Nano Optics and Atomics: Transport of Light and Matter Waves International School of Physics ''Enrico Fermi''

` ITALIANA DI FISICA SOCIETA RENDICONTI DELLA SCUOLA INTERNAZIONALE DI FISICA “ENRICO FERMI” CLXXIII Corso a cura di...

Author: Wiersma D. | Kaiser R. | Fallani L. (eds.)

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` ITALIANA DI FISICA SOCIETA

RENDICONTI DELLA

SCUOLA INTERNAZIONALE DI FISICA “ENRICO FERMI”

CLXXIII Corso a cura di R. Kaiser e D. S. Wiersma Direttori del Corso e di L. Fallani

VARENNA SUL LAGO DI COMO VILLA MONASTERO

23 Giugno – 3 Luglio 2009

Nano-ottica e ﬁsica atomica: trasporto di luce e onde di materia 2011

` ITALIANA DI FISICA SOCIETA BOLOGNA-ITALY

ITALIAN PHYSICAL SOCIETY

PROCEEDINGS OF THE

INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

Course CLXXIII edited by R. Kaiser and D. S. Wiersma Directors of the Course and L. Fallani

VARENNA ON LAKE COMO VILLA MONASTERO

23 June – 3 July 2009

Nano Optics and Atomics: Transport of Light and Matter Waves 2011

AMSTERDAM, OXFORD, TOKIO, WASHINGTON DC

c 2011 by Societ` Copyright a Italiana di Fisica All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISSN 0074-784X (print) ISSN 1879-8195 (online) ISBN 978-1-60750-755-0 (print) (IOS Press) ISBN 978-1-60750-756-7 (online) (IOS Press) ISBN 978-88-7438-061-9 (SIF) LCCN 2011927441

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INDICE

R. Kaiser, D. S. Wiersma and L. Fallani – Preface . . . . . . . . . . . . . . . .

pag. XIII

Gruppo fotograﬁco dei partecipanti al Corso . . . . . . . . . . . . . . . . . . . . . . . . . .

XVI

P. W¨ olfle – Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1. Basic notions of localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Introductory remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Electrons and classical waves in disordered systems . . . . . . . . . . . . . . . 1 3. Strong localization and Anderson transition . . . . . . . . . . . . . . . . . . . . . . 1 4. Weak localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5. One-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6. Quasi–one-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Theory of localization: fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Thouless conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Scaling theory of the conductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Renormalization group equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4. Critical exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. Dynamical scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6. Symmetry classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 7. Non-linear σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8. Fractal structure of critical wave functions . . . . . . . . . . . . . . . . . . . . . . . 2 9. Anderson transition in the kicked rotor model . . . . . . . . . . . . . . . . . . . 3. Theory of localization: diagrammatic approaches . . . . . . . . . . . . . . . . . . . . . . 3 1. Self-consistent theory of localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Results of the self-consistent theory of localization . . . . . . . . . . . . . . . . 3 3. Destruction of localization by decoherence processes . . . . . . . . . . . . . . 4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 3 5 7 7 8 9 9 10 11 13 14 14 15 16 17 17 20 21 22

P. J. Steinhardt – Photonic properties of non-crystalline solids . . . . . . . .

25

1. 2. 3. 4. 5. 6.

Some basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Photonic band gaps in icosahedral quasicrystals . . . . . . . . . . . . . . . . . . . . . . Dependence of band gap width on symmetry in 2D . . . . . . . . . . . . . . . . . . . Finding optimal complete band gaps for 2D photonic quasicrystals . . . . . . Isotropic disordered photonic materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26 31 34 37 41 48 VII

indice

VIII

R. C. Mesquita and A. G. Yodh – Diﬀuse optics: Fundamentals and tissue applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag.

51

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Light transport tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Dynamic light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4. Multiple light scattering in tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. Simple solutions of the photon diﬀusion equation . . . . . . . . . . . . . . . . 3. Diﬀuse Optical Spectroscopy (DOS): monitoring . . . . . . . . . . . . . . . . . . . . . . 4. Diﬀuse Optical Tomography (DOT): imaging . . . . . . . . . . . . . . . . . . . . . . . . 5. Diﬀuse Correlation Spectroscopy (DCS): blood ﬂow . . . . . . . . . . . . . . . . . . . 6. Background on tissue hemodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Validation and clinical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 53 53 53 55 55 58 59 61 62 63 65 71

J. H. Page – Ultrasonic wave transport in strongly scattering media . . . . .

75

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Acoustic wave transport in random media . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Ballistic propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Diﬀusive propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Wave transport in ordered media: phononic crystals in 2D and 3D . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 76 77 81 84 91

J. H. Page – Anderson localization of ultrasound in three dimensions . . . .

95

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Mesoglasses: porous elastic solids with very strong scattering . . . . . . . . . . . 3. Time-dependent transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Transverse conﬁnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Statistical approach to localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Schr¨ odinger and Helmholtz equations in disordered media . . . . . .

96 97 99 102 107 112 112

J. H. Page – Ultrasonic spectroscopy of complex media . . . . . . . . . . . . . . . .

115

1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diﬀusing Acoustic Wave Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probing food biomaterials with ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116 116 124 129

M. Fink and M. Tanter – MultiWave imaging . . . . . . . . . . . . . . . . . . . . . . .

133

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Transcending classical diﬀraction limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Wave-to-wave generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 134 136

indice 4. Wave-to-wave tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Wave-to-wave imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Super-resolution in supersonic shear wave imaging . . . . . . . . . . . . . . . . 5 2. Clinical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Shear wave spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Waves propagation in tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 1. Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 1.1. Low frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 1.2. Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 1.3. Optical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A 2. Mechanical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

pag. 138 139 145 146 146 149 150 150 150 150 151 151

M. Fink, J. de Rosny, G. Lerosey and A. Tourin – Time reversal focusing and the diﬀraction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Basic principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. An ideal time reversal experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Time reversal in free space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Time reversal through heterogeneous medium . . . . . . . . . . . . . . . . . . . . 2 4. An experimental point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Time reversal in complex media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. One-channel time reversal in chaotic cavities . . . . . . . . . . . . . . . . . . . . . 3 2. Time reversal in open systems: random media . . . . . . . . . . . . . . . . . . . 4. Focusing microwaves below the diﬀraction limit . . . . . . . . . . . . . . . . . . . . . . 5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

155 157 158 160 161 162 162 164 170 173 176

L. Fallani and M. Inguscio – Ultracold atoms in bichromatic lattices . . .

179

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Ultracold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Light forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Crystals made of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Monochromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Energy bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Tight-binding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Adding interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Mott insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Bichromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Quasicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. General notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1. Harper and Aubry-Andr´e model . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Superlattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4. Incommensurate lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5. Localization in bichromatic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 180 180 181 182 184 184 187 189 191 195 196 197 198 200 202 203

indice

X

. 4 5.1. Localized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5.2. Spectrum of the localized states . . . . . . . . . . . . . . . . . . . . . . . . . 4 6. Further considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Anderson localization of matter waves in bichromatic lattices . . . . . . . . . . . . 5 1. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Absence of diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Imaging the localized states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Eﬀect of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Strongly interacting atoms in bichromatic lattices . . . . . . . . . . . . . . . . . . . . . . 6 1. Towards a Bose glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Bragg spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. Bloch – Exploring strongly correlated ultracold bosonic and fermionic quantum gases in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Optical dipole force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Optical lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. 1D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2. 2D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3. 3D lattice potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Bose-Hubbard model of interacting bosons in optical lattices . . . . . . . . . . . . 3 1. Ground states of the Bose-Hubbard Hamiltonian . . . . . . . . . . . . . . . . . 3 2. Double-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Multiple-well case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4. Superﬂuid-to-Mott-insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . 4. Multi-orbital quantum phase diﬀusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Probing the energy scales via Fock state heterodyning . . . . . . . . . . . . . 4 4. Experimental setup and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Compressible and incompressible quantum phases of fermionic spin mixtures in optical lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Hubbard Hamiltonian in a trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. Cloud compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Entropy distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Controlled superexchange interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Time-resolved observation of superexchange interactions . . . . . . . . . . 7. Quantum noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Time-of-ﬂight versus noise correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Noise correlations in bosonic Mott and fermionic band insulators . . . 8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 203 205 208 209 210 211 214 215 218 220 223 224 227

233 234 234 234 237 237 237 238 240 241 241 243 244 247 247 248 249 250 250 252 255 256 258 259 259 261 262 264 265 269

indice

XI

Z. Hadzibabic and J. Dalibard – Two-dimensional Bose ﬂuids: An atomic physics perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

pag. 273

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Absence of true long-range order in 2d . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Outline of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The inﬁnite uniform 2d Bose gas at low temperature . . . . . . . . . . . . . . . . . . . 2 1. The ideal 2d Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Interactions in a 2d Bose gas at low T . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. Suppression of density ﬂuctuations and the low-energy Hamiltonian . . 2 4. Bogoliubov analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5. Algebraic decay of correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2d Bose gas . . . . 3 1. The role of vortices and topological order . . . . . . . . . . . . . . . . . . . . . . . . 3 2. A simple physical picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3. Results of the microscopic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. The 2d Bose gas in a ﬁnite box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. The ideal Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2. The interacting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Width of the critical region and crossover . . . . . . . . . . . . . . . . . . . . . . . 4 4. What comes ﬁrst: BEC or BKT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5. The case of anisotropic samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. The 2d Bose gas in a harmonic trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. The ideal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. LDA for an interacting gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3. What comes ﬁrst: BEC or BKT? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4. Width of the crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Achieving a quasi-2d gas with cold atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1. Experimental implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Interactions in a 2d atomic gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Residual excitation of the z-degree of freedom . . . . . . . . . . . . . . . . . . . 7. Probing 2d atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. In situ density distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Two-dimensional Time-of-Flight expansion . . . . . . . . . . . . . . . . . . . . . . 7 3. Three-dimensional Time of Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4. Interference between independent planes . . . . . . . . . . . . . . . . . . . . . . . . 7 5. Interfering a single plane with itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

274 274 275 276 276 279 280 282 285 287 287 289 290 292 292 293 294 295 296 298 298 299 302 303 304 304 305 307 308 308 310 311 313 316 316

Elenco dei partecipanti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Preface

Many fundamental processes in physics involve transport. For a variety of physical systems, e.g. electrons, light, cold atoms and sound, transport mechanisms eventually reduce to diﬀerent manifestations of wave transport. In the last decades, pushed by the spectacular progresses in the control and engineering of matter at the nano-scale, new regimes of wave transport became of strong interest. Indeed, fascinating eﬀects emerge when transport is studied at the “nano” level, when atoms behave like waves and light propagation in nano-engineered structures acquires intriguing behaviors. This book collects contributions from speakers and lecturers of the CLXXIII International School of Physics “Enrico Fermi” which was held in Varenna (Italy) from June 23rd to July 3rd 2009. Diﬀerent aspects of wave transport were covered during the school, from electrons to light propagation, from sound to ultracold atoms. Considering the ubiquitous nature of wave transport phenomena, the school was characterized by a strongly interdisciplinary approach, with speakers, lecturers and students from diﬀerent communities meeting and sharing their knowledge and the often complementary points of view and approaches. Among the diﬀerent media in which waves can travel, periodic and disordered structures surely deserve particular attention. Interference of waves in periodic structures results in the formation of energy bands, which are responsible for the conduction properties of electrons in solids. Periodic structures can be realized also for light and ultra-cold atoms, in the form of photonic crystals or optical lattices, respectively, which allow the observation of eﬀects which have been originally predicted in the context of solid-state physics. The most recent advances in the physics of ultra-cold atoms in optical lattices are discussed in the contribution by I. Bloch concerning quantum simulation of condensed-matter physics. Optical lattices also allow the production of low-dimensional atomic systems, as discussed in the paper by Z. Hadzibabic and J. Dalibard devoted to the investigation of transport and superﬂuidity in 2D bosonic quantum gases. Disordered structures also show fascinating phenomena. Multiple scattering in random media results in the localization of waves predicted by P. W. Anderson ﬁfty years ago for electrons moving in disordered crystals, and reviewed in this book in the opening XIII

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Preface

article by P. Woeﬂe. Also in this case, a phenomenon originally predicted for electrons in crystals has been observed both for propagation of classical waves —light and sound— in disordered media and very recently for ultracold atoms expanding in disordered optical potentials. Transport of sound waves in diﬀerent media, including Anderson localization in disordered structures, is discussed in the contributions by J. Page. At the border between periodic and disordered media, quasicrystals are topological structures showing long-range order and absence of periodicity, which results in intriguing properties that are described in the contribution by P. J. Steinhardt. Optical quasicrystals can be realized for ultra-cold atoms and used to study Anderson localization of matter waves, as discussed in the article by L. Fallani and M. Inguscio. Knowledge of transport properties in complex systems is important not only for fundamental studies, but also for applications. Understanding the propagation of light is extremely important for engineering new devices, as metamaterials and plasmonic materials, and for applications in the ﬁeld of energy, biology and medicine. In this perspective, the article by R. C. Mesquita and A. G. Yodh covers the application of diﬀuse optics to medical imaging. Extending these concepts beyond the ﬁeld of optics, in two diﬀerent contributions to this book, M. Fink and coworkers discuss multi-wave imaging for medical applications and present theory and applications of time-reversal focusing. The success of the Summer School was not only determined by the high quality of the lectures, but also by the enthusiasm of the students and observers who attended the Course. Their active participation resulted in the success of the two poster sessions (the most interesting posters have been upgraded to invited presentations) and of the ﬁnal discussion session on future research perspectives. We would like to warmly thank all the speakers, lecturers, participants and express our gratitude to the organizing team, in particular Barbara Alzani of the Italian Physical Society for her passion and dedication in the Course organization, as well as Ramona Brigatti and Marta Pigazzini for their enthusiasm and assistance in Varenna. We also acknowledge ﬁnancial support from the Italian Physical Society through its president Luisa Cifarelli, and from the European network Intercan. Finally, the Summer School hosted a celebration in memoriam of Franco Bassani (1929-2008), former president of the Italian Physical Society. On this occasion, friends and colleagues Lucio Andreani, Luisa Cifarelli, Massimo Inguscio and Erio Tosatti presented several portraits of his scientiﬁc and personal life. Franco Bassani was an excellent scientist and an outstanding man. We dedicate this book to his memory.

R. Kaiser, D. S. Wiersma and L. Fallani

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VILLA MONASTERO 23 Giugno - 3 Luglio 2009

CLXXIII CORSO - VARENNA SUL LAGO DI COMO

Società Italiana di Fisica SCUOLA INTERNAZIONALE DI FISICA «E. FERMI»

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1) T. Paul 2) A. Ullah 3) D. Elam 4) M. Straticiuc 5) N. Mercadier 6) J. R. Ott 7) B. T. Gebrehiwot 8) M. Thoreson 9) W. K. Hildebrand 10) M. Berritta 11) B. N. S. Bhaktha 12) E. Kot 13) S. Rist

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14) J. Hartwig 15) S. T. Seidel 16) W. Peeters 17) J. J. Saenz 18) S. L. Portalupi 19) S. Gualini 20) S. Kurylchyk 21) M. Larcher 22) P. L. Koswatta 23) L. Ratschbacher 24) E. Nunes-Pereira 25) A. Peña 26) H. Ahlers

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27) P. Barthelemy 28) R. Mupparapu 29) K. Vynch 30) R. Vivekananthan 31) D. Milovanovic 32) J. Armijo 33) F. J. Valdivia Valero 34) M. R. De Saint Vincent 35) T. Bienaimé 36) R. Menchòn Enrich 37) Y. Lahini 38) R. Pugatch

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39) M. Baumert 40) N. Meyer 41) M. Holynski 42) M. C. Pigazzini 43) R. Brigatti 44) T. Wasak 45) P. Szankowski 46) B. Alzani 47) O. Sliusarenko 48) B. Barnes 49) P. Wölfle 50) R. Hulet

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51) A. Aspect 52) N. Fabbri 53) L. Chantada Santodomingo 54) M. Inguscio 55) G. Maret 56) D. S. Wiersma 57) L. Fallani 58) R. Kaiser 59) J. Page 60) P. Steinhardt 61) C. Hofmann

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62) S. Hoinka 63) A. Rakonjac 64) T. Kitagawa 65) C. Wuttke 66) S. Giudicatti 67) T. Hartmann 68) K. Putteneers 69) I. Prieto 70) V. Eremeev 71) A. Benseny Cases 72) E. Vogt 73) N. Ciobanu

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-1

Anderson localization ¨ lfle P. Wo Institut f¨ ur Theorie der Kondensierten Materie, and DFG-Center for Functional Nanostructures, Universit¨ at Karlsruhe - 76131 Karlsruhe, Germany

Summary. — The basic concepts of the theory of Anderson localization are reviewed in the example of electrons in disordered solids. The lectures are organized in three sections. In the ﬁrst, the phenomenon of localization of quantum particles or classical waves is introduced on a qualitative level. The regimes of strong and weak localization are discussed. Sample to sample ﬂuctuations are considered for one-dimensional and quasi–one-dimensional systems. In sect. 2 the scaling theory of the Anderson localization transition is presented. The renormalization group theory is introduced and results and consequences are presented. The classiﬁcation of the Anderson transitions into universality classes is described. Basic concepts of the fractal structure of the wave functions at the critical point are reviewed. In sect. 3 the self-consistent theory of Anderson localization is presented. The eﬀect of the electron-electron interaction in destroying the phase coherence is brieﬂy discussed.

1. – Basic notions of localization . 1 1. Introductory remarks. – The localization of quantum particles or classical waves by a static random potential or random ﬂuctuations of the local parameters determining wave propagation in a disordered medium has been studied for more than ﬁfty years. The very concept of localization was introduced in a seminal paper by P. W. Anderson entitled “Absence of diﬀusion in certain random lattices” [1]. There it was shown that c Societ` a Italiana di Fisica

1

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¨ lfle P. Wo

electrons may be localized by a random potential, so that diﬀusion would be suppressed. That work had been triggered by the experimental observation of the apparent absence of spin diﬀusion as detected in NMR signals of a disordered solid. The fundamental reason for the localizing eﬀect of a random potential is the multiple interference of wave components scattered by randomly positioned scattering centers. The interference eﬀect takes place both with quantum particles and with classical waves, provided the propagation is coherent. It is interesting to note that the ﬁrst application of the idea of localization concerned the spin diﬀusion D of electrons and not the electrical conductivity σ. Anderson considered a tight-binding model of electrons on a crystal lattice, with energy levels at each site chosen from a random distribution. The traditional view had been that scattering by the random potential causes the Bloch waves to lose phase coherence on the length scale of the mean-free path . Nevertheless, the wave function was thought to remain extended throughout the sample. Anderson pointed out that if the disorder is suﬃciently strong, the particles may become localized, in that the envelope of the wave function ψ(r ) decays exponentially from some point r0 in space |ψ(r )| ∼ exp(|r − r0 |/ξ),

(1)

where ξ is the localization length. There exist a number of review articles covering the earlier work on the Anderson localization problem. The most complete account of the early work is given by Lee and Ramakrishnan [2]. The seminal early work on interaction aﬀects is presented in [3]. A complete account of the early numerical work can be found in [39]. A path integral formulation of weak localization can be found in [4]. Several more review articles and books are cited along the way. In the following we will use units with Planck’s constant ¯h and Boltzmann’s constant kB equal to unity, unless stated otherwise. . 1 2. Electrons and classical waves in disordered systems. – The wave function ψ(r ) of a single electron of mass m in a random potential V (r ) obeys the stationary Schr¨ odinger equation (2)

¯2 2 h ∇ + V (r ) − E ψ(r ) = 0. − 2m

In the simplest case V (r ) may be taken to obey Gaussian statistics with V (r )V (r ) = V 2 δ(r − r ). Electrons propagating in the random potential V (r ) will be scattered on average after a time τ . For weak random potential the scattering rate is given by (3)

¯ h = πN (E) V 2 , τ

where N (E) is the density of states at the energy E of the electron. In a metal the electrons carrying the charge current are those at the Fermi energy E = EF . Within the time τ the electron travels a distance = vτ , where v is its velocity.

3

Anderson localization

In close analogy the wave amplitude ψ(r ) of a classical monochromatic wave of frequency ω obeys the wave equation (4)

ω2 2 + ∇ ψ(r ) = 0. c2 (r )

Here c(r ) is the wave velocity at position r in an inhomogeneous medium, assumed to be a randomly ﬂuctuating quantity. A possible realization would be a system of spheres of diameter r0 at random positions ri , with wave velocity c0 < 1 inside the spheres and c = 1 in the surrounding medium. The main diﬀerence between the Schr¨ odinger equation and the wave equation is that in the wave equation the “random potential” 1/c2 (r ) is multiplied by ω 2 , so that disorder is suppressed in the limit ω → 0. By contrast, in the quantum case disorder will be dominant in the limit of low energy E. A further diﬀerence may arise if the wave amplitude is a vector quantity, as e.g. for electromagnetic waves in d = 3 dimensions. In real systems particles or wavepackets are not independent, but interact. Electrons are coupled by the Coulomb interaction, leading to important eﬀects that go much beyond the single-particle model. Some of these eﬀects will be mentioned later. Similarly, wavepackets interact via non-linear polarization of the medium. Apart from these complications, the physics of electronic wavepackets and classical wavepackets is quite similar. In the following we will present most of the discussion in the language of electronic wavepackets. . 1 3. Strong localization and Anderson transition. – The appearance of localized states is easily understood in the limit of very strong disorder: localized orbitals will then exist at positions where the random potential forms a deep well. The admixture of adjacent orbitals by the hopping amplitudes will only cause a perturbation that does not delocalize the particle. The reason is that nearby orbitals will have suﬃciently diﬀerent energies so that the amount of admixture will be small. On the other hand, orbitals close in energy will in general be spatially far apart, so that their overlap is exponentially small. Thus, we can expect the wave functions in strongly disordered systems to be exponentially localized. Whether the particles become delocalized, when the disorder strength is reduced, is a much more complex question. In one dimension, it can be shown rigorously that all states are localized, no matter how weak the disorder [5-7]. In three dimensions, the accepted view is that the particles are delocalized at weak disorder. Localized and extended states of the same energy are not expected to coexist. In the generic situation any small perturbation would lead to hybridization, delocalizing any localized state. We can therefore assume that the localized and extended states of a given energy are separated: as a function of increasing disorder strength η there will be a sharp transition from delocalized to localized states at a critical disorder strength ηc . A qualitative criterion as to when an Anderson transition is expected in 3d systems has been proposed by Ioﬀe and Regel [59]. It states that as the mean free path becomes shorter with increasing disorder, the Anderson transition occurs when is of the order

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¨ lfle P. Wo

Fig. 1. – Phase diagram showing metallic (M ) and insulating (I) regions of the tight-binding model with site diagonal disorder (box distribution of width W ). Dots: numerical study [39]; solid line: self-consistent theory [13]. The remaining lines are bounds on the energy spectrum.

of the wavelength λ of the particle (which amounts to the condition kF ∼ 1 in metals, where kF is the Fermi wave number). As we will see later, in 1d or 2d systems may be much longer than the wavelength and the particles are nonetheless localized. In fact, the relevant mean free path here is the one with respect to momentum transfer. A similar situation exists when we ﬁx the disorder strength, but vary the energy E. Electrons in states near the bottom of the energy band are expected to be localized even by a weakly disordered potential, whereas electrons in states near the band center (in dimension d = 3) will be delocalized, provided the disorder is not too strong. Thus there exists a critical energy Ec separating localized from delocalized states, the so-called mobility edge [8,9]. The electron mobility as a function of energy is identically zero on the localized side (at zero temperature), and increases continuously with energy separation |E − Ec | in the delocalized, or metallic, phase. The continuous character of this quantum phase transition, termed Anderson transition, is a consequence of the scaling theory to be presented in sect. 2. Historically the continuous nature of the metal-insulator transition in disordered solids has been a point of controversy for many years. According to an earlier theory by [8,9], the conductivity changes discontinuously at the transition, such that a “minimum metallic conductivity” exists on the metallic side of the transition. Numerical simulations have shown beyond doubt that the transition is in fact continuous at least in the absence of interactions. In the much more complex situation of interacting electrons the results obtained, e.g. for electrons in the Hubbard model using the Dynamical Mean Field Theory (DMFT) suggest that the metal-insulator transition at ﬁnite temperature is of ﬁrst order, but becomes continuous in the limit T → 0 [10, 11].

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Anderson localization

Fig. 2. – Phase diagram showing regions of localized and extendes states for scalar waves propaging in a random system of point scatterers of average separation a and average dielectric constant ¯ = (r ) = 1/c2 (r ).

The phase boundary separating localized and extended states in a disordered threedimensional system may be determined approximately by a variety of methods. For electrons on a cubic lattice with nearest-neighbor hopping and one orbital per site of random energy i chosen from a box distribution in the interval [−W/2, W/2], the phase diagram has been determined by numerical simulations [12] as shown in ﬁg. 1. Also shown is the result of an analytical approximation to be discussed in more detail in sect. 3 [13]. The agreement is quite satisfactory. In ﬁg. 2. the result of the analogous analytical approximation for the case of scalar waves of fequency ω propagating in a medium with random dielectric constant is shown [14]. Here the region of localized states is much smaller, the reason being that waves of low frequency and correspondingly long wavelength average over the disorder. The longer the wavelength, the smaller the eﬀective disorder strength. Localization is then most likely to appear in the regime of resonant scattering, i.e. when the wavelength is comparable to the extension of the scattering centers. . 1 4. Weak localization. – An electron or a wavepacket moving through a disordered medium will be scattered by the random potential on the average after propagating a distance , the mean free path. On larger length scales the propagation is diﬀusive. Weak localization is a consequence of destructive interference of two wave components starting at some point and returning to the same point after traversing time-reversed paths. Let the probability amplitudes for the wavepacket to move from point r0 along some path C back to r0 be A, and the amplitude for the time-reversed path be Ar , then the transition probability will be w = |A + Ar |2 = wcl + wint ,

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where wcl = |A|2 +|Ar |2 and wint = 2 Re(A∗ Ar ). For any two paths the interference term wint may be positive or negative, and thus averages to zero. However, if the system is time-reversal invariant, A = Ar , and the probability of return w is enhanced by a factor of two compared to the probability wcl of a classical system w = 4|A|2 = 2wcl .

(5)

In that case the probability for transmission is reduced, which leads to a reduced diffusion coeﬃcient and a reduced conductivity. One may estimate the correction to the conductivity in the following qualitative way. The relative change of the conductivity σ by the above interference eﬀect is equal to the probability of interference of two wavepackets of extension λ, the wavelength, after returning to the starting point. The probability of return to the origin in time t of a particle diﬀusing in d dimension is given by (4πDt)−d/2 d3 r, where D is the diﬀusion coeﬃcient. Since the volume of interference in the time interval [t, t + dt] is λd−1 vdt, where λ is the wavelength and v is the velocity of the wavepacket, one ﬁnds the quantum correction to the conductivity δσ, as [34, 15]

(6)

δσ ≈− σ0

τ

τφ

⎧

3 3 λ2 τ ⎪ − 1 − ⎪ ⎪ 2π 4π 2 τφ , ⎪ ⎨ d−1 vλ dt 1 λ = − 2π ln(τφ /τ ), ⎪ (4πDt)d/2 ⎪

⎪ ⎪ τφ ⎩− √1 − 1 , τ π

d = 3, d = 2, d = 1.

Here the expressions D = d1 v 2 τ for the diﬀusion coeﬃcient and = vτ for the meanfree path have been used. The mean-free time between successive elastic collisions is τ . The Drude conductivity σ0 = e2 nτ /m, with n the particle density. The upper limit of the integral is the phase relaxation time τφ , i.e. the average time after which phase . coherence is lost due to inelastic or other phase-shifting processes. In subsect. 3 2 we will present an estimate of 1/τφ . In order for weak-localization processes to exist at all, the inequality τφ τ must hold. We note that the correction in three and two dimensions depends on the ratio of wavelength λ to mean free path , and gets smaller in the limit of weak disorder, where λ/ 1. In two and one dimension, however, the correction grows large in the limit τ /τφ → 0. Since one expects the phase relaxation rate 1/τφ for a system in thermodynamic equilibrium to go to zero in the limit temperature T = 0 (see below), the WL quantum correction will be large in any system in d = 1, 2 no matter how weak the disorder is, in the limit of T → 0. As we will see, this behavior signals the fact that there are no extended states in d = 1, 2 dimensions. The characteristic length wavepacket retains phase coherence is related to τφ by the diﬀusion Lφ over which a coeﬃcient Lφ = Dτφ . In systems of restricted dimension like ﬁlms of thickness a or wires of diameter a the eﬀective dimensionality of the system with respect to localization is determined by the ratio Lφ /a: for Lφ a the system is three dimensional, while for Lφ a diﬀusion over time τφ takes place in the restricted geometry of the ﬁlm or wire and the eﬀective dimension is therefore 2 or 1.

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Anderson localization

. 1 5. One-dimensional systems. – One-dimensional systems in transport theory are characterized by a single transmission channel along a wire or waveguide. Such systems are not mathematical idealization of systems of point particles in strictly one dimension, but are actually realized. The conductance of a wire is given by the transmission probability T for the quantum particle to propagate through the wire (7)

G=

e2 T 2π¯h

as ﬁrst stated by Landauer [16, 17]. The dimensionless conductance of a given random system depends exponentially on its length (8)

g = G/

e2 2π¯h

∼ e−αL ,

where the average value of α is equal to the inverse localization length ξ, α = 1/ξ. Here the angular brackets denote the appropriate weighted average over all realizations of disorder of the system under consideration. The exponential dependence of g on the randomly ﬂuctuating (from sample to sample) quantity α causes g to have an extremely wide probability distribution P (g). The average value of g is then very diﬀerent from the typical value of g (value of the maximum of P (g)). As ﬁrst discussed in [18] the quantity α has a normal (Gaussian) distribution, which in turn implies that the logarithm of g is normally distributed, or

(9)

ξ P (g) ∼ exp − 4L

g L ln + 4 ξ

2 .

(for details see, e.g., ref. [19]). . 1 6. Quasi–one-dimensional systems. – Systems with many transport channels (wires), but of one-dimensional character, such that the mean free path is much longer than the diameter of the wire, are called quasi–one-dimensional. In this case the joint probability distribution of the eigenvalues Ti of the transmission matrix as a function of the length L of the wire may be calculated from a Fokker-Planck-type equation, called DMPK equation [20, 21]. The DMPK equation may be solved exactly in certain cases [22]. The distribution of conductances P (g) may be calculated approximately from the distribution of eigenvalues Ti [23], using the relation g = Σi Ti [17]. Depending on the value of L/ξ, one may distinguish a localized regime (L/ξ 1), a metallic regime (L/ξ 1), and a crossover regime (L/ξ ∼ 1). In the localized regime the eigenvalues Ti are widely spaced and only the largest eigenvalue contributes. Then P (g) is given by eq. (9), the result for the one-dimensional case. In the metallic regime, the eigenvalues are densely packed around the limiting value T = 1. Since the eigenvalues Ti are approximately statistically independent, one expects

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¨ lfle P. Wo

Fig. 3. – Distribution of dimensionless conductance g of a quasi–one-dimensional wire of length L, with N channels and mean-free path L. Solid line: N /L = 0.7; dashed line: N /L = 0.2.

a Gaussian distribution P (g). The density of eigenvalues determines the width of the Gaussian. It turns out that the width is independent of the properties of the system [23] (10)

15

2

P (g) ∼ e− 2 (g−γ) ,

where γ = N /L = ξ/L. This result holds for the simplest symmetry class, namely unitary symmetry (see sect. 2). The sample-to-sample ﬂucuations of the conductance are hence universal [24, 25], and are one of the hallmarks of disordered systems. In the crossover regime P (g) has a very asymmetric shape (see ﬁg. 3). For ξ/L ∼ 1/2, e.g., P (g) is given by a log-normal distribution centered at g ≈ 1/2 at g < 1 and a sharply dropping exponential tail at g > 1, which has been termed “one-sided log-normal distribution” [26]. 2. – Theory of localization: fundamental concepts The Anderson localization transition is a quantum phase transition, i.e. it is a transition at zero temperature tuned by a control parameter (strength of disorder, particle energy, wave frequency). Unlike other quantum phase transitions, the Anderson transition does not appear to have an obvious order parameter. Nontheless, there appears to be a dynamically generated length scale, the localization or correlation length ξ, which tends to inﬁnity as the transition is approached. Therefore, drawing an analogy with magnetic phase transitions, Wegner early on proposed scaling properties [27]. Later, he formulated a ﬁeld-theoretic description of the Anderson transition in the form of a nonlinear sigma model (NLσM) of interacting matrices (rather than vectors, as for magnetic

9

Anderson localization

systems) [28]. The NLσM was later formulated in the mathematically more tractable supersymmetric form [29] (see below). Many rigorous results especially on systems of restricted dimensions or in inﬁnite dimensions have been obtained within this framework (see, e.g., [29, 30]). However, a systematic calculation of the properties of the Anderson transition in d = 3 dimensions within the NLσM has not been possible so far. An approximate theory of the Anderson transition, which takes into account the renormalization of the diﬀusion coeﬃcient in a self-consistent way, accounts well for the non-critical properties, but fails to describe the critical behavior quantitatively [31, 32]. It will be discussed in sect. 3. In the following we will mainly discuss a somewhat diﬀerent approach, the scaling theory, which allows to obtain the main features of Anderson localization without complex mathematical formalism. . 2 1. Thouless conductance. – The idea of scaling theory is to consider localization behavior as a function of the system size L. The ﬁrst studies along these lines were performed by Thouless and collaborators [33]. They envisioned building a sample of size (2L)d by putting building blocks of size Ld (cubes or squares) together. If the building blocks are suﬃciently large, i.e. are characterized by uniform disorder strength, one should expect that the eigenstates of the sample of volume (2L)d should be entirely determined by the properties of the building blocks. The eigenstates of the (2L)d sample are linear combinations of those of the Ld sample and the amount of admixture of states of neighboring blocks depends on the overlap integral and the energy denominator. The energy denominator will be typically of the order of the mean level spacing in the Ld sample, δ = (N0 Ld )−1 , where N0 is the density of the states at the energy of the particles or wavepackets considered (in metals this energy is given by the Fermi energy). To estimate the overlap, Thouless introduced the sensitivity of energy levels to the boundary conditions. The energy shift ΔE obtained when the boundary conditions are changed from periodic to antiperiodic is a measure of the extension of the eigenstates across the volume Ld . Clearly, a state localized inside the Ld sample will have exponentially small ΔE, whereas a delocalized state may have ΔE δ . The dimensionless parameter ΔE/δ is the single parameter that characterizes the wave functions with respect to their localization properties. Thouless further noticed that the conductance G (and not the conductivity) is a dimensionless quantity, when expressed in units of the quantum of conductance GQ = h) e2 /(2π¯ (11)

g = G/(e2 /2π¯h).

Then Thouless conjectured that g should be uniquely determined by the parameter ΔE/δ . . 2 2. Scaling theory of the conductance. – Along a diﬀerent line Wegner [27] argued that the Anderson localization transition should be described in the language of critical phenomena of continuous (quantum) phase transitions. This requires the assumption of

10

¨ lfle P. Wo

a correlation length ξ diverging as a function of disorder strength η at the critical point (12)

ξ(η) ∼ |η − ηc |−ν .

The conductivity is then expected to obey the scaling law (13)

σ(η) ∼ ξ 2−d ∼ (ηc − η)s ;

η < ηc ,

d > 2,

which follows from the fact that σ in units of e2 /(2π¯h) has dimension (1/length)d−2 , and the only characteristic length near the transition is the correlation length ξ. By comparing the conductivity exponent s with the exponent of ξ, one ﬁnds (14)

s = ν(d − 2).

On the other hand, the conductance g of a d-dimensional cube of length L, which for a good metal of conductivity σ is given by g(L) = σLd−2 , must obey the scaling property (15)

g(η; L) = Φ(L/ξ).

This means that g is a function of a single parameter corresponds to a value g.

L ξ,

so that each value of L/ξ

. 2 3. Renormalization group equation. – It follows that g(L) obeys the renormalization group (RG) equation (16)

d ln g = β(g), d ln L

where β(g) is a function of g only, and does not depend on disorder. In a landmark paper, Abrahams, Anderson, Licciardello and Ramakrishnan [34] proposed the above equation and calculated the β-function in the limits of weak and strong disorder. A conﬁrmation of the assumption of scaling was obtained from a calculation of the next-order term [35]. At strong disorder we expect all states to be localized, with average localization length ξ. It then follows that g(L) is an exponentially decreasing function of L (17)

g(L) ∼ exp(−L/ξ).

In comparison with the above (Ohmic) dependence g ∼ Ld−2 , this is a very non-Ohmic behavior. The β-function is then given by (18)

β(g) ∼ ln(g/gc ) < 0.

At weak disorder one ﬁnds from g ∼ Ld−2 (19)

β(g) = d − 2.

11

Anderson localization

The important question of whether the system is delocalized (metal) or localized (insulator) may be answered by integrating the RG equation from some starting point L0 , where g(L0 ) is known. Depending on whether β(g) is positive or negative along the integration path, the conductance will scale to inﬁnity or to zero, as L goes to inﬁnity. In d = 3 dimensions we have β(g) > 0 at large g, but β(g) < 0 at small g. Thus there exists a critical point at g = gc , where β(gc ) = 0, separating localized and delocalized behavior. In d = 1 dimension, on the other hand, β(g) < 0 at large and small g, and by interpolation also for intermediate values of g, so that there is no transition in this case and all states are localized. The dimension d = 2 plays apparently a special role, as in this case β(g) → 0 for g → ∞. In order to ﬁnd out, whether β > 0 or < 0 for large g one has to calculate the scale dependent (i.e. L-dependent) corrections to the Drude result at large g. This is nothing but the weak-localization correction we already considered. For a system of −2 , leading to ﬁnite length L < Lφ we should replace τ1φ = DL−2 φ in eq. (6) with DL (20)

g(L) = σ0 − a ln

L ,

where a quantitative calculation [34] gives a = 2/π and σ0 = /λ (in units of e2 /¯ h) has been used. It follows that (21)

a β(g) = − , g

d = 2,

so that we can expect β(g) < 0 for all g, meaning that again all states are localized. This result is valid for the “usual” type of disorder, i.e. in case all symmetries, in particular time reversal symmetry (required for the weak-localization correction to be present) are preserved. In case time reversal invariance is broken, e.g. by spin-ﬂip scattering at magnetic impurities, the weak-localization eﬀect is somewhat suppressed, but not completely removed. The ﬁrst correction term in the β-function is then ∝ −1/g 2 (see, e.g., [29]) implying that still all states are localized. In the presence of a magnetic ﬁeld the situation is more complex, however, since the scaling of the Hall conductance is coupled to the scaling of g. As a result, one ﬁnds exactly one extended state per Landau energy level, which then gives rise to the quantum Hall eﬀect [36]. On the other hand, if spin-rotation invariance is broken, but time reversal invariance is preserved, as is the case in the presence of spin-orbit scattering, the correction term is ∝ +1/g, i.e. it is anti-localizing. In this case the β-function in d = 2 dimensions has a zero, implying the existence of an Anderson transition [37]. . 2 4. Critical exponents. – In the neighborhood of the critical point at g = gc in d = 3 dimensions we may expand the β-function as 1 g − gc , |g − gc | gc . (22) β(g) = y gc

12

¨ lfle P. Wo

Fig. 4. – Renormalization group β-function in dimensions d = 1, 2, 3 for the orthogonal ensemble, calculated by the self-consistent theory [56].

Integrating the RG equation for g > gc from g() = g0 to β → 1 at large L we ﬁnd g(L) = σL, where (23)

σ∼

1 (g() − gc )y .

Since g() − gc ∝ ηc − η, we conclude that the inverse of the slope of the β-function, y, is equal to the conductivity exponent s = y. Similarly, one ﬁnds on the localized side (g < gc ) (24)

g(L) ∼ gc exp − c(gc − g())y L/ ∼ gc exp(−L/ξ),

from which the localization length follows as (25)

ξ ∼ |η − ηc |−y .

The critical exponent ν governing the localization length is therefore ν = y = s in d = 3 dimensions. Since the critical conductance gc = O(1) in d = 3, analytical methods are not available to calculate the β-function in the critical region in a quantitative way. A perturbative expansion in 2 + dimensions, where gc 1 is available, but the expansion in is not well behaved, so that it cannot be used to obtain quantitative results for s and ν in d = 3.

13

Anderson localization

In ﬁg. 4 the result of a calculation of the β-function using a self-consistent approximation is shown [38]. In that approximation, ν = 1 is found. There exist, however, reliable results on ν from numerical studies, which give s = ν = 1.58 ± 0.02 [39, 40]. . 2 5. Dynamical scaling. – The dynamical conductivity σ(ω) (the a.c. conductivity at frequency ω) in the thermodynamic limit in d = 3 obeys the scaling law [41, 42] (26)

σ(ω; η) =

1 Φ(Lω /ξ), ξ

where the scaling function Φ has been introduced in eq. (15). Here Lω is the typical length a wavepacket travels in the time of one cycle, 1/ω. Since the motion is diﬀusive, Lω = D(ω)/ω. It is important to note that the relevant diﬀusion coeﬃcient is scale dependent D = D(ω; η), which is related to the conductivity via the Einstein relation (27)

σ(ω) = h ¯ N (E)D(ω),

where N (E) is the density of states at the particle energy E. At the Anderson transition, when ξ → ∞, we expect σ(ω) to be ﬁnite. It follows that limξ→∞ Φ(Lω /ξ) ∼ ξ/Lω and consequently (28)

σ(ω; η) ∼

1 , Lω

η = ηc .

This is a self-consistent equation for σ(ω), with solution (29)

σ(ω) ∼ ω 1/3 ,

η = ηc .

To be more precise, ω in the above expressions should be replaced with the imaginary frequency −iω, such that σ(ω) is a complex-valued quantity. In more general notation, introducing the dynamical critical exponent z by σ(ω) ∼ ω 1/z , we conclude that z = 3. Since the dynamical critical exponent determines how the scaling of wave vector q and frequency ω of the critical ﬂuctuations are related, ω ∼ q z , we conclude that the initial scaling of the diﬀusion process, ω ∼ q 2 is modiﬁed at the transition as ω ∼ D(q)q 2 ∼ q 3 , i.e. the wave vector-dependent diﬀusion coeﬃcients scales as D(q) ∼ q. The dynamical scaling is valid in a wide neighborhood of the critical point, deﬁned by ω > τ1 (ξ/)−z ∼ |η − ηc |νz , where νz ≈ 4.8. This scaling regime is accessible in experiment, not only by measuring the dynamical conductivity directly, but also by observing that at ﬁnite temperature the scaling in ω is cut oﬀ by the phase relaxation rate 1/τφ [42, 43]. Therefore, assuming a single temperature power law 1/τφ ∼ T p , one ﬁnds the following scaling law for the temperature-dependent d.c. conductivity: (30)

σ(T ; η) ∼ T p/3 ΦT (ξT p/3 ).

14

¨ lfle P. Wo

Using this scaling law, one may in principle determine the critical exponent ν from the temperature dependence of the conductivity in the vicinity of the critical point. In the case of disordered metals or semiconductors, where studies of this type have been performed, the eﬀect of electron-electron interaction has to be taken into account. One major modiﬁcation in the above is that the Einstein relation is changed: the single-particle density of states, which is not critical is replaced with the compressibility ∂n/∂μ (n = density, μ = chemical potential), which is expected to vanish at the transition, i.e. the system becomes incompressible. Another change is that the frequency cutoﬀ is simply given by the temperature. The critical exponents determined from experiment vary widely, from s = 0.5 [44], s = 1 [45] to s = 1.6 [46, 47] and from z = 2 [46] to z = 2.94 [47]. . 2 6. Symmetry classes. – So far we have mostly considered systems with all the available symmetries. As one may expect from a comparison to conventional thermodynamic phase transitions, the critical behavior will depend on which of the symmetries are present in a given system. The systems may be grouped into symmetry classes and within each class the critical behavior is universal. Symmetry classes for disordered systems were ﬁrst introduced by [48] and [49] in the context of random Hamiltonian matrix models (in short Random Matrix Theory (RMT)). The corresponding scheme considers time reversal (T ) and spin rotation (S) symmetries. There are three possible combinations: 1) T and S preserved: Gaussian orthogonal ensemble (GOE) (the Hamiltonian matrices are of orthogonal character). 2) T is violated: Gaussian unitary ensemble (GUE). 3) T preserved, but S violated: Gaussian symplectic ensemble (GSE). In the presence of a strong magnetic ﬁeld, GUE is realized, while in the presence of spin-orbit interaction GSE has to be applied. Additional symmetries arise, when the eﬀective ﬁeld theory has additional degrees of freedom [50]. For example electrons on a bipartite lattice may be labeled by an isospin index specifying the sublattice. Under certain conditions, e.g. disorder only in the hopping matrix elements, the Hamiltonian matrix acquires a “chiral” symmetry. Three chiral classes may be identiﬁed. A further class of symmetries may arise for the eﬀective Hamiltonian of Bogolyubov quasiparticles in superconductors. One ﬁnds four distinct classes of this socalled “Bogolyubov-de Gennes” type (for a detailed discussion, see Evers and Mirlin [51]). The ten symmetry classes may in fact be related to the known classical symmetric spaces [52]. . 2 7. Non-linear σ-model . – Any continuous phase transition is characterized by critical modes, which interact more and more strongly, as the transition is approached. In the case of the Anderson localization transition the diﬀusion modes play the role of critical modes. An eﬀective ﬁeld theory should therefore be formulated in terms of (bosonic) diﬀusion modes. A ﬁeld-theoretic description of the Anderson model is, however, complicated by the fact that there is only one particle in the system at any time (the contribution

15

Anderson localization

of the many independent electrons in a metal may be added at the end of the calculation). Technically speaking this implies that there are no internal closed loop diagrams allowed. These unwanted contributions, appearing naturally in any quantum ﬁeld theory, may be projected out by one of three known methods. The ﬁrst method, used by Wegner [28] in his pioneering work, is the so-called replica trick. There one considers in addition to the system under consideration N − 1 identical replicas. Since any internal loop contribution is proportional to the number of replicas, N , by taking the limit N → 0 at the end the closed loop contributions are projected out. A mathematically better deﬁned procedure is obtained by adding to each Bose-type ﬁeld a corresponding Fermi-type partner [61,29]. For any closed loop the contributions of Boson and Fermion ﬁelds cancel. In the latter formulation the diﬀusion propagator may be represented as (31)

A bb −S[Q] ( r , r )G ( r , r ) = d[Q]Qbb , Φ(r, r ; ω) = GR 12 Q21 e E+ω/2 E−ω/2

where the dynamics of the 4 × 4 matrix ﬁeld Q is deﬁned by the action of a non-linear σ-model (32)

S[Q] =

πN (E) 4

dd rStr − D(∇Q)2 − 2iωΛQ ,

where D is the diﬀusion coeﬃcient, Λ = diag(1, 1, −1, −1), the matrix is normalized, Q2 = 1, and Qbb 12 denotes the boson-boson element of the retarded-advanced block (for details, see [61, 29]). So far the above non-linear σ-model has been solved for simpler limiting cases such as zero-dimensional and inﬁnite-dimensional systems, weak disorder, and to some extent systems close to the lower critical dimension d = 2, to name a few important examples. . 2 8. Fractal structure of critical wave functions. – The spatial structure of the wave function at the critical point has been studied extensively (for a recent review, see [51]). As early as 1980, Wegner introduced the idea that the critical wave functions have a multifractal structure [53]. This means that the inverse participation ratios (IPR) Pq deﬁned by (33)

Pq =

dd r|ψ(r )|2q ,

q real,

show anomalous scaling, when averaged over disorder: (34)

Pq = Ld |ψ(r )|2q ∼ L−τq ,

where a continous set of exponents τq has been introduced. One may deﬁne the fractal dimensions Dq via Dq = (q − 1)/τq . In a metal one has Dq = d, while Dq = 0 in the insulating phase. The spectrum of fractal dimensions may be characterized by a spectral

16

¨ lfle P. Wo

function f (α). The function f (α) characterizes the probability function of intensities |ψ|2 , which may be expressed as (35)

P (|ψ|2 ) ∼

1 −d+f (− ln |ψ2 |/ ln L) L . |ψ|2

Calculating the moments |ψ|2q of the distribution P , one ﬁnds (36)

Pq ∼

dαL−qα+f (α) .

In the limit of L → ∞ the α-integral may be done using a saddle-point approximation, resulting in (37)

τq = qα0 − f (α0 ),

where α0 is obtained from q = f (α0 ). The spectral function f (α) is a convex function deﬁned for α ≥ 0, with a maximum at α0 , where f (α0 ) = d. The function f (α) has been determined in numerical studies for a number of cases of interest [51]. . 2 9. Anderson transition in the kicked rotor model. – A very useful analogue of localization in a random potential in real space is localization of particles in momentum space. This has been achieved in a system of atoms subject to a one-dimensional spatially periodic potential varying periodically in a pulsed fashion in time (kicked rotor model). The periodic kicking gives rise to exponential localization in momentum space (“dynamical localization”) [62], provided the kicking is suﬃciently strong driving the system into the chaotic diﬀusion domain. Dynamical localization has been shown to be the analogue of Anderson localization in one dimension [63]. The experimental observation of dynamical localization in a system of cold atoms [64] may be considered as the ﬁrst realization of Anderson localization with atomic matter waves. In order to access the Anderson transition in the kicked rotor setup the “dimension” of the system has to be increased. This may be done by introducing additional incommensurate periodicities, as expressed by the following Hamiltonian: (38)

N −1 p2 + K cos x[1 + ε cos(ω2 t) cos(ω3 t)] δ(t − n). H= 2 n=0

Here x, p are the particle position and momentum, K is the pulse intensity, ω2 , ω3 are incommensurate frequencies, and appropriate units of mass, length and time have been chosen. The above system has been shown to be essentially equivalent to the Anderson model in three dimensions, with K and ε playing the role of inverse disorder strength and eﬀective dimensionality [65], the limit ε = 0 corresponding to one dimension. The phase diagram in the ε-K plane shows regions of localized and extended states separated by a phase boundary.

17

Anderson localization

The above model has been realized with cesium atoms in a magneto-optical trap [66]. By monitoring the momentum distribution as a function of time, the two regions in the phase diagram were clearly identiﬁed. It was possible to extract the critical behavior with suﬃcient accuracy to allow for a determination of the critical exponent ν = 1.4 ± 0.3, comparing rather well with the numerically determined value of ν = 1.6 ± 0.05. 3. – Theory of localization: diagrammatic approaches The ﬁeld-theoretic description in terms of the non-linear σ-model mentioned in the beginning of sect. 2 is believed to be an exact framework within which the critical properties of the Anderson transition may be, in principle, exactly calculated. The mapping of the initial microscopic model onto the NLσM requires a number of simpliﬁcations, so that the non-critical properties like the critical disorder ηc , or the behavior in anisotropic systems, or systems of ﬁnite extension are no longer well represented by the NLσM. In addition, as mentioned already, it is not known how to solve the NLσM in cases of major interest, such as in d = 3 dimensions. It is therefore useful to consider approximation schemes, which on one hand keep the information about the speciﬁc properties of the system and on the other hand account approximately for the critical properties at the transition. Such a scheme is available at least for the orthogonal ensemble (in which both, time reversal and spin rotation symmetry are conserved). This approach has been developed in [31] and is reviewed in [32]. It may be termed “self-consistent one-loop approximation” in the language of renormalization group theory, but has in fact been derived following a somewhat diﬀerent logic. . 3 1. Self-consistent theory of localization. – The appropriate language to formulate a microscopic theory of quantum transport or wave transport in disordered media is renormalized perturbation theory in the disorder potential. The building blocks of this theory for the model deﬁned by eq. (2) are: i) the renormalized one-particle retarded (advanced) Green’s functions averaged over disorder (39)

−1 R,A 2 , GR,A k (E) = E − k /2m − Σk (E)

A ∗ where ΣR k (E) = (Σk (E)) is the self-energy, and ii) the random potential correlator 2 V . The self-energy Σ is a non-critical quantity and can be approximated by ΣR k (E) −i/2τ , where 1/τ is the momentum relaxation rate entering the Drude formula of the conductivity (assuming isotropic scattering). The quantity of central interest here is the diﬀusion coeﬃcient D. It follows from very general considerations [54] that the density response function describing the change in density caused by an external space- and time-dependent chemical potential χ = δn/δμ is given by (in Fourier space)

(40)

χ(q, ω) =

D(q, ω)q 2 χ0 , −iω + D(q, ω)q 2

18

¨ lfle P. Wo

where D(q, ω) is a generalized diﬀusion coeﬃcient and NF is the density of states at the Fermi level. The form of χ is dictated by particle number conservation. χ may be expressed in terms of GR,A as χ(q, ω) = −

(41)

ω Φ k k (q, ω) + χ0 , 2πi

k, k

where χ0 = NF is the static compressibility (which is non-critical in the model of noninteracting particles) and Φ may be expressed in terms of the irreducible vertex function U as ⎤ ⎡ A ⎣ U k k (q, ω)Φ k k (q, ω)⎦ , (42) Φ k k (q, ω) = GR k+ Gk− δ k, k +

k

where k± = (k ±q/2, E ±ω/2). In terms of diagrams U is given by the sum of all particlehole irreducible diagrams of the four-point vertex function. By expressing GR GA as A GR k+ Gk− =

(43)

ΔGk ω − k · q/m − ΔΣk

,

A R A where ΔGk = GR k+ − Gk− and ΔΣk = Σk+ − Σk− , one may rewrite eq. (39) in the form of a kinetic equation

(44)

k · q − ΔΣk ω− m

⎡

Φkk = −ΔGk ⎣δ k k +

⎤ U k k Φ k k ⎦ .

k

By summing eq. (42) over k, k one ﬁnds the continuity equation ωΦ(q, ω) − qΦj (q, ω) = 2πiNF ,

(45)

with the density and the current density relaxation functions (46)

Φ(q, ω) =

Φ k k (q, ω);

Φj (q, ω) =

k, k

k · qˆ

k, k

m

Φ k k (q, ω),

where qˆ = q/|q |. Here the Ward identity ΔΣk = k U k k ΔG k has been used [31]. In the hydrodynamic limit, i.e. ωτ 1, q 1, the current density is proportional to the gradient of the density (47)

Φj + iqD(q, ω)Φ = 0.

19

Anderson localization

In fact, multiplying eq. (42) by k · q/m and summing over k and k , one may derive relation (47) and by comparison one ﬁnds (48)

D0 /D(q, ω) = 1 − η

2E A A (k · qˆ)GR q , ω)GR ˆ), Gk (k · q k+ Gk− Ukk ( k+ − mn

k, k

1 where the disorder parameter η = πNF V 2 = 2πEτ and D0 = d1 v 2 τ is the bare diﬀusion coeﬃcient. As the Anderson transition is approached, the left-hand side of eq. (48) will diverge (at q, ω → 0), and therefore the irreducible vertex U has to diverge. The leading divergent contribution to U is given by the set of diagrams obtained by using a property of the full vertex function Γ (the sum of all four-point vertex diagrams) in the presence of time reversal symmetry:

(49)

Γ k k (q, ω) = Γ( k− k + q)/2,( k − k+ q)/2 (k + k , ω).

This relation follows if one twists the particle-hole (p-h) diagrams of Γ so that the lower line has its direction reversed, i.e. the diagram becomes a particle-particle (p-p) diagram. Now, if time-reversal symmetry holds, one may reverse the arrow on the lower Green’s function lines if one lets k → −k at the same time. This operation transforms p-p diagrams back into p-h diagrams, so that an identity is established relating each diagram Γ1 to its transformed diagram ΓT 1 , which is the above relation. The leading singular diagrams of Γ are those leading to the diﬀusion pole (50)

ΓD =

1 1 , 2πNF τ 2 −iω + Dq 2

where D is the true diﬀusion coeﬃcient. These diagrams are of the ladder type and therefore reducible Their transformed counterparts ΓT D are, however, irreducible and contribute to U . We may therefore approximate the singular part of U by (51)

sing Ukk =

1 1 . 2 2πNF τ −iω + D(k + k )2

In low-order perturbation theory U sing is given by the “maximally crossed diagrams”, which when summed up give a result U sing,0 similar to eq. (51), with D replaced with the bare diﬀusion constant D0 . When U sing,0 is substituted as a vertex correction into the conductivity diagram, the result is exactly the weak-localization correction discussed in sect. 1. When U sing from eq. (51) is substituted for U , eq. (48) for the diﬀusion coeﬃcient D(ω) (in the limit q → 0) leads to the following self-consistent equation for D(ω): (52)

k 2−d D0 =1+ F D(ω) πm

1/

dQ 0

Qd−1 . −iω + D(ω)Q2

20

¨ lfle P. Wo

Here we have assumed that a ﬁnite limit limq→0 D(q, ω) = D(ω) exists, and that Q is limited to 1/ in the diﬀusive regime. Equation (52) may be reexpressed as (53)

D(ω) = 1 − ηdkF2−d D0

1/

dQ 0

Qd−1 . −iω/D(ω) + Q2

It is useful to express (52) in position-energy space as [60]

(54)

k 2−d D0 = 1 + 2π F C(r, r ), D(ω) m

where C(r, r ) is a solution of the diﬀusion equation (where a cutoﬀ Q < 1/ has to be applied to the spectrum of Q modes): (55)

[−iω + D(ω)∇2 ]C(r, r ) = δ(r − r ).

The above formulation allows to describe position-dependent diﬀusion processes, as appear near the sample surface in a conﬁned geometry, e.g. transmission through a slab. In that case the diﬀusion coeﬃcient may be taken to be position-dependent, D = D(r, ω), and C(r, r ) obeys the modiﬁed diﬀusion equation [67] (56)

[−iω + ∇D(r, ω)∇]C(r, r ) = δ(r − r ).

The solution is subject to an appropriate boundary condition at the surface of the sample. The theory accounts very well for the localization properties of acoustic waves transmitted through a strongly scattering plate [67]. . 3 2. Results of the self-consistent theory of localization. – In d = 3 dimensions eq. (53) has a solution in the limit ω → 0 1 η , η < ηc = √ , (57) D = D0 1 − ηc 3π indicating that the critical exponent of the conductivity is s = 1 in this approximation. The ω-dependence of D(ω) at the critical point is obtained as [55] (58)

D(ω) = D0 (ωτ )1/3 ,

η = ηc ,

implying a dynamical critical exponent z = 3, which is the exact result. At stronger disorder, η > ηc , all states are localized and the localization length ξ deﬁned by ξ −2 = limω→0 (−iω/D(ω)) is found as −1 η π 1 − , 2 ηc

√ (59)

ξ=

21

Anderson localization

so that the exponent ν = 1. For general d in the interval 2 < d < 4, one ﬁnds Wegner scaling, s = ν(d − 2). In dimensions d ≤ 2, there is no metallic-type solution. The localization length is found as 1/2 1 (60) , d = 2, ξ = exp − 1 η ξ∼ d = 1, = c1 , where the coeﬃcient c1 ≈ 2.6, while the exact result is c1 = 4 [7]. The RG β-function has been derived from the self-consistent equation for the lengthdependent diﬀusion coeﬃcient, where a lower cutoﬀ 1/L has been applied to the Qintegral in eq. (50). The result is [56] in d = 3 dimensions in the metallic regime (61)

β(g) =

g − gc , g

g > gc =

1 π2

and in the localized regime (62)

β(g) = 1 −

x2 1 1 + x −x , e − π 2 g 1 + x2 1+x

g < gc ,

where x = x(g) is the inverse function of (63)

g=

1 1 −x . 1 − x arctan (1 + x)e π2 x

The β-functions in d = 1, 2, 3 obtained in this way are shown in ﬁg. 4. . 3 3. Destruction of localization by decoherence processes. – At any ﬁnite temperature inelastic processes, or more precisely, dephasing processes limit the phase coherence of particles or wavepackets to a ﬁnite time interval τφ or equivalently, a characteristic length Lφ . An important mechanism of dephasing for electrons in disordered metals is the Coulomb interaction between electrons. Its contribution to 1/τφ may be estimated by the following argument [57]: an electron moving through the system is subject to a ﬂuctuating electric potential δV (t) resulting from the Coulomb interaction with all the other electrons. The corresponding energy shift eδV (t) leads to a phase shift in the wave function (64) Δφ(t) = dt1 eδV (t1 ). Assuming δV to be Gaussian distributed, the phase factor averages as ! 1 iΔφ 2 (65) e = exp − (Δφ) , 2 where (66)

(Δφ)2 = e2

t0

dt1 0

0

t0

! ! t dt2 δV (t1 )δV (t2 ) = 2 . τφ

22

¨ lfle P. Wo

The thermal ﬂuctuations of the voltage (Nyquist noise) are given by (67)

! δV (t1 )δV (t2 ) = T RT δ(t1 − t2 ),

where T is the temperature and RT is the resistance of a volume of the size explored by the particle in time h ¯ /T , which for a diﬀusing particle is LT = (D/T )1/2 . In d = 3 dimensions, RT = (σLT )−1 , where σ may be approximated by the Drude conductivity. It follows that (68)

1 1 1 ∼ √ τφ τ EF τ

T EF

3/2 ,

d = 3.

This result should be contrasted with the standard Fermi liquid result for the inelastic scattering rate in the clean system, τ1 ∼ T 2 . In d = 2 dimensions RT = (σa)−1 , where a ∼ kF−1 is the thickness of the ﬁlm. Then (69)

1 1 T ∼ , τφ τ EF

d = 2.

A more quantitative calculation leads to an additional factor of ln(EF τ ) [24]. Finally, in d = 1 dimensions it turns out that LT > Lφ . The relevant volume of the resistor is therefore limited by Lφ , no longer by LT , and RT = kF2 Lφ /σ. As a result, one ﬁnds a self-consistent relation for 1/τφ , the solution of which is given by (70)

1 1 ∼ (T τ )2/3 , τφ τ

d = 1.

We note that in all dimensions 1/τφ tends to zero as T → 0. This appears to be in agreement with observation. In some cases a plateau behavior of 1/τφ has been found in experiment, which gave rise to the speculation that the zero-point ﬂuctuations may cause decoherence. Given a unique ground state, it is hardly possible for a particle in the system to lose its phase coherence. Several physical mechanisms that may lead to a plateau of 1 τφ versus T have been identiﬁed. For a recent discussion of these issues, see [58]. 4. – Conclusion After ﬁfty years of intense and wide-ranging studies Anderson localization is still a lively ﬁeld of research. Most of the current interest is concentrated on systems diﬀerent from the conventional disordered electron system, namely classical waves (light, electromagnetic microwaves, acoustic waves) or ultracold atoms. Even though the fundamental concepts of Anderson transition are probably well understood by now, there still remain a number of open questions. Some of those are related to the analytical theory of critical properties near the Anderson transition. Others concern the quantitative theory for realistic materials, e.g. the question under which conditions precisely light or acoustic

23

Anderson localization

waves become localized. As Anderson localization is a wave interference phenomenon, the limitations of phase coherence are an important subject of study in this context. By now Anderson localization has been observed in many diﬀerent systems beyond doubt. On the other hand, observation of the Anderson transition separating localized and extended states is a much more challenging task. Here the more recently studied classical waves and atomic matter waves oﬀer new perspectives, which will undoubtedly lead to a deeper understanding of the localization phenomenon. ∗ ∗ ∗ I would like to thank my long term collaborators on the subject of Anderson localization, H. Kroha, K. Muttalib, C. Soukoulis and D. Vollhardt, for many fruitful discussions. REFERENCES [1] Anderson P. W., Phys. Rev., 109 (1958) 1492. [2] Lee P. A. and Ramakrishnan T. V., Rev. Mod. Phys., 57 (1985) 287. [3] Altshuler B. L. and Aronov A. G., Electron-Electron Interactions in Disordered Systems, in Modern Problems in Condensed Matter Sciences, Vol. 10, edited by Efros A. L. and Pollak M. (North-Holland, Amsterdam) 1985. [4] Chakravarty S. and Schmid A., Phys. Rep., 140 (1986) 193. [5] Mott N. F. and Twose W. D., Adv. Phys., 10 (1961) 107. [6] Borland R. E., Proc. R. Soc. London, Ser. A., 274 (1963) 529. [7] Berezinskii V. L., Sov. Phys. JETP, 38 (1974) 620. [8] Mott N. F., Metal-Insulator Transitions (Taylor and Francis, London) 1974. [9] Mott N. F. and Davis E. A., Electronic Processes in Non-crystalline Materials (Clarendon Press, Oxford) 1979. [10] Kotliar G. and Vollhardt D., Physics Today, 57 (2004) 53. [11] Byczuk K., Hofstetter W. and Vollhardt D., Phys. Rev. Lett., 94 (2005) 056404. [12] Bulka B., Schreiber M. and Kramer B., Z. Phys. B, 66 (1987) 21. ¨ lfle P., Phys. Rev. B, 41 (1990) 888. [13] Kroha J., Kopp T. and Wo ¨ lfle P., Phys. Rev. B, 47 (1993) 11093. [14] Kroha J., Soukoulis C. M. and Wo [15] Gorkov L. P., Larkin A. I. and Khmelnitskii D. E., JETP, 30 (1979) 248. [16] Landauer R., IMB J. Res. Rev., 1 (1957) 223. [17] Landauer R., Philos. Mag., 21 (1970) 683. [18] Anderson P. W., Thouless E., Abrahams E. and Fisher D. S., Phys. Rev. B, 22 (198) 3519. [19] Altshuler B. L. and Prigodin V. N., JETP Lett., 45 (1987) 687. [20] Dorokhov U. N., JETP Lett., 36 (1982) 318. [21] Mello P. A., Pereyra P. and Kumar N., Ann. Phys. (N.Y.), 181 (1988) 290. [22] Beenakker C. W. J., Rev. Mod. Phys., 69 (1997) 731. ¨ lfle P. W. and Gopar V. A., Ann. Phys. (N.Y.), 308 (2003) 156. [23] Muttalib K. A., Wo [24] Altshuler B. L., JETP Lett., 41 (1985) 648. [25] Lee P. A. and Stone A. D., Phys. Rev. Lett., 55 (1985) 1622. ¨ lfle P. W., Phys. Rev. Lett., 83 (1999) 3013. [26] Muttalib K. A. and Wo [27] Wegner F., Z. Phys. B, 25 (1976) 327. [28] Wegner F., Z. Phys. B, 35 (1979) 207.

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[29] Efetov K. B., Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge) 1997. [30] Mirlin A. D., Phys. Rep., 326 (2000) 259. ¨ lfle P. W., Phys. Rev. B, 22 (1980) 4666. [31] Vollhardt D. and Wo ¨ lfle P. W., Electronic Phase Transitions (Modern Problems in [32] Vollhardt D. and Wo Condensed Matter Sciences), Vol. 32 (North-Holland, Amsterdam) 1992. [33] Thouless D., Phys. Rep., 13 (1974) 93. [34] Abrahams E., Anderson P. W., Licciardello D. and Ramakrishnan T. V., Phys. Rev. Lett., 42 (1979) 673. [35] Gorkov L. P., Larkin A. I. and Khmelnitskii D. E., JETP Lett., 30 (1979) 248. [36] Pruisken A. M. M., in The Quantum Hall Eﬀect, edited by Prange S. and Girvin S. (Springer, Berlin) 1987. [37] Hikami S., Larkin A. I. and Nagaoka Y., Progr. Theor. Phys., 63 (19810) 707. ¨ lfle P., Phys. Rev. Lett., 48 (1982) 699. [38] Vollhardt D. and Wo [39] Kramer B. and MacKinnon A., Rep. Progr. Phys., 56 (1993) 1469. [40] Slevin K. and Ohtsuki T., Phys. Rev. Lett., 82 (1999) 382. [41] Shapiro B. and Abrahams E., Phys. Rev. B, 24 (1981) 4889. [42] Imry Y., Gefen Y. and Bergman D., Phys. Rev. B, 26 (1982) 3436. [43] Thouless D., Phys. Rev. Lett., 39 (1977) 1167. [44] Paalanen M. A. et al., Phys. Rev. Lett., 48 (1982) 1284. [45] Field S. B. and Rosenbaum T. F., Phys. Rev. Lett., 55 (1985) 522. [46] Bogdanovich S., Sarachik M. P. and Bhatt R. N., Phys. Rev. Lett., 82 (1999) 137. ¨ hneysen H., Phys. Rev. Lett., 83 (1999) [47] Waffenschmidt S., Pfleiderer C. and v. Lo 3005. [48] Wigner E., Ann. Math., 53 (1951) 36. [49] Dyson F. J., J. Math. Phys., 3 (1962) 140. [50] Gade R. and Wegner F., Nucl. Phys. B, 360 (1991) 213. [51] Evers F. and Mirlin A. D., Rev. Mod. Phys., 80 (2008) 1355. [52] Zirnbauer M. R., J. Math. Phys., 37 (1996) 4986. [53] Wegner F., Z. Phys. B, 36 (1980) 209. [54] Forster D., Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, Reading) 1975. [55] Shapiro B., Phys. Rev. Lett., 48 (1982) 823. ¨ lfle P., Phys. Rev. Lett., 48 (1982) 699. [56] Vollhardt D. and Wo [57] Altshuler B., Aronov A. G. and Khmelnitskii D. E., J. Phys. C, 15 (1982) 7367. [58] von Delft J., Int. J. Mod. Phys. B, 22 (2008) 727. [59] Ioffe A. F. and Regel A. R., Prog. Semicond., 4 (1960) 237. [60] Skipetrov S. E. and van Tiggelen B. A., Phys. Rev. Lett., 96 (2006) 043602. [61] Efetov K. B., Adv. Phys., 32 (1983) 53. [62] Casati G., Chirikov B. V., Ford J. and Izrailev F. M., Stochastic Behavior of a Quantum Pendulum Under Periodic Perturbation (Springer, Berlin) 1979, p. 334. [63] Grempel D. R., Prange R. E. and Fishman S., Phys. Rev. A, 29 (1984) 1639. [64] Moore F. L., Robinson J. C., Bharucha C. F., Sundaram B. and Raizen M. G., Phys. Rev. Lett., 75 (1995) 4598. [65] Casati G, Guarneri I. and Shepelyansky D. L., Phys. Rev. Lett., 62 (1989) 345. [66] Chabe J., Lemarie G., Gremaud B., Delande D., Szriftgiser P. and Garreau J. C., Phys. Rev. Lett., 101 (2008) 255702. [67] Hefei Hu, Strybulevych A., Page J. H., Skipetrov S. E. and van Tigelen B. A., Nature Phys., 4 (2008) 945.

Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-25

Photonic properties of non-crystalline solids P. J. Steinhardt Department of Physics & Princeton Center for Theoretical Science, Princeton University Princeton, New Jersey, 08544 USA

Summary. — Photonic crystals, periodic arrangements of two or more dielectric materials, have been studied for more than two decades as a means of controlling and manipulating the ﬂow of light. These lectures describe recent progress in designing non-periodic photonic solids. The aim is to ﬁnd arrangements of dielectric materials that produce substantial complete photonic band gaps that block light in all directions and for all polarizations over a range of frequencies. Methods are described for constructing quasicrystalline examples with a wide range of rotational symmetries and a special class of isotropic, disordered photonic band gap materials.

Photonic band gap materials [1-3] are heterostructures composed of two or more materials with diﬀerent dielectric constants arranged in a spatial conﬁguration that forbids the propagation of electromagnetic waves in a certain frequency range. They are the photonic analogue of semiconductors, which are characterized by a ﬁnite electronic band gap, and, hence, are potentially as important for photonic applications as semiconductors are for electronic ones. Since their introduction in 1987, the understanding of photonic band gap materials has evolved dramatically and their unusual properties have been considered for use in eﬃcient radiation sources [4], sensors [5], and optical computer chips [6]. c Societ` a Italiana di Fisica

25

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P. J. Steinhardt

To date, though, the only known dielectric heterostructures with sizable, complete band gaps (Δω/ωC ≥ 10%, say, where Δω is the width of the band gap and ωC is the midpoint frequency) have been crystalline (periodic) [1-3]. Crystalline structures have a limited range of possible rotational symmetries and defect properties critical for controlling the ﬂow of light in applications. For this reason, there has been increasing interest in non-crystalline photonic band gap materials. In these lectures, we will discuss two classes that have been studied: quasicrystals, with long-range quasiperiodic translational order and discrete rotational symmetries that are forbidden for crystals (such as ﬁve-fold symmetry in two dimensions and icosahedral symmetry in three dimensions); and a novel class of isotropic disordered structures known as “hyperuniform.” As will be described below, examples of substantial complete photonic band gaps (all direction and polarizations) have been found in both classes. The disordered case is especially surprising since band gaps are normally associated with translational order. The resulting non-crystalline photonic band gap materials have distinctive properties that are of interest both for basic physics and for applications. For example, because the non-crystalline heterostructures are more isotropic (circular or spherical), they can produce a wider band gap under some conditions. Also, compared to photonic crystals, they generically have a more isotropic band gap, new types of photonic modes and new kinds of defects. All of these distinctive properties may be desirable for some applications. This is a new direction in the study of photonic materials, one that has already oﬀered some surprises and may present more in the future. Also, many of the same principles can be applied to phononics, electronic, plasmonic, shear wave and other physical properties. For these reasons, the subject is opportune for bright students. I caution that my personal expertise is in exploring novel patterns, packings and structures and characterizing the mathematical properties relevant for physics. I am still a novice in the subject of photonics, only slightly ahead of (most of) the students to whom I am lecturing. The lectures represent work in which I have been directly involved over the last few years and surely omit valuable work by other groups with which I am not yet familiar; I apologize for these omissions and encourage students to investigate the literature.

1. – Some basics The physics underlying photonic band gaps is, in many ways, analogous to the physics of electronic band gaps even though photonics involves purely classical wave phenomena and electronic band gaps are quantum in nature. Electronic band gaps in the independent electron approximation arise from solutions to the Schr¨ odinger equation

(1)

h2 2 ¯ ∇ + V (r) ψ(r) = Eψ(r), − 2m

Photonic properties of non-crystalline solids

27

where V (r) is the electron potential created by a conﬁguration of atomic nuclei, ψ is the electron wave function, and E is the energy of an electron eigenstate. An electronic stop gap exists if there is a ﬁnite range of energies for which there is no eigenstate for any wave vector k along some direction. An electronic band gap occurs if there is a ﬁnite range of energies for which is there no eigenstate for any wave vector k, independent of direction. Photonic band gaps arise from solutions of the classical Maxwell equations

ω 2 1 ∇ × H(r) = (2) H(r), ∇×

(r) c

ω 2 × ∇× ∇ E(r) = (r) E(r), c

E} are the magnetic and electric ﬁelds, respectively; (r) deﬁnes the distribuwhere {H, tion of dielectric material in the photonic material; and c is the speed of light. Typically,

(r) takes on two possible values correspond to the two dielectric constants of the dielectric materials used to make the heterostructure. A photonic stop gap occurs of there is a range of ω for which there is no solution for any wave vector k along some direction. A photonic band gap occurs if there is a range of ω for which there is no solution for any wave vector k, independent of direction. Although the structures of the Schr¨ odinger and Maxwell equations are similar, leading to many analogies between electronic and photonic band gaps, let us note here some diﬀerences. First, one equation is quantum, causing the band gap to shrink to zero as ¯h → 0; the Maxwell equations are purely classical. The Schr¨ odinger equation describes a spin-1/2 particle, though ψ is treated as a scalar in this approximation; Maxwell’s equations describe a massless vector quantity that can be expressed as a combination of two polarizations, conventionally labeled TM (transverse magnetic) and TE (transverse electric) models. For two-dimensional photonic materials (or three-dimensional materials with azimuthal symmetry), where the wave vector lies in the plane, the labeling convention is that TM corresponds to modes in which the electric ﬁeld oscillates in and out of the plane; and TE corresponds to modes in which the electric ﬁeld oscillates within the plane. The Schr¨ odinger equation for electron propagation is a single-electron approximation; more precise treatments require inclusion of electron-electron interactions, which make the equations inherently non-linear. Maxwell’s equations are inherently linear: there is no photon-photon interaction. The electron problem involves massive quanta that satisfy a parabolic dispersion relation in vacuo, ω ∝ k 2 . The photonics problem entails light, which has a massless, linear dispersion relation in vacuo, ω ∝ k. For the Schr¨ odinger equation, the ratio h ¯ 2 /m, the electron Compton wavelength, sets a fundamental length scale of atomic dimensions. Hence, an electronic band gap structure is sensitive to the detailed arrangement of atoms and molecules and the interatomic and intermolecular forces, over which there is limited control. For Maxwell’s equations, the equations are scale-invariant (in the limit that the absorption length is negligible). Photonic materials can, therefore, be constructed from macroscopic materials that can be regarded as continuous (e.g., air and dielectric) and can be shaped at will. The

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P. J. Steinhardt

challenge is to ﬁnd the shapes and arrangements with optimal properties. Once the band structure for a given photonic heterostructure is known, the band structure for a rescaled heterostructure is the same, up to a rescaling of frequencies and wave vectors. This is useful experimentally: a trial heterostructure can be constructed on any convenient length scale where it is easy to measure the scattering radiation; and, then, the perfected design can be rescaled for the desired frequency range needed for applications. We will be interested in these lectures in examples of dielectric heterostructures that produce the largest photonic band gaps for a given symmetry of the structure. The largest band gaps occur when the wavelength of the radiation is comparable to the size and/or separation of the scatterers. Two properties are most important in determining the band structure. The ﬁrst is long-range translational order. For practical purposes, the degree of translational order in a structure can be determined by taking the Fourier transform of dielectric density (r). For crystals and quasicrystals, the transform is a discrete sum of Bragg peaks arranged in three-dimensional reciprocal space with the same symmetry as the structure. The periodicity of crystals insures that the wave vectors corresponding to the Bragg peaks for a d-dimensional structure can be expressed as an integer linear combination of d basis wave vectors, where basis vectors are deﬁned to be integer linearly independent (none can be expressed as a integer linear combination of the others). The Bragg wave vectors form a periodic reciprocal lattice with a non-zero minimum spacing between wave vectors along each direction. According to the well-known theorems of crystallography, crystals can have only a limited set of rotational symmetries. Quasicrystals are quasiperiodic structures with rotational symmetries that violate the crystallographic theorems. The ﬁrst examples of quasicrystalline solids were discovered in 1984 by Shechtman et al., who observed icosahedral symmetry with ﬁve-fold rotation axes in the electron diﬀraction pattern of an alloy of Al-Mn [7]. Independently, Levine and Steinhardt [8] introduced the concept of quasicrystals and, then, showed that the Al-Mn alloy ﬁt well the predictions for an icosahedral quasicrystal. See ref. [9] for an introduction and collection of foundational papers. Over the last twenty-ﬁve years, over a hundred other quasicrystalline alloys have been synthesized in the laboratory [10], and, recently, there is evidence of a natural quasicrystal [11]. Also, there have been notable studies of impermanent non-linear photonic quasicrystals [12] produced by optical induction (only existing when arrays of laser beams shine on non-linear optical material) that are beyond the scope of these lectures. For photonic materials, quasiperiodic means that (r) can be expressed as a ﬁnite sum of periodic functions whose periods are incommensurate (the ratio of the periods is an irrational number). Because they are not periodic, quasicrystals evade the crystallographic theorems and can exhibit any crystallographically forbidden symmetry, including ﬁve-fold symmetry in two-dimensions and icosahedral symmetry in three dimensions, as will be illustrated below. Yet, because (r) can be decomposed into a sum of periodic functions, the Fourier transform is a discrete sum of Bragg (δ function) peaks, with some signiﬁcant diﬀerences compared to the crystal case. First, the wave vectors can be expressed as a integer sum of basis wave vectors, but the number of (integer linearly

Photonic properties of non-crystalline solids

29

independent) basis vectors is greater than the number of dimensions. For example, four basis vectors are required for the two-dimensional quasicrystal with ﬁve-fold symmetry (e.g., any four vectors pointing from the center of a pentagon to the vertices), and six basis wave vectors deﬁne the reciprocal lattice for three-dimensional quasicrystals with icosahedral symmetry (e.g., along the six independent ﬁve-fold symmetry axes). As for crystals, the reciprocal space pattern has the same rotational symmetry as the real space structure. However, by constructing this reciprocal lattice, it is easy to see there is no minimum distance between Bragg peaks; between any pair of Bragg peaks there are more Bragg peaks. The heights (or intensities) of the Bragg peaks vary so that there is a clear hierarchy of bright to arbitrarily dim peaks. For crystals, the primitive unit cell of the reciprocal lattice is known as the Brillouin zone. It corresponds to the polygon formed by joining the perpendicular bisectors of the segments between the origin of the reciprocal lattice and the reciprocal lattice points nearest to it. For the quasicrystal, there are no “nearest” points to the origin since between any two points are more points. In practice, one joins the bisectors of the segments between the origin and the brightest points about the origin with G ∼ 2π/a. The irreducible Brillouin zone is the smallest section of the Brillouin zone that contains all the symmetry axes of the complete zone. Figure 1 illustrates the band structure for a photonic crystal composed of silicon with an array of air holes arranged in a diamond (FCC) lattice. The inset shows the Brillouin zone of the diamond structure with various symmetry directions indicated. The band structure plot shows the frequency ω of all the photonic modes as the wave vector k sweeps a path along the Brillouin zone that passes through the various symmetry directions. Along any given direction, there is range of frequencies which contains no modes; the range is the stop gap. There is also the range of frequency (indicated by the shaded strip) which contains no modes along any direction; this is the complete photonic band gap. Long-range translational order and Bragg scattering are one contribution to band gap formation. Another is resonant scattering by individual dielectric elements. The Mie resonance eﬀect, described in other lectures in this School, becomes stronger as the dielectric contrast increases since this concentrates the electric ﬁeld inside the high dielectric components. For certain wavelengths and nearest-neighbor arrangements of dielectric components, the Mie resonances by neighboring dielectric elements cause interference of electromagnetic waves that block propagation. The largest possible photonic band gaps result from the interplay between Bragg and Mie scattering. However, the fact that Mie resonance relies only on the short-range geometric order makes it conceivable that band gaps can occur in photonic materials with no long-range order. Although completely random (Poisson) distribution of dielectric components fails because there is insuﬃcient local order for the Mie resonance to be effective, we will see that there exists a special class of isotropic, disordered “hyperuniform” structures that can have substantial complete band gaps.

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Fig. 1. – (Color online) The photonic band structure (above) for photonic crystal comprised of silicon with spherical air holes arranged in a diamond (FCC) lattice (below). The abscissa indicates the symmetry axes of the Brillouin, as shown in the inset of the upper ﬁgure. The thin strip, which contains no bands, is an example of a complete photonic band gap.

Photonic properties of non-crystalline solids

31

Fig. 2. – (Color online) Experimental photonic structures and their Brillouin zones. (a) Stereolithographically produced icosahedral quasicrystal with 1 cm length rods. (b) Diamond structure with 1 cm rods. (c) Triacontahedron, the eﬀective Brillouin Zones of icosahedral symmetry. (d) Brillouin zone for the FCC/diamond structure. The dashed and dotted lines are trajectories referenced in ﬁgs. 3 and 4.

2. – Photonic band gaps in icosahedral quasicrystals To motivate the theoretical studies in subsequent sections, we begin by describing an experimental investigation of photonic band gaps in three-dimensional icosahedral quasicrystals [13]; this is work done in collaboration with W. Man, M. Megens, and P. Chaikin. Although there had been some studies of 1D and 2D photonic quasicrystals [1417] prior to this investigation where exact (1D) or approximate (2D) band structures can be calculated numerically, analogous calculations for the 3D case require enormous computational resources and have not been performed to date. An alternative is to do a series of physical experiments. For example, using stereolithography, a photonic quasicrystal and a diamond (FCC) crystal with centimeter-scale cells have been constructed and, then, used as the target for a series of microwave transmission measurements. The quasicrystal consists of a quasiperiodic array of oblate and acute rhombohedra [8] that can be constructed by projecting the points from a sixdimensional hypercubic lattice into three dimensions, as described in ref. [9]. Figure 2 shows the eﬀective Brillouin zone of the icosahedral structure with its irreducible Brillouin zone highlighted in yellow. The eﬀective Brillouin zone is constructed

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Fig. 3. – (Color online) Measured transmission for an icosahedral quasicrystal. (a) T (f, θ), transmission as a function of frequency and angle, for a rotation about a 2-fold rotation axis of the quasicrystal (corresponding to the dotted line in the triacontahedron Brillouin zone shown in ﬁg. 2) using two overlapping frequency bands. The dashed line is a 1/ cos (θ) curve characteristic of Bragg scattering from a Brillouin zone face. (b) T (f, θ), for a rotation about a 5-fold rotation axis corresponding to the dashed line in the triacontahedron Brillouin zone. Inset: Schematic of the microwave horn and lens arrangement used for these measurements.

from the shell of brightest peaks nearest to k = 2π/a, where a is the edge length of the rhombohedra (about 1 cm). The Brillouin zone is a rhombic triacontahedron composed of 30 identical rhombi arranged with icosahedral symmetry. The triacontahedron, like the icosahedron, has 5-, 3-, and 2-fold symmetry axes, as illustrated in the indicated irreducible Brillouin zone. For comparison, the Brillouin zone of the diamond (FCC) structure with its irreducible Brillouin zone is also shown. Note that, as a measure of asphericity, along the edge of diamond structure’s irreducible Brillouin zone the magnitude of k increases by 29.1% from L to W. Along the edge of the eﬀective irreducible triacontahedral Brillouin zone of the icosahedral structure, the magnitude of k increases by only 17.5% from the 2-fold to the 5-fold symmetry points. Moreover the triacontahedron’s faces are identical and subtend smaller solid angles. The question to be considered is whether this higher degree of sphericity for the quasicrystal results in advantageous photonic properties. Figure 3 shows the transmission T (f, θ) of microwave radiation through the icosahedral sample as the frequency f and the angle of incidence θ are varied along the trajectories indicated in ﬁg. 2, which passes through the symmetry directions of the Brillouin zone. The rather complex spectra are consistent with Bragg scattering. As explained in the previous section, to lie on the Brillouin zone, a wave vector must satisfy the condition k·G = |G|2 |/2, or equivalently, |k| = |G|/(2 cos θ). To lowest order, the center frequency

Photonic properties of non-crystalline solids

33

Fig. 4. – (Color online) Imaging of Brillouin zone for diamond and icosahedral quasicrystal structures. (a) Brillouin zone for the diamond structure along 4-fold direction as seen in the contour plot of the calculated frequency deviation (δf = f − (c/¯ n)|k|/(2π) vs. k). (b)-(e): Brillouin zone in the plot of the measured T (r = f, θ) for the diamond lattice along the (b) 4-fold and (c) 2-fold axis; and for the quasicrystal along the (d) 5-fold and 2-fold axes. The inner decagon in (d) and the solid and dashed lines in (e) correspond to dashed and dotted lines indicated in the triacontahedral Brillouin zone. The dash-dotted line is a non-triacontahedral zone face.

of a stop gap is therefore fG = (c/¯ n)|G|/(4π cos θ), where c is the speed of light and n ¯ is the Bruggeman eﬀective medium index of the dielectric heterostructure [18]. The dashed curves in ﬁg. 3 correspond to a 1/ cos θ angular dependence consistent with Bragg scattering. The results for the quasicrystal spectrum are simpler than one might expect given that a quasicrystal has a dense set of Bragg spots (of zero measure), which was ignored in constructing the eﬀective Brillouin zone in ﬁg. 2. One might have expected many gaps and zone faces intersecting. On the contrary, the eﬀective Brillouin zone approximation appears to work well since there are only a few well-deﬁned 1/ cos θ curves in ﬁg. 3 and therefore few zone boundaries with sizable gap formation. To visualize the eﬀective Brillouin zone structure, the process is inverted by using the gaps to ﬁnd the zone faces, as shown in ﬁg. 4 for both the diamond and quasicrystal structures. The zone plots are made by making polar plots T (r = f, θ), as described in ref. [13]. The results show that there is a relatively well-deﬁned eﬀective Brillouin zone for the icosahedral quasicrystal, all faces of which are consistent with the quasicrystal Bragg pattern. Second, the Brillouin zone structure is surprisingly simple despite the fact that a quasicrystal has a dense set of Bragg spots. Third, and a key result for photonics, the

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measured Brillouin zone is close to spherical, with the largest diﬀerence in gap center corresponding to 17%, more spherical than for the best photonic crystal (diamond) for the same dielectric constant ratio. Also, our experiments demonstrate that three-dimensional quasicrystals exhibit sizable stop gaps on reasonably well-deﬁned eﬀective Brillouin zone faces. Hence, despite the quasiperiodicity, much of the intuition built up for conventional crystals may be applied. These empirical results are based on materials (polymer and air) with a relatively low dielectric constant ratio and with no attempt at ﬁnding structures with an optimally large band gaps. The results encourage a more systematic investigation of how symmetry and quasiperiodicity aﬀect photonic properties. At present, such studies are only tractable in 2D. The remainder of these lectures will focus on what has been learned in 2D.

3. – Dependence of band gap width on symmetry in 2D Studies of photonic band gaps in two dimensions (or, equivalently, for threedimensional materials with azimuthal symmetry) are amenable to analytic and computational methods, as well as empirical methods. The studies are not purely academic: the designs can be useful for some applications. However, the principal purpose of considering them in these lectures is to investigate how symmetry and the dielectric arrangement can aﬀect band gap properties. The wave vectors k propagate in the plane and the polarization can be conﬁned to purely TM (electric ﬁeld oscillating in and out of the plane), TE (electric ﬁeld oscillating within the plane) or combined polarizations. Unlike the case of three-dimensions, the two polarizations do not mix through scattering, which simpliﬁes the problem. Figure 5 illustrates 2D photonic structures with TM, TE and full photonic band gaps for photonic crystals. Rigorous computational searches show that the optimal TM band gaps for large dielectric constant ratio are obtained for arrays of cylinders of high dielectric embedded in a background (air) of low dielectric constant. In the frequency range of interest, the TM modes conﬁne the large electric-ﬁeld oscillations to the cylinders. For TE polarization, the optimum for photonic crystals is an in-plane network of high dielectric material obtained by making an array of isolated holes of low dielectric. A full photonic band gap (both polarizations) is a compromise between these two structures. This section describes a study [19] designed to identify and compare crystalline and quasicrystalline dielectric with the widest possible TM band gaps. The systematic approach is to treat the spatial conﬁguration of the two dielectric components as a ﬁnite sum of density waves, ρ(x), assigning regions where the sum exceeds a certain threshold a high dielectric constant ε1 and all other regions to the low value ε0 . As a practical matter, the method is most eﬃcient if the number of density waves for the optimal conﬁguration is small. This makes the method best suited for TM polarization for which the optimal conﬁgurations tend to have smooth features of a single length scale, like the examples of

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35

Fig. 5. – (Color online) Examples of optimal structures for 2D photonic crystals (or, equivalently, 3D photonic crystals with azimuthal symmetry) with band gaps (all directions) for TM, TE and both polarizations. Arrays of cylinders with high dielectric constant are optimal for TM only; arrays of holes in high dielectric constant material are optimal for TE only; a compromise is needed to obtain a complete (all polarizations and directions) band gap.

cylinders in air shown in ﬁg. 5. Thus, the dielectric constant at position x is given by (3)

ε(x) =

" ε1 , if Re{ρ(x)} > 1, ε0 , otherwise,

where ρ is a sum of plane waves (4)

ρ(x) =

n R

Ar exp [i(Gr,j · x + φr,j )] ,

r=1 j=1

and Ar is the (real) amplitude of the “ring” of reciprocal lattice vectors (deﬁned below) indexed by r; Gr,j is the j-th reciprocal lattice vector in ring r; φr,j is the phase of the j-th plane wave in ring r; R is the total number of rings employed; and the system has n-fold rotational symmetry. A ring is deﬁned as a set of reciprocal lattice vectors that have equal norms (for crystals) or approximately equal norms (for quasicrystal periodic approximants), and which have a predeﬁned rotational symmetry. In particular, the r-th ring with elements indexed by j and of wave number Gr is the set {Gr,j : |Gr,j | ≈ Gr , Gr,j ∈ {G}}, where {G} is the set of reciprocal lattice vectors. For example, the ﬁrst ring of the 6-fold symmetric hexagonal lattice is composed of six non-zero reciprocal lattice vectors closest to the origin in reciprocal space. The phases φr,j may be chosen to be the same for all j within ring r without loss of generality, since structures are equivalent up to translations and phason shifts if the sum of their phases are equal (modulo 2π). For n even, set¯ ting equal phases φr,j = φ¯ is indistinguishable from rescaling the amplitudes Ar by cos φ;

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0.6

0.5

0.5

ωa/(2πc)

ωa/(2πc)

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0.4

Complete Bandgap

0.3

0.3

0.2

0.2

0.1

0.1

0 Γ

(a)

E V k, Bloch wave vector

Γ

Complete Bandgap

0.4

0 Γ

(b)

E V k, Bloch wave vector

Γ

Fig. 6. – (Color online) Band structure of the 5-fold symmetric optimized structure, (a), and of the 6-fold symmetric optimized structure, (b), both with ε1 /ε0 = 6. The path traversed through the BZ is from the center (Γ) to the center of an edge (E) to a close vertex (V ) and back to Γ. Here, ω is the frequency, a is the length of a quasiunit cell in (a) and a unit cell in (b), and c is the speed of light. The quasicrystal lattice parameter a is smaller by a factor of (1/8) compared to the periodic approximant unit cell. The band gap in (a) is highly isotropic as a result of the high symmetry of the structure. Here, the full gap in the 6-fold case (32.5%) is larger than that in the 5-fold case (30.3%) in spite of the greater isotropy of the latter.

since we are allowing Ar to vary in our optimization scheme, the phases are redundant variables and can be set to zero. In the optimization procedure, Ar and φr that maximize the full photonic band gap between two chosen bands were found using the “steepest ascent” method (although “simulated annealing” [20] may also be used). For each lattice, the full gap is optimized around a Bloch wave number on the order of k = π/a, where a is the lattice parameter (or eﬀective lattice parameter for quasicrystals), i.e., around the ﬁrst (eﬀective) Brillouin zone. See ref. [19] for details of the optimization procedure. The band structures of the optimized 5- and 6-fold conﬁgurations are shown in ﬁg. 6 for dielectric contrast ratio 6.0. The 5-fold and other quasicrystal conﬁgurations are treated as periodic approximants for the purposes of using conventional band structure computational methods. That is, the band structure is computed as if the quasicrystal conﬁguration repeats over some large length scale (a supercell) and the limit is taken as the supercell size gets large. Optimized band gaps for n-fold symmetry are plotted in ﬁg. 7 for each n over the dielectric contrast range 1–20. The slopes of the band gap curves form a monotonically decreasing function of the number of Bragg peaks per ring. This is because the gap at the Brillouin zone edge is (to ﬁrst order) proportional to the scattering amplitude across the zone, and scattering power must be spread over a higher number of peaks per ring for higher n. It is clear from ﬁg. 7 that the higher-symmetry structures have greater gaps at low contrast. These TM-only studies conﬁrm the hypothesis that high symmetry can be advantageous for producing large band gaps under some conditions. In addition, the quasicrystalline structures have gaps that are more isotropic than those of the crystals for all contrasts

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Photonic properties of non-crystalline solids

60

Gap (%)

40

20

4 fold 6 fold 8 fold 5 fold 12 fold

0

-20

0

5

10

Dielectric contrast,

15 1

20

0

Fig. 7. – (Color online) Normalized band gaps of optimized density wave structures for rotational symmetries n = 4, 5, 6, 8 and 12 as a function of the dielectric contrast of the two phases. For reasons discussed in the text, structures of high symmetry tend to have larger band gaps for low contrast, but smaller gaps for high contrast.

due to the fact that their eﬀective Brillouin zones are more circular. A measure of the isotropy is [19] (5)

I=

mink,EBZ {ωh (k)} − maxk,EBZ {ωl (k)} , maxk,EBZ {ωh (k)} − mink,EBZ {ωl (k)}

where the minima and maxima are taken over all k on the boundary of the eﬀective Brillouin zone, and ωl and ωh refer to frequencies of bands just below and just above the gap. It is clear from its deﬁnition that I ∈ [−1, 1]. When I is at its greatest, the gap is perfectly isotropic, and it is at its lowest in a homogeneous material. Isotropy is plotted against dielectric contrast for all rotational symmetries in ﬁg. 8. The quasicrystals all have higher isotropy than the crystals. I increases monotonically as the eﬀective Brillouin zone becomes more circular. 4. – Finding optimal complete band gaps for 2D photonic quasicrystals The previous section explored how the width and isotropy of the optimal TM photonic band gap varies with symmetry. This section tackles the more challenging case of optimizing the complete (TM+TE) band gap. For the case of TM radiation only, the density wave results of the previous section and other rigorous methods suggest that a nearly optimal photonic band gap is obtained

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1 0.8

Isotropy,

0.6 0.4 0.2 4-fold 6-fold 8-fold 5-fold 12-fold

0 -0.2 -0.4

0

5

10

Dielectric contrast,

15 1

20

0

Fig. 8. – (Color online) The degree of isotropy, I (deﬁned in the text), of the optimized dielectric conﬁgurations plotted against dielectric contrast. The quantity I is bounded from below by −1 (for the case of a homogeneous medium) and from above by 1 (for the case where the upper and lower bands are perfectly ﬂat). For all contrasts, quasicrystalline structures have higher isotropy. Note that the 5-fold case has greater isotropy than the 8-fold case because the former has an eﬀective BZ that is closer to a circle than that of the latter. These zones are a decagon and an octagon, respectively.

by placing identical dielectric cylinders centered at each point and adjusting the radius. To obtain the optimal photonic band gaps for TE radiation, where the electric ﬁeld is oriented in the plane of the scatters, necessitates a connected network of dielectric with air pockets in between, as shown in ﬁg. 5. For example, a commonly used conﬁguration for photonic crystals is inverted compared to the optimal structures for TM photonic band gaps, i.e., placing an identical air cylinder at each point so as to produce a connected network of dielectric material. However, in the case of quasicrystalline structures, this method fails to produce sizable TE photonic band gaps. The main reason is that the inverted structure has a very non-uniform distribution of dielectric scattering regions that broadens the distribution of resonances. As noted in the previous section, the density wave methods cannot be used for ﬁnding an optimal TE (or complete) band gap structure because the optimal conﬁgurations in these cases require many more waves. Exhaustive searches over all possible designs is well known to be a daunting computational task, despite the recent development of optimization methods, such as gradient-based approaches, and evolutionary methods [19, 21, 22]. The major diﬃculty in solving this inverse problem comes from the relatively large number of iterations required to achieve an optimal design and the high computational cost of obtaining the band structure for complex distributions of dielectric materials, as needed to simulate heterostructures without long-range order. For instance, the evolutionary algorithms employed in [23] require over 1000 generations of designs to achieve fully convergence.

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Fig. 9. – (Color online) The protocol described in the text joins the vertices of a rhombus tiling (grey points) into a Delaunay triangular lattice (dashed lines) and, then, joins the centroids of the triangles into a network of cells whose vertices are trivalent (black segments and points). To construct a photonic material with a complete band gap, the (black) edges are replaced with a wall of dielectric of ﬁnite thickness and the vertices are replaced with cylinders of ﬁnite radius.

For these reasons, the recent development of a simpler design protocol by Florescu et al. [24] that requires vastly less computational resources to produce a nearly optimal design for a TE and complete band gap heterostructure is signiﬁcant. The protocol can be applied to a crystal or quasicrystal point pattern with any point symmetry. The optimal pattern for TE modes obtained by the protocol is a planar, continuous trivalent network (as in the case of the triangular lattice), which can be obtained from the point pattern using the steps described in ﬁg. 9. Namely, construct a Delaunay tiling [25] from the original two-dimensional point pattern and follow the steps in ﬁg. 9 to transform it into a tessellation of cells. Then decorate the cell edges with walls (along the azimuthal direction) of dielectric material of uniform width w and vary the width of the walls until the maximal TE band gap is obtained. To obtain designs for complete photonic band gaps, the protocol is to optimally overlap the TM and TE band gaps by decorating the vertices of the trivalent network of cell walls with circular cylinders (black circles in ﬁg. 9) of radius r. Then, for any

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Fig. 10. – Optimal photonic crystal and quasicrystal heterostructures and their diﬀraction patterns derived from: (a) a periodic 6-fold symmetric point pattern; (b) a 5-fold symmetric quasicrystal (Penrose) tiling; (c) an 8-fold symmetric quasicrystal tiling.

given set of dielectric materials, the maximal complete photonic band gap is achieved by varying the only two free parameters, w and r. Although a constrained optimization method like this is not guaranteed to produce the absolute optimum over all possible designs, in examples where the absolute optimum is known by rigorous optimization methods [21,22,19], the protocol produces a design whose band gap is the same within the numerical error using exponentially less computational resources. For the optimization of the two degrees of freedom (w and r), the photonic band structure must be computed as parameters are varied. Since the computational requirements are modest, one can use the conventional plane-wave expansion method [26]; we generate the disordered pattern within a periodic box of size L much greater than the average interparticle spacing and take the limit as L becomes large. The photonic band gaps for heterostructures obtained by the protocol turn out to be equivalent to the fundamental band gap in periodic systems in the sense that the spectral location of the TM gap, for example, is determined by the resonant frequencies of the scattering centers [27] and always occurs between band N and N + 1, with N precisely the number of points per unit cell. The examples illustrated here are based on the vertices of a Penrose tiling, [28] a 5-fold symmetric pattern composed of obtuse and acute rhombi and on the vertices of an octagonal tiling, [29] an 8-fold symmetric pattern composed of squares and rhombi. The

Photonic properties of non-crystalline solids

41

protocol-optimized structures, along with a crystal 6-fold structure obtained by the same protocol, is shown in ﬁg. 10. Note that the quasicrystal diﬀraction patterns are all pointlike (Bragg peak) and have the symmetry of the real-space structure, with most of the scattering intensity in a ring of peaks, conditions that are optimal for band gap formation. TE band gaps: The optimal TE band gap designs correspond to the network of walls connecting the centroids of the Delaunay tiling, as shown in ﬁg. 10. In the case of the 5-fold symmetric quasicrystal, the optimum has wall width w/a = 0.103; the TE band gap is Δω/ωC = 42.3%, the largest ever reported for a photonic quasicrystal. For the octagonal quasicrystal, the optimal structure has a dielectric material width w/a = 0.106 along each edge, which produces a TE band gap of Δω/ωC = 39.2%. The TE band gap formation is analogous to the TM case in that it involves an interplay between scattering from individual cells and the Bragg scattering of the quasiperiodic arrangement of scattering planes. As shown in ﬁg. 11, the band gap size is related to the degree of localization of the photonic states. In this case, the magnetic ﬁeld (out of the plane) of the lower band-edge states are concentrated in the air component and in the dielectric for the upper band-edge states. Similar to the case of TM polarized radiation, the band gap for the optimal structures always occurs between bands NP and NP + 1, where NP is now the number of cells in the structure. Complete band gap: For the complete band gap, the optimal structure consists of placing a wall along each edge and a cylinder at each trihedral vertex of the network generated by the protocol. The optimum for the 5-fold symmetric case has cylinder radius r/a = 0.157 and cell wall thickness w/a = 0.042. For the 8-fold symmetric point patterns, the optimum is r/a = 0.167 and w/a = 0.014. The scattering properties of the individual scattering centers and cells are again essential in the band gap opening and the complete band gap occurs always between bands 3NP and 3NP + 1, where NP is the number of points in the periodic approximant point pattern (there are 3NP total scattering units in the system, 2NP dielectric disks and NP dielectric cells). The resulting optimal 5-fold symmetric structure displays complete (TM and TE) photonic band gap of 16.5% —the ﬁrst complete band gap ever reported for a photonic quasicrystal with 5-fold symmetry and comparable to the largest band gap (20%) found for photonic crystals with the same dielectric contrast. The optimal 8-fold symmetric structure has a full photonic band gap of 13.5%.

5. – Isotropic disordered photonic materials The photonic structures considered above and in the literature to date exhibit longrange translational order and Bragg peak scattering that contribute to the formation of a band gap. But are these conditions essential to the formation of a band gap? In this section, we show that the surprising answer is no: it is possible to design isotropic, translationally disordered photonic materials with large complete photonic band gaps using the same protocol described in the previous section beginning from a disordered

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Fig. 11. – (Color online) Panels (a)-(d) show the electric-ﬁeld distribution for a 5-fold symmetric photonic quasicrystal optimized for TM polarized radiation. The structure consists of dielectric cylinders of radius r/a = 0.177 placed at the vertices of a Penrose tiling and displays a TM photonic band gap of 36.5%. Lower (a) and upper (c) band edge modes display a well-deﬁned degree of localization; modes just below the lower band edge (b) and just above the upper band edge (d) display an extended character. Panels (e)-(h) show the magnetic-ﬁeld distribution in 5-fold symmetric quasicrystalline network optimized via the protocol for TE polarized radiation. The structure consists of trihedral network with wall thickness w/a = 0.102 and displays a TE photonic band gap of 42.5%. The lower (e) and upper (g) band edge modes display a high degree of spatial concentration and modes just below the lower band edge (f) and just above the upper band edge (h) display an extended character.

Photonic properties of non-crystalline solids

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point pattern. The examples considered here are two-dimensional, though the same approach can be used in three dimensions. Obtaining complete photonic band gaps in dielectric materials without long-range order is counterintuitive. In fact, the protocol cannot generate a complete photonic band gap structure from an arbitrary point pattern; for example, it fails in the case of a random Poisson pattern. Not all disordered points patterns are alike, though. There is a largely unexplored zoology of disordered patterns distinguished according to how their density ﬂuctuations scale with volume [24, 30]. Of special interest for photonic band gaps are point patterns that are hyperuniform [30], where the number variance σ 2 (R) ≡ NR2 − NR 2 within a spherical sampling window of radius R (in d dimensions) grows more slowly than the window volume for large R, i.e., more slowly than Rd . Among hyperuniform patterns, there is a further subclassiﬁcation. The 2D hyperuniform patterns considered in this section are restricted to the most hyperuniform subclass in which the number variance grows like the window surface area for large R, i.e., σ 2 (R) ∼ AR, up to small oscillations. (For a pattern in d dimensions, σ 2 (R) ∼ Rβ must scale with a power β ≥ d − 1.) The coeﬃcient A measures the degree of hyperuniformity within this subclass: smaller values of A are more hyperuniform. In reciprocal space, hyperuniformity corresponds to having a structure factor S(k) that tends to zero as the wave number k = |k| tends to zero (omitting forward scattering), i.e., inﬁnite wavelength density ﬂuctuations vanish. As shown below, patterns can be further subclassiﬁed according to the behavior of S(k) as k approaches zero, which is important in optimizing their photonic properties. All crystal and quasicrystal point patterns are hyperuniform, but it is considerably more diﬃcult to construct disordered hyperuniform point patterns. Recently, a collective coordinate approach has been devised to explicitly produce point patterns with precisely tuned wave scattering characteristics (that is to say, tuned S(k) for a ﬁxed range of wave numbers k), including a large class of disordered ones [30]. The motivation for considering hyperuniformity for photonics is the strong correlation between the degree of hyperuniformity (smallness of A) for a variety two-dimensional crystal structures as measured in ref. [31] and the resulting band gaps. For example, a triangular lattice of parallel cylinders has the smallest value of A and the largest band gap for light polarized with its electric ﬁeld oscillating normal to the plane, whereas a square lattice of cylinders has a larger value of A and a smaller photonic band gap. Among disordered patterns, the most hyperuniform examples known are stealthy, sonamed because they are transparent to incident radiation (S(k) = 0) for a ﬁnite range of wave numbers k < kC for some positive kC . Stealthy hyperuniform patterns are parameterized by kC or, equivalently, χ, the fraction of wave numbers k within the Brillouin zone that are set to zero; as χ increases, kC and the degree of hyperuniformity increase, thus, decreasing A in our deﬁnition of the number variance. Figure 12 presents four designs of photonic structures with diﬀerent values of χ and their structure factors, S(k). When χ reaches a critical value χC (≈ 0.77 for two-dimensional systems) the pattern develops long-range translational order, so the disordered patterns are restricted to smaller χ.

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Fig. 12. – Four designs of isotropic dielectric heterostructures and their structure factors, S(k). Only panel (d) exhibits a complete photonic band gap. Panel (a) is a disordered network design derived from a Poisson (p = 2, non-hyperuniform) point pattern. Panel (b) shows a network derived from a nearly hyperuniform equiluminous point pattern in which the structure factor S(k → 0) = S0 > 0 for k < kC . Panel (c) shows a network derived from a RSA point pattern in which the structure factor S(k → 0) > 0 but there is more local geometric order than in (b). Panel (d) is derived from an isotropic, disordered, stealthy hyperuniform pattern with S(k) precisely zero within the inner disk. Note the two concentric shells of sharply increased density just beyond the disk. These features sharpen as the ordering parameter χ increases; this trend coincides in real space with the exclusion zone increasing around each particle and the emergence of complete photonic band gaps.

The largest photonic band gaps in hyperuniform patterns occur in the limit of large dielectric contrast; For the purposes of illustration, the results below assume that the photonic materials are composed of silicon (with dielectric constant = 11.56) and air. For this dielectric constrast, a signiﬁcant band gap begins to open for the stealthy hyperuniform designs for suﬃciently large χ ≈ 0.35 (but well below χC ), at a value where there emerges a ﬁnite exclusion zone between neighboring points in the real space hyperuniform pattern. That is, unlike a Poisson pattern which allows the spacing between neighboring points to be arbitrarily small, there is a positive lower bound to the separation in hyperuniform patterns with χ > 0.35. The structures built around stealthy hyperuniform patterns with χ = 0.5 are found to exhibit remarkably large TM (of 36.5%) and TE (of 29.6%) photonic band gaps, making them competitive with many of their periodic and quasiperiodic counterparts. More importantly, there are complete photonic band gaps of appreciable magnitude reaching values of about 10% of the central frequency for χ = 0.5. A striking feature of the photonic band gaps is their isotropy. Using the isotropy measure introduced earlier for quasicrystals, one ﬁnds that the most isotropic crystal band gap has a variation of 20%, compared to less than 0.1% for the hyperuniform disordered pattern in ﬁg. 12d.

Photonic properties of non-crystalline solids

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Even with hyperuniformity, the appearance of photonic band gaps in disordered structures presented is counterintuitive. There is not yet a rigorous explanation, but the current conjecture, based on numerical evidence and physical arguments, is that complete photonic band gaps can occur in disordered systems that exhibit a combination of three mathematical properties: hyperuniformity, uniform local topology, and short-range geometric order. First, consider the evidence provided by numerical experiments to date. Photonic crystals are hyperuniform (an automatic consequence of periodicity) and the known examples with the largest TM, TE and complete photonic band gaps satisfy the other two conditions [22, 32, 33]. Numerical experiments indicate that hyperuniformity is a crucial condition. For example, ﬁg. 12b compares results for the hyperuniform pattern with networks generated from non-hyperuniform Poisson point patterns with p = 2, as in ﬁg. 12a; equiluminous point patterns with S(k → 0) > 0 for k < kC , where the non-zero constant S(0) is made very small, as shown in ﬁg. 12c; and with a random-sequential absorption (RSA) point pattern [34] generated by randomly, irreversibly and sequentially placing equal-sized circular disks in a large square box with periodic boundary conditions subject to a non-overlap constraint until no more can be added. (It has been shown that such two-dimensional RSA packings have S(k → 0) slightly positive at k = 0 and increasing as a positive power of k for small k [35].) The latter two patterns are very nearly hyperuniform presenting similar deviations from hyperuniformity, Se-lum (k → 0) = 0.05 and SRSA (k → 0) = 0.053; and the RSA network in ﬁg. 12c exhibits uniform topological order (trivalency) and well-deﬁned short-range geometric order; furthermore, these two patterns produce TM and TE band gaps separately. Yet, none of the three families of non-hyperuniform patterns has been found to yield a complete photonic band gap. Note also that hyperuniform stealthy patterns with χ < 0.35 (and keeping all other parameters ﬁxed) do not produce sizable complete photonic band gaps while those with χ > 0.35 do. They are all hyperuniform, but they have diﬀerent degrees of shortrange geometric order, the variance in the near-neighbor distribution of link lengths and the distance between centers of neighboring links. This is evidence that short-range geometrical order is essential, as well. In principle, hyperuniformity and short-range geometrical order can be varied independently, but it is notable that patterns with the highest degree of hyperuniformity also possess the highest degree of short-range geometrical order and that, for the case of stealthy patterns, both hyperuniformity and short-range geometric order increase as χ increases. To understand how hyperuniformity and short-range geometric order, when combined with uniform local topology, can lead to a complete photonic band gap, consider that the band gaps arise in the limit of large dielectric constant ratio. In this limit and for the optimal link widths and cylinder radii, the interaction with electromagnetic waves is in the Mie scattering limit. At frequencies near the Mie resonances (which coincide with the photonic band gap lower band edge frequencies), the scattering of TM electromagnetic waves in a heterostructure composed of parallel cylinders is similar to the scattering of electrons by atomic orbitals in cases where the tight-binding approximation can be

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reliably applied [36]. The same applies for TE modes for any one direction k if, instead of parallel azimuthal cylinders, there are parallel thick lines (or walls in the azimuthal direction) in the plane and oriented perpendicular to k; however, to obtain a complete band gap, some compromise must be found to enable band gap for all directions k. A conjecture, based on comparison with rigorous optimization results, numerical experiments, and arguments below, is that uniform local topology is advantageous for forming optimal band gaps. In two dimensions, this is easiest to achieve in disordered structures without disrupting the short-range geometric order if the networks are trivalent. If the arrangement of dielectrics has local geometric order (the variance in link lengths and inter-link distances is small), the propagation of light in the limit of high dielectric constant ratio is described by a tight-binding model with nearly uniform coeﬃcients. In the analogous electronic problem, Weaire and Thorpe [37] proved that band gaps can exist in continuous random tetrahedrally coordinated networks, commonly used as models for amorphous silicon and germanium. In addition to tight binding with nearly uniform coeﬃcients, the derivation required uniform tetrahedral coordination. (Weaire and Thorpe call networks satisfying these conditions topologically disordered.) The analogy in two dimensions is a trivalent network. Although their proof discussed three dimensions and tetrahedral-coordination speciﬁcally, we ﬁnd that it can generalize to other dimensions and networks with diﬀerent uniform coordination. Note that our protocol automatically imposes uniform topology (e.g., trivalency in two dimensions) and limits variation of the tight-binding parameters by imposing local geometric order. To complete their proof, Weaire and Thorpe added a mild stipulation that the density has bounded variation, deﬁned as the condition that the density remains between two ﬁnite values as the volume is taken to inﬁnity. This condition is satisﬁed by any homogeneous system, hyperuniform or not, and thus is much weaker than hyperuniformity, for which σ 2 (R) = AR in the stealth two-dimensional examples. Bounded variation is not suﬃcient for the photonics problem, though. A gap is needed simultaneously for both TM and TE, and the gap centers must have values that allow an overlap. Also, the goal is not simply to have a gap, but to have the widest gap possible. The evidence shows that hyperuniformity is highly advantageous (perhaps even essential) for meeting these added conditions. The comparison to electronic band gaps is also useful in comparing states near the band edges and continuum. For a perfectly ordered crystal (or photonic crystal), the electronic (photonic) states at the band edge are propagating such that the electrons (electromagnetic ﬁelds) sample many sites. If modest disorder is introduced, localized states begin to ﬁll in the gap so that the states just below and just above are localized. Although formally the disordered heterostructures do not have equivalent propagating states, an analogous phenomenon occurs. In the upper four panels of ﬁg. 13, we compare the azimuthal electric-ﬁeld distribution for modes well below or well above the band gap (upper two panels), which one might call extended since the ﬁeld is distributed among many sites; and then modes at the band edges, which are localized. The formation of the TM band gap is closely related to the formation of electromagnetic resonances localized within the dielectric cylinders (ﬁg. 13(a) and (b)) and that

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Fig. 13. – (Color online) (a), (b), (c) and (d): Electric-ﬁeld distribution in hyperuniform disordered structures for TM polarized radiation. The structure consists of dielectric cylinders (radius r/a = 0.189 and dielectric constant = 11.56) in air arranged according to a hyperuniform distribution with χ = 0.5 and displays a TM photonic band gap of 36.5% of the central frequency. (a) Localized and (b) extended modes around the lower photonic band gap edge, and (c) localized and (d) extended modes around the upper photonic band gap edge. (e), (f), (g), and (h): Magnetic-ﬁeld distribution in hyperuniform disordered structures for TE polarized radiation. The structure consists of trihedral network architecture (wall thickness w/a = 0.101 and dielectric constant = 11.56) obtained from a hyperuniform distribution with χ = 0.5 and displays a TE photonic band gap of 31.5% of the central frequency. (e) Localized and (f) extended modes around the lower photonic band gap edge, and (g) localized and (h) extended modes around the upper photonic band gap edge.

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there is a strong correlation between the scattering properties of the individual scatterers (dielectric cylinders) and the band gap location. In particular, the largest TM gap occurs when the frequency of the ﬁrst Mie resonance coincides with the lower edge of the photonic band gap [27]. Analogous to the case of periodic systems, the electric ﬁeld for the lower band edge states is well localized in the cylinders (the high dielectric component), thereby lowering their frequencies; and the electric ﬁeld for the upper band edge states is localized in the air fraction, increasing their frequencies (see ﬁg. 13(c) and (d)). As shown in ﬁg. 13(e),(f),(g) and (h), an analogous behavior occurs for the azimuthal magnetic-ﬁeld distribution for TE modes: for the lower edge state, the azimuthal magnetic ﬁeld is mostly localized inside the air fraction and presents nodal planes that pass through the high index of refraction fraction of the structure, while the upper edge state displays the opposite behavior. 6. – Discussion The principal lesson from these lectures is that complete photonic band gaps are possible for a much wider range of dielectric structural designs than previously considered. The possibilities have been extended here to quasicrystals with arbitrary rotational symmetry and a class of isotropic disordered structures. Although photonic crystals have the largest complete band gaps (consistent with their being the most hyperuniform), quasicrystalline and disordered hyperuniform heterostructures with substantial complete photonic band gaps oﬀer advantages for many applications. Both are signiﬁcantly more isotropic, which is advantageous for use as highly eﬃcient isotropic thermal radiation sources and waveguides with arbitrary bending angle. The properties of defects and channels [38] useful for controlling the ﬂow of light are diﬀerent for crystal, quasicrystal and disordered structures. Quasicrystals, like crystals, have a unique, reproducible band structure; by contrast, the band gaps for the disordered structures have some modest random variation for diﬀerent point distributions. Also, light with frequencies above or below the band edges are propagating modes that are transmitted through photonic crystals and quasicrystals but are likely localized modes in the case of hyperuniform disordered patterns, which give the former advantages in some applications, such as light sources. On the other hand, due to their compatibility with general boundary constraints, photonic band gap structures based on disordered hyperuniform patterns can provide a ﬂexible optical insulator platform for planar optical circuits. Moreover, eventual ﬂaws that could seriously degrade the optical characteristics of photonic crystals and perhaps quasicrystals are likely to have less eﬀect on disordered hyperuniform structures, therefore relaxing fabrication constraints. Finally, we note that the lessons learned here have broader physical implications. One is led to appreciate that all non-crystalline solids are not the same: as methods of synthesizing solids and heterostructures advance, it will become possible to produce diﬀerent types and degrees of hyperuniformity, and, consequently, many distinct classes of materials with novel photonic, electronic, phononic, plasmonic, shear wave and other physical properties. The subject is in its infancy, with many possible directions to explore.

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∗ ∗ ∗ These lectures are based on what I have learned about photonics from working with a wonderful group of collaborators: S. Torquato, M. Florescu, P. Chaikin, M. Rechtsman, W. Man, and M. Megens, who have been crucial in developing the ideas and results described above. The ﬁgures and much of the text draw heavily from our papers together. I would like to thank the Directors, D. Wiersma and R. Kaiser, for inviting me to take part in this Summer School; I learned a great deal from the lectures and from the outstanding students. I would also like to thank B. Alzani and her staﬀ for making my stay in Varenna so comfortable and memorable; and the Societ` a Italiana di Fisica for funding this important and inﬂuential series of schools. This work was supported in part by the National Science Foundation under Grant No. DMR-0606415 and by the National Science Foundation MRSEC program through New York University under Grant No. DMR-0820341. REFERENCES [1] John S., Phys. Rev. Lett., 58 (1987) 2486. [2] Yablonovitch Y., Phys. Rev. Lett., 58 (1987) 2059. [3] Joannopoulos J. D., Villeneuve P. R. and Fan S., Solid State Commun., 102 (1997) 165. [4] Altug H., Englund D. and Vuckovic H., Nature Phys., 2 (2006) 484. [5] EI-Kady I., Taha M. M. R. and Su M. F., Appl. Phys. Lett., 88 (2006) 253109. [6] Chutinan A., John S. and Toader O., Phys. Rev. Lett., 90 (2003) 123901. [7] Schectman D., Blech I., Gratias D. and Cahn J. W., Phys. Rev. Lett., 53 (1984) 1951. [8] Levine D. and Steinhardt P. J., Phys. Rev. Lett., 53 (1984) 2477. [9] Steinhardt P. J. and Ostlund S., The Physics of Quasicrystals (World Scientiﬁc, Singapore) 1987. [10] Janot C., Quasicrystals: A Primer (Oxford University Press, Oxford) 1994. [11] Bindi L., Steinhardt P. J., Yao N. and Lu P., Science, 324 (2009) 1306. [12] Freedman B., Bartal G., Segev M., Lifshitz R., Christodoulides D. N. and Fleischer J. W., Nature, 440 (2006) 1166; Freedman B., Lifshitz R., Fleischer J. W. and Segev M., Nat. Mater., 6 (2007) 776. [13] Man W., Megens M., Steinhardt P. J. and Chaikin P. M., Nature, 436 (2005) 993. [14] Chan Y. S., Chan C. T. and Liu Z. Y., Phys. Rev. Lett., 80 (1998) 956. [15] Dal Negro L., Oton C. J., Gaburro Z., Pavesi L., Johnson P., Lagendijk A., Righini R., Colocci M. and Wiersma D. S., Phys. Rev. Lett., 90 (2003) 055501. [16] Cheng S. S. M., Li L., Chan C. T. and Zhang Z. Q., Phys. Rev. B, 59 (1999) 4091. [17] Kaliteevski M. A., Brand S., Abram R. A., Krauss T. F., Millar P. and De La Rue R. M., J. Phys.: Condens. Matter, 13 (2001) 10459. [18] Zeng X. C., Bergman D. J., Hui P. M. and Stroud D., Phys. Rev. B, 38 (1988) 10970. [19] Rechtsman M. C., Jeong H.-C., Chaikin P. M. and Torquato S., Phys. Rev. Lett., 101 (2008) 073902. [20] Kirkpatrick S., Gelatt C. D. jr. and Vecchi M. P., Science, 220 (1983) 671. [21] Kao C., Osher S. and Yablonovitch E., Appl. Phys. B, 81 (2003) 235. [22] Sigmund O. and Hougaard K., Phys. Rev. Lett., 100 (2008) 153904.

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[23] Preble S., Lipson H. and Lipson M., Appl. Phys. Lett., 86 (2005) 061111. [24] Florescu M., Torquato S. and Steinhardt P. J., Proc. Natl. Acad. Sci. U.S.A., 106 (2009) 20658. [25] Preparata F. R. and Shamos M. I., Computational Geometry: An Introduction (Springer-Verlag, New York) 1985. [26] Johnson S. G. and Joannopoulos J. D., Opt. Express, 8 (2001) 173. [27] Rockstuhl C., Peschel U. and Lederer F., Opt. Lett., 31 (2006) 1741. [28] Penrose R., Bull. Inst. Math. Appl., 10 (1974) 266. [29] Grunbaum B., Grunbaum Z. and Shephard G., Comput. Math. Appl., 12 (1986) 641. [30] Batten R. D., Stillinger F. H. and Torquato S., J. Appl. Phys., 104 (2008) 033504. [31] Torquato S. and Stillinger F. H., Phys. Rev. E, 68 (2003) 041113. [32] Fu H., Chen Y., Chen R. and Chang C., Opt. Express, 13 (2005) 7854. [33] Chan C. T., Datta S., Ho H.-M. and Soukoulis C., Phys. Rev. B, 50 (1994) 1988. [34] Torquato S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer-Verlag, New York) 2002. [35] Torquato S., Uche O. U. and Stillinger F. H., Phys. Rev. E, 74 (2006) 061308. [36] Lidorikis E., Sigalas M. M., Economou E. N. and Soukoulis C. M., Phys. Rev. Lett., 81 (1998) 1405. [37] Weaire D., Phys. Rev. Lett., 26 (1971) 1541; Weaire D. and Thorpe M. F., Phys. Rev. B, 4 (1971) 2508. [38] See, for example, Socolar J. E. S., Lubensky T. C. and Steinhardt P. J., Phys. Rev. B, 34 (1986) 3345; see also ref. [9].

Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-51

Diﬀuse optics: Fundamentals and tissue applications R. C. Mesquita and A. G. Yodh Department of Physics & Astronomy, University of Pennsylvania 209 South 33rd Street, Philadelphia, PA 19104-6396, USA

Summary. — The material in this paper is diﬀerent from the mainstream topics in this summer’s International School of Physics “Enrico Fermi”. It should become apparent, however, that the roots of these biomedical optics research problems share common features with much of the light scattering and transport research taught in the Varenna summer school. Here, our intention is to provide an informal review that establishes the roots of diﬀuse optics, and then demonstrates how diﬀuse optics is ﬁnding application in medicine. This paper will have two main themes. After a brief motivation of the problem, the ﬁrst theme will provide a coherent discussion about light transport in turbid media. The second theme is oriented towards problems in biomedicine. As such, a short discussion of hemodynamics will be followed by representative current work from our lab, particularly with breast and brain.

1. – Introduction The dream of optics for in vivo biopsy has been with us in various contexts for many years, and it continues to pop up in popular culture, particularly science ﬁction. Famous examples come from the TV show Star Trek, wherein Dr. McCoy uses a “tricorder” device to assess the condition of a patient, and from movies such as Minority Report, wherein ﬁber optics pick up brain signals from special patients. In most cases, an instrument c Societ` a Italiana di Fisica

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Fig. 1. – Illustration of a diﬀuse optical measurement on a subject’s arm. One source ﬁber and one detector ﬁber are shown, separated by approximately 2 cm.

shines light into the body and/or collects light from the body in transmission or reﬂection; then a scientist employs the data to make a rapid diagnosis. Without thinking too hard about the details, one might believe that such measurements are possible. For example, if we use light in the near-infrared (700–950 nm) rather than the uv, visible or mid- and far-infrared parts of the spectrum, then human tissue has a window of low absorption. Within this window, the absorption of oxy- and deoxy-hemoglobin is falling to zero, and the absorption of water has not started to increase signiﬁcantly. Thus, light from such near-infrared devices can penetrate deeply in tissue. Furthermore, since each tissue chromophore has distinct spectral features, one can readily envision using light transmission properties as a function of wavelength to acquire sensitivity to tissue physiology, particularly to blood dynamics and edema. It has thus turned out that, besides the obvious convenience of such a device for continuous non-invasive measurement at the bedside, optical contrasts are complementary to other kinds of medical diagnostics such as, for example, X-ray and ultrasound. A typical diﬀuse optical measurement is shown in ﬁg. 1. This image reveals what we mean by “deep tissue”, i.e., big chunks of tissue located millimeters to centimeters below the tissue surface. Typically an optical ﬁber, coupled to a light source such as a diode laser, injects light into the tissue at the air-tissue interface, and a second optical ﬁber collects remitted or transmitted light at another position on the air-tissue surface. In practice, we vary source-detector position, light wavelength, light modulation, and even the mode of light detection in order to derive physiological information about the tissue in real time. Currently, two limiting versions of this basic scheme are employed. One approach is diﬀuse optical imaging. In this case, many source-detector pairs are placed on the tissue surface, and the scientist attempts to reconstruct images of optical and physiological properties in each of many volume elements (or voxels) within the tissue interior based on measurements at the tissue surface. This imaging or tomographic scheme has been particularly prevalent in breast imaging. The second approach is tissue monitoring.

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Here, a probe with few source-detector pairs is typically placed on (or near) tissues of interest, and average properties of the underlying tissues are derived by ﬁtting reﬂection/transmission data to simple light diﬀusion models. This monitoring approach has been employed extensively in clinical studies of brain, muscle and tumor. 2. – Light transport tools We now begin our discussion of light transport in tissues. In particular, we will introduce key physical and physiological parameters, and we will discuss important underlying concepts. We aim to facilitate understanding of the light diﬀusion problem in tissues, with minimal use of mathematics. Readers interested in more formal discussions can consult a variety of recent reviews [1-6] and references therein; a rather complete review from our group, i.e., one that ﬂeshes out the mathematics of many topics to be discussed herein, should be available soon in the journal Reports on Progress in Physics. The natural starting point for our discussion is the optically thin or single-scattering sample. Typically such samples consist of molecules that can absorb light, and relatively larger particle-like objects that scatter light signiﬁcantly. In the typical “traditional optics” experiment, the sample is illuminated with an incident light ﬁeld (or incident light intensity), and we are concerned with how much light remains in the beam after the light traverses a distance L straight through the sample. . 2 1. Absorption. – The best known attenuation eﬀect from traditional optics is light absorption (ﬁg. 2a). Absorption is due to molecular chromophores in the sample and is characterized by an absorption coeﬃcient, μa . The classic result for a transmission experiment of this kind is a law which states that the input light intensity is attenuated exponentially with distance traveled through the sample. The absorption coeﬃcient that characterizes this attenuation depends on the concentration of chromophores in the sample and the chromophore cross-section or extinction coeﬃcient —which in turn depends on incident-light wavelength. Thus, by carrying out such an absorption measurement as a function of input wavelength, one can learn which molecules are present, how many molecules reside in the sample, and one can even learn very subtle details about the molecule’s local environment via spectral shifts or spectral broadening. In the case of tissue optics in the near infrared, the most important endogenous molecules are oxy- and deoxy-hemoglobin, water and lipid. . 2 2. Scattering. – A second important eﬀect in traditional optics is scattering [7] (ﬁg. 2b). The scattering eﬀect is characterized by several variables. The ﬁrst parameter, by analogy with absorption, is the scattering coeﬃcient, μs , which is essentially the exponential decay rate of light intensity with distance traveled in the sample due to scattering eﬀects. The scattering coeﬃcient depends on the concentration of scattering particles and on the total scattering cross-section of these particles. This total scattering cross-section is generally wavelength-dependent (e.g., depending on the particle size, the particle index of refraction, and on light wavelength), although the scattering cross-section usually has a much weaker wavelength dependence than typical molecular

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Fig. 2. – Schematic representation of diﬀerent eﬀects that light encounters when traversing an optically thin sample: (a) absorption, (b) scattering, and (c) dynamic scattering.

absorption spectra. Scattering experiments, however, can oﬀer more information to the experimenter. Though the incident light is not transmitted, it is also not lost. Rather, it is sent oﬀ into diﬀerent angular directions. This angular information can be measured and is quantiﬁed by the so-called diﬀerential scattering cross-section (or scattering amplitude). Note that the integral of the diﬀerential scattering cross-section over all angles equals the total scattering cross-section. The amount of light scattered into a particular solid angle is thus proportional to the product of the incident light beam intensity and the diﬀerential scattering cross-section at that angle. Another factor that will eventually become of interest to us is the so-called anisotropy factor, g. The anisotropy factor is the average value of the cosine of the scattering angle for a typical scattering event. If the anisotropy factor, g, is near unity, then light scattering is nearly forward. If the anisotropy factor is about one half, then scattering is fairly isotropic and each scattering event is said to randomize the initial photon direction. We introduce one more term in this context, because it turns out to be critical for light diﬀusion: the reduced scattering coeﬃcient, μs . The reciprocal of the reduced scattering coeﬃcient is called the photon random walk step or transport-mean-free pathlength; after the incident light beam travels a photon random walk step in the scattering medium, only e−1 of the input light remains in the incident beam. The reduced scattering coeﬃcient is simply related to the scattering coeﬃcient and the anisotropy factor: μs = μs (1 − g). The reduced scattering coeﬃcient is the microscopic tissue scattering parameter that survives the diﬀusion approximations to the linear transport equation, which will be discussed shortly.

Diffuse optics: Fundamentals and tissue applications

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Scattering measurements such as this teach us about the microscopic objects that cause substantial light scattering. Examples of such objects include particles (e.g., polystyrene, silica, PMMA, nanoparticles, etc.), cell organelles (e.g., the mitochondria, nucleus, etc.), and cells (e.g., red blood cells, etc.). In principle one can learn about scatterer concentration, about the ﬂuids that surround the scatterers and whose indices of refraction aﬀect scattering strength, and more. This information is very useful, as was the case of absorption, and it is complementary to the absorption information. . 2 3. Dynamic light scattering. – The ﬁnal traditional optics topic that we will describe is called dynamic or quasi-elastic light scattering (i.e., DLS or QELS, respectively) [8-10]. In these experiments, the ﬂuctuations of scattered light intensity (or electric ﬁeld) are measured as a function of time (ﬁg. 2c). By analyzing these ﬂuctuations, speciﬁcally their temporal autocorrelation functions, we can learn about the motions of the scatterers. In particular we can learn about how fast these particles are moving, how many are moving, and in what manner they are moving (e.g., diﬀusive, ballistic, etc.). In the typical dynamic light scattering (DLS) experiment, a point-like photon detector collects light from the sample at an oﬀ-axis angle, i.e., an angle diﬀerent from the input beam propagation direction. The DLS eﬀect is relatively easy to understand for particle-like scatterers, and we shall adopt this approximation here. In the presence of the input light ﬁeld these particles acquire an oscillating induced-dipole moment. These oscillating dipoles behave as antennas and re-radiate light into many oﬀ-axis angles. When the particle-like objects move, the relative phases of these re-radiated dipole ﬁelds landing at the detector will vary too, and the re-radiated ﬁelds will add both constructively and destructively, depending on the particle conﬁguration in the sample. Thus the total detected electric ﬁeld (and intensity) will vary in time, and intensity ﬂuctuations are readily observed. These ﬂuctuations of the electric ﬁeld strength and light intensity carry information about the dynamic properties of the medium. Speciﬁcally, they carry information about the mean-square displacements of the particles. The normalized temporal autocorrelation function of the scattered electric ﬁeld, or the analogous temporal intensity autocorrelation function, is measured in the DLS experiment. The decay rate of the autocorrelation function depends on the particle motion; larger motions give faster autocorrelation function decay rates. Quantitatively, the decay rate depends on the particle mean-square displacement (during the autocorrelation time interval), the scattering angle, and the incident light wavelength. Often, the DLS autocorrelation function will decay by e−1 when the particles move a distance of about one optical wavelength. . 2 4. Multiple light scattering in tissues. – We have seen that the basic techniques of traditional optics probe both scattering and absorption, and both eﬀects lead to exponential attenuation of the incident beam. Furthermore, the scattered beam can teach us even more about the nature of the scatterer and its motions, e.g., if we measure other quantities such as the temporal autocorrelation function. These traditional techniques are rigorous and have been tested to a very substantial degree. The methods work. Tissues are more complicated. Tissue samples are not optically thin. Tissues multiply scatter light. One can envisage the transport of photons through tissue as a sequence

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Fig. 3. – Pictorial representation of a light path due to multiple scattering in tissue.

of single scattering events (ﬁg. 3). The light traverses the sample like a random walker. Incident light travels some distance into the sample and then scatters, and then it travels some distance along its new propagation direction and scatters again, and so on —many times over before it eventually emerges from the sample. In this case, absorption, scattering and correlation eﬀects are seemingly scrambled together. Furthermore, the total pathlength travelled by the incident light is many times the sample size. Thus, if we want to learn something about these media, then we have to learn how to unscramble this information from the emerging light ﬁelds. The use of Maxwell’s full electromagnetic theory to address such a complex problem, in general, is extraordinarily diﬃcult [11]. Fortunately, there exists a useful approximation called linear transport theory, which can deal with many aspects of the light transport problem in highly scattering media [12, 13]. Linear transport theory is not a perfect theory for the problem. For example, it ignores most (but not all) of the wave aspects of the light ﬁelds. However, it turns out that the theory provides a very useful starting point for many problems of light propagation in tissues. The key physical quantity in the theory is called the radiance. The radiance is basically the light power per area per solid angle at position r and time t in the sample, traveling in a particular direction, Ω. The radiance scales as the absolute square of the light ﬁeld at r, t and traveling in direction Ω. Linear transport theory balances the radiance in each small volume of the multiply scattering medium. This balance produces an equation for radiance, i.e., the transport equation, that can, in principle, be used to solve for radiance throughout the sample given some spatio-temporal distribution of absorption and scattering coeﬃcients, some distribution of diﬀerential scattering cross-section, boundary conditions, and initial conditions. Of course, the equation is non-trivial to solve, and closed-form solutions only exist for very simple geometries and conditions. The physics behind the mathematics in linear transport theory becomes more evident within a ﬁrst-order approximation of the radiance balance equation. In this case the diﬀusive nature of light transport becomes apparent. We will brieﬂy outline some of the steps involved in going from the fully general transport equation to the light diﬀusion equation. In this regime, the two simplest and most important physical quantities associ-

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ated with the radiance are the photon ﬂuence rate and the photon ﬂux. The ﬂuence rate at position r is a scalar function derived by integrating the radiance over a small sphere centered on r. It is the isotropic part of the radiance. The photon ﬂux is a vector. The photon ﬂux is also derived from an integration of the radiance over the solid angles, but in this case it is derived from a vector integration that takes explicit account of the vector nature of the radiance. The photon ﬂux is the lowest-order anisotropic contribution to the radiance. To solve the transport equation, typically one carries out the so-called PN approximation. In the PN approximation the radiance, the source distribution, and the diﬀerential scattering function are all expanded in terms of spherical harmonics or Legendre polynomials. These expansions are then stuﬀed back into the transport equation and a set of equations of diﬀerent order are thus derived. Interestingly, it is straightforward to show that the ﬂuence rate depends only on the zeroth-order spherical harmonic, and the ﬂux depends only on the ﬁrst-order spherical harmonics. Thus, in the lowest order, i.e., in the so-called P1 approximation where the expansions are truncated at ﬁrst order, the radiance can be expressed precisely in terms of the ﬂuence rate and ﬂux. With a little more work, the transport equation reduces to the photon diﬀusion equation for the ﬂuence rate. We write out the photon diﬀusion equation in its full glory below: (1)

∇ · (D(r)∇Φ(r, t)) − vμa (r)Φ(r, t) + vS(r, t) =

∂Φ(r, t) . ∂t

Here Φ(r, t) is the photon ﬂuence rate (Joules/cm2 s), v is the velocity of light in the medium (cm/s), and S(r, t) represents an isotropic source term proportional to the number of photons emitted (at point r and time t) per unit volume per unit time (Joules/cm3 s). The main new parameter in the problem is the photon diﬀusion coeﬃcient D (cm2 /s); D(r) = v/3(μs +μa ), and (for small absorption) it depends primarily on the reduced scattering coeﬃcient, the reciprocal of which is the photon random walk step. The analysis also shows explicitly that the photon ﬂux is proportional to the gradient of the ﬂuence rate, as in standard diﬀusion problems. We therefore arrive at the key mathematical result of this paper. It is worthwhile at this point to consider some of the assumptions that went into this analysis, many of which we have glossed over. First we assumed that the scattering length, (μs )−1 , is much less than the absorption length, (μa )−1 , an assumption that is ﬁne for the vast majority of tissues. Second, we implicitly assumed that the ﬂuence rate is signiﬁcantly larger than the ﬂux; this approximation is generally ﬁne, but it can break down near boundaries and sources. We have assumed isotropic sources, an assumption that is reasonable as long as we do not make measurements within a random walk step of the source. We have assumed that the scattering angle of a typical scattering event does not depend on incident angle, i.e., it depends only on the cosine of the angle between input and output wave vectors. Finally, an assumption about the rate of change of the ﬂux has been made, which amounts to requiring that the time scale of source modulation is much longer than the time between photon scattering events. Generally these assumptions

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Fig. 4. – (a) Experimental setup for measurements in “ideal” turbid media, an aquarium ﬁlled with milky Intralipid. A source ﬁber injects light into the medium near its center, and a movable detector ﬁber measures amplitude and phase of the diﬀusive waves at diﬀerent positions in sample. (b) Constant phase contours of diﬀusive wave in the aquarium. Notice, the contours are circular and emanate from the source. Inset: Measured phase and amplitude (in logarithm) as function of the source-detector separation. (c) Propagation through two diﬀerent media; the medium containing source (S0 ) has larger scattering then the other medium. Refraction eﬀects are demonstrated. (d) Diﬀraction/scattering of diﬀusive wave due to an absorbing sphere of diameter a in the turbid medium [14, 15].

are ﬁne for tissues. The photon random walk step in most tissues is about 1 mm and the absorption lengths are on the order of 5 to 10 cm. It is important, however, that our applications do not demand great precision from our measurements, otherwise the aﬀects of so many relatively innocuous approximations can wash out the eﬀects of the physiological perturbations we seek to probe. . 2 5. Simple solutions of the photon diﬀusion equation. – In order to use the diﬀusion equation for light, we must understand its solutions. We start with ideal case of inﬁnite, homogenous, turbid media. A good example of such a medium is an aquarium full of milk (or Intralipid) and ink (see ﬁg. 4a). This kind of tissue phantom sample can be adjusted to have properties similar to tissues, without the clinical complications. Working in the frequency domain, the source is amplitude-modulated at some frequency ω (e.g., 100 MHz), and we look for solutions that oscillate at this same frequency. In this case, the diﬀusion equation reduces to a very simple standard diﬀerential equation which will give spherical wave solutions with a complex wave number that depends on sample absorption and scattering, and on source modulation frequency. For a point oscillating source, the solution to this diﬀerential equation is the Green’s function of the diﬀusion equation for the inﬁnite homogenous geometry. Notice (in ﬁg. 4b insets) that the wave attenuates exponentially with distance from the source, and that the phase of the wave disturbance has an associated wavelength that depends on scattering, absorption and frequency. The fundamental photon density disturbance is thus a kind of overdamped wave. We call these disturbances diﬀuse photon density waves, or diﬀusive waves. It

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turns out that these disturbances of light energy density behave in many ways like the regular waves we know and love, except that properties such as disturbance wavelength and medium eﬀective index of refraction depend on factors such as the photon random walk step. Some examples of measurements we made to conﬁrm these ideas using point sources and detectors in an aquarium ﬁlled with Intralipid and ink are given in ﬁg. 4b-d. In these experiments, one source ﬁber is employed to launch a diﬀuse photon density wave (or diﬀusive wave) modulated at 200 MHz into the medium. Another ﬁber coupled to a photodetector measures the phase and amplitude of the diﬀusive wave at diﬀerent positions in the sample. In ﬁg. 4b, one can see a nice spherical diﬀusive wave, whose phase and amplitude variation with respect to source-detector separation can be used to obtain the absorption and reduced scattering coeﬃcients of the turbid medium. In the third ﬁgure we divide the tank into two media with diﬀering photon random walk steps. Refraction type eﬀects are apparent in this case. The last ﬁgure shows diﬀraction/scattering eﬀects. Thus far we have used a frequency domain picture to analyze the light propagation problem in simple turbid media. We can also solve the problem in the time domain. In this case, we seek the Green’s function solutions due to a point source emitting a very short duration pulse in the inﬁnite media. The result is a broadened light pulse whose terminal slope is related to sample absorption and whose peak position is related primarily to the light diﬀusion coeﬃcient in the sample. At this point it is worth reﬂecting on what has been gained. In fact, a lot has been gained. We now have an experimental means to separate scattering from absorption in turbid media by measuring the phase and amplitude of diﬀuse photon density waves or by measuring the temporal broadening of a very short light pulse in the medium. Thus we can derive the absorption coeﬃcient and the scattering properties of the medium, even though the medium is turbid. Next we explore the eﬀects of heterogeneity. 3. – Diﬀuse Optical Spectroscopy (DOS): monitoring We ﬁrst consider piecewise continuous turbid media. A most important example in practice is the semi-inﬁnite medium, e.g., a tissue medium with an air-tissue interface. To make progress on this problem, we need boundary conditions for the ﬂuence rate at the interface. The boundary condition is simple to state. We require that the radiance coming into the medium at the boundary is equal to the Fresnel reﬂected radiance that is traveling outward at the boundary. The result of carrying out this analysis is the so-called partial ﬂux boundary condition, which relates the ﬂuence rate at the boundary to its gradient at the same boundary. The proportionality constant in this relationship is a length which can be explicitly derived and which depends on the indices of refraction of the surrounding medium. One can solve the light diﬀusion problem exactly with partial ﬂux boundary conditions (for the semi-inﬁnite medium), or one can invoke a fairly innocuous approximation which leads to an even simpler boundary condition: the so-called extrapolated zero-boundary condition. In this case, the true boundary condition is replaced by a zero-boundary condition at an extrapolated distance Ls above the boundary,

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Fig. 5. – (a) Schematic showing the probe on the baby and the probe with its sources (◦) and detectors (×). (b) Measured amplitude and phase as a function of separation for diﬀerent sourcedetector combinations (black dots). The line in both plots represents the best ﬁt to the data points, and the slopes can be used to derive tissue absorption and scattering coeﬃcients [16].

i.e., the ﬂuence rate vanishes at the extrapolated distance. The zero-boundary condition problem can then be solved by the classic method of images to derive a simple result for the predicted diﬀuse light reﬂectance from a semi-inﬁnite turbid medium. Believe it or not, this light reﬂectance result for semi-inﬁnite homogeneous turbid media is THE workhorse result of the monitoring ﬁeld. It performs fairly well in the clinic, giving average optical properties quite accurately, even though the media are not truly homogeneous or semi-inﬁnite. As an example, ﬁg. 5 shows a measurement of the optical properties of a baby brain. We used a pad with many sources and detectors, and placed it on the baby’s head. Then we measured the diﬀuse photon density wave amplitude and phase as a function of source-detector separation. A ﬁt to the data using the extrapolated zero-boundary condition provides the average tissue scattering and absorption coeﬃcients at a single optical wavelength. If the medium is slab-like instead of semi-inﬁnite, it is straightforward to derive other analytic results. One can also derive results for cylinders, spheres, spheres in slabs, and more. We will not discuss these examples further. Suﬃce to say that for simple monitoring applications, some sort of analytic model can always be found to ﬁt for measured phase and amplitude data, and, therefore, for deriving average scattering and absorption coeﬃcients. Before turning to imaging and tomography, we brieﬂy consider the critically important problem of tissue diﬀuse optical spectroscopy (DOS). Thus far we have discussed the reﬂectance problem at a single optical wavelength. Arguably the most important feature of optics, however, is its potential for spectroscopy. For example, suppose we assume that there are two chromophores in tissue, oxy- and deoxy-hemoglobin; both of these chromophores will contribute to the measured absorption coeﬃcient. The relative amount that each chromophore contributes depends on its extinction coeﬃcient and its concentration. If we derive tissue absorption coeﬃcients at multiple wavelengths —say 780 nm and 805 nm— then we generate two equations with two unknowns. The two unknowns are the concentrations of oxyhemoglobin ([HbO2 ]) and deoxyhemoglobin ([Hb]).

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By solving these equations, using the data and the known chromophore extinction coefﬁcients, we can therefore derive [HbO2 ] and [Hb]. From this information we can further derive total hemoglobin concentration ([THC] = [HbO2 ] + [Hb]) and tissue blood oxygen saturation (St O2 = [HbO2 ]/([HbO2 ] + [Hb])). [THC] and St O2 are important clinical quantities that can thus be derived from relatively simple multi-wavelength, diﬀuse optical reﬂectance measurements. These reﬂectance techniques go by the names Diﬀuse Optical Spectroscopy (DOS) or Near-Infrared Spectroscopy (NIRS) in the biomedical optics community. 4. – Diﬀuse Optical Tomography (DOT): imaging Suppose the media we seek to understand are very heterogeneous. Furthermore, suppose that these heterogeneities are the quantities which we seek to learn about. How does one handle this situation? In this case, some kind of tomography or image reconstruction is desirable [17-19]. Fortunately, it is conceptually straightforward to develop strategies for this goal, because the heterogeneity problem is basically a problem that has already been studied in the context of waves and scattering theory. The only diﬀerence for us is that the waves are now diﬀuse photon density waves. In essence, in our experiments an incident diﬀusive wave is launched from each source ﬁber into the medium and the wave scatters from local optical property heterogeneities. From measurements of scattered waves on the tissue surface, one can set up an inverse problem to work backwards from the “perturbed” wave at the tissue surface to derive the heterogeneous optical properties within the entire medium. The standard approach for setting up the problem is to divide the absorption and light diﬀusion coeﬃcients into background and heterogeneous pieces, and then insert these parameters back into the diﬀusion equation. Then we look for perturbative solutions. Both the Born perturbation approach and the Rytov perturbation approach [20] will work as long as scattering of diﬀusive waves by heterogeneities in the sample is weak. As a concrete example, let us suppose the tissue optical property variation is only due to absorption. In this case the scattered wave depends approximately on the value of the incident wave at the heterogeneity times the strength of the heterogeneity times a propagator that describes the diﬀusive wave transport from the heterogeneity to the detector. The propagator is just the Green’s function for the diﬀusive wave. Within this formulation, there will arise an integral equation for every source-detector pair on the sample surface. The integral is over the entire sample volume. At this point the sample volume is divided up into discrete-volume elements and the inverse problem is formulated readily in the language of matrices. Here, a weight matrix, whose elements are basically the product of Green’s functions, couples a vector of tissue optical properties to a vector of measurements of the scattered diﬀusive waves. The absorption unknowns in each volume element of the sample are represented as a vector; the measured scattered signals for each source-detector pair are also represented by elements of a measurement vector. The result is a set of linear equations that can be inverted, for example, by the singular value decomposition technique [21], to derive an absorption tomogram. In

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practice many other techniques can be used to solve this class of problems, though issues of uniqueness and regularization always crop up. Most people nowadays solve the inverse problem iteratively, a method of choice when the heterogeneities are not weak. DOT has been demonstrated to work very well in phantoms. Its use in the clinic is promising and is a subject of current research. 5. – Diﬀuse Correlation Spectroscopy (DCS): blood ﬂow We will discuss one last formal problem before moving to applications: Diﬀuse Correlation Spectroscopy (DCS). DCS is a multiple-scattering correlation methodology for measuring blood ﬂow. First, let us recall the single-scattering version of this problem: dynamic light scattering. In the DLS experiment, a sample is illuminated. Illuminated particles in the sample act like radiating dipoles. We collect their radiated ﬁelds with a photon detector located oﬀ at some scattering angle. The collected ﬁelds ﬂuctuate and the intensity ﬂuctuates too, because the particles move and the relative phases of the radiated dipole ﬁelds change as a result of this motion. Information about scatterer motions is contained in the electric ﬁeld and intensity temporal autocorrelation functions. The electric ﬁeld autocorrelation function decays exponentially at a rate that depends on how far the scattering particles move in the correlation time interval, i.e., on the particles’ mean-square displacement in the correlation time interval. Thus, we derive motional information by measuring this decay rate. Several schemes can be employed to analyze this problem in the multiple scattering limit. To keep things formally consistent, we will focus on the transport equation methodology. Ackerson and coworkers [22, 23] suggested that it should be possible to understand the propagation of temporal autocorrelation in turbid (and dynamic) media primarily by replacing radiance in the transport equation with the normalized electric ﬁeld temporal autocorrelation function. This replacement is tantamount to exchanging the function E ∗ (r, t, Ω)E(r, t, Ω) in the transport equation with E ∗ (r, t + τ, Ω)E(r, t, Ω); here E(r, t, Ω) is the electric ﬁeld of the diﬀusing light at r, t and traveling in the direction Ω (hereafter we will supress the propagation direction in our notation), and τ is the autocorrelation function time delay or interval. Then, following essentially the same logic and mathematics as before, we arrive at a linear transport equation for correlation, this time with the scattering source term that we have from DLS. We make the P1 approximation, as before, to convert the correlation transport equation to the Correlation Diﬀusion Equation shown below [24, 25]: (2)

α ∇ · (D(r)∇G1 (r, τ )) − v μa (r) + μs k02 Δr2 (τ ) G1 (r, τ ) = vS(r, t), 3

where G1 (r, τ ) = E ∗ (r, t + τ )E(r, t) is the unnormalized temporal electric ﬁeld autocorrelation function; k0 is the wave vector of the ﬁelds in the medium, and Δr2 (τ ) is the mean-square displacement of the scattering particles in time τ . The brackets

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(a)

63

(b)

Fig. 6. – (a) Schematic diagram of the cuﬀ ischemia experiment. (b) Intensity temporal autocorrelation curves measured for diﬀerent cuﬀ pressures. Notice that the decay rate decreases as cuﬀ pressure increases, indicating attenuation of blood ﬂow (obtained from [29]).

denote ensemble averages, or averages, over time. The parameter α represents the fraction of scattering events in the tissue that occur from moving particles, and S(r) is a source term. The correlation diﬀusion equation is a diﬀusion equation for electric ﬁeld temporal autocorrelation! Notice that in the limit that τ goes to zero, this equation reduces to the conventional diﬀusion equation for ﬂuence rate in the zero-frequency limit. Formally, this result (eq. (2)) is the diﬀerential equation form of the diﬀusing wave spectroscopy (DWS) technique [26-28] that Georg Maret invented and has also talked about in this summer school. The diﬀerential equation form is particularly attractive, however, because almost all the formal ideas we have discussed with respect to diﬀuse photon density waves will apply to diﬀusing ﬁeld temporal autocorrelation. That is, the solutions of the correlation diﬀusion equation are formally the same as for ﬂuence rate with only an additional absorption term due to particle motions. Thus, it follows that analogous monitoring and imaging measurements can be carried out with diﬀusing correlation. Figure 6 shows an example of a blood ﬂow measurement, based on cuﬀ ischemia [29]. The measurement of correlation is carried out with source and detector on the forearm, and it gives an autocorrelation function that decays in time. As the cuﬀ pressure is increased, the slope of the autocorrelation decay decreases. Thus the slope of this curve can be used to deﬁne a blood ﬂow index (BFI) for clinical experiments. The correlation function decay rate is used to get the BFI. The BFI, in turn, can be related explicitly to an eﬀective diﬀusion constant (not the Einstein Brownian Diﬀusion constant), that parameterizes the correlation function decay rate, times the factor alpha, which accounts for the fraction of scattering events from moving scatterers. We have found that relative changes in BFI, i.e., rBFI, provide a robust and quite accurate measure of relative blood ﬂow changes in a broad range of clinical scenarios [30-38]. 6. – Background on tissue hemodynamics Before we describe some physiological applications of diﬀuse optics, let us brieﬂy recall the tissue parameters to which these tools are sensitive. Absorption information provides access to the tissue concentrations of endogenous chromophores such as HbO2 ,

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Hb, water and lipid, and also to the concentrations and local environments of exogenous chromophores such as imaging contrast agents and drugs. Scattering information provides access to organelle concentrations, cell concentrations, and subtle changes in the properties of background ﬂuids. Correlation spectroscopy provides information about the movement and ﬂow of scattering “objects” in tissues such as red blood cells. Taken together, the most prevalent application of these tools is for probing tissue hemodynamics. In this case, absorption gives information about total hemoglobin concentration and tissue blood oxygen saturation, and correlation spectroscopy gives information about blood ﬂow down to the tissue microvasculature level. Of course, all of these measurements represent tissue-averaged quantities; the degree of tissue averaging depends on the source-detector geometry. The circulatory system is a network of channels of varying diameter with the purpose to deliver nutrients to tissues and remove waste products of metabolism from tissues [39]. The arterial side of the vasculature is oxygen rich; it takes blood from heart and lungs to tissues. The venial side of the vasculature is oxygen poor; it takes blood back to the heart and lungs. The network is branched. The tubes start big, but break into smaller and smaller tubes until ﬁnally (at the oxygen delivery points), the capillaries bring red blood cells and plasma cells to within a hundred or so microns of every tissue cell in the body. The arterioles, capillaries and venules therefore ﬁll up a very sizable fraction of the tissue space. Thus, it is their responses that we are typically measuring with diﬀuse optics. Each red blood cell carries many molecules of hemoglobin (oxygenated or deoxygenated), the primary oxygen carrier. As we have noted, two forms of hemoglobin are important: oxygenated and deoxygenated. These hemoglobin molecules are typically in chemical equilibrium with dissolved oxygen in the tissue. When dissolved oxygen is low in tissue, then oxygen is unloaded oﬀ the HbO2 molecules and then the oxygen diﬀuses into the tissue. This equilibrium between blood oxygen saturation and tissue pO2 is characterized by the classic “Hill Curve” [39, 40]. Remember that diﬀuse optics measures blood oxygen saturation, which is closely related to tissue pO2 via the Hill Curve. When tissue oxygen is low, then bad things can happen to tissues. Brieﬂy, oxygen is brought into tissues via the arterioles, and then some of this oxygen is used by the tissues for metabolic processes, and the leftover oxygen is removed via the venules. Hypoxia occurs when tissue oxygen is low. This condition can arise from a variety of eﬀects. For example, the amount of breathed oxygen could be low (as on a high mountain), or the delivery of oxygen to the tissue could be impaired (e.g., by blockages in the feeding arteries), or the tissue metabolism could be abnormally high, etc. Tissue hypoxia can therefore reﬂect problematic clinical scenarios such as ischemic stroke, therapy resistant hypoxic tumors, muscle disease, and more. To recapitulate, tissue oxygen dynamics are important in clinical contexts. Diﬀuse optics does not measure pO2 directly, but it does measure blood oxygen saturation, total hemoglobin concentration, and blood ﬂow. All of these hemodynamic ingredients are useful to know, and they permit experimenters to develop a picture of the functioning (or malfunctioning) tissue. In fact, measurement of changes in all three hemodynamic

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quantities permits scientists to compute changes in another fundamental tissue property: oxygen metabolism. Convincing recent work along these lines has been carried out in all-optical functional experiments in brain, which probe changes in cerebral oxygen metabolism [31, 32]. 7. – Validation and clinical examples We will use the remainder of this paper to provide a set of representative illustrations from our laboratory of the diﬀuse optical techniques in clinical and pre-clinical studies. These examples are not intended to be comprehensive; nor are they intended to provide a review of all activity in the ﬁeld. Rather, they are intended to give a ﬂavor for the kinds of measurements that are becoming possible. We will begin with some validation studies. One goal for our research group (as well as for many researchers in biomedical optics) is to validate the diﬀuse optical methods. For example, in ﬁg. 7a we show a measurement of the Hill Curve that relates oxygen partial pressure in tissue to blood oxygen saturation. The data in ﬁg. 7a are derived from a simple tissue phantom experiment. The tissue phantom had optical properties very similar to those of human tissue and contained mouse erythrocytes (i.e., mouse blood). Over the course of sample deoxygenation we measured both oxygen partial pressure with needle electrodes (both Eppendorf histograph and Clark-style electrodes) and blood oxygen saturation with a standard diﬀuse optical spectroscopy (DOS) set-up. It should be apparent that the curve behaves as expected, thereby providing validation for diﬀuse optical HbO2 and Hb quantiﬁcation. In ﬁg. 7b, in vivo mouse tumor experiments show relative changes in DOS-measured St O2 (in ﬁgure, St O2 and SO2 both refer to blood oxygen saturation) as a function of relative changes in Eppendorg histograph pO2 during carbogen breathing. We see that the two techniques are again in quite good agreement. Indeed there is a substantial literature that corroborates DOS (NIRS) measurements of tissue blood oxygen saturation and blood volume concurrently with other medical diagnostics and/or with the literature.

(a)

(b)

Fig. 7. – (a) Oxyhemoglobin dissociation curve obtained from mouse erythrocytes in a tissue phantom. (b) Correlation between relative blood oxygen saturation (SO2 ), as measured by diﬀuse optics, and relative pO2 , as measured by an Eppendorf histograph, in ﬁbrosarcoma tumors in mice during carbogen breathing [41].

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(b) EtCO2 CBF

Room Air

(c ) Capnograph 3 6% CO2

Heart Rate: Pulse-Ox Arterial O 2 : Pusle-Ox Blood Pressure: Arm-Cuff E t CO2 : Capnograph

Fig. 8. – (a) Schematic diagram of the hypercapnia experiment. (b) Measured end-tidal CO2 and CBF, from DCS, for a single subject. (c) Simultaneous skin/scalp ﬂow measured by Laser Doppler for the same subject. Activation occurs between the thick solid lines in the ﬁgure.

Serious validation of diﬀuse correlation spectroscopy (DCS) has only begun relatively recently [32, 36, 38, 42, 43]. In ﬁg. 8 we show the functional blood ﬂow response of an adult to a hypercapnia perturbation [42]. In hypercapnia, the subject breathes excess CO2 and blood ﬂow is increased in his/her brain as a result. In practice we place the DCS probe (with source-detector separation of 2.5 cm on the tissue surface) locally on the head, even though hypercapnia is a whole-brain response. In DOS/DOT and DCS measurements, light penetration depth through the scalp and skull and into the cortex depends on source-detector separation; the penetration depth is typically one-third to one-half of the source-detector separation on the surface of the head. Thus, the light penetrates into approximately 0.5 cm of cortex in the present case. The increase in blood ﬂow with inspired pCO2 is evident from the data and is in quantitative agreement with other measurements from the literature [44-46]. We also conﬁrmed that the DCS measurement is not simply measuring skin/scalp ﬂow, by comparing the DCS response to the much smaller (and noisier) Laser Doppler skin/scalp ﬂow measurement. As a second DCS validation example, we show data from experiments in brain-injured patients in the neuro-intensive care unit [47]. In this case (ﬁg. 9), we compared DCS measurements of cerebral blood ﬂow to ﬂow measurements by xenon-CT. The latter is a standard-of-care ﬂow measurement in the clinic, but it cannot be used for continuous bedside monitoring because of its complexity. Using the Xe-CT images, we can compare

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Fig. 9. – (a) Top: single axial slice from non-contrast CT scan, with the region under DCS probes outlined. Bottom: baseline (left) and after Xe infusion (right) CBF maps obtained from XeCT scan. (b) Correlation between CBF as measured by DCS and xenon CT (color online; for more details, see [47]).

the same tissue volumes during various drug stimulations, such as increased dose of vasopressor drugs. A simple protocol was thus set up to induce cerebral blood ﬂow changes and to follow responses with both techniques. Again, we found that both techniques were quite strongly correlated (ﬁg. 9) [47]. Beyond validation, research in the ﬁeld is oriented towards two classes of brain investigation. The ﬁrst class of study is concerned with the function of normal brain, and the second class of study is clinical, aimed to improve patient care. We will give one example of each class. First, we illustrate functional imaging through the skull. Many functional paradigms have been developed to understand normal responses associated, for example, with vision, with verbal ﬂuency, with motor skills, and more. As a concrete example, consider the classic motor stimuli functional activation problem: ﬁnger tapping. When you tap your thumb against your foreﬁngers, it is well known that a very small part of the motor cortex is activated. In a recent experiment (ﬁg. 10) we investigated whether the full hemodynamic responses associated with ﬁnger tapping could be measured non-invasively in vivo, through the skull. To this end, we built a small diﬀuse optical probe pad with multiple source-detector separations (∼ 2.5 cm), and with the ability to carry out both DOS and DCS measurements concurrently. We placed the probe on the heads of multiple subjects, sometimes with ﬁnger tapping on and sometimes with ﬁnger tapping oﬀ, and sometimes over the activated region of the motor cortex and sometimes a centimeter away from this activation region. Our observations are summarized concisely in ﬁg. 10. With the probe in the “wrong place”, activation was not observed. On the other hand, with the probe placed directly over the motor activation site, we observed large perturbations in the concentrations of oxy-, deoxy-hemoglobin, and in blood ﬂow. Averaging over a small subject population permits comparison with other techniques, and calculation of oxygen metabolism from the all-optical probe! A most exciting aspect of this work is the quantiﬁcation potential

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Fig. 10. – (a) Schematic diagram showing the experimental setup. Hemodynamic responses are measured by diﬀuse optical methods as a function of time. In (b) and (c) data between dashed lines mark activation period. (b) Data with probe located over the somatomotor cortex, and (c) 1 cm oﬀ-center from the activation spot (color online) [32].

of diﬀuse optics. Indeed, many researchers are exploring all sorts of functional activation paradigms with diﬀuse optics at the present time [43, 48-52]. In considering clinical applications for brain, it is important to ﬁrst reﬂect upon the sorts of things that doctors measure at the present time, as well as upon the kinds of physiological information that doctors need/want to know. Most of the current clinical methods focus on metabolism-related problems and issues such as oxygen delivery, pressure diﬀerences in the brain due to, for example, swelling, and ﬂow autoregulation. In fact, many treatment strategies for brain-injured patients basically aim to increase blood ﬂow to the injured parts of brain. For example, the normal brain has a broad range of cerebral perfusion pressure conditions for which the brain autoregulatory apparatus adjusts to keep oxygen delivery optimized. However, if a patient falls out of this normal range due to stroke or head injury, then the vasculature does not respond as well, and the patient might need drugs or other manipulations to ameliorate problems. The situation is exacerbated because the current diagnostic tools available to clinicians tend to be very invasive (e.g., intracranial pressure and oxygen monitors, etc.) and/or very slow, cumbersome and costly (e.g., MRI, Xe-CT, etc.). Thus a golden opportunity for diﬀuse optics is evident: continuous cerebral blood ﬂow and oxygenation monitoring. A recent study, that illustrates the potential of diﬀuse optics for monitoring ﬂow treatment, was carried out in a critically ill population of ischemic stroke patients [37]. The idea of these experiments (ﬁg. 11) was to longitudinally monitor cerebral hemodynamics at the bedside induced by changes in head-of-bed positioning. Our hypothesis

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Fig. 11. – (a) Schematic representation of the head-of-bed (HOB) manipulation. (b) CBF changes in each hemisphere after each HOB manipulation in a healthy population. CBF changes in diseased population showed (c) impaired autoregulation in the injured hemisphere, but about 25% of the population presented a paradoxical response (d), where CBF was decreased at −5 degrees (color online) [37].

was that in response to this challenge, the impaired cerebral autoregulation would lead to larger changes in cerebral hemodynamics in the infarcted hemisphere by comparison to the “healthy”, contralateral hemisphere. To this end, diﬀuse optical measurements were obtained from patients with acute hemispheric ischemic stroke (n = 17, mean age 65 years). The probes were placed on the forehead near the frontal poles. Cerebral blood ﬂow and the hemoglobin concentrations were measured at diﬀerent head-of-bed (HOB) positions of 30, 15, 0, −5 and 0 degrees, and normalized to their values at 30 degrees. A clear diﬀerentiation was observed between two hemispheres that was statistically significant over the whole population. Interestingly, in roughly one-fourth of the patients we observed that cerebral blood ﬂow was not maximized at −5 degrees; rather it was very small at this HOB angle. This paradoxical response was observed in traumatic brain injury patients and was likely a result of a substantial increase in intracranial pressure, a parameter that is not routinely monitored in ischemic stroke patients. This simple example illustrates that diﬀuse optical instrumentation can be deployed at the bed-side of critically ill patients, and that the methodology may be promising for use as a tool to optimize patient care based on real-time cerebral hemodynamic measurements. The last clinical example we will discuss concerns breast cancer detection, diagnosis and monitoring based (primarily) on diﬀuse optical tomography. Even though diﬀuse optics is a relative newcomer to the breast imaging ﬁeld, the methodology could ﬁnd uses in this important ﬁeld. Potential niches for diﬀuse optics include detection/screening in high and intermediate risk populations, diagnosis between malignant and benign among certain classes of call-back patients, and therapy monitoring. Collectively, our view is that the diﬀuse optics ﬁeld is just now at the point where we are starting to obtain higher-ﬁdelity images of breast cancer, and we are beginning to conﬁrm and identify application niches. Here again, rather than provide a comprehensive review of the breast-DOT ﬁeld, we opt to show some illustrative results from our lab [33,53-55]. We have built a parallel-plate soft compression device shown schematically in ﬁg. 12. The instrument carries out measurements in transmission and remission at six optical wavelengths, and it is capable of

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Fig. 12. – Schematic representation of the parallel-plate diﬀuse optical tomography instrument.

both continuous-wave and frequency-modulated measurements. Approximately 105 –106 measurements are eﬀectively performed in ∼ 10 minutes. These measurements are then used as input to the inverse problem, and a three-dimensional (3D) tomogram of breast tissue physiological properties is thereby obtained [56, 57]. A sample image, showing a slice out of the full 3D reconstruction, is given in ﬁg. 13; in this case an invasive ductal carcinoma is found. Notice that the tumor shows up in some physiological variables (e.g. THC), but not in all (e.g. St O2 ). Furthermore, optical indices based on multiple physiological/optical properties can be constructed to improve tumor-to-normal contrast. Images such as these are based on endogenous tissue contrast, which is typically relatively small (1.3× to 1.5×). Exogenous contrast agents, such as Indocyanine Green (ICG), oﬀer the potential for improved contrast. Figure 14 shows recent images based on exogenous contrast via ﬂuorescence-DOT and standard-DOT endogenous contrast of the same tumor; notice that the ICG contrast is substantial and is therefore likely to stimulate development of better molecular markers for tumors in the future. Particularly in breast cancer, we expect that innovations in instrumentation and reconstruction algorithms will continue to be developed and combined to improve image ﬁdelity and resolution. In addition, more in vivo breast cancer data will provide critical insight and guidance for directed algorithm/instrumentation development. In searching for enhanced diﬀerentiation between tumor and normal tissues, groups across the community are employing broader wavelength ranges to explore water, lipid and collagen concentrations, bound water fraction, and refractive index [58-60], and they are even exploring blood ﬂow and metabolic contrast in breast cancer [33,61]. Multi-modal imaging and monitoring approaches can potentially overcome structural resolution limitations of DOT, using the spatial information provided by other imaging modalities to constrain the DOT inverse problem. These multi-modal approaches provide extra physiological information. DOT measurements have been made with concurrent MRI [62, 63], 3D X-ray mammography [64], and ultrasound [65]. Furthermore, advances in diﬀuse optical tomography of breast are critical for exploitation of the advances of molecular imaging [66], an emerging ﬁeld of medicine with promise of new-generation optical-contrast agents.

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Fig. 13. – Caudal-cranial view of diﬀuse optical images in a ﬁeld of view (black arrows) covering a tumoral region, with diﬀerent parameters used as contrast [55].

Fig. 14. – Optical images from a breast with a tumoral region, using diﬀerent parameters as constrast. Notice the high contrast provided by ICG, compared to the others endogenous parameters (see [53] for more details).

8. – Concluding remarks It should be evident that the dream of using optics for in vivo biopsy is being realized. In this informal review we have described these developments broadly, but we have also left a substantial amount of important research out of our discussion. For example, diﬀuse optics diagnostics have found uses in the study of muscle disease [36], in cancer therapy monitoring of human subjects [55] and pre-clinical animal models [67, 68], in studies of osteoarthritis [69] and skin disease [70]. We hope that our discourse will have inspired the reader to explore the ﬁeld further and perhaps even to join in the fun!

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∗ ∗ ∗ The authors would like to acknowledge many fruitful discussions and interactions over the years with colleagues from the biomedical community at the University of Pennsylvania and throughout the world. At UPenn, much of this research was facilitated by sustained collaborations with B. Chance, J. Greenberg, J. Detre, M. Schnall, J. Culver, G. Yu, D. Boas, T. Durduran, R. Choe, and T. Busch. This work was supported by the National Institutes of Health through NS-060653, HL-57835, RR-02305, NS-45839, and CA-126187. REFERENCES [1] Yodh A. and Chance B., Phys. Today, 48 (1995) 34. [2] Hielscher A. H., Bluestone A. Y., Abdoulaev G. S., Klose A. D., Lasker J., Stewart M., Netz U. and Beuthan J., Disease Markers, 18 (2002) 313. [3] Yodh A. G. and Boas D. A., Functional Imaging with Diﬀusing Light, in Biomedical Photonics, edited by Vo-Dinh T. (CRC Press, Boca Raton) 2003, pp. 21/1-45. [4] Obrig H. and Villringer A., J. Cereb. Blood Flow Metab., 23 (2003) 1. [5] Gibson A. and Dehghani H., Philos. Trans R. Soc. A, 367 (2009) 3055. [6] Wang L. V. and Wu H., Diﬀuse Optical Tomography, in Biomedical Optics: principles and imaging (John Wiley & Sons, New Jersey) 2007, pp. 249-282. [7] van de Hulst H. C., Light Scattering by Small Particles (Dover, Mineola) 1981. [8] Clark N. A., Lunacek J. H. and Benedek G. B., Am. J. Phys., 38 (1970) 575. [9] Berne B. J. and Pecora R., Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Krieger, Malabar) 1990. [10] Chu B., Laser Light Scattering: Basic Principles and Practice (Academic, New York) 1991. [11] Arridge S. R. and Schotland J. C., Inverse Problems, 25 (2009) 123010. [12] Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley, Reading) 1967. [13] Ishimaru A., Wave Propagation and Scattering in Random Media (Academic Press, San Diego) 1978. [14] O’Leary M. A., Boas D. A., Chance B. and Yodh A. G., Phys. Rev. Lett., 69 (1992) 2658. [15] Boas D. A., O’Leary M. A., Chance B. and Yodh A. G., Phys. Rev. E, 47 (1993) R2999. [16] Danen R. M., Wang Y., Li X. D., Thayer W. S. and Yodh A. G., Photochem. Photobiol., 67 (1998) 33. [17] Arridge S. R., Inverse Problems, 15 (1999) R41. [18] Boas D. A., Brooks D. H., Miller E. L., DiMarzio C. A., Kilmer R. J. and Zhang Q., IEEE Sign. Proc. Mag., November (2001) 57. [19] Gibson A. P., Hebden J. C. and Arridge S. R., Phys. Med. Biol., 50 (2005) R1. [20] Kak A. C. and Slaney M., Principles of Computerized Tomographic Imaging (IEEE Press, New York) 1988. [21] Golub G. and Kahan W., J. SIAM Numer. Anal., 2 (1965) 205. [22] Ackerson B. J., Dougherty R. L., Reguigui N. M. and Nobbman U., J. Termophys. Heat Trans., 6 (1992) 577. [23] Dougherty R. L., Ackerson B. J., Reguigui N. M., Dorri-Nowkoorani F. and Nobbman U., J. Quantum Spectrosc. Radiat. Transfer, 52 (1994) 713.

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[49] Franceschini M. A., Thaker S., Themelis G., Krishnamoorthy K. K., Bortfeld H., Diamond S. G., Boas D. A., Arvin K. and Grant P. E., Pediatr. Res., 61 (2007) 546. [50] Zeff B. W., White B. R., Dehghani H., Schlaggar B. L. and Culver J. P., Proc. Natl. Acad. Sci. U.S.A., 104 (2007) 12169. [51] Jaillon F., Li J., Dietsche G., Elbert T. and Gisler T., Opt. Express, 15 (2007) 6643. [52] Becerra L., Harris W., Joseph D., Huppert T., Boas D. A. and Borsook D., Neuroimage, 41 (2008) 252. [53] Corlu A., Choe R., Durduran T., Rosen M. A., Schweiger M., Arridge S. R. and Yodh A. G., Opt. Express, 15 (2007) 6696. [54] Konecky S. D., Choe R., Corlu A., Lee K., Wiener R., Srinivas S. M., Saffer J. R., Freifelder R., Karp J. S., Hajjioui N., Azar F. and Yodh A. G., Med. Phys., 35 (2008) 446. [55] Choe R., Konecky S. D., Corlu A., Lee K., Durduran T., Busch D. R., Pathak S., Czerniecki B. J., Tchou J., Fraker D. L., DeMichele A., Chance B., Arridge S. R., Schweiger M., Culver J. P., Schnall M. D., Putt M. E., Rosen M. A. and Yodh A. G., J. Biomed. Opt., 14 (2009) 024020. [56] Arridge S. R. and Schweiger M., Appl. Opt., 34 (1995) 8026. [57] Arridge S. R. and Schweiger M., Opt. Express, 2 (1998) 213. [58] Srinivasan S., Pogue B. W., Brooksby B., Jiang S., Dehghani H., Kogel C., Wells W. A., Poplack S. P. and Paulsen K. D., Technol. Cancer Res. Treat., 4 (2005) 513. [59] Cerussi A., Shah N., Hsiang D., Durkin A., Butler J. and Tromberg B. J., J. Biomed. Opt., 11 (2006) 044005. [60] Srinivasan S., Pogue B. W., Carpenter C., Jiang S., Wells W. A., Poplack S. P., Kaufman P. A. and Paulsen K. D., Antioxidants Redox Signaling, 9 (2007) 1143. [61] Zhou C., Choe R., Shah N., Durduran T., Yu G., Durkin A., Hsiang D., Mehta R., Butler J., Cerussi A., Tromberg B. J. and Yodh A. G., J. Biomed. Opt., 12 (2007) 051903. [62] Ntziachristos V., Yodh A. G., Schnall M. D. and Chance B., Neoplasia, 4 (2002) 347. [63] Carpenter C. M., Srinivasan S., Pogue B. W. and Paulsen K. D., Opt. Express, 16 (2008) 17903. [64] Fang Q., Carp S. A., Selb J., Boverman G., Zhang Q., Kopans D. B., Moore R. H., Miller E. L., Brooks D. H. and Boas D. A., IEEE Trans. Med. Imaging, 28 (2009) 30. [65] Zhu Q., Tannenbaum S., Hegde P., Kane M., Xu C. and Kurtzman S. H., Neoplasia, 10 (2008) 1028. [66] Weissleder R. and Ntziachristos V., Nature Med., 9 (2003) 123. [67] Luckl J., Zhou C., Durduran T., Yodh A. G. and Greenberg J. H., J. Neurosci. Res., 87 (2009) 1219. [68] Zhou C., Eucker S. A., Durduran T., Yu G., Ralston J., Friess S. H., Ichord R. N., Margulies S. S. and Yodh A. G., J. Biomed. Opt., 14 (2009) 034015. [69] Hofmann G. O., Martick J., Grosstuck R., Hoffmann M., Lange M., Plettenberg H. K. W., Braunshweig R., Schilling O., Kaden I. and Spahn G., Pathophysiol., 17 (2010) 1. [70] Bednov A., Ulyanov S., Cheung C. and Yodh A. G., J. Biomed. Opt., 9 (2004) 347.

Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-75

Ultrasonic wave transport in strongly scattering media J. H. Page Department of Physics and Astronomy, University of Manitoba Winnipeg, MB Canada R3T 2N2

Summary. — Ultrasonic experiments are well suited to the investigation of classical wave transport through strongly scattering media, and are playing a role that is often complementary to investigations using light or microwaves. Advantages of ultrasonic techniques are their ability to readily detect the wave ﬁeld (not just the intensity), to perform experiments resolved in both time and space, and to control the properties of the medium being investigated over a wide range of scattering contrasts. This ﬁrst paper reviews what has been learned from ultrasonic experiments over the last 15 years about the ballistic and diﬀusive propagation of classical waves through strongly scattering disordered media. These results are compared with studies of ordered media (phononic crystals), where band gaps and super-resolution focusing have been observed.

1. – Introduction For more than a decade, there has been growing interest in ultrasonic wave transport in strongly scattering media. Just as for other classical waves, such as light and microwaves, much of this interest revolves around the many unusual wave phenomena have been observed at intermediate frequencies, where the wavelengths are comparable to the size of the scatterers [1]. Examples range from strikingly large variations in wave speeds caused by strong resonant scattering (when a pulse can even appear to travel so quickly c Societ` a Italiana di Fisica

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through a sample that its velocity is negative) to the inhibition of wave propagation that can occur in very strongly scattering samples when the waves become localized. In seeking to discover and understand such wave phenomena, ultrasonic experiments have an important role to play, partly because of the relative ease with which the full wave ﬁeld, rather than just the intensity, can be measured. Thus, ultrasonic techniques give direct experimental access to the wave function and/or Green’s function, allowing both phase and amplitude information to be obtained. Ultrasound is also well adapted to pulsed experiments, enabling the path length dependence of the transmission or reﬂection to be resolved in time, usually in the near ﬁeld. Furthermore, the fact that the scattering contrast is governed by diﬀerences in both velocity and density enables the scattering strength to be controlled over a very wide range. As a result, experiments with acoustic or elastic waves can make important contributions to both fundamental studies and practical applications of wave scattering in complex media, and are often complementary to optical and microwave methods for investigating these phenomena. In this paper, I will review the progress that has been achieved over the last 15 years in understanding how ultrasonic waves propagate through both random and ordered media. The regime of interest here is one where multiple scattering dominates, but the scattering is not so strong that the interference eﬀects leading to Anderson localization are present. (The latter is the subject of the second paper in this series, while the third paper discusses applications such as Diﬀusing Acoustic Wave Spectroscopy.) To illustrate the scope of information that is accessible to ultrasonic experiments in random systems, sect. 2 summarizes results obtained for acoustic waves (longitudinal polarization only) in a model system consisting of a suspension of glass beads in a liquid, where a rather complete picture of wave transport has been achieved through transmission measurements. Other types of acoustic scattering systems (plastic spheres and bubbles surrounded by water), which lead to diﬀerent wave behaviour, are also mentioned. By contrast to the diﬀusive transport of energy that is seen in disordered systems, the propagation of multiply scattered waves in ordered media is characterized by a coherent multiply scattered wave ﬁeld, leading to band gaps and unusual focusing phenomena. These eﬀects are described in the last major section of the paper on phononic crystals (sect. 3). 2. – Acoustic wave transport in random media Many features of ultrasound transport in strongly scattering media are demonstrated by acoustic pulse propagation experiments that have been performed in a disordered medium consisting of randomly packed 0.5-mm-radius glass beads immersed in water [2-5]. Strong scattering in this model system is ensured by the large acoustic impedance diﬀerence between glass and water (Zglass /Zwater ≈ 10, where Z = ρvp is the acoustic impedance, ρ is the density and vp is the phase velocity). Although the transmitted signals are dominated by multiply scattered waves at intermediate frequencies, the phase sensitivity of piezoelectric ultrasonic transducers allows measurements to be made of the weak signal that propagates ballistically through the medium without

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Fig. 1. – Comparison of the total ﬁeld (a), average ﬁeld (b) and scattered ﬁeld (c) transmitted through a thin sample (thickness, L = 1.7 mm) of glass spheres in water. The data are normalized with respect to an input pulse of unit peak amplitude.

scattering out of the forward direction. This ballistic signal remains coherent both temporarily and spatially with the input pulse, so that it can be extracted by averaging the transmitted wave ﬁeld over many speckles, a procedure that causes the multiply scattered component to cancel because of the random phase ﬂuctuations from speckle to speckle. The separation of the ballistic pulse from the multiply scattered waves, whose energy transport was found to be well described by the diﬀusion approximation, allows a very complete picture of wave transport in strongly scattering media to be obtained. . 2 1. Ballistic propagation. – Figure 1 shows an example of how the ballistic pulse can be extracted from the total transmitted ﬁeld when the sample is suﬃciently thin. The experiments were performed by enclosing the suspension of glass particles in a cell with thin walls that are transparent to ultrasound, and then placing the sample in a water tank between a plane-wave generating transducer and a subwavelength-diameter hydrophone detector. The hydrophone position was scanned in a plane near the surface of the sample to measure the transmitted signal in many independent coherence areas, or speckles, using a grid separation of approximately a wavelength. The wave ﬁeld averaged over more than 100 speckles reveals the ballistic pulse, shown by the solid curve in ﬁg. 1(b). The multiply scattered waveforms (ﬁg. 1(c)), often called the coda, especially in the context of seismic waves, since they arrive after the ballistic pulse, can then also be obtained by subtracting the ballistic pulse from the total transmitted ﬁeld in each speckle. This demonstration of coherent ballistic pulse propagation provides convincing experimental evidence that a (uniform) eﬀective medium can still be deﬁned

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Fig. 2. – Frequency dependence of (a) the phase velocity, (b) the group and energy velocities, (c) the scattering and transport mean free paths, and (d) the diﬀusion coeﬃcient. The dotted horizontal line in (a) indicates the sound velocity in water. All data (symbols) and theory (solid and dashed curves) were measured/averaged over a small 5% variation in the bead size.

in the intermediate strongly scattering regime, a result that has been inferred less directly in recent optical experiments [6]. The ballistic pulse contains information that is crucial for determining the frequency dependence of the scattering properties of any sample, as it allows both the phase and group velocities [vp = ω/k, vg = dω/dk] as well as the scattering mean free path ls [I(L) = I(0) exp[−L/ls ]], to be determined [5]. Here ω, k, L and I are the angular frequency, wave vector, sample thickness and ultrasound intensity, respectively. Experimentally, vp and ls are determined from the phase diﬀerence Δφ and amplitude ratio A(L)/A(0) of the fast Fourier transforms (FFT) of the ballistic and input pulses (vp = ωL/Δφ, ls = −L/ ln[A(L)/A(0)]2 ). The group velocity is accurately measured by digitally ﬁltering the ballistic and input pulses using a narrow Gaussian ﬁlter, whose bandwidth is chosen to be suﬃciently narrow that pulse distortion due to dispersion is negligible, and measuring the delay tg between their peak arrival times (vg = L/tg ). Results for randomly close packed suspensions of glass beads in water are shown in ﬁg. 2 over a wide frequency range, corresponding to wavelengths from approximately 5a to 0.5a (dimensionless frequency range: 1 kw a 10, where kw is the wavelength in water and a is the bead radius). For kw a > 2, strongly scattering is seen, with the scattering mean free path approaching the bead radius, and the product kls ranging from 3 to 9. Both the phase and group velocities exhibit a considerable frequency dependence, with the group velocity varying by more than a factor of 2. Note the very low values of the group velocity near kw a = 2, where vg is substantially less than the sound velocities in either water or glass (vw = 1.5 mm/μs, vglass = 5.6 or 3.4 mm/μs for longitudinal or transverse polarizations).

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The origin of these large velocity variations can be understood using an eﬀective medium model, based on a Spectral Function Approach (SpFA), which overcomes a fundamental limitation of the traditional Coherent Potential Approximation (CPA) in the intermediate frequency regime [3]. The scattering is calculated by modeling a typical glass bead scatterer as an elastic sphere that is coated with a layer of water and embedded in a homogeneous eﬀective medium, which accounts for the presence of all the other scatterers. The dispersion relation, ω versus k, for acoustic waves in the medium is then determined by identifying the peaks of the spectral function, given by the negative imaginary part of the Green’s function. The simple physical interpretation of the method is that these peaks correspond to the locus of points in the frequency-wavevector plane where the scattering is weakest, so that they delineate the modes that succeed in propagating through the medium and identify the eﬀective medium properties. The approach is accurate so long as kls 2 [5]. This dispersion relation enables vp and vg to be calculated, giving the excellent agreement with the experimental data shown in ﬁg. 2. In addition, the scattering mean free path can be determined from the scattering cross section of the coated elastic sphere [3, 5]. By calculating the energy density of a typical scatter as a function of frequency, the sharp features in the group velocity near kw a ∼ 2 and above kw a ∼ 5 were found to be associated with resonances of the ﬂuid coating and solid spheres, respectively, leading in the ﬁrst case to a slowing down of the velocity by tortuosity of the connected ﬂuid pathways and in the second case to resonant trapping of energy in the solid scatterers [5]. The overall mechanism underlying the frequency dependence of the phase and group velocities can be understood as follows: because of the strong coupling between the resonant scatterers, the uniform eﬀective medium sensed by the coherent ballistic propagation is very strongly renormalized, in much the same way as quantum-mechanical resonances are shifted when there is strong coupling between them. Thus, the ballistic pulse is still able to propagate coherently while being very strongly aﬀected by the scatterers. These experimental and theoretical results also show that the group velocity remains well deﬁned despite the strong scattering [3], thereby addressing a question about the meaning of the group velocity in dispersive media that was raised by Sommerfeld [7] and Brillouin [8] in the ﬁrst part of the 20th century and discussed more recently by Albada et al. [9]. Ultrasonic experiments on other types of suspensions with diﬀerent acoustic properties have also been performed to examine how ballistic pulse propagation is aﬀected by the strength and character of the scattering resonances. One interesting example is a slurry of randomly close-packed plastic spheres in water, where gaps open up in the mode spectrum due to scattering resonances having the character of interfacial or Stoneley waves. These Stoneley-wave-like resonances involve both longitudinal and transverse displacements inside the spheres, and compressional deformations of the surrounding nearby liquid. As a result, a second longitudinal mode with slow velocities, due to the coupling between these Stoneley wave resonances on adjacent spheres, is observed. This slow mode was ﬁrst discovered by Brillouin scattering experiments [10], which probe the modes of the system by measuring the frequencies of the modes at ﬁxed wave vector, in contrast to ultrasonic pulse propagation experiments, which measure the velocities of

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Fig. 3. – (a), (b) Dispersion curves for a suspension of PMMA spheres in water. In (a), experimental results are represented by symbols for two diﬀerent sample thicknesses, and theoretical predictions from the peaks of the spectral function (shown in (b)) are shown by the solid curves. (c) Incident and transmitted pulses through a cloud of bubbles, which are shown in the grey scale background picture behind the graphs. The bubbles were generated by an electrolysis method and had a radius of ∼ 15 μm.

the modes at ﬁxed frequency. Ultrasonic measurements of the dispersion relations are shown in ﬁg. 3(a), and compared with peaks in the spectral function (ﬁg. 3(b)) predicted by the SpFA model. Good overall agreement is found, conﬁrming the basic character of the unusual modes of this system. In contrast to the Brillouin scattering results, the ultrasonic measurements reveal that because of absorption, longitudinal modes still propagate inside the “gap” and interfere with the Stoneley wave branch, leading to rich behaviour that provides a stringent test of the accuracy of the SpFA model. A second example of the eﬀects of very strong scattering is acoustic pulse propagation through a suspension of bubbles. The acoustic properties of bubbly suspensions are dominated by a low-frequency multipole resonance, leading to a wide range of unusual wave phenomena such as anomalous dispersion and superradiance (e.g., see refs. [11-13]). One remarkable consequence is shown in ﬁg. 3(c), which provides compelling experimental evidence that the group velocity is negative near the fundamental bubble resonance frequency [14]. This unusual eﬀect occurs because of pulse reshaping due to the anomalous dispersion, which leads to constructive interference at the leading edge of the pulse and destructive interference at the trailing edge; thus, the peak of the transmitted pulse emerges from the sample before the peak of the input pulse has entered it, so that the pulse transit time and hence group velocity is negative. It is noted that, at a given time, the intensity of the incident wave is always greater than the transmitted one, so that causality is not violated. In this case and in analogous examples for light [15], the group velocity still accurately describes ballistic pulse propagation, providing the bandwidth is suﬃciently narrow, but can no longer correspond to the ballistic energy transport velocity [16].

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Fig. 4. – (a) Time dependence of the transmitted intensity, normalized so that the peak of the corresponding input pulse is unity, for three randomly close-packed glass-bead-in-water samples having diﬀerent thicknesses L. The symbols represent experimental data taken with a point source, and the solid curves are ﬁts to the predictions of the diﬀusion approximation, from which the diﬀusion coeﬃcient and absorption time are determined. (b) Time dependence of the mean square transverse width w2 of the diﬀuse halo for a slab-shaped sample. The linear growth of w2 is characteristic of a diﬀusion process, enabling D to be measured from the slope of the straight line ﬁt (solid line) independent of absorption and boundary conditions.

. 2 2. Diﬀusive propagation. – Transport beyond the scale of the mean free path is dominated by multiply scattered waves (ﬁg. 1(c)). So long as the thickness of the sample is greater than about three mean free paths and the scattering is not so strong that kls ∼ 1, the transport of energy in ultrasonic experiments is well described using the diﬀusion approximation [2, 17]. In this approximation, all phase information is ignored and the quantity of interest is energy transport, which is treated as a random walk process, characterized by the diﬀusion coeﬃcient D = vE l∗ /3. Here vE is the average local velocity of energy transport, and l∗ is the transport mean free path, or distance over which the direction of propagation is randomized. The transport and scattering mean free paths are related by l∗ = ls /(1−cos θ), where θ is the scattering angle, and are therefore equal only when the scattering is isotropic. Dynamic (pulsed) measurements, which probe the distribution of multiply scattered path lengths in the time domain, are sensitive to D, while steady state (continuous wave) experiments, such as the measurement of total energy transmission, are sensitive to l∗ /L. To demonstrate the applicability of the diﬀusion approximation to acoustic wave transport in strongly scattering media, and to measure D, l∗ and vE over a wide frequency range, an extensive series of pulsed and quasi-continuous-wave experiments have been performed on the same glass bead suspensions described above [2, 4, 17]. For examples of other approaches to investigating diﬀusive transport of acoustic waves in 3D and 2D systems, see references [18, 19]. In refs. [2, 4, 17], slab-shaped samples were used and the diﬀusion coeﬃcient was measured from the temporal evolution of the average

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transmitted intensity, I(t), which was determined by averaging the square of the envelope of the scattered sound ﬁeld over a large number of independent speckles. Typical results, which were obtained using a tightly focused incident pulse to create a point source, are shown in ﬁg. 4(a) for three diﬀerent sample thicknesses, ranging from 7 to 30 scattering mean free paths. Comparison of the experimental data with solutions of the diﬀusion equation was facilitated by performing the measurements on slab-shaped samples, with widths at least 10 times the thickness so that edge eﬀects could be ignored (i.e., the samples were excellent approximations to inﬁnite slabs for the range of times over which signals could be detected). Accounting for internal reﬂections at the front and back faces of the slab and the possibility of dissipation inside the sample, the transmitted intensity (ﬂux) for a delta-function diﬀuse source of unit strength in time and position, δ(t)δ(x − x )δ(y − y )δ(z − z ), is given by the solution of the diﬀusion equation with these boundary conditions: (1)

2 ∞ 2 2 e−ρ /4Dt e−t/τa ∂U An e−Dβn t/L , = I(t) = −D ∂z z=L 2πL2 t n=1

where τa is the absorption time, βm are the positive roots of the transcendental equation tan β = 2βK/(β 2 K 2 − 1), K is equal to z0 /L with z0 = (2l∗ /3)(1 + R)/(1 − R) (z0 is known as the extrapolation length, since it is the distance outside the sample where the diﬀuse energy extrapolates to zero), R is the angle-averaged reﬂectivity of diﬀuse sound at the sample boundaries (calculated from the acoustic impedance mismatch), and the coeﬃcients An are given by an analytic function of βn , K and z [2]. Here 2 2 ρ = (x − x ) + (y − y ) is the transverse distance in the plane parallel to the slab at which the intensity is detected relative to the point directly opposite the source. The location of the diﬀuse source inside the sample, z , has been shown by numerical simulations to be equal to l∗ [20]. The solid curves in ﬁg. 4(a) show the results of leastsquares ﬁts of eq. (1) to this data set, with the initial increase of I(t) being sensitive to D and the tail quite strongly inﬂuenced by τa . The good agreement between theory and experiment demonstrates the validity of the diﬀusion approximation for multiply scattered sound, enabling reliable measurements of both D and τa to be made. One advantage of the point source geometry is that it enables the growth of the diﬀuse halo to be measured in the transverse direction parallel to the surface of the slab. This gives a method of measuring D directly, independent of boundary conditions and absorption [2]. Experimentally, the average transmitted intensity at transverse distance ρ (“oﬀ-axis”) and at ρ = 0 (“on-axis”) are measured by averaging over diﬀerent sample positions with source and detector positions ﬁxed relative to each other. From eq. (1), 2 2 2 it can be seen that the ratio I(ρ, t)/I(0, t) is given simply by e−ρ /4Dt = e−ρ /w (t) , so that the transverse width w(t) of the diﬀuse halo grows as the square root of time, as expected for a diﬀuse process. Plotting w2 (t) versus time enables D to be measured from the slope of a straight line ﬁt to the data, as shown in ﬁg. 4(b). The excellent agreement between the values of D measured directly from the transverse width, and from the more cumbersome ﬁts of eq. (1) to the time-of-ﬂight proﬁle, gives additional

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conﬁdence in the accuracy with which the diﬀusion coeﬃcient can be measured in pulsed transmission measurements. Ultrasonic experiments can also be performed using a good approximation to a plane wave source by placing the sample in the far ﬁeld of a planar immersion transducer. The solution of the diﬀusion equation for this experimental geometry can be obtained by integrating eq. (1) over x and y , giving Iplane (t) = 4DtIpoint (ρ = 0, t). Again, accurate measurements of D can be obtained by ﬁtting this expression to the measured time-ofﬂight proﬁles for this geometry (e.g., see ref. [2]). This exact solution of the diﬀusion equation is often approximated by the somewhat simpler expression (2)

∞ nπ(z + z0 ) nπ(z + z0 ) 2e−t/τa −Dn2 π2 t/(L+2z0 )2 sin . e sin U (t) ≈ π(L + 2z0 ) n=1 L + 2z0 L + 2z0

Equation (2) is a good approximation in many experimental situations, especially at long times, but is not accurate for large values of the reﬂectivity R. At long times, in the absence of absorption, eq. (2) is proportional to exp[−t/τD ], with τD = (L+2z0 )2 /(π 2 D). This gives a very simple result for the exponential decay of the time-of-ﬂight proﬁle in terms of the diﬀusion time τD , which is determined by the eﬀective thickness of the sample L + 2z0 and the diﬀusion coeﬃcient. Results for the frequency dependence of the diﬀusion coeﬃcient in the glass bead suspensions are shown in ﬁg. 2(d). A considerable variation, roughly a factor 3, is seen over the range of frequencies investigated. To determine its origin, experiments were also performed for very long pulses to attain quasi-continuous-wave conditions, so that l∗ could be measured from the thickness dependence of the total transmitted intensity, √ I(L) = fn(l∗ /L, α = Dτa ) (see ref. [2] for the complete expression). It was found that l∗ has at most a very weak frequency dependence (ﬁg. 2(c)), being approximately equal to the diameter of the beads in the strong scattering regime. This weak frequency dependence is also shown from calculations of l∗ using the SpFA model, where cos θ is determined from the average of cos θ weighted by the square of the angle-dependent scattering amplitude (solid curve in ﬁg. 2(c)). Hence the strong frequency dependence of D must be due to the variation of vE , which was determined experimentally from the ratio vE = 3D/l∗ using the measured values of D and l∗ . Figure 2(b) compares the measurements of the energy velocity with the group velocity, showing that both vE and vg , which describe the transport of energy through the sample by diﬀusive and ballistic waves, respectively, are remarkably similar in magnitude and frequency dependence. This similarity between vE and vg , which was not anticipated from earlier theoretical work for light [9], appears to hold quite generally except in cases of extreme dispersion, where the group velocity loses its meaning as the ballistic energy transport velocity (even though vg still describes narrow-band coherent pulse propagation accurately in such extreme conditions). The comparison shown in ﬁg. 2(b) suggests a simple physical picture for vE and its relationship to vg . Even in the forward direction, the transport of energy is strongly aﬀected by scattering resonances, which lead to a large scattering delay near the minima in vg . It is reasonable to expect that wave pulses scattered through a non-zero

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scattering angle will experience a similar, but not identical, scattering delay, so that in this case vE should be similar to vg , with the relation between them taking into account the additional angle-average scattering delay of the scattered waves [4]. These ideas can be formulated quantitatively by extending the SpFA model to calculate the additional scattering delay experienced by a wave pulse. In this approach, the angular dependence of the magnitude and phase shift in a typical scattering event is calculated for each frequency component of the wave pulse from the complex scattering amplitude of the coated elastic sphere embedded in the eﬀective medium. By incorporating these frequency-dependent phase and amplitude variations into the Fourier components of the incident Gaussian pulse, and taking the inverse Fourier transform to recover the scattered pulse, the corresponding time delay of the scattered pulse envelope relative to the forward direction can be calculated for each scattering angle. The intensity-weighted angular average of these additional scattering delays, Δtave , can then be used to express the energy velocity in terms of the group velocity, giving vE = l∗ /(l ∗ /vg + Δtave ) = vg /(1 − δm ), where δm = Δtave vg /l∗ . Note that, in this approach, vE , l∗ , vg and δm are all calculated in a renormalized eﬀective medium, which accounts for the eﬀects of the multiple scattering that become especially pronounced for high volume fractions of scatterers. Excellent quantitative agreement between the predictions of this model and the experimental data was found not only vE and l∗ but also for the diﬀusion coeﬃcient that is calculated from them using D = vE l∗ /3. In summary, these ultrasonic experiments in a model system consisting of glass beads in water have enabled a quantitative and comprehensive assessment of the applicability of the diﬀusion approximation to the description of energy transport by multiply scattered acoustic waves. By comparing the parameters that govern diﬀusive and ballistic transport over a wide frequency range, a uniﬁed physical picture of energy transport in strongly scattering media has emerged. In addition, the success of the SpFA model in describing the experimental results for both ballistic and diﬀusive waves highlights the relevance of an eﬀective medium description even in the strongly scattering intermediate frequency regime. The methods developed in this work have facilitated both the search for ultrasonic wave localization in more strongly scattering samples (see paper II, this volume, p. 95) and the development of novel dynamic probes of multiply scattering materials (see paper III, this volume, p. 115). 3. – Wave transport in ordered media: phononic crystals in 2D and 3D The character of ultrasound transport in strongly scattering media is changed dramatically when the scatterers are arranged in an ordered array to form a phononic crystal. These materials are the acoustic and elastic counterparts of photonic crystals for light, and have been the subject of increasing interest since the early 1990s [21]. Because it is relatively easy to control the strength of the scattering contrast between the component materials, phononic crystals may be viewed as ideal media in which to study the profound eﬀects of lattice structure on wave propagation. Much of the initial research concentrated on phononic band gaps, which occur due to Bragg scattering when the wavelength is com-

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Fig. 5. – Left panels: Amplitude transmission coeﬃcient along the [111] direction for a 12-layer fcc phononic crystal made from tungsten carbide beads, and the corresponding band structure. Experimental data are shown by the symbols, and the results of MST calculations by the solid curves. A photo of part of the surface of the crystal is shown in the insert on the left. Right panel: Thickness dependence of the group velocity at a frequency of 0.95 MHz in the ﬁrst band gap.

parable to the lattice constants, leading to frequency bands where wave propagation is forbidden. As a result, much is now known about the conditions under which phononic crystals with compete band gaps can be fabricated, allowing wave transport in this frequency range to be investigated and novel acoustic waveguides and noise blocking devices to be constructed [22-30]. Methods for making compact phononic crystal sound insulators have also been proposed and demonstrated [25]. More recently, attention has turned to wave transport in the pass bands both below and above the band gaps, where unusual negative refraction, diﬀraction and focusing eﬀects have been observed [31-35]. To illustrate the main diﬀerences between ultrasonic wave transport in ordered and disordered structures, consider the results that have been obtained for 3D phononic crystals made from 0.8-mm-diameter tungsten-carbide beads surrounded by water [28,31]. In this case, excellent crystal quality was assured by the availability of extremely monodisperse spheres due to the needs of the ballpoint pen industry, and meticulous handassembly of the spheres in a custom-made mould. In transmission, multiple scattering from the periodic array of scatterers leads to a transmitted pulse in the far ﬁeld with a well-deﬁned, but coherent, coda, so that the entire transmitted pulse can be analysed . by the methods outlined in subsect. 2 1. Thus, ultrasonic pulsed techniques can readily measure all the basic wave properties of the crystal, including the transmission coeﬃcient (from the ratio of the amplitudes of the FFTs of the transmitted and incident pulses) and the band structure (from the phase shift at each frequency in the pulse, yielding the variation of ω with k = Δφ/L). Typical results for the 3D tungsten-carbide/water crystal can be found in ﬁg. 5. The left pair of panels shows the transmission coeﬃcient and the band structure of

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this face-centred-cubic crystal, revealing a wide band gap due to Bragg scattering near 1 MHz (width ∼ 20% of the central frequency), where the spacing between layers of the crystal is approximately equal to half the wavelength in water. In the [111] direction, in which the experimental data were obtained, the gap is even wider, as shown by the broad dip in the transmission coeﬃcient, measured for a crystal consisting of 12 layers. These results illustrate the relative ease with which wide band gaps can be obtained in acoustics relative to optics, because of the large scattering contrast that can be achieved in ultrasound (for this combination of solid spheres and liquid matrix, the longitudinal impedance ratio is 60). Even wider gaps (∼ 100%) are found in solid-solid systems such as steel beads in epoxy, where coupling with a resonance of the spheres enhances the band gap considerably [29, 36]. In ﬁg. 5 the experimental data are compared with predictions of Multiple Scattering Theory (MST, indicated by the solid curves) [24], which is ideally suited to calculating the properties of phononic crystals built from scattering elements having simple geometric shapes such as spheres, where the scattering can be calculated accurately with no adjustable parameters. Excellent agreement between theory and experiment is found. Note that this agreement indicates that the band structure, which is calculated for an inﬁnite crystal, can be accurately measured by transmission experiments in ﬁnite-thickness samples consisting of remarkably few layers. The transmission coeﬃcient in the middle of the gap (at f = 0.95 MHz) is found, both experimentally and theoretically, to decrease exponentially with thickness as exp[−κL], consistent with evanescent modes having imaginary wave vector κ. This suggests that ultrasound is transmitted through the crystal by tunneling, whose dynamics can be investigated through measurements of the group velocity [28]. The right panel of ﬁg. 5 shows that the group velocity increases linearly with sample thickness, an unusual result that is the classic signature of tunneling, implying that the tunneling time is independent of thickness. For the thickest crystals, the magnitude of vg is remarkably fast —see the horizontal arrows in the ﬁgure for the longitudinal velocities in the two constituent materials. The solid and dashed curves in the ﬁgure are calculated using Multiple Scattering Theory both without and with absorption, the latter being taken into account by complex moduli of the constituents. It can be seen that the theory with absorption gives a very satisfactory description of the experimental results, indicating how dissipation, which has no counterpart in the quantum tunneling case, signiﬁcantly aﬀects the measured tunneling time. This eﬀect was interpreted using a so-called two modes model, which allows the role of absorption to be understood in simple physical terms [28]. Absorption in the band gap of a phononic crystal cuts oﬀ the long multiple scattering paths, making the destructive interference that gives rise to the band gap incomplete. As a result, a small-amplitude propagating mode is produced in parallel with the dominant tunneling mode, accounting for the reduction in the measured group velocity relative to the predictions without absorption. This simple model was also found to give a good quantitative explanation of the data [28]. Experiments on the same 3D tungsten-carbide/water phononic crystal were the ﬁrst to demonstrate ultrasound focusing by negative refraction [31] —another area of phononic (and photonic) crystal research that is currently attracting considerable attention. At

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Fig. 6. – (a) Band structure of a phononic crystal of steel rods in water (triangular lattice of 1.02-mm-diameter rods with lattice constant a = 1.27 mm). The solid curves were calculated using Multiple Scattering Theory calculations, and the symbols represent experimental data. (b) Equifrequency contours at the three frequencies, 0.75, 0.85 and 0.95 MHz, in the second pass band. (c) Snapshot the negatively-refracted pulse emerging from a phononic crystal prism (angles 30◦ , 60◦ and 90◦ , as shown) after a narrow-band pulse (central frequency of 0.85 MHz) was normally incident on the shortest face of the prism (in the direction of wide blue arrow). The data were measured by scanning a hydrophone in a rectangular grid, digitally ﬁltering the pulses to narrow the bandwidth, and measuring the wave ﬁeld at a particular moment in time to construct the spatial variation of the ﬁeld at that time.

frequencies in the pass band near 1.6 MHz in ﬁg. 5, the initially diverging beam from a quasi-point source was observed to be sharply focused in a plane that was quite far from the crystal, where the focal spot could be easily measured. As is explained in more detail below, focusing occurs because the group velocity inside the crystal is of opposite sign to the wave vector, and as a result the direction of energy transport (which is given by the group velocity) corresponds to a negative angle of refraction. In terms of a simple ray picture, in which the rays are drawn parallel to the group velocity, the wave vector components of the ﬁeld from the source that are incident at angles ±θ are refracted negatively as they enter the crystal, cross inside the crystal and are then refracted negatively again as they leave the crystal, so that the emergent rays converge to a focus on the far side of the crystal. The data in these initial experiments were interpreted using a Fourier imaging model that accounted for this unusual wave transport through the crystal, giving a quantitative explanation of the observed focusing eﬀect [31].

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To explore the phenomena of negative refraction and focusing in phononic crystals in more detail, a number of experiments and theoretical calculations have been performed on 2D crystals [32-35]. The most direct observations of negative refraction were made by Sukhovich et al. [34], who constructed a prism-shaped phononic crystal of steel rods, arranged in a triangular lattice at a volume fraction of 58% and surrounded by water. This crystal has the advantage of a relatively simple band structure, as shown by the solid curves (MST) and symbols (experiment) in ﬁg. 6(a). The second pass band, between the stop band along ΓM (the [1,1] direction) and the band gap near 1 MHz, has a single branch, which appears quite isotropic. This isotropic behaviour is conﬁrmed by the equifrequency contours (ﬁg. 6(b)), which characterize the variation with direction in the magnitude of the wave vector at a given frequency. The contours are remarkably circular and shrink in radius as the frequency increases, indicating that the wave vector has the same magnitude in all directions, and that the group velocity, vg = ∇ k ω(k), points towards the centre of each circular contour. Thus, for this band, the group velocity and wave vector in the ﬁrst Brillouin zone are antiparallel for all directions of propagation, a situation resembling left-handed behaviour in negative index metamaterials [37]. A consequence of vg and k being antiparallel is that waves arriving at the surface of the crystal at non-normal incidence will be negatively refracted. This eﬀect is demonstrated by the experimental data shown in ﬁg. 6(c), which was obtained by directing a narrowband pulse with central frequency 0.85 MHz towards the shortest face of the prism at normal incidence (see the wide blue arrow) and imaging the ﬁeld that emerged from the longest face using a miniature hydrophone. Since the wave pulse enters the crystal at normal incidence, the pulse continues to travel inside the crystal in the original direction, which is parallel to the group velocity. As the pulse leaves the crystal, the outgoing ﬁeld pattern is seen to bend backwards in the negative direction, showing according to Snell’s law that the wave vector inside the crystal must also point in the negative direction, opposite to the direction of the group velocity, as predicted from the equifrequency contour. To emphasize this point, the directions of the Bloch wave vector and group velocity inside the crystal are also shown in ﬁg. 6(c), as well as the direction of the refracted beam outside, which is perpendicular to the wave fronts. (Note that to measure the direction of k, it is crucial to measure the wave ﬁeld and not just the intensity so that k can be determined from the wave fronts, as the position of maximum intensity in the refracted beam in this pulsed experiment is also inﬂuenced by the time the pulse reached the exit surface of the crystal, with the earlier arrivals being closer to the top of the prism and corresponding to the signals on the top left part of the measurement area.) Furthermore, the measured refraction angle is given within experimental uncertainty by Snell’s law, using the value of the wave vector inside the crystal predicted by MST, providing additional evidence that the data can be quantitatively described in terms of negative refraction. The direct observation of negative refraction in this 2D phononic crystal suggests that it is a good system in which to investigate focusing by negative refraction in ﬂat phononic crystal lenses, and in particular to examine the ultimate image resolution that may be possible. For this purpose, a rectangular-shaped six-layer crystal of steel rods with the same crystal structure was constructed. Each layer contained 60 rods (to avoid edge

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eﬀects), and the layers were stacked in the ΓM direction, i.e. with the base of the triangular unit cell parallel to the surface. To explore the resolution capabilities of the lens, a narrow line source (width 0.55 mm, which is less than the wavelength in water at the frequencies of interest) was built from piezoelectric polymer strips. When the crystal was ﬁlled and surrounded by water, the best image of the source was measured at 0.70 MHz, the lowest frequency at which the equifrequency contours are circular. However, the image resolution, as determined by the Rayleigh criterion (resolution equals half the full width of the peak Δ, i.e. the distance Δ/2 from the maximum to the adjacent minimum (zero)), was only 1.15λ, where λ is the ultrasonic wavelength in water. This is not as good as the diﬀraction limit of λ/2, which is obtained when all propagating components of the ﬁeld from a point source are brought to focus in the image plane, because the equifrequency contours inside and outside the crystal were not matched, cutting oﬀ all angles of incidence greater than 56.8◦ in this case. To overcome this limitation, a second crystal was built with thin transparent walls to enable the liquid inside the crystal to be replaced by methanol, which has a lower sound velocity than water, shrinking the frequency axis of the dispersion curve by 74%. As a result, the size of the equifrequency contours of both the crystal and the water outside were perfectly matched at a frequency of 0.55 MHz in the second band. Thus, all angle negative refraction (AANR) is achieved at this frequency, and all others down to the bottom of the band at 0.50 MHz. The image obtained at 0.55 MHz, when the source was placed 1.6 mm from the opposite surface of the crystal, is shown in ﬁg. 7(a). A good focal pattern is clearly seen, with the focal spot narrowly conﬁned both perpendicular and parallel to the crystal surface. By ﬁtting a sinc function (ﬁg. 7(b)), the transverse width of the image was measured to be 3.0 mm, with a corresponding resolution of 0.55λ. This shows that a ﬂat phononic crystal with equifrequency contours matched to those of the medium outside is capable of producing images with an excellent resolution approaching the diﬀraction limit [34]. To achieve super resolution (better than the diﬀraction limit), it is necessary to capture and amplify evanescent waves from the source —something that clearly did not occur for the data shown in ﬁg. 7(a). However, when the source was brought even closer to the surface of the crystal, 0.1 mm or λ/25 away, it was found that signiﬁcantly improved resolution could be obtained [35]. The best resolution was found at a slightly lower frequency, 0.53 MHz, as shown by the experimental results in ﬁgs. 7(c)(d) and (f), which are compared with Finite Diﬀerence Time Domain (FDTD) simulations in ﬁgs. 7(d)-(f). Both the experimental and theoretical resolutions, 0.37λ and 0.35λ, are clearly better than the diﬀraction limit. The reason why super resolution can be attained for this very small source-crystal separation is that some of the evanescent waves from the source are now able to couple to a bound mode of the crystal, and hence become ampliﬁed suﬃciently to participate in image restoration. Evidence for the excitation of this bound mode can be seen in the ﬁeld patterns of ﬁgs. 7(d) and (e), which show several subsidiary peaks that are largest at the crystal surface; these additional peaks (not seen in (a)) decay rapidly with distance from the crystal, as expected for the evanescent decay of bound crystal modes. Additional evidence for the existence of this bound mode was obtained from FDTD calculations of the band structure of a ﬁnite crystal slab with the

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Fig. 7. – Contour maps of the ultrasonic amplitude (magnitude of the FFT of the wave ﬁeld at the frequencies indicated) on the imaging side of a ﬂat methanol-steel phononic crystal lens for a 0.55-mm-wide line source, and corresponding plots of the amplitude through the focus. (a) Image measured at 0.55 MHz for a source-lens distance of 1.6 mm. (b) Amplitude parallel to the lens surface (circles) through the focus in (a). The data are compared with a sinc function (red line), indicating a resolution Δ/2 = 0.55λ. (c) and (e) Images measured (c) and calculated with FDTD (e) for a frequency of 0.53 MHz when the source-lens distance is only 0.1 mm. Note the appearance of a bound mode of the crystal, which decays evanescently as the distance from the surface (at z = 0) increases. (d) and (f) Comparison of experiment (circles) and theory (solid curves) for the transverse width of the focal spot (d) and its variation with distance from the surface of the crystal (f). Super resolution is evident from the half widths of the primary peaks in (d), give a resolution of 0.37λ and 0.35λ for experiment and theory, respectively.

same number of layers as in the experiment. These calculations revealed a nearly ﬂat band that extends from 0.525 MHz at the water to 0.51 MHz at the zone boundary; as it lies below the water line, this mode is bound to the crystal slab as it cannot propagate in water. The best focusing is seen at 0.53 MHz, as this frequency lies between the frequency for perfectly matched equifrequency contours (0.55 MHz) and the resonance frequencies of the bound mode (0.51–0.525 MHz), but is still close enough to the bound mode that it can be excited. Calculations of the ﬁeld patterns inside the phononic crystal indicate that this bound mode is a slab mode of the crystal, and not a surface mode. This demonstration that super resolution can be achieved in practice with phononic crystal lenses is enabling a detailed study of the many factors that can inﬂuence the optimum resolution. Perhaps the most interesting question concerns the mechanism that sets the resolution limit for this crystal. This is determined by the largest transverse wave vector kmax that the crystal will support, with the most logical choice for kmax being the wave vector at the Brillouin zone boundary of the crystal along the ΓK direction (parallel to the surface of the lens). (Note that since the bound mode that is excited is a slab mode of the crystal, it is the bulk Brillouin zone boundary and not the surface Brillouin zone boundary that sets the resolution limit, allowing better res-

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olution to be achieved for this triangular lattice than would be found for the surface modes that were considered by Luo et al. for photonic crystals.) This condition gives kmax = 4π/3a. If we assume perfect transmission for all transverse wave vectors k⊥ less than kmax , and zero transmission for k⊥ greater than kmax ,# then the image amplik exp[ik⊥ x]dk⊥ | = tude will vary with distance x parallel to the crystal surface as | −kmax max |2 sin(kmax x)/(kmax x)| so that the resolution limit Δmin /2 = π/kmax = 3a/4. This condition gives Δmin /2 = 0.34λ at 0.53 MHz, which is very close to our experimental and FDTD results. In conclusion, these experimental and theoretical results demonstrate the conditions needed to achieve optimal focusing: i) the equifrequency surfaces/contours should be spherical/circular, ii) the equifrequency surfaces in the phononic crystal and in the medium outside should be matched, and iii) the crystal should have a bound mode at a frequency close to the operational frequency, in order to enable ampliﬁcation of evanescent waves from the source, for super resolution to be attained. The analysis of the maximum possible resolution that can be obtained with the 2D methanol-steel phononic crystal will be useful for designing new phononic crystal lenses in which the super resolution may be enhanced. 4. – Conclusions Experiments with ultrasonic waves are playing an increasing important role in probing and understanding the rich diversity of wave phenomena that occur in strongly scattering media. In disordered media, the phase sensitivity of ultrasonic detectors enables pulsed experiments to separate the coherent, forward-scattered signal, which propagates ballistically through the medium, from the multiply scattered coda. Thus, transmission experiments can be used to obtain a very complete set of measurements of wave transport through the medium, allowing the parameters that describe both ballistic and diﬀusive propagation to be compared over a wide frequency range. Such measurements have been performed on a simple model system of glass beads in water, illustrating the potential of ultrasound for gaining useful insights into the character of wave transport in the presence of strong multiple scattering, and laying a useful foundation for future experiments on more complex systems. In the second part of this paper, the properties of ordered acoustic media, or phononic crystals, have been reviewed. The main emphasis has been on focusing by negative refraction, where super-resolution imaging has recently been demonstrated experimentally. This area of research continues to grow, providing complementary information and applications to analogous optical experiments on photonic crystals. ∗ ∗ ∗ I would like to thank the many students and colleagues who have contributed to the research that has been reviewed in this paper. Support from NSERC is also gratefully acknowledged.

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[28] Yang S., Page J. H., Liu Z., Cowan M. L., Chan C. T. and Sheng P., Phys. Rev. Lett., 88 (2002) 104301. [29] Page J. H., Yang S, Liu Z., Cowan M. L., Chan C. T. and Sheng P., Z. Kristallogr., 220 (2005) 859. [30] Mei J., Liu Z., Shi J. and Tian D., Phys. Rev. B, 67 (2003) 245107. [31] Yang S., Page J. H., Liu Z., Cowan M. L., Chan C. T. and Sheng P., Phys. Rev. Lett., 93 (2004) 024301. [32] Zhang X. and Liu Z., Appl. Phys. Lett., 85 (2004) 341. [33] Ke M., Liu Z., Cheng Z., Li J., Peng P. and Shi J., Solid State Comm., 142 (2007) 177. [34] Sukhovich A., Jing L. and Page J., Phys. Rev. B, 77 (2008) 014301. [35] Sukhovich A., Merheb B., Muralidharan K, Vasseur J. O., Pennec Y., Deymier P. A. and Page J. H., Phys. Rev. Lett., 102 (2009) 154301. [36] Sainidou R., Stefanou N. and Modinos A., Phys. Rev. B, 66 (2002) 212301. [37] Pendry J. B., Phys. Rev. Lett., 85 (2000) 3966. [38] Luo C., Johnson S. G., Joannopoulos J. D. and Pendry J. B., Phys. Rev. B, 68 (2003) 045115.

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-95

Anderson localization of ultrasound in three dimensions J. H. Page Department of Physics and Astronomy, University of Manitoba Winnipeg, MB Canada R3T 2N2

Summary. — Some ﬁfty years after Anderson localization was ﬁrst proposed, there is currently a resurgence of interest in this phenomenon, which has remained one of the most challenging and fascinating aspects of wave transport in random media. This paper summarizes recent progress in demonstrating the localization of ultrasound in a “mesoglass” made by assembling aluminum beads into a disordered three-dimensional elastic network. In this system, the disorder is suﬃciently strong that interference leads to trapping of the waves at intermediate frequencies, as demonstrated by studying three diﬀerent fundamental aspects of Anderson localization: time-dependent transmission, transverse conﬁnement of the waves, and the statistics of the non-Gaussian intensity ﬂuctuations. Additional ultrasonic experiments have been performed to reveal the multifractal character of the wave functions near the Anderson transition. This is the ﬁrst time that so many diﬀerent aspects of localization have been studied simultaneously, providing very convincing evidence for the localization of ultrasonic waves in the presence of disorder in three dimensions, and enabling new aspects of Anderson localization to be studied experimentally.

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1. – Introduction During the 1980s, it was realized that Anderson localization [1, 2] —the spatial trapping of waves due to disorder— is not only a quantum eﬀect, but is, more generally, a phenomenon that may occur for any type of wave: quantum or classical. This phenomenon results from the interference of waves that have been multiply scattered in a disordered medium, and therefore should be observable so long as the disorder is suﬃciently strong and coherence is maintained, the latter condition being readily satisﬁed for classical waves such as light, microwaves, sound, elastic waves, and even seismic waves. To appreciate the analogy that exists between quantum and classical wave behaviour in disordered media, one need go no further than to compare the Schr¨ odinger and Helmholtz equations in the presence of disorder, as is outlined in appendix A. This comparison shows that these equations for quantum and classical waves have the same form, but with an important diﬀerence. This diﬀerence implies that it is only possible to localize classical waves at intermediate frequencies where the wavelength is comparable to the size of the scattering inhomogeneities, and not more or less trivially at very low frequencies, as in the case for electrons. This absence of “bound states” for classical waves, and the need to achieve suﬃciently strong scattering, makes the localization of classical waves challenging to observe in practice. Not withstanding these challenges, there are several reasons why classical waves are potentially better adapted to observing the phenomenon of Anderson localization directly. For electrons or other quantum particles (e.g., cold atoms), experiments must be performed at low temperatures to minimize the eﬀects of inelastic scattering, which destroys phase coherence. There is no such restriction for classical waves. Classical waves also have the advantage that the analogue of electron-electron interactions (nonlinearities) can be avoided by suitable choices of materials and power levels. Perhaps most signiﬁcant is the versatility of experiments with classical waves, where measurements as a function of both time and space are feasible, potentially yielding much more information about localization than is possible by simply measuring the total transmittance at a single frequency. The latter is equivalent to measuring the overall sample conductance for electronic systems, the technique that has been used almost exclusively in studies of electron localization. Once these advantages of classical waves were appreciated, experimental work showing strong localization of both acoustic and electromagnetic waves followed in one- and two- dimensional systems (1D and 2D), as well as in quasi-1D waveguides [3-8]. These were signiﬁcant steps forward, as they permitted localized wave functions and their statistical properties to be studied directly, stimulating many new theoretical advances as well. However, the central question in the ﬁeld, whether or not classical waves could be localized in three dimensions (3D), has been more diﬃcult to answer, despite several tour-de-force experiments in optics [9-11]. Three dimensions is especially important, as it is only in 3D that scaling theory predicts the existence of a real transition from propagating to localized modes [12]. In seeking experimental evidence, one of the problems, in addition to the challenges mentioned above, has been absorption, which is always present

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to some extent for classical waves, and which leads to a reduction in total transmission having the same dependence on sample thickness as localization. Thus, to demonstrate 3D localization convincingly, it is necessary to combine a number of experimental approaches that can probe key signatures of localization: e.g., anomalous dynamics (time dependence), spatial conﬁnement of the waves, and the statistics of the large intensity ﬂuctuations. This paper describes the recent progress that has been achieved using ultrasonic experiments, in combination with advances in the self-consistent theory of localization, to unambiguously demonstrate Anderson localization in three dimensions [13]. One key to this success has been the construction of suﬃciently strongly scattering samples, which are described in the next section (sect. 2). The following sections outline the three main experimental approaches that have been exploited to obtain evidence of localization: time-dependent transmission (sect. 3), transverse conﬁnement (sect. 4) and statistics (sect. 5). This last section ends with an example of an aspect of localization in 3D that has not been accessible to experimental study previously, namely the structure of localized wave functions as characterized by their multifractal properties [14]. 2. – Mesoglasses: porous elastic solids with very strong scattering In samples suitable for localization experiments, it is generally important to maximize the scattering strength and minimize the absorption (or dissipation). Despite the very large scattering contrast between solids and liquids for ultrasonic waves, with diﬀerences in acoustic impedance as large as 10 to 60 being readily achievable, suspensions of solid spherical particles in a ﬂuid, as described in [15], were found to have insuﬃciently strong scattering. This is true even in the intermediate frequency regime, where the wavelength is comparable to the size of the scatterers and shape resonances can enhance the scattering. The other problem with such samples is the relatively large dissipation, one important contribution being viscous losses at the interface between the solid and liquid phases. To avoid this diﬃculty, we decided to take a diﬀerent approach and investigate porous, single-component solid systems instead. Our initial experiments were performed on highly porous solid networks of well-sintered glass beads, revealing interesting plateaus in both the diﬀusion coeﬃcient and the density of states [16]. However, in spite of very strong scattering, no evidence of localization was seen in these early experiments. To make better samples for observing Anderson localization, three important steps were taken: the glass particles were replaced by aluminum, thereby further reducing the intrinsic absorption in the constituent particles; the particle radius a was increased, allowing higher eﬀective frequencies (ka) to be accessed using the same ultrasonic transducers; and a new way was developed for joining the particles together into a solid network, allowing for better control of the interparticle contacts, and hence the scattering strength. In this way, a disordered elastic network of aluminum beads was created by brazing the beads together to form weak elastic bonds between the beads while preserving their spherical shape, as shown in ﬁg. 1. Such a structure may be viewed as a “mesoglass” in which the beads are linked by narrow necks to form a disordered material

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Fig. 1. – One of the samples in which the localization of ultrasonic waves was observed in the intermediate frequency regime. The aluminum beads are brazed together with weak elastic links to form a disordered solid network, in which the ultrasonic waves become trapped after strong multiple scattering from the pores.

with mesoscopic particles as the building blocks rather than atoms, and with elastic bonds between the particles rather than interatomic forces. The beads were monodisperse, with a diameter of 4.11 ± 0.03 mm, and the samples had an aluminum volume fraction of approximately 55%, corresponding to random loose packing of the beads before brazing. The samples were slab-shaped, with circular cross sections of diameter much larger than the thickness L, which ranged from 8 mm to 23 mm. To study the samples using ultrasonic immersion techniques [15,17], the samples were ﬁrst waterproofed with very thin plastic walls, so that the samples remained dry when immersed in a water tank between the generating and detecting transducers. This procedure ensured that wave transport occurred through the aluminum network, where the incident acoustic wave from the water (longitudinal polarization only) was converted into an elastic wave in the solid (which supports both longitudinal and transverse polarizations). Measurements were performed over a wide frequency range from 100 kHz to several MHz, since this corresponds to the intermediate frequency regime for this structure, where the wavelength is comparable to the bead and pore sizes and very strong scattering is expected. Initial characterization of the samples was performed by measuring the amplitude transmission coeﬃcient, obtained from the ratio of the fast Fourier transforms of the transmitted and input signals. These measurements conﬁrmed the existence band gaps in this type of system, as ﬁrst reported by Turner and Weaver in 1998 [18]. The gaps occur because the coupled resonances of the aluminum beads broaden to form pass bands, with band gaps forming in between; wide band gaps are observed so long as the coupling is not too strong, which is the case for our samples. For some of the samples, it was possible to extract the coherent ballistic pulse and measure the longitudinal phase and group velocities vp and vg , as well as the scattering mean free path l. Very strong scattering was demonstrated by the observation that, outside the band gaps, the product

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of wave vector and mean free path varied from nearly 1 to 2.5 over this frequency range, thus approaching the Ioﬀe-Regel limit kl = 1. Although we were not able to measure kl for transverse waves in these experiments, previous experiments on sintered glass bead networks have shown similar values of kl for both transverse and longitudinal waves in the strong scattering regime [19]. Localization is expected when kl 1, but the exact critical value klc at which the transition occurs is not known [20], and is likely to be wave and sample dependent; thus, these initial measurements suggest that localization of ultrasonic waves may indeed be possible in these samples. 3. – Time-dependent transmission Our ﬁrst experiment to investigate wave transport by multiply scattered waves in these samples was performed using a short quasi-planar incident pulse and measuring the time-dependent transmitted ﬁeld with a miniature hydrophone. The hydrophone was scanned over a square 55×55 grid parallel to, and within a few wavelengths of, the sample surface. The grid separation was typically equal to the wavelength in water (for more details on the method, see [15] and [17]). A schematic diagram of the setup is shown in ﬁg. 2(a). Several representative waveforms measured at diﬀerent positions in the speckle pattern are shown in ﬁg. 2(b); this example was recorded for a two-cycle input pulse with a central frequency of 0.25 MHz, showing that data for a long range of propagation times, corresponding to progressively longer and longer multiple scattering paths, is observable in these samples. Before determining the time-dependent intensity I(t), the waveforms were digitally ﬁltered to limit the bandwidth to 5% of the central frequency of the pulse. The average transmitted intensity I(t) was then determined by squaring the envelope of the ﬁeld at each position, averaging over each position in the speckle pattern, and normalizing by the peak of the input pulse. Typical time-dependent intensity proﬁles are shown in ﬁgs. 2(c) and (d), revealing the excellent signal-to-noise and large dynamic range obtained at both low and high frequencies in the range of interest between 0.1 and 3 MHz. Note that, even though both compressional (longitudinally polarized) and shear (transversely polarized) waves are excited in these elastic materials, in which shear waves typically propagate at roughly half the speed of longitudinal waves, a single intensity proﬁle I(t) is seen at all frequencies. This occurs because the polarizations mix at each scattering event. Thus, after only a few scatterings, the total energy density becomes equipartitioned between compressive and shear waves [21-23], according to their respective energy densities UL and UT . For weak disorder, (1)

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Fig. 2. – (a) Schematic diagram indicating the experimental setup for measuring the timedependent transmission. (b) The transmitted ﬁeld measured by the hydrophone at three different positions near the sample surface. The source was a short two-cycle pulse with a central frequency of 0.25 MHz. The peak of the input pulse occurs at t = 0. (c) Transmitted intensity I(t) at a representative frequency of 0.2 MHz in the diﬀuse regime for a sample with thickness L = 14.5 mm. The best ﬁt to diﬀusion theory (solid red curve) with R = 0.85 yields D = 3.0 mm/μs and l∗ = 2.5 mm, with τa being to large to be measurable. (d) I(t) for the same sample at a frequency of 2.4 MHz in the localized regime. The data cannot be ﬁtted by diﬀusion theory (dashed blue curve), but is well ﬁtted by the self-consistent theory of localization (solid ∗ red curve) with ξ = 15 mm, lB = 2 mm, DB = 16 mm2 /μs and τa = 160 μs.

waves dominate the energy transport. At suﬃciently low frequencies, one might expect that the transport is diﬀusive and that there are no renormalization eﬀects due to inter∗ /3 and transport mean ference, so that the Boltzmann diﬀusion coeﬃcient DB = vE lB ∗ free path lB are observed. In this regime, providing that the scattering is isotropic (l = l∗ ) and the energy velocity is equal to the group velocity, a simple relation can be derived for the eﬀective energy-density-weighted average diﬀusion coeﬃcient:

(3)

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∗ lL 2lT∗ + 2 2 vp,L vp,T

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and the eﬀective energy velocity by (5)

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.

While the general case is more complex, these relations give some insight into the consequences of energy equipartition, and provide a simple starting point for comparing data with the predictions of the diﬀusion approximation whenever the ballistic parameters can be measured. For our brazed aluminum bead samples, the ultrasonic wave transport was indeed found to be diﬀusive at low frequencies. The evidence for diﬀusive transport can be seen in the exponential decay of the ensemble-averaged transmitted intensity at long times, I(t) ∝ exp[−t/τD ], which was observed for frequencies up to 0.4 MHz. In this frequency range, the entire time dependence of I(t) is well described by diﬀusion theory, as is illustrated in ﬁg. 2(c), which compares diﬀusion theory with data measured at 0.2 MHz. By ﬁtting theory to experiment, the diﬀusion coeﬃcient was determined, and the transport mean free path was estimated from the dependence of the transmitted intensity on boundary conditions. One consequence of the equipartition of elastic energy inside the sample is that the internal reﬂection coeﬃcient R is large, as the outside medium only supports longitudinal waves; nonetheless R can still be reliably determined from the measured ballistic parameters by accounting for the angle-dependent reﬂection coeﬃcients for all polarizations [24], thereby reducing the number of ﬁtting parameters. In this frequency range, D was found to be roughly independent of frequency, consistent with earlier experiments on sintered glass bead networks [16]. Signiﬁcantly, we found that absorption, which attenuates I(t) by the factor exp[−t/τa ], where τa is the absorption time, was too small to measure in this frequency range, consistent with our expectations that absorption would be much much lower for these samples than in most strongly scattering acoustic systems, such as suspensions of particles in a ﬂuid [17]. The success in interpreting these data using diﬀusion theory establishes that multiply scattered ultrasound propagates diﬀusively in the lower part of the intermediate frequency range, which is the diﬀusive regime for this system. In the upper part of the intermediate frequency range (∼ 2 MHz), the time dependence of I(t) shows qualitatively diﬀerent behavior: at long times, I(t) decays more slowly than in the diﬀusive regime, with a non-exponential tail that cannot be explained by the diﬀusion approximation (ﬁg. 2(d)). This behavior has been viewed as a slowing down of the eﬀective diﬀusion coeﬃcient D(t) with propagation time, reﬂecting a time-dependent renormalization of D due to interference eﬀects associated with localization [7,10]. Physically, the waves become trapped by the disorder, but eventually manage to escape from the sample, suggesting that Anderson localization may be occurring in these samples. To interpret these data quantitatively, we exploit recent progress in the self-consistent theory of localization, initially developed for electron localization by Vollhardt and W¨ olﬂe in 1980 [25, 26]. The basic idea in this theory is to describe the renormalization of the diﬀusion coeﬃcient by accounting for constructive interferences between reciprocal

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paths, which lead to an increased probability that a quantum particle or classical wave returns to the same spot. The recent progress [27,28] enables the dynamics of wave transport to be predicted for experiments such as ours, which involve open three-dimensional systems. The new aspect of this theory is the incorporation of boundary conditions selfconsistently, thereby accounting for the fact that the return probability is less reduced near the boundaries, where the wave may escape, so that the renormalization of D by interference is less there. This leads to a position-dependent dynamic diﬀusivity kernel D(r, t − t ). The solid curve in ﬁg. 2(d) is a ﬁt of the self-consistent theory to our data at 2.4 MHz, and gives an excellent description of the experiment at all propagation times. From this ﬁt, we are able to determine the localization length ξ = 15 mm for this sample. This measurement of ξ is feasible since several key parameters (l and vp , and hence kl and R) are known for our sample from independent ballistic measurements, leaving the ∗ , the diﬀusion time τD , and the localization length, the bare transport mean free path lB absorption time τa as ﬁtting parameters. In the self-consistent theory, the localization length is related the ratio of kl to its critical value at the mobility edge, χ = kl/(kl)c, ∗ ∗ 2 2 = [6/(klB )c ]χ /(1 − χ4 ). Thus ξ is positive in the localization regime where by ξ/lB χ < 1, and negative in the diﬀuse regime, where the absolute value of ξ plays the role of a correlation length in the vicinity of a phase transition. The most important points to emerge from the ﬁtting are not only that the self-consistent theory describes the time dependence of the measured I(t) very well, but also that it is only possible to ﬁt to the experimental measurements with the theory when ξ > 0. This gives strong evidence for the dynamic localization of ultrasound in our sintered aluminum bead system. 4. – Transverse conﬁnement The measurements of the time-dependent transmission described in the previous section give only indirect evidence of Anderson localization. Is it possible to observe localization more directly? To answer this question, the quasi-plane wave source was replaced by a point source (approximately a wavelength wide), and the transmitted intensity was measured as a function of both position and time on the opposite face of the sample. The point source was obtained by focusing the input pulse through a narrow aperture onto the sample surface at the point ρ = 0, as shown schematically in ﬁg. 3(a). The transmitted wave ﬁeld was measured with subwavelength resolution using a hydrophone, which was moved over a range of transverse positions ρ for a given position of the source. The transmitted intensity I(ρ, t) was calculated from the measured ﬁeld as indicated in the previous section. To average the intensity for each ρ and t over a large number (typically 552 = 3025) of speckle spots, the position of the sample was scanned in the x-y plane parallel to the surface of the sample. Typical data for I(ρ, t) at 2.4 MHz, measured on the same sample for which I(t) is reported in ﬁg. 2, are shown in ﬁg. 3(b). As expected, data for larger ρ start later because the distance from the source is greater; what was not expected is that the curves for diﬀerent ρ decay at essentially the same rate at long times, i.e. they diﬀer by a time-independent factor at long times. To understand this behaviour, the crucial quantity is the ratio I(ρ, t)/I(0, t), which probes the dynamic spreading of the

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Fig. 3. – (a) Schematic illustration (not to scale) of the setup for measuring the dynamic transverse conﬁnement of the transmitted intensity emitted by a point source in the localization regime. (b) Average time- and position-dependent intensity, I(ρ, t) at several positions ρ for the setup shown in (a). The frequency (2.4 MHz) and sample are the same as in ﬁg. 2(d). (c) Mean square width wρ2 (t) of the intensity ratio I(ρ, t)/I(0, t) as a function of time for the data shown in (b) and for a second sample with a thickness L = 23.5 mm. The frequency is 2.4MHz for both samples. The solid curves are the best ﬁts of the self-consistent theory to the experimental data ∗ (symbols) with lB = 2 mm, DB = 11 mm2 /μs, ξ = 15 mm for the L = 14.5 mm sample, and ξ = 7 mm for the L = 23.5 mm sample. Other parameters used in the calculations, kl = 1.8 and R = 0.82, were determined from independent ballistic measurements. The dashed line shows the linear time dependence of w2 that would occur for diﬀuse waves, using a value for D of 1.25 mm2 /μs. (d) Dependence of the intensity ratio on distance ρ at selected times, showing the non-Gaussian proﬁle that is found both experimentally (symbols) and theoretically (solid curves).

intensity proﬁle in a plane parallel to the surface of the sample. Most importantly, this ratio is independent of absorption, since at each time, the absorption factor exp[−t/τa ] is the same for any ρ and therefore cancels in the ratio. We characterize this dynamic spatial proﬁle of the intensity by its width wρ (t), deﬁned by I(ρ, t)/I(0, t) = exp[−ρ2 /wρ2 (t)]. As shown in ref. [15], wρ2 (t) = 4Dt ∝ t in the diﬀuse regime, providing an accurate method for measuring D that is independent of absorption and boundary conditions. By contrast, in the localized regime, the transverse width wρ (t) exhibits completely diﬀerent behaviour, shown in ﬁg. 3(c) for two samples of diﬀerent thickness. Instead of increasing

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linearly with propagation time, wρ2 (t) saturates, approaching a constant value for each ρ at long times. Furthermore, the transverse spatial proﬁle of the intensity is no longer Gaussian in the localized regime, since wρ2 (t) depends on ρ —another clear departure from diﬀuse behaviour that is especially evident in the thinner sample. These results indicate that the data cannot be explained by assuming a diﬀusion coeﬃcient D(t) that # depends only on time, since in this case the magnitude of w(t) ∼ D(t)dt would be independent of ρ(1 ). The observed non-Gaussian shape of the spatial proﬁles is shown explicitly in ﬁg. 3(d) for a range of times separated by (almost) equal intervals at long times, but by narrower intervals at early times. These data were measured on the thinner sample. A Gaussian curve with width equal to the sample thickness (dashed curve) is also included in this ﬁgure for comparison. Both ﬁgs. 3(c) and (d) show that the intensity proﬁle initially grows with time, but then converges to a constant proﬁle at long times, revealing how the initial propagation of the waves away from the source is brought to a halt by localization. This is exactly what is meant by localization. These data are therefore a very direct demonstration of 3D Anderson localization, and are, to the best of our knowledge, the most direct observations of this phenomenon to date. Additional information about Anderson localization in these samples can be obtained by comparing the data with the predictions of the self-consistent theory. The solid curves in ﬁg. 3(c) demonstrate that the behaviour of the dynamic transverse width is accurately predicted by the self-consistent theory, which gives an excellent ﬁt to the data for all t and ρ, with a single set of parameters for each sample. The ﬁts give ξ = 15 mm for the thinner sample (L = 14.5 mm —the same sample whose the time-dependent transmission is plotted in ﬁg. 2), and ξ = 7 mm for the thicker sample (L = 23.5 mm). These results suggest that the scattering is stronger in the thicker sample due to small diﬀerences in microstructure, showing that ξ is very sensitive to the degree of disorder, quantiﬁed by kl, near the localization threshold, as theory predicts. It is worth emphasizing explicitly that the non-Gaussian character of the experimental intensity proﬁles is quantitatively described by the self-consistent theory, showing the importance of accounting for the position dependence of D in the theory. These measurements of the localization length at 2.4 MHz enable us to estimate the proximity to the mobility edge (kl)c , with kl being only 1% below (kl)c at this frequency. Compared with the plane wave case 2(d), these ﬁts of the self-consistent theory for 2 wρ (t) provide a more accurate determination of the localization length. One reason is the elimination of absorption, so there is one less parameter to ﬁt. In addition, the selfconsistent theory predicts that the transverse width at long times depends predominantly on both the localization length ξ and sample thickness L, and since it is straightforward to measure L, the measurement of wρ2 (t) provides a more direct way of determining ξ. (1 ) Indeed, if one were to use the solutions of the standard diﬀusion equation assuming only that the diﬀusion coeﬃcient varies with time, serious inconsistencies are found when comparing D(t) extracted by this approach for planar and point source geometries, yielding results for D(t) diﬀering by two orders of magnitude [29]. This comparison provides another indication of the inadequacy of this approach for our data.

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Fig. 4. – (a) Time dependence of the transverse width for several frequencies in the strong scattering regime, showing that the width increases as the frequency is lowered. The mean square width is normalized by the sample thickness squared (L = 23.5 mm for these data). The data are plotted for ρ = 25 mm, but the dependence on ρ is not large for this thick sample (see ﬁg. 3). A mobility edge can be inferred to lie at a frequency between 1 and 1.8 MHz for this sample. (b) The same data plotted on doubly logarithmic scales to display the power-law behaviour of the time dependence near the mobility edge. At 1.0 MHz, the data are consistent with a crossover from t2/3 to t1/2 behaviour, as predicted by the dynamic self-consistent theory of localization.

For thick samples (L ξ), the width is no longer inﬂuenced by ρ, i.e. the statistical proﬁle is again Gaussian, and the dependence of w2 at long times on L and ξ has been shown to have a simple scaling form [30]: w2 (t → ∞) ≈ 2Lξ(1 − ξ/L); thus, to leading order, the long time limit of w2 is simply 2Lξ. One of the interesting predictions of the theory is a strong and rapid renormalization of the eﬀective diﬀusion coeﬃcient. As a result, DB cannot be measured directly even at the earliest times at which transmission measurements can be made. The best ﬁts give surprisingly large values of DB , which imply vE > vp . Further theoretical work is needed to understand these apparently very large values of the energy velocity in the localized regime. As the frequency is lowered, the transverse width increases, as is shown in ﬁg. 4. One would expect that in the localized regime, the width wρ (t) will always saturate at long times, whereas in the diﬀuse regime, the width will continue to grow with time as the energy density continues to expand in the transverse direction, albeit slowly near the mobility edge. These expectations are conﬁrmed by the self-consistent theory, where recent calculations for thick samples show that the asymptotic value of w2 (t) as t → ∞ remains ﬁnite not only in the localized regime but even at the mobility edge, where the saturation value is approximately equal to the thickness of the sample, w∞ ≈ L [30]. The experimental results plotted in ﬁg. 4 appear consistent with these predictions. At the lowest frequency shown (0.7 MHz), the width clearly continues to increase without limit, and its increase is almost linear, indicating that at this frequency, the transport is

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subdiﬀusive. At 1.0 MHz, wρ2 (t)/L2 remains less than 1 throughout the range of times where the signal could be measured, but is still increasing at the longest times, suggesting that this frequency is very close to, but on the diﬀusive side of, the critical value fc at the mobility edge. At higher frequencies, wρ2 (t)/L2 remains well below 1, and clearly saturates at long times for frequencies above 1.8 MHz, indicating that localization has set in. These results illustrate the behaviour of the transverse width in the vicinity of the Anderson transition, suggesting that it should be possible to use transverse conﬁnement to measure the variation of the localization length as the mobility edge is approached. Work is currently in progress to examine this behaviour in detail. Another interesting question concerns the time dependence of the transverse width, especially at early times where it characterizes the initial growth of the spatial intensity proﬁle. In the localized regime, one might anticipate that the intensity would spread out from the source at a slower rate than for diﬀuse waves; does this imply that wρ2 (t) still increases as a power law, with a smaller exponent than in the diﬀuse regime, or is the behaviour more complex? To examine possible power-law behaviour more closely, the experimental results for wρ2 (t)/L2 are replotted in a doubly logarithmic scale in ﬁg. 4(b). This ﬁgure shows that such behaviour is exhibited at 1.0 MHz near the mobility edge, but that there appear to be two power-law regimes. The data are consistent with an initial growth of wρ (t)2 /L2 as t2/3 , followed by a slower t1/2 regime, which at this frequency extends up to the maximum time at which signals could be recorded reliably. A t1/2 regime is also seen over a smaller range of times (∼ half a decade) at 1.4 MHz, before wρ2 (t)/L2 starts to level oﬀ towards a constant value (the long-time data are not plotted here because the signal-to-noise ratio is poor at these times, preventing an accurate measurement of how the width levels oﬀ, although it is nonetheless clear that it drops below the t1/2 curve for times t > 220 μs). These power laws have been predicted by the self-consistent theory for the initial expansion of the intensity from a point source located deep inside a thick sample (L l) [30], where a relatively simple estimate of the return probability may be obtained by neglecting the position dependence of D. By solving for D(Ω) at the mobility edge, Cherroret et al. show that the mean-square radius of the 3D intensity proﬁle grows as t2/3 at short times (t L2 /DB ), and as t1/2 at longer times (t L2 /DB ) [30]. Assuming that the mean-square transverse width for a slab sample scales with time in a similar way near the mobility edge, this provides a qualitative explanation of the time dependence seen in their numerical calculations [30] and in our experimental results (ﬁg. 4). However, this simple argument for the initial time dependence of w2 (t) does not explain the saturation at longer times. To end this section, it is worth emphasizing again that these measurements of the dynamic transverse conﬁnement of the intensity due to localization are independent of absorption, which has been a major obstacle to reaching deﬁnitive conclusions in previous experiments [6, 9-11, 31]. The method provides a direct way of observing the trapping of waves by disorder. The behaviour revealed in ﬁgs. 3(c) and (d) is both qualitatively and quantitatively diﬀerent to that seen for diﬀusive waves, and provides unambiguous evidence for the localization of ultrasound in these 3D samples.

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5. – Statistical approach to localization The previous sections have examined the marked changes that occur in the temporal and spatial proﬁles of the average intensity in the localized regime. Localization also leads to very large ﬂuctuations in the transmitted intensity, and the measurement and analysis of their statistical properties can be used to reveal other signatures of localization [6]. To investigate this statistical approach to localization, we have measured the large spatial ﬂuctuations of the intensity that occur in ultrasonic speckle patterns [13]. These were measured by scanning the hydrophone in a plane near the surface of the samples when illuminated on the opposite side with short pulse having a spatial proﬁle that corresponds quite closely to a broad Gaussian beam. By taking the Fourier transform of the measured variations in the transmitted ﬁeld, ψ(x, y, t), the variation of the intensity I(x, y) at each frequency in the bandwidth of the incident pulse was determined, enabling the near-ﬁeld speckle patterns to be plotted, as illustrated in ﬁgs. 5(a) and (b) for frequencies in the diﬀuse and localized regimes, respectively. Even by eye, a clear diﬀerence can be seen between these two cases. In the diﬀuse regime, the speckles overlap and the overall ﬂuctuations are less. By contrast, localized speckle patterns are characterized by a few very intense peaks, which are well separated from each other, so that the ﬂuctuations across the speckle pattern are very much larger. These intensity ﬂuctuations can be quantiﬁed by plotting their distribution functions, ˆ of observing the diﬀerent shown in ﬁgs. 5(c) and (d), where we plot the probability P (I) ˆ ˆ values of the intensity normalized by the mean, I = I/ I . In the diﬀuse regime, P (I) ˆ ˆ is close to the well-known Rayleigh distribution, P (I) = exp[−I], for random wave ﬁelds described by circular Gaussian statistics, such as can be observed for light from a laser ˆ < 10−2 beam scattered oﬀ a rough, random surface. The small deviations seen for P (I) in ﬁg. 5(c) can be explained by the leading-order corrections to Rayleigh statistics calculated by Shnerb and Kaveh [32], and by Nieuwenhuizen and van Rossum [33]. Their ˆ = exp[I][1 ˆ + (Iˆ2 − 4Iˆ + 2)/3g], contains only one parameter, the dimenexpression, P (I) sionless conductance g, and gives an excellent description of the experimental results, with g = 11.4 ± 0.8 1. By contrast, near 2.4 MHz in the localized regime (ﬁg. 5(d)), the intensity distribution function exhibits huge departures from Rayleigh statistics, with greatly enhanced probability of observing large values of the normalized intensity, exˆ tending up to 50 times the average. To improve the accuracy of the measurements, P (I) for the localized regime was determined from data for four equivalent samples over a range of 101 frequencies between 2.35 and 2.45 MHz. Figure 5(d) shows that the data can be extremely well ﬁtted over the entire range of intensities by Nieuwenhuizen and van ˆ yielding a value of g = 0.80 ± 0.08. The theoretical expressions Rossum’s theory for P (I), used in the ﬁts were determined for a broad Gaussian beam incident on a slab-shaped sample, and account for interference processes dominated by the loopless connected diagrams [33]. large intensities, the data can also be ﬁtted by a stretched exponential, For ˆ ˆ that can be exp − 2 g I , with the same value of g, a simple analytic form of P (I) deduced from the complete intensity distribution derived in ref. [33]. These observations

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Fig. 5. – (a),(b) Comparison of the near ﬁeld speckle patterns, showing the spatial variation of the intensity normalized by its average value, I(x, y)/I, at frequencies of 0.20 (a) and 2.4 MHz (b). In (a), the speckle pattern is typical of the diﬀuse regime, with broad overlapping speckle spots, while in (b) the pattern is dominated by narrow intense peaks that are characteristic of Anderson localization. In these two ﬁgures, the colour scale is diﬀerent, but the z-axis scale (perpendicular to the plane) is the same so that the striking diﬀerences in these speckle patterns ˆ at 0.2 MHz (open circles) is can be readily seen. (c) The measured probability distribution P (I) close to the Rayleigh distribution (dashed blue line). The solid magenta curve is best ﬁt of the theory of ref. [33] to the data with g = 11.4. (d) At 2.4 MHz, the probability of observing large intensities relative to the mean is very much greater than for diﬀuse waves. The solid curve ˆ with g = 0.80, and is in excellent agreement with the experimental shows the theory [33] for P (I) data (solid symbols). At large Iˆ 25, the data can also be described by a stretched exponential with the same value of g (dotted curve). The large deviation from Rayleigh statistics with g < 1 provides additional evidence that the Anderson localization of ultrasound has occurred at frequencies near 2.4 MHz. (Nature Phys., 4 (2008) 945.)

ˆ reveals g < 1, provide of very large intensity ﬂuctuations, for which the analysis of P (I) additional evidence [6] that localization has occurred in our samples at high frequencies. This interpretation is consistent with the Thouless criterion that g < 1 implies localizaˆ tion. It is remarkable that such good agreement between theory and experiment for P (I) has been found, as the theory was derived for the intensity in the far ﬁeld and for g > 1; this excellent agreement suggests a universality of the statistics of localized waves.

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A simple way of characterizing the intensity ﬂuctuations is measuring the variance, Iˆ2 . At 0.2 MHz, we ﬁnd Iˆ2 = 1.12 ± 0.02, very close to the value of 1 for Rayleigh statistics, while at 2.4 MHz, a much larger value is found, Iˆ2 = 2.74 ± 0.09. The variance of the normalized speckle intensities can be directly related to the dimensionless conductance, Iˆ2 = 1 + 4/(3g) [6], providing an easier way of determining g. Using our measured values of Iˆ2 , this relation gives g = 11.5 ± 2 at 0.2 MHz, and g = 0.77 ± 0.04 at 2.4 MHz, in good agreement with the values of g determined from ﬁtting the intensity distributions. Note that the localization condition g < 1 implies that localization will be reached when the variance Iˆ2 > 7/3 [6]. The measured variance at 2.4 MHz is larger than the threshold value 7/3, again supporting our conclusions that ultrasound is localized at this frequency. The ability of these ultrasonic experiments to measure the wave functions very near the surface of a localized sample suggests that the spatial structure of localized wave functions can now be investigated experimentally. There is a large body of theoretical and numerical work that predicts that wave functions at the Anderson transition have multifractal character —a striking relation between the spatial structure of wave functions and their large ﬂuctuations at criticality [34]. However, there have been virtually no experimental studies until very recently. The following paragraphs outline recent progress in using our ultrasonic data to examine this remarkable aspect of critical wave functions close to the Anderson transition [14]. Multifractality implies that the moments of the wave function intensity, I(r ) = # |ψ 2 (r )|/ |ψ 2 (r )|ddr depend anomalously on length scale, with each moment scaling as a power law with a diﬀerent exponent. Note that I(r ) is now normalized by the total intensity, rather than the average intensity, and is therefore normalized in the same way as |ψ 2 (r )| for quantum systems. To characterize this length scale dependence experimentally, one can either vary the size L of the samples, or, with a sample of a ﬁxed size, divide the sample into boxes of linear size b, and vary b. The latter is easier to implement in practice, and is therefore used for our experimental data; it allows the size dependence to be expressed in terms of the dimensionless scaling length Lg /b, where Lg is the size of the speckle pattern over which the intensity is normalized. Note that in this analysis, since we can only measure the wave function on or near the surface of the sample, the dimension of the measurement space is d = 2, even though the sample is deﬁnitely three dimensional. This procedure is illustrated in ﬁg. 6, which shows the transmitted intensity for three frequencies near 2.4 MHz for a point source. (The point source geometry has the advantage in this context of being more likely than an extended beam to excite a single wave function.) The length scale dependence of the moments of the intensity is quantiﬁed by the generalized Inverse Participation Ratios (gIPR), which are deﬁned as (6)

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and the summation is performed over all of the n = (Lg /b)d boxes. By deﬁnition, P1 ≡ 1 and P0 ≡ n. The length scale dependence was studied experimentally by determining the “typically averaged” gIPR for a single realization of disorder. In the critical regime, the average gIPR are expected to scale anomalously with Lg /b as (7)

Pq ∼ (Lg /b)−τ (q) ≡ (Lg /b)−d(q−1)−Δq ,

where the exponent τ (q) is written in terms of the normal (Euclidean) dimension d(q − 1) and the anomalous dimension, Δq . Typical results for Pq from the ultrasonic data at 2.4 MHz are shown in ﬁg. 7(a) for integer values of q between −2 and 3. By plotting the data on doubly logarithmic scales, power law behaviour is clearly seen over more than a decade in Lg /b, as shown by the excellent ﬁts of the data to straight lines. The slopes of these linear ﬁts yield τ (q), from which the anomalous exponents Δq were determined by subtracting oﬀ the normal part d(q − 1). Average values of the anomalous exponents were determined by averaging over many frequencies between 2.0 and 2.6 MHz. The q-dependence of the anomalous exponents is shown in ﬁg. 7(b), where the results for localized ultrasonic wave functions are compared with data from an optical speckle pattern for diﬀuse waves. The behaviour seen for these two data sets is obviously very diﬀerent. For the diﬀuse optical data, the open circles in this ﬁgure show that Δq ≈ 0, consistent with expectations for a normal (extended) wave function that Δq = 0 for every q. By contrast, for multifractal wave functions, such as are expected in the critical regime of the Anderson transition, τ (q) and hence Δq are expected to be continuous functions of q, with substantial departures from Euclidean behaviour. This is precisely what ﬁg. 7 shows for Δq determined from the ultrasonic data, clearly indicating that each intensity moment has a diﬀerent fractal exponent. In other words, the q-dependence of Δq in

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ﬁg. 7(b) reveals unambiguous evidence for multifractality of the localized ultrasound wave functions. The ultrasonic data in ﬁg. 7(b) enables a recently predicted symmetry relation for the anomalous exponents Δq to be tested experimentally. This relation, Δq = Δ1−q , was predicted to hold exactly for multifractal wave functions at the Anderson transition [35]. The dashed blue curve in ﬁg. 7(b) represents the experimental results reﬂected about q = 1/2, showing that this symmetry relation is consistent with our data. As the theoretical predictions of this symmetry relation were made in the context of particular symmetry classes of electronic systems, its observation in our ultrasonic experiments provides evidence for the universality of critical properties at the Anderson transition. The solid curve in ﬁg. 7(b) is a ﬁt of the predictions of the parabolic approximation to our data. This analytic approximation, derived in ﬁrst-order perturbation theory for an Anderson transition in 2 + dimensions, gives the simple expression Δq = γq(1 − q). This expression describes the ultrasonic data well, with γ = 0.21. The parabolic approximation also gives simple analytic expressions for two other important measures of multifractality, the probability density function and the so-called singularity spectrum, both of which were also measured for our ultrasonic data, as reported in ref. [14]. The measurement of all three manifestations of multifractality, anomalous exponents, the (lognormal) probability density function and the singularity spectrum, have demonstrated an aspect of Anderson transitions that has not been studied experimentally before. This opens up a number of interesting questions for future work, such as seeking an understanding of why the value of γ for the ultrasonic data is smaller than predicted for the 3D Anderson tight-binding model [34], and exploring the relationship between multifractal properties and the critical exponents governing the Anderson transition [36].

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6. – Conclusions Ultrasonic experiments have several advantages for observing and studying Anderson localization. As for all classical waves, they beneﬁt from the convenience and versatility that is associated with performing experiments at room temperature. More important is the ability to readily investigate not only average wave transport at a single frequency, but also the propagation of the multiply scattered wave ﬁelds resolved in time and space. This has enabled the development of a new approach, transverse conﬁnement, that has permitted the most direct observation so far of Anderson localization in 3D, and provides a valuable method for guiding future investigations of localization for any type of wave. In particular, the measurement of the dynamic transverse conﬁnement is a powerful way, which is not aﬀected by absorption, for assessing whether or not waves are localized; it also enables the localization length to be measured. By combining this approach with measurements of the time-dependent transmission and the statistics of the large non-Gaussian intensity ﬂuctuations, recent ultrasound experiments, in conjunction with theoretical advances, have enabled the most unambiguous demonstration of 3D Anderson localization of classical waves to date. These results suggest that ultrasonic experiments on well-controlled samples may be able to investigate previously unexplored aspects of 3D Anderson localization. One of these aspects, the characterization of the multifractal spatial structure of wave functions near the Anderson transition, has already been realized. It is reasonable to anticipate that an even more complete study of 3D Anderson localization using ultrasound is now within reach. ∗ ∗ ∗ I would like to acknowledge the theoretical contributions made by S. Skipetrov and B. van Tiggelen to the work reviewed in this paper, and to Sanli Faez for suggesting and carrying out the multifractal analysis. None of this research would have been possible without the hard work of H. Hu and A. Strybulevych in my research group at the University of Manitoba, who made the samples and performed the ultrasonic experiments. Support from NSERC and a CNRS PICS grant is also gratefully acknowledged.

Appendix A. Schr¨ odinger and Helmholtz equations in disordered media A quantum particle of energy E in a random potential V (r ) is described by the wave function ψ(r) exp[−iEt/], where ψ(r ) obeys the time-independent Sch¨rodinger equation (A.1)

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ω2 −∇2 + σ(r ) ψ(r ) = 2 ψ(r ). v0

Here σ(r ) = ω 2 /v02 − ω 2 /v 2 (r ). This equation has the same form as the Schr¨odinger equation if one considers σ(r ) to play the role of an eﬀective potential (the analogue of 2mV (r )/2 ), and ω 2 /v02 to play the role of an eﬀective energy (the analogue of 2mE/2 ). This analogy indicates that similar wave phenomena can be expected for quantum particles and classical waves. However, there is an important diﬀerence. Since σ(r ) > ω 2 /v02 always, the classical wave case corresponds to the quantum case only when E > V (r ) for all r. Thus classical waves can never be trapped by disorder in a trivial way analogous to the trapping of a low-energy particle in the bottom of a potential well. For classical waves, lowering the frequency also lowers the eﬀective disorder potential seen by the wave, since σ(r ) is proportional to ω 2 , so that localization is most likely to occur at intermediate frequencies where the wavelength is comparable with the size of the inhomogeneities and the scattering is greatest.

REFERENCES [1] Anderson P. W., Phys. Rev., 109 (1958) 1492. [2] For a recent non-specialist review of Anderson Localization, see Lagendijk A., van Tiggelen B. A. and Wiersma D. S., Phys. Today, 62(8) (2009) 24. [3] He H. and Maynard J. D., Phys. Rev. Lett., 57 (1986) 3171. [4] Weaver R. L., Wave Motion, 12 (1990) 129. [5] Dalichaouch R., Armstrong J. P., Schultz S., Platzman P. M. and McCall S. L., Nature, 354 (1991) 53. [6] Chabanov A. A., Stoytchev M. and Genack A. Z., Nature, 404 (2000) 850. [7] Chabanov A. A., Zhang Z. Q. and Genack A. Z., Phys. Rev. Lett., 90 (2003) 203903. [8] Schwartz T., Bartal G., Fishman S. and Segev M., Nature, 446 (2007) 52. [9] Wiersma D. S., Bartolini P., Lagendijk A. and Righini R., Nature, 390 (1997) 671. ¨ rzer M., Gross P., Aegerter C. M. and Maret G., Phys. Rev. Lett., 96 (2006) [10] Sto 063904. ¨ rzer M. and Maret G., Europhys. Lett., 75 (2006) 562. [11] Aegerter C. M., Sto [12] Abrahams E., Anderson P. W., Licciardello D. C. and Ramakrishnan T. V., Phys. Rev. Lett., 42 (1979) 673. [13] Hu H., Strybulevych A., Page J. H., Skipetrov S. E. and van Tiggelen B. A., Nature Phys., 4 (2008) 945.

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[14] Faez S., Strybulevych A., Page J. H., Lagendijk A. and van Tiggelen B. A., Phys. Rev. Lett., 103 (2009) 155703. [15] Page J. H., this volume, p. 75. [16] Page J. H., Hildebrand W. K., Beck J., Holmes R. and Bobowski J., Phys. Status Solidi C, 1 (2004) 2925. [17] Page J. H., Schriemer H. P., Bailey A. E. and Weitz D. A., Phys. Rev. E, 52 (1995) 3106. [18] Turner J. A., Chambers M. E. and Weaver R. L., Acustica, 84 (1998) 628. [19] Schriemer H. P., Pachet N. G. and Page J. H., Waves in Random Media, 6 (1996) 361. [20] Van Tiggelen B. A., in Diﬀuse Waves in Complex Media, edited by Fouque J. P. (Dordrecht, Kluwer) 1998, pp. 1-63. [21] Weaver R. L., J. Acoust. Soc. Am., 71 (1982) 1609. [22] Ryzhik L., Papanicolaou G. and Keller J. B., Wave Motion, 24 (1996) 327. [23] Hennino R., Tr´ egour` es N., N. M. Shapiro N. M., Margerin L., Campillo M., van Tiggelen B. A. and Weaver R. L., Phys. Rev. Lett., 86 (2001) 3447. [24] Turner J. A. and Weaver R. L., J. Acoust. Soc. Am., 98 (1995) 2801. ¨ lfle P., Phys. Rev. B, 22 (1980) 4666; Phys. Rev. Lett., 48 (1982) [25] Vollhardt D. and Wo 699. ¨ lfle P., this volume, p. 1. [26] Wo [27] van Tiggelen B. A., Lagendijk A. and Wiersma D. S., Phys. Rev. Lett., 84 (2000) 4333. [28] Skipetrov S. E. and van Tiggelen B. A., Phys. Rev. Lett., 96 (2006) 043902. [29] Hu H., Ph.D. Thesis, University of Manitoba (2006). [30] Cherroret N., Skipetrov S. E. and van Tiggelen B. A., arXiv:0810.0767. [31] Scheffold F., Lenke R., Tweer R. and Maret G., Nature, 398 (1999) 206. [32] Shnerb N. and Kaveh M., Phys. Rev. B, 43 (1991) 1279. [33] Nieuwenhuizen Th. M. and van Rossum M. C. W., Phys. Rev. Lett., 74 (1995) 2674; van Rossum M. C. W. and Nieuwenhuizen Th. M., Rev. Mod. Phys., 71 (1999) 313. [34] For a recent review, see Evers F. and Mirlin A. D., Rev. Mod. Phys., 80 (2008) 1355. [35] Mirlin A. D., Fyodorov Y. V., Mildenberger A. and Evers F., Phys. Rev. Lett., 97 (2006) 046803. [36] Grussbach H. and Schreiber M., Phys. Rev. B, 51 (1995) 663.

Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-115

Ultrasonic spectroscopy of complex media J. H. Page Department of Physics and Astronomy, University of Manitoba Winnipeg, MB Canada R3T 2N2

Summary. — Mesoscopic wave physics underpins many of the new developments in ultrasonic spectroscopy for probing the physical properties of complex heterogeneous materials. In this paper, two examples of recent progress are summarized. The ﬁrst is Diﬀusing Acoustic Wave Spectroscopy (DAWS), which is a powerful approach for investigating the dynamics of strongly scattering media, one example being velocity ﬂuctuations in ﬂuidized suspensions of particles. Recent advances in using phase statistics to probe the particle dynamics are shown to give increased sensitivity in some situations; this work has also led to new insights into the meaning of phase for multiply scattered waves. The second topic is the spectroscopy of soft food biomaterials, illustrated by experimental studies of ultrasonic velocity and attenuation in bread dough. Since wheat ﬂour dough contains one of the strongest scatterers of ultrasonic waves (bubbles) dispersed in a viscoelastic matrix that is also very dissipative, appropriate ultrasonic techniques provide an excellent means for investigating its structure and dynamics. In addition to fundamental studies, unraveling the contributions of bubbles and matrix to dough properties is relevant to the baking industry, because the bubbles ultimately grow into the voids that determine the structural integrity of bread —an important quality attribute. The interpretation of ultrasonic experiments on bread dough over three decades in frequency is giving new insights into this complex material, as well as providing the basis for new non-destructive methods of evaluating both dough processing behaviour and the breadmaking potential of diﬀerent ﬂours.

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1. – Introduction The fundamental studies of ultrasonic wave transport in strongly scattering random media, described in refs. [1-5], have facilitated the development of ultrasonic techniques for probing the physical properties of complex materials. Many such materials are mesoscopic, with internal structures on length scales comparable with ultrasonic wavelengths, and it is the structure and dynamics at this mesoscopic scale that determine their macroscopic physical properties. Familiar examples include foams, gels, slurries and a wide range of food biomaterials, all of which are playing an increasing important role in industrial applications, and hence our prosperity. Mesoscopic structure, however, often leads to multiple scattering of ultrasound, making traditional imaging methods impossible and motivating the development and application of new approaches for extracting useful information. This paper reviews two examples of ultrasonic spectroscopy and their application to novel materials characterization methods. The next section outlines Diﬀusing Acoustic Wave Spectroscopy (DAWS), a powerful technique in ﬁeld ﬂuctuation spectroscopy for investigating the dynamics of strongly scattering media. Diﬀerences with the complementary technique of Diﬀusing Optical Wave Spectroscopy are discussed, highlighting the advantages of DAWS in some contexts. DAWS is illustrated with experiments on suspensions of particles and bubbles. Recent progress in probing dynamic properties using the phase of multiply scattered waves, which can readily be measured for ultrasound but less easily for light, is summarized. Diﬀusing Acoustic Wave Spectroscopy, introduced in 2000 [6] and described in detail in ref. [7], has also been reviewed in ref. [8], and more recently in a broader context in Physics Today [9]. The interested reader is encouraged to consult these references for additional information. Section 3 illustrates the characterization of food materials using ultrasound. Many food materials are both strongly scattering and strongly absorbing for ultrasound, and in cases where the multiple scattering coda is suppressed by dissipation, it is not feasible to use techniques such as DAWS to probe their evolution during processing. Nonetheless information on the mechanical properties of foods is important in the preparation and production of foods with appealing texture, which is crucial for making foods palatable to eat. This information can be obtained from spectroscopic techniques that rely on ballistic propagation, and are especially valuable when data over a wide range of ultrasonic frequencies are available. This approach is illustrated with experiments on bread dough [10-19], where processing challenges encountered when incorporating nutritional supplements may make ultrasonic monitoring techniques of particular value to the functional foods industry. 2. – Diﬀusing Acoustic Wave Spectroscopy Diﬀusing Acoustic Wave Spectroscopy (DAWS) determines the dynamics of a strongly scattering medium from the temporal ﬂuctuations of ultrasonic waves that are scattered

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many times before leaving the medium [6,7]. Because multiply scattered waves are used, the technique is extremely sensitive to the motion of the scatterers in the medium, or to the evolution of the properties of the host material in which the scatterers are located; this sensitivity results from the large number of scattering events that are involved, leading to long trajectories over which cumulative changes in the detected waves occur. As the name suggests, there is much in common with the analogous technique of Diﬀusing Optical Wave Spectroscopy (often simply abbreviated DWS) [20, 21], although there are diﬀerences in the way in which the measurements are made and in the range of applications for which the two techniques are well suited. One advantage of DAWS is that the scattered wave ﬁeld is measured, not the intensity, so that the ﬁeld correlation function g1 (τ ) is determined directly. Thus, there is no need to invoke the Siegert relation to interpret measurements of intensity correlation functions using models for the ﬁeld correlation functions. Another advantage of detecting the wave ﬁeld in DAWS is that the phase of the scattered ﬁelds can be exploited, oﬀering the potential of better sensitivity in some cases. The other major technical diﬀerence is the ease with which pulsed measurements can be performed, enabling the detected changes to be monitored for a ﬁxed path length of the multiply scattered waves and therefore simplifying the analysis. Finally, since ultrasonic wavelengths and wave periods are both larger (typically ∼ 1 mm and 1 μs), DAWS is sensitive to dynamics on longer length scales than is possible with light (or X-rays), enabling diﬀerent types of materials and phenomena to be investigated. By varying the frequency, this range of length scales can be extended signiﬁcantly, and can range up to kilometres for seismic applications. Figure 1 shows two contrasting examples of evolving multiply scattered wave ﬁelds that can be used to probe changes in the system under investigation. Figure 1(a) shows a typical experimental setup in DAWS, where a pulsed incident wave from a generating transducer propagates through a sample containing moving particles or bubbles (ﬁg. 1(b)) and is detected by a hydrophone. A typical multiple scattering path is indicated by the red arrows. Two segments of the transmitted ﬁeld are shown in ﬁg. 1(c), showing that at early propagation times, almost no change in the transmitted ﬁeld is seen, while at later times, the wave ﬁeld changes signiﬁcantly as the scatterers move. Note that there are two relevant times in this problem, the propagation time t of the waves in the medium, which sets the sensitivity, and the evolution time T , which sets the time scale over which the dynamics are recorded. In DAWS, the medium can be interrogated repeatedly on a scale set by the pulse repetition time ΔT , which can be varied over a wide range to match the rate at which the system is evolving; in this case, T = mΔT , where m is an integer. Figure 1(c) shows that the waves are decorrelated in both amplitude and phase as the evolution time increases, due to the motion of the bubbles in the suspension. Analysis of the detected ﬁeld ﬂuctuations can be used to probe the velocities of the bubbles. By contrast, the changes in the waveforms shown in ﬁg. 1(d), which were detected on Mount Merapi by a seismograph located 2 km away from the source, at evolution times separated by two weeks, are shifted in propagation time but remain similar otherwise. In this case, there is a uniform change in the medium, and the phase shift is related to a change in the seismic wave speed.

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Propagation time, t (s) Fig. 1. – (a) Typical setup for measuring the dynamics of ﬂuidized suspensions or bubbles using Diﬀusing Acoustic Wave Spectroscopy. The red arrows indicate a multiple scattering path through the sample. (b) Photograph of ∼ 20-μm-diameter bubbles generated by an electrolysis technique. These small bubbles move in complex swirling patterns through water. (c) Two segments of the scattered wave ﬁelds transmitted through a ﬂuidized suspension of glass particles, observed at three evolution times separated by 60 ms. The waveforms are very similar at early propagation times, but exhibit large ﬂuctuations in both phase and amplitude a later times. (d) Waveforms detected by a seismograph on Mount Merapi, the site of an active volcano in Indonesia, after an air gun was used to generate short low frequency pulses 2 km away. The evolution time interval between the two recorded signals (red and blue traces) is two weeks. The shift in the phase of the waves can be most simply measured from a windowed cross correlation function of the ﬁelds at the two times; this phase shift is related to a small change in the seismic velocity, as explained in ref. [9].

To determine the changes in the medium from the evolution of the scattered wave ﬁelds, it is helpful to describe the multiply scattered waves detected at propagation time t and evolution time T as the superposition of waves that have propagated along each scattering path p. This can be shown explicitly by writing the measured ﬁeld ψ(t, T ) as the real part of a complex ﬁeld (the complex analytic signal) (1)

Ψ(t, T ) = A(t, T )ei[ωt+Φ(t,T )] =

ap (t, T )ei[ωt+φp (t,T )] .

p

Here ω is the central frequency of the pulse, Φ(t, T ) is the total phase of the scattered waves at the detector and φp (t, T ) is the phase along a single multiple scattering path,

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which may be conveniently called the “path phase”. As the medium evolves in time, the waves still propagate along these scattering paths, but the path lengths change, so that ψ is a function of both propagation time t and evolution time T . One direct way of relating the changes in ψ(t, T ) to the dynamics of the medium is to take the autocorrelation function of the ﬁeld at a ﬁxed propagation time ts , thereby selecting multiple scattering paths with an average length s = ts /vE and a narrow path length range determined by the source pulse width. The ﬁeld autocorrelation function g1,ts (τ ) is # ψ(T ) ψ(T + τ )dT (2) g1,ts (τ ) = e−i Δφp (τ ) , # 2 |ψ(T )| dT where Δφp (τ ) is the change in phase of a path containing n scattering events during the evolution time interval τ . Here n = vE ts /l∗ , where vE and l∗ are the transport velocity and mean free path of the multiply scattered waves. In general, the phase change for each path can be written as the sum of the ensemble average phase shift Δφpath (τ ) and the deviation from the average value δφpath (τ ), enabling the autocorrelation function (2) to be written as a product of two factors, involving the average phase shift and its variance, respectively: ! 1 2 (3) g1,ts (τ ) cos ( Δφpath (τ ) ) exp − δφpath (τ ) . 2 To obtain this result, the contribution to g1 from the ensemble average of e−i δφpath (τ )

is obtained to leading order using a cumulant expansion [6-8], and the real part of g1 is taken, since this corresponds to the experimental situation. A nonzero average phase shift arises when there is a uniform dilation of the medium seen by the waves, such as can occur if there is a change in wave velocity, which shifts the arrival time of all the scattered waves in the same way (e.g., see ﬁg. 1(d)). In the case of a small wave velocity change Δv, Δφ = −ωtΔv/v, where ω is the angular frequency. When the scatterers are moving, such as for the example of moving bubbles in ﬁg. 1(b),(c), the dominant contribution to the decay of g1 comes from the path phase variance δφ2path . In this case, the path phase variance can be related to the phase ﬂuctuations for each step j along a path, which are given by kj · Δrrel,j where kj is the wave vector of the wave scattered between the j-th to the (j + 1)-th scatterers, and Δrrel,j is their relative displacement [6-8]. When the successive phase shifts along the paths, as well as the directions of kj and Δrrel,j , are uncorrelated, the ﬁeld autocorrelation function is given by ! nk 2 2 ∗ Δrrel (τ, l ) . (4) g1,ts (τ ) ≈ exp − 6 (Here the average phase shift in eq. (3) has been set to zero, as is observed for ﬂuidized particles; then, δφ2path = Δφ2path .) This equation shows that the decay of the correlation function is determined by the relative mean square displacement of the scatterers

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Fig. 2. – (a) Mean square relative displacement of glass beads in a ﬂuidized suspension measured by Diﬀusing Acoustic Wave Spectroscopy. The data are plotted for several volume fractions of beads φvf , showing τ 2 behaviour indicative of ballistic particle motion at short times. (b) Root mean square relative velocity, divided by the ﬂuidization velocity Vf , as a function of the ultrasound transport mean free path l∗ normalized by the bead radius a. The mean free path l∗ determines the average particle separation at which the velocity ﬂuctuations are measured. The solid curves are ﬁts of eq. (5), with CV given by eq. (6), to the data for two volume fractions, enabling the particle velocity correlation length ξ to be measured at each volume fraction. (c) The particle velocity correlation function as a function of the average inter-particle separations R = l∗ at which the relative velocities are measured. The data show a good ﬁt to the exponential function exp[−R/ξ], conﬁrming the form of the correlation function that was assumed in (b). The diﬀerent symbols represent data measured at diﬀerent volume fractions of scatterers, with all the data collapsing onto a common curve when CV is plotted as a function of R/ξ.

that are separated, on average, by the average step length of the multiply scattered waves, l∗ . Measuring the ﬁeld autocorrelation function thus enables the relative motion of the scatterers to be determined on a length scale that can be tuned by the ultrasonic frequency. Typical DAWS results for the dynamics of ﬂuidized particles, suspended by ﬂowing the liquid upwards to counteract sedimentation, are shown in ﬁg. 2. In this example, the scatterers are 1-mm-diameter glass spheres surrounded by a liquid mixture of water and glycerol. At short evolution times, the motion of the particles is ballistic since the relative

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2 2 ∗ mean square displacement grows quadratically with time, Δrrel (τ, l∗ ) = ΔVrel (l ) τ 2 , allowing the variance in the relative particle velocities to be measured directly from the 2 2

(τ, l∗ ) versus τ 2 . The root mean square relative velocity ΔVrel = ΔVrel slope of Δrrel √ determines the characteristic time scale of the motion, τDAWS ≡ 1/[ nkΔVrel (l∗ )], so that, with this deﬁnition, the ﬁeld autocorrelation function can be written in a very simple way as g1 (τ ) = exp[− 16 τ 2 /τDAWS ]. By varying the frequency, the scattering strength and hence also l∗ was varied (see, for example, ref. [1]), enabling the relative particle velocity to be measured over a wide range of inter-particle distances R = l∗ inside the suspension. Figure 2(b) shows that at short distances, the relative particle velocity increases as the square root of distance, but that at √ longer distances it levels oﬀ to the value 2Vrms , where Vrms is the absolute rms particle velocity that can be measured directly in the √ single scattering regime using Dynamic Sound Scattering [6]. The saturation value 2Vrms is the relative velocity of particles that move independently, indicating that all correlations in the motions become lost at large inter-particle separations. These observations can be summarized mathematically as follows:

2 & ∗ ΔV (r + l ) − ΔV (r ) = (r + l∗ ) · ΔV (r ) = 2 ΔV 2 − 2 ΔV !

2 ∗ ΔVrel (l )

(5)

%

2 (1 − CV (l∗ )), = 2 Vrms

where

(6)

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r + R · ΔV (r ) ΔV ΔV 2

= exp[−R/ξ]

is the particle velocity correlation function, whose decay rate is determined by the velocity correlation length ξ. Equations (5), (6) show how particle velocity correlation function can be determined from the relative velocity ﬂuctuations measured in DAWS experiments, yielding the experimental results shown in ﬁg. 2(c). The data in ﬁgs. 2(b) and (c) show that the assumed exponential decay of the velocity correlations with distance is consistent with observations, and enable the correlation length to be measured over a wide range of particle concentrations (with φvf varying from 0.08 to 0.50 in this case). The correlation length ξ measures the range of distances over which the particles move together, and is an important quantity for understanding the physics of ﬂuidized suspensions. Figure 2(c) also reveals a remarkable scaling of the velocity correlations for diﬀerent particle concentrations when CV is plotted as a function of R/ξ. Thus, Diﬀusing Acoustic Wave Spectroscopy can provide information on the dynamics of suspensions that is relevant both to fundamental studies of the motions in suspensions and turbulent ﬂuids, and to practical applications such as monitoring mixing processes, the performance of chemical slurry bed reactors, and slurry ﬂow in industrial processing.

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By capitalizing on the ability of ultrasonic techniques to measure the phase and amplitude of the multiply scattered waves, DAWS has recently been extended to monitor system dynamics by analysing the ﬂuctuations of the phase of the waves in time-varying systems [22]. Not only has this helped to advance our understanding of mesoscopic wave physics, where the role of phase for multiply scattered waves has often been ignored, but it has also provided a new approach for probing evolving media with increased sensitivity in some situations. Experimentally, the wrapped phase Φ(t, T ) ∈ (−π : π] was extracted from the measured ﬁeld at any evolution time T by a numerical technique that is equivalent to taking a Hilbert transform and obtaining the complex analytic signal A(t, T ) exp[i[ωt + Φ(t, T )]]. Information on the dynamics of the medium is contained in the phase diﬀerence over the time interval τ , ΔΦ(τ ) = Φ(T + τ ) − Φ(T ), which, since the phase diﬀerence is still a phase, is best represented between −π and +π, and was therefore re-wrapped ∈ (−π : π]. To relate this phase diﬀerence to the particle dynamics, the relationship between the measured phase shift ΔΦ(τ ) and the evolution of the path phase Δφpath (τ ) was established. The simplest way to do this is via the wrapped phase diﬀerence probability distribution P (ΔΦ(τ )), which can be calculated for random wave ﬁelds described by circular Gaussian statistics from the joint probability distribution of the ﬁelds at two times Ti and Tj = Ti + τ [22]. Results for the same ﬂuidized suspensions of glass particles used for the data in ﬁg. 2 are compared with theory in ﬁg. 3(a)-(d), showing how the statistics of the phase diﬀerence evolve as the scatterers move, with the distribution becoming wider as the relative mean square displacement of the particles

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increases. At long times when the ﬁelds are no longer correlated, P (ΔΦ(τ )) reaches a ﬂat distribution. At short time intervals τ and small ΔΦ, P (ΔΦ(τ )) has the simple form (7)

P (ΔΦ) =

Δφ2path

1 3/2 , 2 Δφ2path + ΔΦ2

showing explicitly how the distribution depends on the path phase variance Δφ2path (τ )

and hence on the relative mean square displacement of the particles. While the general expression for P (ΔΦ(τ )) is more complicated [22], it still only depends on one parameter, the path phase variance, allowing the excellent ﬁts of theory and experiment shown in ﬁg. 3 to accurately measure the mean square relative displacement of the particles. 2 (τ ) from P (ΔΦ) and g1 are in superb Figure 3(e) shows that measurements of Δrrel agreement over a wide range of evolution time intervals τ , validating this phase method. The insert of this ﬁgure also shows an example where measurements of P (ΔΦ) yield more accurate results. In this case, the presence of amplitude noise due to gain ﬂuctuations degrades the ﬁeld correlation measurements of the particle dynamics at short times, but has little eﬀect on the phase statistics, which still give an accurate measurement of the particle motions. Another way of characterizing the dynamics is to measure the variance of the wrapped phase diﬀerence ΔΦ2 (τ ) . The variance of the measured phase shift is very diﬀerent to the path phase variance Δφ2path (τ ) , since the phase of the measured ﬁeld is determined by the superposition of waves along all paths reaching the detector, while the path phase variance is determined by the ﬂuctuations in the phase along a typical path (see eq. (1)). Remarkably, a universal relation has been found between the wrapped phase variance and the path phase variance, as shown in ﬁg. 4(a) by the solid curve. This universal relation means that the particle dynamics can be determined directly from the measured phase variance (open squares in ﬁg. 3) —a simpler procedure than ﬁtting the theoretical expression for P (ΔΦ) to experimental data. Both methods work well for evolution times that are short enough that ΔΦ2 (τ ) is less than its upper limit of π 2 /3, which occurs when P (ΔΦ) has become ﬂat. Information on the dynamics can be followed to longer times by unwrapping the phase, removing the jumps of 2π to determine the evolution of the cumulative phase Φc (τ ). Here, as an example, we consider the cumulative phase shift variance, which is plotted as a function of τ /τDAWS in ﬁg. 4(b). At early times, its increase with time is the same as the wrapped phase variance, but at long times ΔΦ2c (τ ) becomes proportional to time, with ΔΦ2c (τ ) = DΦ τ , enabling the phase diﬀusion coeﬃcient DΦ to be measured. If the particles continue to move in ballistic trajectories at long times, DΦ = 1/τDAWS , but if the relative motion slows down, due to deviations from ballistic particle trajectories due to particle interactions, DΦ is reduced. The solid curve in ﬁg. 4(b) shows a ﬁt to a simple empirical crossover model [6], indicating that the characteristic time τc for such deviations to set in is about 7τDAWS . An interesting general point to emerge from this analysis of the cumulative phase shift variance is that unwrapping the phase destroys the

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Fig. 4. – (a) Relation between the measured phase shift variance ΔΦ2 (τ ) and the path phase variance Δφ2path (τ ). For the wrapped phase, this relationship is universal (solid curve), while unwrapping the phase destroys the universality, giving the cumulative phase shift variance ΔΦ2c (τ ) greater sensitivity to the dynamics at long times. (b) The time dependence of the cumulative phase shift variance ΔΦ2c (τ ), showing a crossover to phase diﬀusion for times longer that τDAWS , with a phase diﬀusion coeﬃcient that is inﬂuenced by the long-time dynamics.

universal relationship between the measured phase variance and the path phase variance; this actually has a positive beneﬁt since it gives the cumulative phase shift variance increased sensitivity to details of the particle motions at long times. The behaviour is shown by the dashed and dotted curves in ﬁg. 4(a). Another advantage of examining the phase statistics has been demonstrated by theory and experiment for the probability distributions of the phase derivatives with evolution time. These distributions have been determined for Φ , Φ , Φ and found to be remarkably sensitive to early time dynamics, allowing the relative particle motions to be determined up to the 6th power in time —something that simply could not be achieved from measurements of the ﬁeld correlation function. Another interesting quantity is the cumulative phase correlation function, where current work is showing that, for evolving systems such as the bubbly liquids, the phase correlation function can be used to investigate motions at remarkably long times, beyond those accessible to ﬁeld correlation measurements. Thus, progress in measuring and understanding the phase statistics and correlations of multiply scattered ﬁelds is continuing to advance the capabilities of Diﬀusing Acoustic Wave Spectroscopy for investigating the dynamics of strongly scattering materials. 3. – Probing food biomaterials with ultrasound Many foods are heterogeneous on length scales that are comparable with the wavelengths of ultrasound in the 100 kHz to 10 MHz range, making ultrasonic spectroscopy of

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Fig. 5. – The ultrasonic attenuation and phase velocity as a function of frequency up to 10 MHz for a typical dough sample with a bubble concentration of 12%. The roman numerals and boxes indicate the three frequency regimes, as discussed in the text.

food materials a promising approach for investigating their mechanical properties, structure and dynamics. Because both scattering and dissipation of ultrasound are generally strong in such materials, most information on their properties comes from ballistic velocity and attenuation measurements. In this section, I focus on one example, bread dough, which contains one of the strongest scatterers of ultrasound, namely bubbles, with the bubbles being dispersed in a viscoelastic matrix, which contributes to the ultrasonic absorption. Thus, the physics of how ultrasound propagates in dough is remarkably rich. Understanding the eﬀect of bubbles on the properties of dough is also critical to controlling the texture of bread, and hence its quality. As a result, ultrasonic characterization of bubbles in bread dough is potentially important to the food industry. Ultrasonic experiments on bread dough reveal diﬀerent properties as the frequency is varied [10-19]. Indeed, there are three important frequency regimes. These are identiﬁed in ﬁg. 5, which shows experiment and theory for the ultrasonic attenuation and phase velocity in bread dough over almost three decades in frequency. At low frequencies, f < 100 kHz, bubbles in dough drastically reduce the sound velocity, due to the large compressibility of the bubbles. There is excellent sensitivity to the presence of bubbles but no information on their sizes. The attenuation is relatively low in this frequency regime, making experiments easier. For frequencies between 100 kHz and 8 MHz, there is a strong resonant interaction between the ultrasonic waves and the bubbles, leading to a very large variation in the velocity and attenuation. Their frequency dependence at these intermediate frequencies depends on the bubble sizes, raising the interesting possibility of extracting information on the bubble size distribution in this opaque medium from

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Fig. 6. – The ultrasonic velocity (a) and attenuation (b) in bread dough as a function of bubble volume fraction at 50 kHz.

ultrasonic measurements. At high frequencies, f > 8 MHz, the ultrasonic attenuation and velocity depend on matrix properties only, enabling structural relaxations of the molecular ingredients of the matrix to be probed. The sensitivity of ultrasound to the concentration of bubbles in the low frequency regime (at f ∼ 50 kHz) is shown in ﬁg. 6. The dough samples were prepared by mixing together a strong Canadian breadmaking ﬂour (CWRS), salt and water, to produce a lean-formula mechanically developed dough [23]. For these experiments, the bubble concentration was adjusted by varying the headspace pressure during mixing. As the bubble concentration is increased, the ultrasonic velocity drops dramatically, reaching values less than the velocity of sound in air for concentrations above 2%. This behaviour can be understood qualitatively in Wood’s approximation for the low frequency compressibility of a bubbly liquid. In this approximation, the average compressibility of the sample κs is simply the volume-fraction-weighted average of the compressibilities of the bubble inclusions i and surrounding matrix m (8)

κs = φvf κi + (1 − φvf )κm ,

of bubbles. Thus, since phase velocity and compresswhere φvf is the volume fraction ibility are related by v = 1/ρκ, where ρ is the density, (9)

1 φvf 1 − φvf = + 2 ρs vs2 ρi vi2 ρm vm

and the average sound velocity for concentrations of bubbles in this range reduces approximately to ' (10)

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ρair . ρs φvf

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The density ratio in the square root factor in this expression shows why the velocity is so much less than the velocity in air (340 m/s); the eﬀective medium behaves as a material with the low compressibility of air, but with a larger density. For dough, this approximation underestimates the velocity at all volume fractions because it neglects the shear modulus of the dough matrix, which can be included in a more complete (but also more complicated) eﬀective medium model [24]. For the higher volume fractions, where the complex shear modulus μm = 0.39 + i0.14 MPa can be reasonably extrapolated from existing lower frequency shear rheology data on dough prepared at ambient pressure [14]; this model gives excellent agreement with experiment. At lower concentrations, however, the measured velocities are larger than this prediction, suggesting that the shear modulus of the dough matrix increases at low volume fractions. Thus the presence of bubbles in the dough enables the shear properties of the dough to be investigated using longitudinal waves —a considerable advantage as longitudinal ultrasonic measurements are easier to perform in lossy materials such as dough. As shown in ﬁg. 6(b), the ultrasonic attenuation increases as the square root of the volume fraction in this low frequency regime, a frequency dependence which is predicted by eﬀective medium theories (solid line) [24,16]. By treating the interaction of ultrasound with bubbles in a viscoelastic medium, it can be shown, at frequencies well below the resonance frequency ω0 of the bubbles, that the attenuation is predicted to have the form ' (11)

α=

3φvf ω 2 Γ . a2 ω03

Here a is the radius of the bubbles, and Γ is the damping rate, which at low frequencies depends on viscous losses and thermal dissipation. If viscous losses dominate, the dependence of α on bubble size in eq. (11) cancels out, since Γviscous = 4μ /ρωa2 and the resonant frequency of the bubbles depends inversely on the bubble radius, ω0 ∝ a−1 . Thus, in this regime, the attenuation is sensitive only to the amount of gas entrained in the bubbles, and not on how the gas is distributed, providing a good indicator of the amount of gas entrained in the dough. The sensitivity of these low-frequency measurements to bubble concentration is enabling the ultrasonic velocity and attenuation to be used to monitor dough mixing, where reliable methods of determining optimum mixing conditions are of considerable value [18]. For doughs prepared with leavening agents, ultrasonic velocity and attenuation can be used to monitor the growth of the bubbles due to incorporation of CO2 [12]. Low frequency velocity measurements can also be used to assess dough quality [25], and since these ultrasonic measurements can be performed on small samples, such measurements are potentially very useful in wheat breeding programs. At intermediate frequencies, the resonant coupling of ultrasound with the bubbles causes the attenuation and phase velocity to exhibit a large frequency dependence, with broad peaks that contain information on the bubble size distribution (ﬁg. 7). To interpret the experimental data, Leroy et al. [16] have used a model that extends the wellestablished model for the resonant interactions of sound with bubbles in liquids [26-28]

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Fig. 7. – Frequency dependence of the ultrasonic attenuation (a1),(a2) and velocity (b1),(b2), showing the broad spectral features characteristic of resonant interactions with bubbles. The grey curves represent experimental data, and the solid black curves are theoretical predictions of the model outlined in the text. (See ref. BIF for more information.) The numbers (1) and (2) identify the times after mixing at which the data were taken: 53 minutes for (1) and 90 minutes for (2). The solid curves in (c1) and (c2) are the bubble size distributions inferred from the ultrasonic data at these two times. The dotted curve in (c2) are the results of X-ray tomography measurements.

to viscoelastic materials, by incorporating a simple correction to the resonant frequency proposed by Alekseev and Rybak [29]. Physically, the eﬀects of ﬁnite shear rigidity are to shift the resonant frequency to higher frequencies and also to weaken the resonance. This model has been tested on transparent agar gels, where the bubble sizes can be measured optically, and found to describe the data well [16]. Applying the same model to dough, as described in ref. [16] and shown by the solid curves in ﬁgs. 7(a),(b), the bubble size distribution can be estimated. The inferred bubble size distributions at two times after mixing are shown in ﬁg. 7(c). At the later time, the evolution of the size distribution had slowed down suﬃciently to enable bench-top X-ray tomography measurements to independently measure the size distribution (dotted curve in ﬁg. 7(c)) [15]. This comparison indicates that the analysis of the ultrasonic data estimates smaller bubble sizes than the X-ray measurement, although comparison between results is not straightforward because the conditions of sample preparation were not the same in both cases. Work is continuing to understand the origin of this discrepancy, so that the ultrasonic technique can be developed to unambiguously determine bubble sizes. Since ultrasonic measurements can be performed quickly, potentially even online, this information has practical relevance for monitoring dough quality during breadmaking. Even though questions remain to be resolved concerning the absolute sizes of the bubbles determined from the ultrasonic velocity and attenuation, the shift of the resonance features in the ultrasonic data to lower frequencies at the later observation time shows the eﬀects of disproportionation in the dough due to Ostwald ripening; this phenomenon leads to an increase in the average size of the bubbles with time as gas diﬀuses from the smaller bubbles to the larger

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ones [30]. Such information on the dynamics of the bubbles in bread dough is valuable for understanding the evolution of the bubble structure. Measurements of ultrasonic velocity and attenuation in dough in the high frequency range (above the bubble resonance regime, f > 8 MHz) reveal information on matrix properties. The frequency dependence of the data show signatures of ultrasonic relaxation phenomena, which can be interpreted using a molecular relaxation model [31, 32]. Diﬀerent fast relaxation times were observed for ambient-mixed dough (5 ns) and vacuummixed dough (1 ns) [33]. These relaxation times may be associated with conformational rearrangements in glutenin —the supermolecular structure of proteins in the gluten matrix— perhaps due to the loop-to-train transition that is thought to play a role in the elasticity of glutenin [34]. Thus data in this frequency range can probe ultrasonic stress-induced changes in the secondary structure of gluten proteins that are important for understanding the viscoelastic properties of this complex food material. This example of ultrasonic spectroscopy shows that both ultrasonic velocity and attenuation are sensitive probes of the gas cell structure of bread dough, enabling new approaches to optimizing loaf quality to be developed. Ultrasound can be used to follow the evolution of the gas cells (bubbles) throughout the entire breadmaking process, from the initial entrainment of gas bubbles in dough during mixing, though the expansion of the gas cells during prooﬁng, all the way up to the ﬁnal foam structure of bread. Ultrasound can also be used to probe changes in the viscoelastic properties of the dough matrix. Remarkably, despite the complexity of dough and bread as mesoscopic materials, their mechanical properties can be elucidated using relative simple physics models. This combination of factors is leading to a new awareness of ultrasound’s potential to provide novel information on technical issues of importance to the cereals processing industry. 4. – Conclusions Ultrasonic spectroscopy is both contributing to and capitalizing on advances in the wave physics of complex mesoscopic materials. As a result, new approaches that exploit the advantages of ultrasonic techniques are being developed to characterize the structure and dynamics of this increasing important class of materials. This paper has discussed two examples. The ﬁrst was Diﬀusing Acoustic Wave Spectroscopy, which is a sensitive technique for monitoring changes in materials in which conventional imaging techniques are impossible due to multiple scattering, and which is complementary to Diffusing Optical Wave Spectroscopy. DAWS is based on direct measurements of the ﬁeld autocorrelation function, and has been extended recently to probe dynamics using the phase of multiply scattered waves. This approach has some advantages practically, as well as being a way of advancing our understanding of phase in mesoscopic wave physics. In this paper, the application of this technique to the investigation of particulate and bubbly suspensions was demonstrated, but many more applications of this technique can be envisaged (e.g., in process control). The second example considered here was the characterization of biological materials of importance in food science. Many such materials have internal length scales that

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are comparable with the wavelength of ultrasound, making ultrasonic spectroscopy a particularly relevant approach. The studies of bread dough that were summarized in this paper demonstrate how advances in physics underpin practical applications. The latter are of considerable economic potential in the rapidly growing functional foods area, where the interaction of functional ingredients with the bubble structure can damage the taste and appearance of food products unless remedial action is taken. By monitoring the properties at an early stage in production and helping to understand the dynamics of these interactions, ultrasonic techniques can help overcome such problems. ∗ ∗ ∗ I would like to thank the many students and colleagues who have contributed to the research that has been reviewed in this paper. Support from NSERC is also gratefully acknowledged. REFERENCES [1] Page J. H., this volume, p. 75. [2] Page J. H., Schreimer H. P., Bailey A. E. and Weitz D. A., Phys. Rev. E, 52 (1995) 3106. [3] Page J. H., Sheng P., Schreimer H. P., Jones I., Jing X. and Weitz D. A., Science, 271 (1996) 634. [4] Schreimer H. P., Cowan M. L., Page J. H., Sheng P., Liu Z. and Weitz D. A., Phys. Rev. Lett., 79 (1997) 3166. [5] Cowan M. L., Beaty K., Page J. H., Liu Z. and Sheng P., Phys. Rev. E, 58 (1998) 6626. [6] Cowan M. L., Page J. H. and Weitz D. A., Phys. Rev. Lett., 85 (2000) 453. [7] Cowan M. L., Jones I. P., Page J. H. and Weitz D. A., Phys. Rev. E, 65 (2002) 066605. [8] Cowan M. L., Page J. H., Weitz D. A. and van Tiggelen B. A., in Wave Scattering in Complex Media: From Theory to Applications, edited by van Tiggelen B. A. and Skipetrov S. E. (NATO Science series, Kluwer Academic Publishers, Amsterdam) 2003, pp. 151-174. [9] Snieder R. and Page J., Phys. Today, 60 (2007) 49. [10] L´ etang C., Piau M., Verdier C. and Lefebvre L., Ultrasonics, 39 (2001) 133. [11] Elmehdi H. M., Ph.D. Thesis, University of Manitoba (2001). [12] Elmehdi H. M., Page J. H. and Scanlon M. G., Trans. Inst. Chem. Eng. Part C: Food Bioprod. Proc., 81 (2003) 217. [13] Elmehdi H. M., Page J. H. and Scanlon M. G., J. Cereal Sci., 38 (2003) 33. [14] Elmehdi H. M., Page J. H. and Scanlon M. G., Cereal Chem., 81 (2004) 504. [15] Bellido G. G., Scanlon M. G., Page J. H. and Hallgrimsson B., Food Res. Int., 39 (2006) 1058. [16] Leroy V., Fan Y., Strybulevych A. L., Bellido G. G., Page J. H. and Scanlon M. G., in Bubbles in Food 2: Novelty, Health and Luxury, edited by Campbell G. M., Scanlon M. G. and Pyle D. L. (AACC Press, St Paul, MN) 2008, pp. 51-60. [17] Scanlon M. G., Page J. H. and Elmehdi H. M., in Bubbles in Food 2: Novelty, Health and Luxury, edited by Campbell G. M., Scanlon M. G. and Pyle D. L. (AACC Press, St Paul, MN) 2008, pp. 217-230.

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[18] Mehta K. L., Scanlon M. G., Sapirstein H. D. and Page J. H., J. Food Science, 74 (2009) E455. [19] Leroy V., Pitura K. M., Scanlon M. G. and Page J. H., J. Non-Newtonian Fluid Mech., 165 (2010) 475. [20] Maret G. and Wolf P. E., Z. Phys. B, 65 (1987) 409. [21] Pine D. J., Weitz D. A., Chaikin P. M. and Herbolzheimer E., Phys. Rev. Lett., 60 (1988) 1134. [22] Cowan M. L., Anache-M´ enier D., Hildebrand W. K., Page J. H. and van Tiggelen B. A., Phys. Rev. Lett., 99 (2007) 094301. [23] Preston K. R., Kilborn R. H. and Black H. C., Can. Inst. Food Sci. Technol. J., 15 (1982) 29. [24] Sheng P., in Homogenization and Eﬀective Moduli of Materials and Media, edited by Ericksen J. L., Kinderlehrer D., Kohn R. and Lions J.-L. (Springer) 1988, p. 196. [25] Scanlon M. G., Mehta K. L., Elmehdi H. M. and Page J. H., unpublished. [26] Foldy L. L., Phys. Rev., 67 (1945) 107. [27] Leighton T. G., The Acoustical Bubble (Academic Press, London) 1994. [28] Prosperetti A., J. Acoust. Soc. Am., 61 (1977) 17. [29] Alekseev V. N. and Rybak S. A., Acoust. Phys., 45 (1999) 535. [30] Van Vliet T., in Bubbles in Food, edited by Campbell G. M., Webb C., Pandiella S. S. and Niranjan K. (Eagan Press, St. Paul, MN) 1999, pp. 121-127. [31] Litovitz T. A. and Davis C. M., in Physical Acoustics, Vol. IIA, edited by Mason W. P. (Academic Press, New York) 1965, pp. 282-350. [32] Marvin R. S. and McKinney J. E., in Physical Acoustics, Vol. IIB, edited by Mason W. P. (Academic Press, New York) 1965, pp. 160-230. [33] Scanlon M. G., Page J. H., Leroy V., Fan Y. and Mehta K. L., in Proceedings of the 5th International Symposium on Food Rheology and Structure, edited by Fischer P., Pollard M. and Windhab E. J. (ETH, Zurich) 2009, pp. 378-381. [34] Belton P. S., J. Cereal Sci., 29 (1999) 103.

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-133

MultiWave imaging M. Fink and M. Tanter Institut Langevin, Ecole Sup´ erieure de Physique et de Chimie de la Ville de Paris CNRS 7587, INSERM - 10 rue Vauquelin, Paris, 75005, France

Summary. — Interactions between waves can be turned into proﬁt to break diﬀraction limits and invent new kinds of medical images. It consists in productively combining two very diﬀerent waves —one to provide contrast, another to provide spatial resolution— in order to build a new kind of image. Contrary to multimodality medical imaging that remains the superposition of two diﬀerent images limited by their respective contrast/resolution couples, MultiWave imaging overcomes this limitation by providing a unique image of the most interesting contrast with the most interesting resolution. MultiWave imaging can beneﬁt from three diﬀerent potential interactions among waves that will be described in this paper.

1. – Introduction The human body supports the propagation of many kinds of waves with very diﬀerent speeds ranging from some m/s for mechanical shear waves to thousands of m/s for ultrasonic compressional waves and some hundreds of thousands km/s for optical and electromagnetic waves. Each of these waves can provide a medical imaging modality whose contrast is depending of the way the wave interacts with tissues (see appendix A). As the image is built from the interaction of only one wave with the medium, spatial resolution at one depth is linked to the order in which three spatial scales appear, i.e. observation depth z, wavelength λ and mean free path l. The ﬁrst situation, z < λ < l, c Societ` a Italiana di Fisica

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corresponds to near-ﬁeld imaging and is encountered in the ﬁeld of near-ﬁeld optics, EMG or EEG imaging and electrical impedance tomography (EIT). The second situation, λ < z < l, corresponds to coherent wave propagation outside of the near ﬁeld of the object and is encountered in the ﬁeld of ultrasonic imaging, optical computed tomography, and CT or X-ray imaging. The third situation, λ < l < z, corresponds to the diﬀusive regime where the wave loses its coherence through tissues interactions. This multiple scattering regime is mainly encountered in deep optical imaging, known as diﬀuse optical tomography [1]. For the ﬁrst situation, spatial resolution is of the order of the observation distance z. For the second one, spatial resolution depends mainly of the wavelength and diﬀraction limits. For the third one, spatial resolution could at best reach the mean free path l, but is generally of the order of the observation distance z. In all these imaging situations, physicists try to reach the optimal limits of the resolution associated to each kind of wave. However, one single wave is sensitive only to one given contrast associated to some physical properties of the medium. For example, shear waves are carrying information on the visco-elasticity of tissues (shear modulus and viscosity); ultrasonic waves are sensitive only to compression modulus and density. For electromagnetic waves, low-frequency waves are sensitive to electrical conductivity while optical waves are sensitive to optical absorption coeﬃcient. Today, after having pushed during decades the technological limits of these modalities, physicists are facing the inherent physical limits of the contrast/resolution couple in each modality. For medical imaging and diagnosis, physicians understood rapidly that the way to overcome these limits was to combine diﬀerent imaging modalities, such as PET/CT, PET/MRI or Ultrasound/X-ray mammography. The basic idea of multimodality imaging [2], such as, for example, the combination PET/CT, is to associate the high-resolution morphological image of a ﬁrst modality (CT) to an image of the second modality (PET) that is poorly resolved, but provides a clinically interesting contrast (i.e. metabolic activity). However, such multimodality imaging remains extremely costly and limited by the inherent physical limits of each separate modality. 2. – Transcending classical diﬀraction limits Is there any other solution than multimodality imaging to improve the diagnostic capabilities? Two diﬀerent scientiﬁc communities proposed new research directions. One approach called molecular imaging was impulsed by chemists and biologists. It diﬀers from traditional imaging in that biomarkers are used to help image particular targets or pathways. These biomarkers interact chemically with their surroundings and in turn increase the image contrast. The other approach was proposed independently by diﬀerent groups in the physicist community. It consists in productively combining two very diﬀerent waves —one to provide contrast, another to provide spatial resolution— in order to build a new kind of image. We propose to unify these approaches under the general concept of MultiWave imaging. Contrary to multimodality imaging that remains the superposition of two images limited by their respective contrast/resolution couples, multiwave imaging

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Fig. 1. – (Colour on-line) Principle of MultiWave imaging. Three possible wave interactions: a) wave-to-wave generation, b) wave-to-wave tagging, c) wave-to-wave imaging. In a) and b) the red wave carries information about the desired contrast, the blue wave provides the resolution.

overcomes this limitation by providing a unique image of the most interesting contrast with the most interesting resolution. The focus of this paper is on MultiWave imaging and will be illustrated using several examples of wave interactions. Multiwave imaging can beneﬁt from three diﬀerent potential interactions between waves (ﬁg. 1): – In a ﬁrst case, the interaction of the ﬁrst wave with tissues during its propagation can generate a second kind of wave. This is the case in thermoacoustic imaging or in magnetoacoustic tomography where any kind of electromagnetic wave is absorbed in some region causing either a transient change in temperature that radiates an ultrasonic wave through thermal expansion or a tissue motion under some Lorentz force. – In a second case, a ﬁrst wave that carries the information about the desired contrast but either completely loses its coherence during propagation through tissues, or has a large wavelength, can be tagged locally by a second kind of wave that remains coherent and well focused. The tagging focal spot can then be steered at diﬀerent locations in order to build a complete image. This is the case of Acousto-optical

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imaging (or acousto-optical tomography) where tissue displacements induced by a focused ultrasound beam modulates the optical speckle pattern of photons travelling through tissues. An image of the optical absorption is built with the submillimetric resolution of the ultrasonic wave. – In a third case, a ﬁrst wave travelling much faster than the second one can be used to produce a movie of the slow wave propagation. This is the case of Transient Elastography, where ultrafast ultrasonic scanners can track the motion of tissues scatterers induced by the propagation of low-speed shear waves. This last case is relatively unique as it allows observing remotely the full movie of the near ﬁeld of the shear wave around each obstacle (even if these obstacles are located in the far ﬁeld of the two waves). A local inversion algorithm performed on this near-ﬁeld movie produces a shear elasticity image relying on a sub-millimetric resolution while the shear wavelength is centimetric. The concept of MultiWave imaging is particularly interesting for the estimation of three physical parameters that remained diﬃcult to map up to recently with a good spatial resolution: optical absorption, that gives access to tissues color; electrical conductivity that depends on ion concentration and mobility in tissue and on the amount of intra- and extra-cellular ﬂuids; and ﬁnally mechanical shear elasticity and viscosity. 3. – Wave-to-wave generation All techniques based on wave/wave generation are related to some dissipative processes that transform one part of a pulsed electromagnetic energy in some transient tissue motion that radiates coherent ultrasonic waves. From the recording of the ultrasonic ﬁeld on an array of piezoelectric transducers, one can deduce an image of the ultrasonic sources. The fact that the ultrasonic speed is practically uniform in all tissue and has a well-known value greatly simpliﬁes the reconstruction process. An image of the sources is then built with the sub-millimetric resolution of the ultrasonic wave. In the thermo-acoustic approach both optical waves [3] and microwaves can be used. An image of the optical absorption or of the tissue conductivity is built with the submillimetric resolution of the radiated ultrasound. Microwave penetration allows deeper exploration and ﬁrst conductivity images of breast have been obtained with this modality, while vascular images on small animals have been obtained with the photo-acoustic approach. Figure 2 illustrates a spectacular application using as a heating source a laser with 532 nm wavelength and a wide-band ultrasonic transducer with a 2.25 MHz central frequency to receive the photo-acoustic wave. Figure 2 shows that blood vessels in the cortical surface of small animals can be imaged with the skin and the skull intact. The imaging depth is limited to 1 cm that is enough to image the entire brain of a small animal. Other modalities have been proposed to improve electrical impedance tomography, known as magnetoacoustic tomography (MAT), where tissues are displaced by an electric or a magnetic stimulation [5,6] to produce ultrasound. In the most interesting technique

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Fig. 2. – a) Non-invasive photo-acoustic image of a superﬁcial lesion (1 mm × 4 mm) in the right cortex on rat’s cerebra acquired through skull. RH is the right cerebral hemisphere, LH the left cerebral hemisphere and L the lesion. The blood vessels are clearly imaged. b) Open skull photograph of the rat surface acquired after the photo-acoustic experiment. See ref. [4].

with magnetic induction (MAT-MI), tissues are put both in a strong static magnetic ﬁeld and in a time-varying magnetic ﬁeld (MHz range). The time-varying magnetic ﬁeld induces eddy currents that interact with the static magnetic ﬁeld to produce a Lorentz force that induced ultrasonic waves, that can be also recorded by ultrasonic transducers. In this approach the acoustic wave amplitude is proportional to the electrical conductivity in the MHz range. The wave equation governing the ultrasonic pressure ﬁeld radiated in all these approaches can be written in a common way as (1)

) ( 1 ∂2 Δ − 2 2 p(r, t) = s(r, t), cp ∂t

where cp is the ultrasound compressional speed (quasi-uniform in soft tissues) and where the source term can correspond either to monopolar sources (dilation) s(r, t) = − cΓ2 ∂H p ∂t for thermo-acoustic imaging [3] or to dipolar source in magnetoacoustic imaging (MATMI) s(r, t) = ∇ · (J × B0 ). In the last source term, B0 is the static magnetic ﬁeld, and J is the induced Eddy current that depends on the pulsed magnetic ﬁeld B1 (t) and on the local conductivity σ(r) according to Faraday’s and Ohm’s Law. For B0 = 1 T and for a typical value of the conductivity σ = 0.2 S/m, ultrasonic pressure ﬁeld of some 10 millibars can be radiated, which is suﬃcient to be measurable for ultrasonic transducers elements. Diﬀerent approaches can be conducted to map the source terms and the inverse problem image reconstruction is typically based on the fact that in a medium with

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constant speed the data recorded on the transducers array are spherical integrals of pressure source. Back-propagation algorithm allows recovering function from integrals over spheres (spherical radon transform). This step can also be accomplished by timereversing and back-propagating the acoustic data in a computer model of constant speed. 4. – Wave-to-wave tagging Acousto-optic imaging combines, thanks to acousto-optic eﬀects, ultrasound and light in a diﬀerent way from photoacoustic imaging that is directly related to a dissipative process. A focused ultrasonic beam induces locally an ultrasonic modulation of a light beam traversing a scattering medium. Light transmitted through an organ contains thus diﬀerent frequency components: the main component (the carrier) is centered at the incident coherent optical beam frequency. It is related to the dﬀused photons that do not interact with ultrasound. The sideband components are shifted by the ultrasound frequency. The sideband photons which results from the interaction between light and ultrasound are called “tagged photons”. The weight of these tagged photons components depends on the optical absorption in the region of interest. Acousto-optic imaging serves in detecting selectively the tagged photons. An image related both to the optical absorption and to optical diﬀusion is then built up in scanning the focused ultrasonic beam over the whole organ. Marks investigated this tagging technique for the ﬁrst time in the early 1990s’ [7]. Since then, many diﬀerent groups have contributed to this ﬁeld [8-12]. Two main mechanisms participate in the ultrasonic modulation of light in a scattering media. One is based on the variation of the optical phase in response to ultrasoundinduced displacements scatterers. The displacement of scatterers modulates the physical path lengths of light traversing the ultrasonic ﬁeld. Multiply scattered light accumulates modulated path lengths. Therefore, the intensity of the speckle associated to multiply scattered light ﬂuctuates with the ultrasonic frequency. A second mechanism is based on the variation of the optical phase in response to ultrasonic modulation. As the result of ultrasonic modulation of the index of refraction, the optical phase between scattering events is modulated and the modulated phase causes also the speckle intensity to vary with ultrasound. Many coherent detection techniques have been proposed to detect the tagged photons. One of the most interesting techniques was the parallel detection scheme that uses a source-synchronized lock-in technique in which a CCD camera works as a detector array [10, 11]. An interesting improvement was proposed to increase the axial resolution of the acousto-optic images which was not as good as the lateral resolution with monochromatic ultrasound. Wang and Ku replaced the ultrasound monochromatic excitation with a frequency swept (chirped signal) that modulated also the gain of the optical detectors [11]. However, the main diﬃculty of this technique in living tissues results from both the motion of the scatterers due to the Brownian motion of the scatterers and to the tissue inner motions (blood ﬂow). This speckle decorrelation broadens the carrier and sideband lines. Typically with 4 cm breast thickness, the speckle decorrelation time is in the ms

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range and yields a 3 kHz broadening. Detector bandwidth in this range is needed. With a mono detector (photodiode) there is of course no problem, but to achieve a good signal to noise one needs a large optical etendue in detector plane (the product of the detector area by the detector acceptance angle) to ﬁt the etendue of the tagged photons source i.e. to maximize the scattered light collection. This is why multidetectors like a CCD camera have been investigated, but they suﬀer low image frequency rate (typically 100 Hz) which is not enough to avoid broadening eﬀect in living tissues. Faster cameras are not sensitive enough. More recently diﬀerent groups [12] proposed very promising tagged-photons detection techniques based on the photorefractive crystal-based interferometry that can give both a large etendue and a detection bandwidth in the kHz range (the response time of GaAs photorefractive material is of the order of 1 ms). It is interesting to note here that the tagging concept could be used not only in acousto-optics but in many other ﬁelds of medical imaging such as, for example, electric impedance tomography tagged by ultrasonic remote vibrations. Here we discuss the concept of wave-to-wave tagging using two kinds of totally different waves. However, although it is not a multiwave technique, MR imaging can also be interpreted as a tagging technique. It uses only one kind of wave combined with a static ﬁeld: a radio frequency electromagnetic wave that causes protons to absorb some of its energy and to release it later at a resonance radio frequency. The spatial tagging is achieved here through the addition of non-uniform magnetic ﬁelds to a static magnetic ﬁeld whose spatial gradient modiﬁes the local Larmor frequency, allowing during the reception mode to get a spatial resolution much better than the RF wavelength through a frequency analysis of the received signal. 5. – Wave-to-wave imaging The last example of multiwave imaging is perhaps the most fascinating. Indeed, the wave interaction is here chiseled such that the near ﬁeld of a wave around each obstacle can be ﬁlmed in the far ﬁeld of the imaging sensors. In this approach, the playground consists in sonic shear waves and ultrasonic waves. These waves interact to produce a quantitative and highly resolved image of deep organs stiﬀness. Stiﬀness is characterized by Young’s Modulus E (in kPa) and is an important parameter in medicine. Stiﬀness changes are often linked to pathology [13] and the signiﬁcant dependence of E on the structural changes in the tissue is the basis for the palpatory diagnosis of various diseases, such as detection of cancer nodules in the breast or prostate. Although it is strongly subjective, manual palpation is not only useful for screening and diagnosis but also during interventions to eﬀectively guide the surgeon toward the pathological area. In order to understand how to map tissue elasticity, we can in a ﬁrst approximation consider soft tissue as an isotropic elastic medium. The mechanical behaviour of such a soft solid is characterized by two parameters, K (inverse of the compressibility κ = 1/K) and μ, respectively, the bulk and shear modulus. The relationship between stiﬀness and

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these parameters is described by the relationship (2)

E=

9Kμ . 3K + μ

The property K μ is a kind of deﬁnition of “soft solids” and human soft tissues belong to this category. It implies straightforwardly a direct link between stiﬀness E and the shear modulus, E = 3μ. Therefore to access stiﬀness properties, an elegant way consists in using shear waves whose speed cs depends simply on the shear modulus cs = μ/ρ. Another important consequence of the big discrepancies between K and μ in soft solid is that the compressionnal wave speed is much larger than the one of the shear waves (from 1540 m/s for cp to some m/s for cs ). This is a unique case where two mechanical waves exhibit totally diﬀerent wave speeds. In conventional “single wave” imaging, only the compressional wave (and consequently the contrast of bulk modulus) is used. This is the successful ﬁeld of medical ultrasound imaging. Now, could we use shear waves to image the shear modulus contrast of tissues? As we have just seen, the MHz frequency range is forbidden due to shear tissue viscosity and shear waves can only propagate on centimetric distances at low sonic frequencies. The typical shear wave frequency ranges between 10 Hz and 1 kHz. For example, to propagate on a 5 cm distance, we are limited to frequencies lower than 100 Hz, corresponding to typical wavelengths of several centimeters. The use of shear waves in a “single wave” imaging approach can only lead to poor results as it will rely on very bad spatial resolution. However, the contrast sensed by shear waves remains very relevant information for the diagnosis. How to solve this problem using a “multiwave” approach? We can take beneﬁt from the huge discrepancy between shear and compressional wave speeds. The idea is to use the compressional waves at ultrasonic frequencies to produce ultrafast images of tissues to image the propagation of low speed sonic shear waves. The goal is to obtain a movie of the shear wave propagating inside organs with millimetric resolution. During their propagation, shear waves induce local tissue displacements of the order of some tens of microns around their equilibrium position. As the typical shear wave speed varies between 1 m/s and 10 m/s, one needs at least to reach 10000 frames per second to follow the shear wavefront millimeter by millimeter. Using such an ultrafast scanner, it could be possible to estimate these local displacements between successive images. Our group developed such an ultrafast scanner [14]. This is the ﬁrst ultrasonic device able to reach more than 10000 frames per second of deep-seated organs. In this device, whose architecture emerged from the concept of time reversal mirrors [15, 16], one transmits several thousand times per second an ultrasonic beam widely spread in the whole area of interest. This imaging sequence is very diﬀerent from the one used in conventional ultrasound scanners that insonicate the medium only using a very thin ultrasonic focused beam that needs to be translated step by step to sequentially map the imaged area (see ﬁg. 3). Such a conventional echographic image results in more than 128 successive insonications. Taking into account the time of ﬂight of backscattered ultrasound (a 20 cm back

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Fig. 3. – From conventional to ultrafast ultrasonic imaging. In conventional ultrasound, the backscattered echoes of hundreds of focused beams successively ﬁred into the medium are needed to produce one ultrasonic image. In ultrafast imaging, a plane wave is ﬁred and insoniﬁes in a single shot the whole medium and backscattered echoes are processed to produce the ultrasonic image.

and forth propagation requires some 130 μs), a typical frame rate of about 50 images per second can be reached for sequential imaging. On the contrary, in an ultrafast scanner, for each transmit beam, the backscattered echoes coming from a very large region of interest are recorded by an array of some hundreds of piezoelectric transducers and stored into large memories. Then, a fast algorithm transforms several thousand times per second the backscattered echoes into an echographic image. Thanks to the fact that ultrasonic wave speed is known and constant, this operation can be obtained through a numerical time reversal refocusing. In order to track the local displacements induced by shear wave propagation, successive ultrasonic images are compared. This is possible because the ultrasonic images are dominated by the so-called “speckle” noise that originates from the random distribution of weak scatterers (Rayleigh scatterers much smaller than the wavelength) that exist everywhere in tissues. Note that in soft tissue, contrary to optics, ultrasonic backscattering is dominated by single scattering process thus ensuring an unambiguous correspondence between the arrival time of the speckle noise and the spatial location of the scatterers distribution. By cross-correlating in the time domain the speckle noise observed from one frame to the other, a motion speckle tracking algorithm enables to reconstruct a complete movie of the tissue displacements

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Fig. 4. – Principle of elasticity imaging: In step 1, the ultrasonic probe generates a pushing force in the focal area of the ultrasonic beam. In step 2, this radiation force generates a low-frequency shear wave. The ultrasonic array switches into an ultrafast imaging mode. In step 3, the resulting ultrasonic images are built and stored into memories. In step 4, successive ultrasonic images are compared using cross-correlation operations in order to image tissue displacement induced by the propagation of the shear wave.

ﬁeld along the ultrasonic beam direction (ﬁg. 4). From this movie, one can locally deduce the shear wave speed and thus the shear modulus μ (ﬁg. 5). How can we generate shear waves in the human body? Such shear waves already exist naturally in our body. Each heart beat creates transient vibrations that propagate near the cardiac muscles and along arteries. Our vocal cords are also producing shear vibrations into nearby organs during speech. These natural waves usually remain conﬁned closed to their source and cannot be used to assess all organs. External vibrators applied at the surface of our body can also produce controlled vibrations that induce shear waves propagating from the surface to deeper regions. Finally, instead of using heavy external vibrators, the most elegant way to create a controlled shear wave source consists in using the radiation force induced by ultrasonic focused beams into tissues. Indeed, by focusing an ultrasonic beam at a given location in the organ, it is possible to create a volumic radiation force localized in the focal spot and oriented along the beam axis. This force is due to the momentum

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Fig. 5. – Generation of a supersonic shear source. a) Bursts of ultrasound are focused at successive depths in the organ. Each burst creates a “pushing” radiation force at focus that induces a shear wave. As the “pushing” force is moved faster than the shear waves it generates, a supersonic regime is reached and shear waves accumulate on a Mach cone. b) Images (40 × 40 mm2 ) of micrometric displacements induced in a tissue mimicking phantom obtained using the ultrasonic ultrafast imaging mode at diﬀerent time steps.

transfer from the ultrasonic wave to the medium caused by nonlinearities, dissipation and reﬂection and is proportional to the square of the ultrasonic pressure ﬁeld. Thus, the time proﬁle of the applied force is linked to the beam intensity spatiotemporal proﬁle. The transmission of a 1 ms burst of focused ultrasound with a 5 MHz carrier frequency will lead to an axial force in the kHz range. While remaining below the FDA limitations, the radiation force of the ultrasonic beam permits to remotely create low-frequency shear displacements of some tens of microns at several centimeters depths. Thus, it is possible to generate the shear wave using the same array of piezoelectric transducers that is used in ultrafast imaging. The use of the ultrasonic radiation force as a remote generator of shear waves was proposed by Sarvazyan et al. in 1998 [17] and is used by several research groups in medical imaging [18-21]. The transducers are used in a ﬁrst step to transmit a long burst of ultrasound focused at the desired location in the imaged area (ﬁg. 4, panel 1). The beam generates a low frequency pushing force at focus. When the transmission (and consequently the pushing force) ends, tissues displaced in the focal spot come back to their equilibrium position while generating a small localized source of transient shear waves. It is the biomedical analog of a small earthquake created by the shear force of a moving tectonic plate. The resulting shear wave begins to propagate in

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the organ whereas the ultrasonic probe instantaneously switches into an ultrafast imaging sequence in order to ﬁlm this propagation (ﬁg. 4, panel 2). The amplitude of the radiated shear wave decreases quite rapidly due to the natural divergence associated to a small shear source. In order to extend the area sensed by the shear wave, we proposed an original solution enabling the generation of weakly diﬀracting shear waves based on the remote creation of a supersonic source radiating shear waves along a Mach cone [18]. This eﬀect is the analog of the Cherenkov electromagnetic radiation emitted by a beam of high-energy charged particles passing through a transparent medium at a speed greater than the speed of light in that medium [19]. In our conﬁguration, Ultrasonic waves are successively focused at diﬀerent focal depths by changing the electronic delays between the signals transmitted by the transducer elements. By moving the resulting shear source faster than the radiated shear waves, one can induce constructive interference along a Mach cone (ﬁg. 5). Such a sonic boom is very eﬃcient to create a shear wave of higher displacement that can travel with minimized diﬀraction in a large area. Compared to the use of external vibrators [22, 23], this technique enables to optimally polarize the shear displacement ﬁeld along the ultrasonic beam axis that corresponds to the most sensitive axis of the speckle tracking algorithm. From the 2D experimental movie of the shear displacements along the ultrasonic beam axis (Oz) uz (x, z, t) induced by the supersonic push, one can have access to the shear modulus map by solving locally the inverse problem of wave propagation. Indeed, the ∂2 elastodynamic wave equation characterizing the shear wave propagation {Δu = μρ ∂t 2 u} can be inverted for each pixel (x, z) in the imaging plane and gives access to the local shear modulus μ(x, z) [18] ∂ 2 uz (x, z, t) ∂t2 , μ(x, z) = 2 ∂ uz (x, z, t) ∂ 2 uz (x, z, t) + ∂x2 ∂z 2 ρ

(3)

where ρ is the medium density (almost constant in soft tissues). Surprisingly, this very simple inversion approach (based on the calculation of second-order derivatives of the displacement ﬁeld both in time and space) reaches very good performances. It is mainly due to the fact that the wave ﬁeld uz (x, z, t) is experimentally measured everywhere in the region of interest contrary to conventional imaging approaches where the ﬁeld is known only at boundaries and so requires complex inverse problem approaches. An experiment conducted in a tissue mimicking phantom (made of Agar gelatin) illustrates the interesting features of this technique. A conventional ultrasonic image of this phantom reveals an almost perfectly homogeneous medium in terms of bulk modulus (ﬁg. 6A) although the phantom contains a centimetric stiﬀer inclusion. After generation of the shear Mach cone, high frame rate images of tissue displacements induced in the phantom are recorded during the resulting shear wave propagation. The shear wave is clearly sensitive to the shear modulus contrast as it strongly accelerates while passing

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Fig. 6. – Supersonic shear imaging in tissue mimicking phantoms. A) Images of local tissue displacements (gray scale ranging from −10 μm to +10 μm) at diﬀerent time steps after the supersonic shear source generation. One clearly sees that the shear wave is sensing the stiﬀness contrast as it is distorted while passing through a 10 mm stiﬀ inclusion. B) On the contrary, the ultrasonic image of the medium does not reveal the inclusion, and C) quantitative image of Young’s modulus deduced from the shear wave movie (A).

through the stiﬀer region (ﬁg. 6B). Then a local estimation of the shear wave speed based on basic time-of-ﬂight estimation enables to quantitatively map Young’s modulus (ﬁg. 6C). Young’s modulus image reveals a highly contrasted spherical inclusion twice harder than the surrounding tissues. . 5 1. Super-resolution in supersonic shear wave imaging. – Interestingly, one can notice in ﬁg. 6C that the resolution of the elasticity map is of the order of the millimeter, despite the fact that the shear wavelength is of the order of a centimeter (as seen in

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ﬁg. 6A). This super-resolution capability of SSI can be explained by the fact that the local displacements induced by the wave propagation are not only recorded on the boundaries of the investigated medium (as, for example, in seismology) but also at every location deep into the investigated medium. The estimation of a local displacement ﬁeld in the ultrasonic images provides as many virtual motion sensors (accelerometers) deep into tissues and this even in the far ﬁeld of the shear wave source. Such ultrasound-based virtual motion sensors give a remote access to the local near ﬁeld of the wave ﬁeld around each obstacle even if these obstacles are located in the far ﬁeld of the two waves. The evanescent waves created by the interaction of the incident shear wave with any shear heterogeneities are recorded in the movie with a spatial sampling of some λ/20 in ﬁg. 6. The inverse algorithm based on the computation of the spatial and time derivative of the ﬁeld at each location (see eq. (3)) extracts this information. Therefore the spatial resolution of the shear modulus image is no more limited by the classical diﬀraction limit of shear wavelength but linked to the much smaller ultrasonic wavelength. We have access to a complete movie of the near ﬁeld. . 5 2. Clinical applications. – A major advantage of this imaging technique, called “Supersonic Shear Imaging” (SSI), is that it can be performed using conventional ultrasonic probes and provides an additional imaging modality on a new generation of ultrasound scanners. It gives spectacular in vivo results as, for example, breast cancer diagnosis [24]. Figure 7 shows two interesting cases obtained on breast cancer diagnosis. The 2D color map of Young’s modulus of a patient breast is superimposed on the conventional ultrasonic image. In ﬁg. 7b, two very small and stiﬀ breast lesions (2 mm diameter invasive ductal carcinomas) are visible on the highly contrasted elasticity image but undetected on the conventional ultrasonic image. Figure 7a corresponds to a very stiﬀ invasive ductal carcinoma. This example is particularly interesting as it shows that the elasticity information provides a new information clearly diﬀerent from the conventional ultrasonic image. The invasive ductal carcinoma presents very stiﬀ boundaries (red color regions) and a very soft core (blue region) corresponding to a necrotic region. These results were conﬁrmed by histology and emphasize the ability of SSI to provide new insights into the radiological characterization of cancer lesions. . 5 3. Shear wave spectroscopy. – Thanks to its ability to image very fast and transient motion of shear waves, SSI can provide even much reﬁned information about the mechanical properties of tissue than just a single estimation of Young’s modulus. Indeed, if soft tissues are considered as a purely elastic medium, the shear wave speed cs is directly linked to shear modulus μ via cs = μ/ρ. As cs does not vary with frequency, the time proﬁle of the shear wave generated by the supersonic source is considered to be undistorted during propagation. This approximation of a purely elastic medium leads to the stiﬀness image provided in SSI by the estimation of the group speed of the shear wave. However, in many organs tissues exhibit shear viscosity and signal processing of the shear wave propagation movie can be reﬁned to study this more complex biomechanical behaviour. Viscosity aﬀects the shear wave propagation speed. The time proﬁle of

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Fig. 7. – In vivo clinical application of the Supersonic Shear Imaging (SSI) technique for breast cancer diagnosis. Case (a) corresponds to an inﬁltrating ductal carcinoma (grade II) and shows the ability of SSI to provide a biomechanical characterization of lesions. The superimposed Young’s modulus image shows a lesion much stiﬀer than the surrounding tissue and a very soft core corresponding to a necrosed area. This result was conﬁrmed by histology. Case (b) corresponds to two very small inﬁltrating ductal carcinomas (Grade I &RH+). The millimetric resolution of Young’s modulus image enables to image these lesions which were not detected by X-ray mammography. (Courtesy 5of Supersonic Imagine, France.)

the plane shear wave is progressively distorted and attenuated during propagation. This distortion is characterized by a frequency dependence of the shear wave speed and attenuation that fully describes the rheological behaviour of tissue. The estimation of the shear wave speed in the Fourier domain enables to provide the dispersion curve of the shear wave phase speed cs (ω). As the radiation force of the ultrasonic burst (typically 100 μs burst) induces a transient excitation, the frequency bandwidth of the generated shear wave is quite large, typically ranging from 100 to 1000 Hz. Such wide-band “shear wave spectroscopy” [25] gives a much reﬁned analysis of the complex mechanical behaviour of tissue. Figure 8 shows an example of this analysis in transcostal liver imaging. Superimposed on the echographic image, a color map presents the quantitative Young’s modulus image of liver (assuming a purely elastic medium). As it requires a better signal to noise

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Fig. 8. – In vivo Young’s modulus image of a healthy volunteer liver using an ultrasonic probe placed in the intercostal space. Superimposed on the ultrasonic image, Young’s modulus image exhibits a stiﬀ region corresponding to the intercostal muscle and a soft homogeneous region corresponding to the liver. Shear wave spectroscopy performed in the two boxes shows a strong dispersion of shear waves in liver (high shear viscosity) and a very low dispersion in the muscle (low shear viscosity).

ratio, shear wave spectroscopy is performed in a larger box (5 × 5 mm2 ) and provides the full dispersion curve cs (ω). As a comparison, the dispersion curve is also estimated in the intercostal muscle (almost purely elastic along the muscle ﬁbers) and exhibits almost no frequency dependence in that region. The slope of the dispersion curve is linked to shear viscosity and many diﬀerent rheological models can then be chosen to describe this complex behaviour with a few determined parameters. Recent works in our lab focus on demonstrating that such shear viscosity at the macroscopic level (some millimeters) is related to the underlying architecture of tissues network at a much smaller scale. In particular, the combined estimation of stiﬀness and shear viscosity could provide a reﬁned estimation of liver ﬁbrosis/steatosis degree in cyrrhotic or Hepatitis C patients. In parallel to the advent of the SSI technology, the quest for stiﬀness imaging led to extensive research eﬀorts in the medical imaging community during the last twenty years.

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The concept of stiﬀness imaging was introduced in the early 90s’ by J. Ophir et al. and named Elastography [26]. Their technique is based on the ultrasonic imaging of tissue deformations induced by a quasi-static compression of organs applied by the operator at the surface of the body. Tissue deformations are obtained by acquiring two precompression and post-compression ultrasonic images of the organ using a conventional ultrasound scanner. So, static elastography is inherently a single-wave approach (based on the single use of ultrasonic waves) that implies some important drawbacks. The comparison of the two images enables only the mapping of the local tissue strain. This strain image, called elastogram, is linked to stiﬀness as soft regions tend to exhibit a higher strain than stiﬀer areas. However, even for a simple one-dimensional model, the underlying link between local strain ζ and stiﬀness E (Young’s Modulus in kPa) is strongly dependent on the local and unknown stress τ via the well-known relation E = τ /ζ. Unfortunately, applying a quasi-static compression at the surface of the body can create a very complex spatial distribution of stress that both prevents the assessment of local stiﬀness and induces image artifacts. Another elasticity imaging technique based on the imaging of shear wave propagation using a MRI system (MR-Elastography) was introduced by Mayo Clinic in 1995 [27]. The shear wave is generated by vibrators located at the surface of the body and a dedicated MR sequence enables the local 3D displacements of tissues during shear wave propagation. Thanks to its MultiWave imaging nature, MRE succeeds in providing 3D quantitative images of Young’s modulus with a millimetric resolution. This approach is nevertheless limited in clinical routine by huge acquisition times (typically tens of minutes). 6. – Conclusion MultiWave imaging is a very general concept that can be extended to geophysics as well as non-destructive testing and remote sensing. The future of MultiWave imaging for clinical and biomedical applications is bright. Parallel to the advent of molecular imaging that requires the injection of clinically approved biomarkers, this general wave physics concept is today a fertile source of new ideas and technologies. Many MultiWave imaging approaches can be performed as stand-alone techniques but could also be coupled to molecular imaging strategies. Among all these smart combinations of waves, the wave-to-wave imaging approach is quite exciting, as it is the only one to provide a quantitative assessment of the desired contrast. The ﬁrst MultiWave imaging application that encounters today clinical success is elasticity imaging. It can be used for breast cancer diagnosis, cardiovascular applications, abdominal or musculo-skeletal imaging and will be soon applied to ophthalmology and dermatology. 2D ultrasonic arrays will also facilitate new clinical applications such as image guidance for minimally invasive surgery or the monitoring of thermal ablation treatments. MultiWave imaging also introduces new scientiﬁc concepts transcending the conventional limits of wave physics, such as (in the case of supersonic shear wave imaging) providing a way to observe the near ﬁeld of waves around heterogeneities from far-ﬁeld detectors. On the technology side, the ﬁrst clinical ultrafast ultrasonic scanner provides

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for the ﬁrst time a way to image in real time many natural transient vibrations occurring (permanently) in our body. This is perhaps the most fascinating kind of MultiWave imaging where one of the waves naturally exists in living organs such as pulsatility vibrations induced in the heart, along arteries or mechanical vibrations induced in nerves, neurons and muscle ﬁbers by electric actions potentials. Appendix A. Waves propagation in tissues Electromagnetic and mechanical waves can propagate through tissues with very different regimes, depending mainly on their frequencies. . A 1. Electromagnetic waves . A 1.1. Low frequency. In Electrical Impedance Tomography (EIT), an image of the conductivity or dielectric permittivity of tissue is inferred from surface electrical measurements. Conducting electrodes are attached to the skin and small alternating currents are applied to some electrodes. The resulting electrical potentials are measured and an inverse problem algorithm allows mapping conductivity and permittivity. In this technique, one generates electromagnetic waves within frequency range of kHz up to few MHz (the so called β dispersion region) but also sometimes as high as 10 MHz and as low as 100 Hz. At those frequencies, the wavelength is very large compared (> 100 m) to the organ dimension. We are in the near-ﬁeld regime and the electromagnetic ﬁeld is dominated by the frequency-dependent impedance of tissues. Biological impedance is characterized by a complex-valued conductivity σ ∗ = σ − iωε that incorporates both the conduction current and the displacement current. The dielectric permittivity ε (farads/meter) results from capacitive energy storage due to the bound (dipolar) charge at cellular membranes, while the conductivity σ (siemens/meters) is dominated by ionic transfer in tissue and by the amount of intra- and extra-cellular ﬂuids. The injected current a low frequency will mainly ﬂow through extracellular water; while at high frequencies the current will ﬂow through both extra- and intra-cellular water. Note that at low frequency the imaginary part of the complex conductivity is small and very precise measurements of the phase are needed to map not only the conductivity but also the permittivity . A 1.2. Microwaves. Microwaves (500 MHz to 20 GHz) can also propagate deeply through tissues. However, compared to LF electromagnetic waves, they are strongly attenuated due to electrical conductivity that induces dissipative eﬀects and heat deposition, while the dielectric permittivity aﬀects the wavelength of the microwave. Typically, around 1 GHz the average dielectric permittivity of normal breast tissue is 12.5, resulting in a wavelength close to 10 cm, and the average ionic conductivity is 0.21 S m2 /mol. These are values quite similar to the one of low-water content fat. In contrast, for some breast cancer carcinoma the average dielectric permittivity was found to be close to 55, while the average conductivity increases to 1 S m2 /mol. This has been attributed to an increase in bound water and sodium in malignant cells. Various radar-based (backscatter response to ultra wide-band radar) and tomographic methods have been recently proposed for breast imaging but they still suﬀer of a lack of spatial resolution.

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. A 1.3. Optical waves. The use of light and near infrared (NIR) wavelength is motivated by its relative low absorption in the so-called optical “therapeutic or diagnostic window” (650 to 1300 nm) and by the existence of optical contrast between healthy and malignant tissues in this spectrum. However in opaque tissue light loses very rapidly its coherence when propagating through tissues heterogeneities such as collagen ﬁbers, cell membrane, organelles, extracellular matrix, etc. Therefore, both multiple scattering between individual heterogeneities and absorption are responsible for the light beam broadening and decay as it travels through tissue. In this regime, light scattering is characterized by two length parameters, i.e. the scattering length ls and the light transport mean free path ls ∗ . The scattering length ls characterizes the memory of the optical phase and corresponds to the average distance that separates two scattering events. The light transport mean free path ls ∗ characterises the memory of the light propagation direction. In tissues ls is typically 50 to 100 μm, while ls ∗ is ten time larger (0.5 to 1 mm). Absorption of light is also characterized by another length: the absorption length la which is in the 1 cm to 10 cm range. Absorption strongly depends on the nature of tissue and is the main optical contrast that one tries to map. X rays and Γ rays are also electromagnetic waves that propagate in tissue but they can be interpreted as rays following geometrical optics approximation. They will not be discussed here . A 2. Mechanical waves. – In a ﬁrst approximation, soft tissues can be considered as an isotropic elastic medium. Therefore two kinds of mechanical waves can propagate in such medium: the well-known P and S waves of seismology. P waves are compressional waves traveling at a speed cp that depends mainly on the compressional bulk modulus K (inverse of the compressibility κ = 1/K) and shear waves (S waves) that propagate at a slower speed cs aﬀected by the shear modulus μ. Soft human tissues are mainly characterized by the fact that K is huge in comparison with μ. K is almost uniform (less than 5% ﬂuctuations) in all soft tissues with typical values of ∼ 1 GPa. Moreover, K does not change signiﬁcantly with pathology as it is deﬁned mainly by molecular composition of tissue and short-range molecular interactions. Since most soft tissues are by about 80% made of water, K is not much diﬀerent from the water bulk modulus. On the contrary, μ is very small with typical values of some kPa and it strongly varies from one organ to the other. μ is deﬁned by cellular and higher levels of structural organization of tissue and consequently is greatly aﬀected by pathological or physiological changes in tissue structure. For example, an inﬁltrating carcinoma can be characterized by a shear modulus more than 100 times higher than surrounding healthy tissues. As the P wave speed cp is given by cp = (K + (2/3)μ)/ρ ≈ K/ρ and the S wave speed cs by μ/ρ, we may notice that compressionnal waves propagate around 1500 m/s corresponding to the sound speed in water, while shear waves propagate at a much smaller speed with speed values ranging from 1 m/s to 10 m/s in soft human tissues. Taking into account the fact that tissues are not only elastic media but also viscoelastic media, one has to introduce viscous eﬀect (or rheological eﬀect) in wave propagation. The main eﬀect is that at ultrasonic frequency shear waves are so strongly attenuated by shear viscosity eﬀects that they cannot be used to penetrate inside tissues. However compressionnal waves are not aﬀected by shear viscosity but by bulk viscosity, which is much weaker and depends on various mechanical relaxation processes. Typically compressional wave attenuation enables to use frequencies ranging between 2 and 50 MHz corresponding to imaging depth ranging, respectively, between 12 cm and 2 mm and associated to wavelengths ranging from 0.8 mm to 30 μm. In this range of frequencies soft tissues be-

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have more like a ﬂuid with a frequency linear attenuation (typically 0.1 dB/cm/MHz). This explains the success of ultrasound imaging in terms of spatial resolution for in-depth imaging. However, the contrast of ultrasound images remains poor as it is related only to the weak heterogeneities of the bulk modulus. Due to weak heterogeneities, ultrasonic compressionnal waves can propagate on long distance without losing their coherence, resulting in a mean free path ls much larger than the observation distance. This means that in pulse echo-mode the backscattering echoes resulting from the distribution of heterogeneities are obtained only from single scattering processes, which greatly simpliﬁed the imaging modality, compared to diﬀuse optical tomography. Moreover the fact that the ultrasonic speed is practically constant and known in all tissues allows very simple processing to recover an image. To observe shear wave in tissues, we have to work at low sonic frequencies (less than 1 kHz). Thus shear attenuation is low and shear waves may propagate deeply in tissues to reveal shear properties. For example, at 100 Hz, shear waves can propagate in glandular breast tissue on nearly 10 cm, with a wavelength of 2 cm, corresponding to a shear speed of 2 m/s. REFERENCES [1] [2] [3] [4] [5] [6] [7]

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Yodh A. and Chance B., Phys. Today, 48 (1995) 34. Cherry S. R., Annu. Rev. Biomed. Eng., 8 (2006) 35. Emelianov S. Y., Li P. C. and O’Donnell M., Phys. Today, 62(8) (2009) 34. Wang X., Pang Y., Ku G., Xie X., Stoica G. and Wang L. H. V., Nature Biotechnol., 21 (2003) 803. Towe B. C. and Islam M. R., IEEE Trans. Biomed. Eng., 35 (1988) 892. Li X., Xu Y. and He B., J. Appl. Phys., 99 (2006) 066112. Marks F. A., Tomlinson H. W. and Brooksby G. W., Photon Migration and Imaging in Random Media and Tissues, Proc. SPIE, edited by Chance B. and Alfano R. R., Vol. 1888, (1993) 500. Wang L.-H. V., Jacques S. L. and Zhao X.-M., Opt. Lett., 20 (1995) 629. Leutz W. and Maret G., Physica B, 204 (1995) 14. Leveque S., Boccara A. C., Lebec M. and Saint-Jalmes H., Opt. Lett., 24 (1999) 181. Wang L.-H. V and Ku G., Opt. Lett., 23 (1998) 975. Gross M., Lesa M., Ramaz F., Delaye P., Roosen G. and Boccara A. C., Eur. Phys. J. E, 28 (2009) 173. Sarvazyan A. P., Acoustical Imaging, Vol. 21 (Plenum Press, New York) 1995, pp. 223– 240. Sandrin L., Tanter M., Catheline S. and Fink M., IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 49 (2002) 426. Fink M., Phys. Today, 20 (1997) 34. Fink M., Montaldo G. and Tanter M., Annu. Rev. Biomed. Eng., 5 (2003) 465. Sarvazyan A. P., Rudenko O. V., Swanson S. D., Fowlkes J. B. and Emelianov S. Y., Ultra. Med. Biol., 20 (1998) 1419. Bercoff J., Tanter M. and Fink M., IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 51 (2004) 374. Bercoff J., Tanter M. and Fink M., Appl. Phys. Lett., 84 (2004) 2202. Nightingale K. R., Soo M. S., Nightingale R. W. and Trahey G. E., Ultra. Med. Biol., 28 (2002) 227.

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[21] Fatemi M. and Greenleaf J. F., Science, 280 (1998) 82. [22] Sandrin L., Tanter M., Gennisson J. L., Catheline S. and Fink M., IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 49 (2002) 436. [23] Catheline S., Thomas J.-L., Wu F. and Fink M., IEEE Trans. Ultrason., Ferroelec., Freq. Contr., 46 (1999) 1013. [24] Tanter M., Bercoff J., Athanasiou A., Deffieux T., Gennisson J.-L., Montaldo G., Muller M., Tardivon A. and Fink M., Ultra. Med. Biol., 34 (2008) 1373. [25] Deffieux T., Montaldo G., Tanter M. and Fink M., IEEE Trans. Med. Imag., 28 (2009) 313. ´spedes I., Ponnekanti H., Yasdi Y. and Li X., Ultrasonic Imaging, 13 [26] Ophir J., Ce (1991) 111. [27] Muthupillari R., Lomas D. J., Rossman P. J., Greenleaf J. F., Manduca A. and Ehman R. L., Science, 269 (1995) 1854.

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-155

Time reversal focusing and the diﬀraction limit M. Fink, J. de Rosny, G. Lerosey and A. Tourin Institut Langevin, Ecole Sup´erieure de Physique et de Chimie Industrielle de la Ville de Paris UMR CNRS 7587 - 10 Rue Vauquelin, 75005 Paris, France

Summary. — Time reversal mirrors refocus an incident-wave ﬁeld to the position of the original source, regardless of the complexity of the propagation medium. TRMs have now been implemented in a variety of physical scenarios from GHz Microwaves to MHz Ultrasonics and to hundreds of Hz in ocean acoustics. Common to this broad range of scales is a remarkable robustness exempliﬁed by observations at all scales that the more complex the medium (random or chaotic), the sharper the focus. A TRM acts as an antenna that uses complex environments to appear wider than it is, resulting, for a broad-band pulse, in a refocusing quality that does not depend on the TRM aperture. Moreover, when the complex environment is located in the near ﬁeld of the source, time reversal focusing opens completely new approaches to super-resolution. We will shown that, for a broad-band source located inside a random metamaterial, a TRM located in the far ﬁeld radiates a time-reversed wave that interacts with the random medium to regenerate not only the propagating but also the evanescent waves required to refocus below the diﬀraction limit.

1. – Introduction Time reversal invariance of the wave equation in acoustics and electromagnetism allows to build time reversal mirrors (TRMs) made of arrays of reversible antenna, allowing an incident broad-band wave ﬁeld to be sampled, recorded, time-reversed and re-emitted. c Societ` a Italiana di Fisica

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Fig. 1. – (a) Recording step: A closed surface is ﬁlled with transducer elements. A point-like source generates a wave front which is distorted by heterogeneities. The distorted pressure ﬁeld is recorded on the cavity elements. (b) Time-reversed or reconstruction step: The recorded signals are time-reversed and re-emitted by the cavity elements. The time-reversed pressure ﬁeld backpropagates and refocuses exactly on the initial source.

TRMs refocus the incident-wave ﬁeld to the position of the original source regardless of the complexity of the propagation medium. The ﬁrst TRMs have been developed in the ﬁeld of Acoustics [1-3]. An acoustic source, located inside a lossless medium, radiates a brief transient pulse that propagates and is potentially distorted by the medium. Time reversal of the acoustic ﬁeld would entail the reversal, at some instant, of every particle velocity in the medium. This kind of instantaneous time reversal in the whole volume is practically impossible to achieve. A more realistic alternative can be developed thanks to the Helmoltz-Kirchoﬀ integral theorem. The acoustic ﬁeld radiated by a source could be measured on every point of an enclosing surface (acoustic retina), and retransmitted in time-reversed order, then the wave will travel back to its source, see ﬁg. 1. Both time reversal invariance and spatial reciprocity [4] are required to reconstruct a time-reversed wave in the entire volume by means of this two-dimensional time reversal operation. From an experimental point of view a closed TRM consists of a twodimensional piezoelectric transducer array that samples the wave ﬁeld over a closed surface. An array pitch of the order of λ/2, where λ is the smallest wavelength of the pressure ﬁeld, is needed to ensure the recording of all the information on the wave ﬁeld. Each transducer is connected to its own electronic circuitry that consists of a receiving ampliﬁer, an A/D converter, a storage memory and a programmable transmitter able to synthesize a time-reversed version of the stored signal.

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In practice, closed TRMs are diﬃcult to realize and the TR operation is usually performed on a limited angular area, thus apparently limiting focusing quality. A TRM consists typically of a small number of elements or time-reversal channels. The major interest of TRM, compared to classical focusing devices (lenses and beam forming) is certainly the relation between the medium complexity and the size of the focal spot. A TRM acts as an antenna that uses complex environments to appear wider than it is, resulting in a refocusing quality that does not depend on the TRM aperture. One spectacular result that is shown in this paper deals with complex environment located in the near ﬁeld of the source. Such environment can be made, for example, of random or periodic distribution of resonating scattererers with a mean distance smaller than the wavelength. It will be shown that, for a broad-band source located inside such random metamaterials, a TRM located in the far ﬁeld radiated a time-reversed wave that interacts with the random medium to regenerate not only the propagating but also the evanescent waves required to refocus below the diﬀraction limit. This focusing process is very diﬀerent from the one developed with superlenses made of negative index material only valid for narrow-band signals. We will emphasize the role of the frequency diversity in time reversal focusing. 2. – Basic principles The basic theory employs a scalar wave formulation p(r, t) and, hence, is strictly applicable to acoustic or ultrasound propagations in ﬂuid. However, the basic ingredients and conclusions apply equally well to elastic waves in solid and to electromagnetic ﬁelds. Let us consider the propagation of an acoustic wave in a heterogeneous and nondissipative medium, whose compressibility κ(r) and density ρ(r) vary in space. By introducing the sound speed c(r) = (ρ(r)κ(r))−1/2 , one can obtain the wave propagation equation for a given pressure ﬁeld p(r, t) (1)

· ∇

∇p(r, t) ρ(r)

−

1 ∂ 2 p(r, t) = 0. ρ(r)c(r)2 ∂t2

One can notice the particular behaviour of this wave equation regarding the time variable t. Indeed, it only contains a second-order time derivative operator. This property is the starting point of the time reversal principle. A straightforward consequence of this property is that if p(r, t) is a solution of the wave equation, then p(r, −t) is also solution of the problem. This property illustrates the invariance of the wave equation during a time reversal operation, the so-called time reversal invariance. However, this property is only valid in a non-dissipative medium. If wave propagation is aﬀected by dissipation eﬀects, odd order time derivatives appear in the wave equation and the time reversal invariance is lost. Nevertheless, one should here note that if the ultrasonic absorption coeﬃcient is suﬃciently small in the frequency bandwidth of the ultrasonic waves used for the experiments, the time reversal invariance remains valid.

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In any propagation experiment, the acoustic sources and the boundary conditions determine a unique solution p(r, t) in the ﬂuid. The goal, in time reversal experiments, is to modify the initial conditions in order to generate the dual solution p(r, T − t), where T is a delay due to causality requirements. Cassereau and Fink [4] and Jackson and Dowling [5] have studied theoretically the conditions necessary to ensure the generation of p(r, T − t) in the entire volume of interest. . 2 1. An ideal time reversal experiment. – Although reversible acoustic retinas usually consist of discrete elements, it is convenient to examine the behavior of idealized continuous retinas, deﬁned by two-dimensional surfaces. In the case of a time reversal cavity, we assume that the retina completely surrounds the source. In a ﬁrst step, let us consider a point-like source located at r0 inside a volume V surrounded by a surface S, emitting a time modulation s(t). The inhomogeneous wave equation is given by (2)

· ∇

∇p(r, t) ρ

−

1 ∂ 2 p(r, t) = −δ(r − r0 )s(t). ρc2 ∂t2

Note that contrary to eq. (1), the right part of this equation describes the source term and this term may contain spatial and time singularities. Considering, for example, an impulsive source s(t) = δ(t) at time 0, the causal solution to eq. (2) reduces to the retarded Green’s function Gret (r, r0 ; t) that takes into account the heterogeneities and the boundaries of the medium. Note that, to respect causality, only the causal Green’s function (retarded) that satisﬁes the Sommerfeld radiation boundary condition at inﬁnity is selected, while the advanced Green’s function (the anti-causal) is neglected. The initial goal of a perfect time-reversed experiment is to generate in the medium this advanced Green’s function Gadv (r, r0 ; t) = Gret (r, r0 ; −t) by modifying the initial conditions on the boundaries of the experiment. This would be an optimal way to obtain super-resolution in a focusing experiment, because the advanced Green’s function converges towards a spatial singularity. In a more realistic way, taking into account any source modulation s(t) with a well-deﬁned bandwidth, we are interested in generating the volume of the experiment p(r, −t). The so-called time reversal cavity approach was developed, by using the fact that a wave ﬁeld at any location inside a volume V (without source) can be predicted from the knowledge of both the ﬁeld and its normal derivative on the surrounding surface S. Therefore a time reversal experiment can be conceived in the following way: During the second step of the time reversal process, the initial source at r0 is removed and we create on the surface of the cavity monopole and dipole sources that correspond to the time reversal of those same components measured during the ﬁrst step. The time reversal operation is described by the transform t → −t and the secondary sources are (3)

" ps (r, t) = G(r, r0 ; −t) ⊗ s(−t), ∂n ps (r, t) = ∂n G(r, r0 ; −t) ⊗ s(−t),

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159

where we use now and in the following the notation G(r, r0 ; t) instead of Gret (r, r0 ; t) and neglect the causal delay T needed to record and re-emit the signals. Due to these secondary sources on S, the time-reversed pressure ﬁeld ptr (r, t) propagates backward inside the cavity. It can be calculated using a modiﬁed version of the Helmoltz-Kirchhoﬀ integral, valid inside a zone without source (4) ptr (r, t) =

+∞ −∞

dt

[G(r, r ; t − t )∂n ps (r , t )−ps (r , t )∂n G(r, r ; t − t )]

S

d2 r . ρ(r )

Instead of directly computing this integral, there is a straightforward way to predict the ﬁeld ptr (r, t). Our initial goal was to radiate inside the volume surrounded by surface S, the ﬁeld p(r, −t) = G(r, r0 ; −t) ⊗ s(−t), with G(r, r0 ; −t) the advanced Green’s function. However, the wave equation veriﬁed by p(r, −t) in the volume V can be obtained by changing t into −t in eq. (2) (5)

· ∇

∇p(r, −t) ρ

−

1 ∂ 2 p(r, −t) = −δ(r − r0 )s(−t). ρc2 ∂t2

Therefore, to obtain a perfect time reversal ﬁeld would require also that the original active source that injected energy into the system in the initial step be replaced with a sink (the time reversal of a source) that corresponds to the right term of eq. (4). This means that to achieve a perfect time reversal, both the source has to be transformed into a sink, and the ﬁeld and its normal derivative on surface S have also to be time-reversed (like in eq. (4)). The superposition of these two ﬁelds will give exactly p(r, −t). Therefore p(r, −t) is given by the following sum: (6)

p(r, −t) = ptr (r, t) + G(r, r0 ; t) ⊗ s(−t).

For a source term with a Dirac excitation, we directly get for the time-reversed ﬁeld (7)

ptr (r, t) = G(r, r0 ; −t) − G(r, r0 ; t).

This important result is, in some way, disappointing, because it means that reversing an acoustic ﬁeld using a closed TRM is not enough to radiate only the advanced wave ﬁeld. Complete time reversal requires not only to time-reverse the source but the original source as well. Equation (7) can be interpreted as the diﬀerence of advanced and retarded waves centered on the initial source position. The converging wave (advanced) collapses at the origin and is followed by a diverging (retarded) wave. Thus the time-reversed ﬁeld observed as a function of time shows two wave fronts of opposite sign. The wave re-emitted by the time reversal cavity looks like a convergent wave ﬁeld during a given period, but a wave ﬁeld does not know how to stop. When the converging wave ﬁeld reaches the location of the initial source location, it collapses and then continues its propagation as a diverging wave ﬁeld.

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To achieve a perfect time reversal, both the ﬁeld on the surface of the cavity has to be time-reversed, and the source has to be transformed into a sink [6, 7]. In this manner one may achieve time-reversed focusing below the diﬀraction limit. The role of the new source term −δ(r−r0 )s(−t) in eq. (5) is to transmit a diverging wave that exactly cancels the outgoing spherical wave. In a monochromatic approach, taking into account the evanescent waves concept, the necessity of replacing a source with a sink in the complete time-reversed operation can be interpreted as follows. In the ﬁrst step a point-like source of size much smaller than a wavelength radiates a ﬁeld that can be described as a superposition of homogeneous plane waves propagating in the various directions k and of decaying, non-propagating, evanescent plane waves [8]. The evanescent waves contain information on ﬁne-scale features of the source; they decay exponentially with distance and do not contribute in the far ﬁeld. If the TRM is located in the far ﬁeld of the source, the time-reversed ﬁeld retransmitted by the mirror does not contain these evanescent components. The role of the sink is to radiate exactly, with good timing, the evanescent waves that have been lost during the ﬁrst step. The resulting ﬁeld contains the evanescent part that is needed to focus below diﬀraction limits. Time reversal below the diﬀraction limit has been experimentally demonstrated in acoustics, using an acoustic sink placed at the focal point. Focal spots of size λ/14 have been observed by de Rosny et al. [6]. One drawback is the need to use an active source at the focusing point to exactly cancel the usual diverging wave created during the focusing process. . 2 2. Time reversal in free space. – For example, in the case of a homogeneous medium, assuming that the retina does not perturb the ﬁeld propagation (free-space assumption), the free-space retarded Green’s function G0 reduces to a diverging spherical impulse wave that depends only on r − r0 and propagates with sound speed c. Thus, neglecting the causal time delay T , the time-reversed ﬁeld can be written as ) ( |r − r0 | 1 |r − r0 | 1 δ t+ − δ t− ⊗ s(−t), (8) ptr (r, t) ≺ 4π|r − r0 | c 4π|r − r0 | c that reduces to the time derivative of the source modulation at the origin (9)

ptr (r = r0 , t) = −

1 s (−t). 2πc

In the case of a narrow band excitation (monochromatic excitation of pulsation ω), the interference between the converging and the diverging ﬁelds leads to the classical diﬀraction limits. Indeed by calculating the Fourier transform of eq. (7) over the time variable t, we obtain (10)

exp (−jk|r − r0 |) exp (jk|r − r0 |) Pˆtr (r, ω) = − 4π|r − r0 | 4π|r − r0 | sin (k|r − r0 |) ˆ − r0 , ω), = −2j = −2j Im G(r 4π|r − r0 |

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where j 2 = −1. The time-reversed ﬁeld at the initial source position is ﬁnite because it is the diﬀerence between a converging and a diverging wave and not the sum (otherwise it will have a discontinuity there). As a consequence, the time-reversed ﬁeld is focused on the initial source position, with a focal spot size limited to one half-wavelength π/k that corresponds to the standard ˆ − r0 , ω) formulation for the complex ﬁeld modulus where k is the wave number and G(r is the monochromatic Green’s function. The point spread function is proportional to the imaginary part of the monochromatic Green’s function. . 2 3. Time reversal through heterogeneous medium. – In the case of a non-dissipative heterogeneous medium surrounding the source, a similar interpretation can be given, but ˆ r0 ; ω) is no more dependent on r − r0 but is now a the retarded Green’s function G(r, function separately of both r and r0 taking into account, for example multiple scattering process between heterogeneities

2 ˆ ∗ (r , r0 ; ω)G(r, ˆ r ; ω) − G ˆ ∗ (r , r0 ; ω)∂n G(r, ˆ r ; ω) d r ∂n G ρ(r ) S ˆ r0 ; ω). = −2j Im G(r,

(11) Pˆtr (r, ω) =

Note that the ﬁeld amplitude at the focal point is directly proportional to the LDOS, the so-called local density of states that depends on the medium complexity around the source point. In situation where the source is located inside a periodic or random metamaterial with scatterers close to the source, it can happen that the LDOS be zero for some source position and therefore the time-reversed wave will have a node at the source point. However, for broad-band excitation, the resulting ﬁeld takes advantage of the frequency diversity. For a broad-band excitation s(t) the time-reversed ﬁeld is given by (12)

ptr (r, t) = −2j

ˆ r0 ; ω)S ∗ (ω) exp(jωt)dω, Im G(r,

where S(ω) is the Fourier transform of the source modulation. For an excitation with a ﬂat bandwidth Δω, the ﬁeld at the collapse time (t = 0) reads (13)

ˆ r0 ; ω)dω. Im G(r,

ptr (r, t = 0) = −2j Δω

Therefore the time-reversed ﬁeld at the focus (source point) and at the collapse time is given by ˆ 0 , r0 ; ω)dω. (14) ptr (r = r0 , t = 0) = −2j Im G(r Δω

Thus, the time-reversed ﬁeld at the source point and at the focal time is directly proportional to the number of modes excited by the source.

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. 2 4. An experimental point of view . – From an experimental point of view a perfect time reversal mirror verifying eq. (11) is an ideal description. Indeed, it is diﬃcult to measure both the ﬁeld and its normal derivative at any point of the surface S. Ideal experiments would be carried out with two transducer arrays that behave either as monopolar or as dipolar transducers and that spatially sample the receiving and emitting surface. If, however, the time reversal mirror is located in the far ﬁeld of source and observation points and heterogeneities, the expression can be simpliﬁed. In this case we may assume that ˆ r ; ω) ≈ jkG(r, ˆ r ; ω). ∂n G(r,

(15)

Thus, the time-reversed ﬁeld can be written as (16)

ω Pˆtr (r, ω) ≈ 2j ρc

ˆ ∗ (r , r0 ; ω)G(r, ˆ r ; ω)d2 r . G

S

If we come back to the time domain, eq. (11) can be written as (17)

ptr (r, t) ≈

2 ∂ ρc ∂t

G(r , r0 ; −t) ⊗ G(r, r ; t)d2 r .

S

From an experimental point of view, it is not easy to measure and re-emit the ﬁeld at any point of a surface S: experiments are carried out with transducer arrays that spatially sample the receiving and emitting surface. Assuming that the time reversal retina consists of discrete elements located at position ri , this allows to replace the integration over S in eq. (11) with a summation over N surface element positions ∂ G(ri , r0 ; −t) ⊗ G(r, ri ; t), ∂t i=1 N

(18)

ptr (r, t) = C

where C is a scaling factor. Taking into account spatial reciprocity, this expression can be written a summation of cross-correlation functions ∂ G(r0 , ri ; −t) ⊗ G(r, ri ; t). ∂t i=1 N

(19)

ptr (r, t) = C

The spatial sampling of surface S by a set of elements may introduce grating lobes. These lobes can be avoided by using an array pitch smaller than λmin /2, where λmin is the smallest wavelength of interest. 3. – Time reversal in complex media It is generally diﬃcult to use acoustic arrays that completely surround the area of interest, so the closed cavity is usually replaced with a TRM of ﬁnite angular aperture.

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163

Fig. 2. – One part of the transducers is replaced with reﬂecting boundaries. In the ﬁrst step (receive mode) the wave radiated by the source is recorded by a set of transducers through the reverberation inside the cavity. In the second step, the recorded signals are time-reversed and re-emitted by the transucers.

This yields an increase of the point spread function that is related to the limited angular size of the mirror observed from the source. In the standard theory of diﬀraction in homogeneous free space, the point spread function is related to the angular spectrum of the aperture. For a closed time reversal mirror, the k vectors of the radiated ﬁeld span the whole 4π solid angle and the focal spot dimension is minimal (λ/2). When a TRM covers a limited solid angle, the spatial diversity of k vectors that interact with the TRM is reduced. Therefore the focal spot size is increased. The main interest of focusing with TRM is that in media with complex structure the spatial diversity of the k vectors captured by a small TRM can be signiﬁcantly increased. Wave propagation in media with complex boundaries or random scattering medium can increase the apparent aperture of the TRM, resulting in a focal spot size smaller than that predicted by classical formulas. The basic idea is to replace one part of the transducers needed to sample a closed time reversal surface with reﬂecting boundaries that redirect one part of the incident wave towards the TRM aperture (see ﬁg. 2). When a source radiates a wave ﬁeld inside a closed cavity or in a waveguide, multiple reﬂections along the medium boundaries can signiﬁcantly increase the apparent aperture of the TRM. Thus spatial information on the k vectors that is usually lost with a ﬁnite aperture TRM is converted into the time domain. The reversal quality then depends crucially on the duration of the time-reversal window, i.e., the length of the recording that is reversed.

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Such a concept is strongly related to a kaleidoscopic eﬀect that appears thanks to the multiple reverberations on the waveguide boundaries. Waves emitted by each transducer are multiply reﬂected, creating at each reﬂection “virtual” transducers that can be observed from the desired focal point. Thus, we create a large virtual array from a limited number of transducers and a small number of transducers is multiplied to create a “kaleidoscopic” transducer array. Two diﬀerent examples will be presented (a chaotic cavity and a multiply scattering medium). . 3 1. One-channel time reversal in chaotic cavities. – In this section, we are interested in multiply reﬂected waves: waves conﬁned in closed reﬂecting cavities with nonsymmetrical geometry. With closed boundary conditions, no information can escape from the system and a reverberant acoustic ﬁeld is created. If, moreover, the geometry of the cavity shows ergodic and mixing properties, one may hope to collect all information at only one point. Ergodicity means that, due to the boundary geometry, any acoustic ray radiated by a point source and multiply reﬂected would pass every location in the cavity. Therefore, all the information about the source can be redirected towards a single time reversal transducer. This is the regime of fully diﬀuse wave ﬁelds that can be also deﬁned as in room acoustics as an uncorrelated and isotropic mix of plane waves of all propagation directions [9, 10]. Draeger and Fink [11-13] showed experimentally and theoretically that in this particular case a time reversal focusing with λ/2 spot can be obtained using only one TR channel operating in a closed cavity. The ﬁrst experiments were made with elastic waves propagating in a 2D cavity with negligible absorption. They were carried out using guided elastic waves in a monocrystalline D-shaped silicon wafer known to have chaotic ray trajectories. This property eliminates the eﬀective regular gratings of the previous section. Silicon was selected also for its weak absorption. Elastic waves in such a plate are akin to Lamb waves. An aluminum cone coupled to a longitudinal transducer generated waves at one point of the cavity. A second transducer was used as a receiver. The central frequency of the transducers was 1 MHz and their relative bandwidth was 100% (Δω = 1 MHz). At this frequency, only three propagating modes are possible (one ﬂexural, one quasi-extensional, one quasi-shear). The source was considered point-like and isotropic because the cone tip is much smaller than the central wavelength. A heterodyne laser interferometer measures the displacement ﬁeld as a function of time at diﬀerent points on the cavity. Assuming that there is no mode conversion at the boundaries between the ﬂexural mode and other modes, we have only to deal with one ﬁeld, the ﬂexural-scalar ﬁeld. The experiment is a “two-step process” as described above: In the ﬁrst step, one of the transducers, located at point r0 (ﬁg. 3), transmits a short omnidirectional signal of duration 0.5 μs. Another transducer, located at rtrm , observes a long randomlooking signal that results from multiple reﬂections of along the boundaries of the cavity. It continues for more than 50 milliseconds, corresponding to some hundred reﬂections at the boundaries. Then, a portion ΔT of the signal is selected, time-reversed and re-emitted by point rtrm . As the time-reversed wave is a ﬂexural wave that induces vertical displacements of the silicon surface, it can be observed using the optical

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165

Fig. 3. – Time reversal experiment conducted in a chaotic cavity with ﬂexural waves. In a ﬁrst step, a point transducer located at point A transmits a 1 μs long signal. The signal is recorded at point B by a second transducer. The signal spreads by more than 30 ms due to reverberation. In the second step of the experiment, a 1 ms portion of the recorded signal is time-reversed and retransmitted back in the cavity.

interferometer that scans the surface on diﬀerent observation points r around the point r0 (see ﬁg. 3). For time reversal windows of suﬃciently long duration ΔT , one observes both an impressive time recompression at point r0 and a refocusing of the time-reversed wave around the origin (see ﬁgs. 4a and b for ΔT = 1 ms), with a focal spot whose radial dimension is equal to half the wavelength of the ﬂexural wave. Using reﬂections at the boundaries, the time-reversed wave ﬁeld converges towards the origin from all directions and gives a circular spot, like the one that could be obtained with a closed time reversal cavity covered with transducers. A complete study of the dependence of the spatiotemporal side lobes around the origin shows a major result [11-13]: a time duration ΔT of nearly 1 ms is enough to obtain a good focusing. For values of ΔT larger than 1 ms, the sidelobes shape and the signal-to-noise ratio (focal peak/sidelobes) does not improve further. There is a saturation regime. Once the saturation regime is reached, point rtrm will receive redundant information. The saturation regime is reached after a time τHeisenberg called the Heisenberg time. It is the minimum time needed to resolve the eigenmodes in the cavity. It can also be interpreted as the time it takes for all a single ray to reach the vicinity of any point in the cavity within a distance λ/2. This guarantees enough interference between all the multiply reﬂected waves to build each

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Fig. 4. – (a) Time-reversed signal observed at point A. The observed signal is 210 μs long; (b) time-reversed wave ﬁeld observed at diﬀerent times around point A on a square of 15 mm × 15 mm.

of the eigenmodes in the cavity. The mean distance δω between the eigenfrequencies is 1 . related to the Heisenberg time, τHeisenberg = δω The success of this time reversal experiment in closed chaotic cavity is particularly interesting with respect to two aspects. Firstly, it proves the feasibility of acoustic time reversal in cavities of complex geometry that give rise to chaotic ray dynamics. Paradoxically, in the case of one-channel time-reversal, chaotic dynamics is not only harmless but even useful, as it guarantees ergodicity and mixing. Secondly, using a source of vanishing aperture, there is an almost perfect focusing quality. The procedure approaches the performance of a closed TRM, which has an aperture of 360◦ . Hence, a one-point time reversal in a chaotic cavity produces better results than a limited aperture TRM in an open system. Using reﬂections at the edge, focusing quality is not aperture limited; the time-reversed collapsing wave front approaches the focal spot from all directions.

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Time reversal focusing and the diffraction limit

Although one obtains excellent focusing, a one-channel time reversal is not perfect, as a weak noise level throughout the system can be observed. There is a saturation regime beyond the Heisenberg time. Residual temporal and spatial sidelobes persist even for time reversal windows of duration larger than the Heisenberg time. They are due to multiple reﬂections passing over the locations of the TR transducer and they have been expressed in closed form by Draeger and Fink. Using an eigenmode analysis of the wave ﬁeld, they explain that, for long time reversal windows, there is a saturation regime that limits the signal-to-noise ratio (SNR). To evaluate the time-reversed ﬁeld for the elastic wave in the one-channel experiment, we can use eq. (15) with a TRM located at unique point rtrm and the vertical component of the displacement ﬁeld ϕtr (r, t) is given by (note that the time derivative of eq. (19) has disappeared because of the dimensionality of the displacement ﬁeld) (20)

ϕtr (r, t) ≺ G(rtrm , r0 ; −t) ⊗ G(r, rtrm ; t).

Taking into account the modal decomposition of the Green’s functions G(r, rtrm ; t) and G(rtrm , r0 ; −t) on each of the eigenmodes uj (r) of the cavity with eigenfrequency ωj , we get (21)

G(r, rtrm ; t) =

j

uj (r)uj (rtrm )

sin(ωj t) ωj

(t > 0).

Under the assumption that the eigenmodes are not degenerated (valid for a chaotic cavity), we calculate ϕtr (r, t) for a time window of duration longer than the Heizenberg time of the cavity and we get (22)

ϕtr (r, t) ∝

1 2 2 ui (r)ui (r0 )ui (rtrm ) cos(ωi t). ω i i

Note that at the focal time t = 0 (collapse) the directivity pattern of the time-reversed wave ﬁeld is (23)

ϕtr (r, t) ≺

1 2 2 ui (r)ui (r0 )ui (rtrm ). ω i i

Note that in a real experiment one has to take into account the limited bandwidth of the source, so a spectral function S(ω) centered on center frequency ωc , with band width Δω, must be introduced and we can write eq. (23) in the form (24)

ϕtr (r, 0) =

1 2 2 ui (r)ui (r0 )ui (rtrm )S(ωi ). ω i i

Thus the summation is limited to a ﬁnite number of modes, which is typical in our experiment of the order of some hundreds. As we do not know the exact eigenmode

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distribution for each chaotic cavity, we cannot evaluate this expression directly. However, due to the ergodic properties of the cavity, one may use a statistical approach and consider the average over diﬀerent realizations, which consist in summing over diﬀerent cavity realizations. So we replace in eq. (24) the eigenmodes product with their expectation values . . .. We use also a qualitative argument proposed by Berry [14,15] to characterize irregular modes in chaotic system. If chaotic rays support an irregular mode, it can be considered as a superposition of a large number of plane waves with random direction and phase. This implies that the amplitude of an eigenmode has a Gaussian distribution with u2i = σ 2 and a short-range isotropic correlation function given by a Bessel function that reads (25)

ui (r)ui (r0 ) = J0 (2π|r − r0 |/λi ),

with λi the wavelength corresponding to ωi . If r and r0 are suﬃciently far apart from rtrm not to be correlated, then (26)

! ! ui (r)ui (r0 )u2i (rtrm ) = ui (r)ui (r0 ) u2i (rtrm ) .

One obtains ﬁnally (27)

ϕtr (r, 0) =

1 2 2 J0 (2π|r − r0 |/λi )σ F (ωi ). ω i i

The experimental results obtained in ﬁg. 4b agree with this prediction and show that in a chaotic cavity the spatial resolution is independent of the time reversal mirror aperture. Indeed, with a one-channel time reversal mirror, the directivity pattern at t = 0 is close to the Bessel function J0 (2π|r − r0 |/λc ) corresponding to the central frequency of the transducers. This means that the one-channel time-reversed ﬁeld is a good estimate of the imaginary part of the Green’s function (see eq. (14)) that was predicted for a closed time reversal cavity made of a large number of antennas. One can also observe, in ﬁg. 4b, a very good estimate of the eigenmode correlation function experimentally obtained with only one realization. A one-channel omnidirectional transducer is able to refocus a wave in a chaotic cavity, and if the bandwidth is very large, we do not have to use a TRM made of many transducers. The focusing process described here is very diﬀerent from the focusing techniques used in monochromatic regime. Here, the frequency diversity is used to concentrate the wave ﬁeld at one time at one location. It is interesting to compare this focusing approach for broad-band signals with phase conjugation of monochromatic signal. Time reversal of p(r, t) is equivalent, for each spectral component Pˆ (r, ω), to complex conjugation. For a single-frequency signal, time reversal is equivalent to complex conjugation of complex amplitude. In a closed cavity, as above, if one works only at a single frequency (say that of one of the eigenmodes ωi ), one constructs only the eigenmode pattern corresponding to the selected frequency. The refocusing process discussed above works only with

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169

Fig. 5. – Schematic representation of a broad-band time reversal operation at the source (right part) and oﬀ the source (left part). Each arrow represents a diﬀerent Fresnel vector corresponding to a frequency component. At the source position, all the phases are set back to 0, the amplitude of the resulting signal rises proportionally to the number of independent frequencies N . Outside the source, the diﬀerent √ contributions are presumably decorrelated, and the standard deviation of their sum rises as N .

broad-band pulses, over a bandwidth that includes a large number of eigenmodes. Here, the averaging process that gives a good focusing is obtained by a sum over the diﬀerent modes in the cavity by assuming that in a chaotic cavity, we have a statistical decorrelation of the diﬀerent eigenmodes, the time-reversed ﬁeld can be computed by adding the various frequency components (each individual mode) and it can be represented as a sum of Fresnel vectors (ﬁg. 5). At the source position, all these phase-conjugated ﬁelds have a zero phase (this comes from the phase conjugation operation that exactly compensates for the forward phase) and even if there is no amplitude focusing for each spectral contributions, there is a constructive interference between all these ﬁelds at the ˆ 2 ˆ focusing time as i |G(r 0 , rtrm ; ωi )| , where G(r0 , rtrm ; ω) is the Fourier transform of G(r0 , rtrm ; t). Thus, the total ﬁeld at the focusing time increases proportionally to the number I of modes (or arrows). Outside the source position, at point r, we observe ˆ ˆ∗ i G(r0 , rtrm ; ωi )G (r, rtrm ; ωi ), the contributions of each individual mode are decorrelated because there is √ no more coherent phase compensation and therefore the total length only increases as I. On the whole, the focusing peak emerges at the focusing time from the noise when the bandwidth is large enough to contain many diﬀerent modes. Ideally, if we could indeﬁnitely expand √the bandwidth, the background level on the directivity patterns should decrease as 1/ I. As the number of eigenfrequencies available in the transducer bandwidth increases, the refocusing quality becomes better and the focal spot pattern becomes closed to the ideal Bessel function. As a conclusion, it must be emphasized that in a closed cavity a one-channel time reversal mirror can focus with λ/2 resolution if the duration of time reversal window is greater than or equal to the cavity Heisenberg time. Longer time windows do not improve √ the focusing quality. However, larger bandwidth Δω reduces the side lobe levels as 1/ Δω. Time reversal in reverberant cavities at audible frequencies has been shown to be an eﬃcient localizing technique in solid objects. The idea consists in detecting acoustic waves

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in solid objects (for example a table or a glass plate) generated by a slight ﬁnger knock. As in a reverberating object, a one channel TRM has the memory of many distinct source locations, and the information location of an unknown source can then be extracted from a simulated time reversal experiment in a computer. Any action, for example, turning on the light or a compact disk player, can be associated with each source location. Thus, the system transforms solid objects into interactive interfaces. Compared to the existing acoustic techniques, it presents the great advantage of being simple and easily applicable to inhomogeneous objects whatever their shapes. The number of possible touch locations at the surface of objects is directly related to the number of independent time-reversed focal spots that can be obtained. For example, a virtual keyboard can be drawn on the surface of an object; the sound made by ﬁngers when a text is captured, is used to localize impacts. Then, the corresponding letters are displayed on a computer screen [16]. . 3 2. Time reversal in open systems: random media. – The ability to focus with a onechannel time reversal mirror is not only limited to experiments conducted inside closed cavity. Similar results have also been observed in time reversal experiments conducted in open random media with multiple scattering [17-19]. Derode et al. carried out the ﬁrst experimental demonstration of the reversibility of an acoustic wave propagating through a random collection of scatterers with strong multiple-scattering contributions. A multiple-scattering sample is immersed between the source and an TRM array made of 128 elements. The scattering medium consists of 2000 randomly distributed parallel steel rods (diameter 0.8 mm) arrayed over a region of thickness L = 40 mm with average distance between rods 2.3 mm. The elastic mean free path in this sample was found to be 4 mm (see ﬁg. 6). A source 30 cm away from the 128 elements TRM transmitted a short (1 μs) ultrasonic pulse (3 cycles of a 3.5 MHz, Δω = 1 MHz). Figure 7a shows one part of the waveform received by one element of the TRM. It spreads over more than 200 μs, i.e. 200 times the initial pulse duration. After the arrival of a ﬁrst wave front corresponding to the ballistic wave, a long diﬀuse wave is observed due to the multiple scattering. In the second step of the experiment, any number of signals (between 1 and 128) is time-reversed and transmitted and a hydrophone measures the time-reversed wave in the vicinity of the source. For a TRM of 128 elements, with a time reversal window of 300 μs, the time-reversed signal received on the source is represented in ﬁg. 7b: an impressive compression is observed, since the received signal lasts about 1 μs, against over 300 μs for the scattered signals. The directivity pattern of the TR ﬁeld is also plotted in ﬁg. 8. It shows that the resolution (i.e. the beam width around the source) is signiﬁcantly ﬁner than it is in the absence of scattering: the resolution is 30 times ﬁner, and the background level is below −20 dB. Moreover ﬁg. 9 shows that the resolution is independent of the array aperture: even with only one transducer doing the time reversal operation, the quality of focusing is quite good and the resolution remains approximately the same as with an aperture 128 times larger. This is clearly the same eﬀect as observed with the closed cavity. High transverse spatial frequencies of arbitrary k that would have been lost in a homogeneous medium are redirected by the scatterers towards the array. Once again this result illustrates the diﬀerence between

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Fig. 6. – Time reversal focusing through a random medium. In the ﬁrst step the source (A) transmits a short pulse that propagates through the rods.The scattered waves are recorded on a 128-element array (B). In the second step, N elements of the array (0 < N < 128) retransmit the time-reversed signals through the rods.The piezoelectric element (A) is now used as a detector, and measures the signal reconstructed at the source position. It can also be translated along the x-axis while the same time-reversed signals are transmitted by B, in order to measure the directivity pattern.

phase conjugation and time reversal. If the experiment had been quasi-monochromatic and the single array element had simply phase-conjugated one frequency component, the conjugated wave ﬁeld would never have focused on the source position. Indeed, whatever its phase, there is no reason for a monochromatic wave emanating from a point source to be focused in a particular place on the other side of a random sample. The phaseconjugated ﬁeld at one frequency in the source plane is perfectly random and veriﬁes the classical speckle distribution. As for a broad-band signal in a closed cavity, an analysis similar to that of the last paragraph can be conducted in order to predict the level of the side lobes around the focal peak. A modal decomposition of the ﬁeld is not directly applicable. However, if we keep in mind that the focusing with one channel occurs only for a broad-band transducer, we identify the number of uncorrelated spectral correlation length δω of the scattered waves. Then there are Δω/δω uncorrelated bits of spectral information in the frequency bandwidth, and the signal-to-noise is expected to vary like Δω/δω. To evaluate the spectral correlation length, one can use the Wiener-Kinchin theorem [18] that gives the spectral correlation function (averaged over the frequency bandwidth) as the Fourier

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Fig. 7. – Experimental results. (a) Signal transmitted through the sample (L = 40 mm) and recorded by the array element n◦ 64, and (b) signal recreated at the source after time reversal.

Fig. 8. – Directivity pattern of the time-reversed waves around the source position, in water (upper line) and through the rods (lower line), with a 16-element aperture. The sample thickness is L = 40 mm. The −6 dB widths are 0.8 and 22 mm, respectively.

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Fig. 9. – Directivity pattern of the time-reversed waves around the source position through L = 40 mm, with N = 128 transducers (bottom curve) and N = 1 transducer (top curve). The −6 dB resolutions are 0.84 and 0.9 mm, respectively.

transform of the “time of ﬂight” distribution. In a multiply scattering medium, in the diﬀusive approximation, it is well known that the typical spreading time (the so-called Thouless time) is equal to τThouless = L2 /D, where D is a diﬀusion coeﬃcient related to the mean free path and L is the thickness sample. Therefore δω = D/L2 so the number of uncorrelated frequencies grows with L2 , provided we can neglect dissipation eﬀects (τdissipat ≥ τThouless ), where τdissipat is the dissipative time. 4. – Focusing microwaves below the diﬀraction limit Super-resolution can be achieved with an acoustic sink but it has a severe drawback. It needs to use an active source at the focusing point to exactly cancel the usual diverging wave created during the focusing process. Since we know that the time-reversal focusing spot at each frequency depends on the imaginary part of the Green function for any heterogeneous medium, another approach consists to surround the focusing point by a microstructured medium with length scales well below the wavelength; strong resonating scatterers were placed in the near-ﬁeld of the source. In this case, the microstructured medium strongly modiﬁes the spatial dependence of the imaginary part of the Green function that now oscillates on scales much smaller than the wavelength. For a broadband pulse with enough frequency diversity, a time reversal will generate at the focal time an interference between the imaginary part of the Green’s function at each frequency (see eq. (13)). At the source point, the time-reversed ﬁeld is directly proportional to the number of modes excited inside the microstructure from the source. While at the other points, the oscillations of the imaginary parts at diﬀerent frequencies cancel their eﬀects. To predict the behaviour of time reversal in such medium, we have to know the ﬁeld correlations. A wave propagating in any medium can be characterized by a spatial

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Fig. 10. – (a) A Time Reversal Mirror (TRM) made of eight commercial dipolar antennas operating at 2.45 GHz (i.e., λ = 12 cm) is placed in a 1 m3 reverberating chamber. Ten wavelengths away from the TRM a subwavelength receiving array is placed, consisting of eight microstructured antennas λ/30 apart from one another. (b) Details of one microstructured antenna. It consists of the core of a coaxial line which comes out 2 mm from an insultating layer and is surrounded by a microstructure consisting of a random distribution of thin copper wires. (c) Photograph of the 8-element subwavelength array surrounded by the random distribution of copper wires. Antennas #3 and #4 are indicated by the red and blue arrows.

correlation length and a spectral correlation length which have a pretty simple meaning. The spatial correlation length of a medium represents, at a given frequency, the smallest distance between two points which exhibits statistically diﬀerent wave ﬁelds, while the spectral correlation frequency δω measures the minimal frequency change that leads to independent wave ﬁelds. If the correlation length of the medium is much smaller than the wavelength, and if we use a bandwidth that contains several spectral correlation lengths, one can achieve a focusing on a scale of the order of the correlation length of the medium.

175

Amplitude(a u )

Time reversal focusing and the diffraction limit 1

(a)

0 1 0

Amplitude(a u )

(b)

200

400

600

1

800

0

200

400

600

(c)

800 (d)

0 1 400

200 0

200

400 400 time(ns)

200 0

200

400

1.2

λ/30

Amplitude(a u )

1

(e)

0.8 0.6 0.4 0.2 0 0

0.05

0.1 0.15 position in λ

0.2

Fig. 11. – In (a) (respectively, (b)) is shown the signal received at one antenna of the TRM when a 10 ns pulse is sent from antenna #3 (respectively, #4) of the subwavelength array. The signals in (a) and (b) look signiﬁcantly diﬀerent although antenna #3 and antenna #4 are distant from λ/30. In (c) (respectively, (d)) is shown the time compression obtained at antenna #3 (respectively, #4) obtained when the eight signals coming from antenna #3 (respectively, antenna #4) are time-reversed and sent back from the TRM. In (e) are shown the focusing spots obtained around antenna #3 and #4. Their width is λ/30. Thus antenna #3 and #4 can be addressed independently.

This is exactly the idea we exploit in the ﬁeld of time reversal with microwaves [20] to create focal spots much thinner than the wavelength. In a recent experiment [21] we consider 8 possible focusing points placed in a strong reverberating chamber (ﬁg. 10a). Eight electromagnetic sources are placed at these 8 locations to be used in the learning step of the TR process. These sources consist of wire antennas used at a central frequency of 2.45 GHz (i.e., λ = 12 cm), with a bandwidth of 100 MHz. The pitch between them is λ/30! These eight antennas form an array which will be referred to as the receiving array. Each antenna in this array is surrounded by a microstructure consisting here of a random distribution of thin copper wires (ﬁg. 10b). The mean distance between the thin copper wires was of the order of 1 mm (correlation length λ/100), while the frequency

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correlation δω was of the order of 30 MHz, resulting in three independent speckle patterns in the whole bandwidth. A TRM made of eight commercial dipolar antennas is placed in the far ﬁeld, ten wavelengths apart from the receiving array. The set “reverberant chamber/TRM” acts as a virtual far-ﬁeld time reversal cavity. When antenna marked #3 in ﬁg. 10 sends a short electromagnetic pulse (10 ns), the 8 signals received at the TRM are much longer than the initial pulse due to strong reverberation in the chamber (typically 500 ns). As an example, the signal received at one of the antennas of the TRM is shown in ﬁg. 11a. When antenna marked #4 is in its turn used as a source, it is remarkable to point out that now the signal received at the same antenna in the TRM (shown in ﬁg. 11b) looks signiﬁcantly diﬀerent although sources #3 and #4 were λ/30 apart from each other. When these signals are time-reversed and transmitted back, the resulting waves converge, respectively, to antenna #3 and #4 where they recreate pulses as short as the initial ones (ﬁg. 11c and d). Measuring the signal received at the other antennas of the receiving array gives access to the spatial focusing around antennas #3 and #4 (ﬁg. 11e). The remarkable result is that the two antennas can be addressed independently since the focusing spots created around them have a size much less than the wavelength (here typically λ/30): the diﬀraction limit is overcome although the focusing points are in the far-ﬁeld of the TRM! Contrary to the acoustic sink experiment, in the microwave time reversal experiment, the source remains passive and high spatial frequency components of the ﬁeld are created upon scattering at the disordered structure. Reciprocity ensures that the time-reversed scattering process creates a sub wavelength focus around the source location [22]. The initial evanescent waves created around the initial wire are converted into propagating waves by the random distribution of wires. In the time-reversed step, these propagating waves are playback, from the far ﬁeld, with reverse k. Spatial reciprocity ensures that each propagating wave with a reverse k interacts with the random distribution of wires to recreate the initial evanescent waves around the focus. Note that this approach can work not only when the experiment is conducted inside a reverberant cavity [23]. Provided the microstructured medium generates enough multiple scattering in the near ﬁeld of the source resulting in a transmitted signal suﬃciently long, the time reversal signal from a small TRM located in free space leads to beat the diﬀractions limit. 5. – Conclusion We have shown that in the presence of multiple reﬂections or multiple scattering, a small size time reversal mirror manages to focus a pulse back to the source with a spatial resolution that beats the diﬀraction limit. The resolution is no more dependent on the mirror aperture size but it is only limited by the spatial correlation of the wave ﬁeld. In these media, due to a sort of kaleidoscopic eﬀect that creates virtual transducers, the TRM appears to have an eﬀective aperture that is much larger that its physical size. Resolution can be improved in reverberating media using this concept. Time reversal focusing opens also completely new approaches to super-resolution. We have show that

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in a medium made of random distribution of sub-wavelength scatterers, a broad-band time-reversed wave ﬁeld interacts with the random medium to regenerate not only the propagating but also the evanescent waves required to refocus below the diﬀraction limit. Focal spots as small as λ/30 have been demonstrated with microwaves. This results in a large increase of the information transfer rate by time reversal in such disordered media. REFERENCES [1] Fink M., IEEE Trans. Ultrason. Ferroelec. Freq. Contr., 39 (1992) 555. [2] Fink M., Phys Today, 50, 3 (1997) 34. [3] Fink M., Cassereau D., Derode A., Prada C., Roux P., Tanter M., Thomas J. L. and Wu F., Rep. Progr. Phys., 63 (2000) 1933. [4] Cassereau D. and Fink M., IEEE Trans. Ultrason. Ferroelec. Freq. Contr., 39 (1992) 579. [5] Jackson D. R. and Dowling D. R., J. Acoust. Soc. Am., 89 (1991) 171. [6] de Rosny J. and Fink M., Phys. Rev. Lett., 89 (2002) 124301. [7] van Manen D-J., Robertsson J. and Curtis A., Phys. Rev. Lett., 94 (2005) 164301. [8] Nieto-Vesperinas M and Wolf E, J. Opt. Soc. Am. A, 2-9 (1985) 1429. [9] Weaver R., J. Acoust. Soc. Am., 71 (1982) 1608. [10] Ebeling K. J., in Physical Acoustics, edited by Mason W. P., Vol. 17 (Academic, New York) 1984, pp. 233–309. [11] Draeger C. and Fink M., Phys. Rev. Lett., 79 (1997) 407. [12] Draeger C. and Fink M., J. Acoust. Soc. Am., 105 (1999) 618. [13] Draeger C., Aime J-C. and Fink M., J. Acoust. Soc. Am., 105 (1999) 611. [14] Berry M. V., Chaotic Behaviour of Deterministic Systems (North Holland, Amsterdam) 1981, pp. 171-271. [15] McDonald S. W. and Kaufman A. N., Phys. Rev. A, 37 (1988) 3067. [16] Ing R. K., Quieffin N., Catheline S. and Fink M., Appl. Phys. Lett., 87 (2005) 204104. [17] Derode A., Roux P. and Fink M., Phys. Rev. Lett., 75, 23 (1995) 4206. [18] Derode A, Tourin A and Fink M, Phys. Rev. E, 64 (2001) 36606. [19] Blomgren P., Papanicolaou G. and Zhao H., J Acoust. Soc. Am., 111, 230. [20] Carminati R., Pierrat R., de Rosny J. and Fink M., Opt. Lett., 32 (2007) 3107. [21] Lerosey G., de Rosny J., Tourin A. and Fink M., Science, 315 (2007) 1120. ´ enz J., Greffet J. and Nieto-Vesperinas M., Phys. Rev. A, 62 [22] Carminati R., Sa (2000) 012712. [23] Li X. and Stockman M. I., Phys. Rev. B, 77 (2008) 195109.

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-179

Ultracold atoms in bichromatic lattices L. Fallani and M. Inguscio LENS, European Laboratory for Nonlinear Spectroscopy Dipartimento di Fisica ed Astronomia, Universit` a di Firenze & INO-CNR Via Nello Carrara 1, I-50019 Sesto Fiorentino (FI), Italy

Summary. — In this paper we illustrate the physics of ultracold atoms in bichromatic optical lattices. The properties of biperiodic systems are presented in detail, with a particular focus on the localization transition for incommensurate lattices. We then present recent work on the experimental investigation of these systems.

1. – Introduction The possibility of using far-detuned laser light to engineer trapping potentials for cold atoms allows an unprecedented control of the forces exerted on a quantum system. In this paper we discuss the physics of ultracold atoms in optical lattices, i.e. periodic potentials created with laser standing waves, which allow the investigation of many fundamental problems in strong connection with the physics of condensed matter [1]. The extremely long coherence times and the absence of defects in the lattice provide the ideal environment for investigating the transport of quantum particles in periodic structures. Furthermore, ultracold quantum gases in optical lattices represent promising resources for studying many-body quantum-correlated systems with important implications in the growing ﬁeld of quantum simulation [2]. c Societ` a Italiana di Fisica

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In this paper we focus on the physics of bichromatic optical lattices produced by the superposition of two standing waves with diﬀerent spacing. We will especially consider the case of incommensurate lattices, aperiodic potentials with discrete Fourier components, which, similarly to quasicrystals, exhibit intriguing properties which interpolate between the physics of ordered and disordered systems. While retaining peculiar properties of periodic lattices, as the existence of energy bands, they also support a quantum phase transition from extended to Anderson-localized states, as in purely disordered systems. This property has been investigated in a recent experiment, where Anderson localization of matter waves has been observed [3]. This paper is structured according to the following scheme. In sect. 2 we will review some basic notions related to the experimental realization of ultracold atomic gases in optical lattices. In sect. 3 we will consider the case of monochromatic lattices, starting from the quantum-mechanical theory of non-interacting particles in periodic potentials and arriving at the physics of strongly interacting many-body systems. In sect. 4 we will introduce the physics of bichromatic lattices, evidencing similarities and diﬀerences from the uniform lattice case, and focusing on the localization transition for non-interacting particles in incommensurate lattices. Section 5 will be devoted to the presentation of recent LENS experiments in which bichromatic lattices have been used to study Anderson localization of non-interacting and weakly interacting Bose gases. Finally, in sect. 6 we will discuss the physics of strongly interacting disordered systems, showing the results of experimental work performed in this regime and discussing possible experimental perspectives. 2. – Optical lattices . 2 1. Ultracold atoms. – In the last decades a constantly growing interest has been devoted to the possibility of cooling and trapping atomic gases. After the ﬁrst spectacular demonstrations of this possibility, culminated in the 1997 and 2001 awards of the Nobel Prize in Physics for laser cooling [4-6] and for the achievement of Bose-Einstein condensation [7,8], it is now clear that ultracold gases can be used as versatile nano-laboratories devoted to the investigation of many diﬀerent aspects and open problems of quantum mechanics. The reason of this possibility lies in the remarkably low decoherence rates and in the high degree of control that can be reached in the manipulation of neutral atoms. Properly chosen conﬁgurations of laser beams permit to cool atomic gases to temperatures of few microkelvin, which are unaccessible by the most advanced criogenic techniques. Optical or magnetic traps can be easily operated, providing conﬁnement of cold atoms in ultrahigh vacuum chambers, with long storage times and no thermal contact with the ambient. With these techniques, it is possible to create quantum gases of either bosonic or fermionic atoms, and eventually reach the degenerate regime, with the formation of Bose-Einstein condensates or quantum-degenerate Fermi gases, respectively. In this paper we will mostly concentrate on the physics of bosonic atoms below the critical temperature for Bose-Einstein condensation [9]. Typical condensates of neutral

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181

alkaline atoms are made of 105 –106 particles at a temperature of few hundreds nanokelvin. Condensates are created in either optical or magnetic traps, and generally have a spatial extension of few tens of microns, which makes them easily observable via standard imaging techniques. Owing to the small densities (typically 1014 atoms/cm3 ) and ultralow temperatures, interactions between particles are limited to binary elastic s-wave collisions. The strength of the interactions can be controlled by changing the density of the gas (which is determined by the trapping potential) or by using a magnetic ﬁeld tuned . in proximity of a Feshbach resonance (see sect. 5 1 for more details). We will not enter here the description of the techniques by which it is possible to create and probe Bose-Einstein condensates. An important resource on this topic is provided by the Proceedings of the CXL International School of Physics “Enrico Fermi” on “Bose-Einstein Condensation in Atomic Gases” [10]. . 2 2. Light forces. – Most laser cooling techniques rely on the near-resonant lightmatter interaction, in which the atoms are repeatedly excited to a higher-energy state through cycles of absorption and spontaneous emission of photons. The mechanical eﬀects of this process originate the radiation pressure force, which, under suitable conditions, provides the dissipative mechanism responsible for cooling. Optical trapping, instead, relies on the oﬀ-resonant interaction, in which absorption is negligible and atoms are not excited to higher-energy states. In this regime, the interaction is purely dispersive and results in a conservative dipole force which can be used for trapping atoms for long times (several seconds) without heating. A complete theoretical analysis of these eﬀects, founded on an elegant and rigorous quantum-mechanical treatment, can be found in [11]. We will now focus on the dipole force, following the semiclassical approach presented in [12], that gives a simple insight into the origin of this force. We consider a simple model in which the atom, treated as a two-level system, is subject to a classical radiation ﬁeld oscillating at frequency ω: (1)

E(r, t) = eˆE(r)e−iωt + c.c.

This ﬁeld induces a polarization of the atom, i.e. a deformation of the electron charge distribution, which can described by an electric dipole moment (2)

p(r, t) = eˆp(r)e−iωt + c.c.

oscillating at the same frequency of the driving ﬁeld and proportional to the latter through the complex polarizability α: (3)

p(r) = αE(r).

This relation holds only in the linear regime, when saturation eﬀects can be neglected and the atomic population is almost entirely in the ground state. The real part of α, describing the component of p oscillating in phase with E, is responsible for the dispersive properties of the interaction, while the imaginary part, describing the out-of-phase

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component of p, is connected with the absorptive properties (radiation pressure). Both these components strongly depend on ω as the frequency of the laser is scanned across the atomic resonance ω0 . In analogy with classical electrodynamics, we deﬁne the optical dipole potential as the interaction energy of the induced dipole p interacting with the driving electric ﬁeld E (4)

1 1 Re(α)I(r), Udip (r) = − p · E = − 2 20 c

where the angle brackets indicate the time average over the optical oscillations and I(r) is the average ﬁeld intensity. Equation (4) describes the potential energy associated to a conservative optical dipole force (5)

Fdip (r) = −∇Udip (r) =

1 Re(α)∇I(r), 20 c

which is nonzero when the intensity of the radiation ﬁeld I(r) is not uniform. Starting with just focusing a single laser beam, by engineering the geometry of the beams in such a way to produce conﬁgurations with non-uniform intensity, it is possible to create almost any kind of potential and to change its strength by tuning the laser parameters. Solving the optical Bloch equations for a two-level system [13], one can easily derive an analytical expression for the atomic polarizability α(ω). In the far-oﬀ resonant regime, in which the detuning Δ = ω − ω0 from the atomic resonance ω0 is much larger than the radiative linewidth of the excited state Γ and of the Rabi frequency Ω, eq. (4) becomes [12] (6)

Udip =

3πc2 2ω03

Γ Δ

I.

From eq. (6) we note that the sign of the dipole potential depends on the sign of the detuning. More precisely, if the light is red-detuned (Δ < 0) the dipole potential is negative, hence maxima of intensity correspond to minima of the potential: as a consequence, the atoms tend to localize in regions of high ﬁeld intensity. On the contrary, if the light is blue-detuned (Δ > 0) the dipole potential is positive, hence maxima of intensity correspond to maxima of the potential and the atoms tend to localize in regions of low ﬁeld intensity. . 2 3. Crystals made of light. – A periodic potential for cold neutral atoms can be easily produced by using laser light. In the previous section we have discussed how the mechanical eﬀects of the non-resonant interaction between radiation and matter can be used to produce conservative optical potentials. What is needed to produce a periodic potential is a periodic modulation of the light intensity, which is produced whenever two laser beams with the same optical frequency cross and interfere. If two beams cross at an angle θ, as sketched in ﬁg. 1, the distance between two adjacent maxima (or minima)

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Fig. 1. – Schematics of the periodically modulated intensity resulting from the intersection of two coherent laser beams propagating at an angle θ.

of the resulting interference pattern is (7)

d=

λ . 2 sin (θ/2)

The simplest and most common experimental setting is provided by two counterpropagating beams, which form a standing wave with an intensity modulation of period d = λ/2. In this counterpropagating conﬁguration, the resulting optical dipole potential can be written in the form (8)

V (x) = V0 cos2 (kx),

where k = 2π/λ is the laser wave number and V0 = 3πc2 ΓI0 /2ω 3 Δ is the depth of the periodic potential, being I0 the maximum intensity of the standing-wave pattern, ω the optical frequency, Γ the linewidth of the atomic transition and Δ the detuning of the laser from the resonance. The height of the optical lattice V0 is often measured in units of the recoil energy ER = 2 k 2 /2m = h2 /8md2 , which physically corresponds to the kinetic energy an atom at rest acquires after absorption of one lattice photon. This optical way to produce periodic potentials for neutral atoms oﬀers several unique features. The ﬁrst is the extreme tunability of the potential: the height of the optical barriers V0 can be manipulated by changing laser intensity and detuning from the atomic resonance, while the period d can be tuned by changing the wavelength or the angle formed by the beams. A second important feature is the intrinsic periodicity of the pattern: approximating the laser beams with plane monochromatic waves, the intensity modulation turns out to be exactly periodic, with no inhomogeneities or defects. A third important property of optical periodic potentials is the absence of phonons. In a real crystal the atoms are arranged “in average” according to the lattice structure (because this is the conﬁguration which minimizes the total energy of the crystal), but they can oscillate around this equilibrium conﬁguration. The coupled oscillations of the

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atoms in the lattice result in a vibration of the whole crystal, which can be quantummechanically described in terms of phonons. The resulting vibration of the lattice potential then aﬀects the motion of the electrons and this can be modeled with an eﬀective electron-phonon interaction. In an optical lattice, instead, vibrations of the lattice can be neglected (as far as instrumental sources of noise, as the laser phase noise and mirror vibrations, are suppressed) and cold atoms can be used to study the physics of a gas of particles in an “ideal” static crystal. The absence of vibrations and defects, as well as the possibility of controlling the interactions between the particles, makes ultracold atoms in optical lattices a wonderful system for the investigation of ideal solid-state phenomena. A remarkable example has been given by the investigation of Bloch oscillations [14-17], which is a purely quantum eﬀect that is hardly observable in real solids, where electrons strongly interact one with each other and decoherence is very fast due to scattering of the electrons with phonons and imperfections. A further example of this possibility is represented by the observation of Anderson localization [18], which will be discussed thoroughly in the following sections. Ultracold atoms have recently allowed a clear observation of this phenomenon [19, 3], that was originally predicted by P. W. Anderson in 1958 in the context of electronic transport as a complete cancellation of conductivity induced by the presence of disorder. This observation has been made possible by the production of strongly inhomogeneous optical potentials, which allow the realization of atomic systems with a controlled amount of disorder 3. – Monochromatic lattices In this section we will brieﬂy review the theory of quantum particles in a periodic potential, to the aim of introducing concepts and notations that will be extensively used throughout the paper. . 3 1. Energy bands. – Periodic potentials are at the heart of the description of electric conduction in crystalline solids, where the atoms are arranged in lattices with well-deﬁned symmetries. The most energetic electrons of the crystal, instead of forming a bound state with individual atoms, experience the combined attraction of all the atoms of the lattice, which can be modeled with a periodic potential having the same discrete translational symmetries of the lattice. In the one-dimensional case the periodicity condition for the potential V (x) takes the form (9)

V (x) = V (x + d),

where d is the distance between diﬀerent lattice sites. The time-independent Schr¨ odinger equation for the wave function Ψ(x) of a particle of mass m moving in a periodic potential

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185

V (x) reads (10)

2 ∂ 2 ˆ + V (x) Ψ(x) = EΨ(x). HΨ(x) = − 2m ∂x2

A fundamental contribution to the solution to this problem was formulated by F. Bloch in 1928 [20]. The Bloch theorem states that, when the symmetry in eq. (9) is applied, no matter what is the particular expression of the potential V (x), the solutions of eq. (10) take the general form (11)

Ψn,q (x) = eiqx un,q (x),

(12)

un,q (x) = un,q (x + d),

which describes plane waves eiqx modulated by functions un,q (x) having the same periodicity of the lattice. These stationary solutions are labelled by two quantum numbers: the band index n and the quasimomentum (or crystal momentum) q. The quantum number q is called quasimomentum because it presents some analogies with the momentum p, which is the good quantum number to describe the eigenstates of the Schr¨ odinger equation in the absence of any external potential. However, since the potential V (x) does not present a complete translational invariance, the solutions in eqs. (11)-(12) are not eigenstates of the momentum operator and q is not the expectation value of the momentum. We note that, because of the discrete invariance of the Hamiltonian under translations x → x + nd (with n integer), the quasimomentum is deﬁned modulus 2π/d, that is the period of the reciprocal lattice. As a matter of fact, the periodicity of the problem in real space induces a periodic structure also in momentum space, in which the elementary cells are the so-called Brillouin zones(1 ). For a given quasimomentum q many diﬀerent solutions Ψn,q (x) with diﬀerent energies En (q) exist. These solutions are identiﬁed with the band index n. The term band refers to the fact that the periodic potential causes a segmentation of the energy spectrum into forbidden zones(2 ) and allowed zones, the so called energy bands, which in solidstate physics are at the basis of the conduction properties of metals and insulators. Figure 2 shows a plot of En (q) as a function of the quasimomentum q for the ﬁrst three energy bands of the periodic potential in eq. (8). The solid thick lines correspond to a representation of the energy spectrum in the so called extended zone scheme, in which diﬀerent energy bands are plotted in diﬀerent Brillouin zones. However, because the quasimomentum is deﬁned modulus a vector of the reciprocal lattice, the spectrum En (q) can be more generally represented in the repeated zone scheme, in which all the (1 ) In higher dimensions the periodicity in momentum space is deﬁned by the vectors of the reciprocal lattice which can be constructed starting from the base vectors of the Bravais lattice in real space [21]. (2 ) With the term “forbidden zones” we refer to intervals of energies in which the density of states ρ(E) vanishes, while in the “allowed zones” ρ(E) > 0.

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L. Fallani and M. Inguscio 1st Brillouin zone

2nd Brillouin zone

3rd Brillouin zone

0.5

1 1.5 2 quasimomentum [q/k]

2.5

10

energy [E/E R ]

8

E3(q)

6 E2(q)

4 E1(q)

2

E0(q)

0 0

3

Fig. 2. – Energy spectrum of a particle in a periodic potential. Solid curves: lowest three energy bands for a particle of mass m in the periodic potential of eq. (8) with V0 = 4ER . Dashed curve: energy spectrum of the free particle. The energies are expressed in natural units ER = 2 k2 /2m.

energy bands are plotted in all the Brillouin zones (both thin and thick solid lines). For comparison, we plot in the same graph also the parabolic spectrum E0 (q) = 2 q 2 /2m of the particle in free space (dashed curve). Some general features of the band structure already emerge from this particular case. 1) At low energies (En V0 ) the bands are almost ﬂat and, for increasing height of the periodic potential, asymptotically tend to the energies of the bound states in the single lattice well. 2) At high energies (En V0 ) the bands are pretty similar to the free particle spectrum (except for a zero-point energy shift) and diﬀer from the latter only close to the boundaries of the Brillouin zones. 3) Near the zone boundaries, in correspondence with the appearance of the energy gap, the bands have null derivative. We now consider the shape of the eigenstates of eq. (10) in the case of the sinusoidal potential in eq. (8). In ﬁg. 3 we plot the squared modulus of the ground-state wave function Ψ0,0 (x) for three diﬀerent heights of the periodic potential. For small potential heights (dotted line) the Bloch states Ψ0,q (x) are similar to the plane waves eiqx , except for a small amplitude modulation with the periodicity of the lattice. Increasing the lattice height (solid line), when the energy of the state becomes much smaller than the maximum height of the potential, the Bloch wave functions are strongly modulated. In any case, the Bloch wave functions are extended states, i.e. their amplitude is nonzero in any position of the lattice, similarly to plane waves which are deﬁned in the entire space. In other words, the stationary states of a particle in a periodic potential are states in which the particle is delocalized.

187

Ultracold atoms in bichromatic lattices V0 V0 V0

ER 5ER 25ER

E E E

0 47ER 1 82ER 4 73ER

1.5

0 0(x)

2

[a u ]

2

1

0.5

1

2

3

4

position [x/d]

Fig. 3. – Squared modulus of the ground-state wave function Ψ0,0 (x) for diﬀerent heights of the sinusoidal potential in eq. (8): V0 = ER (dotted), V0 = 5ER (dashed), V0 = 25ER (solid). Increasing the lattice height, the wave function changes from a weakly modulated plane wave to a function that is strongly localized at the lattice sites. The energy of these states is reported in the inset.

Before concluding this section, we note that, in the case of a ﬁnite-sized system of size L, boundary conditions have to be set. A natural choice is taking the wave function to be periodic at the boundaries: Ψ(0) = Ψ(L). This is the common choice when solving the quantum-mechnical problem of a particle in free space. Imposing such boundary conditions leads to a quantization of the momentum of the particle (i.e. the wave function has to be a standing wave of the system volume). The same eﬀect happens also in the case of the Bloch waves of a periodic potential: by imposing periodic boundary conditions at the edges, the quasimomentum gets quantized according to the relation (13)

qi =

2π i L

with i integer. If the length L contains N lattice sites, i.e. L = N d, it is easy to realize that the ﬁrst Brillouin zone is spanned by quasimomentum states with i ∈ [−N/2, +N/2]. This means that only N individual quasimomentum states can be deﬁned. . 3 2. Tight-binding model. – The problem of a quantum particle in a periodic potential can also be treated in a simpliﬁed tight-binding approach, which has the advantage of providing simple analytic results. This approach is inspired by the observation that for large lattice depths the wave functions are strongly modulated by the periodic potential (i.e. the particles are almost trapped in the lattice sites), and can be more conveniently written as the sum of many wave functions localized at the lattice wells. A generic state

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Fig. 4. – Tight-binding approximation to the motion of one particle in a periodic potential. The probability of hopping from one site |j to a next-neighbor |j ± 1 is expressed by the tunnelling matrix element J.

|Ψ (for simplicity, in the lowest energy band) can be written as a linear combination |Ψ =

(14)

aj |j

j

of Wannier states |j centered at the lattice site j with complex coeﬃcients aj . When the lattice depth is suﬃciently large, the wave functions of the Wannier states become strongly localized around single lattice sites(3 ). The Hamiltonian can be expanded on the basis of maximally localized Wannier states (which provide an alternative basis to the one provided by Bloch waves). By making the tight-binding approximation of small overlap between the diﬀerent Wannier states, a simple equation can be derived: (15)

H = −J

(|j j + 1| + |j + 1 j|) .

j

The physical meaning of eq. (15) is clear: a particle that is in state |j has a ﬁnite probability of tunnelling through the potential barrier separating two lattice sites, hopping into state |j − 1 or |j + 1, as schematically shown in ﬁg. 4. The introduction of this tight-binding approach allows us to obtain a simple expression for the spectrum. It is easy to show that the stationary Schr¨ odinger equation H|Ψ = E|Ψ for the Hamiltonian in eq. (15) is equivalent to the set of equations (16)

−J(aj+1 + aj−1 ) = Eaj ,

∀j.

Owing to the discrete translational invariance of the problem (all the sites are identical) the coeﬃcients aj must have the same modulus, i.e. the particle occupies each site of (3 ) In the limit of an inﬁnitely high lattice the Wannier functions reduce to the Gaussian ground state of the harmonic oscillators obtained expanding the lattice potential around the potential minima.

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the lattice with equal probability, as prescribed by the Bloch theorem. What can be diﬀerent from site to site is the phase: again, owing to the translational invariance, the phase diﬀerence from site to site should be the same all across the lattice. Therefore we can write (17)

aj = eiqdj ,

which means that the amplitudes aj in next-neighboring sites have a phase diﬀerence δφ = qd, where q can be clearly identiﬁed with the quasimomentum. With this deﬁnition, eq. (16) can be rewritten as (18)

E = −2J cos qd,

which is the spectrum of the lowest band of the lattice in the tight-binding approximation. We note that the width of the lowest band, i.e. the range of admitted energies, is 4J. This tight-binding approach is particularly suited in the case of deep lattices, when the total wave function can be expanded into the sum of many strongly localized states. In the case of low lattice heights, when the wave function is just weakly modulated and we cannot neglect the overlap integral between more distant neighboring sites, the tunnelling rate J loses most of its physical meaning. In this case other energy scales J , J , etc. describing longer-range couplings should be considered and the simple form of the energy spectrum in eq. (18) gets more complicated. The phase coherence relation provided by eq. (17) can be veriﬁed experimentally with ultracold atoms in time-of-ﬂight experiments. Let us consider the simpliﬁed case of a noninteracting Bose-Einstein condensate, where all the bosons share the same single-particle wave function. If the BEC atoms occupy the ground state of an optical lattice, q = 0 and the phase of each Wannier state is the same. If the lattice potential is suddenly switched oﬀ, the atomic wave function undergoes a free-space evolution, in which the diﬀerent Wannier wavepackets expand, overlap and eventually interfere, as skematically shown in ﬁg. 5. The phase coherence relation in eq. (17) manifests as an ordered interference pattern, with constructive interference peaks at position 0, ±2ktexp /m ,±4ktexp /m, . . . (where texp is the expansion time and k = π/d), resembling the diﬀraction of light from a material grating [22]. This pattern also provides an “image” of the initial wave function in momentum space. As a matter of fact, for a suﬃciently long time-of-ﬂight texp , the density distribution after free expansion corresponds to the initial momentum distribution in the trap. We will further discuss this detection technique in sect. 5, devoted to the experimental observation of the transition from extended (Bloch-like) to localized (Anderson) states in the presence of an incommensurate bichromatic lattice. . 3 3. Adding interactions. – Until now we have considered the physics of a single particle in a periodic potential. However, ultracold gases are not ideal quantum gases: there are interactions between the atoms, and these atom-atom interactions turn the singleparticle physics of the Bloch bands into a more complex many-body problem. In neutral

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Fig. 5. – Expansion of a Bose-Einstein condensate from an optical lattice. The atoms are trapped in a periodic potential generated with laser light (top) and then released from this trap. The density distribution after expansion (bottom) exhibits resolved peaks originated from the interference of the condensates initially located at the lattice sites. Adapted from [22].

atomic gases no long-range Coulomb force is present and dipole-dipole interactions can be neglected in most of the cases, the dominant interaction term coming from shortrange van der Waals interactions. Since the range of these interactions is much smaller than the average inter-particle distance, in a cold dilute sample one can approximate such interactions with collision events of two atoms occasionally coming into contact one with the other. At ultralow temperatures the collisional channels reduce to only s-wave collisions, whose cross-section is parametrized by just one scalar parameter, the scattering length a (positive for repulsive interactions, negative for attractive interactions). The quantum statistics of the atoms, i.e. their bosonic or fermionic nature, also make a fundamental diﬀerence. As a matter of fact, ultracold identical fermions cannot undergo s-wave collisions, which are forbidden by the Pauli exclusion principle. Bosons, instead, do interact, and interactions are fundamental to describe static and dynamic properties of atomic Bose-Einstein condensates [9]. For not too large atomic densities a mean-ﬁeld approach can be used, and interactions between the atoms of a Bose-Einstein condensate can be described with a nonlinear term in the wave equation for the BEC wave function Ψ(r, t), which is described by the following Gross-Pitaevskii equation [9]: (19)

2 2 ∇ 4π2 a ∂ 2 + V (r) + |Ψ(r, t)| Ψ(r, t). i Ψ(r, t) = − ∂t 2m m

The nonlinear term in Ψ, being analogous to the nonlinear Kerr eﬀect for light propagation, is responsible for the self-nonlinear behavior of matter waves and is at the basis of many intriguing phenomena which are well known in optics, such as solitonic propagation [23] and four-wave mixing [24]. In the context of optical lattices, the presence

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191

Fig. 6. – Sketch of interacting bosons in a periodic potential. Hopping of particles from one site to the next neighbor is described by the tunnelling matrix element J, while repulsive short-range two-body interactions are taken into account by the on-site interaction energy U .

of such nonlinearities leads to a number of new eﬀects deviating from the single-particle Bloch theory, including the observation of diﬀerent kinds of instabilities [25, 26]. In cold-atoms experiments the strength of interactions a can be modiﬁed by using a static magnetic ﬁeld adjusted around a Feshbach resonance [27]. The magnetic ﬁeld allows tuning the energy of a colliding atom pair in proximity of a bound molecular state: as a result, the scattering of the two atoms is modiﬁed and the scattering length can be resonantly tuned. This technique can even allow to cancel atom-atom interactions or switch repulsive interactions into attractive ones. We will discuss the importance of this tool in the following sections devoted to the experimental observation of Anderson localization with non-interacting Bose-Einstein condensates. . 3 4. Mott insulators. – An optical lattice can dramatically increase the eﬀects of interactions between the atoms and change a weakly interacting gas intro a stronglycorrelated many-body state. As a matter of fact, increasing the depth of the optical lattice results in a lower tunnelling rate, i.e. a lower mobility of the atoms in the lattice, which ampliﬁes the eﬀects of interactions. At the same time, the atoms get more squeezed in the lattice potential wells, thereby increasing their interaction energy. The physics of a gas of interacting ultracold bosons in a deep optical lattice is well captured by the Bose-Hubbard model [28], originally introduced to describe the superﬂuidinsulator transition observed in condensed-matter systems [29]. The Bose-Hubbard Hamiltonian is the second-quantization generalization of the tight-binding Hamiltonian in eq. (15) to the case of interacting particles. It takes the form (20)

ˆ = −J H

j,j

ˆb†ˆbj + U n ˆ j (ˆ nj − 1), j 2 j

where ˆbj (ˆb†j ) is the annihilation (creation) operator of one particle in the j-th site, n ˆ j = ˆb†j ˆbj is the number operator, and j, j indicates the sum on nearest neighbors. The two terms on the right-hand side of eq. (20) account for diﬀerent contributions to the total energy of the system. The ﬁrst term, proportional to the same hopping energy . J already introduced in sect. 3 2, describes the tunnelling of bosons from one site to an

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Fig. 7. – Two interacting bosons in a double-well potential. The three possible states of the system are shown.

adjacent site. The second term, proportional to the interaction energy U , arises from atom-atom on-site interactions and gives a nonzero contribution only if more than one particle occupies the same site (as schematically sketched in ﬁg. 6). In the presence of repulsive interactions U > 0, this model supports a quantum phase transition from a superﬂuid state (U J) in which atoms are delocalized occupying extended Bloch states, to an insulating state (U J) where the atom wave functions are localized in individual lattice sites. This behavior can be studied in the simpliﬁed setting represented in ﬁg. 7, where one considers only two lattice sites, which are coupled by a tunnelling probability J. This double-well problem is paradigmatic of a large number of eﬀects where quantum tunnelling is important, e.g. molecular spectra (inversion spectrum of ammonia molecule) and superconducting Josephson junctions. The Hamiltonian describing this situation is the following: (21)

ˆ = −J ˆb† ˆb2 + ˆb† ˆb1 + U [ˆ n1 (ˆ n1 − 1) + n ˆ 2 (ˆ n2 − 1)] . H 1 2 2

We can use an occupation-number representation and describe the state of the system with the notation |Ψ = |mn, which means that m particles are present in the left well and n particles are present in the right well(4 ). In the simplest case of 2 identical √ 2 sin α|20 + cos α|11 + bosons the state of the system can be written as |Ψ = 1/ √ 1/ 2 sin α|02, which is parametrized by the angle α and automatically satisﬁes the normalization condition Ψ|Ψ = 1. One could easily work out that the energy of such state is (22)

E = Ψ|H|Ψ = U sin2 α − 2J sin(2α).

We consider now two interesting limiting cases. When J U the energy can be approximated with E −2J sin(2α), which is minimized by α = π/4, yielding (23)

ΨJU

1 1 1 |20 + √ |11 + |02, 2 2 2

(4 ) We note that this kind of representation automatically takes into account the indistinguishability of the paricles.

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193

which means that each particle is delocalized in the two wells and the number of particles in each well has nonzero ﬂuctuations around unity. When instead U J the energy becomes E U sin2 α, which is minimized by α = 0, yielding ΨU J |11,

(24)

which means that no number ﬂuctuations are present: one particle is present in the left well and one particle is present in the right well. This happens because the energy U saved by localizing each particle in a diﬀerent well is larger than the energy J that would have been gained if the particles were delocalized. This localization behavior, induced by repulsive interaction between the atoms, is ampliﬁed in a lattice, which can be thought as an inﬁnite array of identical potential wells linked by a tunneling coupling between next-neighboring sites. In the case of unit ﬁlling (i.e. number of sites = number of atoms) the two limits are a superﬂuid state for J U , with each particle being delocalized in an extended Bloch wave (25)

ˆb† + ˆb† + . . . + ˆb† |00 . . . 0, ∝ ΨSF JU 1 2 N

and a Mott-insulating state for U J in which each atom is localized in a single site: (26)

I ˆ† ˆ† ˆ† ΨM U J ∝ b1 b2 . . . bN |00 . . . 0.

These two states exhibit remarkably diﬀerent properties. The transition from a superﬂuid to a Mott insulator when the Hamiltonian parameters are changed is example of quantum phase transition [30]. In this kind of transitions the control parameter is not temperature, as in the case of classical phase transitions (e.g. the ferromagnetic transition in the Ising model [31]), but a diﬀerent coupling parameter entering the system Hamiltonian, in this case the ratio U/J. The ﬁrst experimental observation of the quantum phase transition from a superﬂuid to a Mott insulator has been reported by M. Greiner and coworkers in [32], following the initial proposal by [28]. The experiment was performed by trapping a BEC in a threedimensional cubic optical lattice. The control parameter for the Mott transition, i.e. the ratio U/J between interaction energy and hopping energy, was tuned by adjusting the depth V0 of the optical lattice. As a matter of fact, in a deep 3D optical lattice, the following relations hold [33]: '

(27) (28)

3/4 V0 8 , kaER U = π ER ' 3/4 V0 V0 4 , exp −2 J = √ ER ER ER π

and the ratio U/J scales exponentially with the lattice depth V0 .

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Fig. 8. – Interference pattern of an ensemble of interacting ultracold bosons released from a 3D optical lattice for diﬀerent lattice heights V0 = sER . Around V0 = 15ER one observes the disappearance of the interference peaks, indication of entering the Mott Insulator state.

The Mott transition was investigated by monitoring the interference pattern of the atomic cloud released from the 3D lattice as a function of the lattice height. When the lattice conﬁning the atoms is suddenly switched oﬀ, the atom wave functions expand in free space and overlap: if long-range phase coherence exists in the system, the interference of the overlapping wave functions builds up in a regular diﬀraction pattern, similar to the one shown in ﬁg. 5 for a 1D optical lattice. This pattern, resembling the diﬀraction of light from a grating or the diﬀraction pattern of X-rays scattered from a crystalline structure, can be taken as a measure of the degree of coherence in the many-body system. The interference patterns shown in ﬁg. 8, recorded in similar more recent LENS experiments, correspond to diﬀerent lattice depths V0 , going from zero to 22.5ER . For small lattice depths the intensity of the side peaks increases with V0 : the atoms get more tightly conﬁned in the lattice sites and, following the optical analogy, this means that the lattice acts as a grating with narrower slits, resulting in a stronger “diﬀraction” eﬀect(5 ). At a lattice height V0 15ER the visibility of the interference peaks suddenly decreases, eventually going to zero, reﬂecting the loss of coherence of the system after entering the Mott insulator state, where the atoms are localized and the number ﬂuctuations δni vanish. This coherence loss is reversible: by ramping down the height of the optical lattice one can recover the peaked interference pattern peculiar of the superﬂuid state, where atoms are delocalized and on-site coherent states are created. An important property characterizing Mott insulators is the existance of a gap in the excitation spectrum. A Mott insulator is a “rigid” insulating state, in which no current (5 ) We observe that the free-space expansion maps the initial momentum distribution in real space. In the case of phase coherence (superﬂuid) the interference pattern detected is nothing but the momentum distribution of the initial extended Bloch states.

Ultracold atoms in bichromatic lattices

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Fig. 9. – Sketch of a bichromatic lattice.

ﬂow can be sustained and no excitations can be produced, unless an energy gap is broken. This gap coincides with the energy ∼ U that is required for moving a particle from a site to a next-neighbor already occupied site, i.e. for the creation of a particle-hole excitation. A superﬂuid, instead, is characterized by a gapless excitation spectrum, with the lowest branch of excitations corresponding to the creation of phonons inside the superﬂuid: long-wavelength excitations can be created at very low energies , according to the linear dispersion relation of phonons = ck [9]. The existance of a gap in the Mott state was already evidenced in [32], where the authors applied a magnetic potential gradient in order to produce tunnelling excitations. A marked resonance centered at a site-to-site energy shift ∼ U signaled the presence of the Mott gap. More recently, the measurement of the excitation spectrum across the superﬂuid-Mott transition has been carried out by using a lattice modulation technique [34] and inelastic light scattering (Bragg scattering) [35]. We will further discuss these issues in sect. 6 devoted to the study of strongly interacting disordered bosons. 4. – Bichromatic lattices In this section we will review the physics of bichromatic lattices, i.e. potentials obtained from the superposition of two periodic potentials with diﬀerent periods. From an experimental point of view, bichromatic optical lattices can be realized quite easily: starting from an already existing optical lattice, it is suﬃcient to add a second pair of laser beams with diﬀerent wavelength, as sketched in ﬁg. 9. In the one-dimensional case the resulting potential can be written as (29)

V (x) = V1 sin2 (k1 x) + V2 sin2 (k2 x).

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Fig. 10. – One of the simplest examples of 2D quasicrystals is oﬀered by the Penrose tiling, where the non-periodic combination of two elementary cells (light-grey and dark-grey rhomboids) provides a gapless tiling of the plane.

As we will see in the following, an important quantity characterizing this class of potentials is the ratio between the two lattice periods (30)

β = k2 /k1 ,

which quantiﬁes the degree of commensurability between the two lattices. If this number is rational the resulting potential is periodic (commensurate case), if it is irrational there is no periodicity (incommensurate case). The latter case is particularly interesting, since this broken translational invariance has important consequences on the properties of the eigenstates of this potential. Bichromatic incommensurate lattices are a beautiful example of a mathematical structure that can be non-periodic and ordered at the same time. Systems of this kind are extremely interesting since their properties interpolate between the properties of crystalline structures and the ones of disordered systems. This behavior is intimately related to the properties of quasicrystals [36, 37]. . 4 1. Quasicrystals. – The history of quasicrystals is quite recent. Their existence was suggested in the ’60s when mathematical structures showing long-range order but no periodicity were discovered. Quasicrystals are typically composed by a ﬁnite number of elementary cells forming a complete set of tiles that can ﬁll the space with no gaps (as in the case of ordinary crystals) but in a non-periodic way. One of the simplest example of quasicrystal is provided by the Penrose tiling [38] showed in ﬁg. 10. In this 2-dimensional

Ultracold atoms in bichromatic lattices

197

structure only two kinds of tiles exist (light-gray and dark-gray rhomboids) which are able to completely ﬁll the plane with no gaps. This arrangement does not follow any discrete translational invariance, i.e. no translation of the lattice ends up locally in the same conﬁguration. Quasicrystals can also possess extended classes of rotational symmetry in addition to the ones allowed for periodic crystals, e.g. the tiling showed in ﬁg. 10 has 5-fold rotational symmetry, which is forbidden to ordinary crystals. The existence of quasicrystals was demonstrated in [39], where sharp peaks with forbidden rotational symmetry were recorded in electron diﬀraction experiments for rapidlycooled Al-Mn alloys. One of the features of quasicrystals is indeed their ability to produce sharp diﬀraction peaks as crystals do, which is related to the presence of long-range ordering. Very recently, quasicrystalline order has been observed in a mineral sample coming from the Museum of Natural History of the University of Florence [40]: this discovery demonstrates that quasicrystals can spontaneously form in Nature and remain stable under geologic conditions. Similarly to periodic optical lattices, also optical lattices with quasicrystalline order can be created by using suitable arrangements of laser beams. The ﬁrst investigation of cold atoms in optical quasicrystals was reported in laser cooling experiments [41] with the realization of 5-fold rotationally symmetric potentials like the Penrose tiling. Going to the quantum degenerate regime, the investigation of a Bose-Einstein condensate in these quasicrystalline 2D potentials has been proposed theoretically in [42]. Incommensurate 1D bichromatic lattices can be considered as the simplest examples of quasicrystals. In the next sections we will describe their properties, with a particular focus to the energy spectrum and to the localization transition exhibited by their eigenstates. . 4 2. General notations. – Throughout the paper we will mostly consider the limiting case in which one lattice is much weaker than the other, i.e. V2 V1 . With clear meaning of the notation we will refer to these lattices as main lattice and perturbing lattice, as also indicated in ﬁg. 9. With this assumption the resulting quasiperiodicity is mostly compositional (and not topological as in the case of a real quasicrystal), this meaning that the sites of the lattice are still almost periodically displaced as in the case of a periodic lattice and the perturbation only aﬀects the properties (the energy) of the sites. It is easy to realize that in this limit the minima of the potential in eq. (29) sit in the same position xj as the minima of the main lattice (31)

xj = jπ/k1

with j integer. We can also assume that the heights of the potential barriers are not signiﬁcantly changed by the addition of the perturbing lattice (see [43] for more details). Therefore we can still assume that the system can be described by a hopping term J which is uniform across the whole bichromatic lattice. At small energy scales, the bichromatic lattice can be characterized by the values j taken by the potential energy at the lattice

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sites, which are not uniform but change according to (32)

j = V (xj ) = V2 sin2 (k2 xj) = V2 sin2 (βπj).

From the above expression we note that diﬀerent values of β could give rise to the same set of energies j . As a matter of fact, the energies j are the same for any substitution β → ±(β+m), with m integer. This is a consequence of the symmetry of the sin2 function and of the fact that {j } arise from a uniform sampling of a sinusoidal curve. Aliasing eﬀects result in the same sampling for sinusoids whose spatial frequencies diﬀer by a phase 2π on the distance of the main lattice period. It is convenient to consider only one of these equivalent samplings, i.e. the one with the longest-wavelength/smallest-frequency modulation: (33)

γ = min |β − m|. m

As we will see in the following, when discussing the general properties of bichromatic potentials, this deﬁnition is particularly helpful. This choice of γ is also natural since it is the only one satisfying the Nyquist-Shannon sampling theorem, and it automatically implies (34)

γ < 1/2 ;

this means that among all the possible sinusoids that yield the same sampling we choose to consider the longest-wavelength one, which has a modulation period larger than 2 sites (and can be reconstructed by interpolation starting from the {j } without aliasing). We can refer to the period of this modulation as bichromatic period. This situation is exempliﬁed in ﬁg. 11, showing a zoom on the lowest-energy region of a bichromatic lattice (black line): three diﬀerent choices of the perturbing lattice with diﬀerent β yield the same set of energies {j } at the bottom of the potential wells. . 4 2.1. Harper and Aubry-Andr´e model. The problem of a particle in a potential with two periodicities has a long history. It has been studied by P. G. Harper in 1955, who derived an equation describing the states of a crystal electron in a uniform magnetic ﬁeld [44]. Harper considered a 2D square lattice in the plane (x, y) with a static uniform magnetic ﬁeld B = B zˆ, directed out of the plane, described by a vector potential A = Bxˆ y. By applying the canonical momentum substitution p → p − eA/c into the Schr¨ odinger equation and using a tight-binding approach, the following Harper model can be derived: (35)

uj+1 − uj−1 + 2 cos (2πβj + φ)uj = uj ,

where uj is proportional to the amplitude of the wave function in site j along direction x ˆ. In this equation two periodicities are present: the periodicity of the lattice (which is intrinsically present in the tight-binding discretization) and an eﬀective periodic potential 2 cos (2πβj + φ). The ratio between the two periodicities is given by β = (d2 B)/(hc/e),

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Fig. 11. – The solid line shows a zoom of the lowest-energy region of the bichromatic potential in eq. (29). The other √ lines show perturbing √ lattices with diﬀerent√β which yield the same set of energies {j } : β = 3 (dotted), β = 3 − 1 (dashed), β = 1 − 3 (dot-dashed). The latter choice coincides with the value of γ associated to the lowest-wavelength modulation giving rise to the same energy sampling {j }.

which corresponds to the ratio between the magnetic ﬂux through one square lattice cell to the magnetic ﬂux quantum. Despite the simplicity of the physical system, this problem shows a very intriguing phenomenology related to the biperiodic structure. As D. R. Hofstadter wrote in the introduction of [45] (the paper where he discovered the fractal structure known as Hofs. tadter butterﬂy, see sect. 4 6), “the problem of Bloch electrons in magnetic ﬁelds is a very peculiar problem, because it is one of the very few places in physics where the diﬀerence between rational numbers and irrational numbers makes itself felt”. The case of irrational numbers is in fact particularly interesting. An extension of the Harper model has been studied in 1980 by S. Aubry and G. Andr´ e, who focused on the properties of the wave functions in the incommensurate case. The Aubry-Andr´e model [46] is a tight-binding Hamiltonian describing a quantum particle hopping in a 1D bichromatic lattice: (36)

H=

j

|j j + 1| + |j + 1 j| + λ

cos (2πβj)|j j|.

j

This Hamiltonian is closely related to the tight-binding Hamiltonian of eq. (15) for a single lattice, with the presence of an additional term proportional to λ, which parametrizes the strength of the quasiperiodic modulation (in our notations λ = V2 /2J). The derivation of this tight-binding model starting from the analytic form of the bichromatic potential is worked out and thoroughly discussed in [47].

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Fig. 12. – Sketch of the biperiodic tight-binding system described by the Aubry-Andr´e model. Hopping of particles from one site of the main lattice to the next-neighbor is described by the tunnelling matrix element J, while the amplitude of the perturbing lattice is denoted with 2Δ = V2 .

The Aubry-Andr´e model can be thought as an extension of the Harper model to the case in which the amplitude of the spatially dependent perturbation is free to change. Aubry and Andr´e showed that, for incommensurate β this model supports a metalinsulator transition at a critical value of the amplitude of the secondary lattice(6 ). For small strength λ of the perturbing lattice the eigenstates are extended, similarly to the Bloch waves of a periodic potential. For larger depths of the perturbing lattice, above the critical strength λ = 2, the eigenstates undergo Anderson localization and become exponentially localized. The realization of the Aubry-Andr´e model (ﬁg. 12) with ultracold atoms has been proposed in [48] and theoretically studied in [49]. More recently, a complete theoretical investigation on the localization mechanism in optical bichromatic lattices has been presented in [47]. . 4 3. Superlattices. – In order to illustrate some general features of bichromatic lattices, we start considering the commensurate case, often called superlattice, which is deﬁned by a rational value of γ. Under this condition the overall potential is periodic and the situation is not much diﬀerent from what we have shown in sect. 3. The system still has a discrete translational invariance and the Bloch theorem can be applied. Of course, the periodicity of the system now shows up on a diﬀerent length scale, which corresponds to γ −1 times the lattice constant of the main lattice(7 ). Let us consider a simple situation in which γ = 1/4, corresponding to a superlattice period of 4 sites. An example of bichromatic potential for this choice of γ is shown in ﬁg. 13a for V2 = 2.5J. It is evident that the resulting potential is strictly periodic, with a periodicity in reciprocal space (e.g. width of the Brillouin zones) which is γ = 1/4 of (6 ) As we will see in the following, the Aubry-Andr´e model reduces to the Harper model exactly at the critical point λ = 2. (7 ) In the commensurate case γ = m /m with m and m integer numbers. Generally speaking, the superlattice period amounts to g lattice sites, where g = LCM(m , m ) is the least common multiple of m and m , and can be diﬀerent from γ −1 . However, in order to simplify the presentation, we illustrate a simpler case where m = 1 and γ = g −1 .

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Fig. 13. – A bichromatic commensurate lattice with γ = 1/4 and V2 = 2.5J. The plots show: a) the real potential (continuous curve) and the quasiperiodic energy modulation (dotted curve); b) the energy spectrum of the bichromatic lattice (black) and of the main lattice only (gray); c) the square modulus of the ground-state wave function in real space; d) the square modulus of the ground-state wave function in momentum space. The spectrum and the wave functions are calculated numerically on a system length of L = 1000 sites.

the periodicity of the main lattice. According to the Bloch theorem, the eigeinstates are extended plane-wave–like functions with a spectrum that shows the presence of energy bands, as in the case of a single lattice. The spectrum of this potential is shown in ﬁg. 13b (thick dark lines), where for comparison we also show the spectrum for the main lattice only (light thin line), which features a gap at q = π/d = k1 . The introduction of the perturbing lattice modiﬁes the translational symmetry of the system and is responsible for the opening of extra energy gaps at multiples of the quasimomentum γk1 = γπ/d, with the larger gaps opening at γk1 and (1 − γ)k1 . The lowest band of the main lattice is therefore split into a ﬁner structure of γ −1 bands: we can refer to these bands as sub-bands of the main lattice fundamental band. The squared modulus of the eigenfunctions has the same periodicity of the lattice potential, as prescribed by the Bloch theorem eq. (12). This is evident from ﬁg. 13c, where we plot the amplitude of the ground state wave function |Ψj |2 in diﬀerent lattice sites j. Correspondingly, the eigenstate wave functions in momentum space are characterized by δ-peaks occurring at submultiples nγk1 = nγπ/d of the main lattice wave vector, reﬂecting the change in periodicity and the emergence of the new energy gaps. The momentum distribution of the ground state wave function for the same choice of parameters is plotted in ﬁg. 13d.

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. 4 4. Incommensurate lattices. – In the case of an incommensurate lattice the situation gets more subtle. As a matter of fact, since there is no periodicity in the system, the Bloch theorem cannot be applied and the quasimomentum cannot be deﬁned. This lack of translational invariance makes the Schr¨ odinger problem more complicated and no simple theorem helps us in ﬁnding the solution. However, numerical studies show that for weak intensities of the perturbing lattice, the situation is not substantially changed and many concepts that we have introduced for particles in periodic potentials still hold. In particular, we can still think in terms of energy bands and quantities related to the quasimomentum can be introduced. As a matter of fact, the eigenstates of an incommensurate bichromatic potential can be labelled by a progressive index i and, in analogy with eq. (13), we can introduce a pseudo-momentum q˜, deﬁned as (37)

q˜i =

2π i. L

The reason why we introduce such quantity is evident if we calculate the spectrum of the incommensurate potential for V2 < 4J (the reason for this choice will be clear in the following sections). √ In ﬁg. 14a,b we show the potential energy modulation and the spectrum for γ = 2 − 3 and V2 = 2.5J. On a coarse-grained view, the spectrum still shows the emergence of energy bands, similarly to what happens in a periodic lattice, even if the system is not periodic. When the perturbation V2 is introduced, clear gaps open at pseudo-momenta q˜ = γk1 and q˜ = (1 − γ)k1 and no major diﬀerences can be detected in this regime from the commensurate case. Looking at the spectrum with more detail, however, one could recognize the appearance of many weaker gaps that open across the entire band following a self-similar fractal structure induced by the absence of periodicity. The appearance of some (the strongest) of these gaps can be detected in the insets of ﬁg. 14b(8 ). Even if no periodicity is present, for small values of the perturbation V2 the wave functions are extended across the entire system size, similarly to Bloch waves. The ground state wave function for the potential in ﬁg. 14a is plotted in ﬁg. 14c: despite the absence of periodicity, the wave function spreads across the whole lattice and it shows an evident modulation at the bichromatic period γ −1 d, similarly to the commensurate case. The momentum distribution of the ground state for the same potential is shown in ﬁg. 14d, where several δ-peaks are observed, the strongest appearing at γk1 and (1−γ)k1 and the weaker ones following the same dense structure of the energy gaps emerging in the spectrum. The presence of sharp peaks in the momentum distribution (reminding the observation of sharp peaks in the diﬀraction of X-rays or electrons from quasicrystals [37]) is simply related to the fact that the eigenstates are extended, i.e. they occupy all the sites of the lattice, and long-range order is present. Therefore, as prescribed by the . (8 ) This structure, related to the Hofstadter butterﬂy discussed in sect. 4 6, shows up clearly only as the spectrum is observed on a small energy scale and the system size L is increased (see boundary condition in eq. (37)).

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√ Fig. 14. – A bichromatic incommensurate lattice with γ = 2 − 3 and V2 = 2.5J. The insets in b) show the appearance of a structure of weaker energy gaps owing to the absence of periodicity. See caption of ﬁg. 13 for more information on the ﬁgure.

uncertainty relation, the momentum distribution of these states should be made up of δ-peaks, the position of which reﬂects the wavelength of the modulation imposed by the perturbing lattice. . 4 5. Localization in bichromatic lattices. – For increasing height of the quasiperiodic perturbation the situation changes drastically. At a critical perturbation strength a quantum phase transition from extended states to localized states takes place. This localization transition is a manifestation of a much more general phenomenon, known as Anderson localization, that we are going to discuss in the following. . 4 5.1. Localized states. We have seen in sect. 3 that in a periodic potential all the eigenstates are extended states. This means that a perfectly periodic crystal is able to support a stationary current which ﬂows from one side of the crystal to the other. This, in turn, means that the system behaves as a conductor. Fifty years ago, P. W. Anderson conjectured that the presence of disorder in a crystal could turn a metal into a perfect insulator with zero conductivity [18]. The microscopic mechanism for this insulating behavior is the localization of the wave functions: the eigenstates of the system change from extended to localized. Spatially-localized eigenstates imply that no current can ﬂow in the system under stationary conditions. Anderson formulated his localization theory for a tight-binding model of electrons hopping in a disordered lattice where each site has a random potential energy. In

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incommensurate bichromatic lattices the presence of inhomogeneities can also lead to an exponential localization of the eigenstates, as in the case of a truly random potential. A bichromatic potential is not a random potential, because the energy modulation is characterized by long-range correlations and a purely deterministic form of the potential can be written out. These correlations have important consequences on the possibility to observe a direct localization transition. As a matter of fact, while a localization transition does not exist in 1D in a δ-correlated random potential (since all the eigenstates are localized, as demonstrated with scaling arguments in [50]), in a 1D bichromatic potential a localization transition takes place at a well-deﬁned value of the perturbation strength. Anderson localization of electrons in bichromatic lattices was ﬁrst studied by S. Aubry and G. Andr´e [46], who predicted a universal transition for λ = V2 /2J = 2. The existence of a phase transition was conjuctered with an elegant reasoning, inspired by the selfduality of the model in position and reciprocal space. As a matter of fact, eq. (36) can be recasted in momentum space by applying the transformation 1 i2πkβj e |j, |k = √ L j

(38)

that connects the Wannier states |j to the eigenstates of the momentum operator |k. With this substitution one obtains the Hamiltonian in reciprocal space (39)

4 λ |k k + 1| + |k + 1 k| + cos (2πβk)|k k| , H= 2 λ k

k

which has the same form as eq. (36), but the amplitude of the quasiperiodic energy modulation (compared to the tunnelling energy), is 4/λ instead of the real-space value λ. Clearly, if a transition from extended states to localized states exists for the Hamiltonian in eq. (36), it must exist also for the Hamiltonian in eq. (39), and the transition point has to be the same. Hence (40)

λ=

4 = 2. λ

This transition is visualized in the left panel of ﬁg. 15. Here the grayscale represents the amplitude of the ground-state wave function |Ψj |2 , with dark regions corresponding to larger amplitudes. The amplitude is plotted as a function of position (horizontal axis) and strength of the perturbing lattice V2 = 2λJ (vertical axis). Above the transition point the wave function becomes localized on shorter and shorter length scales, the more the strength of the perturbation is increased. For comparison, we show in the right panel what happens in the case of the Anderson model, i.e. a tight-binding model with uncorrelated disorder in which j = V2 Rnd[1], where Rnd[1] is a random real number between 0 and 1. In this case no phase transition is observed and the ground state is localized for any amount of disorder, in agreement with the predictions of the scaling theory for localization [50]. However, the behavior of

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Fig. 15. – Squared modulus of√the ground-state wave function (in greyscale) for the AubryAndr´e model with β = (1 − 5)/2 and for the 1D Anderson model with uncorrelated site energies. This quantity is plotted as a function of position (horizontal axis) and strength of the perturbing potential V2 . Clearly, a localization transition happens in the Aubry-Andr´e case for λ = V2 /2J = 2.

. the wave function in the localized case is pretty similar in the two cases, bichromatic and random. In ﬁg. 16 the ground state in the bichromatic and random potential is plotted for four diﬀerent amplitudes of perturbation: V2 = J (a), V2 = 4J (b), V2 = 7J (c), V2 = 10J (d). While for the lowest value the diﬀerent behavior is evident—exponential localization for the random potential (gray line), almost uniform amplitudes for the bichromatic potential (dark line)—in the localized regime the ground state for both the potentials is exponentially localized. . 4 5.2. Spectrum of the localized states. We ﬁnally examine the energy spectrum of the localized states. This is a beautiful demonstration of how the physics of quasicrystals nicely interpolates between the physics of ordered (periodic) and truly disordered systems. Even above the threshold for localization, the spectrum of the eigenstates shows the presence of energy bands and gaps similarly to the spectrum of the extended states below the critical point. An example of this band structure is reproduced in ﬁg. 17a, which features a sequence of gaps, the stronger ones again located at √ q˜ = γπ/d and q˜ = (1 − γ)π/d. The spectrum reported in ﬁgure refers to γ = 2 − 3 and V2 = 6J slightly above the localization transition. The energy width of the spectrum increases with the perturbation strength(9 ) and approaches V2 in the regime V2 J. (9 ) We recall that for a single lattice in the tight-binding limit the energy width of the lowest band is 4J.

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√ Fig. 16. – Ground-state wave function for the Aubry-Andr´e model with β = (1 − 5)/2 (black line) and for the 1D Anderson model with uncorrelated site energies (grey line). The diﬀerent plots refer to a) V2 = J, b) V2 = 4J, c) V2 = 7J, d) V2 = 10J. Note the log scale.

It is possible to show that the collection of states in the lowest sub-band of width γπ/d can be built up by picking, for each quasiperiod of the lattice, an exponentially localized eigenstate centered close to the lattice site with minimum energy within the quasiperiod. If N is the total number of sites, the lowest sub-band is formed by γN localized states, each occupying a diﬀerent lattice quasiperiod. Going towards larger energies, the gap at q˜ = γπ/d forms when all the quasiperiods have been “occupied” by a localized state and the following state gets localized in a quasiperiod that already has a localized state with lower energy within. This is where the strongest gap at q˜ = γπ/d forms. We show in ﬁg. 17b-e some examples of localized states which clarify this mechanism. Here the horizontal grey segments represent the energies of the sites in the Aubry-Andr´e model, while the dark-grey histogram represents the amplitude of the wave function in the lattice sites. The four plots refer to the four diﬀerent states b)-e) labelled in ﬁg. 17a. The lowest-energy state b) localizes around a lattice site which coincides with the minimum of the quasiperiodic modulation (and hence it is the site with the lowest potential energy of the lattice), as clear from ﬁg. 17b. The highest-energy state in the lowest subband c) is instead the one corresponding to ﬁg. 17c, when two lattice sites are equidistant from the minimum of the quasiperiodic modulation. The following state d) is separated by an energy gap, as shown in ﬁg. 17a. It is clear from ﬁg. 17d that states c) and d) occupy the same quasiperiod and their wave functions show an almost equal population of the two lowest lattice sites within the quasiperiod. What is diﬀerent is the phase of the wave function. This situation is closely related to what happens in the double-well problem, where two degenerate states are spatially separated by a potential barrier. As

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Fig. 17. √ – a) Energy spectrum of localized states in a bichromatic potential with V2 = 6J and γ = 2 − 3. Note the presence of energy gaps in the spectrum, even if the states are localized. b-e) Examples of localized states: the dark grey histograms represent the amplitude of the wave functions in the diﬀerent sites.

a matter of fact, for the perturbation strength considered in ﬁg. 17, the eigenstates are strongly localized and most of the wave function occupies two twin sites. The possible eigenstates of a particle in a double-well are the symmetric and antisymmetric combinations, which are separated in energy by 2J(10 ): this is exactly the energy gap that separates the lowest sub-band from the higher sub-band of the bichromatic potential, as shown in ﬁg. 17. (10 ) A beautiful example of this splitting is given by the molecular spectrum of ammonia NH3 , in which the nitrogen atom can jump between two stable conﬁgurations by “tunnelling” through the triangular plane arrangement of hydrogen atoms [51]. This eﬀect, described by an eﬀective double-well potential, is responsible for the existence of “inversion doublets” of lines separated by energy 2J.

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Fig. 18. – The Hofstadter butterﬂy. It represents the spectrum of the Aubry-Andr´e model at the critical point λ = V2 /2J = 2 as a function of the incommensurate parameter β. Adapted from [45].

. 4 6. Further considerations. – The physics of biperiodic systems shows more interesting features. One particurarly intriguing property manifests at criticality, when the depth of the perturbing lattice matches the critical value for the localization transition in the incommensurate case. At this value the Aubry-Andr´e model reduces to the Harper model and the spectrum shows a characteristic structure, shown in ﬁg. 18, known with the name of Hofstadter butterﬂy. This self-similar structure, demonstrated by D. R. Hofstadter in 1976, is one of the ﬁrst fractal structures discovered in physics. The ﬁgure shows a twodimensional representation of the spectrum of the biperiodic system at the critical point as a function of the energy of the states and of the incommensurate parameter β(11 ). More generally, an intriguing feature of biperiodic potentials is related to how the properties of the system change when the incommensurate parameter β is made “less” or “more” irrational. Actually, the localization transition predicted in the Aubry-Andr´e model gets sharper as the ratio between the lattice periods β is made “more” irrational. This behavior can be observed in ﬁg. 19, taken from [47], where the degree of localization of the states is quantiﬁed by plotting the inverse participation ratio (IPR) as a function # −1 # |Ψ(x)|4 dx/ |Ψ(x)|2 dx and of Δ/J = λ = V2 /2J and of β. The IPR is deﬁned as measures the inverse number of sites occupied by the state: it is macroscopic for localized (11 ) The ﬁgure is obtained after averaging the spectra over an arbitrary initial phase φ to be included in the cos term of eq. (36).

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Fig. 19. – Colour-scale plot of the inverse participation ratio (see text for deﬁnition) as a function of perturbation strength Δ/J and incommensurate parameter β. Note that the localization transition becomes sharper far from simple rational numbers. Taken from [47].

states, while for extended states it vanishes as N −1 (with N number of sites). Figure 19 shows that the IPR increases when Δ/J becomes larger than 2, in a sharper or softer way depending on how far β is from a simple rational number. This behavior is particularly evident close to β = 1 (corresponding to a quasiperiodic perturbation with very long period), where the localization crossover becomes extremely slow and very large values of the perturbation are needed to localize the wave function over distances on the order of the lattice spacing. This discussion on the degree of irrationality of the parameter β is particularly important when one considers ﬁnite-sized systems. Owing to the presence of a trapping potential, the size of ultracold atomic samples is typically limited to few hundreds of lattice sites. The existence of a maximum length scale can spoil the sharpness of the localization transition and, at the same time, release the constraints on the incommensurability of the potential: as we will further discuss in the following section, even a periodic superlattice potential can be “experimentally” considered as incommensurate if the resulting period is larger than the system size. 5. – Anderson localization of matter waves in bichromatic lattices Experiments chasing for Anderson localization with ultracold atoms have been started in 2005 with the ﬁrst realization of a Bose-Einstein condensate in a disordered potential [52]. Two major obstacles for the observation of this phenomenon were the production of strongly inhomogeneous potentials with a suﬃciently small disorder “grain size” and the cancellation of interactions between the BEC atoms, which lead to dephasing and spoil the single-particle Anderson physics [53]. The eventual observation of Anderson localization with ultracold atoms has been recently achieved in two diﬀerent ex-

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Fig. 20. – Sketch of the experimental setup for the observation of Anderson localization in a noninteracting 39 BEC [3]. A 1D bichromatic lattice is realized by superimposing two standing waves with incommensurate periodicities. An additional slightly focused laser beam provides the transverse conﬁnement, realizing an optical waveguide along which the BEC (initially conﬁned to a few lattice sites) can expand. Interactions between the atoms are cancelled by using a pair of Helmoltz coils producing a magnetic ﬁeld tuned in proximity of a Feshbach resonance.

periments, one carried out at LENS [3] and the other carried out in the group of A. Aspect at Institute d’Optique [19], which were both successful in circumventing these obstacles. In [19] these two problems were solved by using optical speckles with short correlation length and reducing considerably the BEC density to neglect the eﬀects of interactions. Following a diﬀerent strategy, in [3] an incommensurate bichromatic lattice was used to create inhomogeneities at the quite small length scale of a few lattice sites, while interactions were cancelled by coupling the atoms to an external magnetic ﬁeld tuned in proximity of a Feshbach resonance. For a discussion on these diﬀerent, yet complementary, approaches the reader can refer to [54]. In the following, we will focus on the LENS experiment [3] where Anderson localization in the bichromatic lattice was investigated. . 5 1. Experimental setup. – The experiment described in [3] has been performed with a Bose-Einstein condensate of 39 K. Interactions between the BEC atoms are cancelled by using a Feshbach resonance centered at a magnetic ﬁeld of 403 G, which features a remarkably broad “zero-crossing”, i.e. a region of magnetic ﬁeld in which the scattering length crosses the value a = 0 with a very small slope da/dB [55]. This small dependence of the scattering length on the applied magnetic ﬁeld allows for a precise and stable tuning of the s-wave interactions to zero. This is the ideal situation for studying single-particle eﬀects as Anderson localization, since the macroscopic BEC can be used to study “in parallel” the behavior of ≈ 105 particles all occupying the same single-particle state Ψ(x). Imaging the atoms then provides a direct measurement of the wave function density n(x) = |Ψ(x)|2 without need of averaging on many experimental realizations. In the experiment, a BEC of 39 K is ﬁrst produced at a ﬁnite value of the scattering length a = 180a0 obtained at a magnetic ﬁeld B = 396 G [56], then interactions are

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tuned to zero by changing the magnetic ﬁeld to B = 350 G. The non-interacting BEC is conﬁned in a quasi-1D geometry by a weakly focused laser beam which creates an “optical waveguide” where the atoms can move (see ﬁg. 20). The bichromatic lattice, aligned on the same direction of the optical waveguide, is realized by superimposing two standing waves created with two diﬀerent lasers operating at λ1 = 1032 nm (primary lattice) and λ2 = 862 nm (secondary lattice). The ratio between the two wavelengths gives an incommensurate β = k2 /k1 = λ1 /λ2 = 1.197 . . . (equivalent to γ = 0.197 . . .). We indicate with Δ the half-amplitude of the quasiperiodic energy modulation: with reference to eq. (32) the relation Δ = V2 /2 holds and the localization transition in the Aubry-Andr´e model is expected to occur at Δ/J = 2. We note that, in the experimental realization of bichromatic lattices, the mathematical deﬁnition of incommensurability (i.e. the wavelength ratio being an irrational number) should be substituted with a more practical deﬁnition. Since the lattice wavelengths can be measured with ﬁnite precision, the ratio λ2 /λ1 is always known as a rational number. From a theoretical point of view, it is important to consider that the ﬁnite size of the systems under investigation releases the constraints on the incommensurability: even a periodic potential (resulting from a commensurate ratio) does not show any periodicity if the system size is smaller than the period. The bichromatic lattice is thus eﬀectively incommensurate provided that the ratio between the wavelengths is far from a ratio between simple integer numbers. More precisely, a bichromatic lattice can be considered incommensurate whenever the resulting periodicity (if any) is larger than the system size. . 5 2. Absence of diﬀusion. – In his 1958 paper [18], Anderson formulated his theory of localization by considering a tight-binding model for an electron hopping in a disordered lattice. In particular, he moved from a relatively simple “gedanken-experiment”: he considered the initial situation in which the electron wave function is localized on a single site of the lattice, then he derived what the asymptotic behavior of the system should be for long evolution times. When the electron is placed in a uniform lattice, its wave function is expected to spread across the entire lattice, with its rms width σ expanding ballistically as σ ∝ t: asymptotically, for t → ∞, the wave function uniformly occupies the entire space, as the eigenstates of the uniform lattice are extended Bloch waves (see sketch in ﬁg. 21a). Instead, when disorder is introduced (depending on disorder strength and dimensionality), after a typical localization time the wave function stops expanding, reaching a stationary state in which σ is constant: the exponential decay of the wave function reﬂects the underlying exponentially-localized states of the disordered potential (see ﬁg. 21b). In the laboratory it is possible to realize a similar situation. As sketched in ﬁg. 20, BEC is initially conﬁned in a few sites of the bichromatic optical lattice by means of a red-detuned focused laser beam (not shown in ﬁgure) which creates an additional optical trap(12 ). When this conﬁnement is removed by switching oﬀ the laser beam, the atomic (12 ) The ground-state of the non-interacting BEC in this trap is a Gaussian with a typical size σ ≈ 5 μm.

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Fig. 21. – Pictorial representation of Anderson localization: the spheres represent the asymptotic probability for an electron, initially placed in the center of the lattice, to be detected at diﬀerent sites. a) In a uniform lattice the wave function expands across the entire lattice; b) in a disordered lattice Anderson localization sets is and the wave function is exponentially localized.

cloud is free to expand along the direction of the bichromatic lattice and can be detected after diﬀerent evolution times with in-situ absorption imaging, which directly gives the wave function density n(x) = |Ψ(x)|2 . The results are shown in ﬁg. 22a, where snapshots of the expansion are presented for diﬀerent evolution times up to 640 ms and for diﬀerent values of the perturbation strength Δ/J ranging from 0 to 7. Three diﬀerent regimes of expansion can be observed. For Δ = 0 the condensate expands ballistically in a monochromatic optical lattice, where the sites are all identical and the eigenstates are extended Bloch waves(13 ). For ﬁnite values of Δ the expansion is initially slowed down by the presence of the quasiperiodic energy modulation: the perturbation strength stays below the Aubry-Andr´e critical value for the localization transition, hence the eigenfunctions are still extended, but the induced wave function modulation results in a smaller “eﬀective” tunnelling rate. Then, for Δ/J = 7, the dynamics is completely halted and no expansion can be observed at all: this happens when the eigenstates of the system becomes spatially localized and no transport can take place on distances larger than the localization length. This transition can be investigated more quantitatively by plotting the rms size of the atomic cloud after a ﬁxed evolution time t as a function of the perturbing lattice strength Δ/J. The results presented in ﬁg. 22b for t = 750 ms show that the size decreases for increasing Δ/J up to a value Δ/J 7 above which the system size stays constant at the same initial value (dashed line). The diﬀerent point-styles in ﬁg. 22b refer to diﬀerent hopping amplitudes J. It is interesting to observe how the onset of localization happens at the same value of Δ/J regardless of the diﬀerent values of J for the diﬀerent datasets. (13 ) Actually a very weak residual harmonic conﬁnement is always present along the lattice direction due to the axial conﬁnment of the optical beams, but the eﬀects of this inhomogeneity can be observed only on much larger time and length scales.

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Fig. 22. – Expansion of a non-interacting BEC in an incommensurate bichromatic optical lattice. a) In situ absorption images of the condensate. The lattice direction is horizontal and the ﬁeld of view is 360 μm along that direction. The data refer to a main lattice with hopping amplitude J = h × 153 Hz. b) Rms size of the BEC as a function of Δ/J for diﬀerent hopping amplitudes J. The dashed line indicates the size at the beginning of the evolution. Taken from [3].

As a matter of fact, the Hamiltonian in eq. (36) has two terms which are proportional, respectively, to the hopping energy J and to the disorder strength Δ, therefore it is natural that the physics should depend only on the ratio of these two energy scales(14 ). The Aubry-Andr´e theory predicts a localization transition to occur at Δ/J = 2 (see . sect. 4 5). As already pointed out before, a sharp transition is expected only for a highly (14 ) This is indeed the behavior shown in ﬁg. 22b, the only diﬀerence between the datasets being the size of the cloud at small values of Δ/J, before the onset of localization. This is not strange, however, since the plot refers to a ﬁxed evolution time t which is the same for the diﬀerent datasets, therefore the evolution time in natural units t/J is not the same.

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Fig. 23. – Exponential localization of a non-interacting BEC in an incommensurate bichromatic optical lattice. The points show the exponent α in eq. (41) which ﬁts the wave function tails after a 750 ms long expansion in the bichromatic lattice for diﬀerent perturbation strength Δ/J. Taken from [3].

√ incommensurate value of γ, e.g. for the golden ratio γ = (3 − 5)/2 0.382 . . .. Here γ = 0.197 . . . and the transition becomes slightly smeared out to larger values of Δ/J. At Δ/J = 7 complete localization occurs and the localization length gets down to few lattice sites. We can learn about the physics underlying this localization behavior by analyzing the shape of the expanding wave function. The tails of the wave function, extracted from ﬁg. 22, have been ﬁtted with a generalized exponential function (41)

Ψ(x) = A exp[−γ(x − x0 )α ].

In ﬁg. 23 we show the ﬁtted exponent α for diﬀerent values of Δ/J. The data clearly show that the exponent changes from 2 to 1 as the system scans the localization crossover. The value α = 2 reﬂects the Gaussian wave function of the BEC initially trapped in a harmonic trap, which stays Gaussian during the whole evolution. The value α = 1 in the localized regime reﬂects the appearance of exponentially decreasing tails, which are a signature of Anderson-localized states. Also here, the crossover is completed at a value Δ/J 7 as before. . 5 3. Imaging the localized states. – The suppression of the expansion and the observation of exponentially decreasing tails in the wave function is a ﬁrst proof of Anderson localization. We can get more insight into the localization transition by observing the eigenstates of the system. To this aim, we take advantage of a trick which is often played in ultracold atoms experiment, i.e. observing the BEC wave function in time-of-ﬂight after switching oﬀ all the external potentials. When the conﬁnment is removed, the atomic wave function undergoes a free-space evolution in the absence of any potential (except for gravity, which only induces an uniform center-of-mass acceleration). In the asymptotic

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limit of long expansion time texp → ∞, the wave function in real space Ψ(x; texp ) maps ˜ 0 (k): the initial wave function in momentum space Ψ (42)

˜ 0 (mx/texp ). Ψ(x; texp ) ∝ Ψ

. We have already discussed in sect. 4 5 that the localization transition can indeed be observed also in momentum space: while the extended states for sub-critical quasiperiodic perturbation have a momentum distribution made by narrow δ-like peaks, the localized states above the transition show a much broader unstructured momentum distribution. This is a direct consequence of the Heisenberg uncertainty principle: the more the wave function is localized in real space, the more it is delocalized in momentum space. ˜ 0 (k)|2 is measured by slowly In the experiment, the momentum distribution P (k) = |Ψ loading the BEC in the bichromatic lattice in order to populate only the ground state (or, at most, just a few localized states) and then switching oﬀ both the lattice and the optical trap. In ﬁg. 24a we show the momentum distributions measured along the lattice direction for diﬀerent values of Δ/J ranging from 0 (top) to 25 (bottom). It is evident how the distribution broadens as the strength of the quasiperiodic modulation is made stronger: from the multipeaked pattern of the monochromatic lattice (top), the momentum distribution changes to a single broad curve in the localized regime (bottom). In ﬁg. 24b we show, for comparison, the theoretical momentum distribution calculated for the ground state wave function, showing an excellent agreement with the experimental data. In ﬁg. 24c-d we study the localization crossover by plotting two quantities which characterize the momentum distribution: the rms width of the zero-momentum peak and the “visibility” of the pattern deﬁned as [P (2k1 ) − P (k1 )]/[P (2k1 ) + P (k1 )]. While the rms width increases with increasing Δ/J (ﬁg. 24c), the visibility suddenly decreases (24d). The localization transition is particularly evident in the visibility, which shows a sharp knee around Δ/J=7. . 5 4. Eﬀect of interactions. – In the previous sections we have focused on the localization of a non-interacting ultracold gas in a bichromatic potential. Starting from the conceptually simple, although yet mathematically rich, Anderson scenario, the next level of investigation naturally involves the consideration of the interacting case. Interactions between particles are indeed a fundamental constituent of matter and the consideration of their eﬀect is of crucial importance to describe the physics of condensed-matter disordered systems. The eﬀect of weak repulsive interactions on the localization of a bosonic gas has been investigated in a very recent LENS experiment [57], where the interactions between the atoms of a 39 K BEC were ﬁnely controlled by means of a Feshbach resonance. Similarly to . the experiments described in sect. 5 3, detailed information on the localization properties was extracted in [57] by analyzing the time-of-ﬂight density distribution of the atoms after sudden release from the trap. Figure 25 shows two diﬀerent plots, in which the width of the momentum distribution and the exponent α ﬁtting the wave function tails are presented as a function of

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Fig. 24. – Momentum-resolved study of the localization of a non-interacting BEC in a bichromatic lattice. a) Time-of-ﬂight measurement of the momentum distribution for diﬀerent perturbation strength Δ/J = 0, 1.1, 7.2 and 25 (from top to bottom). b) Corresponding theoretical distributions. c) Rms width of the central peak of the momentum distribution (circles: experiment, lines: theory). d) Visibility of the peaked structure in the momentum distribution (see text for deﬁnition, circles: experimental data, lines: theory). Taken from [3].

the quasiperiodic perturbation amplitude Δ and of the interaction strength, here denoted with the interaction energy Eint (see [57] for details). These colour-scale maps can be compared with the similar plots shown in ﬁg. 24c and ﬁg. 23 obtained for the non-interacting system. Figure 25 clearly shows that the localization crossover is upshifted towards larger values of Δ when interactions are introduced into the system. As it could be intuitively expected, repulsive interactions ﬁght against disorder and contribute to delocalize the eigenstates. A simple energetic argument suggests that this happens when the interaction energy Eint becomes larger than the average energy sep-

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Fig. 25. – Eﬀect of weak repulsive interactions on the localization of a BEC in a bichromatic lattice. a) Rms width of the momentum distribution (see ﬁg. 24c for comparison). b) The decay exponent of the wave function deﬁned in eq. (41) (see ﬁg. 23). Both the quantities are extracted from the experimental time-of-ﬂight images. The white dashed line indicates the region in which the interaction energy Eint equals the average energy separation between localized states. Adapted from [57].

aration δE between the lowest-lying single-particle localized states [57]. If the system is delocalized by the eﬀect of interactions, it is possible to restore localization by increasing the perturbation strength Δ in order to lift the energy separation between localized states to be larger than Eint . The region where Eint matches δE is indicated as a white dashed line in ﬁg. 25. The nature of the localized phase in the presence of interactions presents some differences with respect to the single-particle Anderson scenario, as suggested by recent theoretical works [58-61]. For weak values of the interactions the localized state takes the form of an Anderson glass, in which the ground state results from the occupation of several single-particle localized states. In this regime interactions are strong enough to push atoms away from the single-particle localized states, but not yet suﬃciently strong to connect these spatially separated states and restore superﬂuidity in the system. For larger values of the interactions, phase coherence starts to be restored and a fragmented

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BEC forms, in which diﬀerent single-particle localized states merge together building up local coherent fragments. The crossover between the Anderson glass and the fragmented BEC regime is explored in [57] with additional experimental observations, including the measurement of the ﬁrst-order correlation function, which gives information on the phase coherence. The physics of interacting disordered systems is an extremely interesting subject of research. Even from a theoretical point of view, the problem is diﬃcult to address and many questions have not yet been answered. In this section we have only introduced the weakly interacting regime, which has been recently investigated also in diﬀerent physical systems (e.g. in the localization of light in photonic crystals with Kerr nonlinearities [62]). The interplay between disorder and interactions gets particularly tricky in the strongly interacting regime, when the interaction energy stops from just being a perturbation of the atoms total energy. When this happens, many-body correlations set in and new physics emerge: the next sections will be devoted to this regime. 6. – Strongly interacting atoms in bichromatic lattices In the previous sections we have discussed the physics of disordered non-interacting or weakly interacting bosonic gases. A theoretical description of the weakly interacting regime can be given in terms of the semiclassical Gross-Pitaevskii equation (19), which describes the propagation of nonlinear matter waves. When interactions are strong, however, this mean-ﬁeld description is not capable to fully explain the behavior of the system. A more appropriate description is provided by a full quantum theory, taking into account quantum correlations between the particles. The quantum state of an interacting gas of identical bosons in a disordered lattice potential is well described by the disordered Bose-Hubbard Hamiltonian [29] (43)

ˆ = −J H

j,j

ˆb†ˆbj + U n ˆ j (ˆ nj − 1) + j n ˆj , j 2 j j

where the hopping energy J and the interaction energy U are deﬁned according to the . deﬁnitions given in sect. 3 4, while j ∈ [−Δ/2, Δ/2] is a site-dependent energy accounting for inhomogeneous external potentials superimposed on the lattice (see sketch in ﬁg. 26)(15 ). . In sect. 3 4 we have already discussed the superﬂuid-Mott transition, which takes place in a uniform lattice when the interaction energy U exceeds the hopping energy J, causing a localization of the atomic wave functions in single lattice sites. The phase diagram of the system depends on the chemical potential μ (related to the atomic density) and shows the existence of MI lobes with integer number of atoms per site. In the left (15 ) In this last section of the paper we will make use of a diﬀerent deﬁnition of Δ: instead of using it to denote the half-amplitude of the disordered distribution of site energies, it will denote the full-amplitude (i.e. for a quasiperiodic system, Δ = V2 ).

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Fig. 26. – Sketch of interacting bosons in a disordered lattice.

graph of ﬁg. 27 we show a qualitative sketch of the phase diagram for a 3D system, as ﬁrst derived by M. P. A. Fisher et al. in [29]. In the presence of a disordered external potential the additional energy scale Δ enters the description of the system and is responsible for the existence of a new quantum phase. In the presence of weak disorder the MI lobes in the phase diagram progressively shrink and a new Bose glass (BG) phase appears (central graph of ﬁg. 27), eventually washing away the MI region for Δ > U (right graph of ﬁg. 27) [29]. In a very simpliﬁed view, the properties of a Bose glass are half-way from a Mott insulator to a superﬂuid: A Bose glass is an insulating state with no long-range phase coherence (as the Mott insulator), although it is compressible and supports gapless excitations (as a superﬂuid). The Bose glass phase has been ﬁrst identiﬁed in [64] in the context of strongly interacting 1D bosonic systems. In the ’90s it was widely studied in connection to the superﬂuid-insulator transition observed in many condensed-matter systems, such as 4 He adsorbed on porous media [65], thin superconducting ﬁlms [66], arrays of Josephson junctions [67] and high-temperature superconductors [68, 69]. The possible realization of a

Fig. 27. – Qualitative zero-temperature phase diagram for a disordered system of lattice interacting bosons. Three phases can be identiﬁed: a superﬂuid (SF), a Mott insulator (MI) and a Bose glass (BG). Taken from [63].

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Fig. 28. – A strong 2D optical lattice along direction yˆ and zˆ creates a two-dimensional array of independent 1D Bose gases along direction x ˆ. A bichromatic optical lattice along direction x ˆ is used to realize a quasidisordered Bose-Hubbard model [63].

Bose glass in a system of ultracold bosons in a disordered lattice has been ﬁrst proposed in [58, 70]. More recently, the phase diagram of this system has been derived in other theoretical papers, considering also ﬁnite-temperature eﬀects [71, 72] and the possible realization of a Bose glass with incommensurate bichromatic lattices [73, 74]. The Bose glass is just the simplest disordered quantum phase that can be realized in the strongly interacting regime. When atoms of diﬀerent species, or diﬀerent internal (spin) states of the same species, are considered, more complicated models can be experimentally realized and new disordered quantum phases can emerge. Atomic Bose/Fermi mixtures, in particular, represent a versatile system in which many diﬀerent disordered models can be realized [75, 76], from fermionic Ising spin glasses to models of quantum percolation. . 6 1. Towards a Bose glass. – Experiments with disordered bosons in the strongly interacting regime started at LENS in 2006. The system under investigation was a collection of 87 Rb 1D ultracold gases in a bichromatic optical lattice, as schematically sketched in ﬁg. 28. The main optical lattice was used to induce the transition from a weakly interacting superﬂuid to a strongly correlated Mott insulator. The non-commensurate perturbing lattice was then used to add controlled quasidisorder to the perfect crystalline structure of the MI phase. . As already introduced in sect. 3 4, the excitation spectrum is an important observable that can be measured in order to characterize the quantum state of the system. While a SF supports the excitation of gapless long-wavelength modes, the excitation spectrum of a MI shows the presence of an energy gap. The latter behavior can be exempliﬁed in the

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Fig. 29. – Excitations in the deep insulating phases. a) In a Mott insulator the tunneling of one boson from a site to a neighboring one has an energy cost ΔE = U . b) In the disordered case the excitation energy is ΔE = U ± Δj , that becomes a function of the position. In the Bose glass state, in which |Δj | > U , an inﬁnite system could be excited at arbitrarily small energies and the energy gap would disappear.

strong MI limit (U/J 1), where it is easy to observe that the lowest-energy excitation— the hopping of a particle from a site to a neighboring one, or, in other words, the creation of a particle-hole pair—has an energy cost U , corresponding to the interaction energy of a pair of mutually repelling atoms sitting on the same site (see ﬁg. 29). By exploiting the possibility of time-modulating the lattice potential, as ﬁrst realized in [34], it is possible to probe the excitation spectrum and study how it is modiﬁed by the presence of disorder. In ﬁg. 30a we show the excitation spectrum of a Mott insulator measured in the LENS experiments [63]. The plot shows a well resolved resonance at energy U , which is distinctive of the MI state, and a second resonance at energy 2U . While the physical origin of the excitation peak at U is the tunneling of particles between sites with the same occupancy, the second peak at 2U can be ascribed to tunneling at the boundary between MI regions with diﬀerent site occupancy (that are present due to the inhomogeneity of the conﬁned sample). When increasing disorder the experimental data in ﬁg. 30b-d show a broadening of the resonance peaks, which eventually become undistinguishable when Δ ≈ U . As a matter of fact, the presence of disorder introduces inhomogeneous energy diﬀerences Δj = (j −j−1 ) ∈ [−Δ, Δ] between neighboring sites (see bottom of ﬁg. 29). As a consequence, the tunneling of a boson through a potential barrier costs U ±Δj , that becomes a function of the position [43]. The excitation energy is not the same for all the bosons, diﬀerently from the pure MI case, and the resonances become inhomogeneously broadened, as can be observed in the experimental spectra at weak disorder (Δ < U ) shown in ﬁg. 30b,c [63]. This broadening is in agreement with a semi-classical model [43] and has been predicted in theoretical works [77] studying the dynamical response of a 1D bosonic gas in a superlattice potential when a periodic amplitude modulation of the lattice is applied. Eventually, when Δ U , one expects that an inﬁnite system can be excited at arbitrarily small energies and that the energy gap would shrink to zero. When this

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Fig. 30. – Excitation spectra of the atomic system in a Mott insulator state for increasing height of the disordering lattice. The resonances are lost and the excitation spectrum becomes ﬂat. The ﬁgure is adapted from [63].

happens, nearby sites become degenerate and regions of local superﬂuidity with shortrange coherence appear in the system, resulting in a Bose glass state where there is no gap but the system remains globally insulating. From the experimental point of view, additional information on the nature of the many-body ground state can be acquired by analyzing the density distribution of the atoms released from the lattice after a time of ﬂight. Long-range coherence in the sample results in a density distribution with sharp interference peaks at a distance proportional . to the lattice wave vector, as discussed in sect. 3 4. The visibility of these peaks provides a measurement of phase coherence. When increasing the height of the main lattice, a progressive loss of long-range coherence has been reported in [63] indicating the transition from a superﬂuid to an insulating state, also in the presence of disorder. The combination of the excitation spectra measurements and the time of ﬂight images indicates that, with increasing disorder, the system realized in [63] goes from a MI to a state with vanishinglong-range coherence and a ﬂat density of excitations. The concurrence of these two properties cannot be found in either a SF or an ordered MI, and strongly suggests the formation of a Bose glass, which is indeed expected to appear for Δ U . From the theoretical side, numerical studies [74,78] have considered the problem of 1D interacting bosons in quasiperiodic lattices, working out the phase diagrams (which include the presence of Bose glass and incommensurate “band insulating”/“charge density wave” regions) and studying how the diﬀerent phases aﬀect experimentally detectable signals. From the experimental side, for an exhaustive characterization of this novel disordered state, new detection schemes have to be implemented in order to have access to additional observables. This necessity is not only restricted to the study of disordered systems, being a more general issue shared by the experimental investigation of diﬀerent strongly interacting lattice systems, including, e.g., systems with magnetic ordering or mixtures of diﬀerent species.

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. . Before discussing in sect. 6 2 and 6 3 the application of novel detection techniques to the study of disordered systems, we note that recently the experimental investigation of disordered ultracold bosons in the strongly interacting regime has been performed also in the group of B. DeMarco at University of Illinois. In [79] the authors used a speckle potential with short correlation length to realize a 3D disordered Bose-Hubbard model and investigate the reduction of condensate fraction as a function of the Hamiltonian parameters in the crossover from the superﬂuid to the insulating state. Transport properties in the same system were investigated in [80] by exciting center-of-mass oscillations of the trapped gas: the authors observed that the addition of disorder drives the system into an insulating state where strong dissipation of the atomic motion sets in, in accordance with the expected behavior for an insulating Bose-glass phase. . 6 2. Noise correlations. – In a recent experiment performed at LENS [81] noise interferometry has been used to study interacting 87 Rb bosons in the bichromatic lattice. This detection technique, originally proposed in [82], is based on the analysis of the spatial density-density correlations of the atomic shot noise after time-of-ﬂight. These correlations are based on the Hanbury Brown & Twiss eﬀect [83]: if two identical particles are released from two lattice sites, the joint probability of detecting them in two separate positions (e.g., imaging them on two separate pixels of a CCD camera) depends on the distance between the detection points. These correlations, arising from quantum interference between diﬀerent detection paths, were ﬁrst observed for bosons in a Mott insulator state [84] and then also for band-insulating fermions [85]. The sign of the correlations depends on the quantum statistics: while bosons show positive correlations (due to their tendency to bunch, i.e. to arrive together at the detectors), fermions exhibit negative correlations (due to the antibunching, consequence of the Pauli exclusion principle). In the case of a bosonic Mott insulator, one observes positive density-density correlation peaks at a distance proportional to the lattice wave vector k1 , as shown in the ﬁrst image of the bottom row of ﬁg. 31 for the recent experiment at LENS. In [81] noise correlations have been measured, starting from a Mott insulator state, for increasing heights s2 of the secondary lattice. The absorption images after time of ﬂight do not present signiﬁcative diﬀerences, as shown in the top row of ﬁg. 31, and demonstrate the absence of ﬁrst-order (phase) coherence of the atomic system in the insulating state, even in the presence of the secondary lattice. However, second-order (density) correlations turn out to be signiﬁcantly diﬀerent with varying s2 , as illustrated in the noise correlation functions plotted in the bottom row. More precisely, with increasing s2 , one observes the appearance of additional correlation peaks at a distance proportional to the wave vector k2 of the secondary lattice and to the beating between the two lattices k1 − k2 . These peaks have to be associated with the redistribution of atoms in the lattice sites as the disordering lattice is strengthened: the MI regions characterized by uniform ﬁlling are destroyed and atoms rearrange in the lattice giving rise to a state with non-uniform site occupation, which follows the periodicity of the secondary lattice. The redistribution of atoms is then quantitatively detected by measuring the height of the additional correlation peaks.

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Fig. 31. – Top: time-of-ﬂight absorption images of atoms initially in a Mott insulator state for increasing height of the secondary lattice s2 . Bottom: density-density correlation functions corresponding to the pictures above. The point in the center corresponds to the correlation function at zero distance (the black ring is an artifact of image processing). The peak marked with k1 corresponds to density correlations on the length scale of a lattice site, while the additional correlation peaks marked as k2 and k1 − k2 arise for large s2 from the destruction of the Mott domains and the redistribution of the atoms in the lattice. The data is adapted from [81].

Noise correlations thus prove to be a tool to extract important information on the lattice site occupation, which is connected to the second-order correlation function of the many-body state. The appearance of similar correlation peaks was predicted in theoretical works for hard-core bosons [86] and soft-core bosons [74] in bichromatic lattices. Future works will study the possibility to use noise interferometry to get additional insight on the nature of the disordered insulating states produced in the experiment, in particular in connection with the realization of a Bose glass phase. . . 6 3. Bragg spectroscopy. – The lattice modulation technique discussed in sect. 6 1 for the measurement of the excitation spectrum of ultracold lattice gases is aﬀected by important limitations. In order to detect an excitation signal in the Mott insulator regime the lattice has to be modulated by a large amount and the atomic response is out of the linear regime. We conclude this paper discussing a diﬀerent experimental method that can be used as a cleaner probe of excitations in a strongly interacting gas. In condensed-matter physics, scattering of particles or radiation is the most common way to extract information on the properties of a material. Inelastic scattering is particularly important to study dynamic properties. If a monochromatic beam of particles with momentum k and energy ω hits the sample and particles are scattered with momentum k and energy ω , momentum and energy conservation require that the sample has gained momentum δk = k − k and energy δω = ω − ω . In principle, if one is able to detect the intensities of all the scattering channels {k , ω }, this could provide a measurement of the

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dynamic structure factor S(δk, δω), which completely characterizes the excitations of the system. This is the quantity measured, for instance, in inelastic scattering of decelerated neutrons, which is a valuable technique for the investigation of magnetic and structural excitations in condensed-matter systems. In the case of light, inelastic scattering processes are commonly known as Raman scattering. In a Stokes (anti-Stokes) process, the frequency of the photons scattered by the sample is lower (higher) than the frequency of the incident photons, owing to the transition to higher- (lower-) energy conﬁgurations. Raman spectroscopy is therefore a precious tool for investigating the internal structure and excitations of complex molecules or aggregates. The use of light scattering as a diagnostic tool for ultracold quantum gases, generally referred to as Bragg scattering, has been initiated ten years ago with seminal experiments at NIST [87] and MIT [88]. In later years, it has been used to study the excitation spectrum of weakly interacting Bose-Einstein condensates (BECs) [89] and, very recently, experiments have started to investigate the regime of strong interactions between the particles, as it happens in BECs close to a Feshbach resonance [90] or in Fermi gases across the BEC/BCS crossover [91]. In a Bragg scattering experiment the atoms are illuminated by two Bragg beams with wave vectors and frequencies (k, ω) and (k , ω ). These two beams induce a stimulated inelastic Raman scattering process, in which the energy and momentum transfer can be selected by changing the relative detuning between the two beams and their relative angle θ, according to (44)

δω = ω − ω ,

(45)

δk 2k sin θ/2.

Instead of detecting the scattered photons, it is possible to measure the amount of energy transferred to the system after thermalization of the excitations, which in cold-atoms systems can be easily obtained by time-of-ﬂight imaging techniques (for more details, see refs. [92-94]). We now consider the application of Bragg spectroscopy to systems of strongly interacting 1D bosonic atoms in the crossover from a superﬂuid to a Mott insulator state [32]. A monochromatic optical lattice is used in a setup similar to the one presented in ﬁg. 28 to control the state of the individual 1D gases. In this 1D conﬁguration the system exhibits a crossover between a correlated superﬂuid and a Mott-insulating state around a critical value U/2J 4.5 [95]. In ﬁg. 32 we show the evolution of the Bragg spectra for diﬀerent lattice heights ranging from s = 0 (U/2J = 0) to s = 15 (U/2J 50) and a momentum transfer δk = 0.96(3)kL along the lattice direction. It is evident that, as the height of the lattice is increased and the gas is driven into the Mott insulating regime, the overall response to the Bragg excitation is strongly suppressed and a ﬁne structure with multiple peaks appears. The transition point can be estimated by measuring the rms width of these spectra, which is shown in ﬁg. 33a. This quantity clearly exhibits a minimum, in good agreement with the theoretical transition point [95], which is shown as a vertical bar around s = 5. This minimum results from a combined eﬀect: while the

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Fig. 32. – Bragg spectroscopy of strongly interacting 1D Bose gases across the SF to MI transition, taken for a momentum transfer δk = 0.96kL and diﬀerent values of lattice height s and U/2J. Lines are guides to the eye. Note the amplitude drop and the appearance of a multipeaked structure when increasing the lattice height (vertical scale). Taken from [93].

response in the superﬂuid regime is shifted to lower energies as the lattice is increased (because the energy band gets narrower), new features at higher energies (which are not resolved in the crossover region) emerge in the Mott insulating regime and lead to an increase of the width. These features are well-resolved and clearly observable in the spectrum in ﬁg. 33b, corresponding to U/2J = 30. Three main components are observable in this spectrum. There is a main peak centered around frequency U/h (vertical dotted line), corresponding to the creation of particle-hole excitations in the Mott insulating domains. These excitations have an energy gap (related to the uncompressible nature of the Mott state) which approaches U at large U/J. A smaller peak is also clearly observable at twice the frequency (close to 2U/h), which can be related to the presence of inhomogeneities and “defects” in the system, either due to the shell structure induced by the trapping potential or to an imperfect adiabatic loading of the lattice. Finally, a low-frequency component is present at energies below the Mott insulator gap. This component can be associated to excitations within the superﬂuid (or normal) domains of the system, which are not gapped and have a bandwidth 4nJ (where n is the ﬁlling factor), represented as the vertical stripe in ﬁg. 33b. Excitations above the upper edge of this region and below the Mott insulator gap could be related to the ﬁnite temperature of the gas and to the strong correlations in the superﬂuid component(16 ). (16 ) A complete theoretical picture including the combined eﬀects of ﬁnite U/J, ﬁnite temperature and trapping potential is not yet available.

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Fig. 33. – Bragg spectroscopy of strongly interacting 1D Bose gases across the SF to MI transition. a) Rms width of the Bragg spectra for diﬀerent values of lattice height: the vertical bar corresponds to the theoretical value for the Mott transition point. b) Bragg spectrum for s = 13 with the indication of the interaction energy U and of the bandwidth of the superﬂuid excitation. Adapted from [93].

An important diﬀerence between Bragg spectroscopy and the lattice modulation tech. nique described in sect. 6 1 is that the former can be performed at non-zero momentum transfer. As a matter of fact, the response of the Mott state is strongly dependent on the momentum transfer, vanishing at δk = 0 and being maximum at δk = kL [96, 97]. While signals from a Mott insulator could be observed with the lattice modulation technique only for large modulation amplitudes of ≈ 20%–30% out of the linear regime of excitation, in the case of the spectra shown in ﬁg. 33b the amplitude of the travelling lattice formed by the Bragg beams is only < 5% of the lattice height. This means that the parameters U and J of the system are not signiﬁcantly changed during the excitation and that the measured spectra can be directly compared to the dynamic structure factor describing the excitations in the perturbative regime where linear-response theory holds. These advantages, here demonstrated for strongly interacting bosons at the superﬂuidMott crossover, could make Bragg spectroscopy an important tool to better characterize the properties of Bose-glass and diﬀerent disordered insulating states as well. 7. – Concluding remarks Besides constituting the focus of contemporary atomic physics, ultracold atoms are important resources for diﬀerent areas of physical research, since they can be used as “quantum simulators” to implement and investigate well-controlled Hamiltonian models originally introduced to describe diﬀerent physical systems. One spectacular example of this possibility is given by ultracold atoms in optical lattices, i.e. perfect crystals made of laser light, in which atoms behave as the conduction electrons in a solid and allow the investigation of transport phenomena typical of solid-state physics.

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In this paper we have described this perspective by illustrating the physics of atoms trapped in bichromatic optical lattices. We have mostly considered the case of incommensurate bichromatic potentials, which are an intriguing example of structures which are ordered and disordered at the same time. In particular, we have focused on their “disordered” nature, which comes from the absence of any periodicity and is responsible for the emergence of localized states. Thanks to these properties, incommensurate lattices have been used as a powerful tool to study the emergence of Anderson localization, which we have demonstrated in recent experiments. This quantum simulation perspective is even more attractive when one considers interactions between the particles. The study of interacting atomic systems is indeed particularly interesting, especially in the regime of strong interactions where highlycorrelated atomic states can be created. The interplay between disorder and interactions has a fundamental role in determining the state of the system, with strong implications in contemporary condensed-matter research. First investigations of this regime have been started and much more is likely to come in the near future, thanks to new tools and detection techniques which are currently under development. ∗ ∗ ∗ This work was ﬁnancially supported by ERC DISQUA Project, EU FP7 Programs, ENTE CRF. We acknowledge the precious eﬀort of all the LENS colleagues contributing to the research works presented in this paper. REFERENCES [1] Morsch O. and Oberthaler M., Rev. Mod. Phys., 78 (2006) 179. [2] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885. [3] Roati G., D’Errico C., Fallani L., Fattori M., Fort C., Zaccanti M., Modugno G., Modugno M. and Inguscio M., Nature, 453 (2008) 895. [4] Chu S., Rev. Mod. Phys., 70 (1998) 685. [5] Cohen-Tannoudji C. N., Rev. Mod. Phys., 70 (1998) 707. [6] Phillips W. D., Rev. Mod. Phys., 70 (1998) 721. [7] Cornell E. A. and Wieman C. E., Rev. Mod. Phys., 74 (2002) 875. [8] Ketterle W., Rev. Mod. Phys., 74 (2002) 1131. [9] Dalfovo F., Giorgini S., Pitaevskii L. and Stringari S., Rev. Mod. Phys., 71 (1999) 463. [10] Inguscio M., Stringari S. and Wieman C. E. (Editors), Bose-Einstein Condensation in Atomic Gases, Proceedings of the International School of Physics, “Enrico Fermi”, Course CXL (SIF, Bologna; IOS, Amsterdam) 1999. [11] Cohen-Tannoudji C., in Fundamental systems in quantum optics, edited by Dalibard J., Raimond J.-M. and Zinn-Justin J. (Elsevier, Amsterdam) 1992, pp. 1-164. ¨ller M. and Ovchinnikov Y. B., Adv. At. Mol. Opt. Phys., 42 [12] Grimm R., Weidemu (2000) 95. [13] Allen L. and Eberly H. J., Optical Resonance and Two Level Atoms (Dover Publications) 1987. [14] Ben Dahan M., Peik E., Reichel J., Castin Y. and Salomon C., Phys. Rev. Lett., 76 (1996) 4508.

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Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-233

Exploring strongly correlated ultracold bosonic and fermionic quantum gases in optical lattices(∗ ) I. Bloch Fakult¨ at f¨ ur Physik, Ludwig-Maximilians-Universit¨ at Schellingstr. 4/II, 80798 M¨ unchen, Germany Max-Planck-Institut f¨ ur Quantenoptik - Hans-Kopfermann Str. 1 85748 Garching b. M¨ unchen, Germany

Summary. — We review several experimental aspects of ultracold bosonic and fermionic quantum gases in optical lattices. After introducing optical lattices, we use the superﬂuid-Mott insulator transition of ultracold bosonic quantum gases, to highlight the physics of strongly correlated quantum systems. We discuss the coherence properties and recent measurements of the shell structure in the Mott insulating phase. Subsequently, dynamical interaction eﬀects in the collapse and revival of the matter wave ﬁeld of a BEC are analyzed, followed by a discussion on interacting fermions, superexchange-mediated spin-spin interactions and quantum noise correlation interferometry in optical lattices.

(∗ ) This paper follows the presentation given by I. Bloch, Strongly correlated quantum phases of ultracold atoms in optical lattices, in Proceedings of the International School of Physics “Enrico Fermi”, Course CLXIV, “Ultracold Fermi Gases”, edited by M. Inguscio, W. Ketterle and C. Salomon (IOS Press, Amsterdam and Societ`a Italiana di Fisica, Bologna) 2007, pp. 715-749. c Societ` a Italiana di Fisica

233

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I. Bloch

1. – Introduction Ultracold quantum gases in optical lattices form almost ideal conditions to analyze the physics of strongly correlated quantum phases in periodic potential [1-3]. Such strongly correlated quantum phases are of fundamental interest in condensed matter physics, as they lie at the heart of topical quantum materials, such as high-Tc superconductors and quantum magnets, which pose a challenge to our basic understanding of interacting many-body systems. Quite generally, such strongly interacting quantum phases arise, when the interaction energy between two particles dominates over the kinetic energy of the two particles. Such a regime can either be achieved by increasing the interaction strength between the atoms via Feshbach resonances [4], or by decreasing the kinetic energy, such that eventually the interaction energy is the largest energy scale in the system. The latter can for example simply be achieved by increasing the optical lattice depth. This paper tries to give an introduction into the ﬁeld of optical lattices and the physics of strongly interacting quantum phases. A prominent example hereof is the superﬂuid-toMott-insulator transition [5-9], which transforms a weakly interacting quantum gas into a strongly correlated many-body system. Dominating interactions between the particles are in fact crucial for the Mott insulator transition. The paper is structured as follows: in sect. 2 the basics of trapping neutral atoms in optical lattice potentials are introduced; in sect. 3 we turn to a discussion of the superﬂuid-to-Mott-insulator transition; sect. 4 analyzes the inﬂuence of the interactions between the particles on the stability of the coherent matter wave ﬁeld of a BEC, leading to a pronounced series of collapses and revivals of its macroscopic wave function; in sect. 5 the physics of repulsively interacting fermionic atoms in optical lattices is discussed; sect. 6 explains how spin-spin interactions arise due to superexchange processes on a lattice. In the same section it is shown how these superexchange interactions can be detected and controlled using ultracold atoms. Finally, sect. 7 explains how quantum noise correlation interferometry can be used to reveal quantum phases on a lattice using Hanbury-Brown & Twiss type noise correlation measurements. 2. – Optical lattices . 2 1. Optical dipole force. – In the interaction of atoms with coherent light ﬁelds, two fundamental forces arise. The so-called Doppler force is dissipative in nature and can be used to eﬃciently laser cool a gas of atoms and relies on the radiation pressure together with spontaneous emission. The so-called dipole force on the other hand creates a purely conservative potential in which the atoms can move. No cooling can be realized with this dipole force, however if the atoms are cold enough initially, they may be trapped in such a purely optical potential [10]. How does this dipole force arise? We may grasp the essential points through a simple classical model, in which we view the electron as harmonically bound to the nucleus with oscillation frequency ω0 . An external oscillating electric ﬁeld of a laser E with frequency

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235

ωL can now induce an oscillation of the electron resulting in an oscillating dipole moment d of the atom. Such an oscillating dipole moment will be in phase with the driving oscillating electric ﬁeld, for frequencies much lower than an atomic resonance frequency and 180◦ out of phase, for frequencies much larger than the atomic resonance frequency. The induced dipole moment again interacts with the external oscillating electric ﬁeld, resulting in a dipole potential Vdip experienced by the atom [10] (1)

1 Vdip = − dE, 2

where · denotes a time average over fast oscillating terms at optical frequencies. From eq. (1) it becomes immediately clear that for a red detuning (ωL < ω0 ), where d is in phase with E, the potential is attractive, whereas for a blue detuning (ωL > ω0 ), where d is in 180◦ out of phase with E, the potential is repulsive. By relating the dipole moment to the polarizability α(ωL ) of an atom and expressing the electric ﬁeld amplitude E0 via the intensity of the laser ﬁeld I, one obtains for the dipole potential (2)

Vdip (r) = −

1 Re(α)I(r). 20 c

A spatially dependent intensity proﬁle I(r) can therefore create a trapping potential for neutral atoms. For a two-level atom a more useful form of the dipole potential may be derived within the rotating wave approximation, which is a reasonable approximation provided that the detuning Δ = ωL − ω0 of the laser ﬁeld ωL from an atomic transition frequency ω0 is small compared to the transitions frequency itself Δ ω0 . Here one obtains [10] (3)

Vdip (r) =

3πc2 Γ I(r), 2ω03 Δ

with Γ being the decay rate of the excited state. Here a red detuned laser beam (ωL < ω0 ) leads to an attractive dipole potential and a blue detuned laser beam (ωL > ω0 ) leads to a repulsive dipole potential. By simply focussing a Gaussian laser beam, this can be used to attract or repel atoms from an intensity maximum in space (see ﬁg. 1). For such a focussed Gaussian laser beam the intensity proﬁle I(r, z) is given by (4)

I(r, z) =

2 2 2P e−2r /w (z) , 2 πw (z)

2 ) is the 1/e2 radius depending on the z-coordinate, zR = where w(z) = w0 (1 + z 2 /zR 2 πw /λ is the Rayleigh length and P is the total power of the laser beam. Around the intensity maximum a potential depth minimum occurs for a red detuned laser beam, leading to an approximately harmonic potential of the form " 2 2 * z r . − (5) Vdip (r, z) ≈ −V0 1 − 2 w0 zR

236

I. Bloch D

E

F

Fig. 1. – (a) Gaussian laser beam together with the corresponding trapping potential for a red detuned laser beam. (b) A red detuned laser beams leads to an attractive dipole potential, whereas a blue detuned laser beam leads to a repulsive potential (c).

This harmonic conﬁnement is characterized by radial ωr and axial ωz trapping fre2 ). quencies ωr = (4V0 /mw02 )1/2 and ωz = (2V0 /mzR Great care has to be taken to minimize spontaneous scattering events, as they lead to heating and decoherence of the trapped ultracold atom samples. For a two-level atom, the scattering rate Γsc (r) can be estimated [10] through 3πc2 Γsc (r) = 2ω03

(6)

Γ Δ

2 I(r).

From eqs. (3), (6) it can be seen that the ratio of scattering rate to optical potential depth can always be minimized by increasing the detuning of the laser ﬁeld. In practice, however, such an approach is limited by the maximum available laser power. For experiments with ultracold quantum gases of alkali atoms, the detuning is typically chosen to be large compared to the excited-state hyperﬁne structure splitting and in most cases even large compared to the ﬁne structure splitting in order to suﬃciently suppress spontaneous scattering events. Typical detunings range from several tens of nm to optical trapping in CO2 laser ﬁelds. A laser trap formed by a CO2 laser ﬁelds can be considered as a quasi-electrostatic trap, where the detuning is much larger than the optical resonance frequency of an atom. One ﬁnal comment should be made about state-dependent optical potentials. For a typical multi-level alkali atom, the dipole potential will both depend on the internal magnetic substate mF of a hyperﬁne ground state with angular momentum F , and on the polarization of the light ﬁeld P = +1, −1, 0 (circular σ ± and linear polarization). One can then express the lattice potential depth through [11, 10] (7)

Vdip (r) =

πc2 Γ 2ω03

2 + P g F mF 1 − P gF mF + Δ2,F Δ1,F

I(r).

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237

Fig. 2. – One-dimensional optical lattice potential. By interfering two counterpropagating Gaussian laser beams, a periodic intensity proﬁle is created due to the interference of the two laser ﬁelds.

Here gF is the Land´e factor and Δ2,F , Δ1,F refer to the detuning relative to the transition between the ground state with hyperﬁne angular momentum F and the center of the excited-state hyperﬁne manifold on the D2 and D1 transition, respectively. For large detunings relative to the ﬁne structure splitting ΔFS , the optical potentials become almost spin independent again. For detunings of the laser frequency between the ﬁne structure splitting, special spin-dependent optical potentials can be created [11-13]. . 2 2. Optical lattice potentials. – A periodic potential can simply be formed by overlapping two counterpropagating laser beams (see ﬁg. 2). Due to the interference between the two laser beams an optical standing wave with period λ/2 is formed, in which the atoms can be trapped. By interfering more laser beams, one can obtain one-, two- and three-dimensional periodic potentials [14], which in their simplest form will be discussed below. Note that by choosing to let two laser beams interfere under an angle less than 180◦ , one can also realize periodic potentials with a larger period. . 2 2.1. 1D lattice potentials. The simplest possible periodic optical potentials is formed by overlapping two counterpropagating focussed Gaussian laser beams, which results in a trapping potential of the form (8)

−2r 2 /w2 (z)

V (r, z) = −Vlat · e

r2 · sin2 (kz), · sin (kx) ≈ −Vlat · 1 − 2 2 w (z) 2

where w0 denotes the beam waist, k = 2π/λ is the wave vector of the laser light and Vlat is the maximum depth of the lattice potential. Note that due to the interference of the two laser beams Vlat is four times larger than V0 if the laser power and beam parameters of the two interfering lasers are equal. . 2 2.2. 2D lattice potentials. Periodic potentials in two dimensions can be formed by overlapping two optical standing waves along diﬀerent directions. In the simplest form one chooses two orthogonal directions and obtains at the center of the trap an optical potential of the form (neglecting the harmonic conﬁnement due to the Gaussian beam proﬁle of the laser beams) (9)

+ , V (y, z) = −Vlat cos2 (kx) + cos2 (ky) + 2e1 · e2 cos φ cos(kx) cos(ky) .

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I. Bloch

(a)

(b)

Fig. 3. – 2D optical lattice potentials for a lattice with (a) orthogonal polarization vectors and (b) with parallel polarization vectors and a time phase of φ = 0.

Here e1,2 denote the polarization vectors of the laser ﬁelds, each forming one standing wave and φ is the time between them. If the polarization vectors are chosen not to be orthogonal to each other, then the resulting potential will not only be the sum of the potentials created by each standing wave, but will be modiﬁed according to the time phase φ used (see ﬁg. 3). In such a case it is absolutely essential to stabilize the time phase between the two standing waves [15], as small vibrations will usually lead to ﬂuctuations of the time phase, resulting in severe heating and decoherence eﬀects of the ultracold atom samples. In such a two-dimensional optical lattice potential, the atoms are conﬁned to arrays of tightly conﬁning one-dimensional tubes (see ﬁg. 4a). For typical experimental parameters the harmonic trapping frequencies along the tube are very weak and on the order of 10–200 Hz, while in the radial direction the trapping frequencies can become as high as up to 100 kHz, thus allowing the atoms to eﬀectively move only along the tube for deep lattice depths [16-20]. . 2 2.3. 3D lattice potentials. For the creation of a three-dimensional lattice potential, three orthogonal optical standing waves have to be overlapped. Here we only consider the case of independent standing waves, with no cross interference between laser beams of diﬀerent standing waves. This can for example be realized by choosing orthogonal polarization vectors between diﬀerent standing wave light ﬁelds and also by using diﬀerent wavelengths for the three standing waves. In this case the resulting optical potential is simply given by the sum of three standing waves (10)

2

V (r) = −Vx e−2(y

2

−Vy e−2(x

2

−Vz e−2(x

2 +z 2 )/wx

+z +y

sin2 (kx)

2

)/wy2

sin2 (ky)

2

)/wz2

sin2 (kz).

Here Vx,y,z are the potential depths of the individual standing waves along the diﬀerent directions. In the center of the trap, for distances much smaller than the beam waist, the trapping potential can be approximated as the sum of a homogeneous periodic lattice potential and an additional external harmonic conﬁnement due to the Gaussian laser

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(a)

(b)

Fig. 4. – Two-dimensional (a) and three-dimensional (b) optical lattice potentials formed by superimposing two or three orthogonal standing waves. For a two-dimensional optical lattice, the atoms are conﬁned to an array of tightly conﬁning one-dimensional potential tubes, whereas in the three-dimensional case the optical lattice can be approximated by a three-dimensional simple cubic array of tightly conﬁning harmonic oscillator potentials at each lattice site.

beam proﬁles (11)

V (r) ≈ Vx · sin2 (kx) + Vy · sin2 (ky) + Vz · sin2 (kz) +

, m+ 2 2 ωx x + ωy2 y 2 + ωz2 z 2 , 2

where ωx,y,z are the eﬀective trapping frequencies of the external harmonic conﬁnement. They can again be approximated by (12)

ωx2 =

4 m

Vy Vz + 2 wy2 wz

;

2 ωy,z = (cyclic permutations).

In addition to this harmonic conﬁnement due to the Gaussian laser beam proﬁles, a conﬁnement due to a magnetic trapping typically exists, which has to be taken into account as well for the total harmonic conﬁnement of the atom cloud. For suﬃciently deep optical lattice potentials, the conﬁnement on a single lattice site is also approximately harmonic. Here the atoms are very tightly conﬁned with typical trapping frequencies ωlat of up to 100 kHz. One can estimate the trapping frequencies

240

I. Bloch

at a single lattice site through a Taylor expansion of the sinusoidally varying lattice potential at a lattice site and obtains ωlat ≈

(13)

2Er Vlat 2 k 4 = 2 Er m

'

Vlat . Er

Here Er = 2 k 2 /2m is the so called recoil energy, which is a natural measure of energy scales in optical lattice potentials. 3. – Bose-Hubbard model of interacting bosons in optical lattices The behavior of bosonic atoms with repulsive interactions in a periodic potential is fully captured by the Bose-Hubbard Hamiltonian of solid state physics [5, 6], which in the homogeneous case can be expressed through H = −J

(14)

i,j

1 a ˆ†i a ˆj + U n ˆ i (ˆ ni − 1). 2 i

ˆi describe the creation and annihilation operators for a boson on the i-th Here a ˆ†i and a lattice site and n ˆ i counts the number of bosons on the i-th lattice site. The tunnel coupling between neighboring potential wells is characterized by the tunnel matrix element (15)

J =−

d3 xw(x − xi )(−2 ∇2 /2m + Vlat (x))w(x − xj ),

where w(x − xi ) is a single-particle Wannier function localized to the i-th lattice site and Vlat (x) indicates the optical lattice potential. The repulsion between two atoms on a single lattice site is quantiﬁed by the on-site matrix element U (16)

U = (4π2 a/m)

|w(x)|4 d3 x,

with a being the scattering length of an atom. Due to the short range of the interactions compared to the lattice spacing, the interaction energy is well described by the second term of eq. (14) which characterizes a purely on-site interaction. Both the tunneling matrix element J and the onsite interaction matrix element can be calculated from a band structure calculation (see ﬁg. 5). The tunnel matrix element J is related to the width of the lowest Bloch band through: (17)

4J = |E0 (q = π/a) − E0 (q = 0)|,

where a is the lattice period, such that q = π/a corresponds to the quasi-momentum at the border of the ﬁrst Brillouin zone. Care has to be taken to evaluate the tunnel matrix element through eq. (15), when the Wannier function is approximated by the

241

Exploring strongly correlated ultracold bosonic and fermionic etc.

(a)

(b) 10-1

0.7

0.5

J (Er)

U (Er)

0.6

0.4 0.3

10-2

0.2 10-3

0.1 0

5

10

15

20

25

0

30

5

10

15

20

25

30

V (Er)

V (Er)

Fig. 5. – Onsite interaction matrix element U for 87 Rb (a) and tunnel matrix element J (b) vs. lattice depth. All values are given in units of the recoil energy Er .

Gaussian ground state wave function on a single lattice site. This usually results in a severe underestimation of the tunnnel coupling between lattice sites. The interaction matrix element can be evaluated through the Wannier function with the help of eq. (16). In this case, however, the approximation of the wannier function through the Gaussian ground state wave function yields a very good approximation. Zwerger et al. [21] have carried out a more sophisticated approximation of the tunnel matrix element, ﬁnding (18)

4 J ≈ √ Er Er π

and for the interaction matrix element ' (19)

U≈

Vlat Er

8 ka π

3/4

Vlat Er

e−2

√

Vlat /Er

3/4 .

The ratio U/J is crucial for determining, whether one is in a strongly interacting or a weakly interacting regime. It can be tuned continuously by simply changing the lattice potential depth (see ﬁg. 6). . 3 1. Ground states of the Bose-Hubbard Hamiltonian. – The Bose-Hubbard Hamiltonian of eq. (14) has two distinct ground states depending on the strength of the interactions U relative to the tunnel coupling J. In order to gain insight into the two limiting ground-states, let us ﬁrst consider the case of a double-well system with only two interacting neutral atoms. . 3 2. Double-well case. – In the double-well system √ the two lowest lying states for noninteracting√particles are the symmetric |ϕS = 1/ 2(|ϕL +|ϕR ) and the anti-symmetric |ϕA = 1/ 2(|ϕL − |ϕR ) states, where |ϕL and |ϕR are the ground states of the leftand right-hand side of the double-well potential. The energy diﬀerence between |ϕS and |ϕA will be named 2J, which characterizes the tunnel coupling between the two wells and depends strongly on the barrier height between the two potentials.

242

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1000

U/J

100

10

1 0

5

10

15

20

25

30

V (Er) Fig. 6. – U/J vs. optical lattice potential depth for 87 Rb. By increasing the lattice depth one can tune the ratio U/J, which determines whether the system is strongly or weakly interacting.

In case of no interactions, the ground state of the two-body system is realized when each atom is in the symmetric ground state of the double-well system (see ﬁg. 7a). Such a situation yields an average occupation of one atom per site with the single-site many-body state actually being in a superposition of zero, one and two atoms. Let us now consider the eﬀects due to a repulsive interaction between the atoms. If both atoms are again in the symmetric ground state of the double well, the total energy of such a state will increase due to the repulsive interactions between the atoms. This higher energy cost is a direct consequence of having contributions where both atoms occupy the same site of the double well. This leads to an interaction energy of 1/2U for this state.

Fig. 7. – Ground state of two interacting particles in a double well. For interaction energies U smaller than the tunnel coupling J the ground state of the two-body system is realized by the “superﬂuid” state (a). If, on the other hand, U is much larger than J, then the ground state of the two-body system is the Mott insulating state (b).

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243

If this energy cost is much greater than the splitting 2J between the symmetric and anti-symmetric ground states of the non-interacting system, the system can minimize its energy when each atom is in√a superposition of the symmetric and antisymmetric ground state of the double well √ 1/ 2(|ϕS ± |ϕA ). The resulting many-body state can then be written as |Ψ = 1/ 2(|ϕL ⊗ |ϕR + |ϕR ⊗ |ϕL ). Here exactly one atom occupies the left and right site of the double well. Now the interaction energy vanishes because both atoms never occupy the same lattice site. The system will choose this new “Mott insulating” ground state when the energy costs of populating the antisymmetric state of the double well system are outweighed by the energy reduction in the interaction energy. It is important to note that precisely the atom number ﬂuctuations due to the delocalized single particle wave functions make the “superﬂuid” state unfavorable for large U . Such a change can be induced by adiabatically increasing the barrier height in the double-well system, such that J decreases exponentially and the energy cost for populating the antisymmetric state becomes smaller and smaller. Eventually it will then be favorable for the system to change from the “superﬂuid” ground state, where for U/J → ∞ each atom is delocalized over the two wells, to the “Mott insulating” state, where each atom is localized to a single lattice site. . 3 3. Multiple-well case. – The above ideas can be readily extended to the multiple-well case of the periodic potential of an optical lattice. For U/J 1 the tunnelling term dominates the Hamiltonian and the ground state of the many-body system with N atoms is given by a product of identical single-particle Bloch waves, where each atom is spread out over the entire lattice with M lattice sites, M N † |0. a ˆi (20) |ΨSF U/J≈0 ∝ i=1

Since the many-body state is a product over identical single-particle states, a macroscopic wave function can be used to describe the system. Here the single site many-body wave function |φi is almost equivalent to a coherent state. The atom number per lattice site then remains uncertain and follows a Poissonian distribution with a variance given by ni . The non-vanishing the average number of atoms on this lattice site Var(ni ) = ˆ ai |φi then characterizes the coherent matter wave ﬁeld expectation value of ψi = φi |ˆ on the i-th lattice site. This matter wave ﬁeld has a ﬁxed phase relative to all other coherent matter wave ﬁelds on diﬀerent lattice sites. If, on the other hand, interactions dominate the behavior of the Hamiltonian, such that U/J 1, then ﬂuctuations in the atom number on a single lattice site become energetically costly and the ground state of the system will instead consist of localized atomic wave functions that minimize the interaction energy. The many-body ground state is then a product of local Fock states for each lattice site. In this limit the ground state of the many-body system for a commensurate ﬁlling of n atoms per lattice site is given by (21)

|ΨMI J≈0 ∝

M n . a ˆ†i |0. i=1

244

I. Bloch C

D

E

F

G

H

I

J

Fig. 8. – Absorption images of multiple matter wave interference patterns after releasing the atoms from an optical lattice potential with a potential depth of a) 0Er , b) 3Er , c) 7Er , d) 10Er , e) 13Er , f) 14Er , g) 16Er and h) 20Er . The ballistic expansion time was 15 ms.

Under such a situation the atom number on each lattice site is exactly determined but the phase of the coherent matter wave ﬁeld on a lattice site has obtained a maximum uncertainty. This is characterized by a vanishing of the matter wave ﬁeld on the i-th ai |φi ≈ 0. lattice site ψi = φi |ˆ In this regime of strong correlations, the interactions between the atoms dominate the behavior of the system and the many-body state is not amenable anymore to a description as a macroscopic matter wave, nor can the system be treated by the theories for a weakly interacting Bose gas of Gross, Pitaevskii and Bogoliubov [22]. For a 3D system, the transition to a Mott insulator occurs around U/J ≈ z · 5.6 [23, 6, 24, 21], where z is the number of next neighbors to a lattice site (for a simple cubic crystal z = 6). . 3 4. Superﬂuid-to-Mott-insulator transition. – In the experiment the crucial parameter U/J that characterizes the strength of the interactions relative to the tunnel coupling between neighboring sites can be varied by simply changing the potential depth of the optical lattice potential. By increasing the lattice potential depth, U increases almost linearly due to the tighter localization of the atomic wave packets on each lattice site and J decreases exponentially due the decreasing tunnel coupling. The ratio U/J can therefore be varied over a large range from U/J ≈ 0 up to values in our case of U/J ≈ 2000. In the superﬂuid regime [25] phase coherence of the matter wave ﬁeld across the lattice characterizes the many-body state. This can be observed by suddenly turning oﬀ all trapping ﬁelds, such that the individual matter wave ﬁelds on diﬀerent lattice sites expand and interfere with each other. After a ﬁxed time-of-ﬂight period the atomic density distribution can then be measured by absorption imaging. Such an image directly reveals the momentum distribution of the trapped atoms. In ﬁg. 8b an interference pattern can be seen after releasing the atoms from a three-dimensional lattice potential. If, on the other hand, the optical lattice potential depth is increased such that the

Visibility V

Exploring strongly correlated ultracold bosonic and fermionic etc.

245

100 4

10-1 10-2

100

101 U/zt

102

Fig. 9. – Visibility of the interference pattern versus U/z t, the characteristic ratio of interaction to kinetic energy. The data is shown for two atom numbers 5.9 × 105 atoms (black circles), and 3.6 × 105 atoms (grey circles). The former curve has been oﬀset vertically for clarity. The lines are ﬁts to the data in the range 14–25Er , assuming a coherent particle hole admixture as in eq. (22) (see ref. [29]).

system is very deep in the Mott insulating regime (U/J → ∞), phase coherence is lost between the matter wave ﬁelds on neighboring lattice sites due to the formation of Fock states [6, 5, 26, 3]. In this case no interference pattern can be seen in the time-of-ﬂight images (see ﬁg. 8h) [7, 8]. For a Mott insulator at ﬁnite U/J one expects a residual visibility in the interference pattern [27, 28], which can be caused on the one hand by the residual superﬂuid shells and on the other hand through an admixture of coherent particle hole pairs to the ideal MI ground state. A more detailed picture for the residual short-range coherence features beyond the SFMI transition is obtained by considering perturbations deep in the Mott insulating regime at J = 0. There, the ﬁrst-order correlation function G(1) (R) describing the coherence properties vanishes beyond R = 0 and the momentum distribution is a structureless Gaussian, reﬂecting the Fourier transform of the Wannier wave function. With increasing tunneling J, the Mott state at J/U → 0 is modiﬁed by a coherent admixture of particlehole pairs. However, due to the presence of a gapped excitation spectrum, such particle hole pairs cannot spread out and are rather tightly bound to close distances. They do, however, give rise to a signiﬁcant degree of short-range coherence (see ﬁg. 9). Using ﬁrst-order perturbation theory with the tunneling operator as a perturbation on the dominating interaction term, one ﬁnds that the amplitude of the coherent particle hole

246

I. Bloch

admixtures in a Mott insulating state is proportional to J/U ,

(22)

|ΨU/J ≈ |ΨU/J→∞ +

J † a ˆi a ˆj |ΨU/J→∞ . U i,j

Within a local density approximation, the inhomogeneous situation in a harmonic trap is described by a spatially varying chemical potential μR = μ(0) − R , with R = 0 at the trap center. Assuming, e.g., that the chemical potential μ(0) at trap center falls into the n ¯ = 2 “Mott-lobe”, one obtains a series of MI domains separated by a SF by moving to the boundary of the trap where μR vanishes. In this manner, all the diﬀerent phases which exist for given J/U below μ(0) are present simultaneously! The SF phase has a ﬁnite compressiblity κ = ∂n/∂μ and a gapless excitation spectrum of the form ω(q) = cq because there is a ﬁnite superﬂuid density ns . By contrast, in the MI phase both ns and κ vanish identically. As predicted by [6], the incompressibility of the MI phase allows to distinguish it from the SF by observing the local density distribution in a trap. Since κ = ∂n/∂μ = 0 in the MI, the density stays constant in the Mott phases, even though the external trapping potential is rising. The existence of such wedding-cake–like density proﬁles of a Mott insulator has been supported by accurate Monte Carlo [27, 30] and DMRG [31] calculations in one, two, and three dimensions. Very recently in-trap density proﬁles have been detected experimentally by [32] and [33]. In the latter case it has been possible to directly observe the wedding-cake density proﬁles and thus conﬁrm the incompressibiltiy of the Mott insulating regions of the atomic gas in the trapping potential (see ﬁg. 10). It should be noted that the in-trap density proﬁles can be used as a sensitive thermometer for the strongly interacting quantum gas. Already for small temperatures around T ≈ 0.2U/kB the wedding-cake proﬁles become completely washed out. Close to the transition point, higher-order perturbation theory or a Green’s function analysis can account for coherence beyond nearest neighbors and the complete liberation of the particle-hole pairs, which eventually leads to the formation of long-range coherence in the superﬂuid regime. The coherent particle-hole admixture and its consequence on the short-range coherence of the system have been investigated theoretically and experimentally in [29,34]. It has been demonstrated that through a quantitative analysis of the interference pattern, one can even observe traces of the shell structure formation in the Mott insulating regime. In addition to the fundamentally diﬀerent momentum distributions in the superﬂuid and Mott insulating regime, the excitation spectrum is markedly diﬀerent as well in both cases. Whereas the excitation spectrum in the superﬂuid regime is gapless, it is gapped in the Mott insulating regime. This energy gap of order U (deep in the MI regime) can be attributed to the now localized atomic wave functions of the atoms [5, 6, 24, 3].

247

Exploring strongly correlated ultracold bosonic and fermionic etc.

Slice Fraction (%)

20 15

c

d

e

f

10 5

Slice Fraction (%)

0

15 10 5 0

-20

-10

0

10

z Position (μm)

20 -20

-10

0

10

20

z Position (μm)

Fig. 10. – Integrated distribution of a superﬂuid (a) and a Mott insulating state (b) calculated for a lattice with harmonic conﬁnement. Grey solid lines denote the total density proﬁles, blue (red) lines the density proﬁles from singly (doubly) occupied sites. A vertical magnetic-ﬁeld gradient is applied which creates almost horizontal surfaces of equal Zeeman shift over the cloud (dashed lines in inset). A slice of atoms can be transferred to a diﬀerent hyperﬁne state by using microwave radiation only resonant on one speciﬁc surface (coloured areas in insets). Spin changing collisions can then be used to separate singly (blue) and doubly occupied sites (red) in that plane into diﬀerent hyperﬁne states. Experimental data: (c) 1.0 × 105 atoms in the superﬂuid regime (V0 = 3Er ), (d) 1.0 × 105 atoms in the Mott regime (V0 = 22Er ), (e) 2.0 × 105 atoms, (f) 3.5 × 105 atoms. The grey data points denote the total density distribution and the red points the distribution of doubly occupied sites. The blue points show the distribution of sites with occupations other than n = 2. The solid lines are ﬁts to an integrated Thomas-Fermi distribution in (c), and an integrated shell distribution for (d) to (f). The n = 2 data points are oﬀset vertically for clarity.

4. – Multi-orbital quantum phase diﬀusion . 4 1. Introduction. – Next we turn to the time dynamics of many-body states when jumping rapidly from a superﬂuid into a Mott insulating parameter regime. Initially, in the superﬂuid state the quantum states on a lattice are close to coherent states. Such coherent quantum states represent the most robust and stable ﬁeld solutions in physics [35]. Due to their correspondence to classical coherent ﬁelds, characterized by a single amplitude and phase, they have found widespread use in physics ranging from the description of laser light to coherent matter waves in superconductors, superﬂuids or atomic Bose-Einstein condensates. Whenever interactions between the underlying particles are present, or —more generally— whenever the phase of the number states that form the coherent state evolve nonlinear in particle number over time, such coherent states can undergo an intriguing sequence of collapses and revivals. The quantum state ﬁrst evolves into a highly correlated and entangled state where at the time of the collapse

248

I. Bloch

the classical ﬁeld vanishes, whereas at a later time the entanglement is unraveled again and the original classical ﬁeld is ideally recreated. Remarkable examples of such collapses and revivals have been observed for a coherent light ﬁeld interacting with a single atom in Cavity Quantum Electrodynamics [36, 37], for a classical oscillation of a single ion held in a trap [38, 39] or for a matter wave ﬁeld of a Bose-Einstein condensate via the non-linear two-body interactions between the atoms [40-42]. In the latter case, one typically assumes the atoms to occupy a single spatial orbital of the system. Atom-atom collisions can however promote particles to higher lying orbitals and even for the case where real occupation of excited vibrational states can be neglected, virtual transitions can still have a profound impact on the system. They can, e.g., modify the spatial wave function of the atoms depending on the ﬁlling, giving rise to renormalized two-particle interactions and the generation of higher-order eﬀective multi-particle interactions that are induced via higher-order virtual transitions of the atoms to excited orbital states [43]. Current experiments have only allowed to observe few cycles of the resulting quantum phase diﬀusion dynamics [40-42] and therefore were not able to reveal the striking consequences of multi-orbital eﬀects. Here, we have been able to observe up to 40 collapses and revivals in long time traces of the quantum evolution of coherent matter wave ﬁelds trapped in a 3D optical lattice potential. . 4 2. Theoretical model. – We consider a single site of an optical lattice ﬁlled with n particles occupying the lowest energy state of the system. For non-interacting particles, this corresponds to the ground state vibrational wave function ψ0 (r). Assuming weak interactions and excluding multi-orbital eﬀects, cold collisions between the atoms lead to #a single-orbital interaction energy given by EnSO = U n(n − 1)/2, where U = 4π2 a/m |ψ0 (r)|4 d3 r denotes the two-particle interaction energy, determined by the on-site wave function ψ0 (r), the s-wave scattering length a and the mass of an atom m. Within the restriction to the lowest vibrational state of the system, U is independent of the ﬁlling n at the lattice site. By taking virtual transitions to higher vibrational states into account, however, interactions modify the shape of the ground-state wave function (ﬁg. 11b) and U itself becomes atom number dependent [44, 43, 45]. For the case where real occupation of excited vibrational states can be neglected, but virtual transitions to these states are important, the multi-orbital Fock state energy of a single lattice well can be approximated by [43] (23)

EnMO =

1 U2 n(n − 1) 2 1 + U3 n(n − 1)(n − 2) 6 1 + U4 n(n − 1)(n − 2)(n − 3) + . . . . 24

In this description, the coherent multi-particle interactions become explicitly visible, where characteristic strengths of the n-particle interactions are given by Un .

Exploring strongly correlated ultracold bosonic and fermionic etc.

a

c0

b

n=2

+ c1

+ c2

+ c3

n=3

249

+ ...

n=4

c Coherence

1.0

0 0

5 t (trev

10 h/U)

Fig. 11. – (Colour on-line) Quantum phase diﬀusion and multi-orbital eﬀects. (a) A BoseEinstein condensate loaded to a weak optical lattice forms a superﬂuid with each atom being delocalized over several lattice sites. The quantum state on each site can be expressed as a superposition of Fock states |n with amplitudes cn . (b) For repulsive onsite interactions virtual transitions to higher lattice orbitals broaden the ground-state wave function of the non-interacting system (grey dashed line), giving rise to coherent multi-particle interactions. (c) A coherent state conﬁned to a deep lattice well undergoes multi-orbital quantum phase diﬀusion (blue solid line, see text). The dynamics are markedly diﬀerent from the monochromatic evolution expected in a single-orbital model with a single two-body interaction energy scale U (grey solid line).

. 4 3. Probing the energy scales via Fock state heterodyning. – An eﬃcient way to experimentally probe the eigenenergies of a Hamiltonian is to monitor the non-equilibrium dynamics of a quantum state prepared in a superposition of diﬀerent eigenstates. In our case, such a superposition state consists of diﬀerent atomic Fock states {|n} and ∞ −iEn t/ forms an atomic matter wave ﬁeld of the general form |ψ(t) = |n. n=0 cn e Experimentally, we create a 3D array of such matter wave ﬁelds by loading a BEC into a shallow 3D lattice potential. Their time evolution can be probed by analyzing the visibility of the atomic interference pattern as observed after rapid switch-oﬀ of the lattice potential and subsequent time-of-ﬂight expansion [29]. For an array of identical states |ψ, the visibility of the interference pattern is proportional to | ψ(t)|ˆ a|ψ(t)|2 /¯ n, where a ˆ denotes the annihilation operator on a lattice site. Hence, the dynamical evolution of

250

I. Bloch

the matter wave ﬁeld, the quantum phase diﬀusion of |ψ(t) is given by (24)

| ψ(t)|ˆ a|ψ(t)|2 =

∞ √ √ n + 1 m + 1 cm c∗n c∗m+1 cn+1 m,n=1

×ei(En −En+1 −Em +Em+1 )t/ ≡ | ˆ a|2 . Multi-orbital eﬀects, however, reach beyond the picture of monochromatic collapses and revivals: the time evolution of | ˆ a|2 contains multiple frequency components since Fock state energies are no longer integer multiples of the U (ﬁg. 11c (blue solid line)). Detection of multi-orbital quantum phase diﬀusion over suﬃciently long times allows for a precise measurement of the individual Fock state energies EnMO . . 4 4. Experimental setup and results. – Our experiments begin with an atomic BoseEinstein condensate of 87 Rb atoms in the |F = 1, mF = +1 state, with variable atom numbers between 1.2×105 and 4.5×105 . The atoms are initially held in a pancake-shaped crossed optical dipole trap. Subsequently a 3D blue-detuned optical lattice (λ = 738 nm) of simple cubic type is ramped up to lattice depths VL between 3Er and 13Er . A sudden increase of the lattice depth from VL to VH ranging between 25Er and 41Er then essentially freezes out the equilibrium atom number distribution at VL on each site through a strong suppression of the tunnel coupling. In this regime, the time evolution of each site is governed by the Hamiltonian of eq. (23) and the quantum phase diﬀusion process is initiated. After letting the system evolve for hold times t in the deep lattice VH , we have monitored the phase coherence by simultaneously switching-oﬀ all trapping potentials and recording an absorption image of the matter wave interference pattern after 10 ms time of ﬂight. The visibility of the interference pattern is used as a measure of the phase coherence of the system [29]. A typical time trace (VL = 8Er ) of the quantum phase diﬀusion is shown in ﬁg. 12a displaying up to 40 revivals. On top of a fast series of collapse and revivals, we observe a slower modulation of the envelope indicating a beat note between at least two diﬀerent energy scales in the system. In a Fourier analysis of the corresponding time trace, we ﬁnd clear evidence for three distinct frequency components present in the time trace, the smallest one originating from sites occupied by up to four atoms. In general, the measurement allows one to reveal even very small Fock state amplitudes cn due to a heterodyning eﬀect with other Fock states |n−1 and |n−2 of typically larger amplitude cn−1 and cn−2 (see eq. (24)). 5. – Compressible and incompressible quantum phases of fermionic spin mixtures in optical lattices Next to bosonic particles, interacting fermions in periodic potentials lie at the heart of condensed matter physics. They present, however, some of the most challenging problems to quantum many-body theory. A prominent example is high-Tc superconductivity in

251

Exploring strongly correlated ultracold bosonic and fermionic etc.

a

40μs

2720μs

5120μs

7600μs

1.0

Visibility

0.8 0.6 0.4 0.2 0

0

2000

4000 6000 Holdtime (μs)

Power spectral density (a u )

b

c 1.0

05

8000

lattice VH 50μs

0.5

VL

0 3 25

35

time

dipole trap

0 3.0

3.5 4.0 Frequency (kHz)

4.5 100ms

time

Fig. 12. – Collapse and revival dynamics of an atom number superposition state. (a) Time trace of observed collapses and revivals in the phase coherence of the system. A BEC of about 1.5 × 105 Rb atoms has been adiabatically loaded to a VL = 8Erec lattice within 100 ms. Phase evolution is induced by a non-adiabatic jump (50 μs) into a VH = 40Erec deep lattice, preserving superposition states with ﬁnite number ﬂuctuations and an average ﬁlling of about n ¯ = 1.5 (c). The evolution of phase coherence shows a beat-note signature resulting from coherent multi-particle interactions with diﬀerent interaction strengths. (b) Spectral analysis of the time traces using a numerical Fourier transform of the data (a) reveals the contributing frequencies.

cuprate compounds [46]. An essential part of the physics in these systems is described by the fermionic Hubbard Hamiltonian [47], which models interacting electrons in a periodic potential [46, 48]. Probing this Hamiltonian in a controllable and clean experimental setting is therefore of great importance. For the case of bosonic particles [5, 6], the signiﬁcance of ultracold quantum gases in this respect has been discussed in the previous sections. For both bosonic and fermionic systems, the entrance into a Mott insulating state is signaled by a vanishing compressibility, which can in principle be probed experimentally by testing the response of the system to a change in external conﬁnement. This is a straightforward way to identify the interaction-induced Mott insulator and to distinguish it, e.g., from a disorder-induced Anderson insulator [49-51]. In a solid, however, the corresponding compressibility can usually not be measured directly, as a compression of the crystalline lattice by an external force does not change the number of electrons per unit cell. In a recent work, non-interacting and repulsively interacting spin mixtures of fermionic atoms deep in the degenerate regime were studied in a three-dimensional optical lattice,

252

I. Bloch

Fig. 13. – Relevant phases of the Hubbard model with an inhomogeneous trapping potential for a spin mixture at T = 0. A schematic is shown in the left column. The center column displays the corresponding in-trap density proﬁles and the right column outlines the distribution of singly and doubly occupied lattice sites after a rapid projection into the zero tunneling limit, with p denoting the total fraction of atoms on doubly occupied lattice sites.

where the interaction strength, the lattice depth and in one experiment the external harmonic conﬁnement of the quantum gas could be varied independently [52, 53]. By monitoring the in-trap density distribution for increasing harmonic conﬁnement [53], we could directly probe the compressibility of the many-body system. This measurement allowed us to distinguish compressible metallic phases from globally incompressible states and revealed the strong inﬂuence of interactions on the density distribution. In previous experiments, a suppression of the number of doubly occupied sites was demonstrated for increasing interaction strength for bosons [54] and fermions [52] at ﬁxed harmonic conﬁnement, signaling the entrance into a strongly interacting regime. Here we compare the experimentally observed density distributions and fractions of doubly occupied sites to numerical calculations using Dynamical Mean Field Theory (DMFT) [55-58]. DMFT is a central method of solid state theory being widely used to obtain ab initio descriptions of strongly correlated materials [56]. The comparison of DMFT predictions with experiments on ultracold fermions in optical lattices constitutes a parameter-free experimental test of the validity of DMFT in a three-dimensional system. . 5 1. Hubbard Hamiltonian in a trap. – Restricting our discussion to the lowest energy band of a simple cubic 3D optical lattice, the fermionic quantum gas mixture can be modeled via the Hubbard-Hamiltonian [47] with an additional term describing the

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0.75

0.7

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Et /12J=0.4

55

R (d)

Renormalized Cloud Size Rsc (d)

a

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45

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b

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80

120

Ramp time (ms)

0.6 c

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0.55

e

0.5 U/12J U/12J U/12J U/12J

0.45

0 05 1 15

0

0.5

1 1.5 Compression Et /12J

2

Fig. 14. – Cloud sizes of the interacting spin mixture versus compression. Measured cloud size Rsc in a Vlat = 8Er deep lattice as a function of the external trapping potential for various interactions U/12J = 0 . . . 1.5. Dots denote single experimental shots, lines the theoretical expectation from DMFT for T /TF = 0.15 prior to loading. The insets (a-e) show the quasi-momentum distribution of the non-interacting clouds (averaged over several shots). (f) Resulting cloud size for diﬀerent lattice ramp times at Et /12J = 0.4 for a non-interacting and an interacting Fermi gas. The arrow marks the ramp time of 50 ms used in the experiment.

underlying harmonic potential: (25)

ˆ = −J H

i,j,σ

+Vt

cˆ†i,σ cˆj,σ + U

n ˆ i,↓ n ˆ i,↑

i

(i2x + i2y + γ 2 i2z ) (ˆ ni,↓ + n ˆ i,↑ ) .

i

Here the indices i, j denote diﬀerent lattice sites in the three-dimensional system (i = (ix , iy , iz )), i, j neighboring lattice sites, σ ∈ {↓, ↑} the two diﬀerent spin states, J the tunneling matrix element and U the eﬀective on-site interaction. The operators cˆi,σ (ˆ c†i,σ ) are the annihilation (creation) operators of a fermion in spin state σ on the i-th lattice site and n ˆ i,σ measures the corresponding atom number. The strength of the harmonic 2 2 d between two adjacent conﬁnement is parameterized by the energy oﬀset Vt = 12 mω⊥ lattice sites at the trap center, with ω⊥ = ωx = ωy = ωz being the horizontal trap frequency, d the lattice constant and m the mass of a single atom. The constant aspect ratio of the trap is denoted by γ = ωz /ω⊥ . The quantum phases of the Hubbard model with harmonic conﬁnement are governed by the interplay between three energy scales: kinetic energy, whose scale is given by the lattice bandwidth 12J, interaction energy U , and the strength of the harmonic

254

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3

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0 0 0.5

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T/TF 0.07 0.10 0.15

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T/TF 0.17 0.20

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1

E /1 1 t 2J

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f

40

) r (d

T/TF

0.15

1

ni,σ

Compress b ty κRsc (d -2)

2

T/TF

e

b

a

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0

0 20

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Et /12J

2.0

0

0 0.5

1

1.5

2.0

E /1 1 t 2J

0

40

) r (d

Et /12J

Fig. 15. – (a-c) Global compressibility κRsc of the atom cloud for various interactions ((a) U/12J = 0, (b) U/12J = 1, (c) U/12J = 1.5). Dots denote the result of linear ﬁts on the measured data, the error bars represent the ﬁt uncertainty. Solid lines display the theoretically expected results for an initial temperature of T /TF = 0.15. The inﬂuence of the initial temperature on the calculated compressibility is shown in (d) for U/12J = 1.5. The corresponding density distributions are plotted in (e, f) with r denoting the distance to the trap center. The red lines mark the region where a Mott-insulating core has formed and the global compressibility is reduced.

conﬁnement, which can conveniently be expressed by the characteristic trap energy Et = Vt (γNσ /(4π/3))2/3 , which denotes the Fermi energy of a non-interacting cloud in the zero-tunneling limit, with Nσ being the number of atoms per spin state (N↓ = N↑ ). The characteristic trap energy depends both on atom number and trap frequency via 2/3 2 E t ∝ ω⊥ Nσ and describes the eﬀective compression of the quantum gas, controlled by the trapping potential in the experiment. Depending on which term in the Hamiltonian dominates, diﬀerent kinds of manybody ground states can occur in the trap center (ﬁg. 13). For weak interactions in a shallow trap U Et 12J the Fermi energy is smaller than the lattice bandwidth (EF < 12J) and the atoms are delocalized in order to minimize their kinetic energy. This leads to compressible metallic states with central ﬁlling n0,σ < 1, where the local ﬁlling factor ni,σ = ˆ ni,σ denotes the average occupation per spin state of a given lattice site. A dominating repulsive interaction U 12J and U Et suppresses the double occupation of lattice sites and can lead to Fermi-liquid (n0,σ < 1/2) or Mott-insulating (n0,σ = 1/2) states at the trap center, depending on the ratio of kinetic to characteristic trap energy. Stronger compressions lead to higher ﬁlling factors, ultimately (Et 12J, Et U ) resulting in an incompressible band insulator with unity central ﬁlling at T = 0.

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Finite temperature reduces all ﬁlling factors and enlarges the cloud size, as the system needs to accommodate the corresponding entropy. Furthermore, in the trap the ﬁlling always varies smoothly from a maximum at the center to zero at the edges of the cloud. For a dominating trap and strong repulsive interaction at low temperature (Et > U > 12J), the interplay between the diﬀerent terms in the Hamiltonian gives rise to a wedding-cake-like structure (ﬁg. 15e, f) consisting of a band-insulating core (n0,σ ≈ 1) surrounded by a metallic shell (1/2 < ni,σ < 1), a Mott-insulating shell (ni,σ = 1/2) and a further metallic shell (ni,σ < 1/2) [57]. The outermost shell remains always metallic, independent of interaction and conﬁnement, only its thickness varies. . 5 2. Experimental setup. – In the experiment, we used an equal mixture of quantum degenerate fermionic 40 K atoms in the two hyperﬁne states |F, mF = | 29 , − 92 ≡ | ↓ and | 92 , − 72 ≡ |↑ in a pancake-shaped optical dipole trap (aspect ratio γ ≈ 4), which is formed by overlapping two elliptical laser beams (λ = 1030 nm) travelling in the horizontal plane. Applying evaporative cooling, ﬁnal temperatures of T /TF = 0.15(3) with 1.5–2.5 × 105 potassium atoms are reached. The temperature is extracted from time-of-ﬂight images using Fermi ﬁts. A Feshbach resonance located at 202.1 G [59] is used to tune the scattering length between the two spin states and thereby control the on-site interaction U . The creation of the spin mixture and the last evaporation step are performed either above the resonance (220 G), giving access to non-interacting (209.9 G) and repulsively interacting clouds with a ≤ 150 a0 (B ≤ 260 G), or below the resonance (165 G), where larger scattering lengths up to a = 300 a0 (191.3 G) can be reached. A further approach to the Feshbach resonance is hindered by enhanced losses and heating in the lattice. After evaporation, the magnetic ﬁeld is tuned to the desired value. Subsequently, a blue-detuned 3D optical lattice (λlat = 738 nm) with simple cubic symmetry is increased to a potential depth of Vlat = 1Er , where Er = h2 /(2mλ2lat ) denotes the recoil energy. The combination of a red-detuned dipole trap and a blue-detuned lattice potential allows us to vary lattice depth and external conﬁnement independently. In this way a wide range of horizontal trap frequencies can be accessed (ω⊥ 2π × (20–120) Hz), especially enabling metallic states with high atom numbers. To monitor the in situ density distribution for diﬀerent external conﬁnements, we ramp the dipole trap depth in 100 ms to the desired external harmonic conﬁnement, followed by a linear increase of the optical lattice depth to Vlat = 8Er within 50 ms. An in situ image of the cloud is taken along the short (vertical) axis of the trap using phase-contrast imaging [60] at detunings of Δ = 2π × (200–330) MHz after a hold time of 12 ms in the lattice. 2 is extracted using adapted 2D Fermi ﬁts. From this picture the cloud size R = r⊥ As phase-contrast imaging modiﬁes the state of the atoms only marginally, the quasimomentum distribution can be measured in the same experimental run using a bandmapping technique [61, 16, 62]. For this, the lattice is ramped down in 200 μs and a standard absorption image is taken after 10 ms time of ﬂight. All experimental data are compared to numerical calculations, in which the DMFT equations of the homogeneous model are solved for a wide range of temperatures and

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chemical potentials using a numerical renormalization group approach [63, 64]. The trapped system can be approximated to very high accuracy by the uniform system through a local density approximation (LDA) even close to the boundary between a metal and an insulator [65,57]. For a comparison with the experimental results it is convenient to express the cloud size R in rescaled units Rsc = R/(γNσ )1/3 , along with the dimensionless compression Et /12J (see ﬁg. 14). In these units, the cloud size depends only on the interaction strength U/12J and the entropy. In all calculations, we use the entropy determined from a non-interacting Fermi gas in a harmonic trap at an initial temperature T /TF and assume adiabatic lattice loading. . 5 3. Cloud compression. – The numerically calculated density distributions, the corresponding column densities and the experimentally measured ones are presented in ﬁg. 13. While for low compression all distributions are metallic (ﬁrst row), we ﬁnd a Mott-insulating core with half-ﬁlling at intermediate compression and strong repulsion (second row). For high compression the non-interacting curve shows a band insulating core and the repulsive curves display a metallic core. In order to compare experiment and theory quantitatively, the measured cloud size Rsc and the numerically calculated one are plotted in ﬁg. 14 as a function of the characteristic trap energy Et . Additionsc (ﬁg. 15) of the system can be ally, the global compressibility κRsc = − R13 ∂(E∂R t /12J) sc extracted from these measurements using linear ﬁts to four consecutive data points to determine the derivative. In the non-interacting case we ﬁnd the cloud sizes (ﬁg. 15, black dots) to decrease continuously with compression until the characteristic trap energy roughly equals the lattice bandwidth (Et /12J ∼ 1). For stronger conﬁnement the compressibility approaches zero (ﬁg. 15a), as almost all atoms are in the band insulating regime while the surrounding metallic shell becomes negligible. The corresponding quasi-momentum distribution (ﬁg. 14a-e) changes gradually from a partially ﬁlled ﬁrst Brillouin zone, characteristic for a metal, to an almost evenly ﬁlled ﬁrst Brillouin zone for increasing compressions, as expected for a band insulator. Note, however, that a bandmapping technique reveals only the relative occupations of the extended Bloch states. For an inhomogeneous system, it therefore provides no information about the real-space density and especially cannot distinguish insulating from compressible states, e.g. nonequilibrium states in which the atomic wave functions are localized to single lattice sites. In contrast, the measurements shown here directly demonstrate the incompressibility of the fermionic band insulator, in excellent agreement with the theoretical expectation for a non-interacting Fermi gas (black line). For moderately repulsive interactions (U/12J = 0.5, 1) (green, blue) the cloud size is clearly bigger than in the non-interacting case but eventually reaches the size of the band insulator. For stronger repulsive interactions (U/12J = 1.5) (red) we ﬁnd the onset of a region (0.5 < Et /12J < 0.7) where the cloud size decreases only slightly with increasing harmonic conﬁnement, whereas for stronger conﬁnements the compressibility increases again. This is consistent with the formation of an incompressible Mott-insulating core, surrounded by a compressible metallic shell, as can be seen in the corresponding intrap density proﬁles (ﬁg. 15e, f). For higher conﬁnements an additional metallic core

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(1/2 < ni,σ < 1) starts to form in the center of the trap and the cloud size decreases again. A local minimum in the global compressibility is in fact a genuine characteristic of a Mott insulator for large U and low temperature. The experimental data, indeed, show an indication of this behavior (ﬁg. 15c) for increasing interactions. For Et /12J 0.5 a minimum in the compressibility is observed, followed by an increase of the compressibility around Et /12J 0.8, slightly earlier than predicted by theory. When the system is compressed even further, all cloud sizes approach that of a band insulating state and all compressibilities tend to zero. In the theory predictions, the repulsively interacting clouds can even become slightly smaller than the non-interacting ones due to Pomeranchuk cooling [66]: At the same average entropy per particle, the interacting system has a considerably lower temperature in the lattice, as the spin entropy is enhanced due to interactions. In the experiment this feature is barely visible, as a second eﬀect becomes important: At very high compressions (Et /12J 2) the second Bloch band gets slightly populated during the lattice ramp up, which leads to smaller cloud sizes for all interactions, as a small number of atoms in a nearly empty band can carry a considerable amount of entropy. Overall, we ﬁnd the measured cloud sizes to be in very good quantitative agreement with the theoretical calculations up to U/12J = 1.5 (B = 175 G). Nevertheless, for repulsive interactions and medium compression (Et /12J ≈ 0.5) the cloud size is slightly bigger than the theoretical expectation. The discrepancies become more prominent for stronger interactions, i.e. on further increasing the scattering length. This could be caused by non-equilibrium dynamics in the formation of a Mott-insulating state for strong interactions or may be an eﬀect not covered by the simple single-band Hubbard model [66] or the DMFT calculations and requires further investigation. To ensure that the used lattice loading time of 50 ms is adiabatic, we have measured the resulting in situ cloud size as a function of ramp time (ﬁg. 14f) in the regime around Et /12J ≈ 0.5, where the diﬀerences between experiment and theory are most pronounced. In this regime a too fast loading would result in a larger cloud and our measurement therefore indicates adiabaticity for the used ramp time. However, a second longer timescale, which could become more relevant for stronger interactions [67], cannot be ruled out. In addition, the temperatures before loading into the lattice and after a return to the dipole trap with a reversed sequence are compared. We ﬁnd a rise in temperature between 0.010(5) T /TF for a non-interacting cloud and 0.05(2) T /TF for a medium repulsion of U/12J = 1 at compressions around Et /12J ≈ 0.5. The good agreement between the experimental data and the numerical calculations, which assume adiabatic loading and an initial temperature of T /TF = 0.15, indicates that our actual initial temperatures lie rather at the lower end of the measured temperature range T /TF = 0.15(3). The theoretical calculations of the compressibility shown in ﬁg. 15d demonstrate that the minimum in the local compressibility, which signals the Mott-insulating state, starts to form at initial temperatures in the range of 0.15 T /TF 0.2. At these temperatures, the entropy per particle is much higher than possible in a Mott insulator in the homogeneous case (< kB ln 2), and even exceeds the maximum possible entropy per par-

258

I. Bloch 2ln(2)

a

b

Et /12J = 0.1

Et /12J = 0.5

1

Entropy (kB)

ln(2) 0.5 T / TF = 0.07 T / TF = 0.15

0

c

d

E /12J = 1 t

Et /12J = 1.8

1 ln(2) 0.5 0 0

0

20

40

20

40

Distance from trap center r (d) Fig. 16. – Calculated entropy distributions in the lattice. Solid (dashed) lines show the entropy per lattice site (per particle) for initial temperatures of T /TF = 0.07 (black) and T /TF = 0.15 (red) and strong repulsive interaction U/12J = 1.5 (see refs. [53, 70]).

ticle of a half-ﬁlled homogeneous Hubbard model (kB ln 4). In the trap, however, a large fraction of the entropy is accumulated in the metallic shells at the edges of the atomic cloud where the diluted atoms have a large conﬁgurational entropy. Therefore, the temperature remains on the order of kB Tlat J U in the Mott insulating regime. This is similar to the results obtained in recent calculations and experiments on the melting of incompressible bosonic Mott insulating shells for increasing temperatures [68, 33]. . 5 4. Entropy distribution. – In ﬁg. 16a the entropy distribution of a purely metallic state with less than half-ﬁlling in the center of the trap is shown. At higher compression (b) a Mott-insulator with unity ﬁlling and kB ln 2 entropy per particle has formed in the center of the trap even in the case of T /TF = 0.15, for which the average entropy per particle is above kB 2 ln 2. This is possible only due to the inhomogeneity of the system, as most of the entropy is carried by the metallic shells where the entropy per particle can diverge. For high compressions (d) a band-insulating core has formed and for low enough temperatures (black) nearly all entropy is carried by the surrounding shells. The small dip at r = 20 is a remnant of the Mott-insulating shell between the two metallic shells. Note that entropy per lattice site and entropy per particle are equal at half-ﬁlling. The inhomogeneous entropy distribution could allow for eﬃcient cooling schemes, by removing entropic metallic layers followed by a subsequent equilibration of the system [69].

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6. – Controlled superexchange interactions Quantum spin systems on a lattice have served for decades as paradigms for condensed matter and statistical physics, elucidating fundamental properties of phase transitions and acting as models for the emergence of quantum magnetism in strongly correlated electronic media. In all these cases the underlying systems rely on a spin-spin interaction between particles on neighboring lattice sites, such as in the Ising or Heisenberg model [71-73]. As initially proposed for electrons by Dirac [74,75] and Heisenberg [72,76], eﬀective spin-spin interactions can already arise due to the interplay between the spinindependent Coulomb repulsion and exchange symmetry and do not require any direct coupling between the spins of the particles. The nature of such spin exchange interactions is typically short-ranged, since it is governed by the wave function overlap of the underlying electronic orbitals. In several topical insulators, such as ionic solids like, e.g., CuO and MnO, however, antiferromagnetic ordering arises even though the wave function overlap between the magnetic ions is practically zero. In this case a “superexchange” interaction can be eﬀective over large distance, as introduced in the seminal works of Kramers and Anderson [77, 78]. Here, the spin-spin interactions are mediated by higher-order virtual hopping processes, which in the case of bosons (fermions) leads to an (anti)-ferromagnetic coupling between particles on neighboring lattice sites [73]. Such superexchange interactions are believed to play an important role in the context of high-Tc superconductivity [46]. Furthermore, they can form the basis for the generation of robust quantum gates similar to recent work in electronic double quantum dot systems [79], and can be employed for the eﬃcient generation of multi-particle entangled states [80], as well as for the production of many-body quantum phases with topological order [81, 82]. . 6 1. Theoretical model. – An isolated system of two coupled potential wells constitutes the simplest conceptual setup for the investigation of superexchange-mediated spin dynamics between neighboring atoms. In the following, we consider a single double-well potential occupied by a pair of bosonic atoms with two diﬀerent spin states denoted by |↑ and |↓. If the vibrational level splitting in each well is much larger than all other relevant energy scales and intersite interactions are neglected, the system can be described in a two-mode approximation by the Hubbard-type Hamiltonian ˆ BH = (26) H

1 † † nσL − n ˆ σR ) +U (ˆ n↑L n ˆ ↓L + n ˆ ↑R n ˆ ↓R ) , −J a ˆσL a ˆσR +ˆ aσR a ˆσL − Δ (ˆ 2

σ=↑,↓

where the operators a ˆ†σL,R and a ˆσL,R create and annihilate an atom with spin σ in the left and right well, respectively, n ˆ σL,R count the number of atoms per spin-state and well, J is the tunnel matrix element, Δ the potential bias or tilt along the double-well # 4 axis and U = U↑↓ = g × wL,R (x)dx the onsite interaction energy between two atoms ↑↓ in |↑ and |↓. Here, g = (4π2 a↑↓ s )/mRb is the eﬀective interaction strength with as being the (positive) scattering length for the spin states used in the experiment and mRb

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+ +α t

4J2 U

U

4J2 U

s t +α +

Fig. 17. – Schematics of superexchange interactions. a) Second-order hopping via |↑↓, 0 and |0, ↑↓ mediates the spin-spin interactions between atoms on diﬀerent sides of the double well. b) Energy levels for Δ = 0 and U J. The evolution in the upper doublet of states corresponds to the correlated tunneling of atom pairs [83], while the superexchange takes place in the lower one. Both doublets are coupled by ﬁrst-order tunneling processes.

the rubidium mass, and wL,R (x) denote the wave functions for a particle localized on the left or right side of the double well. The state of the system can be described as a superposition of the Fock states | ↑, ↓, | ↓, ↑, | ↑↓, 0 and |0, ↑↓, where the left and right side in the notation represent the occupation of the left and right well, respectively, and the states |↑↓, 0 and |0, ↑↓ are spin triplets. In the following, we will focus on the dynamical evolution of the population imbalance x = nL − nR and the spin imbalance m = (n↑L +n↓R −n↑R −n↓L )/2 starting with double wells initially prepared in |↑, ↓. Here n↑,↓;L,R denote the corresponding quantum-mechanical expectation values n↑,↓;L,R = ˆ and nL,R = n↑L,R + n↓L,R . In the limit of strong interactions (U J), when starting in the subspace of singly occupied wells spanned by |↑, ↓ and |↓, ↑, the energetically high lying states |↑↓, 0 and |0, ↑↓ can only be reached as “virtual” intermediate states in second-order tunneling processes. Such processes lead to a long-ranged (super) spin-exchange interaction, which couples the states |↑, ↓ and |↓, ↑ (see ﬁg. 17a). The eﬀective coupling strength for this superexchange can readily be evaluated by perturbation theory up to quadratic order in the tunneling operation and yields Jex = 2J 2 /U . More generally, for an arbitrary spin conﬁguration with equal interaction energies U↑↑ = U↑↓ = U↓↓ , the second-order hopping events are described by an isotropic Heisenberg-type eﬀective spin Hamiltonian (27)

z ˆ eﬀ = −2Jex SˆL · SˆR = −Jex Sˆ+ Sˆ− + Sˆ− Sˆ+ − 2Jex SˆLz SˆR H , L R L R

+ − z where SˆL,R = |↑ ↓|L,R and SˆL,R = |↓ ↑|L,R and SˆL,R = (ˆ n↑L,R − n ˆ ↓L,R )/2 denote the ± corresponding spin operators of the system, with SˆL,R = Sˆx ± iSˆy . When a potential bias Δ > 0 is applied, the degeneracy of the two intermediate states in the superexchange process is lifted (see ﬁg. 17a). For J, Δ U this leads to a modiﬁcation of the eﬀective superexchange coupling with now Jex = J 2 /(U + Δ) + J 2 /(U − Δ) = 2J 2 U/(U 2 − Δ2 ). By tuning the bias to Δ > U , it is possible to change

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the sign of Jex and therefore to switch between ferromagnetic and antiferromagnetic exchange interactions. For J |U −Δ| the picture of an eﬀective coupling via two virtual 2 intermediate states is again valid √ and the full reversal of the sign to Jex = −2J /U is found to be reached for Δ = 2 U . For symmetric double wells (Δ = 0), the Hamiltonian eq. (26) can be diagonalized analytically to give a valid picture for all values of J and U within the single band Hubbard model. A convenient basis to express the eigenstates is given by the spin triplet √ √ and singlet state |t/s = (|↑, ↓ ± |↓, ↑)/ 2 and the states |± ≡ (|↑↓, 0 ± |0, ↑↓)/ 2. Two of the eigenstates are linear combinations of |t and |+, where the one having the larger overlap with |t is the ground state. The spin singlet |s and the state |− are already eigenstates themselves with energy 0 and U , respectively (see ﬁg. 17b). As a direct consequence, |− cannot be reached with the initial state |↑, ↓ = |s + |t and the dynamical evolution in this case is therefore given by only two frequencies ⎛⎞ 2 4J U + 1 ± 1⎠ , (28) ω1,2 = ⎝ 2 U √ which allow for the extraction of J = ω1 ω2 /2 and U = (ω1 − ω2 ). . 6 2. Time-resolved observation of superexchange interactions. – In order to investigate the spin dynamics across a single quantum link, we initially prepare a sample of ultracold neutral atoms with two relevant internal states |↑ and |↓ in a 3D array of double wells with N´eel type antiferromagnetic ordering. The spin dynamics is initiated by rapidly ramping down the short lattice and thereby the double-well barrier in 200 μs, thus signiﬁcantly increasing the tunneling and superexchange couplings. After letting the system evolve for a hold time t, we freeze out the spin-conﬁguration by ramping up the barrier in 200 μs, quenching both J and Jex again. The measurement of the mean values x(t) and m(t) is carried out as described above. Three typical time traces obtained by this procedure are shown in ﬁg. 18. For low barrier depths (J/U > 1), we observe a pronounced time evolution of the spin imbalance m(t) consisting of two frequency components with comparable amplitudes and frequencies (ﬁg. 18a). With increasing interaction energy U relative to J, the frequency ratio increases, leaving a slow component with almost full amplitude and an additional high-frequency but small-amplitude modulation (ﬁg. 18b). For J/U 1, the fast oscillation is completely suppressed and the only process visible is the superexchange oscillation (ﬁg. 18c). For all barrier heights, the population imbalance x(t) stays ﬂat, emphasizing that even though strong spin-currents are present in the system, there is no net mass ﬂow for our initial state. Since the energies of the states |↑, ↓ and |↓, ↑ are hardly aﬀected by small deviations of the tilt from Δ = 0, the damping of the signal due to dephasing is only small, thus allowing us to tune the system far into the regime of strong interactions. The comparison of the results with the theoretical predictions by the simple Hubbard model shows signiﬁcant deviations toward low barriers which cannot be explained by our

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Population and spin imbalance

0.5

a

0.0 -0.5 0

1

2

3

0.5

b

0.0 -0.5 0

2

4

6

0.5

8

c

0.0 Population Spin

-0.5

0

50

100

150

200

t (ms) Fig. 18. – (Colour on-line) Spin and population dynamics in symmetric double wells. The time evolution of the mean spin m(t) (blue circles) and population imbalance x(t) (brown circles) are shown for three barrier depths within the double-well potential: (a) Vshort = 6Er , J/U = 1.25, (b) Vshort = 11Er , J/U = 0.26 and (c) Vshort = 17Er , J/U = 0.048. The measured traces for the spin imbalance are ﬁtted to the sum of two damped sine waves (blue lines). The population imbalance x(t) stays ﬂat for all traces.

uncertainties in the lattice depths. In this region, we ﬁnd an extended Hubbard model to describe the experimental data much more accurately, which can be understood by the fact that the inter-well interaction energy increases with decreasing barrier and thus has to be included in the model [84]. 7. – Quantum noise correlations For now almost 15 years, absorption imaging of released ultracold quantum gases has been a standard detection method for revealing information on the macroscopic quantum state of the atoms in the trapping potential. For strongly correlated quantum states in optical lattices, however, the average signal in the momentum distribution that one usually observes, e.g. for a Mott insulating state of matter, is a featureless Gaussian wave packet. From this Gaussian wave packet one cannot deduce anything about the strongly correlated quantum states in the lattice potential apart from the fact that phase coherence has been lost. Recently, however, the widespread interest in strongly correlated quantum gases in optical lattices as quantum simulators has lead to the prediction of intriguing quantum phases for ultracold atoms, e.g. with anti-ferromagnetic structure, spin waves or charge density waves. So far it has not been clear how one could detect

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Fig. 19. – Hanbury-Brown-Twiss correlations in expanding quantum gases from an optical lattice. For bosonic particles that are detected at distances r (e.g. on a CCD camera), an enhanced detection probability exists due to the two indistinguishable paths the particles can take to the detector. This leads to enhanced ﬂuctuations at special detection distances r, depending on the ordering of the atoms in the lattice. Detection of the noise correlation can therefore yield novel information on the quantum phases in an optical lattice.

those states. Recently a theoretical proposal by Altman et al. [85] has shown that noise correlation interferometry could be a powerful tool to directly visualize such quantum states. Noise correlation in expanding ultracold atom clouds can in fact be seen as a powerful way to read out the quantum states of an optical-lattice–based quantum simulator. The basic eﬀect relies on the fundamental Hanbury-Brown-Twiss correlations [86, 87] in the ﬂuctuation signal of an atomic cloud. For bosons, e.g., a bunching eﬀect of the ﬂuctuations is predicted to occur at special momenta of the expanding cloud, which directly reﬂect the ordering of the atoms in the lattice. Such bunching eﬀects in momentum space can be directly revealed as spatial correlations in the expanding atom cloud. Our goal therefore is to reveal correlations in the ﬂuctuations of the expanding atomic gas after it has been released from the trap. Such correlations in the expanding cloud at distance d can be quantiﬁed through the second-order correlation function (29)

C(d) =

n(x + d/2) n(x − d/2)d2 x .

n(x + d/2) n(x − d/2)d2 x

Here n(x) is the density distribution of a single expanding atom cloud and the angular brackets · denote a statistical averaging over several individual images taken for diﬀerent experimental runs.

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Spatial correlations in the noise of expanding atom clouds arise here due to a fundamental indistinguishability of the particles as is well known from the foundational experiments in quantum optics of Hanbury, Brown and Twiss [86, 87]. Let us for simplicity consider two detectors spaced at a distance d below our trapped atoms and furthermore restrict the discussion two only two atoms trapped in the lattice potential (see ﬁg. 19). As the trapping potential is removed and the particles propagate to the detectors, there are two possibilities for the particles to reach these detectors, such that one particle is detected at each detector. First, the particles can propagate along path A in ﬁg. 19 to achieve this. However, another propagation path exists, which is equally probable, path B in ﬁg. 19. If we fundamentally cannot distinguish which way the particles have been propagating to our detectors, we have to form the sum for bosons or diﬀerence for fermions of the two propagation amplitudes and square the resulting value to obtain the two particle detection probability at the detectors. As one increases the separation between the detectors, the phase diﬀerence between the two propagation paths increases, leading to constructive and destructive interference eﬀects in the two-particle detection probability. The length scale of this modulation in the two-particle detection probability of the expanding atom clouds depends on the original separation of the trapped particles at a distance alat and is given by the characteristic length scale (30)

l=

h t, malat

where t is the time of ﬂight. Such Hanbury-Brown-Twiss correlations in the shot noise of an expanding atom cloud from a Mott insulating state of matter have recently been observed for 3D and 2D Mott insulating states [88, 9]. Similar pair correlations at the shot noise level have also been obtained in the group of D. Jin [89] for dissociated molecular fragments. In the following we will discuss how these noise correlations can be described formally and how they are diﬀerent from typical time-of-ﬂight absorption images. . 7 1. Time-of-ﬂight versus noise correlations. – Let us begin by considering a quantum gas released from a trapping potential. After a ﬁnite time-of-ﬂight time t, the resulting density distribution yields a three-dimensional density distribution n3D (x). If interactions can be neglected during time-of-ﬂight, the average density distribution is related to the in-trap quantum state via (31)

a†tof (x)ˆ atof (x)tof

ˆ n3D (x)tof = ˆ a(k)trap = ˆ n3D (k)trap , ≈ ˆ a† (k)ˆ

where k and x are related by the ballistic expansion condition k = M x/t (a factor (M/t)3 from the transformation of the volume elements d3 x → d3 k is omitted in the equation). Here we have used the fact that for long time-of-ﬂight times, the initial size of the atom cloud in the trap can be neglected. It is important to realize that in each experimental image, a single realization of the density is observed, not an average.

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Moreover, each pixel in the image records on average a substantial number Nσ of atoms. For each of those pixels, however, the number of atoms recorded in a single realization √ of an experiment will exhibit shot noise ﬂuctuations of relative order 1/ Nσ which will be discussed below. As shown in eq. (31), the density distribution after time of ﬂight represents a momentum distribution reﬂecting the ﬁrst-order coherence properties of the in-trap quantum state. This assumption is however only correct, if during the expansion process interactions between the atoms do not modify the initial momentum distribution, which we will assume throughout the text. When the interactions between the atoms have been enhanced, e.g. by a Feshbach resonance, or a high-density sample is prepared, such an assumption is not always valid. Near Feshbach resonances one therefore often ramps back to the zero crossing of the scattering length before expansion. Density-density correlations in time-of-ﬂight images. – Let us now turn to the observation of density-density correlations in the expanding atom clouds [85]. These are characterized by the density-density correlation function (32)

n(x) ˆ n(x )g (2) (x, x ) + δ(x − x ) ˆ n(x),

ˆ n(x)ˆ n(x ) = ˆ

which contains the normalized pair distribution g (2) (x, x ) and a self-correlation term. Relating the operators after time-of-ﬂight expansion to the in-trap momentum operators, using eq. (31), one obtains (33)

ˆ n3D (x)ˆ n3D (x )tof ≈ ˆ a† (k)ˆ a(k)ˆ a† (k )ˆ a(k )trap =

ˆ a† (k)ˆ a† (k )ˆ a(k )ˆ a(k)trap + δkk ˆ a† (k)ˆ a(k)trap .

The last term on the r.h.s. of the above equation is the autocorrelation term and will be dropped in the subsequent discussion, as it only contributes to the signal for x = x and contains no more information about the initial quantum state than the momentum distribution itself. The ﬁrst term, however, shows that for x = x , subtle momentummomentum correlations of the in-trap quantum states are present in the noise-correlation signal of the expanding atom clouds. Let us discuss the obtained results for two cases that have been analyzed in the experiment: 1) Ultracold atoms in a Mott insulating state or a fermionic band insulating state released from a 3D optical lattice and 2) two interfering one-dimensional quantum gases separated by a distance d. . 7 2. Noise correlations in bosonic Mott and fermionic band insulators. – Consider a bosonic Mott insulating state or a fermionic band insulator in a three-dimensional simple cubic lattice. In both cases, each lattice site R is occupied by a ﬁxed atom number nR . Such a quantum gas is released from the lattice potential and the resulting density distribution is detected after a time of ﬂight t. In a deep optical lattice, the ˆ (in-trap) ﬁeld operator ψ(r) can be expressed as a sum over destruction operators a ˆR of

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localized Wannier states and neglecting all but the lowest band. The ﬁeld operator for destroying a particle with momentum k is therefore given by (34)

a ˆ(k) =

3 ˆ e−ikr ψ(r)d r w(k) ˜

e−ikR a ˆR ,

R

where w(k) ˜ denotes the Wannier function in momentum space. For the two states considered here, the expectation value in eq. (33) factorizes into one-particle density matrices ˆ a†R a ˆR = nR δR,R with vanishing oﬀ-diagonal order. The density-density correlation function after a time of ﬂight is then given by (omitting the autocorrelation term of order 1/N ) (35)

ˆ n3D (x)ˆ n3D (x ) = |w(M ˜ x/t)|2 |w(M ˜ x /t)|2 N 2 ⎡ 2 ⎤ 1 ei(x−x )·R(M/t) nR ⎦ . × ⎣1 ± 2 N R

The plus sign in the above equation corresponds to the case of bosonic particles and the minus sign to the case of fermionic particles in a lattice. Both in a Mott state of bosons and in a ﬁlled band of fermions, the local occupation numbers nR are ﬁxed integers. The above equation then shows that correlations or anticorrelations in the density-density expectation value appear for bosons or fermions, whenever the diﬀerence k − k is equal to a reciprocal lattice vector G of the underlying lattice. In real space, where the images are actually taken, this corresponds to spatial separations for which (36)

|x − x | = =

2ht . λM

Such spatial correlations or anticorrelations in the quantum noise of the density distribution of expanding atom clouds can in fact be traced back to the famous Hanbury, Brown and Twiss eﬀect [90, 86, 87] and its analogue for fermionic particles [91-96]. For the case of two atoms localized at two lattice sites, this can be readily understood in the following way: there are two possible ways for the particles to reach two detectors at positions x and x which diﬀer by exchange. A constructive interference for the case of bosons or a destructive interference for the case of fermions then leads to correlated or anticorrelated quantum ﬂuctuations that are registered in the density-density correlation function [85, 87]. The correlations for the case of a bosonic Mott insulating state and anticorrelations for the case of a fermionic band insulating state have recently been observed experimentally [88, 95, 9]. In these experiments several single images of the desired quantum state are recorded after releasing the atoms from the optical trapping potential and observing them after a ﬁnite time-of-ﬂight time (for a single of these images see, e.g., ﬁg. 20a or ﬁg. 21a). These individually recorded images only diﬀer in the atomic shot noise from

267

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a

x10-4

c

6 4 2

-2

b

6

d

0.2

4 2

0.1

0 0

-400 -200

0

200 400

-400 -200

0

200 400

x (μm)

x (μm)

-2

Corr. Amp. (x10-4)

Co umn Dens ty (a.u.)

0

Fig. 20. – Single-shot absorption image including quantum ﬂuctuations and associated spatial correlation function. a) 2D column density distribution of a Mott insulating atomic cloud containing 6 × 105 atoms, released from a 3D optical lattice potential with a lattice depth of 50Er . The white bars indicate the reciprocal lattice scale l deﬁned in eq. (30). b) Horizontal cut (black line) through the centre of the image in a) and Gaussian ﬁt (red line) to the average over 43 independent images each one similar to a). c) Spatial noise correlation function obtained by analyzing the same set of images, which shows a regular pattern revealing the lattice order of the particles in the trap. d) Horizontal proﬁle through the centre of pattern, containing the peaks separated by integer multiples of l. The width of the individual peaks is determined by the optical resolution of our imaging system.

each other. A set of such absorption images is then processed to yield the spatially (2) averaged second-order correlation function gexp (b) (37)

(2) gexp (b)

=

n(x + b/2) · n(x − b/2) d2 x .

n(x + b/2) n(x − b/2) d2 x

As shown in ﬁg. 20, the Mott insulating state exhibits long-range order in the pair correlation function g (2) (b). This order is not connected with the trivial periodic modulation of the average density imposed by the optical lattice after time of ﬂight, which is factored out in g (2) (x, x ) (see eq. (32)). Therefore, in the superﬂuid regime, one expects g (2) (x, x ) ≡ 1, despite the periodic density modulation in the interference pattern after time of ﬂight. It is interesting to note that the correlations or anticorrelations can

268

D

F

E

G

[ P

&G [ S[

I. Bloch

[ P

Fig. 21. – Noise correlations of a band insulating Fermi gas. Instead of the correlation bunching peaks observed in ﬁg. 20, the fermionic quantum gas shows an HBT type antibunching eﬀect, with dips in the observed correlation function [95].

also be traced back to the enhanced ﬂuctuations in the population of the Bloch waves with quasi-momentum q for the case of the bosonic particles and the vanishing ﬂuctuations in the population of Bloch waves with quasi-momentum q for the case of fermionic particles [95]. Note that in general the signal amplitude obtained in the experiments for the correlation function deviates signiﬁcantly from the theoretically expected value of 1. In fact, one typically observes signal levels of 10−4 –10−3 (see ﬁgs. 20 and 21). This can be explained by the ﬁnite optical resolution when imaging the expanding atomic clouds, thus leading to a broadening of the detected correlation peaks and thereby a decreased amplitude, as the signal weight in each correlation peak is preserved in the detection process. Using single-atom detectors with higher spatial and temporal resolution such as the ones used in refs. [97] and [96], one can overcome such limitations and thereby also evaluate higher-order correlation functions.

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8. – Outlook Ultracold atoms in optical lattices have proven to be versatile model systems for the investigation of strongly correlated many-body systems with ultracold atoms. For the future, one of the focus areas of this research will be directed towards the realization of quantum magnetism with ultracold atoms. Both the prospect of reaching low-energy magnetically ordered quantum phases, such as an antiferromagnetically ordered phase in the fermionic Hubbard model or the non-equilbirium spin dynamics of 1D and 2D systems, seem within reach. To observe several of these pheneomena, the thermal energy of the many-body system will have to be lowered below the superexchange interaction energy scale, which currently requires further cooling of the atomic systems by a factor of 2–3. If such antiferromagnets for fermionic particles could eventually be doped or analyzed at non-integer ﬁlling, one could hope to solve one of the long-standing problems of condensed matter physics, namely whether the fermionic Hubbard model with repulsive interactions contains a superconducting phase [98, 46] and what the mechanism behind such a superconducting phase could be. ∗ ∗ ∗ The author would like to acknowledge stimulating discussions with U. Schneider, S. Will, S. Trotzky, B. Paredes, A. Rosch, E. Demler, M. Lukin, A. Polkovnikov and W. Zwerger. This work was supported by the Deutsche Forschungsgemeinschaft, the European Union, EuroQUAM, the US Army Research Ofﬁce with funding from the Defense Advanced Research Projects Agency (Optical Lattice Emulator programme), and the US Air Force Oﬃce of Scientiﬁc Research. REFERENCES [1] Jaksch D. and Zoller P., Ann. Phys. (N.Y.), 315 (2005) 52. [2] Lewenstein M., Sanpera A., Ahufinger V., Damski B., Sen De A. and Sen U., Adv. Phys., 56 (2007) 243. [3] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885. [4] Chin C., Grimm R., Julienne P. and Tiesinga E., Rev. Mod. Phys., 82 (2010) 1225. [5] Fisher M. P. A., Weichman P. B., Grinstein G. and Fisher D. S., Phys. Rev. B, 40 (1989) 546. [6] Jaksch D., Bruder C., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev. Lett., 81 (1998) 3108. ¨nsch T. and Bloch I., Nature, 415 [7] Greiner M., Mandel M. O., Esslinger T., Ha (2002) 39. ¨ ferle T., Moritz H., Schori C., Ko ¨ hl M. and Esslinger T., Phys. Rev. Lett., 92 [8] Sto (2004) 130403. [9] Spielman I. B., Phillips W. and Porto J., Phys. Rev. Lett., 98 (2007) 080404. ¨ller M. and Ovchinnikov Y. B., Adv. At. Mol. Opt. Phys., 42 [10] Grimm R., Weidemu (2000) 95. [11] Jessen P. S. and Deutsch I. H., Adv. At. Mol. Opt. Phys., 37 (1996) 95. [12] Jaksch D., Briegel H. J., Cirac J. I., Gardiner C. W. and Zoller P., Phys. Rev. Lett., 82 (1999) 1975.

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[97] [98]

Proceedings of the International School of Physics “Enrico Fermi” Course CLXXIII “Nano Optics and Atomics: Transport of Light and Matter Waves”, edited by R. Kaiser, D. S. Wiersma and L. Fallani (IOS, Amsterdam; SIF, Bologna) DOI 10.3254/978-1-60750-755-0-273

Two-dimensional Bose ﬂuids: An atomic physics perspective Z. Hadzibabic Cavendish Laboratory, University of Cambridge JJ Thomson Avenue, Cambridge CB3 0HE, UK

J. Dalibard ´ Laboratoire Kastler Brossel, CNRS, UPMC, Ecole Normale Sup´erieure 24 rue Lhomond, 75005 Paris, France

Summary. — We give in this lecture an introduction to the physics of twodimensional (2d) Bose gases. We ﬁrst discuss the properties of uniform, inﬁnite 2d Bose ﬂuids at non-zero temperature T . We explain why thermal ﬂuctuations are strong enough to destroy the fully ordered state associated with Bose-Einstein condensation, but are not strong enough to suppress superﬂuidity in an interacting system at low T . We present the basics of the Berezinskii-Kosterlitz-Thouless theory, which provides the general framework for understanding 2d superﬂuidity. We then turn to experimentally relevant ﬁnite-size systems, in which the presence of residual “quasi–long-range” order at low temperatures leads to an interesting interplay between superﬂuidity and condensation. Finally we summarize the recent progress in theoretical understanding and experimental investigation of ultracold atomic gases conﬁned to a quasi-2d geometry.

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1. – Introduction The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems [1]. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum ﬂuctuations, which favor disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose ﬂuid is particularly fascinating because of its “marginal” behavior. In an inﬁnite uniform 2d ﬂuid thermal ﬂuctuations at any nonzero temperature are strong enough to destroy the fully ordered state associated with Bose-Einstein condensation, but are not strong enough to suppress superﬂuidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual “quasi–long-range” order at low temperatures leads to an interesting interplay between superﬂuidity and condensation in all experimentally relevant ﬁnite-size systems. This behavior is characteristic of a wide range of physical systems which share some generic properties such as dimensionality, form of interactions, and Hamiltonian symmetries. These include liquid-helium ﬁlms [2], spin-polarized hydrogen [3], Coulomb gases [4], ultracold atomic gases [5, 6], exciton [7, 8] and polariton [9, 10] systems. Moreover, the Berezinskii-Kosterlitz-Thouless theory which provides the general framework for understanding 2d superﬂuidity is also applicable to a range of physical phenomena in discrete systems, such as ordering of spins on a 2d lattice [11] and melting of 2d crystals [12, 13]. In this paper we give an introduction to the physics of 2d Bose ﬂuids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultracold atomic gases conﬁned to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other ﬁelds of physics. . 1 1. Absence of true long-range order in 2d. – The most familiar phase transitions in three spatial dimensions (3d), such as freezing of water, ferromagnetic ordering in spin systems, or Bose-Einstein condensation (BEC), are all associated with emergence of true long-range order (LRO) below some non-zero critical temperature. Such order is embedded in a spatially uniform order parameter, e.g. magnetization in a ferromagnet or the macroscopic wave function ψ describing a BEC. Further, in all the above cases emergence of true LRO corresponds to spontaneous breaking of some continuous symmetry of the Hamiltonian. In case of crystallization (freezing), translational symmetry is spontaneously broken. For Heisenberg spins on a (ﬁxed) lattice, the Hamiltonian has a continuous spin rotational symmetry which is spontaneously broken in the ferromagnetic state. In case of BEC, the phase of ψ is arbitrarily spontaneously chosen at the transition. As we will discuss in more detail later, under certain conditions this makes a Bose gas formally equivalent to a system of two-component spins on a lattice, the so-called XY model. Already in 1934 Peierls pointed out that the possibility for a physical system to exhibit true LRO can crucially depend on its dimensionality [14, 15]. Peierls considered

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a 2d crystal at ﬁnite temperature T , and studied the eﬀects of thermal vibrations of atoms around their equilibrium positions in the lattice (i.e. phonons). He found that the uncertainty in the relative position of two atoms diverges with the distance between their equilibrium positions: (1)

r ! (u(r) − u(0))2 ∝ T ln a

for r = |r| a, where u(r) is the atom displacement from its equilibrium position r, a is the lattice spacing, and . . . denotes a thermal average. In contrast, the corresponding result in 3d is ﬁnite, and small compared to a if T is below the melting temperature. The result in eq. (1) is in direct contradiction with the starting hypothesis of long-range crystalline order, since it implies that, based on the positions of atoms in one part of the system, we cannot predict with any certainty the positions of atoms at large distances. The Peierls result (1) is a simple example of the absence of spontaneous symmetry breaking at non-zero T in 2d. The absence of LRO in low-dimensional systems was later more generally and formally studied by Bogoliubov [16], Hohenberg [17], and Mermin and Wagner [18]. The general statement is that LRO is impossible in the thermodynamic limit at any non-zero T in all 1d and 2d systems with short-ranged interactions and a continuous Hamiltonian symmetry. This is now most commonly known as the MerminWagner theorem. In all such systems Hamiltonian symmetry is always restored by lowenergy long-wavelength thermal ﬂuctuations, the so-called Goldstone modes. In the case of an interacting Bose gas Goldstone modes are phonons, while in the case of the XY model on a lattice they are spin-waves. As a direct consequence of the functional form of the density of states in low dimensionality, such modes always have a diverging infrared contribution and destroy LRO. It is however important to stress that the absence of true LRO does not preclude the possibility of any phase transitions in 2d systems, just the symmetry-breaking ones. As we will see later, a phase transition associated with the apparition of a topological order does take place in a uniform 2d gas. . 1 2. Outline of the paper . – In sect. 2 we will explicitly see that Bose-Einstein condensation is impossible in both the ideal and a repulsively interacting inﬁnite uniform 2d Bose gas. The absence of true LRO in these systems is seen in the fact that the ﬁrst-order correlation function (2)

ˆ † (r)Ψ(0) ˆ , g1 (r) ≡ Ψ

ˆ where Ψ(r) is the annihilation operator for a particle at position r, always tends to zero for r → ∞. Note that these two statements are equivalent under the Penrose-Onsager deﬁnition of the condensate density [19]: (3)

n0 ≡ lim g1 (r). r→∞

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However, the weak logarithmic divergence in eq. (1) suggests that the destruction of LRO is only marginal in 2d. The consequence of this weak divergence is that an interacting Bose gas at low T exhibits “quasi–long-range order”, corresponding to g1 (r) which decays only algebraically with distance. This low-T state is also superﬂuid, and the phase transition between the superﬂuid and the high-T normal state is described by the Berezinskii-Kosterlitz-Thouless (BKT) theory [20,21], which we discuss in sect. 3. Further, the slow decay of g1 (r) at low T has important consequences for condensation and symmetry breaking in the experimentally relevant ﬁnite-size systems. We address this issue in sects. 4 and 5, ﬁrst for a ﬁnite box potential, and then for the experimentally most pertinent case of a harmonically trapped gas. In sects. 6 and 7 we introduce the experimental methods of atomic physics used in the current studies of 2d Bose gases. In sect. 6 we give an overview of experimental systems in which (quasi-)2d Bose gases have been realized, and in sect. 7 we discuss the experimental probes of coherence in these systems. We conclude by outlining some research directions which are likely to be of interest in the near future in sect. 8. 2. – The inﬁnite uniform 2d Bose gas at low temperature In this section we discuss thermal ﬂuctuations and the absence of Bose-Einstein con. densation in an inﬁnite uniform 2d gas. In subsect. 2 1 we consider the ideal gas, in which no phase transition occurs. In this case the ﬁrst order correlation function g1 (r) gradually changes from a gaussian function in the high-temperature, non-degenerate regime, to an exponentially decaying function in the degenerate regime. In the repulsively interacting 2d Bose gas the BKT phase transition to a superﬂuid state occurs at a non-zero critical temperature, but the conclusion that BEC transition does not occur remains true. We ﬁrst introduce the description of interactions in an . . atomic 2d gas in subsect. 2 2. In subsect. 2 3 we qualitatively discuss why interactions lead to a strong suppression of density ﬂuctuations in a degenerate gas, so that the lowenergy long-wavelength excitations (phonons) in this system are almost purely phase . ﬂuctuations. The Bogoliubov analysis presented in subsect. 2 4 provides a more quantitative justiﬁcation for this conclusion and also indicates why we expect an interacting . 2d gas to be superﬂuid at very low T . As we show in subsect. 2 5, the long-wavelength phonons still lead to a vanishing g1 (r) at r → ∞. Therefore, in accordance with the Mermin-Wagner theorem, these “soft” Goldstone modes still destroy the long-range order and restore the Hamiltonian symmetry. However, the decay of g1 at large distances is only algebraic with r at very low T . This low temperature, superﬂuid state is said to exhibit “quasi–long-range order”. The BKT phase transition from the superﬂuid state with algebraic correlations to the normal state with exponentially decaying correlations is discussed in the following sect. 3. . 2 1. The ideal 2d Bose gas. – The absence of Bose-Einstein condensation in the ideal 2d Bose gas can straightforwardly be seen by following the standard Einstein’s argument

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which associates condensation with the saturation of the excited single-particle states at some non-zero temperature. In 2d, the density of states for spinless bosons mL2 /(2π2 ) is constant, where m is the particle mass and L → ∞ is the linear size of the system. Assuming no condensation, the total number of particles is then (4)

N=

mL2 2π2

∞

dε eβ(ε−μ)

0

−1

,

where β = 1/(kB T ), and μ ≤ 0 is the chemical potential. Equivalently, the phase-space density D is given by D ≡ nλ = 2

(5)

∞

0

dx = − ln(1 − Z), −1

1 x Ze

√ where n = N/L2 is the 2d number density, λ = h/ 2πmkB T is the thermal wavelength, and Z = eβμ is the fugacity. In 3d, the signature of BEC is that the analogous relationship between the phase space density and fugacity has no solutions for Z when the phase space density is larger than the critical value n3 λ3 ≈ 2.612, where n3 is the 3d number density. Below the condensation temperature, chemical potential is ﬁxed at μ = 0 and the phase space density of particles in the excited states is saturated at ≈ 2.612. However, we see that in 2d a valid solution, 2

eβμ = 1 − e−nλ ,

(6)

always exists. In other words, for any non-inﬁnite phase space density there exists a negative value of μ which allows normalization of the thermal distribution of particles in the excited states to the total number of particles in the system N . This shows that BEC does not occur in the ideal inﬁnite uniform 2d Bose gas. We next look at the ﬁrst-order correlation function g1 (r), which we can write as the Fourier transform of the momentum space distribution function nk : (7)

g1 (r) =

1 (2π)2

∞

nk eik·r d2 k,

with nk =

0

1 eβ( k −μ) − 1

,

k =

2 k 2 . 2m

In the absence of condensation g1 (r) always vanishes at r → ∞. However, it still shows qualitatively diﬀerent behavior at high and low temperature: – In a non-degenerate gas, eq. (6) gives Z ≈ nλ2 1. In this regime |μ| kB T and all momentum states are weakly occupied: (8)

nk ≈ Z e−β k ≈ nλ2 e−k

2

λ2 /4π

1

(∀k).

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In this case g1 (r) is Gaussian, showing only short-range correlations which decay √ on the length scale λ/ π: g1 (r) ≈ n e−πr

(9)

2

/λ2

. 2

– In a degenerate gas with nλ2 > 1, from eq. (6) we get Z ≈ 1 and β|μ| ≈ e−nλ 1, so that D = nλ2 ≈ ln(kB T /|μ|). In this regime the occupation of high-energy states, with βk 1, is still small and given by the Boltzmann factor (10)

nk ≈ e−β k = e−k

2

λ2 /4π

1,

for k 2 4π/λ2 .

However, the low-energy states with βk 1 are strongly occupied: (11)

nk ≈

kB T 1 4π 1, = 2 2 k + |μ| λ k + kc2

for k 2 4π/λ2 ,

where kc = 2m|μ|/. In this case g1 (r) is bimodal. At short distances, up to r ∼ λ, correlations are still Gaussian as in eq. (9). However, the Lorentzian form in eq. (11) corresponds to approximately(1 ) exponential decay of g1 (r) at larger distances, r λ: (12)

g1 (r) ≈ e−r/ ,

with = kc−1 ≈ λ enλ

2

/2

√ / 4π.

We can also estimate the partial phase space densities corresponding to the Gaussian and the Lorentzian parts of the momentum distribution, and see that most particles accumulate in the Lorentzian part corresponding to low momentum states: (13) (14)

∞ λ2 nk d2 k ≈ 1/e D, (2π)2 √4π/λ2 √4π/λ2 λ2 DL ≈ nk d2 k ≈ D. (2π)2 0

DG ≈

We thus see that even though there is no phase transition in this system, the ﬁrst-order correlation function gradually changes from a Gaussian with short-range correlations in the non-degenerate regime, to an exponential in the degenerate regime. Further, 2 for nλ2 > 1, the correlation length ∝ enλ /2 grows exponentially. Therefore, while g1 (r) formally vanishes at r → ∞ at any non-zero T , the length scale over which it decays is exponentially large in the deeply degenerate ideal Bose gas. This has important (1 ) More precisely the Fourier transform of the 2d Lorentzian distribution 1/(k2 + kc2 ) is deﬁned for r = 0 and is proportional to the Bessel function of imaginary argument K0 (kc r), whose √ asymptotic behaviour is e−kc r / r.

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279

consequences for the ﬁnite-sized experimental systems, since for any arbitrarily large ﬁxed system size L there exists a low enough non-zero temperature T for which > L, and correlations span the entire system. This issue will be addressed in sects. 4 and 5. . 2 2. Interactions in a 2d Bose gas at low T . – The interaction between two atoms at positions ri and rj in a 3d Bose gas at low temperature is well characterized by the contact potential(2 ) (15)

4π2 as δ (3d) (ri − rj ), m

V (ri − rj ) =

where as is the 3d s-wave scattering length, and mechanical stability requires repulsive interactions as > 0. In 2d the two-body scattering problem is in general more complicated and the scattering amplitude is energy-dependent [25]. We will address this issue in . more detail in subsect. 6 2. However, in all experimentally relevant situations so far, the analysis of interactions is simpliﬁed by the fact that while the gas is kinematically 2d, the interactions can be described by 3d scattering. The condition for this simpliﬁcation is that the thickness of the sample 0 is much larger than the 3d scattering length as . In all current atomic experiments the ratio 0 /as is larger than 30. In this case we can to a very good approximation write the interaction energy as (16)

Eint =

g 2

n2 (r) d2 r,

where g is the energy-independent interaction strength and n(r) is the local density. Note that here and in the following we treat the density n(r)—and later the phase θ(r)—as a classical function. This will notably simplify the mathematical aspects of our approach, while capturing all important physical consequences related for example to quasi–long-range order and to the normal-superﬂuid transition. In 3d, the interaction strength g (3d) = (4π2 /m)as explicitly depends on the scattering length. However, we see on dimensional grounds that in 2d we can write (17)

g=

2 g˜, m

so that g˜ is a dimensionless coupling constant. The 2d healing length, which gives the characteristic length scale corresponding to the interaction energy, is then given by (18)

√ ξ = / mgn = 1/ g˜n .

(2 ) Strictly speaking, the delta-function in 3d must be regularized by using for example the pseudo-potential [22]. The extension of the notion of zero-range potential to the 2d case is discussed in [23] (see also [24] for a discussion in terms of many-body T -matrix).

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We can anticipate on dimensional grounds that g˜ ∼ as /0 , corresponding to n = n3 0 . Speciﬁcally, if a gas is harmonically conﬁned to 2d, say in the x-y plane, we will see in . subsect. 6 2 that the expression for g˜ is (19)

g˜ =

√

8π

as , az

where in this case 0 = az is the oscillator length along the kinematically frozen direction z. We can qualitatively deﬁne the strongly interacting limit by the value of g˜ for which the interaction energy of N particles, Eint , reaches the kinetic energy EK of N noninteracting particles equally distributed over the lowest N single-particle states. In this limit, we expect the many-body ground state to be strongly correlated. We can estimate the interaction energy with a mean-ﬁeld approximation by setting n(r) = n in eq. (16), which corresponds to neglecting density ﬂuctuations: (20)

Eint =

2 g˜N n. 2m

Using the 2d density of states, mL2 /(2π2 ), the energy of the N -th excited single-particle state is EN = (2π2 /m)n, and so (21)

EK =

1 π2 N EN = N n. 2 m

The strongly interacting limit then corresponds to (22)

Eint = EK ⇒ g˜ = 2π.

For comparison, the value of g˜ in the current experiments on atomic 2d Bose gases varies between ∼ 10−1 [26, 27] and ∼ 10−2 [28], while in the more strongly interacting 4 He ﬁlms [2] it is estimated to be of order 1 [29]. The fact that result (22) is independent of the density of the gas n is a natural consequence of the fact that g˜ is dimensionless. This is in contrast to the 3d case, where the relative importance of interactions is characterized by the dimensionless parameter n3 a3s . . 2 3. Suppression of density ﬂuctuations and the low-energy Hamiltonian. – At strictly T = 0, a weakly interacting 2d Bose gas is condensed and described by a constant macro√ scopic wave function (a uniform order parameter) ψ = neiθ , where n and θ are classical ﬁelds. At any ﬁnite T , both the amplitude and the phase of ψ show thermal ﬂuctuations. However, repulsive interactions will always lead to a reduction of density ﬂuctuations in a # 2 low-temperature gas. The interaction energy is (g/2) n (r) d2 r = (g/2)L2 n2 (r), and so keeping the average density n = n(r) ﬁxed, we see that minimizing the interaction energy is equivalent to minimizing the density ﬂuctuations: (23)

(Δn)2 = n2 (r) − n2 = (g2 (0) − 1)n2 ,

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281

where g2 (r) = n(r)n(0)/n2 is the normalized second-order (density-density) correlation function(3 ). In the ideal Bose gas g2 (0) = 2, while if the density ﬂuctuations are completely suppressed g2 (0) = 1. We can estimate the minimum cost of density ﬂuctuations by equating it with the increase in interaction energy from adding a single particle to the system: (24)

2 ∂Eint = gn = g˜n, ∂N m

where we have set g2 (0) = 1, which is consistent with the density ﬂuctuations being signiﬁcantly suppressed. Comparing this with the thermal energy kB T , we get gn g˜ = D. kB T 2π

(25)

This strongly suggests that at suﬃciently low temperature, given by D 2π/˜ g , any density ﬂuctuations must be very strongly suppressed. Numerical calculations [29] show that they can be signiﬁcantly suppressed already for D 1, with the exact extent of the suppression depending on the strength of interactions g˜. In the limit of strong suppression of density ﬂuctuations, the interaction energy becomes just an additive constant ((1/2)gn2 L2 ) in the Hamiltonian, and the kinetic energy arises only from the variations of θ, the phase of ψ. The system is then often described by an eﬀective low-energy Hamiltonian: (26)

Hθ =

2 ns 2m

(∇θ)2 d2 r,

where one heuristically replaces the total density n with the (uniform) superﬂuid density ns ≤ n. This is physically motivated because one expects only the superﬂuid component to exhibit phase stiﬀness and to ﬂow under an imposed variation of θ, with local velocity of the superﬂuid given by vs = (/m)∇θ. Also note that at T = 0 superﬂuid density is equal to the total density, and at very low T they are similar. In essence, renormalizing n to the lower ns is an eﬀective way of absorbing all the short distance physics, including any residual density ﬂuctuations, and Hθ provides a good description of long-range physics, at distances r ξ, λ. The eﬀective low-energy Hamiltonian Hθ is the continuous version of the Hamiltonian of the XY model of spins on a lattice. It can be used to derive the correct long-range . algebraic decay of g1 (r) in the low-T superﬂuid state (see subsect. 2 5). However, it is important to stress some caveats: 1) Even though at low temperature ns ≈ n, Hθ fundamentally cannot be the correct microscopic Hamiltonian. We can see this on very general grounds since the proper (3 ) Note that following the conventions in the literature on diﬀerent topics we normalize g2 so that it is dimensionless, while g1 has units of density.

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Hamiltonian is by deﬁnition a temperature-independent entity, while Hθ depends on the temperature through ns . More accurately Hθ represents the increase of the free energy of the gas if one imposes the superﬂuid current (/m)∇θ, for instance by setting the gas in slow rotation (see the appendix in [6] for more details). 2) If and only if the density ﬂuctuations are completely absent can the 2d Bose gas be formally mapped onto the XY model. This condition is essentially fulﬁlled for D 2π/˜ g, but it is not satisﬁed at the BKT critical point. As discussed in sect. 3, BKT transition g ), which for the experimentally occurs at a critical phase space density Dc = ln(380/˜ relevant values of g˜ corresponds to Dc ∼ 6–10. At that point density ﬂuctuations are signiﬁcantly suppressed, but cannot be completely neglected. This is one of the key reasons that makes the proper microscopic theory of the BKT transition in the Bose gas diﬃcult. In summary, Hθ can be used as an eﬀective quantum Hamiltonian for the derivation of some key features of long-range physics at low temperature, where ns ∼ n. However Hθ alone is not suﬃcient for the proper derivation of the BKT transition. . 2 4. Bogoliubov analysis. – By performing Bogoliubov analysis near T = 0 we can see more explicitly why the density ﬂuctuations are suppressed at suﬃciently low T , but the phase ﬂuctuations are not, and also why it is natural to expect the low-temperature state to be superﬂuid. The basic idea is that we start with the assumption that the T = 0 state of a weakly √ interacting gas is described by the uniform order parameter ψ = n eiθ , ﬁnd the excitation spectrum, and then consider the eﬀects of the thermal occupation of the excitation modes. This approach may not seem justiﬁed in 2d, because in the end we ﬁnd that thermal ﬂuctuations destroy the order parameter, and thus invalidate our starting assumption. However we can qualitatively argue that it still works well as long as we have a well-deﬁned local order parameter, which is true if the order parameter is destroyed only at very large distances by the long-wavelength phase ﬂuctuations. The applicability of the Bogoliubov approach to 2d quasi-condensates was formally justiﬁed in [30, 31]. Assuming contact interactions, the classical ﬁeld Hamiltonian is given by g 2 (∇ψ ∗ (r))(∇ψ(r)) d2 r + (ψ ∗ (r))2 (ψ(r))2 d2 r. (27) H= 2m 2 The dynamics of ψ(r, t) are governed by the Gross-Pitaevskii equation [32, 33]: 2 2 ∂ψ (28) − ∇ + g|ψ|2 ψ = i , 2m ∂t which we can derive from eq. (27) by treating ψ and ψ ∗ as canonical variables. We now deﬁne the local phase θ(r, t) from ψ(r, t) = |ψ(r, t)| eiθ(r,t) . Assuming that ﬂuctuations are small, we write |ψ(r, t)|2 = n (1 + 2η(r, t)), with η 1 and #the density 2 η d r = 0. Within this approximation we get, up to the additive constant gn2 L2 /2, (29)

H=

2 n 2m

(∇θ(r))2 d2 r +

2 n (∇η(r))2 + 2gn2 (η(r))2 d2 r, 2m

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and a set of coupled linear equations for the time evolution of θ(r, t) and η(r, t): ∂θ gn = ∇2 η − (1 + 2η), ∂t 2m ∂η =− ∇2 θ. ∂t 2m

(30) (31)

It is now convenient to Fourier expand the phase and the density ﬁelds: (32)

θ(r, t) =

ck (t) eik·r ,

η(r, t) =

k

dk (t) eik·r ,

k

where k = 2π(jx , jy )/L, with jx , jy integers, is a discrete variable since we consider for the moment a sample of ﬁnite size L2 . We will let L → ∞ at the end of the calculation. The functions θ and η are real, #which implies c∗k = c−k and d∗k = d−k . In addition the conservation of particle number η d2 r = 0 leads to d0 = 0. This yields the Hamiltonian (33)

H = nL

2

2 k 2 k

2m

|ck | + 2

2 k 2 2 + 2gn |dk | , 2m

and the coupled equations of motion (for k = 0):

k 2 2gn + 2m

(34)

c˙k = −

(35)

k 2 d˙k = ck . 2m

dk ,

For k = 0 the equation of motion deduced from eq. (30), c˙0 = −gn/, simply gives the time evolution of the global phase of the gas. In this formulation ck and dk are conjugate dimensionless quadratic degrees of freedom corresponding to the phase and the density ﬂuctuations, respectively. At each k we have a harmonic-oscillator–like Hamiltonian, and from the equations of motion we can read oﬀ the eigenfrequencies (36)

ωk =

k 2 2m

k 2 2gn + , 2m

which is the well-known Bogoliubov result. At low k, the eigenmodes are phonons with ωk = ck, where c = gn/m. At high k we have free particle modes with ωk = k 2 /(2m)+ √ gn/. The crossover between the two regimes is at k ∼ 1/ξ = g˜n. From this analysis, and speciﬁcally the results in eqs. (33) and (36), we can draw several conclusions: 1) From the dispersion relation ωk we see that the excitation modes in this system have a non-zero minimal speed c. From the Landau criterion, we thus expect it to be

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superﬂuid with a critical velocity vc = c. This argument is identical to the 3d case, and relies just on the existence of a (reasonably) well-deﬁned local order parameter, rather than true LRO. We note however that the Landau criterion is a necessary, but not a suﬃcient condition for the existence of a superﬂuid state, because it does not address the question of metastability of the superﬂuid ﬂow (see e.g. [34]). 2) Both in the classical and in the quantum regime the harmonic oscillator at thermal equilibrium obeys the virial theorem, mω 2 x2 = p2 /m, where x and p are position and momentum, respectively. Reading-oﬀ the coeﬃcients in front of |ck |2 and |dk |2 in eq. (33), we see that in our case this corresponds to 2 k 2 /2m

|dk |2 = 2 2 . 2

|ck | k /2m + 2gn

(37)

We thus explicitly see that long-wavelength phonons involve only phase ﬂuctuations, since |dk |2 |ck |2 for k → 0. On the other hand, the high-k free particles involve both phase and density ﬂuctuations in equal parts, since |dk |2 = |ck |2 for k → ∞. 3) We also explicitly see that density ﬂuctuations are not “soft” Goldstone modes, because their energy cost does not vanish in the k → 0 limit. The eﬀect of interactions is to suppress (compared to the ideal gas) the density ﬂuctuations at length scales > ξ. The ﬂuctuations on short length scales (< ξ, corresponding to k > 1/ξ) are not suppressed by interactions, but those in any case do not have a diverging contribution. 4) On the other hand, the energy cost of phase ﬂuctuations (phonons) vanishes for k → 0. This conclusion would be the same in 3d, but the crucial diﬀerence is that in 2d the density of states at low k leads to a diverging eﬀect of these ﬂuctuations and the . destruction of true LRO, as shown in the next subsect. 2 5. 5) We can use this Bogoliubov analysis to provide an estimate of the relative density ﬂuctuations Δn2 /n2 = 4 η 2 . In principle the thermal equilibrium distribution of the Bogoliubov modes is given by the Bose-Einstein distribution. For simplicity we approximate it in the following way; we suppose that the modes with frequency ωk lower than kB T / have an average energy kB T /2 (classical equipartition theorem) and that the higher-energy modes have a negligible population. Therefore we take(4 ) (38)

nL

2

" kB T /2, if ωk < kB T, 2 k 2 2 + 2gn |dk | = 2m 0, if ωk > kB T.

In addition we assume that kB T is notably higher than the interaction energy gn, which is the case in most experiments realized with cold atoms so far. The cutoﬀ ωk = kB T then lies in the free-particle part of the Bogoliubov spectrum, at the wave vector kT ≈ (4 ) We use here that ck and dk are complex amplitudes, and that the modes k and −k are correlated, so that d−k = d∗k .

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285

√ mkB T /. We can now estimate the relative density ﬂuctuations (39)

Δn2 L2 2 = 4

|d | ≈ k n2 4π 2 k

k
kB T /2 4 d2 k nL2 2 k 2 /2m + 2gn

and we ﬁnd 2 Δn2 ≈ ln n2 nλ2

(40)

kB T 2gn

.

For realistic values of the ratio kB T /gn (i.e. not exponentially large), we recover here the previously announced result that density ﬂuctuations are notably suppressed when nλ2 1. Based on these arguments, with some more quantitative justiﬁcation we arrive at the same conclusions as before. First, for eﬀectively describing the low-temperature state of the gas, the most important part of the Hamiltonian in eq. (33) is the term corresponding to the phase ﬂuctuations. Second, if we keep only that term, we also have to keep in mind that we are neglecting short-distance physics and eﬀectively introducing a momentum cutoﬀ at kmax = 1/ξ. As before we can heuristically incorporate short-distance physics by replacing n → ns to obtain (41)

H ≈ ns L

2

2 k 2 2 2 |ck | = ns (∇θ(r))2 d2 r, 2m 2m

k k<ξ −1

which coincides with the Hamiltonian Hθ given in eq. (26). . 2 5. Algebraic decay of correlations. – To derive the low-T behavior of the one-body correlation function g1 (r) = ψ ∗ (r)ψ(0) at large distances, r ξ, λ, we start with the √ Hamiltonian Hθ , and the wave function with no density ﬂuctuations ψ(r) = ns eiθ(r) . Note that this normalization of ψ leads in r = 0 to the incorrect value g1 (0) = ns , whereas it should be g1 (0) = n > ns . However, as discussed earlier, density ﬂuctuations at short distances lead to a more complicated decay of g1 , and at large r replacing n → ns is the appropriate normalization. The long-distance (r ξ) behavior of g1 (r) essentially depends on the population of phonon modes with wave vector k ξ −1 , which coincides with the momentum cutoﬀ introduced in Hθ . The occupation of eigenmodes is in general given by the standard Bose result (exp(βωk )−1)−1 . Assuming as in eq. (40) that kB T > gns , the phonon modes are in the regime ωk kB T and the occupation number simpliﬁes into kB T /(ωk ) which leads to (classical equipartition theorem) (42)

phonon modes: ns L2

2 k 2 kB T

|ck |2 = . 2m 2

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Introducing the real and imaginary part of the Fourier coeﬃcients ck = ck + ick , we have (43)

! ! |ck |2 = |ck |2 =

1 π . ns λ2 L2 k 2

We recall that ck and ck are independently ﬂuctuating variables ( ck ck = 0) and that the mode k and −k are correlated because θ is real: ck = c−k and ck = −c−k . We want to calculate (44) g1 (r) = ψ ∗ (r)ψ(0) = ns ei(θ(r)−θ(0)) , where (45)

θ(r) − θ(0) =

ck (cos(k · r) − 1) − ck sin(k · r).

k 1

2

For each independent Gaussian variable u, eiu# = e− 2 u . Using eq. (43), and transforming the discrete sum over k into L2 /(4π 2 ) d2 k we obtain:

(46)

1 g1 (r) = ns exp − 2πns λ2

1 − cos(k · r) 2 d k . k2

The integral in the exponent has signiﬁcant contributions only from modes k > 1/r so that 1 − cos(k · r) ∼ 1. Since we restrict our analysis to r λ, this is not inconsistent with the classical ﬁeld approximation which requires k < 1/λ. The upper limit of the integral is set by the short-distance cutoﬀ kmax =#1/ξ. We thus expect the integral to be ∼ ln(r/ξ). More formally, we can note that ∇2 (1 − cos(k · r))k −2 d2 k = (2π)2 δ(r), from which we infer r 1 − cos(k · r) 2 . d k = 2π ln (47) k2 ξ This leads to (48)

g1 (r) = ns

1/(ns λ2 ) ξ . r

Note that depending on the relative size of λ and ξ we can also set the upper limit of the integral to 1/λ, but this diﬀerence in the short-distance cutoﬀ does not aﬀect the main conclusion about the power law decay of correlations at large distances. To summarize, we have shown that in an interacting 2d Bose gas at low T , the ﬁrst-order correlation function g1 (r) decays algebraically with r at large distances. The conclusion that g1 (r) vanishes for r → ∞ is consistent with the Mermin-Wagner theorem, i.e. the absence of BEC and true LRO at any non-zero T . However the decay of g1 (r) is very slow and the system exhibits a “quasi–long-range order”. Further, as discussed

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in the following sect. 3, the exponent 1/(ns λ2 ) is never larger than 1/4 in the superﬂuid state, making the decay of g1 (r) extremely slow. The superﬂuid state with suppressed density ﬂuctuations can be viewed as a superﬂuid “quasi-condensate”, i.e. a condensate with a ﬂuctuating phase [35-37]. 3. – The Berezinskii-Kosterlitz-Thouless (BKT) transition in a 2d Bose gas Our analysis so far does not explain how the phase transition from the lowtemperature superﬂuid state to the high-temperature normal state takes place. This transition is unusual because it does not involve any spontaneous symmetry breaking in the superﬂuid state, and in the usual classiﬁcation of classical phase transition is termed “inﬁnite order”, suggesting that most thermodynamic quantities (except for example superﬂuid density) vary smoothly at the transition. There is no true LRO on either side of the transition, but the functional form of the decay of g1 (r) changes from algebraic in the superﬂuid state (corresponding to quasi-LRO) to exponential in the normal state (corresponding to no LRO). The microscopic theory of the 2d superﬂuid transition was developed by Berezinskii [20] and Kosterlitz and Thouless [21] (see [4] for a more recent review). The transition takes place in the degenerate regime, where the density ﬂuctuations in an interacting gas are signiﬁcantly suppressed. We therefore expect that the transition can still be at least qualitatively explained by considering only phase ﬂuctuations. However, a sudden transition with a well-deﬁned critical point cannot be explained by considering only phonons, since we have seen in eq. (48) that, while they destroy true LRO at any non-zero T , their eﬀect grows smoothly with temperature. . 3 1. The role of vortices and topological order. – The key conceptual ingredient of the BKT theory is that in addition to phonons described by the Hamiltonian (26), another natural source of phase ﬂuctuations are vortices, points at which the superﬂuid density vanishes, and around which the phase θ varies by a multiple of 2π. For our purposes we can consider only “singly-charged” vortices with phase winding ±2π, which are energetically stable. Around an isolated single vortex, centered at the origin, the velocity ﬁeld ∇θ/m varies as /(mr), corresponding to angular momentum per particle. The two signs of the vortex charge correspond to the two senses of rotation around the vortex. The size of the vortex core (hole in the superﬂuid density) is set by the healing length ξ, so their presence is not inconsistent with the picture that the density ﬂuctuations are suppressed at length scales r ξ. In fact, it makes sense to speak of well-deﬁned individual vortices only if away from the vortex cores the density ﬂuctuations are suppressed on the length scale ξ; otherwise we simply have a fully ﬂuctuating thermal gas. As we will illustrate below, once one considers vortices as another source of phase ﬂuctuations, one can explain the microscopic mechanism behind the superﬂuid-to-normal phase transition. Below a well-deﬁned critical temperature TBKT , vortices can exist only in the form of bound (“dipole”) pairs of vortices with opposite circulations ±2π. Since they do not have any net charge, such pairs do not create any net circulation along closed

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Fig. 1. – The BKT mechanism at the origin of the superﬂuid transition. Below the transition temperature (left ﬁgure), vortices exist only in the form of bound pairs formed with vortices of opposite circulation. When approaching the transition point the density of pairs grows and the average size of a pair diverges. Just above the transition point (right ﬁgure), a plasma of free vortices is formed and the superﬂuid density vanishes.

contours larger than the pair size, which for a tightly bound pair is also of order ξ(5 ). Such pairs therefore have only a short-range eﬀect on the phase θ and the velocity ﬁeld, and do not have a large eﬀect on the behavior of g1 (r) at large distances. They fall under “short-range physics” of the system and together with the residual density ﬂuctuations they lead to some renormalization of ns , but do not qualitatively alter the phenomenology of the long-range physics discussed in sect. 2. On the other hand, above TBKT unbinding of vortex pairs and proliferation of free vortices becomes energetically favorable. Free vortices then form a disordered gas of phase defects and completely “scramble” the phase θ (see ﬁg. 1). This destroys the quasi-LRO and suppresses superﬂuidity. At even higher temperature where density ﬂuctuations are strong, the notion of individual vortices becomes physically irrelevant. In hindsight, we can associate superﬂuidity with presence of a “topological order”. Long-wavelength phase ﬂuctuations (phonons) destroy true LRO, but do not alter the topology of the system. In other words, phonons lead to smooth local variations of the ﬁeld ψ which can be eliminated (or “ironed out”) by continuous deformations. The same argument holds for bound vortex pairs which can be annihilated. Therefore, the superﬂuid quasi-condensate with no free vortices is topologically identical to the BEC with true LRO. On the other hand, an isolated free vortex cannot be unwound and eliminated from the system by continuous deformations of ψ; it aﬀects the phase θ non-locally, at arbitrary large distances. The annihilation argument also does not work for a plasma of free vortices, because if we consider a closed contour of arbitrary large size it will in general not contain equal number of vortices with opposite charges. Therefore at any length scale the free vortex plasma is topologically diﬀerent from an ordered BEC. Although we could not have necessarily anticipated this, we can deduce that topological order, rather than true LRO, is a suﬃcient condition for superﬂuidity in an interacting 2d Bose gas. (5 ) For a dipole ﬁeld |∇θ| ∼ 1/r2 , so the circulation

H

∇θ · dr vanishes for large contours.

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. 3 2. A simple physical picture. – The full thermodynamic description of the 2d gas, including the role of vortices, is a diﬃculy task. It ﬁrst requires to introduce the velocity ﬁelds [38] or the mass-current densities [39] in the ﬂuid. The normal and superﬂuid components can then be extracted from the spatial correlation functions of this masscurrent density [39]. Finally an analysis using renormalization group arguments leads to the so-called universal jump for the superﬂuid density: This density takes the value ns = 4/λ2 on the low-temperature side of the transition point, and ns = 0 on the high temperature side. The existence and the value of the universal jump in the superﬂuid density was formally derived by Nelson and Kosterlitz [38], and it was ﬁrst conﬁrmed to high accuracy in experiments with liquid He ﬁlms by Bishop and Reppy [2]. This full description is outside the scope of this set of lectures. Here we simply illustrate how vortices drive the BKT transition, starting from a superﬂuid with ﬁnite ns , and then considering the free energy associated with spontaneous creation of a single free vortex. Without any loss of generality, in order to simplify the calculations we consider a circular geometry, with R → ∞ the radius of the system. The kinetic energy cost of a single vortex placed at the origin is simply given by (49)

E= ξ

R

1 ns 2

mr

2

2 π ns ln d r= m

2

R ξ

,

where as always we assume that only the superﬂuid component rotates under the inﬂuence of the vortex. The normal component does not have any phase stiﬀness and its motion is not aﬀected by the presence of the vortex. The entropy associated with a single vortex core is given by the number of distinct positions where a vortex of radius ξ can be placed in a disc of radius R: (50)

S = kB ln

R2 π ξ2π

= 2kB ln

R ξ

.

Note that in the above calculations we ignore the “edge eﬀects” such as the correction to the energy for an oﬀ-centered vortex. One can check that these eﬀects are negligible for R ξ. Combining eqs. (49) and (50), we get for the free energy F = E − T S (51)

1 βF = (ns λ2 − 4) ln 2

R ξ

.

We thus see that the free energy associated with a free vortex changes sign at ns λ2 = 4. Since ln(R/ξ) diverges with the size of the system, this point separates two qualitatively diﬀerent regimes. For ns λ2 > 4, F is very large and positive, so the superﬂuid is stable against spontaneous creation of a free vortex. On the other hand, for ns λ2 < 4, the large and negative F signals the instability against proliferation of free vortices. Appearance of ﬁrst free vortices reduces ns and makes the appearance of further free vortices even easier, and this avalanche eﬀect renormalizes the superﬂuid density to zero. We thus ﬁnd

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that in contrast to 3d, where below the BEC critical temperature the superﬂuid density grows smoothly, in 2d the superﬂuid density cannot have any value between 4/λ2 and 0. Even though it does not explicitly address the microscopic origin of the vortex (the breaking of a vortex pair), this simple calculation correctly predicts the result for the universal jump in the superﬂuid density which takes place at the transition temperature TBKT : (52)

ns λ2 = 4.

This success relies on the fact that the above derivation is a powerful self-consistency argument. Whatever the origin of the vortex, and the relation between ns and total density n, it shows that it is inconsistent to suppose that we have a system with superﬂuid density which is non-zero, but smaller than 4/λ2 . Remarkably, this result also does not depend on the strength of interactions g˜, even though we know that the phase transition is mediated by interactions, since it does not occur in the ideal gas. Recent classical ﬁeld Monte Carlo calculations performed with parameters relevant for atomic gases [40-42] have conﬁrmed the proliferation of vortices around the critical point characterized by eq. (52), although the transition was rounded oﬀ by ﬁnite-size eﬀects. If we repeat the above arguments for tightly bound vortex pairs, we ﬁnd that a ﬁnite density of pairs is present in the gas at any non-zero temperature.# The energy of a pair is ﬁnite since the velocity ﬁeld decays as v ∝ 1/r2 and the integral v 2 d2 r is convergent. On the other hand, the entropy is still divergent and essentially identical to the result of eq. (50), since the size of a tightly bound pair is of the same order as the size of a single vortex. The free energy for vortex pairs is therefore always negative. At any non-zero T pairs are continuously created and annihilated through thermal ﬂuctuations. As the temperature is increased, but still kept below TBKT , the density of pairs grows and also thermal ﬂuctuations result in pairs of increasing size. As the average size of the pairs becomes comparable to the distance between the pairs, they start to overlap and this leads to eﬀective screening of the attraction between two bound vortices, making it easier for ﬂuctuating pairs to grow to even larger sizes. We can use an analogy with a Coulomb gas: Two nominally paired but well-separated vortices create a ﬁeld which polarizes the more tightly bound vortex dipoles between them. This results in an eﬀective dielectric constant which reduces the attraction between two oppositely charged vortices. As TBKT is approached from below, this creates an avalanche eﬀect which eventually leads to breaking up of pairs and creation of a plasma of free vortices. Within the Coulomb gas analogy, this plasma provides a perfect screening at the transition point: A charge added at a given point does not change the ﬂux of the electric ﬁeld across a large radius circle centred on this point. Proper BKT calculation identiﬁes the phase transition with the temperature at which the average size of the pairs, or equivalently the screening dielectric constant, diverges. . 3 3. Results of the microscopic theory. – The relation (52) between the superﬂuid density and the temperature TBKT at the critical point is elegant and universal, in the

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291

sense that it does not depend on the interaction strength g˜. However this self-consistent result alone does not allow us to predict the value of TBKT in a given system. It just tells us that whatever TBKT is, the superﬂuid density jumps accordingly to 4/λ2 at the transition. Calculating the actual value of TBKT in terms of the bare system properties n and g˜ is a diﬃcult problem, because it depends on the short-distance physics, such as density ﬂuctuations which control the relationship between ns and n at the transition. In the weak coupling limit, g˜ 1, a combination of analytical [43] and numerical [29,44] eﬀorts gives the value for the critical phase-space density: (53)

Dc = (nλ2 )c = ln(C/˜ g ),

where the dimensionless constant C = 380 ± 3 is obtained by a classical ﬁeld Monte Carlo simulation [29]. The calculation leading to eq. (53) is formally valid only in the weak-coupling limit g˜ 1. We can set one obvious bound on its validity by noting that at the transition n ≥ ns . Setting Dc > 4 we obtain g˜ ≤ 7. This result is remarkably close to our estimate of the strong-coupling limit g˜ = 2π (eq. (22)). The numerical simulation of [29] provides another quantity of interest, which characterizes the reduction of density ﬂuctuations due to interactions. The authors of [29] introduce the quasi-condensate density deﬁned as (54)

+ !,1/2 . nqc ≡ 2n2 − n2 (r)

When interactions are negligible, n2 = 2 n2 and nqc = 0. On the other hand if density ﬂuctuations are completely suppressed, n2 = n2 and nqc = n. According to [29], at the critical point (55)

nqc 7.16 = . n ln(C/˜ g)

This result indicates that nqc is of the order of the total density n at the transition point, unless the interaction strength g˜ is exponentially small. In other words, for realistic parameters density ﬂuctuations are notably reduced in the vicinity of the BKT transition, which justiﬁes the simpliﬁed Hamiltonian (26) used above. Actually this result sets a stronger constraint than eq. (53) on the applicability of the classical Monte Carlo analysis: the condition n ≥ nqc requires g˜ ≤ 0.3. Note that the terminology quasi-condensate density can sometimes be misleading. The quantity nqc takes a non-zero value even above the critical temperature for the BKT transition. It refers only to the properties of the density distribution in the gas, and not to the phase distribution as the word condensate might suggest. For example, we can have nqc ∼ n at T > TBKT , but this does not imply that a large contrast interference would be observed if one would superpose a pair of 2d gases prepared in the non-superﬂuid regime (see sect. 7).

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Finally, we note that as the transition temperature is approached from above, the length scale characterizing the exponential decay of correlations g1 (r) ∼ e−r/ in the normal state diverges as (56)

= λ exp

√ a TBKT √ , T − TBKT

where a is a model-dependent dimensionless constant. Diverging correlation length is a very general property of phase transitions, but while in the case of most conventional (3d) second-order phase transitions the divergence of the correlation length is polynomial, in the case of the BKT transition it is exponential. This makes the critical region above TBKT larger, and has implications for the broadening of the transition in ﬁnite size . systems (see, e.g., subsect. 4 3). 4. – The 2d Bose gas in a ﬁnite box It is well known that ﬁnite-size eﬀects can play a signiﬁcant role in the quantitative analysis of the phase transitions that are observed experimentally. In the two-dimensional situation of interest here, this is even more the case because the thermodynamic limit is reached only when ln(R/ξ) 1 (see for example eq. (51)), which can be only marginally true for realistic systems. The analysis of these ﬁnite-size eﬀects is therefore crucial for understanding the observed phenomena, and we will do so for the cases of a ﬂat box potential (this section) and a harmonic trap (next section). We consider in this section a gas of N particles conﬁned in a ﬂat box of area L2 . The . box is supposed to be square (Lx = Ly = L) except in the last subsection (subsect. 4 5) where we discuss possible eﬀects due to an anistropic conﬁnement (Lx = Ly ). The conﬁnement introduces a natural energy scale E0 = 2 /(mL2 ) in the problem and makes it possible to reach a true Bose-Einstein condensate at non-zero temperature, in contrast to the inﬁnite case. In this section we ﬁrst review the results that can be derived for the ideal gas case, and then discuss what happens for an interacting system. We will assume that the size L is much larger than the thermal wavelength λ so that E0 kB T . . 4 1. The ideal Bose gas. – In the absence of interactions, the statistical description of a Bose gas in a square box of size L is straightforward. Let us choose for simplicity periodic boundary conditions so that the single-particle eigenstates are plane waves eik·r /L, of energy k = 2 k 2 /2m, where the momentum k = (jx , jy )(2π/L), with jx,y positive, zero or negative integers. The chemical potential μ is always negative so that the fugacity Z = eβμ lies in the interval 0 < Z < 1. Three regimes can be identiﬁed, the ﬁrst two being identical to what we have met for the inﬁnite case: – The non-degenerate, high-temperature regime with a phase space density D = nλ2 1. This corresponds to a negative chemical potential such that |μ| kB T (Z 1). The one-body correlation function is a Gaussian function in this regime, 2 2 g1 (r) = n e−2πr /λ , and is vanishingly small for distances r λ, which are much

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smaller than the box size L. The conﬁnement has no signiﬁcant consequence on the coherence of the gas in this regime. – The degenerate, but non-condensed regime, where the√ momentum distribution is bimodal, with a Lorentzian shape for small k (kλ 4π) and a Gaussian shape for large k. The one-body correlation function √ decays exponentially at large r, with a characteristic decay length = λeD/2 / 4π. No signiﬁcant condensed fraction appears as long as is small compared to the size L of the sample, i.e. when D < ln(4πL2 /λ2 ) or equivalently |μ| E0 /2. – The condensed regime, which occurs when the characteristic decay length of g1 is larger than the system size L. This occurs when the phase space density D reaches the value ln(4πL2 /λ2 ) (or equivalently |μ| ≤ E0 /2). A signiﬁcant phase coherence then exists between any two points in the gas. . 4 2. The interacting case. – We now turn to the interacting case with repulsive interactions and discuss what can be expected in the vicinity of the BKT transition. For now we assume that the size of the sample is large enough so that D ln(4πL2 /λ2 ) at the point where the phase space density D is equal to Dc , the critical phase space density for the BKT transition in an inﬁnite system (see eq. (53)). This condition has the following physical meaning: suppose that we increase the density of particles n at ﬁxed temperature; the point where the superﬂuid transition occurs is reached well before the point at which the Bose-Einstein condensation due to the ﬁnite system size would . occur in the absence of interactions (see subsect. 4 1 above). We ﬁrst recall the nature of the superﬂuid transition in an inﬁnite, homogenous sample. When D is notably below Dc , but larger than 1, one expects that no superﬂuid component is present and g1 (r) decays exponentially. When D reaches Dc the superﬂuid transition occurs and g1 (r) decays algebraically (eq. (48)): g1 (r) ≈ ns (ξ/r)α for r > ξ, with α = 1/(ns λ2 ) and ns λ2 ≥ 4. According to the Penrose-Onsager criterion, no condensate is expected in an inﬁnite system since g1 (r) vanishes at inﬁnity for any non-zero temperature. In sharp contrast with the inﬁnite case, we now show that the BKT transition in a realistic ﬁnite system is always accompanied by the appearance of a signiﬁcant condensed fraction, deﬁned as the largest eigenvalue Π0 of the one-body density matrix. The basic reason for this eﬀect is that the algebraic decay of g1 (r) is extremely slow. To prove this result we proceed in two steps: ﬁrst we give a general relation between Π0 and the value of g1 (r) for distances r comparable to the size L of the box; then we discuss what a realistic value of Π0 can be for a typical atomic gas. Let us denote by Πj and φj (r) the eigenvalues and eigenstates of the one-body density matrix. The condensed fraction Π0 is associated with the eigenstate φ0 (r) = 1/L. We now consider the general expansion of g1 (57)

g1 (r) = N

j

Πj φ∗j (0)φj (r)

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and integrate this expression over the area L2 centered on the origin. For simplicity we √ integrate the left-hand side over a disk of radius R = L/ π and the right-hand side over a square of side L. This simpliﬁcation cannot signiﬁcantly aﬀect our conclusions. The left-hand side gives

(58)

g1 (r) d2 r = 2π ns ξ

R

(ξ/r)α r dr L2

2 π α/2 g1 (L). 2−α

On the right-hand side, only the contribution of j = 0 is non zero and gives N Π0 . All the φj ’s with j = 0 are orthogonal to φ0 , so their integral over L2 is zero. We thus get g1 (L)/n ∼ Π0 . Using eq. (18) we can also write this result for the condensed fraction as Π0 ∼ (ns /n)˜ g −α/2 N −α/2 . For α ≤ 1/4, in practice we have g˜−α/2 ∼ 1 and ns ∼ n, so just below the transition temperature Π0 ∼ N −1/8 . Taking N = 105 as a typical value for cold atom experiments, we get Π0 ∼ 0.25.(6 ). We therefore meet here a paradoxical situation: the appearance of a non-zero condensed fraction may be used as a signature of the BKT transition, whereas the BKT mechanism was presented (for an inﬁnite system) as a feature that takes place in a 2d interacting gas instead of the usual BEC of 3d Bose ﬂuids. Note that in a “true” BEC the condensed fraction Π0 should not explicitly depend on N . However, for α ≤ 1/4 this distinction becomes experimentally irrelevant, and in order to observe a BKT transition with no signiﬁcant BEC one would need to consider unrealistically large samples. This was pointed out by the authors of [45] who wrote the famous statement (in the context of 2d magnetism): “With a magnetization at the BKT critical point smaller than 0.01 as a reasonable estimate for the thermodynamic limit, the sample would need to be bigger than the state of Texas for the Mermin-Wagner theorem to be relevant!”. A similar remark holds in the context of superﬂuid helium ﬁlms [2]. . 4 3. Width of the critical region and crossover. – The intricate mixing between the BKT mechanism and the emergence of a signiﬁcant degree of coherence exists even at . temperatures slightly above the BKT transition point. We mentioned in subsect. 3 3 that in the normal state the characteristic decay length of g1 (r) diverges exponentially in the vicinity of the critical point: ln(/λ) ≈ (aTBKT /(T − TBKT ))1/2 , where a is a model-dependent coeﬃcient (see eq. (56)). The critical region where becomes larger than λ as a precursor of the BKT transition is therefore very broad, (T − TBKT ) ∼ TBKT . Further, there clearly exists a temperature close to (but still above) TBKT for which exceeds the system size. At this temperature a signiﬁcant condensed fraction appears in the system. Because of the exponential variation of with T − TBKT , the temperature (6 ) We get an equivalent estimate from Π0 ∼ g1 (L)/n and ns ∼ n in the algebraic decay regime. In cold atom gases, the typical values of λ and ξ are 0.1–1 μm, while the maximal system size is L ∼ 100 μm. At the transition point we have Π0 ∼ (10−3 )1/4 to (10−2 )1/4 ∼ 0.2 to 0.3.

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295

range where L, and condensation gradually sets in, can be signiﬁcant(7 ): (59)

ΔD a ΔT = ∼ . TBKT Dc (ln(L/λ))2

Although cold atom systems are so far commonly conﬁned in harmonic rather than box-like potentials, it is interesting to provide an estimate for this case. Taking a = 1 and reasonable values for L/λ between 10 and 100, the BKT transition is expected to become a crossover with a relative width ΔT /TBKT ranging from 5% to 20%. The above analysis explicitly concerns only the emergence of a non-zero condensed fraction, but it also suggests broadening of the universal jump in the superﬂuid density. . For example, using standard Bogoliubov argument (see subsect. 2 4) we can deduce that ﬁnite condensed fraction in an interacting system also implies a ﬁnite superﬂuid density. Also, if we take the deﬁnition of superﬂuid density which associates it with the energy cost of twisting the phase of the wave function at the edge of the system [46,47], we again conclude that L implies a ﬁnite superﬂuid density. In the critical region, quantitative conclusions might actually depend on what theoretical deﬁnition of superﬂuid density we accept, but the qualitative conclusions will not change. . 4 4. What comes ﬁrst: BEC or BKT? – This is an often raised and subtle question. We have so far discussed the case of a large system such that Dc ln(4πL2 /λ2 ), where g ) is the critical phase space density for the BKT transition in an inﬁnite Dc = ln(380/˜ system. For such large systems, the ﬁrst relevant mechanism that occurs when increasing the phase space density is a BKT transition. However, we have seen that the approach of the BKT threshold always results in the appearance of a signiﬁcant condensed fraction. We can qualify this “BKT-driven” condensation as “interaction-enhanced”, since it would not take place in an ideal gas with the same density and temperature. Experimentally, the strength of interactions in cold atom systems can be dynamically controlled using a Feshbach resonance [48, 49]. One can therefore imagine preparing a non-interacting gas at phase space density where no condensation occurs and then driving the condensation via the BKT mechanism by turning on the interactions. In the opposite regime of a small system where the BKT transition would require Dc > ln(4πL2 /λ2 ), the ﬁrst phenomenon that is encountered when the phase space density is increased is “conventional” BoseEinstein condensation as for an ideal Bose gas. As in the 3d case, in the presence of weak repulsive interactions the formation of a condensate is accompanied by the apparition of a superﬂuid fraction with a comparable value. It would be very interesting to study cold atomic gases in a (quasi-)uniform potential and vary the experimental parameters so as to explore both regimes. However we point out that this will not be an easy task if we require that the criteria for the two types of transitions are well separated, for example by more than the crossover width discussed in (7 ) Here we can assume that λ under the logarithm on the right-hand side is constant over the range ΔT .

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. subsect. 4 3. To analyze the system requirements for reaching the two diﬀerent regimes, it is convenient to ﬁx the ratio L/λ and write ln(4πL2 /λ2 ) = γDc . Now γ is a dimensionless parameter such that γ > 1 means that (as particle number is increased) condensation occurs via the BKT mechanism. The critical number for the BKT transition is then (60)

Nc =

L2 1 Dc eγDc , Dc = λ2 4π

while the critical number for condensation in an ideal gas is γNc . For illustration purposes we may deﬁne the BKT regime by γ ≥ 1.5, and the BEC regime by γ ≤ 0.5. (For values of γ close to 1, the two eﬀects are diﬃcult to disentangle experimentally.) The experiments with cold atoms have so far been performed at coupling strength g˜ ∼ 10−2 –10−1 . Taking g˜ = 0.1, γ = 1 corresponds to Nc ≈ 2.5 × 103 , and the BKT regime γ = 1.5 corresponds to Nc ≈ 1.5 × 105 , which is easily achievable. However the opposite BEC regime of γ = 0.5 corresponds to Nc ≈ 40; studying such a small particle number is experimentally very challenging, although it might become feasible with the development of single-atom detection [50-52]. For a more weakly interacting gas with g˜ = 0.01, γ = 1 corresponds to Nc ≈ 3 × 104 , and γ = 0.5 to Nc ≈ 160, which might be easier to explore. On the other hand, γ = 1.5 corresponds to Nc ≈ 6 × 106 , which would be experimentally challenging. It would therefore generally be diﬃcult to explore both the large (BKT) and the small (BEC) system regime using the same value of g˜, and reaching the BEC regime may require a more weakly interacting quasi-2d atomic gas than has so far been studied. . 4 5. The case of anisotropic samples. – So far we have assumed that for phase space densities D larger than the critical value Dc for the BKT transition, the functional form of g1 found in the inﬁnite case (algebraic decay) remained valid for a ﬁnite-size system. This assumption is reasonable for square samples (Lx = Ly ), but may not be valid for anisotropic samples, with a width along one direction (say x) much larger than the other one: Lx Ly . We brieﬂy review the expected properties in this regime, which is relevant for several of the previous or current experimental setups. Since we are interested in the regime D > Dc , we assume that a superﬂuid component is present in the sample and we use the Hamiltonian Hθ given in (41) to estimate the amplitude of phase ﬂuctuations and their consequence on the one-body correlation function g1 . We start from the result (46) obtained in the inﬁnite case. In a ﬁnite-size system the integral over k is replaced by a discrete sum over k = 2π(jx /Lx , jy /Ly ) times the constant prefactor 4π 2 /(Lx Ly ): (61)

ln(g1 (r)/ns ) = −

1 − cos(k · r) 2π . ns λ2 Lx Ly kx2 + ky2 k

We are interested here in the decay of g1 along the long axis of the sample and we choose r = xux , where ux is the unit vector along the x direction. We take x Lx so that the ﬁniteness of the sample along x has no relevance here. On the contrary the size Ly

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Two-dimensional Bose fluids: An atomic physics perspective

( Lx ) will play an important role since we can choose x either small or large compared to Ly . We now show that these two cases lead to diﬀerent decaying regimes for g1 . Since we assume |x| Lx , the discrete sum over kx can always be replaced by an integral and we get (62)

ln(g1 (x)/ns ) = −

1 − cos(kx x) 1 − e−|xky | π 1 dk = − . x ns λ2 Ly kx2 + ky2 ns λ2 Ly |ky | ky

ky

We single out the contribution of ky = 0 in the sum, introduce a cutoﬀ kmax for large ky jmax ∞ and use j=1 1/j ≈ ln jmax and j=1 ζ j /j = − ln(1 − ζ) for 0 < ζ < 1. We then get (63)

ln(g1 (x)/ns ) = −

1 ns λ2

(

) π|x| kmax Ly

. + ln 1 − e−2π |x|/Ly Ly 2π

Two regimes clearly appear in this expression. If π|x| Ly then the logarithm on the right-hand side is the dominant term, and we recover the algebraic decay that holds for an inﬁnite system: (64)

−1 kmax |x| Ly : g1 (x) ≈

ns , (|x|kmax )α

with α =

1 . ns λ2

This result is intuitive: as long as we probe the coherence of the system on a distance shorter than the smallest size of the sample, the anisotropy introduces no signiﬁcant deviation with respect to an inﬁnite system. The situation is dramatically diﬀerent when |x| Ly . In this case the dominant contribution on the right-hand side of (63) is the linear term π|x|/Ly . This term, which originates from the contribution of the ky = 0 mode to the sum (61), leads to an exponential decay of g1 : (65)

Ly |x| : g1 (x) ≈ ns e−|x|/d ,

with d = ns λ2 Ly /π.

It is quite remarkable that when we probe the coherence of this anisotropic system on distances x ≥ d > Ly , we obtain an exponential decay as if the system was not superﬂuid. At the same time, the characteristic distance over which g1 decays, d, explicitly depends on the superﬂuid density. The physical interpretation of this counterintuitive result is that over such distances the system acquires a quasi–one-dimensional character: the phase stiﬀness between the origin and the point at coordinate x is decreased with respect to an inﬁnite plane because there is a severe reduction in the number of independent paths connecting these two points. Ultimately, for very large |x|, only the channel ky = 0 contributes signiﬁcantly to the connection between these two points. This explains why, although the system is superﬂuid, the decay of g1 turns to an exponentially decaying function, that is characteristic of 1d degenerate gases (see also [53] for a similar discussion for elongated atomic 3d gases).

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5. – The 2d Bose gas in a harmonic trap Up to now, experiments performed with (quasi-)2d atomic gases used harmonic trapping in the xy-plane. We discuss in this section how the presence of the harmonic trapping potential modiﬁes the above results. We shall see that it can lead to a dramatic change of the properties of the system, through the modiﬁcation of the density of states of the single-particle Hamiltonian. In particular, in this case “conventional” Bose-Einstein condensation, as deﬁned through the saturation of excited states at some non-zero temperature Tc , can occur in the ideal Bose gas even in the thermodynamic limit. However, in the presence of repulsive interactions, and in large enough systems, this type of condensation is suppressed and replaced by the BKT normal to superﬂuid transition. . 5 1. The ideal case. – Consider for simplicity an isotropic 2d harmonic potential V (r) = mω 2 r2 /2. The single-particle energy levels are Ej = (j + 1)ω, with j positive or zero integer, and each level having a degeneracy gj = j + 1. The maximum number of atoms Nc that can be placed in all excited states (j > 0) at a given temperature T is obtained by choosing a chemical potential μ equal to the ground-state energy: (66)

Nc(id) (T ) =

+∞ j=1

gj , eζj − 1

where ζ = ω/(kB T ). Assuming ζ 1, the discrete sum can be replaced by an integral and one obtains (67)

Nc(id) (T )

π2 ≈ 6

kB T ω

2 .

For an atom number N > Nc (T ) there must be at least N − Nc atoms occupying the single-particle ground state j = 0. Equivalently, for a given atom number N placed in the trap, there must be a signiﬁcant fraction of the atoms that occupy the ground state j = 0 if the temperature is reduced below the critical value (68)

√ 6 √ ω N . kB Tc = π

Since the ground state is separated from the ﬁrst-excited state by a non-zero gap ω, this Bose-Einstein condensation can be viewed as a natural consequence of the ﬁnite size of the system [54], similar to the condensation of the ideal gas in a ﬁnite box. However, one can see that the condensation of the ideal gas in a 2d harmonic trap is a more interesting phenomenon by considering the appropriately deﬁned thermodynamic limit for the harmonic conﬁnement. This limit is obtained by taking N → ∞ and ω → 0, while keeping T and N ω 2 constant. For a gas described by Boltzmann statistics, this ensures that the central density n0 = N mω 2 /(2πkB T ) remains constant. Equation (68)

Two-dimensional Bose fluids: An atomic physics perspective

299

leads to a non-zero critical temperature in the thermodynamic limit, contrarily to what happens in the uniform case. This result can be understood by noticing that the density of states has a diﬀerent functional form in a box (ρ(E) constant) and in a 2d harmonic potential (ρ(E) ∝ E) [55]. The vanishing density of states at E = 0 for a 2d harmonic potential leads to a similar situation to the 3d uniform case, hence the possibility for a genuine Bose-Einstein condensation in the ideal gas (for a discussion of small logarithmic anomalies in the compressibility of the trapped ideal 2d gas, see [56]). Quite remarkably the result (67) can be recovered by starting from the uniform result given in eq. (6): n = −λ−2 ln(1 − eβμ ) and using a local density approximation (LDA). This approximation amounts to replacing the uniform chemical potential μ by the local one μ − V (r), which gives the following expression for the total atom number: (69)

N = −λ−2 =−

ln(1 − eβ(μ−V (r)) ) 2πr dr

kB T ω

2

+∞

2 ln 1 − Ze−R /2 R dR,

0

where we set R = r/rT with rT2 = kB T /mω 2 . For μ = 0, the result coincides with eq. (67). Therefore in spite of the fact that LDA leads to a diverging spatial density at the center of the trap for μ = 0 [n(r) ∝ − ln(r)], it provides the same upper bound Nc as eq. (66) for the total number of atoms, assuming no macroscopic occupation of the single-particle ground state. . 5 2. LDA for an interacting gas. – In order to take into account repulsive interactions for a trapped gas, we use again the local density approximation. We will start with an analytical mean-ﬁeld treatment based on the Hartree-Fock approximation. We will then use the numerical results of a classical ﬁeld Monte Carlo approach [44] that will provide a more precise determination of the BKT transition. In the mean-ﬁeld Hartree-Fock approach when no condensate is present, interactions are taken into account by adding the energy 2gn(r) to the external potential [57,58]. The local chemical potential is now μ − V (r) − 2gn(r) so that the local phase space density D(r) = n(r)λ2 is the solution of the implicit equation (70)

D(r) = − ln {1 − Z exp[−βV (r) − g˜D(r)/π]} .

Putting R = r/rT as above, we can write the total atom number as N

(71)

(id)

Nc

6 = 2 π

+∞

D(R) R dR, 0

where D is solution of (72)

4 3 D(R) = − ln 1 − Z exp −R2 /2 − g˜D(R)/π .

300 Centra phase space dens ty

Z. Hadzibabic and J. Dalibard 0.01

14

14

0.03

0.03

0.05 0.1

12

0.1

12

0.2 0.3

10

10

0.3

8

8

(a)

6

(b)

6

4

4

2

2 0

0 0

1

2

3

N/

4 (id) Nc

5

6

0

1

2

3

(id)

N / Nc

Fig. 2. – Variation of the central phase space density as a function of the atom number, nor(id) for the ideal gas in the same potential and at the malized by the critical atom number Nc same T . The value of the interaction strength g˜ is given for each curve. The black squares indicate the values of N at which the BKT criterion of eq. (53) is met at the center of the trap. (a) Results obtained using the mean-ﬁeld Hartree-Fock approach. (b) Results obtained using the bulk results of [44] and the local density approximation.

Note that the solution D(R) depends only on the fugacity Z and the interaction strength g˜. The trap frequency and the temperature do not appear explicitly so that the scaling of the atom number with ω and T (at ﬁxed Z) is identical to the result (69) for the ideal gas. Interactions, when treated at the mean-ﬁeld level, dramatically change the nature of the solution of eqs. (71)-(72). For a given trapping frequency ω and temperature T , and for any non-zero g˜, the atom number N obtained from eq. (71) can be made arbitrarily large by choosing properly the fugacity Z. The condensation phenomenon that was obtained in the ideal gas case does not occur anymore. This can be understood qualitatively. For an ideal gas, the saturation of the atom number occurs when the central density in the trap becomes inﬁnite. In the presence of repulsive interactions, this singular point cannot be reached and the mean-ﬁeld treatment provides a solution for any atom number [59]. We have plotted in ﬁg. 2a the prediction of the mean-ﬁeld approach for the central phase space density D(0) as a function of the total number of atoms in the trap, for various values of g˜. As expected, D(0) is a monotonically increasing function of the atom number N : more atoms in the trap lead to a larger central density. The other expected feature is that, for a given atom number, D(0) decreases as the repulsion between atoms is increased. In particular the divergence of D(0) that is found in the ideal case for (id) N = Nc does not show up anymore in the presence of repulsive interactions. One could try to push further the mean-ﬁeld analysis of the equilibrium state, and look for dynamical or thermodynamical instabilities that could appear above some critical atom number [60-63]. However we will rather follow the spirit of LDA and assume that the normal to superﬂuid BKT transition occurs at the center of the trap when the phase

Two-dimensional Bose fluids: An atomic physics perspective

301

space density at this point exceeds the critical value Dc (eq. (53)) predicted for the uniform system [64]. Within the mean-ﬁeld Hartree-Fock analysis, one can show that, to a very good approximation, the number of atoms that have to be placed in the trap so that the central phase space density reaches the critical value Dc is [65] (mf)

(73)

Nc

(id) Nc

=1+

3˜ g 2 D . π3 c

An obvious consequence of this result is that for a given trap and a given temperature, the BKT threshold in the presence of interactions requires a larger atom number than the BEC of the ideal gas. Equivalently, for a given atom number, the superﬂuid transition temperature in the presence of interaction is lower than the ideal gas condensation tem(mf) for various perature. In ﬁg. 2a we have indicated with black squares the value of Nc interaction strengths. So far we have relied on the mean-ﬁeld approximation to obtain the relationship between the density n(r) and the local chemical potential μ − V (r) − 2gn(r). Although this gives a good feeling for the scaling laws that appear in the problem, it cannot provide a very accurate description of the transition. Indeed the mean-ﬁeld expression 2gn(r) for the interaction energy can only be valid at relatively low density, where the density ﬂuctuations are important, so that n2 = 2 n2 . When the density increases and/or the temperature decreases, density ﬂuctuations are reduced and one eventually reaches a situation at very low temperature where those ﬂuctuations are frozen out and

n2 = n2 . At zero temperature, one expects a quasi-pure condensate in the trap with a density proﬁle given by the Thomas-Fermi law gn(r) = μ − V (r) (whereas the Hartree-Fock approximation would lead to replacing g by 2g in this equation). To capture the reduction of density ﬂuctuations as the phase space density increases, we now use the numerical results of [44] obtained using a classical ﬁeld Monte Carlo analysis. They provide the value of the phase space density D as a function of μ/kT in the vicinity of the BKT critical point for a uniform system. Injecting this numerical prediction in the LDA scheme, we obtain the results shown in ﬁg. 2b for the central density as a function of the total atom number. As expected this ﬁgure is qualitatively similar to the one obtained using the mean-ﬁeld Hartree-Fock approach. However the classical Monte Carlo results lead to a noticeable reduction of the critical atom number with respect to the mean-ﬁeld treatment. For example, for g˜ = 0.15 (as in the ENS experiment [27], see below) the critical atom number for reaching the BKT threshold (id) (mf) ∼ is expected to be ∼ 1.4 Nc using the numerical predictions of [44] instead of Nc (id) 1.9 Nc using the Hartree-Fock approximation. One might wonder if the classical Monte Carlo simulations of [44], which assume g˜ 1, remain accurate for the relatively large interaction strength g˜ = 0.15. The (positive) answer was given in [96], which provides a detailed comparison between the predictions of [44] and those of a quantum Monte Carlo simulation of an assembly of trapped bosons for the interaction strength and trapping geometry of the ENS experiment.

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Z. Hadzibabic and J. Dalibard

. 5 3. What comes ﬁrst: BEC or BKT? – In the previous section devoted to the study of a square potential we have explained that the distinction between a conventional BEC transition and a BKT transition is subtle, and we introduced two related concepts: – “BKT-driven condensation”, meaning that if the phase space density is increased at constant g˜ the ﬁrst many-body mechanism encountered is the BKT transition, but due to the resulting slow decay of g1 this transition is accompanied by the appearance of a ﬁnite condensed fraction. – “Interaction-enhanced condensation”, meaning that there exists a range of phase space densities for which no condensation occurs in an ideal gas but condensation via the BKT mechanism can be induced by increasing the interaction strength from 0 to g˜. In the case of a box potential these two concepts are equivalent. We identiﬁed a range of parameters, such that Dc ln(4πL2 /λ2 ), for which both eﬀects occur. In the opposite regime of a small system and/or small g˜ neither of the two eﬀects occurs. The question “what comes ﬁrst” is even more subtle in the case of a harmonically trapped gas because of the inhomogeneous density proﬁle. In this case the notions of BKT-driven and interaction-enhanced condensation are not equivalent, and the answer depends on what we keep constant in an experiment, i.e. which path we follow in the phase diagram. Since in a harmonic trap ideal gas condensation occurs even in the thermodynamic limit (i.e. if we neglect the discreteness of single-particle energy levels in the trap), we start by analyzing that case, and separately consider two diﬀerent experimental paths: – The critical phase space density Dc for a BKT transition (at a ﬁxed non-zero g˜) is ﬁnite, while the critical phase space density for ideal gas condensation is inﬁnite. Therefore in a standard experiment where g˜ is kept constant and phase space density is increased, BKT-driven condensation always occurs. In this sense, in practice “BKT always comes ﬁrst”. – While the critical phase space density for the BKT transition (at non-zero g˜) is lower than for the ideal gas condensation, the critical atom number at ﬁxed T (id) (BKT) and Nc scale similarly with the is higher. The critical atom numbers Nc (BKT) (id) 2 /Nc is temperature and the trap frequency (∝ (kB T /ω) ), and the ratio Nc always larger than 1. This can be seen from the mean-ﬁeld result of eq. (73) or from the Monte Carlo data shown in ﬁg. 2b. This means that, at ﬁxed N , interactions always reduce the transition temperature. Therefore, contrary to the case of a square box potential, we can never have interaction-enhanced condensation. Note that it is not inconsistent that the BKT transition occurs at a lower critical density but higher critical number than the ideal gas BEC, because in a harmonic trap with ﬁxed N and T the peak (phase space) density in a repulsively interacting gas is lower than in an ideal gas. Also note that if we work within the BKT theory and then formally

Two-dimensional Bose fluids: An atomic physics perspective

303

take the g˜ → 0 limit, we exactly recover the criterion for ideal-gas condensation, which is usually derived from a conceptually completely diﬀerent viewpoint of the saturation of single-particle excited states. We can therefore think of the BEC transition as a special non-interacting limit of the more general BKT theory. This connection naturally emerges when analyzing the case of a harmonically trapped gas, but it could not be made in a uniform system, where the critical temperature for both transitions vanishes in the g˜ → 0 limit. Finally, we brieﬂy comment on the case of a realistic experimental harmonic trap, where the spacing of the single-particle energy levels is non-zero. The results for the critical atom numbers for the BKT and the ideal gas BEC transition are essentially unaﬀected by the non-zero level-spacing. It therefore remains true that interactionenhanced condensation is not possible. However, the ideal gas BEC in this case occurs at a ﬁnite phase space density DBEC in the trap center, which can in principle be lower than Dc for some values of g˜. In this case BKT-driven condensation would also no longer occur, and “BEC would come ﬁrst” no matter what path we take in the phase diagram. This scenario is however not relevant for the currently realistic experiments. The value of DBEC is not universal and depends on the details of the trapping potential, but we have evaluated it numerically for a typical trap used in the ENS experiments [26,27], and obtained DBEC ≈ 13 [66]. This means that the condition DBEC < Dc can be fulﬁlled only in an extremely weakly interacting gas with g˜ < 10−3 . Experimentally, this regime is essentially indistinguishable from the g˜ → 0 limit, where the BKT and the BEC transition are no longer distinct. . 5 4. Width of the crossover . – The divergence of the correlation length that we already . discussed in the case of a square box potential (see subsect. 4 3), must also be taken into account. Suppose that one lowers the temperature of a cold gas in a trap until the BKT threshold density is reached right at the center of the trap. Since the density is everywhere lower than the critical threshold for BKT, LDA would imply that no signiﬁcant superﬂuid fraction is present in the gas at this stage. However a signiﬁcant part of the gas may exhibit a certain degree of coherence as we show now. The critical length (56) is now position dependent, since the critical temperature is a function of the local density. Since (r) is a monotonically decaying function of the distance from the trap center r, we can self-consistently assume that the gas is coherent over a region of radius rc such that (rc ) = rc . We can also provide a crude estimate of rc : rc ≈ rT (ln(rT /λ))−1/2 . For practical parameters (rT /λ ∼ 10–100), we ﬁnd rc ∼ rT , which means that this coherence actually extends over a signiﬁcant fraction of the cloud when D = Dc at the center of the trap. We can also estimate the width of the cross-over over which the condensed fraction becomes signiﬁcant. If instead of taking D = Dc at the center of the trap, we take D = 0.7 Dc then ≈ 5λ at the center of the cloud and no signiﬁcant coherence exists at this point. The above analysis is conﬁrmed at least qualitatively by numerical simulations performed using a classical ﬁeld Monte Carlo analysis. These simulations indeed indicate the emergence of an extended coherence over the cloud at temperatures 10% to 20% above the one for which the bulk BKT criterion is met at the trap center [42].

304

Z. Hadzibabic and J. Dalibard

6. – Achieving a quasi-2d gas with cold atoms The experimental realization of a 2d atomic Bose gas is based on a strongly anisotropic trap with one very tightly conﬁning direction, say z, and two more loosely conﬁned degrees of freedom, x and y. The z degree of freedom can be considered as frozen from the thermodynamic point of view if the energy gap Δz between the ground state and the ﬁrst-excited state of the z motion is much larger than both kB T and the interaction energy gn (both being typically on the order of one to a few kHz). Since the conﬁnement along z is usually harmonic, with frequency ωz , the gap is Δz = ωz . The z degree of freedom is thermodynamically frozen when the√extension of the ground state of the z-motion, az = /mωz , is such that az ξ, λ/ 2π. . 6 1. Experimental implementations. – Conceptually, the simplest scheme to produce a 2d gas is to use a single Gaussian light beam that is red-detuned with respect to the atomic resonance. The beam propagates along the x-direction, with waists along y and z such that wy wz , so that it forms a horizontal light sheet. The dipole potential created by this light sheet attracts the atoms towards the focal point, and ensures a strong conﬁnement in the z-direction. This technique was used at MIT to produce the ﬁrst atomic gas (of sodium atoms) in a quasi-2d regime [67]. More recently it has been implemented at NIST to study the coherence properties of the 2d gas [28]. One can also produce a 2d gas by using an evanescent light wave at the surface of a glass prism [68, 69], so that the atoms are trapped at a distance of a few micrometers from the horizontal glass surface. The conﬁnement in the horizontal xy-plane is provided by an additional laser beam or by a magnetic ﬁeld gradient. The fact that the conﬁnements in the xy-plane and along the z-axis have diﬀerent origins is an interesting feature because it oﬀers the possibility, by releasing only the planar conﬁnement, to study the . ballistic expansion of the atoms in the xy-plane only (see subsect. 7 2). Another experimental system providing independent conﬁnement in the xy-plane and along z has been investigated at Oxford, where a blue detuned, single node, Hermite Gaussian laser beam traps atoms along the z-direction, and the conﬁnement in the xy-plane is provided by a magnetic-ﬁeld gradient [70]. Two-dimensional conﬁning potentials that are not based on light beams have also been investigated. One possibility discussed in [71] consists in trapping paramagnetic atoms just above the surface of a magnetized material that produces an exponentially decaying ﬁeld. The advantage of this technique lies in the very large achievable frequency ωz , typically in the MHz range. One drawback is that the optical access in the vicinity of the magnetic material is not as good as with optically generated trapping potentials. Another appealing technique to produce a single 2d sheet of atoms uses the so-called radio-frequency dressed state potentials [72-74]. A 1d optical lattice setup, formed by the superposition of two running laser waves, is a very convenient way to prepare stacks of 2d gases [75-79]. The 1d lattice provides a periodic potential along z with an oscillation frequency ωz that can easily exceed the typical scale for interaction energy and temperature. The simplest lattice geometry is

305

Two-dimensional Bose fluids: An atomic physics perspective

formed by two counter-propagating laser waves, and provides the largest ωz for a given laser intensity. One drawback is that the lattice period is small (λ0 /2, where λ0 is the laser wavelength) so that many planes are generally populated and the addressing of a single plane is diﬃcult. Therefore practical measurements only provide averaged quantities. Another interesting geometry consists in forming a lattice with two beams crossing at an angle θ smaller than 180◦ [80]. In this case the distance λ0 /(2 sin(θ/2)) between adjacent planes is adjustable, and each plane can be individually addressable if this distance is large enough [50,81]. Furthermore the tunneling matrix element between planes can be made completely negligible, which is important if one wants to achieve a truly 2d geometry and not a periodically modulated 3d system. Finally, while here we are primarily interested in continuous 2d gases of spinless Bosons, two related experiments on 2d physics also need to be mentioned: First, an experiment performed in Boulder constitutes a direct implementation of the XY model [82]. There, an array of parallel elongated (quasi-)condensates is created in a 2d optical lattice, and tunneling matrix element J provides a Josephson-type coupling between the neighboring lattice sites. In this system proliferation of vortices is observed when the temperature is increased. Vortices are detected by turning oﬀ the optical lattice and allowing the quasi-condensates trapped on diﬀerent sites to overlap and interfere. The measured surface density of vortices as a function of the ratio J/T is in good agreement with the BKT theory applied to this system. Second, in an experiment at Berkeley 2d physics was studied in a spinor Bose-Einstein condensate of Rb atoms with total spin F = 1 and weak ferromagnetic spin-dependent interactions [83]. This system is anisotropic, but still 3d with respect to the density degrees of freedom, i.e. the healing length ξ is shorter than the shortest extension of the cloud, along z. However, weak spin-dependent interactions correspond to a longer healing length ξs , so that the system is 2d with respect to the spin degrees of freedom. In this case the magnetization transverse to the quantization axis has a role analogous to the phase of the wave function in a spinless Bose gas. At low T ferromagnetic interactions favor spontaneous symmetry breaking but spin-vortex structures are also observed. . 6 2. Interactions in a 2d atomic gas. – To address the role of interactions in these gases, we start with some considerations concerning the quantum scattering of two atoms when the z motion is strongly conﬁned. In a strictly 2d problem and at low energy, the scattering state between two identical bosonic particles with relative wave vector k is [25] ' (74)

ψk (r) ∼ e

ik·r

−

i eikr f (k) √ , 8π kr

f (k) ≈

4π , − ln(k 2 a22 ) + iπ

where a2 is the 2d scattering length. One should notice that contrarily to the 3d case, the scattering amplitude f (k) does not tend to a non-zero ﬁnite value when k tends to 0. In the experimental implementations of 2d gases that have been achieved so far, the conﬁnement along z was still relatively weak from a collisional point of view, in the sense that the thickness az of the gas remained notably larger than the 3d scattering

306

Z. Hadzibabic and J. Dalibard

length as . The scattering problem in this conﬁned geometry has been discussed in [84,85] (see also [86]); the general expression (74) for the scattering state ψk remains valid and the scattering amplitude can be written (75)

4π f (k) ≈ √ 2πaz /as − ln(κ k 2 a2z ) + iπ

with κ ≈ 3.5, corresponding to the 2d scattering length (76)

a2 = az

√

' π az . κ exp − 2 as

√ Now for all experiments realized so far, the ﬁrst term 2πaz /as in the denominator of eq. (75) is large compared to 1, and dominates over the logarithmic term ln(κ k 2 a2z ) and the imaginary term iπ. We can then take a constant scattering amplitude (as in 3d) to √ describe the collisions in the gas: f (k) ≡ g˜ ≈ 8πas /az . With this approximation the interaction energy of the gas with density n(r) in the xy plane is

(77)

Eint =

2 g˜ 2m

n2 (r) d2 r.

The corresponding values for the 2d scattering length are extremely small, due to the exponential factor in eq. (76). Taking for example az = 200 nm and as = 5 nm (87 Rb atoms), we ﬁnd a2 = 6 10−29 m. Typical surface densities are in the range 1013 m−2 , and the dimensionless parameter na22 that is relevant for perturbative expansions of the equation of state of the 2d Bose gas [87-93] is also extremely small: na22 ∼ 4 10−44 for the numbers given above. The expression (77) can also be obtained by starting from the 3d interaction energy

(78)

Eint,3d =

2π2 as m

n23 (r) d3 r,

in which we plug directly n3 (x, y, z) = n(x, y) exp(−z 2 /a2z )/ πa2z . However the validity condition as az remains hidden in this procedure. To summarize, the collision dynamics in the experiments performed so far is still dominated by 3d physics. The 3d scattering length as is much smaller than the thickness of the gas and the scattering amplitude is nearly k-independent. This regime is often referred to as “quasi-2d”. It is important to note that the term “quasi-2d” is also used to describe another aspect of the 2d gases: very often the temperature of the gas (and possibly the interaction energy) is not small compared to ωz , but comparable or even a bit larger. We discuss in the next subsection how to handle this problem.

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. 6 3. Residual excitation of the z-degree of freedom. – For a quantitative analysis of experiments performed with 2d gases, in particular for the determination of the temperature, it is important to take into account the residual excitation of the z-degree of freedom. This was ﬁrst pointed out in [94], where a quantum Monte Carlo simulation gave an estimate for the distortion of the density proﬁle due to this residual excitation. Several possible ways were subsequently proposed to take this excitation into account [94, 65, 66, 95, 96]. The simplest method consists in renormalizing the interaction strength g˜ to account for the density proﬁle of the gas along the z direction [94]. The predictions derived with this method were compared with quantum Monte Carlo results in [65] and later analyzed in detail in [95]. In the following we outline the slightly more elaborate treatment of [66] which has the advantage of taking into account not only the thermal excitation of the z degree of freedom, but also the possible deformation of the ground state of the z-motion due to atomic interactions. The method used in [66] is a direct implementation of the Hartree-Fock approximation (see, e.g., [97]) and we ﬁrst present it for a gas which is uniform in the xy-plane. We choose a 3d trial density proﬁle n3 (z) uniform in the xy-plane and varying along the strongly conﬁned z direction. We then consider the Hamiltonian with the mean-ﬁeld energy H=−

(79)

2 2 1 ∇ + mωz2 z 2 + 2g (3d) n3 (z). 2m 2

The single-particle eigenfunctions of this Hamiltonian can be written ψk,j (x, y, z) = ϕj (z) ei(kx x+ky y) /2π, with energy Ek,j = 2 k 2 /(2m) + j , where k 2 = kx2 + ky2 . The normalized functions ϕj (z) and the energies j of the z-motion of course depend on the choice of the trial density proﬁle n3 (z). In the Hartree-Fock approximation the average occupation of the single particle level ψk,j is given by the Bose factor f (Ek,j ) = (exp(β(Ek,j − μ)) − 1)−1 . We calculate the corresponding 3d density proﬁle, which is still uniform in xy and has the following z-dependence: (80)

n3 (z) =

j

d2 k |ψk,j |2 f (Ek,j ) = −

+ , 1 |ϕj (z)|2 ln 1 − Ze−β j . 2 λ j

The self-consistency of the Hartree-Fock approximation requires that n3 (z) and n3 (z) coincide, which can be achieved by iterating the solution of the above set of equations until a ﬁxed point is reached. With this method, we fulﬁll two goals: i) We take into account the residual thermal excitation of the levels in the z-direction. ii) Even at zero temperature we take into account the deformation of the z ground state due to interactions. When interactions can be neglected, the eigenstates ϕj (z) are the Hermite functions and j = ωz (j + 1/2). Using the local density approximation, the above method can be straightforwardly adapted to the case where a trapping potential V⊥ is present in the xy-plane. The trial density distribution n3 (r) is now a function of all three spatial coordinates. At any point

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(x, y), we treat quantum mechanically the z motion and solve the eigenvalue problem for the z variable 2 2 − d + Veﬀ (r) ϕj (z|x, y) = j (x, y) ϕj (z|x, y), (81) 2m dz 2 # where Veﬀ (r) = V⊥ (x, y) + mωz2 z 2 /2 + 2g (3d) n3 (r) and |ϕj (z|x, y)|2 dz = 1. Treating semiclassically the xy degrees of freedom, we obtain a new spatial density (82)

n3 (r) = −

1 2 −β j (x,y) . |ϕ (z|x, y)| ln 1 − Ze j λ2 j

Again the Hartree-Fock prediction is obtained by iterating this calculation until the spatial density n3 (r) reaches a ﬁxed point. Since ϕj is a normalized function of z at any point (x, y), the total 2d density is (83)

n(x, y) =

n3 (r) dz = −

1

−β j (x,y) . ln 1 − Ze λ2 j

In the limit where only the ground state j = 0 of the z motion is populated, the result of this Hartree-Fock approach coincides with the solution of (70). The method used in [95] is similar to this approach, but the deformation of the eigenstates due to mean-ﬁeld interaction was neglected. The density proﬁles predicted by this method have been compared with the results of a quantum Monte Carlo simulation [96]. An important outcome for the analysis of experimental data is the excellent agreement between the two approaches as long as nλ2 < 2. This agreement holds for the temperature regime (kB T ≤ 2ωz ) and interaction strength (˜ g 0.15) relevant for current experiments. The Hartree-Fock approach is therefore well suited for ﬁtting the wings of the experimental density proﬁles of a quasi2d gas to extract the temperature and chemical potential. 7. – Probing 2d atomic gases This section is devoted to the presentation of some methods that have been used for the experimental study of 2d Bose gases. We start with the conceptually simplest approach, which consists in the measurement of the steady-state distribution of atoms in a trap. We then turn to the information that can be acquired in a Time-of-Flight expansion. Finally we discuss two schemes that give access to the phase coherence of the gas. . 7 1. In situ density distribution. – Conceptually the simplest information that can be obtained on a 2d gas is a picture of the sample along the direction that is strongly conﬁned. Since this degree of freedom is supposed to be frozen out, there is no loss of information due to integration along the line-of-sight. This is in contrast to what happens

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in 3d, where one has to resort to a non trivial transformation to reconstruct the spatial distribution [98] (see also [99] for a review). Assuming that local density approximation (LDA) is valid, the density distribution in the trap n(r) can be obtained from the equation of state of the homogeneous system. The general form of this equation of state is (84)

nλ2 = F (μ, kB T, a2 ),

where F is at this stage an unknown function and a2 is the 2d scattering length. Within LDA the density n(r) in the trap is calculated by replacing μ by μ − V (r), where V (r) is the trapping potential. In the quasi-2d regime that is of practical interest (as az ), we have seen in sub. sect. 6 2 that the interactions in√the gas are characterized to a good approximation by the dimensionless number g˜ = 8π as /az 1. In this case eq. (84) can be simpliﬁed using dimensional analysis; the expression of the phase space density D = nλ2 must take the functional form (85)

D = G(α, g˜)

with α =

μ . kB T

For a gas that is trapped in a harmonic potential mω 2 r2 /2, the in situ density proﬁle is then given by

(86)

r2 n(r)λ = G α − 2 , g˜ , 2rT 2

where we set as above mω 2 rT2 = kB T . This expression clearly shows a scale invariance for a given interaction strength g˜. Suppose that diﬀerent density proﬁles n(r) are recorded for various temperatures T and various atom numbers N (hence diﬀerent chemical potentials μ). According to eq. (86) the proﬁles can all be superimposed on the same curve G(α, g˜), provided they are plotted as a function of r2 /rT2 and translated along the x-axis by the dimensionless quantity α = μ/kB T . This scale invariance behaviour has been checked with excellent accuracy by M. Holzmann and W. Krauth using quantum Monte Carlo simulations [100]. These simulations were performed for g˜ = 0.15, which corresponds to the interacting strength in ENS experiments with Rb atoms. For g˜ 1, various asymptotic forms of the function G(α, g˜) have been given earlier. When interactions can be neglected (˜ g = 0), the equation of state is (cf. eq. (6)): D = − ln(1 − eα ). In the presence of interactions and for small phase space densities, the mean-ﬁeld Hartree-Fock method amounts to replace μ by μ − 2gn into the ideal-gas result, which leads to the implicit equation D = − ln(1 − eα−˜gD/π ), from which one can extract D as a function of α and g˜ (see also eq. (70)). In the strongly degenerate limit, where μ kB T and D 1, density ﬂuctuations are strongly reduced and one expects μ = gn, which can be written as D = (2π/˜ g )α. In the intermediate regime, in particular close to the BKT transition point, one can use the results of the classical

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(a)

Phase space dens ty

20

15

g˜ = 0 .15

g˜ = 0 .15

15

10

10

5

5

0

-1.0

(b)

20

0 0

-0.5

μ/k B T

+0.5

0.0

0.5

1.0

r/rT

1.5

Fig. 3. – (a): Phase space density as a function of α = μ/kB T for g˜ = 0.15. Continuous line: Total phase space density D = nλ2 ; dashed line: superﬂuid phase space density ns λ2 . The dotted and dash-dotted lines represent the asymptotic regimes for low and high phase space densities, respectively. (b) In situ density proﬁles in a trap deduced from the left panel using the local density p approximation. The plot is made for μ/kB T = 0.5 so that the Thomas-Fermi radius rTF = 2μ/mω 2 is equal to rT .

ﬁeld Monte Carlo analysis of [44]. The resulting function D = F (α, g˜) is represented in ﬁg. 3 for g˜ = 0.15. The two asymptotic regimes that we just described are indicated with dotted and dash-dotted lines. A remarkable characteristic of the function F (α, g˜) is precisely the absence of signiﬁcant features at the critical point for the BKT transition (corresponding to α ≈ 0.2 and D ≈ 8 for g˜ = 0.15). This is due to the inﬁnite order of the BKT transition, that does not cause any singularity in the dependance of the total density n on T or μ. On the other hand, the superﬂuid density ns (plotted as a dashed line in ﬁg. 3) is discontinuous at the transition point, but this quantity is not directly accessible from an in situ measurement. For a detailed comparison between the results of the mean-ﬁeld Hartree-Fock approach and those obtained from a quantum Monte Carlo simulation and from a renormalization group treatment, see [96] and [63], respectively. Finally we note that the analysis of individual images requires a proper knowledge of the temperature and the chemical potential. These are usually obtained by ﬁtting the wings of the distribution with the appropriate function for the quasi non-degenerate gas (see the discussion following eq. (83)). An interesting alternative consists in using in situ density ﬂuctuations to determine these thermodynamic quantities [101]. This promising method that relies on the ﬂuctuation-dissipation theorem for a non-uniform system has not yet been implemented experimentally for a 2d Bose gas. . 7 2. Two-dimensional Time-of-Flight expansion. – Generally speaking, a Time-ofFlight (TOF) procedure consists in switching oﬀ abruptly the potential conﬁning the atoms, letting the cloud expand for an adjustable time, and then measuring the density proﬁle. If the role of interactions is negligible during the expansion, the density proﬁle after a long TOF is proportional to the in-trap momentum distribution. For a two-dimensional system, two types of TOF can be considered. One can switch oﬀ the

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potential conﬁning the atoms in the xy-plane, while keeping the strong conﬁnement along the frozen direction z; we will call this procedure a “2d TOF”. Alternatively, one can switch oﬀ simultaneously the potential in the xy-plane and the conﬁnement along z, corresponding to a “3d TOF”. We discuss 2d TOF in this section, and 3d TOF in the following one. We consider here the case of an isotropic harmonic trap in the xy plane V (r) = mω 2 r2 /2. A 2d gas is initially at thermal equilibrium in this trap, with a density proﬁle neq (r). Suppose that this potential is suddenly switched oﬀ at time t = 0 whereas the conﬁnement along the z direction remains unchanged. Using the Bogoliubov approach, it was predicted in [102] that the subsequent evolution of the density distribution is given by the scaling law (87)

n(r, t) = ηt2 neq (ηt r),

ηt = (1 + ω 2 t2 )−1/2 .

This means that the global form of the spatial distribution is preserved during the TOF. As the expansion proceeds, the interaction energy that was initially present in the gas is converted into kinetic energy in such a way that the density proﬁle at time t is obtained using a scaling transform of the initial one. We emphasize that this remarkable result is stronger than its 3d counterpart [103,102] which holds only in the Thomas-Fermi regime: In the 2d case the scaling behavior is valid both for the superﬂuid component and for the thermal Bogoliubov excitations. This scaling behavior has been recently observed by the ENS group [104]. The scaling invariance in the expansion of a 2d interacting gas has been explained in terms of the SO(2, 1) symmetry group for a whole class of interaction potentials U (r) between two particles [105]: it is suﬃcient that U is a homogeneous function of degree 2, U (αr) = U (r)/α2 . When this is the case, the result of eq. (87) holds for an arbitrary initial state of the 2d gas, irrespective of its temperature. The 2d contact interaction potential, which is implicitly assumed in eq. (16), belongs to this class of functions. We note however that a true contact interaction is singular in 2d because it leads to ultraviolet divergences at the level of quantum ﬁeld equations. A real interatomic potential has a ﬁnite range which provides a UV cut-oﬀ that eliminates the divergences. This regularization will occur if one uses the more precise treatment of atomic interactions given in eq. (75). It will lead to deviations with respect to the universal law (87), which remain to be evaluated and characterized. . 7 3. Three-dimensional Time of Flight. – In a 3d TOF both the trapping potential in the xy-plane and the strong conﬁnement along the z-direction are switched oﬀ simultaneously. The physics is then very diﬀerent from that of a 2d TOF. Along the initially strongly conﬁned direction z, the atom cloud expands very fast since the momentum width Δpz ∼ /Δz is large. If the atoms are initially in the ground state of a harmonic potential with frequency ωz along this axis, the extension of the cloud is multiplied by √ 2 in a time t = ωz−1 . The time scale for the expansion of the gas in the xy-plane is much longer; it is given by ω −1 , where ω ωz is the trapping frequency in this

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plane. Therefore it is a good approximation to decompose a 3d TOF into two phases. During the ﬁrst phase, whose duration is a few ωz−1 (typically 1 ms if ωz /2π = 3 kHz), the thickness of the gas along z increases by a factor much larger than 1, but the xy spatial distribution is nearly not modiﬁed. At the end of this ﬁrst phase, the interactions between atoms have become negligible. During the subsequent phase the expansion in the xy-plane becomes signiﬁcant, but on a much longer time scale. It corresponds to the expansion of an ideal gas, whose initial state is equal to the state of the system in the xy-plane before the beginning of the TOF. We now focus on the evolution of the xy degrees of freedom during the second phase, which is essentially governed by single-particle physics. The evolution of the density distribution in the xy-plane can be determined from the initial one-body density matrix g1 (r, r ) = r|ρ(1) |r , or from its Fourier transform Π(p) with respect to the variable r − r , which represents the momentum distribution in the xy-plane. In the absence of any extended coherence in the gas, g1 (r, r ) tends to zero when |r − r | increases, with a characteristic decay length given by the thermal wavelength λ. The corresponding momentum width is Δp ∼ /λ and the spatial distribution after TOF will reﬂect the initial momentum distribution if the TOF duration t is such that Δp t/m rT , where rT is the initial size of the gas. For a harmonic conﬁnement in the xy-plane, this “far ﬁeld” regime corresponds to ωt 1. Taking ω/2π = 30 Hz as a typical value, the far ﬁeld regime (say ωt > 3) is reached for t > 15 ms. This corresponds to a typical value for TOF experiments, which thus give access to the momentum distribution in this non (strongly) degenerate regime. The situation is very diﬀerent if a signiﬁcant condensed fraction is present in the gas, as expected in the vicinity and below the BKT transition temperature. In this case we . have seen in subsect. 5 4 that the size rc of the coherent region of the cloud is rc ∼ rT . The momentum width Δpc = /rc of this coherent component is then extremely narrow, and it would require a very long TOF to reach the “far ﬁeld” regime for this coherent component. Taking rc = rT as a typical value, we ﬁnd that the time t required for a signiﬁcant expansion of this component, i.e. Δpc t/m = rc , is such that ωt = kB T /ω. For ω/2π = 30 Hz and T = 100 nK, this gives t > 300 ms, which is too long in practice for a TOF. Therefore in the regime where a relatively strong coherence of the gas is present, a 3d TOF of a realistic duration gives access to a hybrid information. The high-energy fraction of the gas is in the far-ﬁeld regime and the wings of the density proﬁle after TOF give access to the large momentum part of the initial state. On the contrary the central feature corresponding to the condensed, superﬂuid fraction, has not yet undergone a signiﬁcant expansion. The detailed study of the border between these two components is still a matter of debate. In experiments with rubidium atoms [27, 106], the density proﬁle after a 3d TOF is well modeled by a two-component distribution and ﬁts with the line of reasoning we just presented. In contrast, in the experiments performed at NIST with a sodium gas [28], an intermediate third component was introduced in order to obtain a good description of the density proﬁles after a 3d TOF. This component corresponds to a phase with a spatial scale of coherence that is intermediate between the

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microscopic length λ and the macroscopic one rT , and it is qualiﬁed as a “non-supeﬂuid quasi-condensate” in [28]. It is interesting to note that 3d TOF is the most common and natural experimental method used in the studies of 3d atomic gases. However, in hindsight, its availability is a non-trivial and rather serendipitous feature of atomic systems for studies of BKT physics. In combination with the ﬁnite-size induced condensation, the ability to suddenly turn oﬀ the interactions through the fast z-expansion provides a much more striking signature of the BKT transition [27, 28, 106] than one might have theoretically expected. Thinking strictly in 2d, the transition is extremely smooth and one would not naturally expect to see such a dramatic signature in any quantity except the superﬂuid density. As we . discussed in subsect. 7 2, in 2d TOF the observed density distribution indeed varies smoothly across the transition. So far we have discussed the “average” density proﬁle in 3d TOF, which theoretically corresponds to the average of a large number of images obtained under same conditions. It is also interesting to consider the density noise in individual images, which can be related to the phase noise of the gas before expansion. This connection has been exploited for quasi-1d gases since 2001 [107]. For the 2d case, it has been shown theoretically in [108] that the two-point density correlation function after TOF can provide information on the in situ g1 function, at least in the superﬂuid regime. This method is also speciﬁc to 3d TOF, where the phase noise evolves into density noise in interaction-free ballistic expansion. . 7 4. Interference between independent planes. – Since an important aspect of the physics of 2d Bose gases is related to phase properties, it is natural to investigate measurement schemes based on interferometry. We start with the proposal by Polkovnikov et al. [109] which showed how a single experimental procedure could characterize both the normal regime (exponential decay of g1 ) and the superﬂuid regime (algebraic decay of g1 ) (see also [110] for a more complete review). Consider two independent, inﬁnite planar gases located at za = +dz /2 and zb = −dz /2. They are prepared in identical conditions, i.e. they have the same temperature and the same density. We perform a 3d time-of-ﬂight of duration t, that is chosen such that the ﬁnal extension along z of each cloud is large compared to the initial separation dz between the planes. The two clouds thus overlap and we want to extract information about the one-body correlation function g1 from their interference pattern (ﬁg. 4a). The state of each plane is described by the wave function ψa/b (x, y). After expansion, the spatial atomic density n is modulated along any line parallel to the z axis with the period Dz = ht/mdz [111]: (88)

n ∝ |ψa |2 + |ψb |2 + ψa ψb∗ ei2πz/Dz + c.c. .

For simplicity we have omitted in the above equation a global envelope factor giving the variation of the density along the z-axis. Also we have neglected the expansion in the xy-plane during the TOF. As explained above there exists a range of TOF duration

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Fig. 4. – (a) Principle of an experiment giving access to the interference between two independent planar gases, observed after time-of-ﬂight. (b-d): Examples of interference patterns measured with the experimental setup described in [26]. The imaging beam is propagating along the y-axis. The pattern (b) is obtained with very cold gases, whereas (c) corresponds to a larger temperature. The dislocation in (d) is the signature for the presence of a vortex in one of the two gases.

where this is valid, if the trapping frequency ω in this plane is much smaller than ωz . We see from eq. (88) that the local (complex) contrast of the density modulation is ψa ψb∗ . Experimentally one cannot measure this quantity along a single line, and one rather has access to the average contrast over a region of ﬁnite area A in the xy-plane. In particular if one performs absorption imaging along the y-axis (ﬁg. 4b-d), the image involves an integration of the local contrast ψa ψb∗ along the y-direction(8 ). Averaging the result of this contrast measurement over a large number of realizations, one can deﬁne the average contrast C(A): (89)

1 C (A) = 2 A 2

5 2 6 ψa (r)ψb∗ (r) d2 r . A

Using the fact that the ﬂuctuations of the wave functions ψa and ψb are uncorrelated and taking advantage of the translational symmetry of the system, we ﬁnd 1 (90) C 2 (A) = |g1 (r)|2 d2 r, A A where g1 (r) = ψj∗ (r)ψj (0) for j = a, b. Suppose for simplicity that the area A is a −r/ , with a square and consider the two √ cases of an exponentially decaying g1 (r) ∝ e characteristic length A (normal ﬂuid), and an algebraically decaying g1 (r) ∝ r−α , with an exponent α < 1/4 (superﬂuid regime). In the ﬁrst case, the integral is nearly independent of A and C 2 (A) ∝ A−1 . In the second case we ﬁnd C 2 (A) ∝ A−2α which corresponds to a decay always slower than A−1/2 . This method is very appealing in the sense that the measurement of a single number, i.e. the exponent η characterizing the variation of C ∝ A−η , is suﬃcient to identify the two possible regimes of a 2d Bose gas, and obtain the value of ns λ2 = 1/η in the superﬂuid case. (8 ) The length over which the line-of-sight integration occurs can be adjusted by a proper “slicing” of the cloud just before the imaging process, as in [111].

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A measurement scheme inspired by this method was implemented at ENS [26], and it indeed revealed a relatively rapid variation of the exponent η, in qualitative agreement with what is expected near the BKT transition point in the center of the trap. However some notable deviations with respect to the original proposal must be stressed. First, the measurement was performed with anisotropic samples, with lengths Ly Lx . The imaging beam was propagating along y and the measured contrast involved a line-of-sight integration over the full length Ly (9 ), which formally breaks the translational invariance that we used to prove eq. (90). Also the presence of a trapping potential in the experiment causes an additional softening of the transition, because of the inhomogeneity of the density along the line-of-sight of the imaging beam. Finally we note that even deep in the superﬂuid regime where ns λ2 1, the anisotropy of the sample adds some . complexity as discussed in subsect. 4 5. At large distances (Δx > ns λ2 Ly ), g1 starts to decay exponentially, which complicates the analysis of the dependence of C 2 on Δx . In summary the rapid increase in coherence that occurs in the vicinity of the BKT transition point is suﬃciently robust to be revealed experimentally in the average contrast of the interference pattern, but it is diﬃcult to provide a quantitative analysis of the experimental measurements for the variations of C 2 (A) over a large range of Δx . A subsequent experiment at ENS has compared the conditions for observing a significant interference contrast between the planes and for measuring a clear bimodal density proﬁle after a 3d TOF [27]. The onsets of the two phenomena were found to coincide within experimental error. Furthermore the spatial part of the gas that gives rise to a visible interference signal coincides with the central, “non expanding” component of the TOF proﬁle. An important outcome of the experiments on the interference between two planes is a direct evidence for thermally activated vortices. At low temperatures, long-wavelength phase ﬂuctuations (phonons) result in smooth variations of the phase of the interference fringes, such as seen in ﬁg. 4c. However, if a single isolated vortex is present in one of the two planes while the phase proﬁle of the other plane is smooth, the interference pattern exhibits a sharp dislocation at the coordinate x of the vortex core. Such dislocations have been observed experimentally [81, 26, 69] and an example is shown in ﬁg. 4d. The occurrence probability of these dislocations has been measured as a function of temperature [26]. The number of dislocations increases with T , until one reaches the temperature at which no interference is visible anymore. Moreover, the relatively sharp increase in the probability of dislocations experimentally coincides with the increase in the exponent η characterizing the decay of g1 [26]. Such dislocations also appear in a classical ﬁeld simulation mimicking the interference between two planar gases [40]. They result from the thermal activation of a vortex pair for which the two members are suﬃciently separated from each other. In principle one should also observe in the interference patterns tightly bound vortex pairs where the two members are separated by ∼ ξ. However these pairs (9 ) The area A was varied by changing the integration distance Δx along x. In this case, the BKT transition causes a crossover from η = 1/2 for an exponentially decaying g1 function (with a decay length Δx ), to η = 1/4 for a superﬂuid state.

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only shift the fringe pattern by a small fraction of the fringe spacing, which is below the current sensitivity of the experiments. . 7 5. Interfering a single plane with itself . – An interesting alternative to the two-plane interference described above consists in preparing a single plane of atoms and looking at its “self-interference”, using a Ramsey-like method [28]. The gas is initially prepared in an internal state |1. Half of the atoms are coherently transferred into another internal state |2 by a stimulated laser Raman process (π/2 pulse) that also provides a momentum kick k0 to the atoms. After this process, the part of the cloud in |1 is still globally at rest and the part in |2 moves with the global velocity v0 = k0 /m. After an adjustable time t a second π/2 Raman pulse remixes the amplitudes of |1 and |2 and provides a momentum kick k0 − k1 . Immediately after this second Raman pulse, one measures the spatial density distribution in |2. This distribution exhibits a modulation along the direction k1 , resulting from the interference between the initial state of the cloud and the state displaced by the distance R = v0 t: (91)

+ , n(r) ∝ |ψ(r)|2 + |ψ(r − R)|2 + ψ(r)ψ ∗ (r − R)eik1 ·R + c.c. .

Note that we assume here that no collision occurred during the time t between the part of the cloud at rest in |1 and the part moving at velocity v0 in state |2. This is a valid assumption for the weakly interacting sodium gas of [28]. The modulated density proﬁle in eq. (91) gives a direct access to the function g1 (r, r − R). It can be observed with an imaging beam along the z-direction, so that its measurement does not involve any lineof-sight integration. This method can then reveal ﬁner details than the one presented . in subsect. 7 4. In particular the authors of [28] could observe a gradual increase of the coherence length of the cloud, as expected from eq. (56). For small phase space densities the measurement gives ∼ λ, and increases to much larger values when the temperature decreases towards the critical temperature TBKT . When T < TBKT a signiﬁcant interference contrast is observed for all values of R within the size of the central superﬂuid region. 8. – Conclusions and outlook We have reviewed in these notes the theoretical basis for the understanding of the physics of 2d quantum ﬂuids, and discussed some recent experiments performed with atomic gases. These experiments have given access to some aspects of 2d physics that were previously hidden or not measurable in other physical systems, such as the existence of thermally activated individual vortices or the spatial variation of the ﬁrst-order correlation function g1 (r). However a number of issues is still open in the physics of 2d quantum gases, and we outline below some topics which are likely to be of experimental and theoretical interest in the future. Higher-order correlation functions. – The matter-wave interference between two statistically similar, but independent quasi-condensates (such as shown in ﬁg. 4), can reveal

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a wealth of information on the correlations within each individual 2d gas. So far only a fraction of this information has been harnessed, with the study of the average contrast of the interference pattern integrated over some area of interest. A convenient tool for extracting more complete information on g1 as well as higher-order correlation functions is the full statistical distribution of interference contrasts. Two limiting cases can easily be characterized: i) If the two independent ﬂuids are fully condensed, each image shows a 100% contrast, with the position of the fringes ﬂuctuating randomly from shot to shot. ii) If each cloud exhibits only short-ranged correlations, the observed interference results from many uncorrelated fringe patterns along the light of sight, and the distribution of contrasts is an exponential function. For 1d gases, it is possible to describe quantitatively the transition between these two limiting cases [112], and the experimental results [113] are in good agreement with the predictions. In the 2d case, the evolution of the contrast distribution through the BKT transition is still an open problem. Out-of-equilibrium dynamical eﬀects. – Throughout this paper we restricted our discussion to the equilibrium properties of a 2d Bose ﬂuid. The study of dynamical eﬀects, such as transient regimes, can reveal additional information about the system. For example Burkov et al. [114] have studied the dynamics of decoherence between two planar Bose gases, assuming that their local phases are initially locked together, and then the two gases are allowed to evolve independently. This can be achieved experimentally by having a weak potential barrier and hence large tunnel coupling between the two planes for t < 0, and then suddenly raising the barrier at t = 0. The contrast of the interference between the two gases gives access to the evolution of the phase distribution under the inﬂuence of thermal ﬂuctuations. In [114] this contrast was shown to decay algebraically at long time, as t−ζ , with the exponent ζ proportional to the ratio T /TBKT . Therefore, in addition to being a stringent test of thermal decoherence in a quantum many-body system, this out-of-equilibrium study could constitute a novel thermometry method. A related phenomenon occurs in 1d systems, where the interference contrast is predicted to decay as exp(−(t/t0 )2/3 ) (with t0 constant) [114], and this prediction is nicely conﬁrmed in the experiments by the Vienna group [115]. Transition from 2d to 3d behavior . – The possibility to vary the tunnel coupling between two or more planar gases can also be used to study the so-called “deconﬁnement transition” [116], corresponding to a gradual evolution from 2d to 3d behavior. The phase coherence between the planes will build up as the strength of the coupling is increased, creating a situation that is reminiscent of the high-Tc cuprate superconductors. For a large number of parallel planes, the deconﬁnement transition should give rise to a true Bose-Einstein condensate [116]. The two-plane situation is also very interesting, and can lead to the observation of the Kibble-Zurek mechanism [117]: the superﬂuid transition temperature is higher for two coupled planes than for a single one, so that sudden switching on of the coupling between the planes (initially in the normal state but close to the single-plane critical temperature) constitutes a quench of the system, and one could observe the subsequent dynamical apparition of a macroscopic quantum (quasi-)coherence.

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Tunable interactions. – As we have seen throughout the paper, interactions between particles play a crucial role in our understanding of the superﬂuid phase transition and condensation in 2d ﬂuids. In contrast to the conventional 3d BEC of an atomic gas, where the critical temperature can to a good approximation be predicted using the ideal gas model, the BKT transition is fundamentally interaction-driven. The strength of interactions also aﬀects a variety of other phenomena such as the suppression of density ﬂuctuations in the normal state and the connection between the 2d Bose ﬂuid and the XY model. It would therefore be interesting to revisit the various eﬀects described in these notes while continuously tuning the strength of interactions with a Feshbach resonance [48, 49]. In the weak coupling regime (˜ g < 10−1 ) we expect a gradual change from the BKT-dominated to the BEC-dominated behavior, as discussed in sects. 4 and 5. Further, it would be very interesting to explore the strong-coupling regime (˜ g > 1), which is closer to liquid helium ﬁlms. This regime, which is outside the domain of validity of the Monte Carlo results [29, 44], corresponds to the case where the scattering length as becomes comparable the thickness of the sample along the kinematically frozen direction, . az (see 6 2). There the very nature of two-body interactions is expected to change from 3d to 2d [84, 85, 118, 119]. Therefore, experimentally reaching the condition as ≥ az would correspond to producing a “truly 2d” as opposed to a quasi-2d Bose gas. Superﬂuid density. – Generally speaking, studies of coherence and correlation functions in a 2d ﬂuid, which are well suited to experimental tools of atomic physics, are a natural complement to the “traditional” studies of superﬂuidity based on transport measurements, which are well suited to other physical systems such as liquid helium ﬁlms [2]. For example, we have so far assumed that the two types of measurements probe the same . superﬂuid density (see, e.g., subsect. 7 4). However this correspondence may in fact depend on the theoretical model and the exact deﬁnition of the superﬂuid density, and be valid only within the eﬀective low-energy theories. It is therefore important to stress that superﬂuidity in the traditional transport sense has not yet been directly observed in atomic 2d Bose gases (see, e.g., [116]). Establishing atomic 2d gases as experimental systems in which both coherence and transport measurements of superﬂuidity could be performed would be an important advance, as it would allow experimental scrutiny of the theoretical connections between the two types of probes, and a direct comparison of the diﬀerent deﬁnitions of superﬂuidity. Two promising schemes for a direct measurement of the superﬂuid density (as traditionally deﬁned through transport properties [46]) in an atomic gas have recently been proposed [120, 121]. The ﬁrst scheme [120] is based on extracting the superﬂuid density from the in situ density proﬁles of a rotating 2d gas. The second scheme [121] is based on using a vector potential generated by Raman laser beams to simulate slow rotation of a gas [122], and allows direct spectroscopic measurement of the superﬂuid density. Note added in proofs. A measurement of the equation of state of a 2d Bose gas for various interaction strengths has just been reported in [123].

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∗ ∗ ∗ We warmly thank the directors of the school R. Kaiser and D. Wiersma, as well as the scientiﬁc secretary L. Fallani, for organizing this very successful meeting. Many colleagues helped us with discussions and interactions and the list of those we would like to thank is too long to ﬁt here, but we mention in particular E. Altman, N. Cooper, E. Cornell, E. Demler, B. Douc ¸ ot, T. Giamarchi, T.-L. Ho, M. Holzmann, M. ¨ hl, W. Krauth, W. Phillips, A. Polkovnikov, G. Shlyapnikov, D. StamperKo Kurn, W. Zwerger, as well as the past and present members of the ENS cold atoms group. ZH is supported by EPSRC Grant No. EP/G026823/1. JD is supported by R´egion Ile de France IFRAF, CNRS, the French Ministry of Research, ANR (Grant ANR-08-BLAN-65 BOFL), and the E.U. project SCALA. Laboratoire Kastler Brossel is a mixed research unit n◦ 8552 of CNRS, Ecole normale sup´erieure, and Universit´e Pierre et Marie Curie. REFERENCES [1] Peierls R. E., Surprises in Theoretical Physics (Princeton University Press) 1979. [2] Bishop D. J. and Reppy J. D., Phys. Rev. Lett., 40 (1978) 1727. [3] Safonov A. I., Vasilyev S. A., Yasnikov I. S., Lukashevich I. I. and Jaakkola S., Phys. Rev. Lett., 81 (1998) 4545. [4] Minnhagen P., Rev. Mod. Phys., 59 (1987) 1001. [5] Posazhennikova A., Rev. Mod. Phys., 78 (2006) 1111. [6] Bloch I., Dalibard J. and Zwerger W., Rev. Mod. Phys., 80 (2008) 885. [7] Snoke D., Science, 298 (2002) 1368. [8] Butov L. V., J. Phys. Condens. Matter, 16 (2004) R1577. [9] Kasprzak J., Richard M., Kundermann S., Baas A., Jeambrun P., Keeling J. M. J., Marchetti F. M., Szymanska M. H., Andre R., Staehli J. L., Savona V., Littlewood P. B., Deveaud B. and Dang L. S., Nature, 443 (2006) 409. [10] Amo A., Lefrere J., Pigeon S., Adrados C., Ciuti C., Carusotto I., Houdre R., Giacobino E. and Bramati A., Nature Phys., 5 (2009) 805. [11] Kosterlitz J. M., J. Phys. C: Solid State Physics, 7 (1974) 1046. [12] Nelson D. R. and Halperin B. I., Phys. Rev. B, 19 (1979) 2457. [13] Strandburg K. J., Rev. Mod. Phys., 60 (1988) 161. [14] Peierls R. E., Helv. Phys. Acta, 7 (1934) 81. [15] Peierls R. E., Ann. Inst. Henri Poincar´e, 5 (1935) 177. [16] Bogoliubov N. N., Physica, 26 (1960) S1. [17] Hohenberg P. C., Phys. Rev., 158 (1967) 383. [18] Mermin N. D. and Wagner H., Phys. Rev. Lett., 17 (1966) 1307. [19] Penrose O. and Onsager L., Phys. Rev., 104 (1956) 576. [20] Berezinskii V. L., Sov. Phys. JETP, 34 (1971) 610. [21] Kosterlitz J. M. and Thouless D. J., J. Phys. C: Solid State Physics, 6 (1973) 1181. [22] Huang K., Statistical Mechanics (Wiley, New York) 1987. [23] Olshanii M. and Pricoupenko L., Phys. Rev. Lett., 88 (2002) 010402. [24] Al Khawaja U., Andersen J. O., Proukakis N. P. and Stoof H. T. C., Phys. Rev. A, 66 (2002) 013615. [25] Adhikari S. K., Am. J. Phys., 54 (1986) 362.

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International School of Physics “Enrico Fermi” Villa Monastero, Varenna Course CLXXIII 23 June – 3 July 2009

“Nano Optics and Atomics: Transport of Light and Matter Waves” Directors Diederik S. WIERSMA European Laboratory for Non-linear Spectroscopy (LENS) and INFM-CNR Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 457 2492 - 2452 fax: +39 055 4572451 [email protected] Robin KAISER Institut Non Lin´eaire de Nice CNRS/UMR 6618 1361, Route des Lucioles F-06560 Valbonne France tel.: +33 (0)4 92 96 73 91 fax: +33 (0)4 92 96 73 33 [email protected]

Scientiﬁc Secretary Leonardo FALLANI European Laboratory for Non-linear Spectroscopy (LENS) and Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 457 2458 fax: +39 055 457 2451 [email protected] c Societ` a Italiana di Fisica

Lecturers and Seminar Speakers Alain ASPECT Groupe d’Optique Atomique Laboratoire Charles Fabry UMR 8501 du CNRS Institut d’Optique Th´eorique et Appliqu´ee CNRS et Universit´e de Paris-Sud XI Campus Polytechnique, RD 128 F-91127 Palaiseau cedex France tel.: +33 1 64 533103 fax: +33 1 64 533118 [email protected] Bill BARNES School of Physics University of Exeter Stocker Road Exeter EX4 4QL UK tel.: +44 1392 264135 fax: +44 1392 264111 [email protected] Immanuel BLOCH Institut f¨ ur Physik Johannes Gutenberg Universit¨ at Staudingerweg 7 D-55128 Mainz Germany tel.: +49 6131 39 26234 -22279 Fax: +49 6131 39 25179 [email protected] 323

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Jean DALIBARD Laboratoire Kastler Brossel 24 rue Lhomond F-75005 Paris France tel.: +33 1 44 32 25 34 fax: +33 1 44 32 34 34 [email protected] Matthias FINK Institut Langevin Laboratoire Ondes et Acoustique Ecole Sup´erieure de Physique et de Chimie Industrielle de la Ville de Paris Universit´e D. Diderot - UMR CNRS 7587 10 Rue Vauquelin F-75005 Paris France tel.: +33 1 713 348 6087 fax: +33 1 713 348 5492 [email protected] Randall HULET Fayez Saroﬁm Professor of Physics Rice University Physics and Astronomy Department MS-61 6100 Main Street Houston TX 77251 USA tel.: +1 713-348-6087 fax: +1 713-348-5492 [email protected] Massimo INGUSCIO European Laboratory for Non-linear Spectroscopy (LENS) and Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 457 2465-2461-2462 fax: +39 055 457 2451 [email protected]

Elenco dei partecipanti Jossie KLAFTER School of Chemistry Tel Aviv University Tel Aviv 69978 Israel tel.: +972 3 6408254 [email protected] Georg MARET Fachbereich Physik Universit¨ at Konstanz D-78457 Konstanz Germany tel.: +49 7531 88 4151 fax: +49 7531 88 4151 [email protected] John PAGE Ultrasonics Research Laboratory Department of Physics and Astronomy University of Manitoba Winnipeg MB Canada R3T 2N2 tel.: +1 204 474-9852 fax: + 1204 474-7622 [email protected] Paul STEINHARDT Center for Theoretical Science and Princeton University Princeton NJ 08544-0708 USA tel.: +1 609 258 1509 [email protected] ¨ Peter WOLFLE Institut f¨ ur Theorie der Kondensierten Materie Universit¨ at Karlsruhe Wolfgang-Gaede-Str. 1 D-76128 Karlsruhe Germany tel.: +49 721 608 3590-3367 fax: +49 721 608 7779 [email protected]

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Elenco dei partecipanti Arjun YODH Department of Physics and Astronomy University of Pennsylvania Philadelphia PA 19104-6396 USA tel.: +1 215 898-6354 fax: +1 215 898-2010 [email protected]

Students Holger AHLERS Institut f¨ ur Quantenoptik Leibniz Universit¨ at Hannover Welfengarten 1 D-30167 Hannover Germany tel.: +49 511 762 4406 fax: +49 511 762 2211 [email protected] Julien ARMIJO Laboratoire Charles Fabry Institut d’Optique Campus Polytechnique, RD 128 F-91127 cedex Palaiseau France tel.: +33 1 6453 33 59 [email protected] Pierre BARTHELEMY LENS - Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 4572477 fax: +39 055 4572451 [email protected]

Mathis BAUMERT School of Physics and Astronomy University of Birmingham Edgbaston B15 2TT Birmingham UK tel.: +44 (0)121 414 4672 fax: +44 (0)121 414 4693 [email protected] Albert BENSENY CASES Grup d’Optica Universitat Aut` onoma de Barcelona Ediﬁci CC - Campus UAB E-08193 Bellaterra (Cerdanyola del Vall`es) Barcelona Spain tel.: +34 935814914 / +34 615957453 fax: +34 935812155 [email protected] Marco BERRITTA MATIS CNR-INFM Viale A. Doria 6 I-95125 Catania Italy tel.: +39 095 7382811 fax: +39 095 333231 [email protected] Shivakiran BHAKTHA B. N. LPMC-CNRS-UMR 6622 Universit´e de Nice Sophia Antopolis Parc Valrose F-06108 cedex02 Nice France tel.: +33 492076798 fax: +33 492076754 [email protected] [email protected]

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´ Tom BIENAIME Institut Non Lin´eaire de Nice 1361 Route des Lucioles F-06560 Valbonne France tel.: +33 492 967336 [email protected]

Laura CHANTADA SANTODOMINGO Department of Physics University of Fribourg Perolles CH-1700 Fribourg Switzerland tel.: +41 263009132 [email protected]

Nellu CIOBANU Academy of Sciences of Moldova Institute of Applied Physics Str. Academiei 5 MD-2028 Chisinau Moldova tel.: +373 22 739907 fax: +373 22 738149 nellu [email protected]

Martin Robert DE SAINT VINCENT Laboratoire Charles Fabry Institut d’Optique Campus Polytechnique, RD 128 F-91127 Palaiseau cedex France tel.: +33 0164533329 martin.robert-de-saint-vincent @institutoptique.fr

Elenco dei partecipanti David ELAM jr. Department of Physics One UTSA Circle University of Texas San Antonio TX 78249 San Antonio USA tel.: +1 210 216 0293 fax: +1 210 458 4919 [email protected]

Vitalie EREMEEV LPMMC Maison des Magist`eres - CNRS 25 Avenue des Martyrs, BP166 F-38042 Grenoble France tel.: +33 476887984 fax: +33 476887983 [email protected]

Nicole FABBRI LENS - Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 4572458 fax: +39 055 4572451 [email protected]

Berihu Teklu GEBREHIWOT Dipartimento di Fisica Universit` a di Milano Via Celoria 16 I-20133 Milano Italy tel.: +39 02 50317686 fax: +39 02 50317269 [email protected]

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Elenco dei partecipanti Silvia GIUDICATTI Dipartimento di Fisica “A. Volta” Universit` a di Pavia Via Bassi 6 I-27100 Pavia Italy tel.: +39 0382987693 [email protected]

Silvio GUALINI Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129, Torino Italy tel.: +39 011 5644120-4125 fax: +39 0115644099 [email protected]

William Kurt HILDEBRAND Department of Physics and Astronomy University of Manitoba 301 Allen Bldg. 30A Sifton Rd. R3T 2N2 MB Winnipeg Canada tel.: +1 2044748415 fax: +1 2044747622 [email protected] Christoph HOFMANN Physics Department University of Heidelberg Philosophenweg 12 D-69120 Heidelberg Germany tel.: +49 6221549205 fax: +49 6221475733 [email protected]

Timo HARTMANN Institut f¨ ur Theoretische Physik Universit¨ at Regensburg D-93040 Regensburg Germany tel.: +49 9419432016 fax: +49 9419434382 timo.hartmann @physik.uni-regensburg.de

Sascha HOINKA Department of Physics University of Otago 730 Cumberland Street Science Building III NZ Dunedin 9016 New Zealand tel.: +64 34797789 fax: +64 34790964 [email protected]

Jonas HARTWIG Institut f¨ ur Quantenoptik Leibniz Universit¨ at Hannover Welfengarten 1 D-30167 Hannover Germany tel.: +49 51176219192 fax: +49 5117622211 [email protected]

Michael HOLYNSKI School of Physics and Astronomy University of Birmingham Edgbaston B15 2TT Birmingham UK tel.: +44 (0)1214144672 fax: +44 (0)1214144644 [email protected]

328

Takuya KITAGAWA Physics Department Harvard University 17 Oxford Street MA 02138 Cambridge USA tel.: +1 6179904979 [email protected]

Priyaranga L. KOSWATTA Birck Nanotechnology Center Purdue University 1205 West State st. IN 47907 West Lafayette USA tel.: +1 7655862920 fax: +1 7654962018 [email protected]

Eran KOT Niels Bohr Institute Blegdamsvej 17 DK-2100 Copenhagen Denmark tel.: +45 35325426 fax: +45 35325016 [email protected]

Siarhei KURYLCHYK Istitute for Optical Materials and Technologies Belarus National Technical University 65 Nezavisimosti Avenuer Building 17 BY-220013 Minsk Belarus tel.: +37 5297535774 fax: +37 5172926286 [email protected]

Elenco dei partecipanti Yoav LAHINI Department of Physics of Complex Systems The Weizmann Institute of Science Rehovot 76100 Israel tel. 972-8-9342058 fax. 972-8-9344109 [email protected]

Marco LARCHER CNR-INFM BEC Center and Dipartimento di Fisica Universit` a di Trento Via Sommarive 14 I-38100 Povo (TN) Italy [email protected]

´ ENRICH Ricard MENCHON Grup d’Optica Departament de F´ısica Universitat Aut` onoma de Barcelona Ediﬁci CC - Campus UAB Cerdanyola del Vall`es E-08193 Bellaterra (Barcelona) Spain tel.: +34 935814914 / +34 647155161 fax: +34 935812155 [email protected]

Nicolas MERCADIER Institut Non Lin´eaire de Nice - CNRS 1361 Route des lucioles F-06560 Valbonne France tel.: +33 492967374 fax: +33 492967333 [email protected]

329

Elenco dei partecipanti Nadine MEYER School of Physics and Astronomy University of Birmingham Edgbaston B15 2TT Birmingham UK tel.: +44 (0)121 414 4672 fax: +44 (0)121 414 4693 [email protected] [email protected]

Rajeshkumar MUPPARAPU LENS - Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy [email protected]

Johan Raunkjær OTT DTU-Fotonik Institute of Photonics Engineering Ørsted Plads, Building 343 DK-2800 Kongens Lyngby Denmark tel.: +45 26176113 [email protected]

Tobias PAUL Institut f¨ ur Theoretische Physik University of Heidelberg Philosophenweg 19 D-69120 Heidelberg Germany tel.: +49 6221549399 fax: +49 6221549331 [email protected]

Wouter PEETERS Quantum Optics and Quantum Information Leiden Institute of Physics Leiden University P.O. Box 9504 NL-2300 RA Leiden The Netherlands tel.: +31 (0)715275926 fax: +31 (0)715275819 [email protected] Abe PENA Department of Physics One UTSA Circle University of Texas San Antonio TX 78249 San Antonio USA tel.: +1 9563575686 fax: +1 2104584919 abee [email protected] Simone Luca PORTALUPI Dipartimento di Fisica “A. Volta” Universit` a di Pavia Via Bassi 6 I-27100 Pavia Italy tel.: +39 0382 987693 fax: +39 0382 987563 [email protected] Ivan PRIETO Instituto de Microelectronica de Madrid Centro Nacional de Microelectronica Consejo Superior de Investigaciones Cientiﬁcas Isaac Newton 8 - PTM E-28760 Tres Canto, Madrid Spain tel.: +34 91 806 0700 fax: +34 91 806 0701 [email protected] [email protected]

330

Rami PUGATCH Department of Physics of Complex Systems The Weizmann Institute of Science Rehovot 76100 Israel tel.: 972-8-9343543 - 9342675 [email protected]

Katrijn PUTTENEERS TQC - Department of Physics Campus Groenenborger Universiteit Antwerpen Groenenborgerlaan 171 B-2020 Antwerpen Belgium tel : +32 32653529 fax: +32 32653318 [email protected]

Ana RAKONJAC Department of Physics University of Otago PO Box 56 NZ Dunedin 9016 New Zealand tel.: +64 34797789 fax: +64 34790964 [email protected] Lothar RATSCHBACHER AMOP Group Cavendish Laboratory University of Cambridge JJ Thomson Avenue CB3 0HE Cambridge UK tel.: +44 1223766464 [email protected]

Elenco dei partecipanti Stefan RIST Departament de Fisica Universitat Autonoma de Barcelona E-08193 Bellaterra Spain tel.: +34 669263241 fax: +34 935812155 [email protected] Marina SAMOYLOVA Faculty of Physics St.-Petersburg State University Ulyanovskaya 5, Petrodvoretz St.-Petersburg Russia tel.: +7 (812) 4284339 [email protected] [email protected] Stephan Tobias SEIDEL Institut f¨ ur Quantenoptik Leibniz Universit¨ at Hannover Welfengarten 1 D-30167 Hannover Germany tel.: +49 5117624887 fax: +49 5117622211 [email protected] Oleksii SLIUSARENKO Akhiezer Institute for Theoretical Physics NSC Kharkiv Institute of Physics and Technology Akademichna str.1 61108, Kharkiv Ukraine tel.: +38 057 335 65 23 fax: +38 057 335 26 83 [email protected]

331

Elenco dei partecipanti Mihai STRATICIUC Horia Hulubei National Institute of Physics and Nuclear Engineering 407 Atomistilor Str. CP 077125 Magurele, jud. Ilfov Romania tel.: +40 214042342 fax: +40 214574440 [email protected]

Piotr SZANKOWSKI Institute of Theoretical Physics Faculty of Physics University of Warsaw, Hoza 69 PL-00-681 Warsaw Poland tel.: +48 225532000, +48 226283396 fax: +48 226219475 [email protected] [email protected]

Mark THORESON Birck Nanotechnology Center Purdue University 1205 West State st. IN 47907 West Lafayette USA tel.: +1 7654963317 [email protected]

Arif ULLAH Department of Physics University of Auckland Private Bag NZ-92019 Auckland New Zealand tel.: +64 93737599 ext. 88845 fax: +64 93737445 [email protected]

Francisco Javier VALDIVIA VALERO Instituto de Ciencia de Materiales de Madrid Consejo Superior de Investigaciones Cientiﬁcas C/ Sor Juana In´es de la Cruz, 3 E-28049 Cantoblanco (Madrid) Spain tel.: +34 913721420 fax: +34 913720623 [email protected]

Radhalakshmi VIVEKANANTHAN LENS - Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 4572477 fax: +39 055 4572473 [email protected]

Enrico VOGT Cavendish Laboratory University of Cambridge JJ Thomson Avenue CB3 0HE Cambridge UK tel.: +44 1223766464 [email protected]

Kevin VYNCK LENS - Universit` a di Firenze Via Nello Carrara 1 I-50019 Sesto Fiorentino (FI) Italy tel.: +39 055 4572477 [email protected]

332

Tomasz WASAK Faculty of Physics University of Warsaw Hoza 69 PL-00681 Warsaw Poland tel.: +48 663381531 [email protected] Christian WUTTKE Institut f¨ ur Physik - AG QUANTUM Johannes-Gutenberg-Universit¨ at Staudingerweg 7 D-55128 Mainz Germany tel.: +49 61313925918 fax: +49 61313923428 [email protected] [email protected]

Elenco dei partecipanti Dubravka MILOVANOVIC Department of Physical Chemisty VINCA Institute of Nuclear Sciences Belgrade Serbia tel.: +381 11 2453 967 cell. +381 64 2184435 [email protected]

Observers

Eduardo NUNES-PEREIRA Centro de Fisica Escola de Ciencias Universidade do Minho Campus de Gualtar P-4710-057 Braga Portugal tel.: +35 1253604336 fax: +35 1253604061 [email protected]

Andrea CAMPOSEO National Nanotechnology Laboratory CNR-INFM Distretto Tecnologico ISUFI Universit` a del Salento via Arnesano 73100 Lecce Italy tel.: +39 0832298147 fax: +39 0832298146 [email protected]

´ Juan Jos´e SAENZ Dept. Fisica Materia Condensada Universidad Aut´ onoma de Madrid E-28049 Madrid Spain tel.: +34 914973804 fax: +34 914973961 [email protected]

PROCEEDINGS OF THE INTERNATIONAL SCHOOL OF PHYSICS “ENRICO FERMI”

Course I (1953) Questioni relative alla rivelazione delle particelle elementari, con particolare riguardo alla radiazione cosmica edited by G. Puppi Course II (1954) Questioni relative alla rivelazione delle particelle elementari, e alle loro interazioni con particolare riguardo alle particelle artiﬁcialmente prodotte ed accelerate edited by G. Puppi Course III (1955) Questioni di struttura nucleare e dei processi nucleari alle basse energie edited by C. Salvetti Course IV (1956) Propriet` a magnetiche della materia edited by L. Giulotto Course V (1957) Fisica dello stato solido edited by F. Fumi Course VI (1958) Fisica del plasma e relative applicazioni astroﬁsiche edited by G. Righini

Course XII (1959) Solar Radioastronomy edited by G. Righini Course XIII (1959) Physics of Plasma: Experiments and Techniques ´n edited by H. Alfve Course XIV (1960) Ergodic Theories edited by P. Caldirola Course XV (1960) Nuclear Spectroscopy edited by G. Racah Course XVI (1960) Physicomathematical Aspects of Biology edited by N. Rashevsky Course XVII (1960) Topics of Radiofrequency Spectroscopy edited by A. Gozzini Course XVIII (1960) Physics of Solids (Radiation Damage in Solids) edited by D. S. Billington

Course VII (1958) Teoria della informazione edited by E. R. Caianiello

Course XIX (1961) Cosmic Rays, Solar Particles and Space Research edited by B. Peters

Course VIII (1958) Problemi matematici della teoria quantistica delle particelle e dei campi edited by A. Borsellino

Course XX (1961) Evidence for Gravitational Theories edited by C. Møller

Course IX (1958) Fisica dei pioni edited by B. Touschek

Course XXI (1961) Liquid Helium edited by G. Careri

Course X (1959) Thermodynamics of Irreversible Processes edited by S. R. de Groot

Course XXII (1961) Semiconductors edited by R. A. Smith

Course XI (1959) Weak Interactions edited by L. A. Radicati

Course XXIII (1961) Nuclear Physics edited by V. F. Weisskopf

Course XXIV (1962) Space Exploration and the Solar System edited by B. Rossi Course XXV (1962) Advanced Plasma Theory edited by M. N. Rosenbluth Course XXVI (1962) Selected Topics on Elementary Particle Physics edited by M. Conversi Course XXVII (1962) Dispersion and Absorption of Sound by Molecular Processes edited by D. Sette Course XXVIII (1962) Star Evolution edited by L. Gratton Course XXIX (1963) Dispersion Relations and their Connection with Casuality edited by E. P. Wigner Course XXX (1963) Radiation Dosimetry edited by F. W. Spiers and G. W. Reed Course XXXI (1963) Quantum Electronics and Coherent Light edited by C. H. Townes and P. A. Miles Course XXXII (1964) Weak Interactions and High-Energy Neutrino Physics edited by T. D. Lee Course XXXIII (1964) Strong Interactions edited by L. W. Alvarez Course XXXIV (1965) The Optical Properties of Solids edited by J. Tauc Course XXXV (1965) High-Energy Astrophysics edited by L. Gratton Course XXXVI (1965) Many-body Description of Nuclear Structure and Reactions edited by C. L. Bloch Course XXXVII (1966) Theory of Magnetism in Transition Metals edited by W. Marshall

Course XXXVIII (1966) Interaction of High-Energy Particles with Nuclei edited by T. E. O. Ericson Course XXXIX (1966) Plasma Astrophysics edited by P. A. Sturrock Course XL (1967) Nuclear Structure and Nuclear Reactions edited by M. Jean and R. A. Ricci Course XLI (1967) Selected Topics in Particle Physics edited by J. Steinberger Course XLII (1967) Quantum Optics edited by R. J. Glauber Course XLIII (1968) Processing of Optical Data by Organisms and by Machines edited by W. Reichardt Course XLIV (1968) Molecular Beams and Reaction Kinetics edited by Ch. Schlier Course XLV (1968) Local Quantum Theory edited by R. Jost Course XLVI (1969) Physics with Intersecting Storage Rings edited by B. Touschek Course XLVII (1969) General Relativity and Cosmology edited by R. K. Sachs Course XLVIII (1969) Physics of High Energy Density edited by P. Caldirola and H. Knoepfel Course IL (1970) Foundations of Quantum Mechanics edited by B. d’Espagnat Course L (1970) Mantle and Core in Planetary Physics edited by J. Coulomb and M. Caputo Course LI (1970) Critical Phenomena edited by M. S. Green Course LII (1971) Atomic Structure and Properties of Solids edited by E. Burstein

Course LIII (1971) Developments and Borderlines of Nuclear Physics edited by H. Morinaga

Course LXVII (1976) Isolated Gravitating Systems in General Relativity edited by J. Ehlers

Course LIV (1971) Developments in High-Energy Physics edited by R. R. Gatto

Course LXVIII (1976) Metrology and Fundamental Constants edited by A. Ferro Milone, P. Giacomo and S. Leschiutta

Course LV (1972) Lattice Dynamics and Intermolecular Forces edited by S. Califano

Course LXIX (1976) Elementary Modes of Excitation in Nuclei edited by A. Bohr and R. A. Broglia

Course LVI (1972) Experimental Gravitation edited by B. Bertotti

Course LXX (1977) Physics of Magnetic Garnets edited by A. Paoletti

Course LVII (1972) History of 20th Century Physics edited by C. Weiner

Course LXXI (1977) Weak Interactions edited by M. Baldo Ceolin

Course LVIII (1973) Dynamics Aspects of Surface Physics edited by F. O. Goodman

Course LXXII (1977) Problems in the Foundations of Physics edited by G. Toraldo di Francia

Course LIX (1973) Local Properties at Phase Transitions ¨ller and A. Rigamonti edited by K. A. Mu

Course LXXIII (1978) Early Solar System Processes and the Present Solar System edited by D. Lal

Course LX (1973) C*-Algebras and their Applications to Statistical Mechanics and Quantum Field Theory edited by D. Kastler

Course LXXIV (1978) Development of High-Power Lasers and their Applications edited by C. Pellegrini

Course LXI (1974) Atomic Structure and Mechanical Properties of Metals edited by G. Caglioti Course LXII (1974) Nuclear Spectroscopy and Nuclear Reactions with Heavy Ions edited by H. Faraggi and R. A. Ricci

Course LXXV (1978) Intermolecular Spectroscopy and Dynamical Properties of Dense Systems edited by J. Van Kranendonk Course LXXVI (1979) Medical Physics edited by J. R. Greening

Course LXIII (1974) New Directions in Physical Acoustics edited by D. Sette

Course LXXVII (1979) Nuclear Structure and Heavy-Ion Collisions edited by R. A. Broglia, R. A. Ricci and C. H. Dasso

Course LXIV (1975) Nonlinear Spectroscopy edited by N. Bloembergen

Course LXXVIII (1979) Physics of the Earth’s Interior edited by A. M. Dziewonski and E. Boschi

Course LXV (1975) Physics and Astrophysics of Neutron Stars and Black Hole edited by R. Giacconi and R. Ruffini

Course LXXIX (1980) From Nuclei to Particles edited by A. Molinari

Course LXVI (1975) Health and Medical Physics edited by J. Baarli

Course LXXX (1980) Topics in Ocean Physics edited by A. R. Osborne and P. Malanotte Rizzoli

Course LXXXI (1980) Theory of Fundamental Interactions edited by G. Costa and R. R. Gatto Course LXXXII (1981) Mechanical and Thermal Behaviour of Metallic Materials edited by G. Caglioti and A. Ferro Milone Course LXXXIII (1981) Positrons in Solid-State Physics edited by W. Brandt and A. Dupasquier Course LXXXIV (1981) Data Acquisition in High-Energy Physics edited by G. Bologna and M. Vincelli Course LXXXV (1982) Earthquakes: Observation, Theory and Interpretation edited by H. Kanamori and E. Boschi Course LXXXVI (1982) Gamow Cosmology edited by F. Melchiorri and R. Ruffini Course LXXXVII (1982) Nuclear Structure and Heavy-Ion Dynamics edited by L. Moretto and R. A. Ricci Course LXXXVIII (1983) Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics edited by M. Ghil, R. Benzi and G. Parisi Course LXXXIX (1983) Highlights of Condensed Matter Theory edited by F. Bassani, F. Fumi and M. P. Tosi Course XC (1983) Physics of Amphiphiles: Micelles, Vesicles and Microemulsions edited by V. Degiorgio and M. Corti Course XCI (1984) From Nuclei to Stars edited by A. Molinari and R. A. Ricci Course XCII (1984) Elementary Particles edited by N. Cabibbo Course XCIII (1984) Frontiers in Physical Acoustics edited by D. Sette Course XCIV (1984) Theory of Reliability edited by A. Serra and R. E. Barlow

Course XCV (1985) Solar-Terrestrial Relationship and the Earth Environment in the Last Millennia edited by G. Cini Castagnoli Course XCVI (1985) Excited-State Spectroscopy in Solids edited by U. M. Grassano and N. Terzi Course XCVII (1985) Molecular-Dynamics Simulations of Statistical-Mechanical Systems edited by G. Ciccotti and W. G. Hoover Course XCVIII (1985) The Evolution of Small Bodies in the Solar System ˇ Kresa `k edited by M. Fulchignoni and L. Course XCIX (1986) Synergetics and Dynamic Instabilities edited by G. Caglioti and H. Haken Course C (1986) The Physics of NMR Spectroscopy in Biology and Medicine edited by B. Maraviglia Course CI (1986) Evolution of Interstellar Dust and Related Topics edited by A. Bonetti and J. M. Greenberg Course CII (1986) Accelerated Life Testing and Experts’ Opinions in Reliability edited by C. A. Clarotti and D. V. Lindley Course CIII (1987) Trends in Nuclear Physics edited by P. Kienle, R. A. Ricci and A. Rubbino Course CIV (1987) Frontiers and Borderlines in Many-Particle Physics edited by R. A. Broglia and J. R. Schrieffer Course CV (1987) Confrontation between Theories and Observations in Cosmology: Present Status and Future Programmes edited by J. Audouze and F. Melchiorri Course CVI (1988) Current Trends in the Physics of Materials edited by G. F. Chiarotti, F. Fumi and M. Tosi

Course CVII (1988) The Chemical Physics of Atomic and Molecular Clusters edited by G. Scoles

Course CXIX (1991) Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CVIII (1988) Photoemission and Absorption Spectroscopy of Solids and Interfaces with Synchrotron Radiation edited by M. Campagna and R. Rosei

Course CXX (1992) Frontiers in Laser Spectroscopy ¨nsch and M. Inguscio edited by T. W. Ha

Course CIX (1988) Nonlinear Topics in Ocean Physics edited by A. R. Osborne Course CX (1989) Metrology at the Frontiers of Physics and Technology edited by L. Crovini and T. J. Quinn Course CXI (1989) Solid-State Astrophysics edited by E. Bussoletti and G. Strazzulla Course CXII (1989) Nuclear Collisions from the Mean-Field into the Fragmentation Regime edited by C. Detraz and P. Kienle

Course CXXI (1992) Perspectives in Many-Particle Physics edited by R. A. Broglia, J. R. Schrieffer and P. F. Bortignon Course CXXII (1992) Galaxy Formation edited by J. Silk and N. Vittorio Course CXXIII (1992) Nuclear Magnetic Double Resonsonance edited by B. Maraviglia Course CXXIV (1993) Diagnostic Tools in Atmospheric Physics edited by G. Fiocco and G. Visconti Course CXXV (1993) Positron Spectroscopy of Solids edited by A. Dupasquier and A. P. Mills jr.

Course CXIII (1989) High-Pressure Equation of State: Theory and Applications edited by S. Eliezer and R. A. Ricci

Course CXXVI (1993) Nonlinear Optical Materials: Principles and Applications edited by V. Degiorgio and C. Flytzanis

Course CXIV (1990) Industrial and Technological Applications of Neutrons edited by M. Fontana and F. Rustichelli

Course CXXVII (1994) Quantum Groups and their Applications in Physics edited by L. Castellani and J. Wess

Course CXV (1990) The Use of EOS for Studies of Atmospheric Physics edited by J. C. Gille and G. Visconti

Course CXXVIII (1994) Biomedical Applications of Synchrotron Radiation edited by E. Burattini and A. Balerna

Course CXVI (1990) Status and Perspectives of Nuclear Energy: Fission and Fusion edited by R. A. Ricci, C. Salvetti and E. Sindoni

Course CXXIX1 (1994) Observation, Prediction and Simulation of Phase Transitions in Complex Fluids edited by M. Baus, L. F. Rull and J. P. Ryckaert

Course CXVII (1991) Semiconductor Superlattices and Interfaces edited by A. Stella

Course CXXX (1995) Selected Topics in Nonperturbative QCD edited by A. Di Giacomo and D. Diakonov

Course CXVIII (1991) Laser Manipulation of Atoms and Ions edited by E. Arimondo, W. D. Phillips and F. Strumia 1 This

Course CXXXI (1995) Coherent and Collective Interactions of Particles and Radiation Beams edited by A. Aspect, W. Barletta and R. Bonifacio

course belongs to the NATO ASI Series C, Vol. 460 (Kluwer Academic Publishers).

Course CXXXII (1995) Dark Matter in the Universe edited by S. Bonometto and J. Primack Course CXXXIII (1996) Past and Present Variability of the Solar-Terrestrial System: Measurement, Data Analysis and Theoretical Models edited by G. Cini Castagnoli and A. Provenzale Course CXXXIV (1996) The Physics of Complex Systems edited by F. Mallamace and H. E. Stanley Course CXXXV (1996) The Physics of Diamond edited by A. Paoletti and A. Tucciarone Course CXXXVI (1997) Models and Phenomenology for Conventional and High-Temperature Superconductivity edited by G. Iadonisi, J. R. Schrieffer and M. L. Chiofalo Course CXXXVII (1997) Heavy Flavour Physics: a Probe of Nature’s Grand Design edited by I. Bigi and L. Moroni Course CXXXVIII (1997) Unfolding the Matter of Nuclei edited by A. Molinari and R. A. Ricci Course CXXXIX (1998) Magnetic Resonance and Brain Function: Approaches from Physics edited by B. Maraviglia Course CXL (1998) Bose-Einstein Condensation in Atomic Gases edited by M. Inguscio, S. Stringari and C. E. Wieman Course CXLI (1998) Silicon-Based Microphotonics: from Basics to Applications edited by O. Bisi, S. U. Campisano, L. Pavesi and F. Priolo Course CXLII (1999) Plasmas in the Universe edited by B. Coppi, A. Ferrari and E. Sindoni Course CXLIII (1999) New Directions in Quantum Chaos edited by G. Casati, I. Guarneri and U. Smilansky

Course CXLIV (2000) Nanometer Scale Science and Technology edited by M. Allegrini, N. Garc´ıa and O. Marti Course CXLV (2000) Protein Folding, Evolution and Design edited by R. A. Broglia, E. I. Shakhnovich and G. Tiana Course CXLVI (2000) Recent Advances in Metrology and Fundamental Constants edited by T. J. Quinn, S. Leschiutta and P. Tavella Course CXLVII (2001) High Pressure Phenomena edited by R. J. Hemley, G. L. Chiarotti, M. Bernasconi and L. Ulivi Course CXLVIII (2001) Experimental Quantum Computation and Information edited by F. De Martini and C. Monroe Course CXLIX (2001) Organic Nanostructures: Science and Applications edited by V. M. Agranovich and G. C. La Rocca Course CL (2002) Electron and Photon Conﬁnement in Semiconductor Nanostructures ´dran, edited by B. Deveaud-Ple A. Quattropani and P. Schwendimann Course CLI (2002) Quantum Phenomena in Mesoscopic Systems edited by B. Altshuler, A. Tagliacozzo and V. Tognetti Course CLII (2002) Neutrino Physics edited by E. Bellotti, Y. Declais and P. Strolin Course CLIII (2002) From Nuclei and their Constituents to Stars edited by A. Molinari, L. Riccati, W. M. Alberico and M. Morando Course CLIV (2003) Physics Methods in Archaeometry edited by M. Martini, M. Milazzo and M. Piacentini Course CLV (2003) The Physics of Complex Systems (New Advances and Perspectives) edited by F. Mallamace and H. E. Stanley

Course CLVI (2003) Research on Physics Education edited by E.F. Redish and M. Vicentini

Course CLXV (2006) Protein Folding and Drug Design edited by R. A. Broglia, L. Serrano and G. Tiana

Course CLVII (2003) The Electron Liquid Model in Condensed Matter Physics edited by G. F. Giuliani and G. Vignale

Course CLXVI (2006) Metrology and Fundamental Constants ¨nsch, S. Leschiutta, edited by T. W. Ha A. J. Wallard and M. L. Rastello

Course CLVIII (2004) Hadron Physics edited by T. Bressani, U. Wiedner and A. Filippi Course CLIX (2004) Background Microwave Radiation and Intracluster Cosmology edited by F. Melchiorri and Y. Rephaeli Course CLX (2004) From Nanostructures to Nanosensing Applications edited by A. D’Amico, G. Balestrino and A. Paoletti Course CLXI (2005) Polarons in Bulk Materials and Systems with Reduced Dimensionality edited by G. Iadonisi, J. Ranninger and G. De Filippis Course CLXII (2005) Quantum Computers, Algorithms and Chaos edited by G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti

Course CLXVII (2007) Strangeness and Spin in Fundamental Physics edited by M. Anselmino, T. Bressani, A. Feliciello and Ph. G. Ratcliffe Course CLXVIII (2007) Atom Optics and Space Physics edited by E. Arimondo, W. Ertmer, W. P. Schleich and E. M. Rasel Course CLXIX (2007) Nuclear Structure far from Stability: New Physics and New Technology edited by A. Covello, F. Iachello, R. A. Ricci and G. Maino Course CLXX (2008) Measurements of Neutrino Mass edited by F. Ferroni, F. Vissani and C. Brofferio Course CLXXI (2008) Quantum Coherence in Solid State Physics ´dran, edited by B. Deveaud-Ple A. Quattropani and P. Schwendimann

Course CLXIII (2005) CP Violation: From Quarks to Leptons edited by M. Giorgi, I. Mannelli, A. I. Sanda, F. Costantini and M. S. Sozzi

Course CLXXIV (2009) Physics with Many Positrons edited by R. S. Brusa, A. Dupasquier and A. P. Mills jr.

Course CLXIV (2006) Ultra-Cold Fermi Gases edited by M. Inguscio, W. Ketterle and C. Salomon

Course CLXXV (2009) Radiation and Particle Detectors edited by S. Bertolucci, U. Bottigli and P. Oliva

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