Quantum imaging with a small number of transverse modes Vincent Delaubert To cite this version: Vincent Delaubert. Quantum imaging with a small number of transverse modes. Atomic Physics [physics.atom-ph]. Université Pierre et Marie Curie - Paris VI, 2007. English. tel-00146849 HAL Id: tel-00146849 https://tel.archives-ouvertes.fr/tel-00146849 Submitted on 15 May 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire Kastler Brossel Universit¶ePierre et Marie Curie Th`ese de doctorat de l'Universit¶eParis VI en co-tutelle avec l'Australian National University Sp¶ecialit¶e: Laser et Mati`ere pr¶esent¶eepar Vincent Delaubert Pour obtenir le grade de DOCTEUR DE L'UNIVERSITE¶ PIERRE ET MARIE CURIE sur le sujet : Imagerie quantique `apetit nombre de modes tranvserses Soutenue le 8 mars 2007 devant le jury compos¶ede : M. Claude FABRE . Co-Directeur de th`ese M. Hans BACHOR . Co-Directeur de th`ese M. Claude BOCCARA . Rapporteur M. Joseph BRAAT . Rapporteur M. John CLOSE . Examinateur M. Nicolas TREPS . Invit¶e 2 Acknowledgements The work presented in this thesis has been undertaken both at the Laboratoire Kastler Brossel, in Claude Fabre's group, and at the Australian National University, within the Australian Research Council Center of Excellence for Quantum-Atom Optics (ACQAO), in Hans Bachor's group. I ¯rst want to stress how lucky I have been to have had the opportunity to work within such a close collaboration. This collaboration was already existing at the beginning of my co-tutelle and, I believe, even strengthened during the course of my PhD. Being the hyphen between such two productive groups was a source of great motivation and gave me the opportunity to equally bene¯t from ideas, supervision and support from both parts. Moreover, far from dispersing me, changing work environment has allowed me to be more e±cient and more independent. I particularly appreciated how genuine the relations between the two groups were, which gave me the impression that I was actually always in the same group. Yet in a world of constant competition, transparency, collaboration and friendship would characterize best my research experience. My ¯rst thanks obviously go to Nicolas Treps, Hans Bachor and Claude Fabre, whom I owe everything, and even more. I am ¯rst extremely grateful to Nicolas for his great kindness and patience, especially when I was asking obscure questions, which I often did not understand myself... Moreover, being awaken in the morning by a "freak" having problems with an optical cavity on the other side of the globe became an imposed routine for him, and it never seemed to alter his enthusiasm whatsoever. I thank Hans for his ongoing support, for the trust he has always put in me, and for the wonderful time we had in scienti¯c discussions as well as during outdoor activities. His ability to interact with people and to smoothly solve problems of all kind with a simple e-mail or letter still amazes me. This surely was a major condition to the success of this co-tutelle. I thank Claude Fabre for his remarkable kindness in all circumstances, and for his depth-of-knowledge which I will always admire. I would also like to thank the other members of the jury, Claude Boccara, Joseph Braat and John Close, who seemed to have survived the painful reading of the 200-odd pages of this thesis, and the successive directors of both labs, Frank Lalo¶eand Paul Indelicato, David McClelland and Aidan Byrne for giving me the great opportunity to work within their institutions. During the course of my PhD, having the chance to work within two friendly and productive environments (surprisingly similar although diametrally located), I have met a number of friends for whom I would be ready to cross oceans. From the LKB, special thanks to Juju, hoping that he will come back from the "grand i ii nord" soon, Brahim, for always waiting for me for after-lab metro discussions, Sylvain, for teaching me all his experimental skills in such a cheerful way, Francesco and Martino, for reminding me that even one ear was more than enough to live in Italy, Charles, Sophie, Florence, FX, EOL, Olivier, Ines, Csilla, Alberto, Tristan, PF, for pleasant and relaxing lunch sessions, Beno^³t, GaÄelle, Giuseppe, Laurent, Virginia, Agn`es and Thomas, the group I had the pleasure to work with, and "bon vent" to Jean-Fran»cois! I also would like to warmly thank Monique and Laeticia, for their patience and great understanding with my peculiar skills for administrative tasks, and Annick for dealing so well with messy physicists. From the ANU, special thanks to my PhD "mates" Magnus, for sharing explicit scien- ti¯c lyrics and culinary experiences, Jiri, my best dual optical and soccer table partner, Mikael, my almost quite perfectly danish friend, Mister Gabriel and Vio, far best hotel in town, Julien, for humiliating me in tennis, Andrew, Nicolaij, Christian, Cristina, Nick, Oliver, for good moments and interesting discussions, and special farewell to Kate who has the hard task of des-entangling the spatial entanglement setup. I am particularly grateful to Ping Koy, Warwick and Charles for their friendly and dedicated scienti¯c support. My stay in Canberra would have been very di®erent without the "winning team", Lout, Mar- ion, Modo, Rio, Cl¶elia, Anne-C, Sese and Sab. Thanks also to my flatmates Nic, Brett and Ian, who outperformed by far my last house-sharing experience..., and ¯nally to my great Ozzy friends Liv, Adam, Caroline, Alan, and Jill, with whom I almost felt like not being an "imported product". My last acknowledgements go to all my friends, from here and there, who kept asking for news of the kangaroos, and to my parents and grandparents who always trusted and encouraged me. And most importantly, thank you Anne-Sophie for your unconditional support and love. My only regret during these four years was not to have you on my side while I was in Canberra. Contents 1 Tools for Quantum Imaging 5 A Generalities on optical images ......................... 7 A.1 Image description in the transverse plane ............... 7 A.1.1 Local electric ¯eld operator .................. 7 A.1.2 Local quadrature operators .................. 8 A.1.3 Local number of photons and local intensity ........ 9 A.1.4 Total number of photons and beam power .......... 10 A.2 Transverse modal decomposition of the electromagnetic ¯eld .... 10 A.2.1 Modal creation and annihilation operators ......... 11 A.2.2 Modal quadratures ....................... 12 A.2.3 Number of photons and intensity in a mode ......... 13 A.2.4 Description relative to the mean ¯eld ............ 13 A.2.5 Changing the transverse basis ................ 14 A.2.6 The Hermite-Gauss basis ................... 15 A.3 Gaussian quantum states of light .................... 17 A.3.1 Covariance matrix ....................... 17 A.3.2 Coherent states ........................ 18 A.3.3 Squeezed states ........................ 19 A.3.4 Entangled states ........................ 21 B Single/Multi-mode criterium .......................... 28 B.1 Classical approach ............................ 28 B.2 Quantum approach ............................ 29 B.2.1 Single Mode quantum light .................. 29 B.2.2 Multi-mode quantum light .................. 31 B.3 Towards an experimental criterium ................... 33 C Image detection in the transverse plane .................... 38 C.1 Noise-modes of detection ........................ 38 C.2 "Bucket" detection ............................ 39 C.3 Interference detection .......................... 40 C.4 Homodyne Detection ........................... 41 C.5 Array Detection ............................. 43 C.5.1 Measured signal ........................ 44 C.5.2 Di®erence measurements ................... 45 C.5.3 General linear measurement ................. 50 iii iv Contents C.6 Homodyne array detection ........................ 52 D Conclusion .................................... 52 2 Quantum study of optical storage 55 Overview on optical data storage ........................... 55 Article 1 : Optical storage of high-density information beyond the di®raction limit: a quantum study ............................. 56 Article 2 : A quantum study of multibit phase coding for optical storage ..... 64 3 Optimal information extraction from an optical image 73 A Spatial optical information carried by transverse modes ........... 75 A.1 Displacement and Tilt of Gaussian beams ............... 75 A.1.1 Classical description ...................... 75 A.1.2 Quantum operators : position and momentum ....... 78 A.1.3 Displacement and tilt of other beams ............ 80 A.2 Waist position and size mismatch .................... 81 A.2.1 Waist-size mismatch ...................... 82 A.2.2 Waist-position mismatch ................... 82 A.2.3 General relation for mode-mismatch ............. 82 A.3 Orbital angular momentum ....................... 83 A.3.1 Rotation of a Hermite Gauss beam about its propagation axis ............................... 83 A.3.2 Rotation of a Laguerre Gauss beam about its propagation axis ..............................