{"kind":"Article","sha256":"e701fbd14ae2996cab89dfd11f202850f19176a257804facdaa3720d3c0d5314","slug":"correlations-intro","location":"/experiment_entanglement/correlations_intro.md","dependencies":[],"frontmatter":{"title":"Observation of quasi-particles entanglement","short_title":"Observation of quasi-particles entanglement","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"}},"exports":[{"format":"md","filename":"correlations_intro.md","url":"/~gondret/phd_manuscript/build/correlations_intro-914e8be706a00accaa69687a42561754.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":10,"column":1},"end":{"line":10,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"This chapter presents the main result of this work: the observation of non-separability of the ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"nCG42i49jE"},{"type":"inlineMath","value":"(k, -k)","position":{"start":{"line":11,"column":1},"end":{"line":11,"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><mi>k</mi><mo separator=\"true\">,</mo><mo>−</mo><mi>k</mi><mo stretchy=\"false\">)</mo></mrow><annotation encoding=\"application/x-tex\">(k, -k)</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 mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"mpunct\">,</span><span class=\"mspace\" style=\"margin-right:0.1667em;\"></span><span class=\"mord\">−</span><span class=\"mord mathnormal\" style=\"margin-right:0.03148em;\">k</span><span class=\"mclose\">)</span></span></span></span>","key":"vgQ3vDoJ4T"},{"type":"text","value":" quasi-particle modes. We start this section with an analysis of the ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"NIUhUr9WdG"},{"type":"crossReference","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"density","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"TTmFxMUZql"}],"identifier":"analyse_densite","label":"analyse_densite","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"analyse-densite","remote":true,"url":"/correlations-1method","dataUrl":"/correlations-1method.json","key":"f4MICIS5IL"},{"type":"text","value":" of the dataset that we use throughout this chapter. We then review the key aspects necessary to measure the correlation signal: the ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"VNVkqlSVDx"},{"type":"crossReference","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"deflection","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"t51AEHLz3z"}],"identifier":"bragg_diffaction_method","label":"bragg_diffaction_method","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"bragg-diffaction-method","remote":true,"url":"/correlations-1method","dataUrl":"/correlations-1method.json","key":"Towc92cGTQ"},{"type":"text","value":" of the ","key":"HKIA1iIC76"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"jzaDP3Wdfc"}],"key":"WPZC6pSZoJ"},{"type":"text","value":", its ","key":"z5ARo3wTRJ"},{"type":"crossReference","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"stability","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"GDnDqOQ2Ru"}],"identifier":"method_stability_bec","label":"method_stability_bec","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"method-stability-bec","remote":true,"url":"/correlations-1method","dataUrl":"/correlations-1method.json","key":"FIykMROBrl"},{"type":"text","value":" and its influence on the correlation signal, as well as the ","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"hu75xWhnnF"},{"type":"crossReference","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"adiabatic","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"VX9VMRt4OG"}],"identifier":"method_adiabatic_opening","label":"method_adiabatic_opening","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"method-adiabatic-opening","remote":true,"url":"/correlations-1method","dataUrl":"/correlations-1method.json","key":"Eo7YGVnHBJ"},{"type":"text","value":" opening of the trap to better map the phonon basis onto the atomic one.","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"giwyKRFZ2G"}],"key":"ZXxiNWfVsz"},{"type":"paragraph","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"children":[{"type":"text","value":"The second and third sections of this chapter use two different methods to measure the local and cross-correlation signals. The first method computes what we call integrated momentum correlations, by constructing a 3D histogram of atomic pairs between two large regions. In this approach, we lose information about the specific momentum of the modes that contribute to the correlation signal. The next section, however, retains the mode position, allowing for the measurement of a well-defined momentum mode. Both methods yield similar results.","position":{"start":{"line":14,"column":1},"end":{"line":14,"column":1}},"key":"uYMmHnILH4"}],"key":"hkSfiDY4xW"},{"type":"comment","value":"The second and third section of this chapter use two different methods to measure the local and cross-correlation signal. The first method compute what we call integrated momentum correlations. We compute the 3D histogram of the atomic pairs between two large regions. In this sense, we lost the information about the specific omentum of the modes: correlations are integrated over the whole region. The next section keeps track of the position of the mode, and allows for the measurement of a well-defined momentum mode. Both methods give similar results.","position":{"start":{"line":17,"column":1},"end":{"line":17,"column":1}},"key":"uOwtOaMrcn"},{"type":"comment","value":"This chapter presents the main result of this work which is the observation of the non-separability of the $(k,-k)$ quasi-particles. In the first section, we recall the main points that allow us to measure the correlation signal: the [deflection](#bragg_diffaction_method) of the BEC, its [stability](#method_stability_bec) and the influence it has on the correlation signal and the [adiabatic](#method_adiabatic_opening) opening of the trap to better map the phonon basis onto the atomic one. We finish this section analyzing the [density](#analyse_densite) of the dataset that we use throughout this chapter.","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"jnTX3RCEtt"},{"type":"paragraph","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"children":[{"type":"text","value":"In the last section, we further study the statistics of each mode to verify they can be modeled by a thermal Gaussian state. This is important to use the criterion derived in the second chapter. We then measure the population and the 4-body correlation function. We conclude on the observation of entanglement. We also discuss on the value of the relative number squeezing and reconstruct the state taking into account the quantum efficiency of the detector that we estimate around 25(10)%. Finally, we report on the measurement of the correlations ","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"FwqJITuzPN"},{"type":"emphasis","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"children":[{"type":"text","value":"via","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"eHb780b8ig"}],"key":"ATq2vuULYj"},{"type":"text","value":" the Cauchy-Schwarz ratio and relative number squeezing for various excitation durations.","position":{"start":{"line":21,"column":1},"end":{"line":21,"column":1}},"key":"NIFn3GE7Nc"}],"key":"v5lq9eAbQs"},{"type":"comment","value":" \nThis chapter goes describe the measurement of the entanglement of the quasi-particles. In the [first section](#bragg_diffaction_method), we summarize the key points that allowed the measurement of the non-separability. In the second chapter, we showed that for thermal Gaussian state, the measurement of the 2- and 4-body correlation functions allows quantifying entanglement. The [second section](#local_correlation_function) of this chapter studies the *local* correlations to probe the covariance matrix and the displacement of each single mode. We measure the correlation length in [section](#full_counting_stat) [%s](#full_counting_stat). We show that our measurement are indeed compatible with a thermal Gaussian state. \n\n\n\nWe define the non-normalized and normalized second order correlation functions \n```{math}\nG^{(2)}(k, k') :=\\braket{\\hat{a}_k^\\dagger\\hat{a}_{k'}^\\dagger\\hat{a}_k\\hat{a}_{k'}}\\quad \\quad g^{(2)}(k, k') := \\frac{G^{(2)}(k, k')}{n_k n_{k'}}\n``` \nwhere $n(k)=\\braket{\\hat{a}_k^\\dagger\\hat{a}_{k}}$ is the mean population.\n ","key":"gfxfltbGEd"}],"data":{"part":"abstract"},"key":"tsr3sY8ntY"},{"type":"block","position":{"start":{"line":37,"column":1},"end":{"line":37,"column":1}},"children":[{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"What we knew, what is new ?","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"rh7jJqV5tx"}],"key":"vhgCqVnfGD"},{"type":"paragraph","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"children":[{"type":"text","value":"The experiments reported in this chapter were conducted as part of this PhD work and represent original research.","position":{"start":{"line":44,"column":1},"end":{"line":44,"column":1}},"key":"v5jreRcsnS"}],"key":"vgzlxBrVEZ"}],"key":"O9RenYWNsT"}],"key":"Oslx4fPGGI"}],"key":"orT0QkBzyv"},"references":{"cite":{"order":[],"data":{}}},"footer":{"navigation":{"prev":{"title":"Exponential creation of phonons","url":"/cosqua-2exponential-creation","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Experimental method","url":"/correlations-1method","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}