In parametric down-conversion, we already have two correlated photons. The third photon can be introduced either by gated detection of another pair of photons from parametric down-conversion or from a weak coherent state. We will consider only the former method. There are two schemes to realize |2H , 1V with parametric down-conversion. The ﬁrst one uses a type-II process, while the second one uses a type-I process. Fig.9.7 shows the scheme with the type-II process, which is very similar to Fig.7.2.2 for stimulated emission. The diﬀerence is in the detection scheme. We use a three-photon NOON-state projection measurement scheme here.

Pump

TH

NOON State Projection

TV

Crystal#2 H

4π/3

V

Crystal#1 H

A Trigger

Pump

V

C

2π/3

PBS

D

B

Coincidence Counter

Fig. 9.7. The schematics for generating a |2H , 1V state from two pairs of photons in type-II parametric down-conversion.

For simplicity, let us assume the two parametric processes are identical. The quantum state for the NOON-state projection is the product of the two

9.7 Experimental Realization for the Cases of |2H , 1V , |2H , 2V

215

states from the two crystals: ˆ†H (ω1 )ˆ a†V (ω2 ) + ... |ΦII = 1 + ξ dω1 dω2 Φ(ω1 , ω2 )eiω1 TH a ˆ†H (ω1 )ˆ a†V (ω2 ) + ... |vac. × 1 + ξ dω1 dω2 Φ(ω1 , ω2 )eiω2 TV a (9.139) Here, the two H-photons are aligned together, but with a relative delay of TH between them. A relative delay of TV is also introduced on the V-photon from the second crystal. The V-photon from the ﬁrst crystal (denoted by V ) is the triggering photon detected by Detector D. The role of the triggering photon is to make sure that the third photon is from the ﬁrst crystal, so that we can control the overlap between the two H-photons. The higher order terms correspond to the situation when the third photon is from the second crystal and can be neglected. Like Eq.(9.63), the ﬁeld operators in front of the three detectors can be ˆH (t), EˆV (t) as expressed in terms of E ⎧ √ ˆA (t) = [E ˆH (t) − E ˆV (t)]/ 6 + ... ⎨E √ ˆH (t) − ei2π/3 EˆV (t)]/ 6 + ... ˆ (t) = [E (9.140) E √ ⎩ ˆB ˆV (t)]/ 6 + .... EC (t) = [EˆH (t) − ei4π/3 E The vacuum ports are not explicitly written. Thus, the four-photon time correlation function is (4)

ΓABCD (t1 , t2 , t3 , t4 ) † † ˆ † (t1 )E ˆA (t1 )E ˆB (t2 )EˆC (t3 )E ˆV (t4 ). (9.141) (t3 )EˆB (t2 )E = EˆV† (t4 )EˆC A ˆC (t3 )EˆV (t4 ). For brevity, we Now, we expand the product of EˆA (t1 )EˆB (t2 )E ˆ set Hj = EH (tj ), etc. Then the expansion becomes: (H1 − V1 )(H2 − V2 ei2π/3 )(H3 − V3 ei4π/3 )V4 = −H1 H2 V3 V4 ei4π/3 − H1 V2 H3 V4 ei2π/3 − V1 H2 H3 V4 + ...

(9.142)

For the state in Eq.(9.139), it is straightforward to ﬁnd: H1 H2 V3 V4 |ΦII = dω1 dω1 dω2 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )ei(ω1 TH +ω2 TV ) × e−iω1 t1 −iω1 t2 + e−iω1 t2 −iω1 t1 e−iω2 t3 −iω2 t4 |vac = G(t1 , t3 − TV )G(t2 − TH , t4 ) (9.143) +G(t2 , t3 − TV )G(t1 − TH , t4 ) |vac, with G(t, t ) =

1 2π

dω1 dω2 Φ(ω1 , ω2 )e−iω1 t−iω2 t .

(9.144)

216

9 Temporal Distinguishability of a Multi-Photon State

So, the four-photon time correlation function is: 2 (4) ˆA (t1 )EˆB (t2 )EˆC (t3 )EˆV (t4 )|ΦII ,(9.145) (t1 , t2 , t3 , t4 ) = vac|E Γ ABCD

with ˆA (t1 )E ˆB (t2 )E ˆC (t3 )E ˆV (t4 )|ΦII −vac|E = [G(t1 , t3 − TV )G(t2 − TH , t4 ) + G(t2 , t3 − TV )G(t1 − TH , t4 )]ei4π/3 +[G(t1 , t2 − TV )G(t3 − TH , t4 ) + G(t3 , t2 − TV )G(t1 − TH , t4 )]ei2π/3 +[G(t2 , t1 − TV )G(t3 − TH , t4 ) + G(t3 T, t1 − TV )G(t2 − TH , t4 )] = [G(t2 , t3 − TV )ei4π/3 + G(t3 , t2 − TV )ei2π/3 ]G(t1 − TH , t4 ) +[G(t1 , t3 − TV )ei4π/3 + G(t3 , t1 − TV )]G(t2 − TH , t4 ) +[G(t1 , t2 − TV )ei2π/3 + G(t2 , t1 − TV )]G(t3 − TH , t4 ). (9.146) The four-photon coincidence rate P4 is then proportional to the time integral of the correlation function in Eq.(9.145) and becomes, after some lengthy calculation: P4 = 6A + 6E(TH , 0) − 3A(TV ) − 3E(TH − TV , 0) − 6[E(TH , TV )], (9.147) where

A= A(τ ) =

dω1 dω2 dω1 dω2 |Φ(ω1 , ω2 )Φ(ω1 , ω2 )|2 ,

(9.148)

dω1 dω2 dω1 dω2 |Φ(ω1 , ω2 )|2 ×Φ∗ (ω1 , ω2 )Φ(ω2 , ω1 )ei(ω2 −ω1 )τ ,

E(τ1 , τ2 ) =

(9.149)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )ei(ω1 −ω1 )τ1

×Φ(ω1 , ω2 )Φ(ω1 , ω2 )ei(ω1 −ω2 )τ2 ,

(9.150)

where we assumed symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ) in all the E formulae. There are two diﬀerent cases corresponding to the scenarios of 1V 2H and 1V 1H + 1H. When TH = 0, the two H-photons overlaps. This becomes the 1V 2H case. And, we have: P4 = 6A + 6E − 3A(TV ) − 3E(TV , 0) − 6[E(0, TV )],

(9.151)

where E = E(0, 0). When we scan TV , a dip will appear with a visibility of V3 (1V 2H) =

A + 3E . 2(A + E)

Note that maximum visibility of one is achieved when E = A.

(9.152)

9.7 Experimental Realization for the Cases of |2H , 1V , |2H , 2V

217

On the other hand, when TH >> Tc , the two H-photons are well-separated, corresponding to the 1V 1H + 1H case. Then, we have: P4 = 6A − 3A(TV ) − 3E(TV − TH , 0).

(9.153)

There will be two dips with respective visibility of V3 (1V 1H + 1H) = 1/2, V3 (1V 1H + 1H) = E/2A.

H H

V

Pump

2H

PBS

4π/3

V

C

2π/3

BS

TV 1V

Crystal#1

(9.155)

NOON State Projection

TH

Crystal#2

Pump

(9.154)

Trigger D

A

B

Coincidence Counter

Fig. 9.8. The schematics for generating a |2H , 1V state from two pairs of photons in type-I parametric down-conversion.

The scheme with the type-I parametric down-conversion is shown in Fig.9.8, where the two H-photons with adjustable overlap are obtained with a Hong-Ou-Mandel interferometer. One of the vertical photons is aligned with the two H-photons via a polarization beam splitter while the other V-photon serves as a triggering photon. This scheme of producing a |2H , 1V state was ﬁrst considered by Sanaka et al. [9.12] for N-photon state ﬁltering. The quantum state for this system is given by 2 |ΦI = ... + ξ dω1 dω2 Φ(ω1 , ω2 )eiω1 TV a ˆ†Vs (ω1 )ˆ a†V (ω2 )× i † † × dω1 dω2 Φ(ω1 , ω2 )eiω2 TH a ˆHs (ω1 )ˆ aHi (ω2 )|vac. (9.156) Here, like before, we introduced the time variables TH , TV for the delay between the two H-photons and between the H and V-photons. The derivation for the visibility in this case is similar to the previous case. We will leave the details to the readers and present only the answer as V(1V 2H) = E/A; V(1V 1H + 1H) = E/2A.

(9.157) (9.158)

The 1V 1H + 1H scenario has two dips, but with same visibility given in Eq.(9.158).

218

9 Temporal Distinguishability of a Multi-Photon State

In the ideal case when E = A, we have V3 (1V 2H) = 1 and V3 (1V 1H + 1H) = 1/2 in both schemes, which is consistent with the extreme cases in Eqs.(9.72, 9.76) in Sect.9.3. The deviation from the ideal values is caused by two factors: the ﬁrst is from the mode match between the V-photon and the H-photon and the second is from the mode match between the two H-photons. Both causes are from the same reason, that is, the three photons are originated from two independent pairs of photons. Therefore, it is not surprising to ﬁnd that the imperfection depends on the ratio E/A, which characterizes the degree of the temporal overlap between the two independent pairs. There is, however, one exception. In Eq.(9.154), the visibility is always 1/2 independent of E/A. This is because the V-photon and the H-photon that overlap are from the same crystal (the second one) and they already have perfect mode match with the assumption of permutation symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ). The experimental demonstration was performed by Liu et al. [9.13] and the experimental data veriﬁed the predictions in Eqs.(9.152, 9.154, 9.155) as well as the pictorial depiction in Fig.9.3 for the N = 2 case. 9.7.2 Distinguishing 4 × 1 Case from 2 × 2 Case for the State of |2H , 2V Let us consider now an input state of |2H , 2V for the NOON-state projection measurement. This state can be easily produced in nondegenerate parametric down-conversion (NPDC). According to the previous discussion, we should have P4 (|2H , 2V ) = 0 because |2H , 2V is orthogonal to the NOON-state. In practice, however, we do not exactly have the state |2H , 2V in NPDC. Simple Pictures for Two Independent Pairs and Four Entangled Photons In Sect.9.1, we learned that depending on the value of E/A, we may have either the so-called 2 × 2 case (2 × HV ) with E/A = 0 or the 4 × 1 case (2V 2H) with E/A = 1. Although we already obtained V4 (2 × HV ) = 1/3 and V4 (2V 2H) = 1 in Sect.9.5.2 by the complicated formula in Eq.(9.118), we will present here a simple argument with the single-mode picture shown in Sect.8.6.2. Let us now label the times at which the two pairs are generated as τ1 , τ2 , respectively. Referring to Fig.9.9, for the case in Fig.9.9a (2 × 2 case), we have |τ1 − τ2 | >> Tc , but for Fig.9.9b (4 × 1 case), |τ1 − τ2 | << Tc . We can then write the quantum state of the two pairs as |Φ = |φ(τ1 ) ⊗ |φ(τ2 ),

(9.159)

|φ = |1H , 1V .

(9.160)

with

9.7 Experimental Realization for the Cases of |2H , 1V , |2H , 2V

V (a)

τ2

τ1

H

V (b)

τ1 = τ2

H V (c)

H

τ1

τ4

τ3 τ2

219

Fig. 9.9. Three scenarios with two pairs of photons from nondegenerate parametric down-conversion: (a) the 2 × 2 case; (b) the 4 × 1 case; (c) the 1 × 4 case. Reprinted ﬁgure with permission from F. W. Sun, Z. Y. Ou, and G. C. Guo, Phys. Rev. A c 73, 023808 (2006). 2006 by the American Physical Society.

Consider the NOON-state projection scheme in Fig.9.2 for the N = 4 case. The four-photon coincidence probability P4 is proportional to the time integral of the four-photon correlation function:

where

ˆ1 (t1 )Eˆ2 (t2 )Eˆ3 (t3 )E ˆ4 (t4 )|Φ||2 , Γ (4) (t1 , t2 , t3 , t4 ) = ||E

(9.161)

⎧

ˆH (t) + EˆV (t) /2 + ..., ˆ1 (t) = E ⎪ E ⎪

⎪ ⎨ ˆ ˆH (t) − EˆV (t) /2 + ..., E2 (t) = E

ˆH (t) + iE ˆV (t) /2 + ..., ˆ3 (t) = E ⎪ E ⎪ ⎪

⎩ ˆ ˆH (t) − iE ˆV (t) /2 + .... E4 (t) = E

(9.162)

Here, we omit the vacuum input modes from the beam splitters and EˆH,V (t) = a ˆH,V (t) for the single-mode treatment. When applying the above to the state in Eq.(9.159), we ﬁnd six nonzero terms in Eq.(9.161) contributing to the time integral of Γ (4) (t1 , t2 , t3 , t4 ), that is, when (i) t1 = t2 = τ1 , t3 = t4 = τ2 or t1 = t2 = τ2 , t3 = t4 = τ1 ; (ii) t1 = t3 = τ1 , t2 = t4 = τ2 or t1 = t3 = τ2 , t2 = t4 = τ1 ; (iii) t1 = t4 = τ1 , t2 = t3 = τ2 or t1 = t4 = τ2 , t2 = t3 = τ1 . Case (i) can be calculated as ˆ1 (τ1 )Eˆ2 (τ1 )E ˆ3 (τ2 )E ˆ4 (τ2 )|Φ E ˆ1 (τ1 )E ˆ2 (τ1 )|φ(τ1 ) ⊗ E ˆ3 (τ2 )E ˆ4 (τ2 )|φ(τ2 ) =E = (1/4)(1 − 1) × (1/4)(i − i)|0 = 0.

(9.163)

ˆ2 (τ2 )Eˆ3 (τ1 )E ˆ4 (τ1 )|Φ. The zero result in this case It is the same for Eˆ1 (τ2 )E stems from the two-photon Hong-Ou-Mandel eﬀect between E1 and E2 and between E3 and E4 . Similarly, case (ii) gives: ˆ1 (τ1 )Eˆ2 (τ2 )E ˆ3 (τ1 )E ˆ4 (τ2 )|Φ E ˆ ˆ ˆ2 (τ2 )E ˆ4 (τ2 )|φ(τ2 ) = E1 (τ1 )E3 (τ1 )|φ(τ1 ) ⊗ E = (1/4)(1 + i) × (1/4)(1 + i)|0.

(9.164)

220

9 Temporal Distinguishability of a Multi-Photon State

ˆ1 (τ2 )Eˆ2 (τ1 )E ˆ3 (τ2 )E ˆ4 (τ1 )|Φ and case (iii) then It yields the same result for E gives: ˆ1 (τ1 )Eˆ2 (τ2 )E ˆ3 (τ2 )Eˆ4 (τ1 )|Φ E = (1/4)(1 − i) × (1/4)(1 − i)|0 ˆ2 (τ1 )Eˆ3 (τ1 )E ˆ4 (τ2 )|Φ. = Eˆ1 (τ2 )E

(9.165)

When the two pairs are separated, i.e., |τ1 −τ2 | >> Tc , all six contributions are distinguishable (2 × 2 case) and we add their absolute values to give P4 : 1 + i 4 1 − i 4 = 1. (9.166) + 2 P4 (2 × 2) ∝ 2 × 0 + 2 4 4 16 On the other hand, when the two pairs overlap and become indistinguishable, i.e., |τ1 − τ2 | << Tc (4 × 1 case), we add the six amplitudes before taking the absolute value:

2

2 2 1+i 1 − i +2 (9.167) P4 (4 × 1) ∝ 2 × 0 + 2 = 0. 4 4 The complete disappearance of P4 (4 × 1) is a result of the orthogonality of |2H, 2V with the NOON-state. Thus, the projection measurement gives a null result. Notice that even when |τ1 −τ2 | >> Tc , there is still two-photon interference (2×2 case). This is the reason that the contribution from case (i) is zero. So, we expect that the value in Eq.(9.166) will be smaller than the situation when all four photons are well-separated in time (1 × 4 case), as in Fig.9.9c. In this situation, no two-photon interference occurs and we add absolute values instead of amplitudes in evaluating the contribution from ||E1 (τ1 )Eˆ2 (τ1 )|φ(τ1 )||2 , which gives (1/4)2 + (−1/4)2 = 1/8 for the two-photon case. So, for the four-photon case, we ﬁnd that all six contributions are of the same value of (1/8) × (1/8) = 1/64. Overall, we have P4 (1 × 4) ∝ (1/64) × 6 = 3/32. As can be seen, P4 (1 × 4) > P4 (2 × 2) > P4 (4 × 1). This means that when we adjust the path diﬀerence between H and V, P4 will drop from P4 (1 × 4) to P4 (2 × 2) for the 2× 2 case and to P4 (4 × 1) for the 4× 1 case. This is similar to the Hong-Ou-Mandel eﬀect but for four photons, two from each side. Hence, the visibility of this generalized Hong-Ou-Mandel dip is V4 (2 × 2) = [P4 (1 × 4) − P4 (2 × 2)]/P4 (1 × 4) = (3/32 − 1/16)/(3/32) = 1/3,

(9.168)

for the 2 × 2 case. For the 4 × 1 case, because P4 (4 × 1) = 0, we always have V4 (4 × 1) = 1. This result is consistent with the values in Table 9.1 in Sect.9.5.2. The single-mode argument presented above is simple and straightforward and can be applied to other scenarios in Tables 9.1-9.5. Indeed, it can be shown that the simple argument leads to the same values shown in those tables.

9.7 Experimental Realization for the Cases of |2H , 1V , |2H , 2V

221

Although the above picture is straightforward and easy to understand, it is not rigorous and it only applies to some extreme scenarios in Tables 9.1-9.5. In the following, we will use a multi-mode theory of parametric down-conversion to accurately calculate the four-fold coincidence rate and cover the case of partial distinguishability for the input state of |2H , 2V . Multi-mode treatment The quantum state from type-II parametric down-conversion was derived in Sect.2.3.2 and has the form of 1 |Ψ = |0 + ξ|Φ2 + ξ 2 |Φ4 , 2 with

|Φ2 =

and

|Φ4 =

(9.169)

dω1 dω2 Φ(ω1 , ω2 )eiω2 ΔT a ˆ†H (ω1 )ˆ a†V (ω2 )|0,

(9.170)

dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 )ei(ω2 +ω2 )ΔT ×ˆ a†H (ω1 )ˆ a†V (ω2 )ˆ a†H (ω1 )ˆ a†V (ω2 )|0,

(9.171)

where we add the time delay ΔT between the H- and the V-photons. |Φ2 is a two-photon state and |Φ4 is a four-photon state. For simplicity of calculation, we assume the symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ), i.e., the photons within one pair are temporally indistinguishable. The delay ΔT may be added after the pair is created. Now, let us calculate the four-photon correlation function Γ (4) in Eq.(9.161) with the state in Eq.(9.169). After expanding the product in Eq.(9.161) with Eq.(9.162), we ﬁnd that only the following combinations are non-zero: ˆ3 (t3 )Eˆ4 (t4 )|Φ Eˆ1 (t1 )Eˆ2 (t2 )E = (1/16) (HHV V − V V HH) + i(V HV H + HV HV )− −i(HV V H + V HHV ) |Φ, (9.172) ˆH , V = EˆV , and we keep the time ordering. For a multi-mode where H= E ˆH , E ˆV are expressed in multi-mode as state in Eq.(9.169), E 1 ˆ EH,V (t) = √ dωˆ aH,V (ω)e−iωt , (9.173) 2π where a ˆH,V (ω) is the annihilation operator satisfying the commutation relation: [ˆ a† (ω), a ˆ(ω )] = δ(ω − ω ).

(9.174)

222

9 Temporal Distinguishability of a Multi-Photon State

We can see that only the four-photon term in Eq.(9.169) will contribute to Eq.(9.172). The ﬁrst term in Eq.(9.172) can be easily calculated as HHV V |Φ ξ2 dω1 dω2 dω1 dω2 Φ(ω1 , ω2 )Φ(ω1 , ω2 ) e−iω1 t1 −iω1 t2 + e−iω1 t2 −iω1 t1 = 2 ×ei(ω2 +ω2 )ΔT e−iω2 t3 −iω2 t4 + e−iω2 t4 −iω2 t3 |0 ≡ ξ 2 G (4) (t1 , t2 , t3 , t4 )|0,

(9.175)

with G (4) (t1 , t2 , t3 , t4 ) ≡ G(t1 , t3 − ΔT )G(t2 , t4 − ΔT ) +G(t1 , t4 − ΔT )G(t2 , t3 − ΔT ), where G(t, t ) =

1 2π

dω1 dω2 Φ(ω1 , ω2 )e−iω1 t−iω2 t .

(9.176)

(9.177)

Note the permutation symmetry: G (4) (t1 , t2 , t3 , t4 ) = G (4) (t2 , t1 , t3 , t4 ) = G (4) (t1 , t2 , t4 , t3 ).

(9.178)

The rest of the terms in Eq.(9.172) have the following forms: V V HH|Φ = ξ 2 G (4) (t3 , t4 , t1 , t2 )|0, 2 (4)

(9.179)

HV HV |Φ = ξ G (t1 , t3 , t2 , t4 )|0, V HV H|Φ = ξ 2 G (4) (t2 , t4 , t1 , t3 )|0,

(9.180) (9.181)

HV V H|Φ = ξ 2 G (4) (t1 , t4 , t2 , t3 )|0, V HHV |Φ = ξ 2 G (4) (t2 , t3 , t1 , t4 )|0.

(9.182) (9.183)

Then Eq.(9.161) becomes: Γ (4) (t1 , t2 , t3 , t4 ) = (|ξ|/4)4 G (4) (t1 , t2 , t3 , t4 ) − G (4) (t3 , t4 , t1 , t2 ) +i G (4) (t2 , t4 , t1 , t3 ) + G (4) (t1 , t3 , t2 , t4 ) 2 −i G (4) (t1 , t4 , t2 , t3 ) + G (4) (t2 , t3 , t1 , t4 ) . (9.184) The four-photon coincidence probability is proportional to an integral of Γ (4) with respect to all times: +∞ dt1 dt2 dt3 dt4 Γ (4) (t1 , t2 , t3 , t4 ). (9.185) P4 (ΔT ) ∝ −∞

We obtain, after some lengthy calculation:

9.7 Experimental Realization for the Cases of |2H , 1V , |2H , 2V

P4 (ΔT ) ∝ (|ξ|/4)4 8A + 12E + 4E (2) (ΔT ) + 4A[1 − q(ΔT )]2 − (1) (1) (1) −8E1 (ΔT ) − 8E2 (ΔT ) − 8E3 (ΔT ) ,

223

(9.186)

where

with

A(τ ) ≡

and

(1)

E1 (τ ) =

q(τ ) = A(τ )/A(0),

(9.187)

2 dω1 dω2 Φ(ω1 , ω2 ) ei(ω2 −ω1 )τ ,

(9.188)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )×

×Φ(ω1 , ω1 )Φ(ω2 , ω2 )ei(ω2 −ω2 )τ ,

(1)

E2 (τ ) =

(9.189)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )× ×Φ(ω1 , ω1 )Φ(ω2 , ω2 )ei(ω2 −ω1 )τ ,

(1)

E3 (τ ) =

(9.190)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )×

×Φ(ω1 , ω1 )Φ(ω2 , ω2 )ei(ω2 −ω1 )τ , E (2) (τ ) =

(9.191)

dω1 dω2 dω1 dω2 Φ∗ (ω1 , ω2 )Φ∗ (ω1 , ω2 )×

×Φ(ω1 , ω1 )Φ(ω2 , ω2 )ei(ω1 −ω2 )τ ei(ω2 −ω1 )τ .

(9.192)

All the quantities in Eqs.(9.187-9.192) are the same as the quantities given (1) (1) (1) in Eqs.(8.84-8.88). Notice that q(0) = 1 and E1 (0) = E2 (0) = E3 (0) = (2) E (0) ≡ E, and, because of the symmetry Φ(ω1 , ω2 ) = Φ(ω2 , ω1 ), we have (1)∗ (1) (1)∗ (1) E1 (τ ) = E2 (τ ), E3 (τ ) = E3 (τ ), and E (2)∗ (τ ) = E (2) (τ ). So, we have, for ΔT = 0: P4 (0) ∝ 8(|ξ|/4)4 (A − E).

(9.193)

When the delay ΔT is much larger than the coherence time, or the reciprocal of the bandwidth of Φ(ω1 , ω2 ), there is no overlap among all four photons. This corresponds to the 1 × 4 case and all the ΔT -dependent terms in Eq.(9.186) are zero. Hence, we have P4 at large delay as P4 (∞) ∝ 12(|ξ|/4)4 (A + E).

(9.194)

224

9 Temporal Distinguishability of a Multi-Photon State

The visibility of the generalized Hong-Ou-Mandel dip is then: V4 ≡

A + 5E P4 (∞) − P4 (0) = . P4 (∞) 3(A + E)

(9.195)

Note that E ≤ A by Schwartz inequality. The equality stands if and only if Φ(ω1 , ω2 ) is factorized as Φ(ω1 , ω2 ) = ψ1 (ω1 )ψ2 (ω2 ). When E = 0, we have: V4 = 1/3,

(9.196)

which is exactly the same as Eq.(9.168) and corresponds to the situation when the two pairs of down-converted photons are well-separated and independent of each other (the 2 × 2 case or the 1HV + 1HV case in Table 9.1). But when E = A, Eq.(9.195) becomes: V4 = 1.

(9.197)

In this situation, the two pairs of down-converted photons are overlapped to form an indistinguishable four-photon entangled state (4 × 1 case). The experimental demonstration of the 2×2 and 4×1 cases was performed by Xiang et al. [9.10] and conﬁrmed the visibility predicted in Eq.(9.195).

References 9.1 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). 9.2 R. J. Glauber, Phys. Rev. 130, 2529 (1963); Phys. Rev. 131, 2766 (1963). 9.3 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, (Cambridge University Press, New York, 1995). 9.4 R. Loudon, The Quantum Theory of Light (Clarendon, Oxford, 1st ed., 1973; 3rd ed., 2000). 9.5 D. F. Walls and G. J. Milburn, Quantum Optics (Springer-Verlag, Berlin, 1994). 9.6 D. Stoler, Phys. Rev. D 1, 3217 (1970). 9.7 H. P. Yuen, Phys. Rev. A 13, 2226 (1976). 9.8 H. J. Kimble, in Fundamental Systems in Quantum Optics (Les Houches Lectures, LIII), J. Dalibard, J. M. Raimond and J. Zinn-Justin, eds. (Elsevier Science Publishers, Amsterdam, 1992). 9.9 H. F. Hofmann, Phys. Rev. A 70, 023812 (2004). 9.10 G. Y. Xiang, Y. F. Huang, F. W. Sun, P. Zhang, Z. Y. Ou, and G. C. Guo, Phys. Rev. Lett. 97, 023604 (2006). 9.11 K. Sanaka, T. Jennewein, J.-W. Pan, K. Resch, and A. Zeilinger, Phys. Rev. Lett. 92, 017902 (2004). 9.12 K. Sanaka, K. J. Resch, and A. Zeilinger, Phys. Rev. Lett. 96, 083601 (2006). 9.13 B. H. Liu, F. W. Sun, Y. X. Gong, Y. F. Huang, Z. Y. Ou, and G. C. Guo, Europhys. Lett. 77, 24003 (2007); quant-ph/0606118.

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

The multi-photon interference eﬀects discussed in the previous chapter rely on multi-photon coincidence to extract the signal, which, according to the simple interpretation of multi-photon interference in Sect.1.2, is what causes the difference between the situations of diﬀerent photon numbers, i.e., between, say, two-photon interference and three-photon interference. Therefore, the number of photons needs to be speciﬁed before we can study the interference eﬀect. In this chapter, we will study the multi-photon interference eﬀects from another perspective, i.e., homodyne detection. Because of the diﬀerent detection scheme, the interference eﬀect involves diﬀerent photon numbers at play at the same time. What motivates this study is the fact that homodyne detection of a singlephoton state is completely diﬀerent from that of a vacuum state, a fact pointed out by Yurke and Stoler [10.1] and conﬁrmed by Vogel and Grabow [10.2]. Since homodyne detection involves a strong classical coherent ﬁeld, it is surprising that the existence of only a single photon can totally change the outcome that is supposed to be dominated by the classical ﬁeld. Therefore, one may guess that such a dramatic nonclassical eﬀect must be the manifestation of some fundamental principles of quantum mechanics. It turns out that it is the quantum multi-photon interference eﬀect that gives rise to the dramatic result in the homodyne detection of a single-photon state. Since any classical ﬂuctuation of the local oscillator is canceled in the homodyne detection scheme [10.3], the interference eﬀect should persist even if the strong coherent ﬁeld is replaced by a ﬁeld in arbitrary state, as will be shown here.

10.1 Interference with a Single-Photon State and an N-Photon State at a Symmetric Beam Splitter We start by revisiting the interference involving a single-photon state and an N -photon state via a 50:50 beam splitter (Fig.10.1). But instead of the probability of one particular term, as discussed in the previous chapters, we

226

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

are interested in the whole probability distribution. Let us ﬁrst look at the situation without the single-photon state. It is well-known that when a number of particles, say N , enter a 50:50 lossless beam splitter from one input port, each particle is randomly sent to the two output ports with equal chance (50%), resulting in the simple Bernoulli binomial probability distribution of P0 (N1 , N2 ) =

N! δN1 +N2 ,N . 2N N1 !N2 !

(10.1)

N1 is the number of particles exiting from port 1, while N2 is for port 2. In the case of the photon, the above result suggests that photons act independently as classical particles.

|0 2 or |1 |N a^1

2

a^2 N1

1

50:50 ^ A2

N2

^

A1

Fig. 10.1. Homodyne detection of a single-photon state or a vacuum state with an N -photon state as the local oscillator.

What will happen then if we let a single-photon state enter the other input port (port 2) of the beam splitter (Fig.10.1)? The outcome from classical particle theory is not much diﬀerent from Eq.(10.1). Because the single photon is independent of the other N photons when it acts as a classical particle, we simply add the probabilities to obtain: 1 N! 1 N! + 2 2N (N1 − 1)!N2 ! 2 2N (N2 − 1)!N1 ! (N + 1)! , = N +1 2 N1 !N2 !

P1cl (N1 , N2 ) =

(10.2)

which is in the exactly same form as that in Eq.(10.1). Therefore the existence of the single photon at the other port does not inﬂuence the photon probability distribution at all. The single photon from port 2 acts as if it were part of the N + 1 photons from port 1. On the other hand, the outcome is totally diﬀerent if we treat photons as quantum particles. To demonstrate the principle of quantum interference, let us consider the special case when N is an odd integer, so that N1 = N2 = (N + 1)/2 is an integer. The principle of quantum mechanics requires us to add not the probabilities but the probability amplitudes, which have two contributions here (see Fig.10.2): (a) the single photon goes to output port 2 while N1 = (N + 1)/2 photons go to output port 1 and N2 − 1 = (N − 1)/2 photons to port 2 or (b) the single photon input at port 2 goes to output port 1 while N1 − 1 = (N − 1)/2 from the N photons go to output port 1

10.1 Interference with a Single-Photon State and an N-Photon State

(b)

(a)

227

Fig. 10.2. Two possible ways to obtain equal photon number output for N = odd case.

and N2 = (N + 1)/2 photons to port 2. From Eq.(10.1), we ﬁnd that these two possibilities have equal probability, thus their probability amplitudes have equal absolute value. For their phases, however, because there is a π phase shift for the reﬂected ﬁeld of input port 1 and no phase shift for the transmitted one for a lossless beam splitter, the total phase shift for the N + 1 photons at the output ports will be diﬀerent for the two possibilities. Referring to Fig.10.2 for evaluating the phases, we ﬁnd that the total phase shift of the N + 1 photons for the ﬁrst possibility is ϕ1 = ϕt (N + 1)/2 + ϕr (N − 1)/2 + ϕt while for the second possibility, it is ϕ2 = ϕt (N − 1)/2 + ϕr (N + 1)/2 + ϕr . Thus, the phase diﬀerence between the two cases is ϕ1 − ϕ2 = ϕt + ϕt − ϕr − ϕr and it is π from Eq.(A.3) in Appendix A. So the two probability amplitudes will cancel each other, resulting in zero probability for N1 = N2 = (N + 1)/2. This result is completely diﬀerent from that of a classical particle theory in Eq.(10.2). As seen above, the probability cancellation at N1 = N2 results from the quantum interference of N + 1 particles. A special case of N = 1 was the famous Hong-Ou-Mandel interference eﬀect discussed in Chapt.3. For the other cases when N1 = N2 , a similar quantum interference eﬀect persists, but because the probabilities for the two cases are not equal, they do not completely cancel each other. We can ﬁnd the probability distribution P1 (N1 , N2 ) by using the formula [10.4]: & ' (Aˆ†1 Aˆ1 )N1 −Aˆ† Aˆ1 (Aˆ†2 Aˆ2 )N2 −Aˆ† Aˆ2 e 1 e 2 : , P1 (N1 , N2 ) = : (10.3) N1 ! N2 ! where √ √ Aˆ1 = (ˆ a1 + a ˆ2 )/ 2, Aˆ2 = (ˆ a2 − a ˆ1 )/ 2

(10.4)

are the annihilation operators for the output modes for a 50:50 lossless beam ˆ2 are in the state of |Φ = splitter. The input modes represented by a ˆ1 , a |N 1 |12 . After some lengthy calculation, we arrive at P1 (N1 , N2 ) =

N !(N1 − N2 )2 δN1 +N2 ,N +1 . 2N +1 N1 !N2 !

(10.5)

Eq.(10.5) can also be derived from the general formula in Eq.(A.18) of Appendix A for arbitrary numbers {M, N } of input photons, with the setting of

228

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

√ t = r = 1/ 2, M = 1. Notice that when N is an odd integer, P1 (N1 , N2 ) = 0 for N1 = N2 = (N + 1)/2, which is exactly the same as predicted from the simple argument in the previous paragraph. Actually, quantum probability cancellation makes the whole probability distribution in Eq.(10.5) diﬀerent from that in Eq.(10.1), as seen in Fig.10.3. The quantum interference plays a crucial role here in the diﬀerence between quantum mechanics and classical mechanics. 0.20

0.15

Probability

(a)

(b)

0.15

0.10 0.10

0.05

0.05

0.00

0.00 0

2

4

6

8

10

12

14

Photon Number

16

18

0

2

4

6

8

10

12

14

16

18

20

Photon Number

Fig. 10.3. Output photon probability distribution for N -photon state input at port 1 with (a) vacuum state or (b) single-photon state input at port 2 (N = 19). Reprinted ﬁgure with permission from Z. Y. Ou, Quan. Semicl. Opt. 8, 315 (1996). c 1996 by the Institute of Physics.

10.2 Interference of a Single-Photon State and an Arbitrary State: Homodyne Detection of a Single-Photon State The above quantum probability cancellation eﬀect due to a single photon state is not restricted to an N -photon state. Since the eﬀect is based on quantum interference between a single-photon state and an arbitrary N -photon state, it should exist for an arbitrary state which consists of an arbitrary number n of photons with probability Pnin . The following is our argument: since the vacuum state and the single-photon state are completely incoherent in the sense that they have a totally random phase distribution, the output ﬁelds due to interference of one of these states with any other state will lose all the coherence information of the input. Therefore, the output photon distribution of the beam splitter will depend only on the photon statistics Pnin of the input state at port 1. So, combining this fact with Eqs.(10.1, 10.5), we have the output photon distributions in the form of P0 (N1 , N2 ) =

(N1 + N2 )! P in 2N1 +N2 N1 !N2 ! N1 +N2

(10.6)

10.2 Interference of a Single-Photon State and an Arbitrary State

229

for vacuum input at port 2 and (N1 + N2 − 1)! (N1 − N2 )2 PNin1 +N2 −1 2N1 +N2 N1 !N2 !

P1 (N1 , N2 ) =

(10.7)

for single-photon state input at port 2. Of course, we may rigorously derive the output photon distribution by following the procedure leading to Eq.(10.5). It can be shown that Eqs.(10.6, 10.7) are, indeed, the correct form for the output photon distribution. Eq.(10.6) is also a special case of Eq.(45) of Ref.[10.11]. By comparing Eqs.(10.6, 10.7), we easily ﬁnd that P1 (N1 = N2 ) = 0 for single-photon state input at port 2 while P0 (N1 = N2 ) =

∞ N1 =0

(2N1 )! P in = 0 22N1 (N1 !)2 2N1

(10.8)

for vacuum input. Therefore, the eﬀect of probability cancellation exists even for an arbitrary input state at port 1. In an actual experiment, however, it is diﬃcult to measure the complete distribution P (N1 , N2 ), but the distribution P (N1 −N2 = M ) can be measured by balanced homodyne detection [10.5]. From Eqs.(10.6, 10.7) we ﬁnd that P0 (M ) =

∞ N1 =M

P1 (M ) = M 2

(2N1 − M )! P in 22N1 −M N1 !(N1 − M )! 2N1 −M

∞ N1 =M

(2N1 − M − 1)! P in 2N 2 1 −M N1 !(N1 − M )! 2N1 −M−1

(10.9)

(10.10)

for M ≥ 0. When M < 0, the symmetry between N1 , N2 in Eqs.(10.6, 10.7) leads to P (M ) = P (−M ). Next, let us evaluate P0 (M ), P1 (M ) for some special states. For an N photon state input, we have Pnin = δn,N , and Eqs.(10.9, 10.10) give results similar to Eqs.(10.1, 10.5): 2 2 N! e−M /2N ≈√ + M/2)!(N/2 − M/2)! 2N π M 2N ! P1 (M ) = N +1 2 (N/2 + M/2 + 1/2)!(N/2 − M/2 + 1/2)! 2 M 2 −M 2 /2N ≈ √ e 2N π N

P0 (M ) =

2N (N/2

(10.11)

(10.12)

for N >> 1, M . The extra normalization factor of 2 in the approximated expressions in Eqs.(10.11, 10.12) is because P (M ) = 0 for every other value of M . For a coherent state, Pnin = n ¯ n e−¯n /n!, with n ¯ being the average photon number, and we have, from Eqs.(10.9, 10.10), that n) P0 (M ) = e−¯n IM (¯ 2 M −¯n e IM (¯ P1 (M ) = n), n ¯

(10.13) (10.14)

230

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

where IM (¯ n) is the Bessel function with a purely imaginary argument. Similar results as Eqs.(10.13, 10.14) were obtained in Refs.[10.2, 10.5]. For large n ¯, 2 1 e−M /2¯n , P0 (M ) ≈ √ 2¯ nπ 1 M 2 −M 2 /2¯n e P1 (M ) ≈ √ , ¯ 2¯ nπ n

(10.15) (10.16)

which has the same form as Eqs.(10.11, 10.12) for large N besides the factor of 2. This is not surprising if we consider the fact that when the photon number is large, the interference scheme discussed above becomes the homodyne detection scheme. Since both a vacuum state and a single-photon state have random phase distribution, homodyne detection with an N -photon state (N >> 1) and coherent state as local oscillators are equivalent. As a matter of fact, the output photon distributions will always have the form of Eqs.(10.15, 10.16) for any state as local oscillator, provided that its average photon number is large and the photon number ﬂuctuation is much less than the average photonnumber ( Δn2 << n ¯ ). We can see this point from Eqs.(10.9, 10.10): in 2 when Δn << n ¯ , Pn has a narrow peak around n ¯ and is a fast changing function, as compared with other terms in the summation, therefore the contribution to the summation comes only from the few terms near n ¯ , so that we can pull the slowly changing terms out of the sum, that is, P0 (M ) ≈

2n¯ (¯ n/2

≈ √

n ¯! P in /2 − M )!(¯ n/2 + M )! n n

2 1 e−M /2¯n 2¯ nπ

when n ¯ >> 1,

(10.17)

and, similarly, M 2 /¯ n −M 2 /2¯n P1 (M ) ≈ √ e 2¯ nπ

when n ¯ >> 1.

(10.18)

We can also understand this result from the fact that any ﬂuctuation in local oscillator is canceled in a balanced homodyne detection scheme [10.3]. Furthermore, if we set n ¯ → ∞, we √ can replace the discrete variable M with a continuous one deﬁned by x = M/ n ¯ and the probability distributions in Eqs.(10.17, 10.18) lead to probability densities of continuous variable x as 2 1 P0 (x) = √ e−x /2 , 2π x2 −x2 /2 e P1 (x) = √ 2π

(10.19) (10.20)

which correspond to the square of the absolute value of the wave function for the vacuum state and single-photon state, respectively. Thus, by measuring

10.2 Interference of a Single-Photon State and an Arbitrary State

231

P (M ) in homodyne detection, we can deduce the wave function of the input state at port 2 besides a phase factor which can be ﬁxed by the technique of optical tomography [10.6, 10.7]. However, there is an exception to the above. It is well-known that for thermal light, ¯ (¯ n + 1), (10.21) Δn2 = n ¯ and we cannot use the approximation in Eqs.(10.17, so that Δn2 ≈ n 10.18). For thermal light, Pnin = n ¯ n /(¯ n + 1)n+1 , so from Eq.(10.9, 10.10), we have: √ P0 (M ) = q M / 2¯ n+1 (M ≥ 0), (10.22) M n (M ≥ 0), (10.23) P1 (M ) = M q /¯ √ with q = 1 + 1/¯ n − 2¯ n + 1/¯ n. Therefore, the output photon distribution for thermal light input is diﬀerent from that of coherent state input even when n ¯ >> 1. But, the general trend in the change of the shape from P0 (M ) to P1 (M ) is similar in both states (Fig.10.4) and the quantum interference eﬀect due to a single-photon is still there.

0.025 0.020

0.04

P0

(a)

0.015

P(M)

P1

0.010

P0

0.02

P1

0.01

0.005 0.000

(b)

0.03

- 60

- 40

- 20

0

M

20

40

60

0.00

- 60

- 40

- 20

0

20

40

60

M

Fig. 10.4. Probability distribution P0,1 (M ) for balanced homodyne detection of vacuum state and single-photon state with (a) coherent state or (b) thermal state as local oscillator. n ¯ = 300. Reprinted ﬁgure with permission from Z. Y. Ou, Quan. c Semicl. Opt. 8, 315 (1996). 1996 by the Institute of Physics.

It is interesting to note that the quantum interference eﬀect studied here has similarities with another type of intensity-independent interference effect where two interfering ﬁelds are substantially diﬀerent in intensity (see Sect.1.3). In both cases, the presence of the weak ﬁeld can dramatically change the outcome of the result. However, the underlying principles are quite different in the two cases. Here, all the N photons participate in the interference (N + 1-particle interference), whereas in Sect.1.3, only two photons are involved and the rest of the photons are not counted. Even though the nonclassical ﬁeld is weak here, the result is very nonclassical in the sense that the probability of detecting equal intensities at the two outputs is zero

232

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

[P1 (M = 0) = 0]. It can be proved that in the similar situation (one ﬁeld is weak and the other is strong), classical wave theory predicts that the probability is largest for equal intensity output at the two ports.

10.3 Multi-Mode Consideration The analysis in the previous section is based on a single-mode model in which we assumed that there is a perfect spatial and temporal mode match between the input ﬁeld and the local oscillator (LO) ﬁeld. Normally, spatial mode can be matched easily if the input ﬁeld is generated in a well-deﬁned cavity with Gaussian mode. The temporal mode, on the other hand, is determined by many factors and is harder to match. An important fact is that we cannot use ﬁlters for mode match because they will introduce vacuum noise in homodyne detection and are equivalent to losses. With vacuum noise, the probability distribution will be a combination of P0 and P1 and if the loss is large enough, we will lose the double peak feature in P1 . In the following, we will consider the same multi-photon interference eﬀect for the case of a pulsed single-photon state as the input and a pulsed coherent state as the LO, and explore the possibility of observing this eﬀect with a down-converted ﬁeld as the singlephoton state. First of all, let us consider a non-stationary single-photon state described by ˆ† (ω)|vac, (10.24) |Ψ = dωΨ (ω)ejωt0 a with normalization condition this state has the form of

dω|Ψ (ω)|2 = 1. The intensity of the ﬁeld in

ˆ † (t)E(t) ˆ I(t) = E = |A(t)|2 , with 1 A(t) = √ 2π

dωΨ (ω)e−jω(t−t0 ) .

(10.25)

(10.26)

It can be easily checked that the quantity γ(t1 , t2 ) = 1 for the state in Eq.(10.24). Thus, it is a transform-limited single-photon pulse centered at t = t0 . It is straightforward to show from Eq.(10.25) that the total photon number is ∞ ∞ dτ I(τ ) = dτ |A(τ )|2 = 1. (10.27) −∞

−∞

From the discussion in Sect.7.1.2, this state can be obtained from a pulsepumped parametric down-conversion process by conditional detection, as long as narrow ﬁltering is performed on the gating ﬁeld.

10.3 Multi-Mode Consideration

ε (t) E(t)

233

Coherent Local Oscillator

Signal D2

D1

∆i

Fig. 10.5. Homodyne detection with a strong coherent state as the local oscillator (LO).

Next, we mix this single-photon ﬁeld with a strong coherent pulse by a 50:50 beam splitter, as shown in Fig.10.5. Let us denote the ﬁeld operator for ˆ the coherent ﬁeld by E(t), which can be expressed in terms of the annihilation ˆ operator b(ω) as ˆ = √1 (10.28) E(t) dω ˆb(ω)e−jωt . 2π For coherent pulses, the quantum states |B can be described by ˆb(ω)|B = β(ω)ejωT0 |B,

(10.29)

where β(ω) is a slowly varying function of ω with a deﬁnite phase relation for a diﬀerent ω, in order to form a coherent pulse whose width is determined by the reciprocal of the bandwidth Δω of β(ω). The coherent pulse is centered at t = T0 . The intensity of the coherent pulse is given by ˆ = |B(t)|2 , IE (t) = Eˆ† (t)E(t) with 1 B(t) = √ 2π

dωβ(ω)e−jω(t−T0 ) .

(10.30)

(10.31)

So the pulse is also transform-limited and is centered at t = T0 . The output ﬁelds of the beam splitter are described by the ﬁeld operators: √ ˆ + E(t)]/ ˆ Eˆ1 (t) = [E(t) √2, (10.32) ˆ − E(t)]/ ˆ Eˆ2 (t) = [E(t) 2. We will be interested in the probability distribution for photon number difference between the two output ports. It can be calculated from the joint probability PN1 N2 of ﬁnding N1 photons in port 1 and N2 photons in port 2, which is given by [10.4] ( ) ˆ N1 (T ) ˆ ˆ N2 W ˆ 2 (T ) 1 −W1 (T ) W2 (T ) −W : , (10.33) PN1 N2 = T : e e N1 ! N2 !

234

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

ˆ 1,2 (T ) = T dτ Iˆ1,2 (τ ). T is the time ordering and : : is the normal with W 0 ordering. However, this quantity is not easy to calculate for the multi-mode states in Eqs.(10.24, 10.29). On the other hand, photo-detectors make a quantum measurement of the photon number of optical ﬁelds, that is, the photoelectrons have the same statistics as the photons that fall on the detector [10.6, 10.12]. So, we can equivalently check the photocurrent ﬂuctuations for the two detectors located at the outputs of the beam splitter and calculate the probability distribution of the photocurrents. The general formula for the characteristic function of the photocurrent ﬂuctuations has been calculated in Ref.[10.12]. Under the large intensity condition, it has the following form for two detectors {Eq.(44) of Ref.[10.12]}: Ci1 ,i2 (r * 1 , r2 )

+ exp{jr1 i1 + jr2 i2 } & ∞ ∞ ≈ T : exp jαr1 dτ Iˆ1 (t − τ )Q(τ ) + jαr2 dτ Iˆ2 (t − τ )Q(τ )− 0 0 % ' αr22 ∞ ˆ αr12 ∞ ˆ 2 2 dτ I1 (t − τ )[Q(τ )] − dτ I2 (t − τ )[Q(τ )] − : , 2 0 2 0 (10.34)

=

where α and Q(τ ) are the quantum eﬃciency and the response function for the two detectors, respectively. For simplicity, we assume the two detectors are identical. In the detection of a non-stationary ﬁeld, especially for ultra-fast laser pulses, the response of the detectors is usually slow, so that their action is simply an average of the photo-current over a long period of time T (longer than optical pulse width). Hence, we can choose the response function as for 0 < τ < T, i0 (10.35) Q(τ ) = 0 for other τ, where i0 is the average photocurrent in the detectors during the period T . For a balanced homodyne detection scheme, as shown in Fig.10.5, the photocurrents from the two detectors are subtracted to produce a diﬀerence current Δi = i1 − i2 , which represents the photon number diﬀerence between the two outputs of the beam splitter. Next, we ﬁnd the probability distribution for the photocurrent diﬀerence Δi. Its characteristic function can be obtained from Eq.(10.34) by setting r1 = −r2 ≡ r. With Eq.(10.35) for Q(τ ), we have: CΔi (r) = exp{jrΔi} & T

≈ T : exp jrαi0 dτ Iˆ1 (t − τ ) − Iˆ2 (t − τ ) − % ' 0

r2 αi20 T ˆ dτ I1 (t − τ ) + Iˆ2 (t − τ ) : . (10.36) − 2 0 From Eq.(10.32), we ﬁnd:

10.3 Multi-Mode Consideration

ˆ ˆ † Eˆ − Eˆ† E, Iˆ1 − Iˆ2 = E †ˆ †ˆ ˆ ˆ ˆ ˆ I1 + I2 = E E + E E.

235

(10.37)

So, Eq.(10.36) becomes: & T

CΔi (r) = T : exp − rαi0 dτ B(t − τ )Eˆ † − B ∗ (t − τ )Eˆ − 0 % ' 2 2 T r2 αi20 T ˆ − τ ) : exp − r αi0 dτ I(t dτ |B(t − τ )|2 , − 2 2 0 0 (10.38) where we replace the operator Eˆ with the c-function B(t) for the coherent pulse because of the normal ordering : :. It is now straightforward to calculate CΔi (r). By expanding the exponential function inside the angle brackets in Eq.(10.38) and using Eq.(10.24) for the single-photon state, we ﬁnd: & T 2 '%

1 2 2 2 † ∗ ˆ ˆ CΔi (r) = 1 + r α i0 : B(t − τ )E − B (t − τ )E dτ : 2 0 T 1 dτ |B(t − τ )|2 , (10.39) × exp − r2 αi20 2 0 where higher-order terms in the expansion are zero due to the single-photon ˆ because we assume property of the state |Ψ , and we neglect the term with I(t) the coherent ﬁeld is much stronger than the single-photon ﬁeld. With the state |Ψ in Eq.(10.24), the quantity inside the angle brackets can be easily calculated and we have: CΔi (r) = (1 − ar2 ) e−br

2

/2

,

(10.40)

with a≡

α2 i20

0

T

2 2 dτ B (t − τ )A(t − τ ) , b ≡ αi0

T

∗

0

dτ |B(t − τ )|2 . (10.41)

The probability distribution for the photocurrent diﬀerence Δi can be easily obtained from the characteristic function in Eq.(10.40) by a Fourier transformation. Hence, 1 PΔi (x) = drCΔi (r)e−jrx 2π

2 1 a ax2 = √ (10.42) 1 − + 2 e−x /2b . b b 2πb The distribution PΔI (x) has a twin peak feature with a minimum at x = 0. This is the multi-photon quantum interference eﬀect when a single photon state and a strong coherent state are superposed. When a = b, complete cancellation of probability is achieved at x = 0. However, by using Schwartz inequality, we have:

236

10 Homodyne of a Single-Photon State: A Special Multi-Photon Interference

a ≤ αb

0

T

dτ |A(t − τ )|2 .

(10.43)

The equal sign occurs when B(τ ) = CA(τ ) with C a constant number, that is, the shape of the coherent pulse matches exactly with that of the single-photon state. Because α ≤ 1 and T ∞ dτ |A(t − τ )|2 ≤ dτ |A(t − τ )|2 = 1, (10.44) 0

−∞

we always have a ≤ b, which ensures the positiveness of the probability distribution PΔi (x) for arbitrary x. Therefore, in order to observe complete probability cancellation, we need a 100% quantum eﬃciency for the detectors and a long integration time and, most importantly, the overlap of the temporal modes of the two interfering ﬁelds. Of course, these are the ideal conditions that can never be met in practice. Any imperfection in the experimental set-up will result in a/b < 1, which can be equivalent to a less-than 100% quantum eﬃciency for the detectors (αe ≡ a/b). Although no perfect cancellation can be achieved, there is still a minimum at x = 0 unless αe ≤ 1/3 ≈ 33.3%, for which the twin peak in PΔI (x) will merge into a single peak and the signature for multi-photon quantum interference is completely lost. 33.3% quantum eﬃciency for a detector is easily achievable (quantum eﬃciency higher than 90% has been reported). However, what makes αe small is the temporal mode mismatch as a result of Schwartz inequality in Eq.(10.43). This will be the most important factor in the experiment investigation of this eﬀect. Experimentally, the multi-photon interference eﬀect discussed in this chapter was observed by Lvovsky et al. [10.13], with a single-photon state produced from parametric down-conversion by gated detection.

References 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13

B. Yurke and D. Stoler, Phys. Rev. A 36, 1955 (1987). W. Vogel and J. Grabow, Phys. Rev. A 47, 4227 (1993). H. P. Yuen and V. W. S. Chan, Opt. Lett. 8, 177 (1983). L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York 1995). S. L. Braunstein, Phys. Rev. A 42, 474 (1990). K. Vogel and H. Risken, Phys. Rev. A40, 2847 (1989). D. T. Smithey, M. Beck, M. G. Raymer, and A. Faridani, Phys. Rev. Lett. 70, 1244 (1990). R. J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766 (1963). E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A 40, 1371 (1990). Z. Y. Ou and H. J. Kimble, Phys. Rev. A 52, 3126 (1995). A. I. Lvovsky, H. Hansen, T. Aichele, O. Benson, J. Mlynek, and S. Schiller, Phys. Rev. Lett. 87, 050402 (2001).

A Lossless Beam Splitter

An optical beam splitter plays an important role in optical interference. It usually acts as a device that splits the amplitude of an incoming wave or combines two waves for interference. Therefore, it is a 4-port device: 2 input ports and 2 output ports, where a beam splitter then connects the four coupled modes. Its behavior with waves is well-documented in classical electromagnetic wave theory [A.1]. Quantum mechanically, the relationship between the four input and output operators has been studied by a number of researchers [A.2, A.3, A.4, A.5, A.6] and it has a simple relation for the annihilation operators of the four modes in Heisenberg picture as ˆb1 = tˆ a1 + rˆ a2 , (A.1) ˆb2 = t a ˆ2 + r a ˆ1 . t, r, t , r are the complex amplitude transmissivity and reﬂectivity of the beam splitter, respectively. From the commutation relations [ˆbk , ˆb†l ] = δkl , (k, l = 1, 2)

(A.2)

we obtain |t|2 +|r|2 = 1, |t |2 +|r |2 = 1, |t| = |t |, |r| = |r |, and t∗ r +r∗ t = 0, which leads to ϕt + ϕr − ϕt − ϕr = π.

(A.3)

The above phase relation is universal and is independent of the speciﬁcs of the beam splitter. This relation can also be derived via input-output energy conservation from classical wave theory [A.7]. In general, t and r are complex numbers. However, by carefully choosing the reference point, we may arbitrarily change ϕt , ϕr , ϕt , ϕr within the restriction in Eq.(A.3). For simplicity, we let ϕt , ϕr , ϕt be zero and ϕr = −π, according to Eq.(A.3). Therefore, we have the beam splitter relation: ˆb1 = tˆ a1 + rˆ a2 , (t, r > 0) (A.4) ˆb2 = tˆ a2 − rˆ a1 .

238

A Lossless Beam Splitter

The argument above holds for one polarization. When the incident ﬁelds have diﬀerent polarizations, we need to decompose them into polarizations parallel (labeled by x) and perpendicular (labeled by y) to the incident plane. The phase relations between these two polarizations can be derived for a dielectric beam splitter from the Fresnel formulae (see, for example, Ref.[A.1]) and they have the form: ϕrx − ϕry = π.

(A.5)

for an incident angle smaller than the Brewster angle. Therefore, we have at near normal incidence: ˆb1x = tx a ˆ1x + rx a ˆ2x , (tx , rx > 0), (A.6) ˆb2x = tx a ˆ2x − rx a ˆ1x ,

ˆb1y = ty a ˆ1y − ry a ˆ2y , ˆb2y = ty a ˆ2y + ry a ˆ1y .

(ty , ry > 0).

(A.7)

The operator relationship in Eq.(A.4), together with the input state, is usually enough to determine the properties at the output ports. However, the above approach with operators lacks the visual connections to such interesting phenomena in quantum information as quantum entanglement and other nonclassical eﬀects in the output ports. So, to see what emerges from the beam splitter, it is better to work in the Schr¨ odinger picture and ﬁnd the output state of the beam splitter. For this purpose, there are at least a couple of papers [A.5, A.8] in the literature dealing with the output state for a general input state. In these papers, a general formalism is given that connects the input state with output state. For example, Ou et al. [A.5] used the GlauberSudashan P-representation [A.9, A.10] for the connection. This presentation, however, is best used to describe states with coherent state connection. For the number state input, the P-representation becomes very complicated. To circumvent this, Campos et al. [A.8] made use of the theory of angular momentum to derive a general formula for the output density operator in the photon number base. But this approach is complicated and requires some eﬀort to review the theory of angular momentum. Here, we will present a more direct method to obtain the output state for the input of number states or their corresponding superposition states. We start by working in the Heisenberg picture, in which the output operators are connected to the input operators by a unitary transformation: ˆb1 = U ˆ †a ˆ = tˆ ˆ1 U a2 , a1 + rˆ (A.8) ˆb2 = U ˆ †a ˆ = tˆ ˆ2 U a1 . a2 − rˆ ˆ is a function of a ˆ Here, U ˆ1 and a ˆ2 . (We will give a simple derivation of U at the end of this Appendix.) The state is unchanged and is the same as the input state in this picture:

A Lossless Beam Splitter

|Ψ = |φin .

239

(A.9)

All the properties at the output can be calculated by averaging the operators ˆb1 , ˆb2 in Eq.(A.8) over the state in Eq.(A.9) [A.5]. On the other hand, all these properties can also be equivalently calculated in the Schr¨ odinger picture in which the output operators ˆb1 , ˆb2 are the same as the input operators a ˆ1 , a ˆ2 , but the output state is connected to the input by ˆ |φin . |Ψ out = U

(A.10)

Generally, in order to ﬁnd the output state, we need the explicit form of ˆ . However, for special states such as a number state or a coherent state, it is U not necessary, as illustrated below. For simplicity, let us ﬁrst consider a single photon state input at port 1: |φin = |11 ⊗ |02 = (ˆ a†1 |01 ) ⊗ |02 .

(A.11)

The output state then becomes: ˆa ˆa ˆ † U|0 ˆ 1 ⊗ |02 , |Ψ out = U ˆ†1 |01 ⊗ |02 = U ˆ†1 U

(A.12)

ˆ = 1 between a ˆ †U ˆ1 and |01 . where we insert the unitary relation U ˆ It is easy to see that U |01 ⊗ |02 = |01 ⊗ |02 , that is, vacuum input gives vacuum output for a beam splitter. (This can be easily conﬁrmed directly ˆ at the end of this Appendix.) To ﬁnd when we have the explicit form of U † ˆ ˆ ˆ1 , a ˆ2 in terms of ˆb1 , ˆb2 : Ua ˆ1 U , we invert Eq.(A.4) to relate a ˆ ˆb1 U ˆ † = tˆb1 − rˆb2 , a ˆ1 = U ˆ ˆb2 U ˆ † = tˆb2 + rˆb1 . a ˆ2 = U So, the principle of reversibility gives: ˆa ˆ † = tˆ U ˆ1 U a1 − rˆ a2 , ˆ ˆ † = tˆ a2 + rˆ a1 . Ua ˆ2 U

(A.13)

(A.14)

ˆ given The above relation can be derived directly from the explicit form of U later. Therefore, we have, for the output state: |Ψ out = (tˆb†1 − rˆb†2 )|01 ⊗ |02 = t|1, 0 − r|0, 1.

(A.15)

ˆ2 by the same output operators Note that we replaced the input operators a ˆ1 , a ˆb1 , ˆb2 in the Schr¨ odinger picture. Likewise, we can ﬁnd the output state of an input state of |1, 1 in the Hong-Ou-Mandel interferometer (see Chapt.3): ˆ |11 |12 = U ˆa ˆa |Ψ out = U ˆ†1 |01 a ˆ†2 |02 = U ˆ†1 a ˆ†2 |0

240

A Lossless Beam Splitter

ˆ †U ˆa ˆ † U|0 ˆ ˆa ˆ † )(U ˆ † )|0 ˆa ˆa ˆ†2 U = (U ˆ†1 U =U ˆ†1 U ˆ†2 U † † † † a2 )(tˆ a2 + rˆ a1 )|0 = (tˆ a1 − rˆ 2 2 † † a1 a ˆ2 |0 √ + tr(ˆ a†2 ˆ†2 = (t − r )ˆ 1 −a 2 )|0 = (t2 − r2 )|1, 1 + 2tr(|2, 0 − |0, 2),

(A.16)

and for a 50:50 beam splitter we obtain the two-photon entangled state: √ (A.17) |Ψ out = (|2, 0 − |0, 2)/ 2. For a general input state of |M, N , we ﬁnd the output state as ˆ 1 |N 2 = √ 1 ˆa ˆ†N |Ψ out = U|M U ˆ†M 1 |01 a 2 |02 M !N ! 1 ˆ † ˆ ˆ†N U ˆ † U|0 ˆ ˆa = √ U ˆ†M 1 U Ua 2 M !N ! 1 ˆ † )M (U ˆ † )N |0 ˆa ˆa (U ˆ†1 U = √ ˆ†2 U M !N ! 1 (tˆ a†1 − rˆ a†2 )M (tˆ a†2 + rˆ a†1 )N |0 = √ M !N ! √ N M (−1)m M !N !tM+N −m−n rm+n †M−m+n †N −n+m a ˆ1 a ˆ2 |vac = (M − m)!m!(N − n)!n! m=0 n=0 M N (−1)m M !N !(M − m + n)!(N − n + m)! = (M − m)!m!(N − n)!n! m=0 n=0 ×tM+N −m−n rm+n |M − m + n, N − n + m,

(A.18)

which can be regrouped as |Ψ out =

M+N

ck |k, M + N − k,

(A.19)

k=0

where ck collects the coeﬃcients of the common terms of |k, M + N − k in Eq.(A.18) and is in a very complicated form for the general case. But for some special cases, we can derive its explicit form. For example, for a 50:50 beam splitter with M = N , the above is simpliﬁed as 1 N (ˆ a†2 − a ˆ†2 2 ) |0 N !2N 1 N †2(N −k) 1 a ˆ†2k a ˆ = N |0 (−1)N −k 1 2 2 k!(N − k)! k=0 N (2k)!(2N − 2k)! 1 N −k (−1) = N |2k, 2N − 2k. 2 k!(N − k)!

|Ψ out =

(A.20)

k=0

√ Another special case of M = 1 with t = r = 1/ 2 is given in Eq.(10.5) of Chapt.10.

A Lossless Beam Splitter

241

The above formalism can also be used in a coherent state representation, such as the Glauber-Sudarshan P-representation [A.9, A.10], to derive a general relation between the input and the output states for an arbitrary case. In the Glauber-Sudarshan P-representation, the input and output states are described by the density matrices as (A.21) ρˆin = d2 α1 d2 α2 Pin (α1 , α2 )|α1 , α2 α1 , α2 |, ρˆout = d2 α1 d2 α2 Pout (α1 , α2 )|α1 , α2 α1 , α2 |, (A.22) where Pin/out (α1 , α2 ) is a quasi-probability distribution and can completely describe the incoming/outgoing ﬁelds at the beam splitter. |α1 , α2 is the coherent state base. Our goal is to ﬁnd the connection between Pin and Pout . From Eq.(A.10), the output density matrix is then given by ˆ ˆ† ρˆout = U ρˆin U ˆ |α1 , α2 α1 , α2 |U ˆ †. = d2 α1 d2 α2 Pin (α1 , α2 )U

(A.23)

ˆ |α1 , α2 is the output state corresponding to a coherent state Obviously, U input state of |α1 , α2 , and, from classical optics and Eq.(A.4), we know the output is also a coherent state of the form: ˆ |α1 , α2 = |β1 , β2 , U with

β1 = tα1 + rα2 β2 = tα2 − rα1 .

(A.24)

(A.25)

The above relation can also be derived with the method discussed before. For a coherent state, we have [A.9]: ˆ |α = D(α)|0,

(A.26)

ˆ D(α) = exp(αˆ a − α∗ a ˆ† ).

(A.27)

with

Then, we have: ˆ |α1 , α2 = U ˆD ˆ 1 (α1 )D ˆ 2 (α2 )|0, 0 U ˆ ˆ ˆ 2 (α2 )U ˆ † |0, 0. = U D1 (α1 )D But, with ˆD ˆ 1 (α1 ) D ˆ 2 (α2 )U ˆ† U

(A.28)

242

A Lossless Beam Splitter

ˆ exp(α1 a ˆ† U ˆ1 − α∗1 a ˆ†1 + α2 a ˆ2 − α∗2 a ˆ†2 )U † † ∗ † ˆa ˆ − α1 U ˆa ˆ + α2 U ˆa ˆ † − α∗2 U ˆa ˆ †) exp(α1 U ˆ1 U ˆ1 U ˆ2 U ˆ†2 U † † a1 − rˆ a2 ) − α∗1 (tˆ a1 − rˆ a2 ) exp[α1 (tˆ a2 + rˆ a1 ) − α∗2 (tˆ a†2 + rˆ a†1 )] +α2 (tˆ ∗ ∗ † a1 − (tα1 + rα2 )ˆ a1 = exp[(tα1 + rα2 )ˆ +(tα2 − rα1 )ˆ a2 − (tα∗2 − rα∗1 )ˆ a†2 ] ˆ 1 (β1 )D ˆ 2 (β2 ), = D (A.29)

= = =

we have Eq.(A.24) and Eq.(A.25). Substituting Eq.(A.24) into Eq.(A.23) and making a change of variables from α to β by Eq.(A.25), we ﬁnd the output state as ρˆout = d2 β1 d2 β2 |β1 , β2 β1 , β2 |Pin (tβ1 − rβ2 , tβ2 + rβ1 ). (A.30) Therefore, we have: Pout (β1 , β2 ) = Pin (tβ1 − rβ2 , tβ2 + rβ1 ).

(A.31)

This is exactly the same function as in Ref.[A.5]. ˆ Derivation of an Explicit Expression for U ˆ bears some resemblance to angular momentum operThis derivation of U ators in rotation. If we note that t2 + r2 = 1, then we can assign t as cos θ and r as sin θ. And Eq.(A.8) becomes: ˆb1 = U ˆ †a ˆ = cos θˆ ˆ1 U a2 , a1 + sin θˆ (A.32) ˆb2 = U ˆ †a ˆ = cos θˆ ˆ2 U a1 , a2 − sin θˆ which is similar to the transformation of the two-dimensional rotation of angle θ. As in any transformation, we consider an inﬁnitesimal transformation of δθ << 1 and make a linear approximation of ˆ (δθ) ≈ 1 + iδθI, ˆ U

(A.33)

ˆU ˆ† = U ˆ †U ˆ = 1, ˆ2 to be determined. Because U where Iˆ is a function of a ˆ1 , a † ˆ ˆ ˆ we have I = I or I is a Hermitian operator. For the transformation of ﬁnite θ, we have: ˆ (θ) = U ˆ (θ/N )N U ˆ = lim (1 + iIθ/N )N N →∞

ˆ = exp(iθI).

(A.34)

References

243

Substituting Eq.(A.33) into Eq.(A.32), we obtain: ˆ = −iˆ ˆ = iˆ a2 , [ˆ a2 , I] a1 . [ˆ a1 , I]

(A.35)

From the commutators: [ˆ a1 , a ˆ†1 ] = [ˆ a2 , a ˆ†2 ] = 1, we have: ˆ†1 + iˆ a1 a ˆ†2 + f (ˆ a1 , a ˆ2 ). Iˆ = −iˆ a2 a

(A.36)

But Iˆ is an Hermitian operator, hence f (ˆ a1 , a ˆ2 ) = 0 and the ﬁnal expression ˆ is for U ˆ = exp[θ(ˆ ˆ†1 − a ˆ1 a ˆ†2 )], U a2 a

(A.37)

with t = cos θ, r = sin θ. The above expression is the same as that in Ref.[A.8], which is based on angular momentum theory.

References A.1 M. Born and E. Wolf, Principle of Optics, (Pergamon, Oxford, 1st ed., 1959; 7th ed., 1999). A.2 A. Zeilinger, Am. J. Phys., 49, 882 (1981). A.3 B. Yurke, S. L. McCall, and J. R. Klauder, Phys. Rev. A33, 4033 (1986). A.4 S. Prasad, M. O. Scully, and W. Martienssen, Opt. Commun. 62, 139 (1987). A.5 Z. Y. Ou, C. K. Hong, and L. Mandel, Opt. Commun. 63, 118 (1987); see also L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995). A.6 H. Fearn and R. Loudon, Opt. Commun. 64, 485 (1987); H. Fearn and R. Loudon, J. Opt. Soc. Am. B6, 917 (1989). A.7 Z. Y. Ou and L. Mandel, Am. J. Phys. 57, 66 (1989); K. Smiles Mascarenhas, Am. J. Phys. 59, 1150 (1991). A.8 R. A. Campos, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A40, 1371 (1989). A.9 R. J. Glauber, Phys. Rev. 130, 2529 (1963); 131, 2766 (1963). A.10 E. C. G. Sudarshan, Phys. Rev. Lett. 10, 277 (1963). A.11 C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987).

B Derivation of the Visibility for |kH , NV

B.1 The Case of |2H , NV Let us consider the next complicated case after |1H , NV , i.e., two H-photons in the input state of |2H , NV . There are two distinct scenarios: (1) the two Hphotons are in the same temporal mode and they are indistinguishable from m V-photons but are completely separated from other N − m V-photons; (2) the two H-photons are well-separated in time with one H-photon indistinguishable from the m V-photons and the other H-photon indistinguishable from the other n V-photons. The remaining N − m − n V-photons are completely separated from the m + n + 2 H- and V-photons. These two scenarios will give rise to diﬀerent visibility, as we calculate below. B.1.1 The Scenario of 2HmV + (N − m)V This is the ﬁrst scenario. The quantum state for the input to the NOON state measurement is given by |2HmV = dω1 dω2 ...dωN +2 Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ) a ˆ†H (ω1 )a†H (ω2 )ˆ a†V (ω3 )...ˆ a†V (ωN +2 )|vac,

(B.1)

Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ) = Φ(P {ω1 , ..., ωm+2 }, ωm+3 , ..., ωN +2 ),

(B.2)

with

where P is an arbitrary permutation operation. This expression is from the indistinguishability among the two H-photons and m V-photons. But because other N − m V-photons are well-separated from the 2H and mV photons, we have the orthogonal relation dωk dωj Φ∗ (ω1 , ω2 ; ω3 , ..., ωN +2 )Φ(Pkj {ω1 , ω2 ; ω3 , ..., ωN +2 }) = 0, (B.3)

246

B Derivation of the Visibility for |kH , NV

where Pkj interchanges ωk with ωj and 1 ≤ k ≤ m + 2, m + 3 ≤ j ≤ N + 2. Or, we can write similar relations in time domain: G(t1 , t2 ; t3 , ..., tN +2 ) = G(P {t1 , ..., tm+2 }, tm+3 , ..., tN +2 );

dtk dtj G∗ (t1 , t2 ; t3 , ..., tN +2 )G(Pkj {t1 , ..., tN +2 }) = 0,

(B.4)

(B.5)

where G(t1 , t2 ; t3 , ..., tN +2 ) is given in Eq.(9.34). The probability of joint photo-detection in the (N+2)-photon NOON state projection measurement is similar to Eq.(9.91). The ﬁeld operators are also similar to Eq.(9.92), but the phases are now changed to δj = 2π(j −1)/(N +2) with j = 1, ..., N + 2. After the expansion of the product, we write down only the terms that give non-zero contributions on the state in Eq.(B.1): ˆH (tk )...E ˆH (tj )...E ˆV (tN +2 )|2V mH. ei(δk +δj ) EˆV (t1 )...E (B.6) j>k

The part of the operators acting on the input state is given by ˆV (t1 )...EˆH (tk )...EˆH (tj )...EˆV (tN +2 )|2V mH E = G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 })|vac, where Pk1 exchanges tk with t1 and Pj2 exchanges tj with t2 , and G(t1 , t2 ; P {t3 , ..., tN +2 }), G2 (t1 , t2 ; t3 , ..., tN +2 ) = 2

(B.7)

(B.8)

P

where we used Eq.(B.4) for the exchange between t1 , t2 . Note that the Gfunction is symmetric among the variables {t3 , ..., tN +2 }. The joint (N + 2)photon detection probability is then proportional to 2 PN +2 = dt1 ...dtN +2 ei(δk +δj ) G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 }) . =

j>k j >k

j>k

e

i(δk +δj −δk −δj )

dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) ×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ).

The sum can be broken up into three parts as = A1 + A2 + A3 , j >k j>k

with the diagonal term equal equal to

(B.9)

(B.10)

B.1 The Case of |2H , NV

A1 =

,

247

(B.11)

j =j>k =k

and the cross terms to A2 =

+

j>k j >k j =j k =k

+

j>k j >k k =k j =j

+

j>k j >k j =k

,

(B.12)

j>k j >k k =j

and to

A3 =

.

(B.13)

j>k j >k j =j=k =k

The ﬁrst diagonal sum is straightforward: 2 A1 = dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) j>k

= 4NN +2 N !(N + 2)(N + 1)/2.

(B.14)

Here, NN +2 is similar to NN +1 in Eq.(9.100), but with Φ(ω1 ; P {ω2 , ..., ωN +1 }) replaced by Φ(ω1 , ω2 ; P {ω3 , ..., ωN +2 }). The second part A2 consists of four sums that all have only two equal indices among (k, j, k , j ), with one from {k, j} and the other from {k , j }. Therefore, the time integral part in Eq.(B.12) is the same as that in Eq.(9.105) and gives 4NN +2 m(N − 1)!. The sum over the phases can be evaluated as a2 ≡ ei(δk −δk ) + ei(δj −δj ) + j>k j >k j =j k =k

+

=

j>k j >k k =k j =j

ei(δj −δk ) +

j>k>2 j >k >2 j =k

ei(δk2 −δk1 ) = N

k3 =k2 =k1

ei(δk −δj )

j>k>2 j >k >2 k =j

ei(δk2 −δk1 ) .

(B.15)

k2 =k1

But similar to Eq.(9.106), we have: ei(δk2 −δk1 ) = −(N + 2),

(B.16)

k2 =k1

where the indices in the sum now run up to N + 2. So, we obtain: a2 = −N (N + 2), and

(B.17)

248

B Derivation of the Visibility for |kH , NV

A2 = 4NN +2 m(N − 1)!a2 = −4NN +2 m(N − 1)!N (N + 2). (B.18) The last term A3 involves a time integral with all four indices (k, j, k , j ) unequal. The time integral part can be evaluated as dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , tN +2 ) =4

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) dt1 ...dtN +2 G(tk , tj ; P {t3 , ..., t1 , ..., t2 , tN +2 }) P G∗ (tk , tj ; P {t3 , ..., t1 , ..., t2 , ..., tN +2 }). (B.19) × P

Because of the permutation properties in Eq.(B.4) and tk , tj = tk , tj , nonzero results for the time integration require tk , tj in the ﬁrst m location in P {t3 , ..., t1 , ..., t2 , ..., tN +2 }. The rest is arbitrary. This leads to a total of m(m − 1)(N − 2)! terms that are nonzero and equal in the sum over P . So, the time integral in Eq.(B.19) is 4NN +2 m(m − 1)(N − 2)!. On the other hand, the part of the sum over the phases is, for A3 : ei(δk +δj −δj −δk ) a3 ≡ j>k j >k j =j=k =k

=

ei(δk +δj −δj −δk ) −

j >k j>k

ei(δk2 −δk1 ) −

k3 =k2 =k1

1

j>k

= 0 − [−N (N + 2)] − (N + 2)(N + 1)/2 = (N + 2)(N − 1)/2, where we used j>k ei(δk +δj ) = 0. Hence, we obtain A3 as A3 = 4NN +2 m(m − 1)(N − 2)!a3 = 4NN +2 m(m − 1)(N − 2)!(N + 2)(N − 1)/2.

(B.20)

(B.21)

Finally, we have, after combining Eqs.(B.14, B.18, B.21): PN +2 (2HmV ) = 4NN +2 N !(N + 2)(N + 1)/2 − m(N − 1)!N (N + 2)+ +m(m − 1)(N − 2)!(N + 2)(N − 1)/2 (B.22) = 2NN +2 (N + 2)! 1 − VN +2 (2HmV ) , with VN +2 (2HmV ) =

m(m − 1) 2m − . N + 1 N (N + 1)

(B.23)

Again, the baseline for calculating the visibility is obtained when we set m = 0.

B.1 The Case of |2H , NV

249

B.1.2 The Scenario of 1HmV + 1HnV + (N − n − m)V In this scenario, there are three groups: the ﬁrst one consists of one H-photon and m V-photons that are indistinguishable; the second one is formed by the other H-photon and n V-photons and they are temporally indistinguishable; the rest of the V-photons make up the third group. Between diﬀerent groups, photons are well-separated in time and are completely distinguishable. The quantum state in this case is given in Eq.(B.1), but with the permutation symmetry: Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ) = PI PII Φ(ω1 , ω2 ; ω3 , ..., ωN +2 ),

(B.24)

where PI , PII are permutation operations that act on the groups of I = {ω1 ; ω3 , ..., ωm+2 } and II = {ω2 ; ωm+3 , ..., ωm+n+2 }, respectively. This symmetry relation indicates that one H-photon and m V-photons are completely indistinguishable in time and another H-photon and other n V-photons are also indistinguishable in time. The third group of frequency variables is denoted by III = {ωm+n+3, ..., ωN +2 }. Then, we may impose the orthogonal relation dωk dωj Φ∗ (ω1 , ω2 ; ω3 , ..., ωN +2 )Φ(Pkj {ω1 , ω2 ; ω3 , ..., ωN +2 }) = 0, (B.25) where Pkj interchanges ωk with ωj , and k, j belong to diﬀerent groups of I, II, III, respectively. This orthogonal relation ensures that photons of different groups are completely distinguishable in time. We can also write similar relations in time domain:

and

G(t1 , t2 ; t3 , ..., tN +2 ) = PI PII G(t1 , t2 ; t3 , ..., tN +2 ),

(B.26)

dtk dtj G∗ (t1 , t2 ; t3 , ..., tN +2 )G(Pkj {t1 , ..., tN +2 }) = 0,

(B.27)

where G(t1 , t2 ; t3 , ..., tN +2 ) has the form similar to Eq.(9.34). The permutation operators now act on time variables. Similar to Eq.(B.9), we may write the joint (N + 2)-photon detection probability, which is proportional to 2 ei(δk +δj ) G2 (Pk1 Pj2 {t1 , t2 ; t3 , ..., tN +2 }) . PN +2 = dt1 ...dtN +2 j>k i(δk +δj −δk −δj ) e = dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 }) j>k j >k

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 }).

But, diﬀerent from Eq.(B.8), we have, instead:

(B.28)

250

B Derivation of the Visibility for |kH , NV

G2 (t1 , t2 ; t3 , ..., tN +2 ) G(t1 , t2 ; P {t3 , ..., tN +2 }) + G(t2 , t1 ; P {t3 , ..., tN +2 }) . ≡

(B.29)

P

Similar to Eq.(B.10), the sum in Eq.(B.28) can be broken up into three parts as = A1 + A2 + A3 . (B.30) j>k j >k

The ﬁrst sum is straightforward: 2 dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) A1 = j>k

= 2NN +2 N !(N + 2)(N + 1)/2.

(B.31)

Note that because of the orthogonal relation in Eq.(B.27), the cross terms in the expansion of the absolute value in Eq.(B.31) give zero result. Similar to A2 , A2 consists of four sums that all have only two equal indices, with one from {k, j} and the other from {k , j }. From Eq.(B.29) for G2 , we then calculate the time integral part as dt1 ...dtN +2 G2 (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G2∗ (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) G(ts , tr ; P {...}) + G(tr , ts ; P {...}) dt1 ...dtN +2 P × G∗ (ts , tr ; P {...}) + G∗ (tr , ts ; P {...}) , (B.32) P

with r, r , s = k, j, k , j and r = r = s. Because of the orthogonal relation in Eq.(B.27), the time integrals of G(ts , tr ; P {...})G∗ (tr , ts ; P {...}) and G(tr , ts ; P {...}) G∗ (ts , tr ; P {...}) are zero. So, we have: dt1 ...dtN +2 G2 (ts , tr ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G ∗ (t , t ; t , ..., t1 , ..., t2 , ..., tN +2 ) 2 s r 3 G(ts , tr ; P {...}) G∗ (ts , tr ; P {...}) dt1 ...dtN +2 P

+

P

G(tr , ts ; P {...})

P

= (m + n)(N − 1)!NN +2 .

Hence, similar to Eq.(B.18), we obtain: A2 = NN +2 (m + n)(N − 1)!a2

G∗ (tr , ts ; P {...})

P

(B.33)

B.2 The Case of |3H , NV

= −NN +2 (m + n)(N − 1)!N (N + 2).

251

(B.34)

Noting that all four indices (k, j, k , j ) are unequal in the sum in A3 , we evaluate the time integral part as dt1 ...dtN +2 G2 (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) =

×G2∗ (tk , tj ; t3 , ..., t1 , ..., t2 , ..., tN +2 ) dt1 ...dtN +2 G(tk , tj ; P {...}) + G(tj , tk ; P {...}) P × G∗ (tk , tj ; P {...}) + G∗ (tj , tk ; P {...}) P

= 4mn(N − 2)!NN +2 .

(B.35)

Using a3 in Eq.(B.20) for the sum, we obtain A3 as A3 = 2NN +2 mn(N − 2)!(N + 2)(N − 1).

(B.36)

Finally, we have, after combining Eqs.(B.31, B.34, B.36): PN +2 (1HmV + 1HnV ) = NN +2 (N + 2)! − (m + n)(N − 1)!N (N + 2)+

+2mn(N − 2)!(N + 2)(N − 1)

= NN +2 (N + 2)!(1 − VN +2 ),

(B.37)

with VN +2 (1HmV + 1HnV ) =

2mn m+n − . N + 1 N (N + 1)

(B.38)

B.2 The Case of |3H , NV We now consider the even more complicated situation of three H-photons and N V-photons for the NOON state measurement. We start with the scenario when all three H-photons are indistinguishable. B.2.1 The Scenario of 3HmV + (N − m)V The quantum state for this case is given by |3HmV = dω1 dω2 ...dωN +3 Φ(ω1 , ω2 , ω3 ; ω4 , ..., ωN +3 ) a†V (ω4 )...ˆ a†V (ωN +3 )|vac, a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ with

(B.39)

B Derivation of the Visibility for |kH , NV

252

Φ(ω1 , ω2 , ω3 ; ω4 ..., ωN +3 ) = Φ(P {ω1 , ..., ωm+3 }, ωm+4 , ..., ωN +3 ),

(B.40)

where P is an arbitrary permutation operation. This expression is from the indistinguishability among the three H-photons and m V-photons. But, because other N − m V-photons are well separated from the 3HmV photons, we have the orthogonal relation (B.41) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj , and 1 ≤ k ≤ m + 3, m + 4 ≤ j ≤ N + 3. Or, we can write similar relations in time domain: G(t1 , t2 , t3 ; t4 , ..., tN +3 ) = G(P {t1 , ..., tm+3 }, tm+4 , ..., tN +3 ),

dtk dtj G∗ (t1 , t2 , t3 ; t4 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.42)

(B.43)

where G(t1 , t2 , t3 ; t4 , ..., tN +3 ) is similar to Eq.(9.34). The NOON state projection measurement has now (N+3) detectors and the phase shifts are δj = 2π(j−1)/(N +3) with j = 1, ..., N +3. The expression for PN +3 is similar to Eq.(9.91). After the expansion of the operator product, the terms with non-zero contributions are: ˆH (tl )...EˆH (tk )...EˆH (tj )...EˆV (tN +3 )|3HmV ˆV (t1 )...E ei(δk +δj +δl ) E j>k>l

(B.44) with ˆV (t1 )...EˆH (tl )...EˆH (tk )...E ˆH (tj )...E ˆV (tN +3 )|3HmV E = G3 (Pl1 Pk2 Pj3 {t1 , t2 , t3 ; t4 , ..., tN +3 })|vac, (B.45) where Pl1 exchanges tl with t1 , etc., and we have, due to Eq.(B.42): G(t1 , t2 , t3 ; P {t4 , ..., tN +2 }). G3 (t1 , t2 , t3 ; t4 , ..., tN +3 ) = 6

(B.46)

P

The joint (N + 3)-photon detection probability is then proportional to 2 PN +3 = dt1 ...dtN +3 ei(δl +δk +δj ) G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) j>k>l dt1 ...dtN +2 G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) = j >k >l j>k>l

×G3∗ (Pl 1 Pk 2 Pj 3 {t1 , ..., tN +3 })ei(δl +δk +δj −δl −δk −δj ) . (B.47)

B.2 The Case of |3H , NV

The sum can be broken up into four parts now as = C1 + C2 + C3 + C4 ,

253

(B.48)

j >k >l j>k>l

with

C1 =

,

(B.49)

j =j>k =k>l =l

C2 =

,

(B.50)

,

(B.51)

.

(B.52)

j>k>l j >k >l two pairs equal

C3 =

j>k>l j >k >l one pair equal

C4 =

j>k>l j >k >l no equal indices

The ﬁrst sum is straightforward: 2 dt1 ...dtN +3 G3 (tl , tk , tj ; t4 , ..., t1 , ..., t2 , ..., t3 , ..., tN +3 ) C1 = j>k>l

= 36NN +3 N !(N + 2)(N + 1)(N + 3)/6.

(B.53)

Here NN +3 is in a similar form as that of NN +1 in Eq.(9.100) and NN +2 in Eq.(B.14), but with Φ replaced by that in Eq.(B.39). In the sum of the second part C2 , the indices have two equal pairs among l, k, j, l , k , j with j > k > l, j > k > l , leaving two unequal indices, with one from {l, k, j} and the other from {l , k , j }. So, the time integral part is similar to that in Eq.(9.105) and gives 36NN +3 m(N − 1)!. On the other hand, for the sum of the phase terms, we have: ei(δl +δk +δj −δl −δk −δj ) = ei(δk2 −δk1 ) c2 ≡ j>k>l j >k >l two pairs equal

= [N (N + 1)/2]

k4 >k3 =k2 =k1

ei(δk2 −δk1 )

k2 =k1

= [N (N + 1)/2] × [0 − (N + 3)] = −N (N + 1)(N + 3)/2.

(B.54)

254

B Derivation of the Visibility for |kH , NV

So, we arrive at C2 = 36NN +3 m(N − 1)!c2 = −18NN +3 m(N − 1)!N (N + 1)(N + 3). (B.55) The third term C3 has one pair of equal indices and the time integral part has the form: dt1 ...dtN +3 G3 (ts , tr , tq ; t4 , ..., tN +2 )G3∗ (ts , tr , tq ; t4 , ..., tN +3 ) G(ts , tr , tq ; P {t4 , ..., tN +3 }) = 36 dt1 ...dtN +2 P × G∗ (ts , tr , tq ; P {t4 , ..., tN +3 }), (B.56) P

where q = q = r = r . Because of the permutation properties in Eq.(B.42) and tq = tq = tr = tr , non-zero results for the time integration require tq , tr in the ﬁrst m location in P {t4 , ..., tN +3 }. The rest is arbitrary. There are m(m−1)(N −2)! terms that are nonzero and equal in the sum over P . So, similar to Eq.(B.19), the time integral in Eq.(B.56) is 36NN +3 m(m − 1)(N − 2)!. On the other hand, we have, for the sum of the phase terms in C3 : ei(δl +δk +δj −δl −δk −δj ) = ei(δk4 +δk3 −δk2 −δk1 ) c3 ≡ j>k>l j >k >l one pair equal

k4 >k3 ,k2 >k1 k5 =k4 =k3 =k2 =k1

= (N − 1)

ei(δk4 +δk3 −δk2 −δk1 )

k4 >k3 ,k2 >k1

k3 =k2 =k1 k4 = = (N − 1) ei(δk4 +δk3 −δk2 −δk1 ) −

1

k4 =k2 >k3 =k1

k4 >k3 k2 >k1

−

ei(δk4 +δk3 −δk2 −δk1 )

%

k4 >k3 ,k2 >k1 one pair equal

= (N − 1){0 − (N + 3)(N + 2)/2 − [−(N + 3)(N + 1)]} = (N + 3)N (N − 1)/2.

(B.57)

Hence, we obtain C3 as C3 = 36NN +3 m(m − 1)(N − 2)!c3 = 18NN +3 m(m − 1)N !(N + 3).(B.58) Since all time variables are diﬀerent, the time integral in the last sum C4 simply gives 36NN +3 m(m − 1)(m − 2)(N − 3)!. The sum can be calculated, similar to Eq.(B.20), as % i(δl +δk +δj −δl −δk −δj ) e = c4 ≡ − c3 − c2 − c1 j>k>l j >k >l no equal indices

j >k >l j>k>l

B.2 The Case of |3H , NV

= 0 − (N − 1)N (N + 3)/2 − [−(N + 3)(N + 1)N/2] −(N + 3)(N + 2)(N + 1)/6 = −(N + 3)(N − 1)(N − 2)/6,

255

(B.59)

so that we have: C4 = −6NN +3 m(m − 1)(m − 2)(N − 3)!(N + 3)(N − 1)(N − 2). (B.60) Finally, we have, after combining Eqs.(B.53, B.55, B.58, B.60): PN +3 (3HmV ) = [(N + 3)! − 3m(N + 1)!(N + 3) + 3m(m − 1)N !(N + 3) −m(m − 1)(m − 2)(N − 1)!(N + 3)]6NN +3 = 6N (N + 3)!(1 − VN +3 ), (B.61) with VN +3 (3HmV ) =

3m 3m(m − 1) m(m − 1)(m − 2) − + . N + 2 (N + 1)(N + 2) N (N + 1)(N + 2)

(B.62)

B.2.2 The Scenario of 2HmV + 1HnV + (N − m − n)V The next scenario has one H-photon separated from the other two H-photons. Similar to the case in Sect.B.1.2, we can divide the photons into three groups: The ﬁrst (I) has the two H-photons and m V-photons and the second (II) has the one H-photon and n V-photons. Photons in each of the two groups are indistinguishable in time. The third group consists of the rest of the V-photons which are well-separated from the photons in groups I and II. The quantum state for this case is given by |2HmV 1HnV = dω1 dω2 ...dωN +3 Φ(ω1 , ω2 ; ω3 ; ω4 , ..., ωN +3 ) a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ a†V (ω4 )...ˆ a†V (ωN +3 )|vac,

(B.63)

with Φ(ω1 , ω2 ; ω3 ; ω4 ..., ωN +3 ) = Φ(PI PII {ω1 , ..., ωN +3 }),

(B.64)

where PI is a permutation operation acting on I ≡ {ω1 , ω2 ; ω4 , ..., ωm+3 } and PII on II ≡ {ω3 ; ωm+4 , ..., ωm+n+3 }, and the orthogonal relation (B.65) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj and ωk,j are not to be simultaneously in any one of subsets I, II, III with III ≡ {ωm+n+4 , ..., ωN +3 }. Or, we can write similar relations in time domain: G(t1 , ..., tN +3 ) = G(PI PII {t1 , ..., tN +3 }),

(B.66)

256

B Derivation of the Visibility for |kH , NV

dtk dtj G∗ (t1 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.67)

where G(t1 , ..., tN +3 ) is similar to Eq.(9.34). Similar to Eq.(B.44), to calculate joint (N+3)-photon detection probability, we need to evaluate ei(δk +δj +δl ) EˆV (t1 )...EˆH (tl )...EˆH (tk )... j>k>l

ˆH (tj )...EˆV (tN +3 )|2HmV 1HnV , ×E

(B.68)

where δj = 2π(j − 1)/(N + 3) and the operator part can be deduced as ˆV (t1 )...EˆH (tl )...E ˆH (tk )...EˆH (tj )...EˆV (tN +3 )|2HmV 1HnV E = G3 (Pl1 Pk2 Pj3 {t1 , t2 , t3 ; t4 , ..., tN +3 })|vac, (B.69) where Pl1 exchanges tl with t1 , etc., and we have, due to Eq.(B.66): G3 (t1 , t2 , t3 ; t 4 , ..., tN +3 ) =2 G(t1 , t2 ; t3 ; P {t4 , ..., tN +2 }) + G(t1 , t3 ; t2 ; P {t4 , ..., tN +2 }) P

+G(t3 , t2 ; t1 ; P {t4 , ..., tN +2 }) . (B.70) The joint (N + 3)-photon detection probability is then proportional to 2 ei(δl +δk +δj ) G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) PN +3 = dt1 ...dtN +3 j>k>l = dt1 ...dtN +2 G3 (Pl1 Pk2 Pj3 {t1 , ..., tN +3 }) j>k>l j >k >l

×G3∗ (Pl 1 Pk 2 Pj 3 {t1 , ..., tN +3 })ei(δl +δk +δj −δl −δk −δj ) . (B.71)

Similar to Eq.(B.48), the sum can be broken up into four parts as = C1 + C2 + C3 + C4 .

(B.72)

j >k >l j>k>l

We calculate each terms as follows: C1 = [(N + 3)(N + 2)(N + 1)/6] dt1 ...dtN +3 |G (t1 , ..., tN +3 )|2 = [(N + 3)(N + 2)(N + 1)/6]4 × 3(N !)NN +3 = 2NN +3 (N + 3)!. (B.73) C2 = c2

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.74)

B.2 The Case of |3H , NV

257

with q = q . With Eq.(B.70) for G3 (ts , tr , tq ; ...), we have the time integral as dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) G(ts , tr ; tq ; P {...}) + G(ts , tq ; tr ; P {...})+ = 4 dt1 ...dtN +3 P G∗ (ts , tr ; tq ; P {...})+ +G(tq , tr ; ts ; P {...}) P +G∗ (ts , tq ; tr ; P {...}) + G∗ (tq , tr ; ts ; P {...}) G(ts , tr ; tq ; P {...}) G∗ (ts , tr ; tq ; P {...})+ = 4 dt1 ...dtN +3 P P + G(ts , tq ; tr ; P {...}) G∗ (ts , tq ; tr ; P {...})+ P P ∗ G(tq , tr ; ts ; P {...}) G (tq , tr ; ts ; P {...}) . + P

P

(B.75) The ﬁrst term requires an exchange between tq , tq within the subset II while the second and third terms are the same and require exchange within the subset I. So, the time integral becomes: dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = 4(2m + n)(N − 1)!NN +3 .

(B.76)

With c2 in Eq.(B.54), we have: C2 = [−N (N + 1)(N + 3)/2] × 4(2m + n)(N − 1)!NN +3 = −2NN +3 (2m + n)(N + 1)!(N + 3). For C3 , we have: C3 = c3

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.77)

(B.78)

where ts = tr = tq = tr = tq . Expanding G (ts , tr , tq ; ...), we have the time integral as dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) G(ts , tr ; tq ; P {...}) + G(ts , tq ; tr ; P {...})+ = 4 dt1 ...dtN +3 P G∗ (ts , tr ; tq ; P {...})+ +G(tq , tr ; ts ; P {...}) P ∗ +G (ts , tq ; tr ; P {...}) + G∗ (tq , tr ; ts ; P {...})

258

B Derivation of the Visibility for |kH , NV

=4

dt1 ...dtN +3 4 G(ts , tr ; tq ; P {...}) G∗ (ts , tr ; tq ; P {...})+ P P G(tr , tq ; ts ; P {...}) G∗ (tq , tr ; ts ; P {...}) + P

= [4mn + m(m − 1)](N − 2)!4NN +3 .

P

(B.79)

Hence, from Eq.(B.57) for c3 , we obtain: C3 = [(N + 3)N (N − 1)/2] × 4[4mn + m(m − 1)](N − 2)!NN +3 = 2NN +3 [4mn + m(m − 1)]N !(N + 3). (B.80) For C4 , we have: C4 = c4

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.81)

where none of the time variables are equal. Substituting G3 from Eq.(B.70) and making an expansion of the product, we ﬁnd that all the nine terms are the same and equal to m(m − 1)n(N − 3)!NN +3 so that the integral is 4 × 9m(m − 1)n(N − 3)!NN +3 . With Eq.(B.59) for c4 , we obtain: C4 = [−(N + 3)(N − 1)(N − 2)/6]4 × 9m(m − 1)n(N − 3)!NN +3 = −6m(m − 1)n(N − 1)!(N + 3). (B.82) Combining Eqs.(B.73, B.77, B.80, B.82), we have: PN +3 (2HmV 1HnV ) = 2NN +3 [(N + 3)! − (2m + n)(N + 1)!(N + 3) +[4mn + m(m − 1)]N !(N + 3) −3m(m − 1)n(N − 1)!(N + 3)] = 2NN +3 (N + 3)![1 − VN +3 (2HmV 1HnV )],

(B.83)

with VN +3 (2HmV 1HnV ) 2m + n 4mn + m(m − 1) 3nm(m − 1) = − + . N +2 (N + 1)(N + 2) N (N + 1)(N + 2)

(B.84)

B.2.3 The Scenario of 1HmV + 1HnV + 1HpV + (N −m−n−p)V The three H-photons in this scenario are all well-separated. Depending on the temporal overlap, we divide all the photons into four groups with the ﬁrst one (I) containing m V-photons with one H-photon in one temporal mode, the second one (II) having n V-photons with one H-photon, and the third one (III) p V-photons with one H-photon. The rest of the V-photons belong to the fourth group (IV). The V-photons in the fourth group (IV) may or may not be indistinguishable in time.

B.2 The Case of |3H , NV

259

As before, we write the quantum state for this case as |1HmV 1HnV 1HpV = dω1 dω2 ...dωN +3 Φ(ω1 ; ω2 ; ω3 ; ω4 , ..., ωN +3 ) a ˆ†H (ω1 )a†H (ω2 )a†H (ω3 )ˆ a†V (ω4 )...ˆ a†V (ωN +3 )|vac,

(B.85)

with Φ(ω1 ; ω2 ; ω3 ; ω4 ..., ωN +3 ) = Φ(PI PII PIII {ω1 , ..., ωN +3 }),

(B.86)

where the permutation PI acts on the ﬁrst group I ≡ {ω1 ; ω4 , ..., ωm+3 }, PII on the second group II ≡ {ω2 ; ωm+4 , ..., ωm+n+3 }, and PIII on the third group III ≡ {ω3 ; ωm+n+4 , ..., ωm+n+p+3 }. Distinguishability among diﬀerent groups is guaranteed by the orthogonal relation (B.87) dωk dωj Φ∗ (ω1 , ..., ωN +3 )Φ(Pkj {ω1 , ..., ωN +3 }) = 0, where Pkj interchanges ωk with ωj and ωk,j are not to be simultaneously in any one of groups I, II, III, IV with IV ≡ {ωm+n+p+4 , ..., ωN +3 }. Or, we can write similar relations in time domain: G(t1 , ..., tN +3 ) = G(PI PII PIII {t1 , ..., tN +3 });

(B.88)

dtk dtj G∗ (t1 , ..., tN +2 )G(Pkj {t1 , ..., tN +3 }) = 0,

(B.89)

where G(t1 , ..., tN +3 ) is similar to Eq.(9.34). The joint (N+3)-photon detection probability from the NOON state projection can be evaluated in the same way as in Eqs.(B.47, B.71), but with the following G3 function: G3 (t1 , t2 , t3 ; t4 ,..., tN +3 ) G(t1 ; t2 ; t3 ; P {t4 , ..., tN +2 }) + G(t1 ; t3 ; t2 ; P {t4 , ..., tN +2 }) = P

+G(t2 ; t3 ; t1 ; P {t4 , ..., tN +2 }) + G(t2 ; t1 ; t3 ; P {t4 , ..., tN +2 }) +G(t3 ; t2 ; t1 ; P {t4 , ..., tN +2 }) + G(t3 ; t1 ; t2 ; P {t4 , ..., tN +2 }) . (B.90) Again, we break up the sum into four parts as = C1 + C2 + C3 + C4 .

(B.91)

j >k >l j>k>l

We calculate each terms as follows: C1 = [(N + 3)(N + 2)(N + 1)/6]

dt1 ...dtN +3 |G (t1 , ..., tN +3 )|2

260

B Derivation of the Visibility for |kH , NV

= [(N + 3)(N + 2)(N + 1)/6]6N !NN +3 = NN +3 (N + 3)!.

(B.92)

Note that the time integral is diﬀerent due to the diﬀerent G3 and, similarly: C2 = c2 dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...), (B.93) with q = q . With Eq.(B.90) for G (ts , tr , tq ; ...), it is straightforward to derive the time integral using the method described previously. Then, it becomes dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = (2m + 2n + 2p)(N − 1)!NN +3 ,

(B.94)

so that we have, with Eq.(B.54) for c2 : C2 = [−N (N + 1)(N + 3)/2] × (2m + 2n + 2p)(N − 1)!NN +3 = −NN +3 (m + n + p)(N + 1)!(N + 3). (B.95) For C3 , we have: C3

= c3

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.96)

where ts = tr = tq = tr = tq . The time integral can be evaluated in a way similar to Eq.(B.79), but with six terms for G3 in Eq.(B.90). The result is dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...) = 4(mn + mp + np)(N − 2)!NN +3 .

(B.97)

Hence, from Eq.(B.57) for c3 , we obtain: C3 = [(N + 3)N (N − 1)/2] × 4(mn + mp + np)(N − 2)!NN +3 = 2NN +3 (mn + mp + np)N !(N + 3). (B.98) For C4 we have C4 = c4

dt1 ...dtN +3 G3 (ts , tr , tq ; ...)G3∗ (ts , tr , tq ; ...),

(B.99)

where none of the time variables are equal. The 36 terms in the expansion of the product of the G3 function in Eq.(B.90) are all same. So, we obtain: C4 = [−(N + 3)(N − 1)(N − 2)/6] × 36mnp(N − 3)!NN +3 = −6mnp(N − 1)!(N + 3)NN +3 , (B.100) where Eq.(B.59) is used for c4 . Combining Eqs.(B.92, B.95, B.98, B.100), we have:

B.3 The General Case of |kH , NV

261

PN +3 (1HmV 1HnV 1HpV ) = NN +3 [(N + 3)! − (m + n + p)(N + 1)!(N + 3) +2(mn + mp + np)N !(N + 3) −6mnp(N − 1)!(N + 3)] = NN +3 (N + 3)![1 − VN +3 (1HmV 1HnV 1HpV )], (B.101) with VN +3 (1HmV 1HnV 1HpV ) 6mnp m + n + p 2(mn + mp + np) − + . (B.102) = N +2 (N + 1)(N + 2) N (N + 1)(N + 2)

B.3 The General Case of |kH , NV We start with the case where all k H-photons are indistinguishable in the state of |kHmV (N − m)V . Before we deal with arbitrary k, let us go one step further from the current result of k = 3 to a k = 4 case. This step is straightforward from Sect.B.2.1. We present only the result as follows. The (N+4)-photon joint detection probability can be derived as (B.103) PN +4 (4HmV ) = 4!(N + 4)!NN +4 1 − VN +4 (4HmV ) , with the visibility as VN +4 (4HmV ) =

6m(m − 1) 4m(m − 1)(m − 2) 4m − + − N + 3 (N + 3)(N + 2) (N + 3)(N + 2)(N + 1) m(m − 1)(m − 2)(m − 3) − . (B.104) (N + 3)(N + 2)(N + 1)N

If we check the coeﬃcients in each term in Eqs.(B.23, B.62, B.104) for k = 2, 3, 4, we ﬁnd that they are simply the binomial coeﬃcients of Ckj = k!/(k − j)!j!. So, we can generalize the expression to an arbitrary k and the (N+k)photon joint detection probability of the NOON state measurement thus has the form of (B.105) PN +k (kHmV ) = (N + k)!k!NN +k 1 − VN +k (kHmV ) , with the visibility as VN +k (kHmV ) =

k l=1

(−1)l−1 Ckl

m(m − 1)...(m − l + 1) . (N + k − 1)...(N + k − l)

(B.106)

Actually, the derivation of the above result is relatively straightforward if we follow the case of k = 3 in the previous section. Like the k = 3 case, the

B Derivation of the Visibility for |kH , NV

262

(N + k)-photon joint detection probability can be broken up into k + 1 parts and is written as PN +k (kHmV ) =

k

dk−l Tk−l ,

(B.107)

l=0

where dl is the sum of the phase part with l pairs of equal indices, while Tl is its corresponding time integral part. It is easy to see that for the state of |kHmV , we obtain Tk−l as Tk = N !(k!)2 NN +k

(l = 0, or k pairs of equal indices),

(B.108)

and Tk−l = m(m − 1)...(m − l + 1)(N − l)!(k!)2 NN +k

(l ≥ 1). (B.109)

For dk−l with l = 0, we have, obviously: k dk = CN +k = (N + k)(N + k − 1)...(N + 1)/k!.

(B.110)

However, the calculation of dk−l with l ≥ 1 is not so trivial. We can obtain it following the same line of argument that leads to the values of the a and c coeﬃcients in previous sections, but we present only its value here: dk−l = (−1)l (N + k)

(N + k − l − 1)! . (N − l)!(k − l)!l!

(B.111)

Substituting the above expressions into Eq.(B.107), we obtain the expression in Eq.(B.104). Notice that Tk−l is state-dependent while dk−l is not. Therefore, it all comes down to ﬁnding Tk−l for the states in diﬀerent scenarios. The most general scenario is when the k H-photons are separated into r groups by the relation k = k1 + k2 + .. + kr with each group having mi (i = 1, ..., r) V-photons overlapping in time (m1 + m2 + ... + mr ≤ N ). It is very complicated to derive Tk−l for this general case. We will derive it only for the r = 2 case as an example and present the general result at the end. B.3.1 The Scenario of k1 V mH + k2 V nH In this scenario, we can group the variables in the Φ-function into three subsets, as in Sect.B.2.2: the ﬁrst includes the k1 H-photons and the m V-photons, the second includes the k2 H-photons and the n V-photons, and the third includes the remaining V-photons. Because of the permutation symmetry within the ﬁrst and the second groups, we may write the Gk -function as Gk (t1 , ..., tk 1 ; tk 1 +1 , ..., tk1 +k2 ; tk+1 , ..., tk+N ) ≡ G(Pk {t1 , ..., tk1 ; tk1 +1 , ..., tk1 +k2 }; PN {tk+1 , ..., tk+N }) Pk PN

B.3 The General Case of |kH , NV

= k1 !k2 !

G(Ck1 k2 {t1 , ..., tk1 +k2 }; PN {...}),

263

(B.112)

Ck1 k2 PN

where Pk is the permutation on the k variables for the k H-photons and PN for the N V-photons. Ck1 k2 is the combination operation of taking k1 variables out of the ﬁrst k = k1 + k2 variables in the G-function and putting them the ﬁrst k1 positions of the G-function and remaining variables in the remaining positions. There are a total of (k1 + k2 )!/k1 !k2 ! diﬀerent terms in the sum of . It is straightforward to ﬁnd for l = 0: Ck k 1 2

2 dt1 ...dtk+N Gk (t1 , ..., tk+N ) 2 2 (k1 + k2 )! dt1 ...dtk+N = (k1 !k2 !) G(...; PN {...}) k1 !k2 !

Tk =

= NN +k N !k1 !k2 !(k1 + k2 )!

PN

(l = 0),

(B.113)

where NN +k is similar to NN +1 in Eq.(9.100), but with N + k variables in the Φ-function. Tk−1 is the time integral part of the term in the sum in Eq.(B.107) with only one unequal pair {s, s } of indices in its corresponding phase sum, similar to that in Eq.(B.50). It has the form of Tk−1 = dt1 ...dtk+N Gk∗ (..., ts , ...; tk+1 , ..., tk+N )Gk (..., ts , ...; tk+1 , ..., tk+N ) 2 dt1 ...dtk+N = (k1 !k2 !) ∗ ∗ × G (..., ts ; ...; PN {...}) + G (...; ts , ...; PN {...}) PN

×

PN

C(k1 −1)k2

C(k1 −1)k2

Ck1 (k2 −1)

G(..., t

s

; ...; PN {...})

+

G(...; t

s

,

, ...; PN {...})

Ck1 (k2 −1)

(B.114) where we write Gk∗ , Gk in two parts: ts( ) among the ﬁrst k1 variables of the G function and ts( ) among the next k2 variables of the G function. After expanding the product, all the cross terms are zero because the indices in these terms belong to groups I and II, respectively, and they are orthogonal. The non-zero direct product terms are evaluated in the same way as those in Eqs.(9.105,B.33), although results of the integration are diﬀerent for s( ) in group I and in group II. The ﬁnal result is then, for l = 1: (k1 + k2 − 1)! Tk−1 = (k1 !k2 !)2 m(N − 1)!NN +k (k1 − 1)!k2 ! (k1 + k2 − 1)! n(N − 1)!N + N +k (k2 − 1)!k1 ! = (k1 + k2 − 1)!k1 !k2 !(k1 m + k2 n)(N − 1)!NN +k . (B.115)

264

B Derivation of the Visibility for |kH , NV

For l = 2, we have: Tk−2 = dt1 ...dtk+N Gk∗ (..., ts , ..., tr , ...; tk+1 , ..., tk+N ) ×Gk (..., ts , ..., tr , ...; tk+1 , ..., tk+N ),

(B.116)

with s = s = r = r . Because of Eq.(B.112), we may break Gk in Eq.(B.116) into three parts, similar to Eq.(B.114): Gk∗ (..., ts , ...,tr , ...; tk+1 , ..., tk+N ) G(..., ts , tr ; ...; PN {...}) = PN

C(k1 −2),k2

+

G(..., ts ; tr , ...; PN {...}) + G(..., tr ; ts , ...; PN {...})

C(k1 −1),(k2 −1)

+

% G(...; ts , tr , ...; PN {...}) .

(B.117)

Ck1 ,(k2 −2)

Substituting Eq.(B.117) into Eq.(B.116), we ﬁnd again that only the direct product terms give non-zero integration, which is diﬀerent for terms in diﬀerent parts. The evaluations of the three diﬀerent time integrals are similar to those in Eqs.(B.19, B.35). We then obtain: (k1 + k2 − 2)! Tk−2 = (k1 !k2 !)2 m(m − 1)(N − 2)!NN +k (k1 − 2)!k2 ! (k1 + k2 − 2)! 4mn(N − 2)!NN +k + (k1 − 1)!(k2 − 1)! (k1 + k2 − 2)! n(n − 1)(N − 2)!NN +k + k1 !(k2 − 2)! = [k1 (k1 − 1)m(m − 1) + 4k1 k2 mn + k2 (k2 − 1)n(n − 1)] ×(k1 + k2 − 2)!k1 !k2 !(N − 2)!NN +k . (B.118) Similarly, we have, for l = 3: (k1 + k2 − 3)! (k1 + k2 − 3)! (3) m + (C 1 )2 m(2) n(1) Tk−3 = (k1 !k2 !)2 (k1 − 3)!k2 ! (k1 − 2)!(k2 − 1)! 3 (k1 + k2 − 3)! (k1 + k2 − 3)! (3) (C32 )2 m(1) n(2) + n + (k1 − 1)!(k2 − 2)! k1 !(k2 − 3)! (B.119) ×(N − 3)!NN +k , where m(i) ≡ m(m − 1)...(m − i + 1) and m(0) = 1, and furthermore, for arbitrary l ≤ k = k1 + k2 : Tk−l = (N − l)!NN +k (k1 !k2 !)2 2 l l! (k1 + k2 − l)! × m(i) n(l−i) , (B.120) (k1 − i)!(k2 − l + i)! i!(l − i)! i=0

where, as before, the sum comes directly from the break-up of the Gk -function.

B.3 The General Case of |kH , NV

265

B.3.2 The Most General Scenario It is straightforward to generalize Eq.(B.120) to the most general scenario k1 Hm1 V...kr Hmr V , as described right before Sect.B.3.1. We present only the result as (k1 + ... + kr )! NN +k N ! k1 !...kr ! = NN +k N !k1 !k2 !...kr !(k1 + ... + kr )! (l = 0),

Tk = (k1 !k2 !...kr !)2

(B.121)

and for l ≥ 1, l

Tk−l = (k1 !k2 !...kr !)2

i1 ...ir i1 +...+ir =l

2 (k1 + ... + kr − l)! l! (k1 − i1 )!...(kr − ir )! i1 !...ir ! (i )

r) ×m1 1 ...m(i r (N − l)!NN +k .

(B.122)

After substituting the above results into Eq.(B.107), we obtain: PN +k (k1 Hm1 V...kr Hmr V ) = (N + k)!k1 !...kr !NN +k [1 − VN +k ], (B.123) with the visibility as VN +k =

k l=1

(−1)l−1

l i1 ...ir i1 +...+ir =l

(i ) (i ) Cki11 ...Ckirr m1 1 ...mr r l! . (B.124) i1 !...ir ! (N + k − 1)...(N + k − l)

Index

asymmetric beam splitter, 159, 169, 208–213 beam-like PDC, 33 Bell measurement, 69 Bell states, 68 de Broglie wavelength four-photon, 171 N-photon, 163, 174 three-photon, 169 two-photon, 95 dispersion cancellation, 55–56 distinguishability N-photon, 189 three-photon, 200 two pairs, 132, 149–155, 182, 187 two-photon, 186 entanglement swapping, 147 evolution operator, 18, 24–26 beam splitter, 167, 238, 242 parametric ampliﬁer, 102, 165 exchange symmetry, 181, 188, 189, 200, 201, 205 four-photon interference, 156 four-wave mixing, 19, 20 Franson interferometer, 89–95, 99 ghost fringes, 78–81 Glauber-Sudarshan P-representation, 11, 241

indistinguishability N-photon, 189 three-photon, 200 two pairs, 132, 182, 187 two-photon, 186 NOON state four-photon, 166 N-photon, 163, 166–168 three-photon, 163–165 two-photon, 95 NOON state projection measurement, 172, 173, 195 normalization factor, 188–190, 192, 202 orthogonality of exchange, 182, 189, 200, 201 phase matching, 20, 21, 27–32, 41 extended, 31, 54 type-I, 19, 24, 27, 28, 30, 41 type-II, 19, 24, 27, 30, 31, 33, 41 photon anti-bunching eﬀect, 67, 101–104, 106 photon bunching eﬀect multi-photon, 177–181, 195 PDC, 122–125 two pairs, 159, 194, 195 two-photon, 60, 128, 142, 154 quantum state of PDC, 18, 26 broad band, 39 entangled, 33 four-photon, 26

268

Index

frequency entangled, 70 narrow band, 34 polarization entangled, 3, 19, 63 time entangled, 89 stimulated emission, 177–181 pair, 121, 125 teleportation, 142–147 three-wave mixing, 19 transform-limitedness, 115–118 single-photon, 118–121, 232, 235 two-photon wave, 7, 86, 88–92, 94–96, 99

visibility of interference classical maximum, 12 four-photon, 142, 147, 148, 154, 162, 171, 177, 248 N-photon, 204–208, 212, 213, 251, 255, 258, 261, 265 three-photon, 170, 199 two-photon, 50, 55, 57, 60, 66, 77, 85, 87, 99, 127, 129, 130, 140 coherent state, 8 Franson interferometer, 94 thermal source, 8, 127, 130, 140 type-I, 50 type-II, 51 wide band pumping, 52–55