{"kind":"Article","sha256":"625528518f772cf5ab7b014cffe442889de4ffd34c6bbe1ef86936966817098f","slug":"intro","location":"/intro.md","dependencies":[],"frontmatter":{"title":"Introduction","numbering":{"heading_1":{"enabled":true},"heading_2":{"enabled":true}},"authors":[{"nameParsed":{"literal":"Victor Gondret","given":"Victor","family":"Gondret"},"name":"Victor Gondret","orcid":"0009-0005-8468-161X","email":"victor.gondret@normalesup.org","affiliations":["Université Paris-Saclay, CNRS"],"url":"http://www.normalesup.org/~gondret/","id":"contributors-myst-generated-uid-0","corresponding":true}],"license":{"content":{"id":"CC-BY-NC-SA-4.0","name":"Creative Commons Attribution Non Commercial Share Alike 4.0 International","CC":true,"url":"https://creativecommons.org/licenses/by-nc-sa/4.0/"}},"github":"https://github.com/QuantumVictor","keywords":[],"affiliations":[{"id":"Université Paris-Saclay, CNRS","name":"Université Paris-Saclay, CNRS"}],"abbreviations":{"MOT":"Magneto-Optical Trap","BEC":"Bose-Einstein Condensate","MCP":"Micro-Channel Plate","DCE":"Dynamical Casimir Effect","HBT":"Hanbury-Brown and Twiss","CFD":"Constant Fraction Discriminator","TDC":"Time-to-Digital Converter","FPGA":"Field Programmable Gate Array","AOM":"Acousto-Optics Modulator","RF":"Radio-frequency","ODT":"Optical Dipole Trap","IGBT":"Insulated-Gap Bipolar Transistor","MPQ":"Max Planck Institute of Quantum Optics","PPT":"Positive Partial Transpose","SSR":"SuperSelection Rule","LN":"Logarithmic Negativity","UV":"UltraViolet","TOF":"Time-Of-Flight","TF":"Thomas-Fermi","CMB":"Cosmic Background Radiation"},"settings":{"myst_to_tex":{"codeStyle":"minted"}},"thumbnail":"/~gondret/phd_manuscript/build/guth_beamline2-8d353e40fde2d9fd15171dea4c42db74.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/guth_beamline2-8d353e40fde2d9fd15171dea4c42db74.webp","exports":[{"format":"md","filename":"intro.md","url":"/~gondret/phd_manuscript/build/intro-471acf806e3d6251ac2796916357ac38.md"}]},"mdast":{"type":"root","children":[{"type":"block","children":[{"type":"comment","value":" \n## Resume\n Lorsqu'un champs est excité périodiquement, certains modes sont excités, En particulier, lorsque l'état initial est le vide, les particules créées dans des modes d'impulsions intriquées. \n","key":"mHVjsse1rt"},{"type":"comment","value":" \n\n## Résumé\n### Résumé de la thèse\n\nCe mémoire de thèse traite de la non-séparabilité de paires de quasi-particules excitées par résonance paramétrique. Le dispositif expérimental utilisé pendant cette thèse permet de produire un condensat de Bose-Einstein d'hélium métastable. L'utilisation d'un gaz d'atomes ultra-froid permet d'atteindre des températures suffisamment basses afin de pouvoir observer des phénomènes intrinsèquement quantiques : la non-séparabilité de l'état. Dans ce travail, nous utilisons le condensat comme un réservoir cohérent permettant de peupler deux modes d'impulsions. L'avantage de l'hélium métastable est sa grande énergie interne, qui permet la détection électronique de particules uniques. Nous mesurons donc la position et le temps d'impact des particules après un temps de vol de 308 ms, ce qui permet de reconstruire la distribution en impulsion dans le piège. Dans la première contribution théorique de ce travail, nous démontrons que la mesure des fonctions de corrélation à 2- et 4-corps permet de quantifier la non-séparabilité d'un état gaussien. Nous dérivons également un critère permettant d'attester la séparabilité de l'état via la seule mesure la fonction de corrélation à deux corps. Dans la partie expérimentale, nous améliorons la machine permettant de produire notre gaz ultra-froid, ainsi que sa stabilité. Par ailleurs, nous mettons en œuvre des techniques originales afin de dévier une partie des atomes et éviter la saturation de notre détecteur. Ces améliorations nous permettent ainsi de mettre en évidence la non-séparabilité de l'état.\n### Résumé vulgarisé\nLorsqu'on excite périodiquement un fluide unidimensionnel avec une fréquence f, on observe la formation d'un motif dont la fréquence spatiale est de f/2c, où c est la vitesse du son dans le fluide. Ce phénomène a été décrit en 1831 et est appelé instabilité de Faraday. Nous réalisons cette expérience avec un superfluide : un condensat de Bose-Einstein d'hélium métastable. Nous décrivons l'onde de matière de nombre d'onde k comme étant constituée de deux phonons d'impulsion k et -k. Lorsque la température initiale du système est suffisamment basse, il a été prédit que les modes d'impulsion k et -k sont intriqués, et que l'état est dit non-séparable. Dans ce manuscrit de thèse, nous démontrons comment quantifier la non-séparabilité de l'état à l'aide des fonctions de corrélation. Nous décrivons ensuite le dispositif expérimental et un détecteur original qui nous permet de mesurer des atomes individuels. Enfin, nous détaillons le protocole qui nous permet d'observer la non-séparabilité de l'état.\n\n## Resume \n### General sumary\nThis thesis focuses on the non-separability of pairs of quasi-particles excited by parametric resonance. The experimental setup used here allows the production of a Bose-Einstein condensate of metastable helium. The use of an ultra-cold atomic gas makes it possible to reach sufficiently low temperatures to observe intrinsically quantum phenomena: the non-separability of the state. In this work, we use the condensate as a coherent reservoir to populate two momentum modes. The advantage of metastable helium is its high internal energy, which allows the electronic detection of single particles. We therefore measure the position and the time of impact of the particles after a time of flight of 308 ms, which allows us to reconstruct the in-trap momentum distribution. In the first theoretical contribution of this work, we demonstrate that measuring the two- and four-body correlation functions not only attests to, but also quantifies the non-separability of a Gaussian state. We also derive a new entanglement witness using only the local correlation function. In the experimental part, we improve the machine used to produce our ultra-cold gas and enhance its stability. We implement original techniques to deflect part of the atoms and avoid the saturation of our detector. These improvements allow us to highlight the non-separability of the state.\n\n### Vulgarisation \nWhen a one-dimensional fluid is periodically excited with a frequency f, a pattern forms with a spatial frequency of f/2c, where c is the speed of sound in the fluid. This phenomenon was described in 1831 and is called the Faraday instability. We perform this experiment with a superfluid: a Bose-Einstein condensate of metastable helium. We describe the matter wave with wavenumber k as consisting of two phonons with momenta k and -k. When the initial temperature of the system is sufficiently low, it has been predicted that the momentum modes k and -k are entangled, and the state is said to be non-separable. In this thesis manuscript, we demonstrate how to quantify the non-separability of the state using correlation functions. We then describe the experimental setup and an original detector that allows us to measure individual atoms. Finally, we detail the protocol that enables us to observe the non-separability of the state. ","key":"T1xXTL9jCj"},{"type":"heading","depth":2,"position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"children":[{"type":"text","value":"Context","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"ktBkMEiqZH"}],"identifier":"context_section","label":"context_section","html_id":"context-section","enumerator":"1","key":"aG5xjfLP1A"},{"type":"paragraph","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"children":[{"type":"text","value":"This thesis lies at the intersection of various fields of physics, drawing analogies between the Dynamical Casimir effect and the preheating scenario in the early Universe. In particular, it explores the well-known Faraday wave effect with a particular focus on quantum aspects. It belongs to the so-called field of quantum field theory in curved space-time (QFTCST), also known as ","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"key":"lag1w7IHuL"},{"type":"emphasis","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"children":[{"type":"text","value":"analog gravity","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"key":"ui5eOIKOMj"}],"key":"YQj8OgC952"},{"type":"text","value":". This field aims to reproduce on table-top experiments systems that behave like unaccessible observables. It also intends to use theoretical tools developed in general relativity or quantum field theory to model condensed matter systems. It was pioneered by a seminal paper by ","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"key":"cY4DXaW35H"},{"type":"cite","identifier":"unruh_experimental_1981","label":"unruh_experimental_1981","kind":"narrative","position":{"start":{"line":33,"column":678},"end":{"line":33,"column":702}},"children":[{"type":"text","value":"Unruh (1981)","key":"m2NRS5E6g2"}],"enumerator":"1","key":"BnqOtjwQrI"},{"type":"text","value":", titled “Experimental Black-Hole Evaporation”. Central to his idea is to observe (analog) Hawking radiations. In such experiments, the horizon of the analog black holes is defined by the interface at which a liquid flow changes from subsonic to supersonic. The photons of the black holes are replaced by the phonons, which cannot go upstream when the water flow is supersonic, as a photon cannot escape once crossed the black hole horizon. 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(b) Evolution of the radius of the (currently) observed Universe, reproduced from ","position":{"start":{"line":42,"column":1},"end":{"line":42,"column":1}},"key":"GA5eZ0nIze"},{"type":"cite","identifier":"guth_1997_beamline","label":"guth_1997_beamline","kind":"narrative","position":{"start":{"line":42,"column":147},"end":{"line":42,"column":166}},"children":[{"type":"text","value":"Guth (1997)","key":"Cf2YmXoeH0"}],"enumerator":"11","key":"WbHb2dtsHr"},{"type":"text","value":". Without inflation, two opposite regions in the sky are causally separated. (c-d) Two toy models for the decay of the inflaton field. Initially, the field is in the a false vacuum state and later decays into a true vacuum state, in which it starts to oscillates, creating pairs of particles in a product of two-mode squeezed states. ©Figures (a-b) from ","position":{"start":{"line":42,"column":1},"end":{"line":42,"column":1}},"key":"ERtn1IGbD2"},{"type":"cite","identifier":"guth_1997_beamline","label":"guth_1997_beamline","kind":"narrative","position":{"start":{"line":42,"column":520},"end":{"line":42,"column":539}},"children":[{"type":"text","value":"Guth (1997)","key":"WIg9R2F9D0"}],"enumerator":"11","key":"FJv5djZJ1K"},{"type":"text","value":".","position":{"start":{"line":42,"column":1},"end":{"line":42,"column":1}},"key":"wXzVrICSAe"}],"key":"EidQUHJNq8"}],"key":"NBHBYihw9h"}],"enumerator":"1","html_id":"guth-beamline","key":"QEefsyknee"},{"type":"comment","value":"The concept of *inflation* was triggered by the need to solve several issues: the so-called \"horizon\" and \"flatness\" problems.","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"key":"IM3PHjxug9"},{"type":"paragraph","position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"children":[{"type":"text","value":"Standard cosmology greatly succeeded in predicting the existence of the Cosmic Background Radiation (","key":"OuXjcDBevJ"},{"type":"abbreviation","title":"Cosmic Background Radiation","children":[{"type":"text","value":"CMB","key":"y1eDbyNIap"}],"key":"rAKyycJHq7"},{"type":"text","value":") and the universe expansion (Hubbles law) ","key":"zho4k8Rkpy"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"children":[{"type":"cite","identifier":"watson_exposition_2000","label":"watson_exposition_2000","kind":"parenthetical","position":{"start":{"line":47,"column":149},"end":{"line":47,"column":172}},"children":[{"type":"text","value":"Watson, 2000","key":"qWIOEqnMBr"}],"enumerator":"12","key":"o6AA9cwYeM"}],"key":"Rwcboaa8vm"},{"type":"text","value":". Before a specific time called decoupling (see ","position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"key":"SeMpqQpBku"},{"type":"crossReference","position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"children":[{"type":"text","value":"Figure ","key":"TCcmWSipDE"},{"type":"text","value":"1","key":"eMbyzwLgC1"}],"identifier":"guth_beamline","label":"guth_beamline","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"guth-beamline","key":"XyuxUf8122"},{"type":"text","value":"(a)), the universe was a hot, dense plasma of photons, electrons, and protons, where light was continuously scattered by free electrons. At decoupling, about 380,000 years after the Big Bang, the universe cooled enough for electrons and protons to form neutral hydrogen atoms, allowing photons to travel freely through space. This moment created what is called the “last scattering surface”, which is a snapshot of the universe at decoupling. The light from this process is now observed as the ","key":"KZqMPbkBei"},{"type":"abbreviation","title":"Cosmic Background Radiation","children":[{"type":"text","value":"CMB","key":"BlxXxJr1i7"}],"key":"kK7f4Tv4sq"},{"type":"text","value":" which is the resource used to compare cosmological models. One of the long-standing problems in cosmology, known as the horizon problem, arises from the observation that the ","key":"kjH4HvBdcw"},{"type":"abbreviation","title":"Cosmic Background Radiation","children":[{"type":"text","value":"CMB","key":"Axyp88gysq"}],"key":"RgnqfTYbCF"},{"type":"text","value":" is remarkably isotropic and homogeneous across vast distances. 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Such period of parametric creation of particles is known as ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"cHrktqmHqC"},{"type":"emphasis","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"children":[{"type":"text","value":"preheating","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"LRiI8Fo5qN"}],"key":"wMpHel7xfv"},{"type":"text","value":".","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"YgzuQo7ZfE"},{"type":"break","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"E300NvxYgs"},{"type":"text","value":"While the number of created particles increases, backreaction with the inflaton field cannot be neglected and the oscillations of the inflaton decrease. 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I think that their conclusion summarizes well the goal of QFTCST which is not to exactly reproduce inflaton physics but rather to demonstrate “universality and robustness of theoretical models tackling the thermalization in the Early Universe and its distinct stages”.","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"sAli138eWK"},{"type":"break","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"MtpPSB84lJ"},{"type":"text","value":"After inflation the Universe is empty: it is vacuum fluctuations that trigger the exponential growth of bosonic modes ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"epoyH07HkJ"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"children":[{"type":"cite","identifier":"parentani_inflationary_2003","label":"parentani_inflationary_2003","kind":"parenthetical","position":{"start":{"line":53,"column":1201},"end":{"line":53,"column":1229}},"children":[{"type":"text","value":"Parentani, 2003","key":"wc8EhkoXG4"}],"enumerator":"20","key":"iIMM559lbG"}],"key":"sLpQr5aguX"},{"type":"text","value":". 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On the right, the histogram shows the number of detected atoms as a function of time: the central peak corresponds to the ","key":"Mkx7mnnUWC"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"IFoRhInjon"}],"key":"qvWZR4TdhN"},{"type":"text","value":" and the two side-bands to the phonon peaks.","key":"nFjgBZAsYm"}],"key":"yzVEgwlo82"},{"type":"heading","depth":2,"position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"children":[{"type":"text","value":"Content of this manuscript","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"key":"K0YnjlhfCE"}],"identifier":"content_manuscript","label":"content_manuscript","html_id":"content-manuscript","enumerator":"2","key":"O5BM8wMSM5"},{"type":"paragraph","position":{"start":{"line":96,"column":1},"end":{"line":96,"column":1}},"children":[{"type":"text","value":"The first chapter of this thesis is a literature review of the quasi-particle production process. 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We introduce a Gaussian Ansatz to model the gas transverse profile that we use throughout this manuscript. The second section introduces collective oscillations, with a special focus on the ","key":"mVsoSA88X6"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"GuWdWKe8Ly"}],"key":"xdK0ysd9t8"},{"type":"text","value":" breathing mode. In particular, we study the response of the ","key":"Ch0epbPfqR"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"rosVXeC7FT"}],"key":"A01AEtQzOb"},{"type":"text","value":" transverse radius to an arbitrary time-dependent potential and propose a protocol to better control its oscillation. 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When the 4-body correlation measurement is not possible or too noisy, we also provide a lower bound on the 2-body correlation function to assess entanglement. We finally discuss the range of applicability of this criterion and its experimental implementation.","position":{"start":{"line":99,"column":1},"end":{"line":99,"column":1}},"key":"rlJK9xmQFH"}],"key":"tvsLs8TY9w"},{"type":"paragraph","position":{"start":{"line":101,"column":1},"end":{"line":101,"column":1}},"children":[{"type":"text","value":"The third chapter introduces the ","key":"jll67rSXww"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"LYlviXACud"}],"key":"F5qU1g3diJ"},{"type":"text","value":" machine on which I have been working. 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