Two-photon interference : spatial aspects of two-photon entanglement, diffraction, and scattering
Peeters, W.H.
Citation
Peeters, W. H. (2010, December 21). Two-photon interference : spatial
aspects of two-photon entanglement, diffraction, and scattering. Casimir PhD Series. Retrieved from https://hdl.handle.net/1887/16264
Version: Not Applicable (or Unknown)
License: Licence agreement concerning inclusion of doctoral thesis in the Institutional Repository of the University of Leiden
Downloaded from: https://hdl.handle.net/1887/16264
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Spatial aspe ts of two-photon entanglement,
dira tion, and s attering
Copyright c 2010 by W. H. Peeters, Leiden, the Netherlands.
Publisher: Casimir Research School, Lorentzweg 1, 2628 CJ Delft, The Netherlands.
ISBN: 97890 8593 089-1
Print: Gildeprint Drukkerijen - The Netherlands Cover Design: Martijn van Leipsig
Cover: The background is an artist impression of two photons having both wave-like and particle-like properties. The figures are experimental two-photon interference patterns recorded in the far field of a double slit. The color scale corresponds to the rate of simultaneous photon detections by two single-photon detectors (black is zero). The horizontal and vertical axes correspond to the positions of the detectors measured along a transverse line perpendicular to the orientation of the slits.
Spatial aspe ts of two-photon entanglement,
dira tion, and s attering
PROEFSCHRIFT
ter verkrijging van
de graad van doctor aan de Universiteit Leiden,
op gezag van rector magnificus prof. mr. P. F. van der Heijden, volgens besluit van het College voor Promoties
te verdedigen op dinsdag 21 december 2010 klokke 13:45 uur
door
Wouter Herman Peeters
geboren te Berg en Terblijt, Nederland
op 27 februari 1980
Promotiecommissie:
Promotor: Prof. dr. J. P. Woerdman Universiteit Leiden Copromotor Dr. M. P. van Exter Universiteit Leiden Leden: Prof. dr. C. W. J. Beenakker Universiteit Leiden Dr. M. J. A. de Dood Universiteit Leiden
Prof. dr. A. Lagendijk Universiteit van Amsterdam (UvA) en Universiteit Twente
Prof. dr. G. Nienhuis Universiteit Leiden Prof. dr. J. M. van Ruitenbeek Universiteit Leiden Prof. dr. J. P. Torres Universitat Polit`ecnica
de Catalunya (UPC)
Dr. V. Zwiller Technische Universiteit Delft
Paranimfen: ir. R. van Melle en drs. O. A. Tuinenburg
The research described in this thesis is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the
Netherlands Organisation for Scientific Research (NWO).
The research has been carried out at the
Quantum Optics and Quantum Information group, which is part of the Leiden Institute of Physics (LION) of the Faculty of Science of Leiden University.
Additionally, the academic environment has been supported by the Casimir Research School, which is supported by LION at Leiden University and
the Kavli Institute of Nanoscience at Delft University of Technology.
An electronic version of this dissertation is available at the Leiden University Repos- itory (https://openaccess.leidenuniv.nl).
Casimir PhD series, Delft-Leiden, 2010-29
Voor Ton, Wilma
en Janneke
1 Introduction 1
1.1 Interference in optics . . . 1
1.1.1 Young’s experiment: interference of waves . . . 1
1.1.2 One-photon and two-photon interference . . . 2
1.1.3 Spontaneous parametric down-conversion: a source of pairs of entangled photons . . . 4
1.2 Research topics in this theses . . . 5
1.2.1 Two-photon interference: three themes . . . 5
1.2.2 Theme 1: Orbital angular momentum entanglement . . . 5
1.2.3 Theme 2: Two-photon diffraction from a double slit . . . 6
1.2.4 Theme 3: Two-photon scattering . . . 8
1.3 Quantum description of SPDC light . . . 9
1.3.1 Motivation . . . 9
1.3.2 Two approaches to describe SPDC light . . . 9
1.3.3 Two-photon field . . . 10
1.3.4 Expression for the two-photon field . . . 12
1.3.5 Klyshko picture . . . 13
1.4 Thesis outline . . . 14
2 Orbital angular momentum analysis of high-dimensional entangle- ment 17 2.1 Introduction . . . 18
2.2 Continuous two-photon amplitude . . . 19
2.2.1 Generated two-photon amplitude . . . 19
2.2.2 Interference after image rotation . . . 20
2.3 Discrete modal analysis . . . 24
2.3.1 Schmidt decomposition of the detected two-photon amplitude 24 2.3.2 Modal decomposition and the Schmidt number . . . 27
2.3.3 The physical significance of V (θ) . . . 28
2.4 Experimental results . . . 29
2.4.1 Experimental setup . . . 29 vii
2.4.2 Spiral phase plate . . . 32
2.4.3 Alignment . . . 33
2.4.4 Experimental results for detection through circular apertures 34 2.4.5 Experimental results for l = 1 detection . . . 38
2.5 Concluding discussion . . . 40
2.6 Acknowledgements . . . 41
3 Optical characterization of periodically poled crystals 43 3.1 Introduction . . . 44
3.2 Phase matching in a periodically poled crystal . . . 45
3.3 Experimental apparatuses . . . 47
3.4 Experimental results . . . 51
3.4.1 Maker fringes in angular intensity pattern of SPDC light . . . 51
3.4.2 Maker fringes in spectrum of SPDC light . . . 55
3.4.3 Maker fringes in temperature dependence of SHG . . . 55
3.5 Interpretation of Maker fringes in terms of poling quality . . . 57
3.5.1 Fourier analysis of small and slowly varying deformations of the poling structure . . . 57
3.5.2 Analysis of Maker fringes in terms of poling quality . . . 61
3.6 Conclusions . . . 63
3.7 Acknowledgements . . . 64
4 Engineering of two-photon spatial quantum correlations behind a double slit 65 4.1 Introduction . . . 66
4.2 Theory: Two-photon state engineering . . . 68
4.2.1 Electromagnetic field behind double slit . . . 68
4.2.2 Quantum state engineering . . . 69
4.2.3 Incident two-photon state . . . 71
4.2.4 Near-field imaging scheme . . . 73
4.2.5 Far-field imaging scheme . . . 75
4.3 Theory: Interference behind the double slit . . . 76
4.3.1 State determination by two-photon interference . . . 76
4.3.2 Interpretation of the two-qubit Bloch sphere . . . 78
4.3.3 One-photon interference . . . 79
4.3.4 Duality between unconditional one-photon interference and two-photon interference . . . 80
4.4 Experimental apparatus . . . 81
4.4.1 General information . . . 81
4.4.2 Experimental apparatus in front of double slit . . . 81
4.4.3 Experimental apparatus behind double slit . . . 83
4.5 Experimental results . . . 85
4.5.1 General information . . . 85 viii
4.5.2 Analysis and tuning of two-photon interference patterns . . . 85
4.5.3 Near-field imaging at ϕ = 0 . . . 88
4.5.4 Far-field imaging at ϕ = 0 . . . 89
4.5.5 Near-field imaging at nonzero curvature phase . . . 91
4.5.6 Far-field imaging at nonzero curvature phase . . . 92
4.6 Conclusion . . . 93
4.7 Discussion: tuning in high-dimensional Hilbert space . . . 94
4.8 Acknowledgements . . . 95
4.9 APPENDIX: Validity of the model for various phase-matching ge- ometries . . . 95
5 Observation of two-photon speckle patterns 99 5.1 Introduction . . . 100
5.2 Theory . . . 100
5.3 Experimental setup . . . 103
5.4 Experimental results . . . 105
5.5 Discussion . . . 106
5.6 Conclusion . . . 107
5.7 Acknowledgements . . . 107
5.8 APPENDIX: Derivation of the two-photon speckle autocorrelation function . . . 107
6 Spatial pairing and antipairing of photons in random media 111 6.1 Introduction . . . 112
6.2 Random unitary scattering of labeled photons . . . 112
6.3 Experiments . . . 116
6.3.1 Experimental scheme . . . 116
6.3.2 Details of experimental apparatus . . . 118
6.3.3 Experimental results . . . 120
6.4 Concluding discussion . . . 123
6.5 Outlook . . . 124
6.6 Acknowledgements . . . 125
6.7 APPENDIX: Scattered density matrix . . . 125
Bibliography 129
List of publications 139
Nederlandse samenvatting 141
Nawoord 147
Curriculum vitae 149
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