{"kind":"Article","sha256":"42aa1d230c1d2ffa1ec926e45c4daa2fbc2b29d78e89b18d314598e24eac24a9","slug":"bec-bragg","location":"/bec/bec_bragg.md","dependencies":[],"frontmatter":{"title":"Bragg diffraction","short_title":"Bragg diffraction","subtitle":"Shaping a Bragg pulse to get rid of saturation effects","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/bragg_setup-6f1e750cdcc99819587d9a04de61f33e.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/bragg_setup-6f1e750cdcc99819587d9a04de61f33e.webp","exports":[{"format":"md","filename":"bec_bragg.md","url":"/~gondret/phd_manuscript/build/bec_bragg-674b222c4db4af494674b2d9dd05a0c0.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"children":[{"type":"text","value":"As we shall see in the next chapter, when an atom enters a channel of the detector, it triggers an electronic cascade. However, it takes a few milliseconds for the channel to re-charge back and be able to detect another atom. In our case, the ","key":"a9LKTAt9DR"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"CocxvT7WUD"}],"key":"OcfruO3A85"},{"type":"text","value":" and the pairs are separated by approximately 1 ms. If the atom of the pair that arrives first is unaffected by the saturation, this is not the case of the one that arrives after the ","key":"HzHdAD4cDy"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"jdlInvy9gn"}],"key":"TOk0Li2BaF"},{"type":"text","value":". The solution would be to remove the ","key":"w6U5DgwLOJ"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"sAsQSJLmQJ"}],"key":"x3d8hLZtsk"},{"type":"text","value":", or to kick it upwards so that it arrives after the pairs. This section introduces Bragg diffraction which will allows us to do so.","key":"WtiooDPyDE"}],"key":"kheG909Fty"},{"type":"comment","value":"us to *deflect* a velocity class of atoms.","position":{"start":{"line":13,"column":1},"end":{"line":13,"column":1}},"key":"aKtaD38Aoi"},{"type":"comment","value":"Note that this section will absolutely not introduce Bragg diffraction in a self-consistent and complete way. I will introduce the key ingredients that are needed to understand the shaping of the light pulses to optimize the BEC deflection. The interested reader will find in the PhD manuscript of Charlie @leprince_phase_2024 much more details about the physics of Bragg diffraction, the experimental setup and the interferometer.","position":{"start":{"line":15,"column":1},"end":{"line":15,"column":1}},"key":"RwitN1Rp7Y"},{"type":"paragraph","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"text","value":"The pulse-shaping techniques reported here were initially developed to realize a 2-particle 4-modes interferometer to perform a Bell test using massive particles entangled in momentum ","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"YY7SsyMqab"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"children":[{"type":"cite","identifier":"leprince_2024_coherent","label":"leprince_2024_coherent","kind":"parenthetical","position":{"start":{"line":16,"column":186},"end":{"line":16,"column":209}},"children":[{"type":"text","value":"Leprince ","key":"GxTjImSf3c"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"hEFqMxZB0M"}],"key":"aY8C0ArDqB"},{"type":"text","value":", 2024","key":"nfqjzpixm3"}],"enumerator":"1","key":"wsDkjrbvcS"}],"key":"rEJ9sP6jNe"},{"type":"text","value":". In this work, we take advantage of this state-of-the-art atomic interferometer to simply “kick-off” the ","key":"i2Bs4S6zC4"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"ghBxQisQ3P"}],"key":"E08dV3d2CM"},{"type":"text","value":" outside the region of interest. I participated in the experimental implementation and characterization of this set-up, but this project was mainly investigated by ","key":"StAtrzRuie"},{"type":"cite","identifier":"leprince_phase_2024","label":"leprince_phase_2024","kind":"narrative","position":{"start":{"line":16,"column":483},"end":{"line":16,"column":503}},"children":[{"type":"text","value":"Leprince (2024)","key":"yBAnPVF7nG"}],"enumerator":"2","key":"LdQudjJsvw"},{"type":"text","value":", to which the interested reader is referred for further details.","position":{"start":{"line":16,"column":1},"end":{"line":16,"column":1}},"key":"Kvfd6pv5Xr"}],"key":"Vb8ImcEgdn"}],"data":{"part":"abstract"},"key":"xXZdvE6ceq"},{"type":"block","position":{"start":{"line":18,"column":1},"end":{"line":18,"column":1}},"children":[{"type":"comment","value":"This section introduces Bragg diffraction that will be used to avoid saturation of the detector.","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"Xl8efycUIK"},{"type":"heading","depth":2,"position":{"start":{"line":22,"column":1},"end":{"line":22,"column":1}},"children":[{"type":"text","value":"Introduction","position":{"start":{"line":22,"column":1},"end":{"line":22,"column":1}},"key":"m5EememCdq"}],"identifier":"bragg_definition","label":"bragg_definition","html_id":"bragg-definition","enumerator":"1","key":"h2VcwG996L"},{"type":"paragraph","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"children":[{"type":"text","value":"Consider an atom illuminated by two lasers 1 and 2, characterized by their electric field ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"qj5F32Dhnm"},{"type":"inlineMath","value":"E_j","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"EjE_jEj​","key":"eItgO8YkRC"},{"type":"text","value":" their frequency ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"ZhDGuvE06r"},{"type":"inlineMath","value":"\\omega_j","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"ωj\\omega_jωj​","key":"fHyr3RLyWl"},{"type":"text","value":", with a relative half-angle that we note ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"sHSNlDJW0e"},{"type":"inlineMath","value":"\\theta_B","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"θB\\theta_BθB​","key":"twI4XVr31F"},{"type":"text","value":". As illustrated in ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"TpnEvgoibk"},{"type":"crossReference","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"children":[{"type":"text","value":"Figure ","key":"OJa0DVXs29"},{"type":"text","value":"1","key":"v5cdAiLkLf"}],"identifier":"bragg_setup","label":"bragg_setup","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"bragg-setup","key":"TNYjR2wCKD"},{"type":"text","value":", the two lasers interfere and create a periodic potential along the vertical direction ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"lx9CfsuMCB"},{"type":"inlineMath","value":"z","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"zzz","key":"ixBW5iDIUz"},{"type":"text","value":". The two lasers are detuned by ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"yEDKcXmpcG"},{"type":"inlineMath","value":"\\delta \\omega_{las} = \\omega_1 - \\omega_2","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"δωlas=ω1−ω2\\delta \\omega_{las} = \\omega_1 - \\omega_2δωlas​=ω1​−ω2​","key":"tp5MO65ItY"},{"type":"text","value":", creating a standing light wave that scatters atoms.","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"UMwRQJmlfU"}],"key":"yQsIBLmx8M"},{"type":"comment","value":"a scheme of the experimental setup: the two lasers (1 yellow and 2 red) create an optical lattice along the $z$ direction.\nFirst experimental realization of Bragg scattering was realized by @martin_bragg_1988.","position":{"start":{"line":24,"column":1},"end":{"line":25,"column":1}},"key":"ucIEaeeluK"},{"type":"paragraph","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"Bragg diffraction can be seen as a two-photon process where the atoms absorb one photon from a laser and re-emits it in a stimulated way into the second one ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"pvXOZPyRvX"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"cite","identifier":"martin_bragg_1988","label":"martin_bragg_1988","kind":"parenthetical","position":{"start":{"line":27,"column":159},"end":{"line":27,"column":177}},"children":[{"type":"text","value":"Martin ","key":"GF4h5BQH8w"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"m51SUN61Vl"}],"key":"sVMvBT9YNJ"},{"type":"text","value":", 1988","key":"OONcE3PBbd"}],"enumerator":"3","key":"RCaXJxuFah"}],"key":"aJdEDky31q"},{"type":"text","value":". This interpretation is easily understood with the scheme on the right panel of ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"caHGoVk4th"},{"type":"crossReference","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"children":[{"type":"text","value":"Figure ","key":"fyMIzeLkJ9"},{"type":"text","value":"1","key":"Rb5EsWOIKR"}],"identifier":"bragg_setup","label":"bragg_setup","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"bragg-setup","key":"Aq0KzpxGmm"},{"type":"text","value":". The atom absorbs a photon from one laser (labeled 1, yellow) and re-emits it in the second laser (labeled 2, red). When the lasers do not co-propagate but have an angle ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"I0MiYw60Dx"},{"type":"inlineMath","value":"2\\theta_B","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"html":"2θB2\\theta_B2θB​","key":"LA1HxzpKbY"},{"type":"text","value":", the atom is kicked by a momentum ","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"d3puQKAcxb"},{"type":"inlineMath","value":"k_b = 2\\frac{2\\pi}{\\lambda}\\sin \\theta_B","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"html":"kb=22πλsin⁡θBk_b = 2\\frac{2\\pi}{\\lambda}\\sin \\theta_Bkb​=2λ2π​sinθB​","key":"PyecCRD3pa"},{"type":"text","value":".","position":{"start":{"line":27,"column":1},"end":{"line":27,"column":1}},"key":"mx9NFjBsLn"}],"key":"CWVcMc1N1q"},{"type":"paragraph","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"text","value":"On the experiment, the Bragg speed was measured to be ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"TAWwI1PZOX"},{"type":"inlineMath","value":"v_b=\\hbar k_b/m","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"vb=ℏkb/mv_b=\\hbar k_b/mvb​=ℏkb​/m","key":"QkJ5tRVRLY"},{"type":"text","value":"=49.58(3) mm/s. The corresponding wavevector is ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"Cdpfhm0vcr"},{"type":"inlineMath","value":"k_b","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"kbk_bkb​","key":"aVQjgR56QT"},{"type":"text","value":"=3.145(2) µm","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"y7BtAwDQ54"},{"type":"superscript","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"children":[{"type":"text","value":"-1","key":"qgHVOzkUDA"}],"key":"WiYJyticsL"},{"type":"text","value":" and the associated recoil frequency is ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"IydC5ZBFOJ"},{"type":"inlineMath","value":"\\omega_b = \\hbar k_b^2/2m","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"html":"ωb=ℏkb2/2m\\omega_b = \\hbar k_b^2/2mωb​=ℏkb2​/2m","key":"NNrM49TzJz"},{"type":"text","value":" is 12.41(3) kHz. The lattice is not perfectly aligned on the vertical direction but experiences a (relatively) small angle along x of 0.08(3)° and 0.04(2)° along y.","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"PEjj3bGXPc"}],"key":"WcwKTaxh0H"},{"type":"container","kind":"figure","identifier":"bragg_setup","label":"bragg_setup","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/bragg_setup-6f1e750cdcc99819587d9a04de61f33e.png","alt":"Setting up the Bragg beams.","width":"75%","align":"center","key":"kSS3UIRyki","urlSource":"images/bragg_setup.png","urlOptimized":"/~gondret/phd_manuscript/build/bragg_setup-6f1e750cdcc99819587d9a04de61f33e.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"bragg_setup","identifier":"bragg_setup","html_id":"bragg-setup","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"gGpm38tsyV"},{"type":"text","value":"1","key":"gbkYWfXD1F"},{"type":"text","value":":","key":"Q7pcl1MMEh"}],"template":"Figure %s:","key":"QBaJQsMlHP"},{"type":"text","value":"A) Experimental scheme of the laser setup. Two lasers with a 2.8 mm waist interfere at the position of the cloud, creating a standing wave whose speed is set by the frequency difference between the two. The lattice interfringe is 2 µm, that is much smaller than the size of the ","key":"CimZUSKWW7"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"ohejdTO4W9"}],"key":"qshYZ5D4mK"},{"type":"text","value":" wave-function which is typically a hundred of µm. B) Illustration of the two-photon process. The two ","key":"MCcF0xs4ps"},{"type":"text","value":"π","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"Qd4YTAppFd"},{"type":"text","value":" polarized lasers are detuned from the ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"ipPjrDTODH"},{"type":"inlineMath","value":"2^3P_0","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"23P02^3P_023P0​","key":"AzuOXWSAzX"},{"type":"text","value":" level by ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"XecBfjhHoV"},{"type":"text","value":"Δ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"e6qQxHzlJ8"},{"type":"text","value":"=800 MHz.","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"ZDRkYKZFyZ"}],"key":"ZKelSIiup4"}],"key":"YhseXkCfMj"}],"enumerator":"1","html_id":"bragg-setup","key":"GaVZB6ezIy"},{"type":"paragraph","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"children":[{"type":"text","value":"Formally, the dipole atom-light interaction is characterized by the Rabi frequency ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"gWKNJDB31o"},{"type":"inlineMath","value":"\\hbar\\Omega_j:=-\\braket{g|\\hat{\\mathbf{d}}\\cdot\\mathbf{E}|e}","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"html":"ℏΩj:=−⟨g∣d^⋅E∣e⟩\\hbar\\Omega_j:=-\\braket{g|\\hat{\\mathbf{d}}\\cdot\\mathbf{E}|e}ℏΩj​:=−⟨g∣d^⋅E∣e⟩","key":"VDoD1hHMpH"},{"type":"text","value":". Here, ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"DOmfxWbhTQ"},{"type":"inlineMath","value":"\\hat{\\mathbf{d}}","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"html":"d^\\hat{\\mathbf{d}}d^","key":"SrIcEmoPa9"},{"type":"text","value":" represents the reduced atomic dipole of the transition ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"RcBGPlaBUd"},{"type":"inlineMath","value":"g\\rightarrow e","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"html":"g→eg\\rightarrow eg→e","key":"A0QfnRcAmS"},{"type":"text","value":" between the ground state ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"x0LoT9rOuB"},{"type":"inlineMath","value":"2^3S_1","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"html":"23S12^3S_123S1​","key":"BRCcPJB3aI"},{"type":"text","value":" and the excited state ","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"TBvG1ev06i"},{"type":"inlineMath","value":"2^3P_0","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"html":"23P02^3P_023P0​","key":"EjvJPpZLHM"},{"type":"text","value":". When the lasers are far detuned from resonance, the population of the excited level can be neglected","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"hS6WwiYwob"},{"type":"footnoteReference","identifier":"adiabatic_elimination","label":"adiabatic_elimination","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"number":1,"enumerator":"1","key":"QUaqIdyH9A"},{"type":"text","value":". The Hamiltonian of the system can then be written using the two-photon Rabi frequency","position":{"start":{"line":40,"column":1},"end":{"line":40,"column":1}},"key":"km6BQmwb4r"}],"key":"LocNHNVZpO"},{"type":"math","identifier":"equation_rabi_freq","label":"equation_rabi_freq","value":"\\Omega_R = \\frac{\\Omega_1\\Omega_2^\\star }{2\\Delta}","tight":true,"html":"ΩR=Ω1Ω2⋆2Δ\\Omega_R = \\frac{\\Omega_1\\Omega_2^\\star }{2\\Delta}ΩR​=2ΔΩ1​Ω2⋆​​","enumerator":"1","html_id":"equation-rabi-freq","key":"vqfHBYY0lk"},{"type":"paragraph","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"children":[{"type":"text","value":"and the Bragg wavevector ","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"key":"GBUqIuHQmt"},{"type":"inlineMath","value":"k_b = 2\\frac{2\\pi}{\\lambda}\\sin \\theta_B","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"html":"kb=22πλsin⁡θBk_b = 2\\frac{2\\pi}{\\lambda}\\sin \\theta_Bkb​=2λ2π​sinθB​","key":"fOYIqkF9Zr"},{"type":"text","value":". As we shall see, the Rabi frequency ","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"key":"fJwahn18q9"},{"type":"inlineMath","value":"\\Omega_R","position":{"start":{"line":45,"column":1},"end":{"line":45,"column":1}},"html":"ΩR\\Omega_RΩR​","key":"xxKXszIaMS"},{"type":"text","value":" plays a central role in determining the different regimes of the light-matter interaction. The laser beam waists are sufficiently large (4 mm) compared to the cloud (200 µm) to assume that the Rabi frequency does not depend on space. 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In practice, if the value of the Rabi frequency, that defines the coupling strength, is low enough, we can restrict the analysis to a two level system ","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"key":"YQwRuaduKf"},{"type":"inlineMath","value":"\\{\\ket{p}, \\ket{p +\\hbar k_b} \\}","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"html":"{∣p⟩,∣p+ℏkb⟩}\\{\\ket{p}, \\ket{p +\\hbar k_b} \\}{∣p⟩,∣p+ℏkb​⟩}","key":"TNxtVMhBcf"},{"type":"text","value":". Such hypothesis requires the Rabi frequency to be much smaller than the recoil frequency. ","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"key":"y5nA4qZdmF"},{"type":"cite","identifier":"leprince_phase_2024","label":"leprince_phase_2024","kind":"narrative","position":{"start":{"line":55,"column":427},"end":{"line":55,"column":447}},"children":[{"type":"text","value":"Leprince (2024)","key":"PDtQvktZHW"}],"enumerator":"2","key":"IbV0Tycuar"},{"type":"text","value":" showed numerically that a factor of 4 is enough to suppress the influence of the other levels. He also discusses other regime for which other levels must be taken into account ","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"key":"zVxng52ckK"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"children":[{"type":"cite","identifier":"beguin_characterisation_2022","label":"beguin_characterisation_2022","kind":"parenthetical","position":{"start":{"line":55,"column":625},"end":{"line":55,"column":654}},"children":[{"type":"text","value":"Béguin ","key":"LD2VpLy2J5"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"am0Iej57Ee"}],"key":"sxoOUaLfac"},{"type":"text","value":", 2022","key":"nlbNSZYb5v"}],"enumerator":"4","key":"vOj2JHFEOK"}],"key":"EhmEtoqSLE"},{"type":"text","value":". In this work, we will use low enough Rabi frequencies (lower than 3 kHz) so that we couple only two different momentum classes.","position":{"start":{"line":55,"column":1},"end":{"line":55,"column":1}},"key":"iZBjo0KLNS"}],"key":"fMswDuh17x"},{"type":"paragraph","position":{"start":{"line":57,"column":1},"end":{"line":57,"column":1}},"children":[{"type":"text","value":"Second, by tuning the frequency difference between the two lasers, one can tune the velocity classes that are resonant with the momentum transfer. 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The right panel of the figure represents the number of transferred atoms as a function of the laser detuning. At fixed laser detuning ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"iG9CTooF2E"},{"type":"inlineMath","value":"\\delta \\omega_{las}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"δωlas\\delta \\omega_{las}δωlas​","key":"LUOzm6Sclz"},{"type":"text","value":", the momentum class that is resonant with the two-photon process is the one for which ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"o8DUWayaPL"},{"type":"inlineMath","value":"p","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"ppp","key":"czhK5LGDJy"},{"type":"text","value":" satisfies equation ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"vNmCSP4vEQ"},{"type":"crossReference","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"(","key":"rrQPg0YTB8"},{"type":"text","value":"3","key":"Kubd3wkDhP"},{"type":"text","value":")","key":"WNGQ5wg9ID"}],"identifier":"detuning_bragg_equation","label":"detuning_bragg_equation","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"detuning-bragg-equation","key":"o8mLR4VFF9"},{"type":"text","value":". The ","key":"ZrNLLDtq2n"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"EXVsbMn5Z8"}],"key":"nU1ZmLopaK"},{"type":"text","value":" initial state is ","key":"KaXCaVg9hY"},{"type":"inlineMath","value":"\\ket{p=0}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"∣p=0⟩\\ket{p=0}∣p=0⟩","key":"wyummenvmG"},{"type":"text","value":": it is therefore resonant with the two-photon process for ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"T1ynjbTzZs"},{"type":"inlineMath","value":"\\delta \\omega_{las} =\\pm \\hbar k_b^2/2m","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"δωlas=±ℏkb2/2m\\delta \\omega_{las} =\\pm \\hbar k_b^2/2mδωlas​=±ℏkb2​/2m","key":"nh2SS0bpfi"},{"type":"text","value":". In the case of a positive detuning, the ","key":"tZLbBQ6oE4"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"D1zsixA8RX"}],"key":"slUX2y3k0r"},{"type":"text","value":" is transferred to the ","key":"PA7qLpjLRm"},{"type":"inlineMath","value":"\\ket{p=+\\hbar k_b}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"∣p=+ℏkb⟩\\ket{p=+\\hbar k_b}∣p=+ℏkb​⟩","key":"rCoy61SYfC"},{"type":"text","value":". If the detuning is negative, the atoms are coherently transferred to the ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"grZTzVw4br"},{"type":"inlineMath","value":"\\ket{-\\hbar k_b}","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"html":"∣−ℏkb⟩\\ket{-\\hbar k_b}∣−ℏkb​⟩","key":"imXucDRVOS"},{"type":"text","value":" state. Still, if the detuning is not exactly the one given by ","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"Kb0PpQ53nY"},{"type":"crossReference","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"children":[{"type":"text","value":"(","key":"CO1mlmnDsR"},{"type":"text","value":"3","key":"G2o5kHWTtR"},{"type":"text","value":")","key":"F5GDSpNxEx"}],"identifier":"detuning_bragg_equation","label":"detuning_bragg_equation","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"detuning-bragg-equation","key":"SDzXpE7OCb"},{"type":"text","value":", a transfer can still happen. To estimate it, we write the detuning as","position":{"start":{"line":62,"column":1},"end":{"line":62,"column":1}},"key":"RQNEygN3gq"}],"key":"jev2BwG9r3"},{"type":"math","identifier":"detuning_delta_equation","label":"detuning_delta_equation","value":"\\delta = \\delta \\omega_{las} - \\frac{k_b}{2m}\\left(\\hbar k_b - 2p\\right)","tight":true,"html":"δ=δωlas−kb2m(ℏkb−2p)\\delta = \\delta \\omega_{las} - \\frac{k_b}{2m}\\left(\\hbar k_b - 2p\\right)δ=δωlas​−2mkb​​(ℏkb​−2p)","enumerator":"4","html_id":"detuning-delta-equation","key":"y4aSrrCejZ"},{"type":"paragraph","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"children":[{"type":"text","value":"When ","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"dbITo2PAnd"},{"type":"inlineMath","value":"\\delta=0","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"html":"δ=0\\delta=0δ=0","key":"F4nEdJsPcU"},{"type":"text","value":", the process is at resonance. However if ","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"YzNS7ePEV0"},{"type":"inlineMath","value":"\\delta\\neq 0","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"html":"δ≠0\\delta\\neq 0δ=0","key":"lMdnZjG0AM"},{"type":"text","value":", the transfer might occur if the Rabi frequency is large enough, that is if ","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"yvjrUjNJmF"},{"type":"inlineMath","value":"\\delta /\\Omega_R <1","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"html":"δ/ΩR<1\\delta /\\Omega_R <1δ/ΩR​<1","key":"xiw2WSYEND"},{"type":"text","value":". Note however that the transfer will be less efficient and oscillate faster. This off-resonance process explains why the resonance curve on ","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"aQM2Yja0BX"},{"type":"crossReference","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"children":[{"type":"text","value":"Figure ","key":"S5bblbxFe9"},{"type":"text","value":"2","key":"Zkfe421fIj"}],"identifier":"detuning_scan_bragg_figure","label":"detuning_scan_bragg_figure","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"detuning-scan-bragg-figure","key":"q8aQiVOJh4"},{"type":"text","value":" is not a Dirac function: its width is given by the Rabi frequency.","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"fdBbZsS3jv"}],"key":"eF0cge5hCa"},{"type":"comment","value":"As we shall see, the natural unit to compare time scale and detuning is the Rabi frequency.","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"key":"hMoznEuwMB"},{"type":"comment","value":"When the detuning of the laser is the equal to the one given by equation [](#detuning_bragg_equation), for $p=-\\hbar k_b$, the momentum class $\\ket{p}$ is transferred coherently to the $\\ket{p-\\hbar k_b}$. For the BEC peak, the laser detuning must be equal to $\\hbar k_b^2/2m$[^note_raman]. If it is not the case, the transfer is less efficient.","position":{"start":{"line":70,"column":1},"end":{"line":70,"column":1}},"key":"liyngFw5eQ"},{"type":"comment","value":"and the two lasers couple states separated by $k_b$.","position":{"start":{"line":72,"column":1},"end":{"line":72,"column":1}},"key":"l8tuAyGu8a"},{"type":"container","kind":"figure","identifier":"detuning_scan_bragg_figure","label":"detuning_scan_bragg_figure","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/bragg_detuning-1655f7680f45711c156e60d424a8ac66.png","alt":"Setting up the Bragg beams.","width":"100%","align":"center","key":"ojWtb8zR09","urlSource":"images/bragg_detuning.png","urlOptimized":"/~gondret/phd_manuscript/build/bragg_detuning-1655f7680f45711c156e60d424a8ac66.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"detuning_scan_bragg_figure","identifier":"detuning_scan_bragg_figure","html_id":"detuning-scan-bragg-figure","enumerator":"2","children":[{"type":"text","value":"Figure ","key":"Gb78LuljCf"},{"type":"text","value":"2","key":"aP97OyZJXn"},{"type":"text","value":":","key":"rChrEUp8D2"}],"template":"Figure %s:","key":"mawo2eKfqT"},{"type":"text","value":"Left: Density of the momentum distribution of the cloud after the Bragg transition. The colorscale represents the atomic density, the darker being the denser. On the first picture, the frequency difference is not set to address the ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"pgP34YBMfL"},{"type":"inlineMath","value":"v_z=0","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"html":"vz=0v_z=0vz​=0","key":"OAVzH830cF"},{"type":"text","value":" ","key":"rM50jZXrvc"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"FcOwaCVcmk"}],"key":"EsEaM6XIcd"},{"type":"text","value":" mode. On the second picture, the detuning is resonant with the ","key":"SGYTIlJ1fd"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"bIdCm3uOOL"}],"key":"t5G1nKd1C0"},{"type":"text","value":" mode and the zero momentum mode has been transferred to ","key":"NUAsUFw94w"},{"type":"inlineMath","value":"-v_b=-50","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"html":"−vb=−50-v_b=-50−vb​=−50","key":"MPwiF05FTb"},{"type":"text","value":" mm/s. Right: number of detected atoms as a function of the laser detuning. Each dot is the number of atoms for a single realization. We clearly observe that the resonance is around -14 kHz. 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When the laser difference frequency is at resonance with one mode ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"miUoVjK2Z8"},{"type":"inlineMath","value":"k","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"kkk","key":"Cfl2zPApQ5"},{"type":"text","value":", atoms coherently transfer between states ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"w3dwZAYR3e"},{"type":"inlineMath","value":"\\ket{k}","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"∣k⟩\\ket{k}∣k⟩","key":"DbNWG5W76b"},{"type":"text","value":" and ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"zlIfzmVEfF"},{"type":"inlineMath","value":"\\ket{k+ k_b}","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"∣k+kb⟩\\ket{k+ k_b}∣k+kb​⟩","key":"kkBlPuLKtY"},{"type":"text","value":" with frequency ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"AO2KcsdGr3"},{"type":"inlineMath","value":"\\Omega_R","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"ΩR\\Omega_RΩR​","key":"V2uQbUwhUp"},{"type":"text","value":". When the momentum class is not perfectly at resonance, that is ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"ezrkW9Og2b"},{"type":"inlineMath","value":"\\delta ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"δ\\delta δ","key":"sFM6wzSLuP"},{"type":"text","value":" defined by equation ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"yTnkrKaL3O"},{"type":"crossReference","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"children":[{"type":"text","value":"(","key":"q0pC4iyuoi"},{"type":"text","value":"4","key":"Ln34xut7fg"},{"type":"text","value":")","key":"LlRWxIsv9M"}],"identifier":"detuning_delta_equation","label":"detuning_delta_equation","kind":"equation","template":"(%s)","enumerator":"4","resolved":true,"html_id":"detuning-delta-equation","key":"jxboefaE5r"},{"type":"text","value":" is not exactly zero, the oscillation frequency between the two classes and the transfer efficiency is lower. The probability ","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"sdnMfp1dhr"},{"type":"inlineMath","value":"\\mathcal{P}","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"html":"P\\mathcal{P}P","key":"T0N8LNMH5q"},{"type":"text","value":" for an atom to be transferred reads","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"key":"ovTe43fb2E"}],"key":"cJ2ijReTOB"},{"type":"math","identifier":"oscillation_frequency_rabi_off_resonance_equation","label":"oscillation_frequency_rabi_off_resonance_equation","value":"\\mathcal{P}(t) = \\frac{\\Omega_R}{\\Omega}\\sin^2(\\Omega t/2), \\quad \\quad\n\\Omega = \\sqrt{\\Omega_R^2+\\delta^2}.","tight":true,"html":"P(t)=ΩRΩsin⁡2(Ωt/2),Ω=ΩR2+δ2.\\mathcal{P}(t) = \\frac{\\Omega_R}{\\Omega}\\sin^2(\\Omega t/2), \\quad \\quad\n\\Omega = \\sqrt{\\Omega_R^2+\\delta^2}.P(t)=ΩΩR​​sin2(Ωt/2),Ω=ΩR2​+δ2​.","enumerator":"5","html_id":"oscillation-frequency-rabi-off-resonance-equation","key":"kJy9EsXGYr"},{"type":"paragraph","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"children":[{"type":"text","value":"This is illustrated by the left panel of ","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"key":"YQ4qtWQ6qa"},{"type":"crossReference","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"children":[{"type":"text","value":"Figure ","key":"p2fHJL5Leu"},{"type":"text","value":"3","key":"A3kPkwaKd1"}],"identifier":"rabi_oscillation_bragg","label":"rabi_oscillation_bragg","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rabi-oscillation-bragg","key":"hC1sbuOtsg"},{"type":"text","value":" in which the coherent oscillation of two momentum classes is represented. 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Experimentally, the oscillation frequency of the resonant class gives access to the Rabi frequency (and therefore the intensity of the beams). 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It is also possible to transfer atoms with a 1/2 probability and therefore to realize a ","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"key":"eTRMTvnkem"},{"type":"emphasis","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"children":[{"type":"text","value":"beam-splitter","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"key":"UxIRnUAvzb"}],"key":"ZCkHl19PfJ"},{"type":"text","value":" with a ","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"key":"yDYPSnhHcV"},{"type":"inlineMath","value":"\\pi/2","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"html":"π/2\\pi/2π/2","key":"czFsocprFN"},{"type":"text","value":" light pulse ","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"key":"jnR3pD217r"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":94,"column":1},"end":{"line":94,"column":1}},"children":[{"type":"cite","identifier":"berman_pump_probe_spectro_1997","label":"berman_pump_probe_spectro_1997","kind":"parenthetical","position":{"start":{"line":94,"column":347},"end":{"line":94,"column":378}},"children":[{"type":"text","value":"Berman & Bian, 1997","key":"PDcmeW5Zcw"}],"enumerator":"5","key":"nD46GezTgX"}],"key":"iifoVTPCaD"},{"type":"text","value":". 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Here the resonant frequency is -14 kHz. Right: ratio of the transferred population between two momentum modes as a function of time. The blue circles represent the oscillation of the resonant ","position":{"start":{"line":101,"column":1},"end":{"line":101,"column":1}},"key":"etwHtz9dEi"},{"type":"inlineMath","value":"k=0","position":{"start":{"line":101,"column":1},"end":{"line":101,"column":1}},"html":"k=0k=0k=0","key":"ednkaDlFOF"},{"type":"text","value":" momentum class while the red squares represent an off resonant velocity class centered at 1.5 mm/s (which represents a 0.8 kHz detuning). The resonant class oscillates at 1.21(2) kHz, which gives the Rabi frequency. 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We observe a damping of the Rabi oscillation for which the typical decay time is 12(5) ms. ®Data taken on the 24/04/23.","position":{"start":{"line":101,"column":1},"end":{"line":101,"column":1}},"key":"QIAVTXj9Mp"}],"key":"v4hbqj1OYk"}],"key":"ZPlPgaSUWH"}],"enumerator":"3","html_id":"rabi-oscillation-bragg","key":"cb6vgOAxdV"},{"type":"paragraph","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"children":[{"type":"text","value":"Equation ","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"key":"hWYwTEQ4bs"},{"type":"crossReference","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"children":[{"type":"text","value":"(","key":"xIfqNrhfo5"},{"type":"text","value":"5","key":"Zpq50jEtJK"},{"type":"text","value":")","key":"J06G3wEaZo"}],"identifier":"oscillation_frequency_rabi_off_resonance_equation","label":"oscillation_frequency_rabi_off_resonance_equation","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"oscillation-frequency-rabi-off-resonance-equation","key":"rw89zIpWXD"},{"type":"text","value":" does not take into account the obvious damping of the oscillation observed on ","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"key":"CYn7oDaCym"},{"type":"crossReference","position":{"start":{"line":103,"column":1},"end":{"line":103,"column":1}},"children":[{"type":"text","value":"Figure ","key":"YFVQigAUP0"},{"type":"text","value":"3","key":"Zu5YNS7aMK"}],"identifier":"rabi_oscillation_bragg","label":"rabi_oscillation_bragg","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"rabi-oscillation-bragg","key":"ZnbbpZgPkt"},{"type":"text","value":". This damping has two origins. The first one is due to the integration volume. 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This value is consistent with the one we observe","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"key":"bZsJrDCG01"},{"type":"footnoteReference","identifier":"note_cornell","label":"note_cornell","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"number":2,"enumerator":"2","key":"FDHCkewmUs"},{"type":"text","value":".","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"key":"oLLoozlQn4"}],"key":"GL5qLDOz8G"},{"type":"comment","value":"[^note_power_increase].","position":{"start":{"line":110,"column":1},"end":{"line":110,"column":1}},"key":"ivZGFip6vn"},{"type":"heading","depth":2,"position":{"start":{"line":119,"column":1},"end":{"line":119,"column":1}},"children":[{"type":"text","value":"Mirror, Mirror on the wall, who’s the fairest of them all?","position":{"start":{"line":119,"column":1},"end":{"line":119,"column":1}},"key":"Stru6zK1Am"}],"identifier":"pi_pulse_optimization","label":"pi_pulse_optimization","html_id":"pi-pulse-optimization","enumerator":"4","key":"UE4Lwt86Jj"},{"type":"paragraph","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"children":[{"type":"text","value":"In the next chapter, ","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"aS8DcrsRc1"},{"type":"crossReference","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"xbZpAo4lU9"}],"identifier":"saturation_of_the_detector","label":"saturation_of_the_detector","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"saturation-of-the-detector","remote":true,"url":"/mcp-physics","dataUrl":"/mcp-physics.json","key":"huJejPGNSZ"},{"type":"text","value":" ","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"SU52Dz8Ztv"},{"type":"crossReference","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"children":[{"type":"text","value":"3","key":"cQ9cxlBQJM"}],"identifier":"saturation_of_the_detector","label":"saturation_of_the_detector","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"saturation-of-the-detector","remote":true,"url":"/mcp-physics","dataUrl":"/mcp-physics.json","key":"iWAJflIHWU"},{"type":"text","value":", we show that the ","key":"dJ7Y2WAbfP"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"PR9N546h2O"}],"key":"VBWACsKy9W"},{"type":"text","value":" saturates the detector which degrades the observation of the second pair. We therefore aim to transfer the ","key":"dDwsjcMGkO"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"qutuqTsbpH"}],"key":"fYsMyoNqgV"},{"type":"text","value":" ","key":"Rug3vkAgQF"},{"type":"inlineMath","value":"v=0","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"html":"v=0v=0v=0","key":"IpScNAPNMb"},{"type":"text","value":" mm/s peak while keeping the ","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"fIZH4BvOve"},{"type":"inlineMath","value":"v=\\pm 10","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"html":"v=±10v=\\pm 10v=±10","key":"S0zXTIEbNn"},{"type":"text","value":" mm/s pairs untouched. From now on, we will therefore use pure deflector (","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"OrhTAbudHa"},{"type":"text","value":"π","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"WToQuhJYWV"},{"type":"text","value":"-pulses, or mirror) and study which Rabi frequency allows us to only deflect the ","key":"UQHw3dpA6x"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"nWlB4KBcMn"}],"key":"ADcpNiP7aQ"},{"type":"text","value":". To characterize the reflectivity properties, we will use the notion of reflectivity profile, which gives the probability to deflect each momentum class. The reflectivity profile for different Rabi frequencies is shown on the left panel of ","key":"qLiIv68Vf6"},{"type":"crossReference","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"children":[{"type":"text","value":"Figure ","key":"mJRKwEyqUf"},{"type":"text","value":"4","key":"ux85bHHp9w"}],"identifier":"mirror_different","label":"mirror_different","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"mirror-different","key":"xFu20vglp4"},{"type":"text","value":". One can see that the smaller the Rabi frequency (the longer the light pulse), the smaller the width in momentum of the Bragg deflector. The reflectivity profile of the smallest Rabi frequency (1.5 kHz) has a good width to transfer entirely the ","key":"f1Ks1Ncd8B"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"bP88RWPVIw"}],"key":"EZ9dWHPI5N"},{"type":"text","value":". However, it exhibits small wings: the reflectivity is not null for the ","key":"wyH3mUFg0z"},{"type":"text","value":"±","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"vZsHqPaP4U"},{"type":"text","value":" 10 mm/s velocity class. This means that we would also deflect the pairs which is, of course, not what we look for.","position":{"start":{"line":120,"column":1},"end":{"line":120,"column":1}},"key":"QcK8VHdsrZ"}],"key":"uLcOtSMk0S"},{"type":"paragraph","position":{"start":{"line":122,"column":1},"end":{"line":122,"column":1}},"children":[{"type":"text","value":"A Fourier-like analogy offers a good picture to optimize the reflectivity profile. Even though it is not totally correct, one can think of the reflectivity profile in momentum (hence in detuning) to be the Fourier transform of the light pulse (in time). For the constant pulses we considered so far (the laser intensity is constant), the reflectivity profile looks like a cardinal sine, whose width is fixed by the duration of the pulse.","position":{"start":{"line":122,"column":1},"end":{"line":122,"column":1}},"key":"ZRd8GwW2hf"}],"key":"Hc2xmahMju"},{"type":"container","kind":"figure","identifier":"mirror_different","label":"mirror_different","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/profile_mirror-ed1b629de97a7c7b53f36363de56b18f.png","alt":"Mirror with constant and sinc Rabi frequencies","width":"100%","align":"center","key":"v7P66vcsaO","urlSource":"images/profile_mirror.png","urlOptimized":"/~gondret/phd_manuscript/build/profile_mirror-ed1b629de97a7c7b53f36363de56b18f.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":131,"column":1},"end":{"line":131,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"mirror_different","identifier":"mirror_different","html_id":"mirror-different","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"IqTXdG3LRi"},{"type":"text","value":"4","key":"m9O6N1aUm3"},{"type":"text","value":":","key":"bQRBY8p34S"}],"template":"Figure %s:","key":"C1eGdzgDuA"},{"type":"text","value":"Left: reflectivity profile of constant light pulse for different Rabi frequencies ","position":{"start":{"line":131,"column":1},"end":{"line":131,"column":1}},"key":"gNyUMqQRrw"},{"type":"inlineMath","value":"\\Omega_R=","position":{"start":{"line":131,"column":1},"end":{"line":131,"column":1}},"html":"ΩR=\\Omega_R=ΩR​=","key":"Cv34gWHNnZ"},{"type":"text","value":" 1.5 kHz (solid green), 2.5 kHz (dashed orange) and 5 kHz (dashed dotted violet). 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Even though the instantaneous Rabi frequency ","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"key":"XFxkSS8tpu"},{"type":"inlineMath","value":"\\Omega_R(t)","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"html":"ΩR(t)\\Omega_R(t)ΩR​(t)","key":"glAskvFxOG"},{"type":"text","value":" now depends on ","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"key":"Rpcu7AWYoY"},{"type":"inlineMath","value":"t","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"html":"ttt","key":"P3ikl628j5"},{"type":"text","value":", we can still define an average Rabi frequency ","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"key":"v6fbbvGhcU"},{"type":"inlineMath","value":"\\bar{\\Omega}_R","position":{"start":{"line":134,"column":1},"end":{"line":134,"column":1}},"html":"ΩˉR\\bar{\\Omega}_RΩˉR​","key":"JPGdxYn1GC"},{"type":"text","value":". This quantity defines the intensity and phase profile of the lasers, hence the width of the reflectivity profile of the pulse. 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Here, the equivalent Rabi frequency ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"uF1hRaujCq"},{"type":"inlineMath","value":"\\bar{\\Omega}_R","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"ΩˉR\\bar{\\Omega}_RΩˉR​","key":"XijvKpzgFt"},{"type":"text","value":" is the Rabi frequency of the equivalent constant ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"Do3abwDPUZ"},{"type":"text","value":"π","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"o6ZnDy4BM2"},{"type":"text","value":" pulse that corresponds to this sinc-shaped pulse. Note we also introduced another parameter, which is the duration of the pulse ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"VaCaJxgY0S"},{"type":"inlineMath","value":"T","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"TTT","key":"ZLjAK6T73w"},{"type":"text","value":". It must be greater than ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"GHgTDdQocK"},{"type":"inlineMath","value":"2\\pi/\\bar{\\Omega}_R","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"html":"2π/ΩˉR2\\pi/\\bar{\\Omega}_R2π/ΩˉR​","key":"EhFlS7dDR5"},{"type":"text","value":" so that the Rabi frequency is sometimes negative. The duration is typically 2 ms for the reflectivity profile shown in ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"pnwDhcD5yR"},{"type":"crossReference","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"children":[{"type":"text","value":"Figure ","key":"AjXQHstFtj"},{"type":"text","value":"4","key":"r5LyMOhMWD"}],"identifier":"mirror_different","label":"mirror_different","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"mirror-different","key":"etk235T21L"},{"type":"text","value":". On the right panel, we see that the wings of the reflectivity profile around ","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"yh0gF2bH3j"},{"type":"text","value":"±","position":{"start":{"line":139,"column":1},"end":{"line":139,"column":1}},"key":"IjHHmkeVhO"},{"type":"text","value":"10 mm/s are removed for the 1.5 and 2.5 kHz Rabi frequencies, compared to the constant pulse scenario. This means we can safely remove the ","key":"VpoIY5iUNX"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"TL2Tz8c8Dw"}],"key":"Pojr6MKHpE"},{"type":"text","value":" while keeping the pairs. 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(a) constant Rabi frequency, (b) sinc shape and (c) reburp. Circles show experimental measurement and lines are the theoretical curve, with no fit parameter. ®July & August 24. ©Theoretical profile courtesy of Ch. Leprince.","position":{"start":{"line":149,"column":1},"end":{"line":149,"column":1}},"key":"PaZxfB8NKx"}],"key":"nUXEhskIAP"}],"key":"FuEHofsFw8"}],"enumerator":"5","html_id":"reflectivity-profile-experiment","key":"EamVuA5KTV"},{"type":"paragraph","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"children":[{"type":"text","value":"One can see, however, that the reflectivity profile of the sinc pulse is not a perfect square-like function. This is because our Fourier analogy is not accurate when the transferred population becomes larger than typically 10%. More exotic, yet still analytical, pulse shapes were developed and recently implemented in our experiment. 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We clearly see here that the reflectivity profile is null at ","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"itilLFDSfV"},{"type":"text","value":"±","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"gxuLj5DEjh"},{"type":"text","value":" 10 mm/s for the ","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"OzG7sMtdea"},{"type":"emphasis","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"children":[{"type":"text","value":"sinc","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"jOiJzdlE8p"}],"key":"UGB536V7UV"},{"type":"text","value":" and ","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"EzTlxRctWf"},{"type":"emphasis","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"children":[{"type":"text","value":"reburp","position":{"start":{"line":153,"column":1},"end":{"line":153,"column":1}},"key":"znd6xRsbYF"}],"key":"g8JsdQDdo2"},{"type":"text","value":" pulses. 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A more detailed comparison of the pulse shaping can be found in @leprince_phase_2024.\n @alway_arbitrary_2007","position":{"start":{"line":158,"column":1},"end":{"line":159,"column":1}},"key":"ZaDWsdvPHB"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Summary","position":{"start":{"line":164,"column":1},"end":{"line":164,"column":1}},"key":"IrHNjZpAxW"}],"key":"swqYmdhuhi"},{"type":"paragraph","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"children":[{"type":"text","value":"This section introduces ","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"key":"xO6WMeFtUV"},{"type":"crossReference","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"children":[{"type":"text","value":"Bragg","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"key":"Ld0oxwbM2k"}],"identifier":"bragg_definition","label":"bragg_definition","kind":"heading","template":"Section %s","enumerator":"1","resolved":true,"html_id":"bragg-definition","key":"zM0gNBVMar"},{"type":"text","value":" diffraction as a tool to get rid of the saturation of the ","key":"J2Kg6MqY3r"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"VeGDoCCDj5"}],"key":"uQDts7nyBz"},{"type":"text","value":". By ","key":"ILfitCGnCo"},{"type":"crossReference","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"children":[{"type":"text","value":"shaping","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"key":"iIuck5EijV"}],"identifier":"pi_pulse_optimization","label":"pi_pulse_optimization","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"pi-pulse-optimization","key":"IuJTUBvwe3"},{"type":"text","value":" the lasers intensity and the phase difference between them, we are able to realize sharp deflectors to remove the ","key":"kbg3sYs5Yk"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"WhBT1wuRXK"}],"key":"GyWpcJaNAk"},{"type":"text","value":" without affecting the pairs.","key":"gqfGmuFqc5"}],"key":"UzYCDGxNAy"}],"key":"wwWulKtN2U"},{"type":"footnoteDefinition","identifier":"adiabatic_elimination","label":"adiabatic_elimination","position":{"start":{"line":165,"column":1},"end":{"line":165,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"children":[{"type":"text","value":"The full derivation of the adiabatic elimination of the excited state can be found in ","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"key":"V6hoa2ub0D"},{"type":"cite","identifier":"perrier_interferences_2020","label":"perrier_interferences_2020","kind":"narrative","position":{"start":{"line":112,"column":87},"end":{"line":112,"column":114}},"children":[{"type":"text","value":"Perrier (2020)","key":"ufYd4ioQ4n"}],"enumerator":"9","key":"L3fVWOqWOz"},{"type":"text","value":".","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"key":"TLWyajkrpB"}],"key":"dR0hkZXoqb"}],"number":1,"enumerator":"1","key":"vwk4E6niY9"},{"type":"footnoteDefinition","identifier":"note_cornell","label":"note_cornell","position":{"start":{"line":112,"column":1},"end":{"line":112,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"children":[{"type":"text","value":"Note that I am writing this paragraph after Eric Cornell’s talk at the ICAP24 conference, where he reported the observation of an astonishing coherence time of 8 seconds ","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"key":"oal2efYAuR"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":107,"column":1},"end":{"line":107,"column":1}},"children":[{"type":"cite","identifier":"wang2024long","label":"wang2024long","kind":"parenthetical","position":{"start":{"line":107,"column":172},"end":{"line":107,"column":185}},"children":[{"type":"text","value":"Wang ","key":"CU5gyEHHIm"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"hc5NGM4hOh"}],"key":"WL19hXeazK"},{"type":"text","value":", 2024","key":"WfaPkxb0tt"}],"enumerator":"10","key":"GDer8qem0C"}],"key":"fKd6WBlmow"},{"type":"text","value":". Afterward, he mentioned that “because the theoretical coherence time is 20 seconds, [his] students should not be satisfied until they would have reached it”. 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It is defined in terms of a Fourier series as ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"qKHYv9Gscj"},{"type":"inlineMath","value":" \\Omega_{\\mathrm{R}}(t)=\\Omega_{\\mathrm{M}}\\left[A_0+\\displaystyle\\sum_{n=1}A_n\\,\\mathrm{cos}(n\\Omega_{\\mathrm{S}}t)\\right]","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"ΩR(t)=ΩM[A0+∑n=1An cos(nΩSt)] \\Omega_{\\mathrm{R}}(t)=\\Omega_{\\mathrm{M}}\\left[A_0+\\displaystyle\\sum_{n=1}A_n\\,\\mathrm{cos}(n\\Omega_{\\mathrm{S}}t)\\right]ΩR​(t)=ΩM​[A0​+n=1∑​An​cos(nΩS​t)]","key":"OATOhZ5djU"},{"type":"text","value":" for ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"r7dQmySlmw"},{"type":"inlineMath","value":"0\\leq t\\leq 2\\pi/{\\Omega_{\\mathrm{S}}}","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"0≤t≤2π/ΩS0\\leq t\\leq 2\\pi/{\\Omega_{\\mathrm{S}}}0≤t≤2π/ΩS​","key":"pO6N08Y6qx"},{"type":"text","value":", where ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"jodqIlxGrY"},{"type":"inlineMath","value":"\\Omega_{\\mathrm{S}}=2A_0\\Omega_{\\mathrm{M}}","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"ΩS=2A0ΩM\\Omega_{\\mathrm{S}}=2A_0\\Omega_{\\mathrm{M}}ΩS​=2A0​ΩM​","key":"RNwGg0X1n9"},{"type":"text","value":" and the ","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"DcOfMmVNPN"},{"type":"inlineMath","value":"A_n","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"html":"AnA_nAn​","key":"Ipu3HOniKV"},{"type":"text","value":" coefficients, experimentally implemented up to the 15th order.","position":{"start":{"line":156,"column":1},"end":{"line":156,"column":1}},"key":"xahm1Ba9fe"}],"key":"Tk2XFNNigk"}],"number":3,"enumerator":"3","key":"fyMWvPIlDP"}],"key":"QJguB9VdBk"}],"key":"INpen9wYEq"},"references":{"cite":{"order":["leprince_2024_coherent","leprince_phase_2024","martin_bragg_1988","beguin_characterisation_2022","berman_pump_probe_spectro_1997","mcdonald_uses_1991","geen_1991_band_selective","luo_contrast_2016","perrier_interferences_2020","wang2024long"],"data":{"leprince_2024_coherent":{"label":"leprince_2024_coherent","enumerator":"1","html":"Leprince, C., Gondret, V., Lamirault, C., Dias, R., Marolleau, Q., Boiron, D., & Westbrook, C. I. (2024). Coherent coupling of momentum states: selectivity and phase control. Submitted to Phys. Rev. A. https://arxiv.org/pdf/2411.09284","url":"https://arxiv.org/pdf/2411.09284"},"leprince_phase_2024":{"label":"leprince_phase_2024","enumerator":"2","html":"Leprince, C. (2024). Phase control and pulse shaping in Bragg diffraction for quantum atom optics: From matter-wave interferences to a Bell’s inequality test [Phdthesis, Université Paris-Saclay]. https://theses.fr/s247200","url":"https://theses.fr/s247200"},"martin_bragg_1988":{"label":"martin_bragg_1988","enumerator":"3","doi":"10.1103/PhysRevLett.60.515","html":"Martin, P. J., Oldaker, B. G., Miklich, A. H., & Pritchard, D. E. (1988). Bragg scattering of atoms from a standing light wave. Phys. Rev. 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The European Physical Journal D, 70(12), 262. 10.1140/epjd/e2016-70428-6","url":"https://doi.org/10.1140/epjd/e2016-70428-6"},"perrier_interferences_2020":{"label":"perrier_interferences_2020","enumerator":"9","html":"Perrier, M. (2020). Interférences multiples avec atomes froids [Phdthesis]. Université Paris-Saclay."},"wang2024long":{"label":"wang2024long","enumerator":"10","html":"Wang, A., Park, S. Y., Hartman, A., Hector Hernandez, P., Ng, K. B., Schlossberger, N., Wright, T., Kompella, R., Ye, J., & Cornell, E. (2024). Long coherence time of the JILA Generation III eEDM experiment. Bulletin of the American Physical Society."}}}},"footer":{"navigation":{"prev":{"title":"Bose-Einstein condensation in an optical dipole trap","short_title":"BEC in ODT","url":"/bec-odt","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Measuring the BEC properties","short_title":"Measuring the BEC properties","url":"/bec-mesure","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}