{"kind":"Article","sha256":"d72a9fb714f7afda3392decb74ddeb16326e90c538ceb329c24fde8c6a34d3bb","slug":"conclusion","location":"/conclusion.md","dependencies":[],"frontmatter":{"title":"Conclusion","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/correlations_high_or-f35575ebc5539312d29bc4b0e179b925.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/correlations_high_or-f35575ebc5539312d29bc4b0e179b925.webp","exports":[{"format":"md","filename":"conclusion.md","url":"/~gondret/phd_manuscript/build/conclusion-2a46b09b7ed2dd8ededaf91d08e05a23.md"}]},"mdast":{"type":"root","children":[{"type":"block","children":[{"type":"paragraph","position":{"start":{"line":10,"column":1},"end":{"line":10,"column":1}},"children":[{"type":"text","value":"This thesis reports on the observation of the production and entanglement of quasi-particles pairs in a time-modulated Bose-Einstein condensate (","key":"MY0rHE1ott"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"rEjkVfZevR"}],"key":"HSfZHDKERx"},{"type":"text","value":"). To summarize the content of the manuscript without repeating the introduction, we take here the historical perspective of this PhD.","key":"pyOiOnhIAL"}],"key":"AWfoiIiCBs"},{"type":"heading","depth":2,"position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"children":[{"type":"text","value":"Conclusion","position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"key":"t4QleRoyEs"}],"identifier":"conclusion","label":"Conclusion","html_id":"conclusion","implicit":true,"enumerator":"1","key":"f3haU6zmWe"},{"type":"paragraph","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"children":[{"type":"text","value":"The first year of my PhD was dedicated to experimental work focused on repairing the He","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"key":"zRBhaQOKAw"},{"type":"superscript","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"children":[{"type":"text","value":"⋆","key":"JfBBUYIpFd"}],"key":"yDyMk582rX"},{"type":"text","value":" ","key":"a0Td2QXONp"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"gyGZe3znwN"}],"key":"aRkzM92u7k"},{"type":"text","value":" apparatus. After a year of modifications, we obtained our first ","key":"MdUMNPAlwj"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"A3nD6nvHrY"}],"key":"ncH8BVIqxb"},{"type":"text","value":" in a crossed dipole trap in March 2022. Just a few weeks later, we produced our first pairs of atomic (entangled?) particles using a blue-detuned lattice, a technique well-established within the team ","key":"q20rG6e3aq"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"children":[{"type":"cite","identifier":"bonneau_tunable_2013","label":"bonneau_tunable_2013","kind":"parenthetical","position":{"start":{"line":14,"column":370},"end":{"line":14,"column":391}},"children":[{"type":"text","value":"Bonneau ","key":"bayd3gBTFQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"l7yy0DYERU"}],"key":"DHDFZP7tOo"},{"type":"text","value":", 2013","key":"avt2ztdx4X"}],"enumerator":"1","key":"duMWCtrmf1"}],"key":"kv6tmsy24E"},{"type":"text","value":". Although this did not constitute new scientific results, it marked a team milestone: for the first time in four years, we successfully produced pairs over a long acquisition time. However, the extended period of machine downtime, along with numerous technological upgrades (including changes to the detection process), interrupted the transmission of knowledge between PhD students. This meant that we could not directly benefit from prior expertise in data analysis and existing efficient code. Following the good practices introduced by Python enthusiasts Alexandre and Quentin, we developed ","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"key":"yxhLWg8cSN"},{"type":"emphasis","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"children":[{"type":"text","value":"heliumtools","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"key":"KHzAdSJPZA"}],"key":"fWZmLXsuCW"},{"type":"text","value":", a collaborative Python module that is object-oriented, efficient, and user-friendly.","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"key":"k7RMenqf4E"}],"key":"JZ6WeqdhDL"},{"type":"comment","value":"The first year of my PhD thesis has been dedicated to experimental work to repair the He$^\\star$ BEC machine. After a year of changes, we obtain our first BEC in a crossed dipole trap in March 2022. A few weeks later, we produced our first (entangled?) atomic pairs of atoms using a blue detuned lattice, which techniques was well-established in the team [@bonneau_tunable_2013]. Even no new science was done, it was a first team victory: it was for the first time since 4 years that we were able to produce pairs for a long acquisition time. However, due to such long time of defectueuse machine and the many technology changes, including the detection process, the transmission chain between PhD students broke. We could not use the past knowledge on the data analysis and the efficient code developed. Building upon the good method \"impulsées\" by the Pythonistas Alexandre Dareau and Quentin Marolleau, we tried our best to code in an objected oriented, efficient and user-friendly way a collaborative Python module *heliumtools*.","position":{"start":{"line":17,"column":1},"end":{"line":17,"column":1}},"key":"zew6eTR2Jf"},{"type":"paragraph","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"The first experiment resembling the Dynamical Casimir Effect (","key":"Uat56Hj8EV"},{"type":"abbreviation","title":"Dynamical Casimir Effect","children":[{"type":"text","value":"DCE","key":"DcO7VNDAuI"}],"key":"OKCUzDgK2O"},{"type":"text","value":") was conducted in June 2022, and I presented these first experimental results at the ","key":"KHyL5b7rvK"},{"type":"emphasis","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"Optique Nice","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"EVlWwgtUB9"}],"key":"C3WNwZlcrC"},{"type":"text","value":" conference. At that time, the stability of the ","key":"KC4FMjaXew"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"uLDKe92y3a"}],"key":"Foi7ZZJoQj"},{"type":"text","value":" arrival time was not yet sufficient to observe clear opposite-momentum correlations. Our progress in both data analysis and the fast cooling of He","key":"icYjrsmVWY"},{"type":"superscript","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"⋆","key":"MLt3PckXUk"}],"key":"ir7neAmoEj"},{"type":"text","value":" to degeneracy were stopped in the end of 2022. We encountered new technical issues among which a vacuum leak that took five months to fully resolve. As “les emmerdes, ça vole toujours en escadrille”, the dipole trap laser failed soon after (though this was quickly fixed). Once these issues were resolved, we focused on the Bragg interferometer and shaping the pulses, exploring a variety of configurations, which are detailed in the manuscript of Charlie ","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"XDOn6NDQ4K"},{"type":"cite","identifier":"leprince_phase_2024","label":"leprince_phase_2024","kind":"narrative","position":{"start":{"line":20,"column":829},"end":{"line":20,"column":849}},"children":[{"type":"text","value":"Leprince (2024)","key":"Hwj2n7gq2l"}],"enumerator":"2","key":"StYhEGvkQB"},{"type":"text","value":". A publication has been submitted to present these results ","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"J5IJznE1Zu"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"cite","identifier":"leprince_2024_coherent","label":"leprince_2024_coherent","kind":"parenthetical","position":{"start":{"line":20,"column":910},"end":{"line":20,"column":933}},"children":[{"type":"text","value":"Leprince ","key":"lJ30hlgmQi"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"GgUSwPSJZG"}],"key":"eS4gOxY3pH"},{"type":"text","value":", 2024","key":"gAy3WUC1iK"}],"enumerator":"3","key":"uuCtfFDZnN"}],"key":"cgQRQgAolK"},{"type":"text","value":". In May 2023, we gathered two weeks of ","key":"uZTkDaTYkN"},{"type":"abbreviation","title":"Dynamical Casimir Effect","children":[{"type":"text","value":"DCE","key":"Q0DVTDd2nm"}],"key":"P9UwwGFa4w"},{"type":"text","value":" data. Although the correlation signal did not reach the expected value, this period marked the successful measurement of the Bogoliubov dispersion relation. These results were presented at the ","key":"sy2a6Bb6xu"},{"type":"emphasis","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"Analog Gravity","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"BXxQcPoUc8"}],"key":"LJ7EdHqNOv"},{"type":"text","value":" summer school and conference that I attended in Benasque, Spain.","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"zUlsFaEZp4"}],"key":"cVwSypzp7i"},{"type":"comment","value":"The first Dynamical Casimir Effect-like (DCE) experiment was carried in June 2022 and the experimental results were presented at a conference *Optique Nice*. At that time, the stability of the BEC arrival time was not sufficient to observe nice opposite momentum correlations. We however faced new experimental issues in the end of 2022: the water-cooling was defecteux and replaced [^note_matos_plomberie_standard], and we faced a vacuum leaks, whose consequences last 4 months. As \"les problèmes, c'est comme les emmerdes, ça vole toujours en escadrille\", the dipole trap laser broke down right after (but this was quickly fixed). Once those issues were fixed, we worked on the Bragg interferometer and the shaping of the pulses. We explored many configurations, which are described in the manuscript of Charlie @leprince_phase_2024. In May 2023, we took two weeks of DCE data. If the correlation signal did not reach the expected value, it is at that time that we measured the Bogoliubov dispersion relation. This series of experiments were also motivated for the *Analog Gravity* summer school and conference that I attended in Benasque, Spain.","position":{"start":{"line":22,"column":1},"end":{"line":22,"column":1}},"key":"PfquSx3TWU"},{"type":"paragraph","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"children":[{"type":"text","value":"After this conference, we shifted our focus and began a series of Hong-Ou-Mandel atomic interferometry experiments in the summer of 2023, varying the input state properties of the interferometer and adjusting the pulse shapes for the mirror and beam-splitter. This successful series led to our first Bell inequality test in October 2023. Unfortunately, this attempt did not yield the expected oscillations in the Bell parameter. Returning briefly to the ","key":"JGaowjHHOS"},{"type":"abbreviation","title":"Dynamical Casimir Effect","children":[{"type":"text","value":"DCE","key":"gg47sefGsj"}],"key":"GPoIi9KXli"},{"type":"text","value":" project, we observed relative number squeezing in November 2023. However, the position and size of the pinholes that filter out unwanted modes needed to be well-adjusted in order to observe this sub-shot-noise signal. Following a relatively minor series of breakdowns and upgrades in the winter of 2024, we dedicated a full month to ","key":"HYpLpbiHgA"},{"type":"abbreviation","title":"Dynamical Casimir Effect","children":[{"type":"text","value":"DCE","key":"GiY9a5YiTC"}],"key":"goigmdvffM"},{"type":"text","value":"-like experiments in the spring of 2024. We observed a reproducible and clear violation of the classical Cauchy-Schwarz inequality and relative number squeezing which led us to start to write an article.","key":"JHP5wnfyYn"}],"key":"K3F5Zyb82D"},{"type":"paragraph","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"We therefore ask ourselves: can relative number squeezing and/or violation of the classical Cauchy-Schwarz inequality enable us to draw conclusions about entanglement? This question, along with the preparation of this manuscript, led me to explore these correlation witnesses in greater detail, as well as the distinction between particle entanglement and mode entanglement. Violation of the classical Cauchy-Schwarz inequality has been shown to serve as a ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"AFfCfK9bRF"},{"type":"emphasis","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"particle","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"d7SnromQzl"}],"key":"U5x0cfIUAE"},{"type":"text","value":" entanglement witness, while in this work, we focus on uncovering ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"YYEvrfl3UU"},{"type":"emphasis","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"mode","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"Qpx33FkPHG"}],"key":"pRTCP3N6Fb"},{"type":"text","value":" entanglement.","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"MWwBXEHwzM"}],"key":"Ojhj7hJRK7"},{"type":"paragraph","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"children":[{"type":"text","value":"Turning to mode entanglement, I focused on the hypothesis ","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"hftgLPtYy7"},{"type":"inlineMath","value":"\\braket{\\hat{a}_k\\hat{a}_{-k}^\\dagger} = 0","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mo stretchy=\"false\">⟨</mo><mrow><msub><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mi>k</mi></msub><msubsup><mover accent=\"true\"><mi>a</mi><mo>^</mo></mover><mrow><mo>−</mo><mi>k</mi></mrow><mo>†</mo></msubsup></mrow><mo stretchy=\"false\">⟩</mo></mpadded><mo>=</mo><mn>0</mn></mrow><annotation encoding=\"application/x-tex\">\\braket{\\hat{a}_k\\hat{a}_{-k}^\\dagger} = 0</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.3267em;vertical-align:-0.3596em;\"></span><span class=\"minner\"><span class=\"mopen\">⟨</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6944em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"mord mathnormal\">a</span></span><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"></span><span class=\"accent-body\" style=\"left:-0.25em;\"><span class=\"mord\">^</span></span></span></span></span></span></span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.967em;\"><span style=\"top:-2.3987em;margin-left:0em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span><span style=\"top:-3.1809em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">†</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3596em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">0</span></span></span></span>","key":"PHzfgIIWtE"},{"type":"text","value":". One possible way to test this assumption is by implementing an atomic interferometer that mixes the ","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"ieEyWotAO5"},{"type":"inlineMath","value":"k","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi></mrow><annotation encoding=\"application/x-tex\">k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span></span></span></span>","key":"fpBtmwsgyr"},{"type":"text","value":" and ","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"k15DeyPFWj"},{"type":"inlineMath","value":"-k","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo>−</mo><mi>k</mi></mrow><annotation encoding=\"application/x-tex\">-k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7778em;vertical-align:-0.0833em;\"></span><span class=\"mord\">−</span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span></span></span></span>","key":"D1lsXpMyod"},{"type":"text","value":" modes. The output of the interferometer oscillates with an amplitude given by the modulus of this coherence term. However, this approach would require adjusting the Bragg beams angle, which is not straightforward due to optical access constraints. Furthermore, the team’s primary objective remains the realization of a (second) Bell test, for which the beams angle must remain unchanged. While drafting the “entanglement” chapter of this manuscript, I delved into Gaussian state formalism - first introduced to me during the 2023 Benasque summer school. This study led first to the derivation of the ","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"b3DbU42KHr"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>g</mi><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></mrow></msup></mrow><annotation encoding=\"application/x-tex\">g^{(2)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0824em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.888em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">2</span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span></span></span></span>","key":"fUBB7Z4ViF"},{"type":"text","value":" bound to assess entanglement, and then to the ","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"eoGCGekwob"},{"type":"inlineMath","value":"g^{(2)}/g^{(4)}","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>g</mi><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></mrow></msup><mi mathvariant=\"normal\">/</mi><msup><mi>g</mi><mrow><mo stretchy=\"false\">(</mo><mn>4</mn><mo stretchy=\"false\">)</mo></mrow></msup></mrow><annotation encoding=\"application/x-tex\">g^{(2)}/g^{(4)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.138em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.888em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">2</span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span><span class=\"mord\">/</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.888em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">4</span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span></span></span></span>","key":"Kv1TWMEyfd"},{"type":"text","value":" criterion, which are now at the core of the second chapter of this work. A publication is in preparation on this result.","position":{"start":{"line":31,"column":1},"end":{"line":31,"column":1}},"key":"Sh9557KrMI"}],"key":"kgMEwVuFOI"},{"type":"paragraph","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"children":[{"type":"text","value":"In the data we collected, the violation of the Cauchy-Schwarz inequality was permitted by a decrease of the local correlation function, rather than an increase above 2 of the cross-correlation function. After completing the initial chapters of this manuscript, I returned to the experiment. We began by investigating the growth process, which highlighted a strong alignment with theoretical predictions. Furthermore, the pronounced oscillation in atom number during exponential growth indicated that mapping the phonon basis to the atom basis relying only on the natural transverse expansion was insufficient. Consequently, we implemented the adiabatic opening of the trap, which, combined with meticulous experimental preparation to ensure stability, allowed us to reveal mode entanglement in October 2024. A publication is in preparation on these results.","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"oLMzbYYatj"}],"key":"CTEjlsULyj"},{"type":"heading","depth":2,"position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"children":[{"type":"text","value":"Outlooks","position":{"start":{"line":47,"column":1},"end":{"line":47,"column":1}},"key":"lMROY1ZIjo"}],"identifier":"outlooks","label":"Outlooks","html_id":"outlooks","implicit":true,"enumerator":"2","key":"KifrHcURbm"},{"type":"comment","value":" ### Bell inequality test ","key":"KWwSZn5Rj2"},{"type":"paragraph","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"children":[{"type":"text","value":"Our proof for non-separability lies on the assumption that the state is Gaussian. If this is not the case, Wick theorem does not apply, and we are unable to relate correlation functions to mode entanglement. Some critics might argue that the Gaussian assumption is too strong, as non-locality with massive particles entangled in external degrees of freedom has yet to be demonstrated. A natural outlook of this work is therefore to wonder if this pair creation process could lead to an experimental violation of Bell inequalities.","position":{"start":{"line":50,"column":1},"end":{"line":50,"column":1}},"key":"onjqtPLBLt"}],"key":"QePQJWYr5m"},{"type":"paragraph","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"children":[{"type":"text","value":"Such experiment ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"RcCHQx2Fvt"},{"type":"emphasis","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"children":[{"type":"text","value":"à la Aspect","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"BaHsH2u1j7"}],"key":"XD7XQnZEBl"},{"type":"text","value":" requires the use of ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"OlQjQJMla5"},{"type":"inlineMath","value":"2\\times 2","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">2\\times 2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">2</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">×</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6444em;\"></span><span class=\"mord\">2</span></span></span></span>","key":"SKvwRITC0G"},{"type":"text","value":" modes ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"C1L1rAvHcR"},{"type":"emphasis","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"XmXNIjq4XC"}],"key":"uRbVeXg03q"},{"type":"text","value":" a product of two-mode squeezed state ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"dYrFI6eGB0"},{"type":"inlineMath","value":"\\ket{\\psi_{k,-k}}\\otimes\\ket{\\psi_{k',-k'}}","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mpadded><mi mathvariant=\"normal\">∣</mi><msub><mi>ψ</mi><mrow><mi>k</mi><mo separator=\"true\">,</mo><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy=\"false\">⟩</mo></mpadded><mo>⊗</mo><mpadded><mi mathvariant=\"normal\">∣</mi><msub><mi>ψ</mi><mrow><msup><mi>k</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup><mo separator=\"true\">,</mo><mo>−</mo><msup><mi>k</mi><mo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′</mo></msup></mrow></msub><mo stretchy=\"false\">⟩</mo></mpadded></mrow><annotation encoding=\"application/x-tex\">\\ket{\\psi_{k,-k}}\\otimes\\ket{\\psi_{k&#x27;,-k&#x27;}}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"mpunct mtight\">,</span><span class=\"mord mtight\">−</span><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span><span class=\"mbin\">⊗</span><span class=\"mspace\" style=\"margin-right:0.2222em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1.0361em;vertical-align:-0.2861em;\"></span><span class=\"minner\"><span class=\"mord\">∣</span><span class=\"mord\"><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ψ</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6828em;\"><span style=\"top:-2.786em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span><span class=\"mpunct mtight\">,</span><span class=\"mord mtight\">−</span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.6828em;\"><span style=\"top:-2.786em;margin-right:0.0714em;\"><span class=\"pstrut\" style=\"height:2.5em;\"></span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">′</span></span></span></span></span></span></span></span></span></span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.2861em;\"><span></span></span></span></span></span></span></span><span class=\"mclose\">⟩</span></span></span></span></span>","key":"CC39ow4Eoo"},{"type":"text","value":". With photons, this experiment was realized by ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"onQ2eGPSIF"},{"type":"cite","identifier":"rarity_1990_bell","label":"rarity_1990_bell","kind":"narrative","position":{"start":{"line":52,"column":206},"end":{"line":52,"column":223}},"children":[{"type":"text","value":"Rarity & Tapster (1990)","key":"WwwUMiFrKS"}],"enumerator":"4","key":"WqP17cIADl"},{"type":"text","value":". With atoms, such violation was not yet evidenced even though two experiments gave preliminary promising results ","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"iRTRNf56Zk"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"children":[{"type":"cite","identifier":"dussarrat_two_particle_2017","label":"dussarrat_two_particle_2017","kind":"parenthetical","position":{"start":{"line":52,"column":338},"end":{"line":52,"column":366}},"children":[{"type":"text","value":"Dussarrat ","key":"HjfvRUkz67"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"owaDrzCmTR"}],"key":"jUL8xLC0PG"},{"type":"text","value":", 2017","key":"pLgziNICWc"}],"enumerator":"5","key":"v8kCpxoYrs"},{"type":"cite","identifier":"thomas_matter_wave_2022","label":"thomas_matter_wave_2022","kind":"parenthetical","position":{"start":{"line":52,"column":367},"end":{"line":52,"column":391}},"children":[{"type":"text","value":"Thomas ","key":"pjr3lfh1wm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"AlOjm1zkt0"}],"key":"REIeMzWxT2"},{"type":"text","value":", 2022","key":"ZJyGlmrUHV"}],"enumerator":"6","key":"MKh50dNqPf"}],"key":"YNmVSxOAqy"},{"type":"text","value":". In order to use the source we described in this manuscript for such experiment, we need to improve two important elements.","position":{"start":{"line":52,"column":1},"end":{"line":52,"column":1}},"key":"vwjyhpBWyw"}],"key":"V663HZ7mbH"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":53,"column":1},"end":{"line":55,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"children":[{"type":"text","value":"The interferometer described in the previous references requires a product of two-mode squeezed states. With our pair creation process, this could be  ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"Iyi7eZjmKf"},{"type":"emphasis","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"children":[{"type":"text","value":"in principle","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"SkWnGfsRXF"}],"key":"Xrmld6e9Cj"},{"type":"text","value":"  possible by modulating the trap power with two frequencies ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"xBql38DelL"},{"type":"inlineMath","value":"\\omega_1","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ω</mi><mn>1</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\omega_1</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"MBvBMc7cGq"},{"type":"text","value":" and ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"s3MOVQmSGF"},{"type":"inlineMath","value":"\\omega_2","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ω</mi><mn>2</mn></msub></mrow><annotation encoding=\"application/x-tex\">\\omega_2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"pTV5RjcOjT"},{"type":"text","value":". This would result in the parametric excitation of two modes ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"kb0PRxlwGx"},{"type":"inlineMath","value":"(k_1, -k_1)","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo separator=\"true\">,</mo><mo>−</mo><msub><mi>k</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(k_1, -k_1)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">−</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>","key":"tawOSF9ypL"},{"type":"text","value":" and ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"yIsQ6qv4Ai"},{"type":"inlineMath","value":"(k_2, -k_2)","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mo stretchy=\"false\">(</mo><msub><mi>k</mi><mn>2</mn></msub><mo separator=\"true\">,</mo><mo>−</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(k_2, -k_2)</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" 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class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span></span></span></span>","key":"NHCGClKKTz"},{"type":"text","value":" such that ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"nz4o8pdszH"},{"type":"inlineMath","value":"\\omega(k_1) = \\omega_1/2","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>ω</mi><mo stretchy=\"false\">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi>ω</mi><mn>1</mn></msub><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\omega(k_1) = \\omega_1/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">1</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">/2</span></span></span></span>","key":"AOkwQM0OKz"},{"type":"text","value":" and ","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"Vtpqo29Jkd"},{"type":"inlineMath","value":"\\omega(k_2) = \\omega_2/2","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>ω</mi><mo stretchy=\"false\">(</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy=\"false\">)</mo><mo>=</mo><msub><mi>ω</mi><mn>2</mn></msub><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">\\omega(k_2) = \\omega_2/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"mopen\">(</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0315em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mclose\">)</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3011em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">2</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">/2</span></span></span></span>","key":"sCHCkfm6LI"},{"type":"text","value":". However, given the difficulties reported in chapter 5 to excite a well-controlled Bogoliubov mode, this option seems unreasonable.","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"ZXxL38RZAo"}],"key":"JYuZpSX6yE"},{"type":"listItem","spread":true,"position":{"start":{"line":54,"column":1},"end":{"line":55,"column":1}},"children":[{"type":"text","value":"The Bell inequality derived with this specific Mach-Zenhder interferometer involves 4-modes but only two-particles. It means that the probability to have 4 particles in the interferometer must be really low: the mean population of the two-mode squeezed states must be lower than 0.14 atoms per mode. In this work, the initial thermal population is 0.6 ","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"key":"lsqvPo6VQj"},{"type":"emphasis","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"key":"eM5tW5vbSw"}],"key":"qXOJ3dMeE9"},{"type":"text","value":" even before squeezing, the state population is too large. The momenta of the two-mode squeezed states must therefore be much larger than ","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"key":"HJJ3uUR7FM"},{"type":"inlineMath","value":"\\xi^{-1}","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>ξ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding=\"application/x-tex\">\\xi^{-1}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.04601em;\">ξ</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.8141em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">−</span><span class=\"mord mtight\">1</span></span></span></span></span></span></span></span></span></span></span></span>","key":"lknkOdNZfH"},{"type":"text","value":" so that the initial thermal seed is negligible.","position":{"start":{"line":54,"column":1},"end":{"line":54,"column":1}},"key":"bXHA7xcCIM"}],"key":"LRheaJKgFO"}],"key":"xoKQSLnryO"},{"type":"paragraph","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"children":[{"type":"text","value":"One possibility to overcome these problems is to take advantage of the already installed blue-detuned lattice to parametrically excite Bogoliubov modes ","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"TcSgJP4ndW"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"children":[{"type":"cite","identifier":"kramer_parametric_2005","label":"kramer_parametric_2005","kind":"parenthetical","position":{"start":{"line":56,"column":154},"end":{"line":56,"column":177}},"children":[{"type":"text","value":"Krämer ","key":"zdlq4mF6Vc"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"FMhyuzb2OO"}],"key":"e26x6bflBl"},{"type":"text","value":", 2005","key":"sh5NqsVE1i"}],"enumerator":"7","key":"iycyIgxqKg"},{"type":"cite","identifier":"lellouch_2017_parametric","label":"lellouch_2017_parametric","kind":"parenthetical","position":{"start":{"line":56,"column":178},"end":{"line":56,"column":203}},"children":[{"type":"text","value":"Lellouch ","key":"bHKp6sEXwf"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"xRg8p0Ah6H"}],"key":"HKpkLT2tIG"},{"type":"text","value":", 2017","key":"xLgepC2qLi"}],"enumerator":"8","key":"FH2UBdPEAG"}],"key":"BLOZCAEJiH"},{"type":"text","value":".","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"P0FKXFnKgs"}],"key":"cpYkiXFJlL"},{"type":"comment","value":"However, this controlled parametric creation still needs to be tested experimentally.","position":{"start":{"line":57,"column":1},"end":{"line":57,"column":1}},"key":"mLVHU18Cww"},{"type":"comment","value":"Such an excitation process could address both issues mentioned previously.","position":{"start":{"line":59,"column":1},"end":{"line":59,"column":1}},"key":"DfF3QgASx2"},{"type":"paragraph","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"Future work will also investigate deviations from Bogoliubov-de Gennes theory. Specifically, when driving the system further out of equilibrium, we observe correlations between Bogoliubov modes. A typical example is shown in ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"bm5lkwil7R"},{"type":"crossReference","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"Figure ","key":"YxrnpllzAE"},{"type":"text","value":"1","key":"cCZnyVjenJ"}],"identifier":"bogohighorder","label":"bogohighorder","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"bogohighorder","key":"TgY1yCQAF5"},{"type":"text","value":". In the density profile, a strong peak appears at ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"bEPlBSFzru"},{"type":"inlineMath","value":"k = \\pm 9","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi><mo>=</mo><mo>±</mo><mn>9</mn></mrow><annotation encoding=\"application/x-tex\">k = \\pm 9</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">±</span><span class=\"mord\">9</span></span></span></span>","key":"Avjmxy8HqE"},{"type":"text","value":" mm/s and a smaller one around ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"ogYk1NsedG"},{"type":"inlineMath","value":"2k = \\pm 18","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mi>k</mi><mo>=</mo><mo>±</mo><mn>18</mn></mrow><annotation encoding=\"application/x-tex\">2k = \\pm 18</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">=</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.7278em;vertical-align:-0.0833em;\"></span><span class=\"mord\">±</span><span class=\"mord\">18</span></span></span></span>","key":"smo2hhuR0d"},{"type":"text","value":" mm/s. Such secondary resonance is expected in periodically driven systems at frequency ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"YZldqymDUQ"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msub><mi>ω</mi><mi>d</mi></msub></mrow><annotation encoding=\"application/x-tex\">\\omega_d</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span></span></span></span>","key":"f2CWKa3Vne"},{"type":"text","value":", where Floquet analysis reveals a series of resonances ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"TOIES4cA9q"},{"type":"inlineMath","value":"n\\omega_d/2","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>n</mi><msub><mi>ω</mi><mi>d</mi></msub><mi mathvariant=\"normal\">/</mi><mn>2</mn></mrow><annotation encoding=\"application/x-tex\">n\\omega_d/2</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\"></span><span class=\"mord mathnormal\">n</span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">ω</span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.3361em;\"><span style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">d</span></span></span></span><span class=\"vlist-s\">​</span></span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.15em;\"><span></span></span></span></span></span></span><span class=\"mord\">/2</span></span></span></span>","key":"HYdZWx0ZbZ"},{"type":"text","value":" for ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"YjnwQ3bpuG"},{"type":"inlineMath","value":"n \\in \\mathbb{R}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant=\"double-struck\">R</mi></mrow><annotation encoding=\"application/x-tex\">n \\in \\mathbb{R}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\"></span><span class=\"mord mathnormal\">n</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span><span class=\"mrel\">∈</span><span class=\"mspace\" style=\"margin-right:0.2778em;\"></span></span><span class=\"base\"><span class=\"strut\" style=\"height:0.6889em;\"></span><span class=\"mord mathbb\">R</span></span></span></span>","key":"W9xj9gXSII"},{"type":"text","value":". Microscopically, assuming a linear dispersion relation, this excitation process is explained by the annihilation of two excitations with momentum ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"zek1YBE4Er"},{"type":"inlineMath","value":"k","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mi>k</mi></mrow><annotation encoding=\"application/x-tex\">k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span></span></span></span>","key":"UmtTwFEf1A"},{"type":"text","value":" and the creation of excitations with momentum ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"LHMpybmirv"},{"type":"inlineMath","value":"2k","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding=\"application/x-tex\">2k</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.6944em;\"></span><span class=\"mord\">2</span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span></span></span></span>","key":"jMwdHVt8FE"},{"type":"text","value":". The presence of such excitations has already been reported in the literature ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"KajfoT5joC"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"cite","identifier":"nguyen_parametric_2019","label":"nguyen_parametric_2019","kind":"parenthetical","position":{"start":{"line":62,"column":822},"end":{"line":62,"column":845}},"children":[{"type":"text","value":"Nguyen ","key":"dOgAmjpfO8"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ekIOwlXQTD"}],"key":"jRSOxMfTUx"},{"type":"text","value":", 2019","key":"eQhsCLXOoS"}],"enumerator":"9","key":"gStyvxi9PO"},{"type":"cite","identifier":"hernandez_rajkov_faraday_2021","label":"hernandez_rajkov_faraday_2021","kind":"parenthetical","position":{"start":{"line":62,"column":846},"end":{"line":62,"column":877}},"children":[{"type":"text","value":"Hernández-Rajkov ","key":"LQdw56xJDc"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"E083HYgUvZ"}],"key":"vpoMHh2yu4"},{"type":"text","value":", 2021","key":"A12BLlGdz6"}],"enumerator":"10","key":"G2pUmqrO5e"}],"key":"QGaLYv0TW6"},{"type":"text","value":"; however, it would be valuable to track the growth of correlations between different modes. Such correlations naturally raise the question of whether the state is multimode entangled and how to reveal it.","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"aA4WP70mrU"}],"key":"j7eofo8nbz"},{"type":"container","kind":"figure","identifier":"bogohighorder","label":"bogohighorder","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/correlations_high_or-f35575ebc5539312d29bc4b0e179b925.png","alt":"beyond Bogoliubov","width":"70%","align":"center","key":"FPVlB0bFJ5","urlSource":"images/correlations_high_order.png","urlOptimized":"/~gondret/phd_manuscript/build/correlations_high_or-f35575ebc5539312d29bc4b0e179b925.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":73,"column":1},"end":{"line":73,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"bogohighorder","identifier":"bogohighorder","html_id":"bogohighorder","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"RLmLLotUnc"},{"type":"text","value":"1","key":"imnof65SXd"},{"type":"text","value":":","key":"UrIT1bfONd"}],"template":"Figure %s:","key":"aaCNDgE9Ae"},{"type":"inlineMath","value":"g^{(2)}","position":{"start":{"line":73,"column":1},"end":{"line":73,"column":1}},"html":"<span class=\"katex\"><span class=\"katex-mathml\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mrow><msup><mi>g</mi><mrow><mo stretchy=\"false\">(</mo><mn>2</mn><mo stretchy=\"false\">)</mo></mrow></msup></mrow><annotation encoding=\"application/x-tex\">g^{(2)}</annotation></semantics></math></span><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.0824em;vertical-align:-0.1944em;\"></span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right:0.03588em;\">g</span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.888em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"></span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mopen mtight\">(</span><span class=\"mord mtight\">2</span><span class=\"mclose mtight\">)</span></span></span></span></span></span></span></span></span></span></span></span>","key":"esaINQ4oRY"},{"type":"text","value":" correlation map after long time and strong excitation process.","position":{"start":{"line":73,"column":1},"end":{"line":73,"column":1}},"key":"ra8XOi3oOg"}],"key":"QhPTdJoUkO"}],"key":"ProKcGpE3E"}],"enumerator":"1","html_id":"bogohighorder","key":"YaBUElfQWs"},{"type":"comment","value":"Still assuming Gaussianity of the state, we can in principle assess wether or not the state","position":{"start":{"line":77,"column":1},"end":{"line":77,"column":1}},"key":"IpNaSKyKFZ"},{"type":"comment","value":" \nIn the quest of this observation, our journey began with theoretical considerations. In the first chapter, alongside characterizing the BEC, we reviewed the recent contributions of Robertson *et. al.* in modeling the dynamics of the quasi-particles pair production. These theoretical advances provided a deeper understanding of the entanglement dynamics and decoherence processes, which were essential to fine-tune the experimental parameters of the excitation process.\n\nThe second step of this manuscript was to investigate how to probe entanglement with a single particle detector. Within Bogoliubov-de Gennes formalism, the Hamiltonian is of second order in creation and annihilation operators which means that the state is Gaussian. Indeed, simulations by @robertson_nonlinearities_2018 showed that in our range of parameters, Gaussianity of the state is preserve. It is only for longer excitation time - or for stronger excitations - that deviation to Bogoliubov theory are expected. In the second chapter, we introduced Gaussian state and its formalism. Central to our investigation was the question: how to certify entanglement of Gaussian state with correlation function ? To assess entanglement, the Wick expanded second order correlation function is often used:\n```{math}\n:label: g2conclu\ng^{(2)}_{k,-k} = \\frac{n_kn_{-k}+ |\\braket{\\hat{a}_k\\hat{a}_{-k}}|^2 + |\\braket{\\hat{a}_k\\hat{a}_{-k}^\\dagger}|^2}{n_kn_{-k}}.\n```\nViolation of $n_kn_{-k}\\geq|\\braket{\\hat{a}_k\\hat{a}_{-k}}|$ implies entanglement which means that, *assuming* the last term in Eq. [](#g2conclu) is null, one can relate an observation of $g^{(2)}> 2$ to entanglement. In this chapter, we demonstrate that this last term of Eq. [](#g2conclu) can in fact be measured with the 4-body correlation function. If the latter cannot be measured with enough precision, we also provide a bound above which the value $g^{(2)}$ assesses entanglement. When the population is bigger than $1/\\sqrt{2}$, this bound is equal to 2. This result however only applies for Gaussian thermal states. A publication is in preparation on this result.\n\n\nThe experiment described in this manuscript uses a metastable helium BEC machine. Metastable helium offers the advantage of being detectable with a micro-channel plate which, coupled to delay lines, allows us to record both the impact time and position of individual atoms. When measured after a time-of-flight, it gives access to the *in situ* momentum distribution at the single-particle level. Given our BEC properties, we are able to measure up to 100 particles per mode. In these chapters, we reported on the consequent improvement of the machine and its stability, allowing to precisely measure opposite momentum correlations. Furthermore, we developed a state-of-the-art pulse-shaping techniques for Bragg diffraction, which we used to realize an efficient and sharp momentum-selective deflector that allowed getting rid of the BEC peak that saturates the detector.\n\n\nIn the last part of this work, we observed the exponential growth of the quasi-particle number. The remarkable agreement with theoretical predictions further validates the accuracy of the description of the first chapter. We showed our ability to measure the difference between the theoretical undamped creation dynamics as a proof of concept for future measurement. The results and their interpretation we presented are still preliminary, but promising. Future works will further investigate this growth as well as the observed shift of the quasi-particle wave-vector. Finally, the last chapter demonstrated non-separability of the $(k,-k)$ state, relying on the theoretical works of the two first chapter and the experimental capabilities described in the two following chapters. \n\n ","key":"u5WBeEQsym"},{"type":"comment","value":"Experimental works often use the second order correlation function $g^{(2)}_{k,-k}$","position":{"start":{"line":101,"column":1},"end":{"line":101,"column":1}},"key":"vgITnXSU78"},{"type":"comment","value":"At short time, *ie.* time for which entanglement can be measured, which is our range of parameters, @robertson_nonlinearities_2018 showed that the","position":{"start":{"line":104,"column":1},"end":{"line":104,"column":1}},"key":"conA5dbxUg"},{"type":"comment","value":"Within Bogoliubov-de Gennes formalism, the Hamiltonian is of second order in creation and annihilation operators which means that the state is Gaussian. Such approximation is motivated by truncated Wigner approximation simulations by @robertson_nonlinearities_2018 which shows that, at short time, Gaussianity of the state is preserved.\nIn fact, the authors also showed that entanglement visibility was lost before non-gaussian effects started to induce decoherence effects.","position":{"start":{"line":108,"column":1},"end":{"line":109,"column":1}},"key":"U4mRfGwXEh"}],"key":"ognLe4cuPM"}],"key":"hOJC2onB0e"},"references":{"cite":{"order":["bonneau_tunable_2013","leprince_phase_2024","leprince_2024_coherent","rarity_1990_bell","dussarrat_two_particle_2017","thomas_matter_wave_2022","kramer_parametric_2005","lellouch_2017_parametric","nguyen_parametric_2019","hernandez_rajkov_faraday_2021"],"data":{"bonneau_tunable_2013":{"label":"bonneau_tunable_2013","enumerator":"1","doi":"10.1103/PhysRevA.87.061603","html":"Bonneau, M., Ruaudel, J., Lopes, R., Jaskula, J.-C., Aspect, A., Boiron, D., & Westbrook, C. I. (2013). Tunable source of correlated atom beams. <i>Physical Review A</i>, <i>87</i>(6), 061603. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1103/PhysRevA.87.061603\">10.1103/PhysRevA.87.061603</a>","url":"https://doi.org/10.1103/PhysRevA.87.061603"},"leprince_phase_2024":{"label":"leprince_phase_2024","enumerator":"2","html":"Leprince, C. (2024). <i>Phase control and pulse shaping in Bragg diffraction for quantum atom optics: From matter-wave interferences to a Bell’s inequality test</i> [Phdthesis, Université Paris-Saclay]. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://theses.fr/s247200\">https://theses.fr/s247200</a>","url":"https://theses.fr/s247200"},"leprince_2024_coherent":{"label":"leprince_2024_coherent","enumerator":"3","html":"Leprince, C., Gondret, V., Lamirault, C., Dias, R., Marolleau, Q., Boiron, D., & Westbrook, C. I. (2024). <i>Coherent coupling of momentum states: selectivity and phase control</i>. Submitted to Phys. Rev. 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E., Río-Lima, A. D., Gutiérrez-Valdés, A., Poveda-Cuevas, F. J., & Seman, J. A. (2021). Faraday waves in strongly interacting superfluids. <i>New Journal of Physics</i>, <i>23</i>(10), 103038. <a target=\"_blank\" rel=\"noreferrer\" href=\"https://doi.org/10.1088/1367-2630/ac2d70\">10.1088/1367-2630/ac2d70</a>","url":"https://doi.org/10.1088/1367-2630/ac2d70"}}}},"footer":{"navigation":{"prev":{"title":"Conclusion on entanglement","url":"/correlations-4entanglement","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Appendix","short_title":"Appendix","url":"/appendix","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}