{"kind":"Article","sha256":"59c44d54c52e36a13f99a0177d450c191dd9ec18f40d44cd9f20accd76240d2f","slug":"cosqua-2exponential-creation","location":"/experiment_controlling/cosqua_2exponential_creation.md","dependencies":[],"frontmatter":{"title":"Exponential creation of phonons","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/exponential_creation-9af50e501973cdd258d86dc11c2b5c05.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/exponential_creation-9af50e501973cdd258d86dc11c2b5c05.webp","exports":[{"format":"md","filename":"cosqua_2exponential_creation.md","url":"/~gondret/phd_manuscript/build/cosqua_2exponential_-3bd56e76c184dccc520463217a8fc0e5.md"}]},"mdast":{"type":"root","children":[{"type":"block","children":[{"type":"heading","depth":2,"position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"children":[{"type":"text","value":"Exponential creation of phonons","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"Sl2GuRfOA9"}],"identifier":"creation_phonon_exp","label":"creation_phonon_exp","html_id":"creation-phonon-exp","enumerator":"1","key":"PehrLqN9sy"},{"type":"paragraph","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"Within the resonant window, the number of created phonons is expected to be exponential. To observe this exponential creation of phonons, we excite the 2.1 kHz breathing mode of the ","key":"xUWjEB3XVu"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"Q8hIQlU74P"}],"key":"cjjtVgatTr"},{"type":"text","value":" for 4 periods, with a laser amplitude of 20%. The cloud is then kept in the trap for an additional duration, ranging from 2 to 6. The amplitude of the breathing is shown on the first panel of ","key":"DI9Rtok73m"},{"type":"crossReference","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"children":[{"type":"text","value":"Figure ","key":"S163jJUR0J"},{"type":"text","value":"1","key":"xSvJetOkky"}],"identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"exponential-creation-oscillation","key":"ns3nHBP88m"},{"type":"text","value":".","position":{"start":{"line":11,"column":1},"end":{"line":11,"column":1}},"key":"seDjYYHIKd"}],"key":"FBeNhDEz7S"},{"type":"comment","value":"Here time is expressed in periods of the breathing mode $N_{osc}$, that is not necessarily an integer.","position":{"start":{"line":12,"column":1},"end":{"line":12,"column":1}},"key":"RruncyD2DZ"},{"type":"container","kind":"figure","identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/exponential_creation-9af50e501973cdd258d86dc11c2b5c05.png","alt":"Observation of the breathing mode","width":"100%","align":"center","key":"ZAQnu2ZE7O","urlSource":"images/exponential_creation_phonon_figure_3.png","urlOptimized":"/~gondret/phd_manuscript/build/exponential_creation-9af50e501973cdd258d86dc11c2b5c05.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"exponential_creation_oscillation","identifier":"exponential_creation_oscillation","html_id":"exponential-creation-oscillation","enumerator":"1","children":[{"type":"text","value":"Figure ","key":"liS4DZepzO"},{"type":"text","value":"1","key":"PqwdIby5M2"},{"type":"text","value":":","key":"i9MyiVqjHH"}],"template":"Figure %s:","key":"c3MtwDPYpS"},{"type":"text","value":"Left: oscillation of the ","key":"tPguCr6YOF"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"jF35H7TAmQ"}],"key":"R8RrtlgoOX"},{"type":"text","value":" radius (top) and the effective 1D coupling constant (bottom). The solid line is a sine fit with a 30% (top) and 50% (bottom) amplitude and gives the breathing mode frequency of 2.08 kHz. Middle: Number of phonons created as a function of time in the positive velocity peak (orange squares) and negative peak (blue circles). The ","key":"ldBrcdtIZD"},{"type":"inlineMath","value":"y","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"html":"yyy","key":"QzGKdK8Mdl"},{"type":"text","value":"-axis is in log scale. Error bars represent the standard deviation over the square root of the repetition number: the error on the measured population (see the ","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"S2pNti78zb"},{"type":"crossReference","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"appendix","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"Cqb0rzCuPy"}],"identifier":"thermal_poisson_distrib_error","label":"thermal_poisson_distrib_error","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"thermal-poisson-distrib-error","remote":true,"url":"/codes","dataUrl":"/codes.json","key":"jAOS8vlo8m"},{"type":"text","value":" ","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"EgLTbKTGyw"},{"type":"crossReference","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"children":[{"type":"text","value":"3","key":"UU6bg5OvQq"}],"identifier":"thermal_poisson_distrib_error","label":"thermal_poisson_distrib_error","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"thermal-poisson-distrib-error","remote":true,"url":"/codes","dataUrl":"/codes.json","key":"dOXwvHaPon"},{"type":"text","value":"). Right: 2D-histogram of the atom-density as a function of time. The box size chosen for the middle plot is 0.8 mm/s centered on ","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"TIUOTGohIb"},{"type":"text","value":"±","position":{"start":{"line":20,"column":1},"end":{"line":20,"column":1}},"key":"tzhNdMmgBK"},{"type":"text","value":"8.3 mm/s. The color-scale saturates the high density ","key":"LnDUFjY2pz"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"CXmUnc5d3W"}],"key":"ZZdWFpYYqq"},{"type":"text","value":" and its asymmetry is due to the Bragg deflector whose velocity selection was badly set and to the saturation of the detector. ®Dataset taken on the 26th of July 24.","key":"xTowCMlRgP"}],"key":"COJxOXjFzG"}],"key":"UCwpvTy1nx"}],"enumerator":"1","html_id":"exponential-creation-oscillation","key":"bT3YzfKbeJ"},{"type":"paragraph","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"children":[{"type":"text","value":"The number of phonons as a function of time is shown on the middle panel of ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"uqQz76QHgW"},{"type":"crossReference","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"children":[{"type":"text","value":"Figure ","key":"oi59rnIFmP"},{"type":"text","value":"1","key":"TPa5QDL8RP"}],"identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"exponential-creation-oscillation","key":"vCDzW2H3Ec"},{"type":"text","value":". On the right panel is shown the histogram of the atomic density on grey scale as a function of the atom speed (","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"BLQ1gK0AmI"},{"type":"inlineMath","value":" y","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"y yy","key":"fEMiGdUH1e"},{"type":"text","value":"-axis) and the time ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"B2YRiz0oYt"},{"type":"inlineMath","value":"N_{osc}","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"NoscN_{osc}Nosc​","key":"w7b4JNRkHw"},{"type":"text","value":". For each slice the ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"UkWpWmg9o8"},{"type":"inlineMath","value":"x ","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"html":"xx x","key":"DrGBtp4DeY"},{"type":"text","value":"-axis (each time), the color scale shows the longitudinal density of the atomic signal. The horizontal black spot in the middle is due to the ","key":"hpa6rQAIfU"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"WaILjoqDOJ"}],"key":"EENnPp3aKV"},{"type":"text","value":", whose signal saturates the color scale chosen to depict the phonon peaks. The asymmetry of the ","key":"nedyKBh7oG"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"VYubud1Q82"}],"key":"hBi57pSVw9"},{"type":"text","value":", with a higher density on the positive velocity side, is due to incorrect settings of the Bragg deflector, whose frequency was misconfigured. This histogram helps to visualize the width of the phonon peak, quite visible on the right of the graph. In particular, we see that width of the peak is of the order of 1-2 mm/s, near the size of one mode. We now focus on the exponential growth of the phonon population. Following ","key":"nNkp3c2VMs"},{"type":"cite","identifier":"micheli_phonon_2022","label":"micheli_phonon_2022","kind":"narrative","position":{"start":{"line":23,"column":954},"end":{"line":23,"column":974}},"children":[{"type":"text","value":"Micheli & Robertson (2022)","key":"VdP7VNRdbD"}],"enumerator":"1","key":"WmAD34tzbH"},{"type":"text","value":", we fit the experimental data with the following function:","position":{"start":{"line":23,"column":1},"end":{"line":23,"column":1}},"key":"poeizEjbFl"}],"key":"hLqGaWIIpI"},{"type":"math","identifier":"formula_gain_temps","label":"formula_gain_temps","value":"n_k(t) = \\left(n_k^{(in)} + (2n_k^{(in)}+1)\\text{sinh}\\left[ \\frac{G_k}{2}(t - t_0)\\right]^2 \\right)\\times \\left[1 + \\alpha_k\\text{cos}(2\\pi ft+\\phi)\\right].","tight":true,"html":"nk(t)=(nk(in)+(2nk(in)+1)sinh[Gk2(t−t0)]2)×[1+αkcos(2πft+ϕ)].n_k(t) = \\left(n_k^{(in)} + (2n_k^{(in)}+1)\\text{sinh}\\left[ \\frac{G_k}{2}(t - t_0)\\right]^2 \\right)\\times \\left[1 + \\alpha_k\\text{cos}(2\\pi ft+\\phi)\\right].nk​(t)=(nk(in)​+(2nk(in)​+1)sinh[2Gk​​(t−t0​)]2)×[1+αk​cos(2πft+ϕ)].","enumerator":"1","html_id":"formula-gain-temps","key":"X3kGa6XYrq"},{"type":"paragraph","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"children":[{"type":"text","value":"In the following, we will discuss the expected and measured value of those fit parameters.","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"key":"uODufYrlRr"}],"key":"f8BBW4J2eG"},{"type":"heading","depth":2,"position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"children":[{"type":"text","value":"Initial thermal seed and initial time of the squeezing","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"key":"X1shKwWLzx"}],"identifier":"initial_size","label":"initial_size","html_id":"initial-size","enumerator":"2","key":"InqL0VuFlP"},{"type":"paragraph","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"children":[{"type":"text","value":"The initial phonon population ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"N4hHhEB1c2"},{"type":"inlineMath","value":"n^{(in)}_k","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"nk(in)n^{(in)}_knk(in)​","key":"OoMgFqNY6I"},{"type":"text","value":" and the initial time ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"i3o1QPTQlD"},{"type":"inlineMath","value":"t_0","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"t0t_0t0​","key":"QbjHYkTHi2"},{"type":"text","value":" play somehow the role of the offset and the amplitude in the exponential growth of phonons. The parameter ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"NM0vEJQeKX"},{"type":"inlineMath","value":"n^{(in)}_k","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"nk(in)n^{(in)}_knk(in)​","key":"aN3jQ1Xoq4"},{"type":"text","value":" in Eq. ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"Akrvk5V6J5"},{"type":"crossReference","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"children":[{"type":"text","value":"(","key":"KlSNISPJ88"},{"type":"text","value":"1","key":"B97NyIkSqC"},{"type":"text","value":")","key":"Uakkq7NTKR"}],"identifier":"formula_gain_temps","label":"formula_gain_temps","kind":"equation","template":"(%s)","enumerator":"1","resolved":true,"html_id":"formula-gain-temps","key":"knfoeXdQ9e"},{"type":"text","value":" is the thermal population of phonons before the modulation: it is the thermal seed of the two-mode squeezed state. We measure a 44 nK temperature which means that we expect an initial thermal population of ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"cOd7hkMkzc"},{"type":"inlineMath","value":"n^{(in)}","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"n(in)n^{(in)}n(in)","key":"a0Qbu3OVUS"},{"type":"text","value":"= 0.5. Note however that this population is the number of atoms for a single mode. Here, we use an integration volume of 1 mm/s along ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"hZ1eQj3ABN"},{"type":"inlineMath","value":"z","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"zzz","key":"haaC2nUjGZ"},{"type":"text","value":", and 60 mm/s along ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"HQHxpcHg6j"},{"type":"inlineMath","value":"x","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"xxx","key":"ySzQJ7s34P"},{"type":"text","value":" and ","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"itddEK3UNL"},{"type":"inlineMath","value":"y","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"html":"yyy","key":"LEjazSRvuK"},{"type":"text","value":". We find an initial thermal population of 3(1) and 2(2) atoms per mode for the negative and positive peak, which is higher than the expected one. The fit results assume a 0.5 quantum efficiency of the detector.","position":{"start":{"line":34,"column":1},"end":{"line":34,"column":1}},"key":"HGeVyMBWl5"}],"key":"EUQtFUSh7S"},{"type":"paragraph","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"children":[{"type":"text","value":"The second parameter ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"LJWPnR8sus"},{"type":"inlineMath","value":"t_0","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"t0t_0t0​","key":"vGDTMhTfYB"},{"type":"text","value":" is the initial time of the squeezing. If the ","key":"lnCztFvOQd"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"n62bwOjHFD"}],"key":"yFOeKQPdKO"},{"type":"text","value":" was instantaneously excited at ","key":"wERinJvk6f"},{"type":"inlineMath","value":"N_{osc}=0","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"Nosc=0N_{osc}=0Nosc​=0","key":"XMovXDJPqq"},{"type":"text","value":", it should be null. However, the excitation process last 4 periods hence we can expect that ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"Ai2X6vIyvv"},{"type":"inlineMath","value":"t_0","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"t0t_0t0​","key":"hbnPIAElO0"},{"type":"text","value":" ranges from 0 to ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"WqnsiN3aQN"},{"type":"inlineMath","value":"- 4","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"−4- 4−4","key":"EmaQbuSydN"},{"type":"text","value":", in units of the breathing mode frequency ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"hCnL3KQCmX"},{"type":"inlineMath","value":"2\\omega_\\perp","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"2ω⊥2\\omega_\\perp2ω⊥​","key":"C2rpzzQiqL"},{"type":"text","value":". Here, we find that ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"S0SAYKFOQn"},{"type":"inlineMath","value":"t_0 = 0.4\\pm 0.3\\, T_{osc}","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"t0=0.4±0.3 Tosct_0 = 0.4\\pm 0.3\\, T_{osc}t0​=0.4±0.3Tosc​","key":"kdbXVkvjTZ"},{"type":"text","value":" and ","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"eoC7xyArvt"},{"type":"inlineMath","value":"t_0 = 0.1\\pm 0.6\\, T_{osc}","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"html":"t0=0.1±0.6 Tosct_0 = 0.1\\pm 0.6\\, T_{osc}t0​=0.1±0.6Tosc​","key":"laRlg6xvn4"},{"type":"text","value":". This result is consistent with our expectations.","position":{"start":{"line":36,"column":1},"end":{"line":36,"column":1}},"key":"oYtAQBvlkR"}],"key":"WdxDu3urTs"},{"type":"heading","depth":2,"position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"children":[{"type":"text","value":"Oscillation of the occupation number","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"PQJU0gZeaB"}],"identifier":"oscillation_atom_vs_phonon_basis","label":"oscillation_atom_vs_phonon_basis","html_id":"oscillation-atom-vs-phonon-basis","enumerator":"3","key":"Bkrzxj34uT"},{"type":"container","kind":"figure","identifier":"amplitude_oscillation_adiabaticite","label":"amplitude_oscillation_adiabaticite","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/amplitude_oscillatio-482cdbedccbf683c14f5e73ae1f99a65.png","alt":"Num. 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The inset shows the time profile of the trapping laser power. ®Data taken on the 09/09/24.","position":{"start":{"line":53,"column":1},"end":{"line":53,"column":1}},"key":"NPMtJMaNTr"}],"key":"tUCacCQvOZ"}],"key":"uwoJEwyEAS"}],"enumerator":"2","html_id":"amplitude-oscillation-adiabaticite","key":"VLuoa42i6T"},{"type":"paragraph","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"children":[{"type":"text","value":"The last term in ","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"key":"HmsOoTkUVP"},{"type":"crossReference","position":{"start":{"line":56,"column":1},"end":{"line":56,"column":1}},"children":[{"type":"text","value":"(","key":"nGYx03Y3ua"},{"type":"text","value":"1","key":"GPU8IcVCdi"},{"type":"text","value":")","key":"JbTbvloav0"}],"identifier":"formula_gain_temps","label":"formula_gain_temps","kind":"equation","template":"(%s)","enumerator":"1","resolved":true,"html_id":"formula-gain-temps","key":"UqpzcEqWcL"},{"type":"text","value":" takes into account the oscillations of the measured population at twice the frequency of the trap. Here, we observe a quite large amplitude of the occupation number (around 50%), which is not expected in the phonon population. 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The amplitude of the oscillation in ","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"key":"TRqYumcPo7"},{"type":"crossReference","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"children":[{"type":"text","value":"Figure ","key":"NyBf4ov9Ja"},{"type":"text","value":"1","key":"XYeoaIp4Lb"}],"identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"exponential-creation-oscillation","key":"pyJhve74aA"},{"type":"text","value":" are only 0.50(3) and 0.55(5). The difference between those two values is due to the expansion of the trap: the interactions are not abruptly switched off. The mapping between the phonon and the atom basis is neither adiabatic nor instantaneous. As a result the oscillation amplitude we observe are weaker. If it is not possible to switch faster the oscillations, we can open adiabatically the trap in order to better map the phonon basis to the atomic basis. To do so, we open the trap by ramping down the transverse confinement in 1.5 ms. The laser trap power is shown in the inset ","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"key":"AcCrpbx4Lb"},{"type":"crossReference","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"children":[{"type":"text","value":"Figure ","key":"SLbNwX1kzS"},{"type":"text","value":"2","key":"nSkVAWoDHR"}],"identifier":"amplitude_oscillation_adiabaticite","label":"amplitude_oscillation_adiabaticite","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"amplitude-oscillation-adiabaticite","key":"MBs9GqepXo"},{"type":"text","value":": see the pink dotted curve compared to the green dashed one when the trap is suddenly switched off. On the main plot of ","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"key":"p7XsacXnJR"},{"type":"crossReference","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"children":[{"type":"text","value":"Figure ","key":"V61UbMy8Pj"},{"type":"text","value":"2","key":"XlkIHaYNZq"}],"identifier":"amplitude_oscillation_adiabaticite","label":"amplitude_oscillation_adiabaticite","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"amplitude-oscillation-adiabaticite","key":"kQi9RYJ5Iu"},{"type":"text","value":", we show the number of atoms with this adiabatic opening (pink squares) and with the sudden opening (green circles). We clearly see that the oscillation is less visible and almost suppressed, as expected.","position":{"start":{"line":68,"column":1},"end":{"line":68,"column":1}},"key":"gyIDb5K90Y"}],"key":"jloIGJxZKq"},{"type":"comment","value":"The frequency is exactly the breathing mode frequency. The relative amplitude of the oscillation is 52(5)% which is much higher than we expect. See for example the expected phonon number in the","position":{"start":{"line":72,"column":1},"end":{"line":72,"column":1}},"key":"J38kJo18Vq"},{"type":"heading","depth":2,"position":{"start":{"line":74,"column":1},"end":{"line":74,"column":1}},"children":[{"type":"text","value":"Growth rate of the phonon occupation","position":{"start":{"line":74,"column":1},"end":{"line":74,"column":1}},"key":"g1tgOmwnWY"}],"identifier":"measuring_growth_rate","label":"measuring_growth_rate","html_id":"measuring-growth-rate","enumerator":"4","key":"PiuAfU4Beu"},{"type":"heading","depth":3,"position":{"start":{"line":77,"column":1},"end":{"line":77,"column":1}},"children":[{"type":"text","value":"Theoretical growth rate","position":{"start":{"line":77,"column":1},"end":{"line":77,"column":1}},"key":"GPML73TT2J"}],"identifier":"theoretical-growth-rate","label":"Theoretical growth rate","html_id":"theoretical-growth-rate","implicit":true,"key":"CWlbpBkeM5"},{"type":"paragraph","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"children":[{"type":"text","value":"The third parameter of equation ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"mhtK90VC0H"},{"type":"crossReference","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"children":[{"type":"text","value":"(","key":"teaAGW5uv6"},{"type":"text","value":"1","key":"fhdKBKZhUi"},{"type":"text","value":")","key":"tZjCUMVne1"}],"identifier":"formula_gain_temps","label":"formula_gain_temps","kind":"equation","template":"(%s)","enumerator":"1","resolved":true,"html_id":"formula-gain-temps","key":"fxZfE2l1Pk"},{"type":"text","value":" is the growth rate ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"RIETKwndK5"},{"type":"inlineMath","value":"G_k ","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"html":"GkG_k Gk​","key":"QXicIWWGVQ"},{"type":"text","value":". 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The bottom left panel of ","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"nNZCFVL79J"},{"type":"crossReference","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"children":[{"type":"text","value":"Figure ","key":"eP5GLkW8hA"},{"type":"text","value":"1","key":"U5PFumhH2u"}],"identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"exponential-creation-oscillation","key":"zXtolTByDj"},{"type":"text","value":" represents the variation of the 1D coupling constant.","position":{"start":{"line":83,"column":1},"end":{"line":83,"column":1}},"key":"F1dDeMeQsj"}],"key":"LSTJp0wPNO"},{"type":"paragraph","position":{"start":{"line":86,"column":1},"end":{"line":86,"column":1}},"children":[{"type":"text","value":"At first, to extract this value, I fitted with a sine function. 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The solid grey line is a ","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"key":"lKFK7eTfwf"},{"type":"inlineMath","value":"y=x","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"html":"y=xy=xy=x","key":"Fq6iL0JjDI"},{"type":"text","value":" line as a guide to the eye. The closer the experimental points to this line, the better the agreement between the experimental gain and ","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"key":"uI2zbjRhMR"},{"type":"crossReference","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"children":[{"type":"text","value":"(","key":"gXievqLnEc"},{"type":"text","value":"3","key":"SJ4e2cA9r7"},{"type":"text","value":")","key":"vE40vVk3Jf"}],"identifier":"gain_phonons","label":"gain_phonons","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"gain-phonons","key":"hsyrrsVCIw"},{"type":"text","value":". The blue circles and orange square represent the growth rate of the negative and positive peaks. The uncertainty along the ","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"key":"X55svR72lq"},{"type":"inlineMath","value":"y","position":{"start":{"line":102,"column":1},"end":{"line":102,"column":1}},"html":"yyy","key":"NVngRHuXiH"},{"type":"text","value":" direction is a combination of the uncertainty of the fit and the result of the fit for various integration volumes. All points lie below the theoretical curve, and the higher the growth rate, the higher the difference. 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The solid grey line is a ","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"key":"cw8kqU7s6C"},{"type":"inlineMath","value":"y=x","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"html":"y=xy=xy=x","key":"UGU0JrQUQt"},{"type":"text","value":" line as a guide to the eye. The blue circles and orange squares represent the measured growth rate for the ","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"key":"oZgFygS1tf"},{"type":"inlineMath","value":"-k","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"html":"−k-k−k","key":"uCUG6PQkIo"},{"type":"text","value":" and ","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"key":"zl4H7IjEYh"},{"type":"inlineMath","value":"+k","position":{"start":{"line":111,"column":1},"end":{"line":111,"column":1}},"html":"+k+k+k","key":"V99WIUm8BI"},{"type":"text","value":" peak. 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In their work, the authors provide an analytical formula and write the ","position":{"start":{"line":114,"column":1},"end":{"line":114,"column":1}},"key":"FzcEVKwug7"},{"type":"emphasis","position":{"start":{"line":114,"column":1},"end":{"line":114,"column":1}},"children":[{"type":"text","value":"corrected","position":{"start":{"line":114,"column":1},"end":{"line":114,"column":1}},"key":"dKZzo1s0er"}],"key":"IIblf1yRcK"},{"type":"text","value":" growth rate as","position":{"start":{"line":114,"column":1},"end":{"line":114,"column":1}},"key":"JEKmwMZRVw"}],"key":"UO4pO8lNA5"},{"type":"math","identifier":"gain_minus_decay","label":"gain_minus_decay","value":"G_k' = G_k^{th} - \\Gamma_k","tight":true,"html":"Gk′=Gkth−ΓkG_k' = G_k^{th} - \\Gamma_kGk′​=Gkth​−Γk​","enumerator":"4","html_id":"gain-minus-decay","key":"nGnYRHIroC"},{"type":"paragraph","position":{"start":{"line":119,"column":1},"end":{"line":119,"column":1}},"children":[{"type":"text","value":"where the phonon decay rate is","position":{"start":{"line":119,"column":1},"end":{"line":119,"column":1}},"key":"aMRLl0nkhS"}],"key":"z5ugqaye15"},{"type":"math","identifier":"gamma_loss_amau","label":"gamma_loss_amau","value":"\\Gamma_k = \\frac{c_s}{\\xi}\\frac{k_BT}{mc_s^2}\\frac{1}{n_1\\xi}\\left(f_+(k\\xi) + f_-(k\\xi)\\right)","tight":true,"html":"Γk=csξkBTmcs21n1ξ(f+(kξ)+f−(kξ))\\Gamma_k = \\frac{c_s}{\\xi}\\frac{k_BT}{mc_s^2}\\frac{1}{n_1\\xi}\\left(f_+(k\\xi) + f_-(k\\xi)\\right)Γk​=ξcs​​mcs2​kB​T​n1​ξ1​(f+​(kξ)+f−​(kξ))","enumerator":"5","html_id":"gamma-loss-amau","key":"zfUoxZPOkF"},{"type":"paragraph","position":{"start":{"line":124,"column":1},"end":{"line":124,"column":1}},"children":[{"type":"text","value":"and","position":{"start":{"line":124,"column":1},"end":{"line":124,"column":1}},"key":"eZ1KZo5nuL"}],"key":"Nqs2jnior1"},{"type":"math","value":"f_\\pm(k\\xi) =\\frac{1}{2}\\frac{k^2\\xi^2}{\\left(\\omega_k/c_sk\\right)^2}\\frac{\\left(\\omega_k/c_sk \\pm 1/2\\right)^2}{v_k^{gr}/c_s\\mp 1}.","tight":"before","html":"f±(kξ)=12k2ξ2(ωk/csk)2(ωk/csk±1/2)2vkgr/cs∓1.f_\\pm(k\\xi) =\\frac{1}{2}\\frac{k^2\\xi^2}{\\left(\\omega_k/c_sk\\right)^2}\\frac{\\left(\\omega_k/c_sk \\pm 1/2\\right)^2}{v_k^{gr}/c_s\\mp 1}.f±​(kξ)=21​(ωk​/cs​k)2k2ξ2​vkgr​/cs​∓1(ωk​/cs​k±1/2)2​.","enumerator":"6","key":"Hu4Z6dBkzw"},{"type":"paragraph","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"children":[{"type":"text","value":"Here ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"u4dTrFzlgs"},{"type":"inlineMath","value":"\\omega_k=\\omega_\\perp","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"ωk=ω⊥\\omega_k=\\omega_\\perpωk​=ω⊥​","key":"Nfce9XzAkl"},{"type":"text","value":" is the Bogoliubov energy of the quasi-excitation and ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"eOiinkhNAv"},{"type":"inlineMath","value":"v_k^{gr}=\\text{d}\\omega_k/\\text{d}k","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"vkgr=dωk/dkv_k^{gr}=\\text{d}\\omega_k/\\text{d}kvkgr​=dωk​/dk","key":"WrFck2dbeC"},{"type":"text","value":" is the group velocity of the quasi-excitation at momentum ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"vTQqO48Pdu"},{"type":"inlineMath","value":"k","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"kkk","key":"n5VooAyUuB"},{"type":"text","value":". In addition to the gas 1D parameters ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"TviSsaP70b"},{"type":"inlineMath","value":"\\xi,\\, c_s","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"ξ, cs\\xi,\\, c_sξ,cs​","key":"GowDaobrCp"},{"type":"text","value":", numerical evaluation of Eq. ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"Lxdu373ZdR"},{"type":"crossReference","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"children":[{"type":"text","value":"(","key":"bd1zYyc5Yw"},{"type":"text","value":"5","key":"dKYZRPYk1J"},{"type":"text","value":")","key":"J03w3zJ45M"}],"identifier":"gamma_loss_amau","label":"gamma_loss_amau","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"gamma-loss-amau","key":"ZGUfWuXL1a"},{"type":"text","value":" requires the temperature. To measure the temperature, we fit the tails of the momentum distribution. To have enough statistics, we concatenate data with different excitation time. For two datasets, the “temperature” depends strongly on the maximal excitation time used to measure the temperature. Indeed, as the system is far for equilibrium, measuring the temperature can be misleading. 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Even though preliminary work and numerical simulations by ","key":"NQvoYeW5w1"},{"type":"cite","identifier":"micheli_twa_2024","label":"micheli_twa_2024","kind":"narrative","position":{"start":{"line":133,"column":149},"end":{"line":133,"column":166}},"children":[{"type":"text","value":"Micheli & Robertson (2024)","key":"nAygjXFZm0"}],"enumerator":"3","key":"f2f0LI3cAv"},{"type":"text","value":" showed however that ","position":{"start":{"line":133,"column":1},"end":{"line":133,"column":1}},"key":"hNYcKD5YZ6"},{"type":"crossReference","position":{"start":{"line":133,"column":1},"end":{"line":133,"column":1}},"children":[{"type":"text","value":"(","key":"e7hGRlg1ke"},{"type":"text","value":"5","key":"eN1aVJgbKG"},{"type":"text","value":")","key":"mjY0LowiZK"}],"identifier":"gamma_loss_amau","label":"gamma_loss_amau","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"gamma-loss-amau","key":"lv7jzs5sSZ"},{"type":"text","value":" could be extended to elongated ","key":"x7f8qTzGVY"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"MeuRJlP9vr"}],"key":"zlN10jr93W"},{"type":"text","value":"s, further checks are needed. Second, we never took into account the harmonic trap hence the fact that the ","key":"Cv9b9y4uYA"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"ykPx0qhjck"}],"key":"P9gvYGfL70"},{"type":"text","value":" is not homogeneous along ","key":"EasA0cjlwp"},{"type":"inlineMath","value":"z","position":{"start":{"line":133,"column":1},"end":{"line":133,"column":1}},"html":"zzz","key":"J16H6CS8Q2"},{"type":"text","value":".","position":{"start":{"line":133,"column":1},"end":{"line":133,"column":1}},"key":"koa1ffFh82"}],"key":"dLbmAcQNnW"},{"type":"paragraph","position":{"start":{"line":136,"column":1},"end":{"line":136,"column":1}},"children":[{"type":"text","value":"Our ","key":"KW7XvS0WMn"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"FasSAY2a9N"}],"key":"cxtSY73NsZ"},{"type":"text","value":" is not a 1D gas : as introduced in the first chapter, Landau decay of Bogoliubov excitations in 3D gases were also studied by ","key":"m2cpPzLcFm"},{"type":"cite","identifier":"pitaevskii_landau_1997","label":"pitaevskii_landau_1997","kind":"narrative","position":{"start":{"line":136,"column":135},"end":{"line":136,"column":158}},"children":[{"type":"text","value":"Pitaevskii & Stringari (1997)","key":"Da42YkBL1W"}],"enumerator":"4","key":"Zl9sZaS5eJ"},{"type":"text","value":" as","position":{"start":{"line":136,"column":1},"end":{"line":136,"column":1}},"key":"By3tAqaJuu"}],"key":"bcT7rfvLWc"},{"type":"math","value":"\\Gamma_k =\\frac{27\\pi}{17}\\omega_k \\times \\frac{2\\pi^2(k_b T)^4}{45\\hbar^3c_s^5mn}","tight":true,"html":"Γk=27π17ωk×2π2(kbT)445ℏ3cs5mn\\Gamma_k =\\frac{27\\pi}{17}\\omega_k \\times \\frac{2\\pi^2(k_b T)^4}{45\\hbar^3c_s^5mn}Γk​=1727π​ωk​×45ℏ3cs5​mn2π2(kb​T)4​","enumerator":"7","key":"te4tQMxur6"},{"type":"paragraph","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"key":"MvUDSXVgLS"},{"type":"inlineMath","value":"n","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"html":"nnn","key":"nObJYZpOFp"},{"type":"text","value":" is the density. We reported this decay rate as black square on ","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"key":"Iuz119JjGG"},{"type":"crossReference","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"children":[{"type":"text","value":"Figure ","key":"JQvu4oaJQ7"},{"type":"text","value":"3","key":"SMOwjVcPUM"}],"identifier":"growth_rate_comparison","label":"growth_rate_comparison","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"growth-rate-comparison","key":"aLYtjt8Tgy"},{"type":"text","value":". Because of the fourth power dependance with the temperature, the error-bars are much bigger for two points. We do not observe a better agreement with experimental points. Note that it was also derived for homogeneous condensate.","position":{"start":{"line":140,"column":1},"end":{"line":140,"column":1}},"key":"xUMVJvA7SM"}],"key":"XNKEPonihL"},{"type":"paragraph","position":{"start":{"line":142,"column":1},"end":{"line":142,"column":1}},"children":[{"type":"text","value":"To conclude this part, as predicted by ","position":{"start":{"line":142,"column":1},"end":{"line":142,"column":1}},"key":"LvjEFlKkja"},{"type":"cite","identifier":"micheli_phonon_2022","label":"micheli_phonon_2022","kind":"narrative","position":{"start":{"line":142,"column":40},"end":{"line":142,"column":60}},"children":[{"type":"text","value":"Micheli & Robertson (2022)","key":"kePJexgI7W"}],"enumerator":"1","key":"Qjz14z62q5"},{"type":"text","value":" we observed the slowing down of the exponential growth. However, the decay rate that we measure is not capture by the theoretical prediction. These results are however really preliminary and will lead to further investigations, both experimentally and theoretically.","position":{"start":{"line":142,"column":1},"end":{"line":142,"column":1}},"key":"TH6kcFRn41"}],"key":"vC2lIMkjDg"},{"type":"comment","value":" rho_n = 2 * np.pi**2*(kb*bec._temperature)**4/(45 * hbar**3 * bec._c_s**5)\n rho = bec._m * bec._n1p / bec._sigma0**2 ## It is not clear in the article that rho is the mass times density but I see only that.\n gamma_pita = 27/16*np.pi * bec._omega_perp * rho_n /rho ","key":"aDRVWonQqo"},{"type":"comment","value":"For small growth rate, the decay rate [](#gamma_loss_amau) is in good agreement with the measured value. However, the higher the growth rate, the less accurate is the prediction. Future works should try to explain such difference.","position":{"start":{"line":149,"column":1},"end":{"line":149,"column":1}},"key":"TxbWExCu5J"},{"type":"comment","value":"Even though the error-bars are quite large, we observe a good agreement between [](#gamma_loss_amau) and experimental data[^test_decay_rate].\nHere again, the quite large error bars prevent us from conclude to the accuracy of the decay rate expression [](#gamma_loss_amau), but this formula gives *at least* the good order of magnitude[^test_decay_rate].","position":{"start":{"line":151,"column":1},"end":{"line":152,"column":1}},"key":"WawBEB7My4"},{"type":"comment","value":"Another drawback must be emphasized: it was shown that the decay rate [](#gamma_loss_amau) was accurate for quasi-BEC. In their work, authors explain that other effects must be taken into account when the coherence length of the qBEC become of the order of the BEC size. In our situation, we are closer to the TF regime than the quasi-1D regime: we do not observe phase fluctuations and the 1D parameter is $a_sn_1\\sim 2$ ($\\chi \\sim 30$). It means that the correlation radius is much larger than the TF radius and the theoretical formula [](#gamma_loss_amau) does not strictly apply. However, numerical simulations following this experiment confirmed that the decay rate should accurately predict the slowing of the exponential growth [@micheli_twa_2024].","position":{"start":{"line":155,"column":1},"end":{"line":155,"column":1}},"key":"y6JYiROwGX"},{"type":"comment","value":"Also, this formula was derived for a quasi-BEC. Here, we have $a_sn_1\\sim 2$ and no phase fluctuation hence we are outside the range of application of the derivation of the decay rate.","position":{"start":{"line":158,"column":1},"end":{"line":158,"column":1}},"key":"D4kWZubDWI"},{"type":"comment","value":" \n### On the interest of such measurement\n\nLet me emphasize briefly the interest of such measurement. Contrary to 3D systems, the decay of phonons in a 1D Bose gas is still an open question and triggers many interests. The integrability of the Lieb-Liniger model implies that the quasi-particles lifetime is infinite, the latter being eigenstates of the Hamiltonian. For example, recently, @bouchoule_relaxation_2023 related the phonon mode occupation in terms of the *real* quasi-particles and predicted phonon relaxation to occur. In their work, authors argue that such relaxation occurs because phonons are in fact not the exact eigenstates of the Lieb-Liniger Hamiltonian. Note however that they do not expect the final mode occupation distribution to reach a thermal distribution. In that sense, it is not a decay of the phonon in contact of a thermal bath. Still, some authors derived a decay rate for phonons in 1D systems: see @ristivojevic_decay_2014 ($\\Gamma_k \\propto k^7$) and @andreev_hydrodynamics_1980 ($\\Gamma_k\\propto k^{3/2}$). The latter result triggered many interests since the paper by @kulkarni_finite_temperature_2013 which relates the phonon damping to the \"Kardar-Parisi-Zhang universality class of dynamical critical phenomena\". For two reasons, we cannot plug and use the decay rate derived by @andreev_hydrodynamics_1980. First, the pre-factor of the decay rate was shown not accurate by a factor 5 by @kulkarni_finite_temperature_2013. Authors explain that they do not expect this pre-factor to be accurate because of the \"uncontrolled nature\" of Andreev's approximation. However, the core of their argument is the scaling of the decay rate $\\Gamma_k\\propto k^{3/2}$. Second, this scaling was shown accurate for $k\\xi<10^{-2}$ while, in our case, we work with $k\\xi\\sim 1$. In fact, this quasi-particle regime ($k\\xi>1$) is the one for which the decay rate of @micheli_phonon_2022 was shown more accurate. On the other hand, our system is not strictly 1D while it is one of the assumption of the authors. However, preliminary numerical simulations by @micheli_twa_2024 showed that this result might be extended to BECs and not only quasi-BECs. ","key":"p2X2YHT7wo"},{"type":"comment","value":"Even though our system is not completely 1D, preliminary simulations by @micheli_twa_2024 showed that the decay rate still holds hence an experimental confirmation would be interesting.","position":{"start":{"line":168,"column":1},"end":{"line":168,"column":1}},"key":"JQjE7qyVkK"},{"type":"comment","value":"The decay rate from @micheli_phonon_2022 triggered many debates within the community. In fact, I think also that the terminology \"phonons\" does not help. Indeed, in their work authors refers to phonons as any quasi-particles (any $k$) while they observe their decay rate to be more accurate on the quasi-particle branch. Finally, to be totally transparent on the measurement of this growth rate, note data were taken blindly: it is only once we remarked the discrepancy between $G_k^{th}$ and $\\Gamma_k$ in left panel of [](#growth_rate_comparison) that we tried to explain it with various decay rates.","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"key":"BRgvdbUzJJ"},{"type":"comment","value":" \n### Decay rate of phonons in a Bose gas\n\nContrary to 3D systems, the decay of phonons in a 1D Bose gas is still an open question and triggers many interests. The integrability of the Lieb-Liniger model implies that the quasi-particles lifetime is infinite, the latter being eigenstates of the Hamiltonian. Still, some authors proposed a decay rate for phonons in 1D systems: see @ristivojevic_decay_2014 ($\\Gamma_k \\propto k^7$) and @andreev_hydrodynamics_1980 ($\\Gamma_k\\propto k^{3/2}$). The latter result triggered many interests since the paper by @kulkarni_finite_temperature_2013 which relates the phonon damping to the \"Kardar-Parisi-Zhang universality class of dynamical critical phenomena\". For two reasons, we cannot plug and use the decay rate derived by @andreev_hydrodynamics_1980:\n * The prefactor of the decay rate was shown not accurate by a factor 5 by @kulkarni_finite_temperature_2013. Still, authors explain that they do not expect this prefactor to be accurate because of the \"uncontrolled nature\" of Andreev's approximation. However, the core of their argument is the scaling of the dacay rate $\\Gamma_k\\propto k^{3/2}$.\n * The $ k^{3/2}$ scaling was shown accurate for $k\\xi<10^{-2}$. In our case, we work with $k\\xi\\sim 1$. \n\nRecently, @bouchoule_relaxation_2023 related the phonon mode occupation in terms of the *real* quasi-particles and predicted phonon relaxation to occur. In their work, authors argue that such relaxation occurs because phonons *are not* the exact eigenstates of the Lieb-Liniger Hamiltonian. Note however that the authors do not expect the final mode occupation distribution to reach a thermal distribution. In that sense, it is not a decay of the phonon in contact to a thermal bath.\n%However, the numerical simulation by @kulkarni_finite_temperature_2013 expects the decay rate to be of the order of a fraction of the chemical potential at $k\\sim \\xi^{-1}$, which is consistent with the analytical formula @gamma_loss_amau and our result. \n\n\nIn our case, the BEC is not totally in the 1D regime: we should check if this decay rate can be explained by 3D damping rate reported in the literature. The Landau decay rate from @pitaevskii_landau_1997 is also consistent with our observations for some points (0.1 ms$^{-1}$ ). For each dataset, the temperature was slightly different. The damping rate scales with $T^4$ which result in really decay rate (0.02 ms$^{-1}$) for the coldest dataset. When evaluating the decay rate provided by @fedichev_damping_1998, I found 0.006 ms$^{-1}$, which is also too small by a factor 10.\n\n\nTo conclude this subsection, we saw that we successfully described the growth rate of the phonon occupation number. Our measured growth rate is compatible with the theoretical growth rate [](#gain_phonons) minus the decay rate [](#gamma_loss_amau) derived by @micheli_phonon_2022. \n","key":"YCukxPIvVb"},{"type":"heading","depth":2,"position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"children":[{"type":"text","value":"Saturation of the phonon growth","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"CzDJyX3yaD"}],"identifier":"saturation_growth_section","label":"saturation_growth_section","html_id":"saturation-growth-section","enumerator":"5","key":"PQk8Fx1jlA"},{"type":"container","kind":"figure","identifier":"saturation_effect_growth","label":"saturation_effect_growth","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/saturation_growthnew-d5ef6d46e5ea49891833ccc97ab2959d.png","alt":"Saturation of the exponential growth","width":"100%","align":"center","key":"k2gSsSO3gW","urlSource":"images/saturation_growthnew.png","urlOptimized":"/~gondret/phd_manuscript/build/saturation_growthnew-d5ef6d46e5ea49891833ccc97ab2959d.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":200,"column":1},"end":{"line":200,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"saturation_effect_growth","identifier":"saturation_effect_growth","html_id":"saturation-effect-growth","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"bPaNo7FTni"},{"type":"text","value":"4","key":"VdrNh6xbdE"},{"type":"text","value":":","key":"OatAyEuGO4"}],"template":"Figure %s:","key":"CF2r1Do2PS"},{"type":"text","value":"Left: number of detected atoms as a function of time in a integration volume of 3 mm/s. Right: 2D histogram plot of the atomic density as a function of time. ®Dataset taken on the 04/09/24.","position":{"start":{"line":200,"column":1},"end":{"line":200,"column":1}},"key":"T84IJThmnE"}],"key":"iqTQWBGwds"}],"key":"fsuUxSSOgW"}],"enumerator":"4","html_id":"saturation-effect-growth","key":"sbhvp3Cbls"},{"type":"comment","value":"If the decay rate [](#gamma_loss_amau) accurately predicts the slowing of the exponential growth, it does not capture the saturation of the growth process.","position":{"start":{"line":202,"column":1},"end":{"line":202,"column":1}},"key":"evfNEjTwca"},{"type":"paragraph","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"children":[{"type":"text","value":"In ","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"ccPKwMHnh5"},{"type":"crossReference","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"children":[{"type":"text","value":"Figure ","key":"rtkxXrm1FW"},{"type":"text","value":"4","key":"yydlZYcyi6"}],"identifier":"saturation_effect_growth","label":"saturation_effect_growth","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"saturation-effect-growth","key":"rZ0A1p9V0f"},{"type":"text","value":", we show the evolution of the phonon population over a wider range than previously observed. We take an integration size of 2 mm/s, a bit larger than the 1.5 mm/s mode size. Of course, the larger the integration volume, the larger the detected maximal population. The right panel represents the 2D histogram of the momentum density as a function of time.","position":{"start":{"line":203,"column":1},"end":{"line":203,"column":1}},"key":"EseMmFD8tg"}],"key":"cgCJeQeJ0H"},{"type":"paragraph","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"children":[{"type":"cite","identifier":"pylak_influence_2018","label":"pylak_influence_2018","kind":"narrative","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":22}},"children":[{"type":"text","value":"Pylak & Zin (2018)","key":"pXsAoocUSu"}],"enumerator":"5","key":"zBVjwLzMmf"},{"type":"text","value":" studied numerically (classical field simulations) the growth process. In their work, they show that for a temperature of 30 nK, the phonon occupation population is expected to saturate at a few hundred. Their theoretical expectation is coherent with our observation. In a different study, ","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"srsjdNr7xD"},{"type":"cite","identifier":"robertson_controlling_2017","label":"robertson_controlling_2017","kind":"narrative","position":{"start":{"line":206,"column":312},"end":{"line":206,"column":339}},"children":[{"type":"text","value":"Robertson ","key":"aIlCA3FPRG"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"ZZSycuswva"}],"key":"HWwFLaDaMs"},{"type":"text","value":" (2017)","key":"gOKdba9QhM"}],"enumerator":"6","key":"SiPDvpsUO1"},{"type":"text","value":" showed that non-linear effects can no longer be neglected when the phonon occupancy reaches the value ","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"bX11hS4fz4"},{"type":"inlineMath","value":"n_1/10\\delta k","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"html":"n1/10δkn_1/10\\delta kn1​/10δk","key":"PdxSKZjW2i"},{"type":"text","value":". Here, ","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"oSZYpqpwb8"},{"type":"inlineMath","value":"\\delta k","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"html":"δk\\delta kδk","key":"wzdraRN6yj"},{"type":"text","value":" is the momentum resonance width and ","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"YB8dwd3kW8"},{"type":"inlineMath","value":"n_1","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"html":"n1n_1n1​","key":"k8pnk8xqp8"},{"type":"text","value":" is the 1D density. In our configuration, this corresponds again to a few hundreds of particles.","position":{"start":{"line":206,"column":1},"end":{"line":206,"column":1}},"key":"APLoZJDvcm"}],"key":"dnFzEeDLMJ"},{"type":"comment","value":"Although, according to eq. [](#formula_gain_temps), the phonon population seems to increase with temperature (as parametric amplification squeezes both the thermal state and the vacuum), they observe that the higher the temperature, the smaller the asymptotic phonon occupation and the earlier the growth stops.","position":{"start":{"line":208,"column":1},"end":{"line":208,"column":1}},"key":"ERkchp4iSX"},{"type":"paragraph","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"children":[{"type":"text","value":"Within a linear stability analysis, ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"dfy85wBL9A"},{"type":"cite","identifier":"de_valcarcel_faraday_2002","label":"de_valcarcel_faraday_2002","kind":"narrative","position":{"start":{"line":213,"column":37},"end":{"line":213,"column":63}},"children":[{"type":"text","value":"Valcarcel (2002)","key":"jTE664F5DT"}],"enumerator":"7","key":"f0IFBP9fsa"},{"type":"text","value":" also studied the growth dynamics of a mode. From his work, we obtain the expected saturation of the population which was also derived by ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"MBtDFxlLtc"},{"type":"cite","identifier":"liebster_emergence_2023","label":"liebster_emergence_2023","kind":"narrative","position":{"start":{"line":213,"column":201},"end":{"line":213,"column":225}},"children":[{"type":"text","value":"Liebster ","key":"W1UKvV09PD"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"eh87HBDMTT"}],"key":"GNChaZjlLO"},{"type":"text","value":" (2023)","key":"DEIA0LDVvm"}],"enumerator":"8","key":"n90QYlpftX"}],"key":"Jt94OPwoL7"},{"type":"math","identifier":"formule_saturation","label":"formule_saturation","value":"N_{max} = N_0\\frac{ \\sqrt{G_k^2 - \\Gamma_k^2 }}{mc_s^2/\\hbar}\\frac{E_{k}}{5\\epsilon_{k} + 3mc_s^2}","tight":true,"html":"Nmax=N0Gk2−Γk2mcs2/ℏEk5ϵk+3mcs2N_{max} = N_0\\frac{ \\sqrt{G_k^2 - \\Gamma_k^2 }}{mc_s^2/\\hbar}\\frac{E_{k}}{5\\epsilon_{k} + 3mc_s^2}Nmax​=N0​mcs2​/ℏGk2​−Γk2​​​5ϵk​+3mcs2​Ek​​","enumerator":"8","html_id":"formule-saturation","key":"N4pua9Y8Zs"},{"type":"paragraph","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"children":[{"type":"text","value":"where ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"ktu1p2h4kw"},{"type":"inlineMath","value":"\\epsilon_{k} = \\hbar^2k_{ph}^2/2m","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"ϵk=ℏ2kph2/2m\\epsilon_{k} = \\hbar^2k_{ph}^2/2mϵk​=ℏ2kph2​/2m","key":"dotoaVuj4N"},{"type":"text","value":" is the phonon kinetic energy and ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"RdESiKax1q"},{"type":"inlineMath","value":"E_{k}^2=\\epsilon_k^2+2mc_s^2\\epsilon_k","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"Ek2=ϵk2+2mcs2ϵkE_{k}^2=\\epsilon_k^2+2mc_s^2\\epsilon_kEk2​=ϵk2​+2mcs2​ϵk​","key":"qPAaLOJGos"},{"type":"text","value":" its energy. Here ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"AfhdUXPvyZ"},{"type":"inlineMath","value":"G_k","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"GkG_kGk​","key":"AROjNLUGUY"},{"type":"text","value":" refers to the theoretical gain and ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"aJcxlGWtau"},{"type":"inlineMath","value":"\\Gamma_k","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"Γk\\Gamma_kΓk​","key":"eMCxqcmJta"},{"type":"text","value":" the decay rate that we discussed above. We measure ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"UUCtbdekyn"},{"type":"inlineMath","value":"G_k=0.92(4)","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"Gk=0.92(4)G_k=0.92(4)Gk​=0.92(4)","key":"H8woanH4kd"},{"type":"text","value":" ms","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"y0PaWaDyT0"},{"type":"superscript","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"children":[{"type":"text","value":"-1","key":"qd2EQyxAgH"}],"key":"qxN5fwxs0C"},{"type":"text","value":" using Eq. ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"aw3svFS4dC"},{"type":"crossReference","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"children":[{"type":"text","value":"(","key":"LZYAozmdql"},{"type":"text","value":"3","key":"eTGH2aWPrJ"},{"type":"text","value":")","key":"bfK7BZZpXL"}],"identifier":"gain_phonons","label":"gain_phonons","kind":"equation","template":"(%s)","enumerator":"3","resolved":true,"html_id":"gain-phonons","key":"sccjXvRhkH"},{"type":"text","value":" and extract ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"PsLeBV9C3b"},{"type":"inlineMath","value":"\\Gamma_k=0.12(4)","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"html":"Γk=0.12(4)\\Gamma_k=0.12(4)Γk​=0.12(4)","key":"kqcvqSCkub"},{"type":"text","value":" using the exponential growth fit, shown as a solid line in ","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"iZUZPY67xM"},{"type":"crossReference","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"children":[{"type":"text","value":"Figure ","key":"zmqbylS7E2"},{"type":"text","value":"4","key":"McAJov5kEI"}],"identifier":"saturation_effect_growth","label":"saturation_effect_growth","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"saturation-effect-growth","key":"RhaAO4dQtC"},{"type":"text","value":". Here again, the higher uncertainty comes from the ","key":"SIGCcB46Bm"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"sCDGMpTcpa"}],"key":"i0XGcC2zwR"},{"type":"text","value":" atom number. We define the ","key":"iUAoMj0Tm4"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"AFTSelDo19"}],"key":"BpcOnQfqKN"},{"type":"text","value":" properties using the phonon speeds which is in between 8 mm/s (short time) and 10 mm/s (long time). We estimate the maximal population to 420(50). In ","key":"nUNNCCV4uv"},{"type":"crossReference","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"children":[{"type":"text","value":"Figure ","key":"MSorcMBCWY"},{"type":"text","value":"4","key":"i71OIXzMfT"}],"identifier":"saturation_effect_growth","label":"saturation_effect_growth","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"saturation-effect-growth","key":"wEzjH423TU"},{"type":"text","value":", we observe the atom number saturates around 150 (integration volume of 1.5 mm/s) and 200 (for a 2 mm/s integration volume). The difference between the two can be explained by the efficiency of our detector, which ranges between 20% and 50%.","position":{"start":{"line":218,"column":1},"end":{"line":218,"column":1}},"key":"ETAdVk4ITV"}],"key":"JrwryfAO0g"},{"type":"paragraph","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"children":[{"type":"strong","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"children":[{"type":"text","value":"Conclusion:","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"key":"aoVJGvpU7W"}],"key":"VzXBlVqmE0"},{"type":"text","value":" Different theoretical works expect a saturation of the phonon production once the population reaches a few hundred quasi-particles. Here, we also observe such saturation; however, a quantitative study seems complicated. First, the atom number approaches the saturation limit of our detector, and second, our high uncertainty on the detection efficiency clearly limits our knowledge.","position":{"start":{"line":221,"column":1},"end":{"line":221,"column":1}},"key":"POXaK2gvHU"}],"key":"nGJdKcqTVs"},{"type":"heading","depth":2,"position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"children":[{"type":"text","value":"Shift of the density peaks","position":{"start":{"line":224,"column":1},"end":{"line":224,"column":1}},"key":"eoJl0nJW7u"}],"identifier":"shift_section","label":"shift_section","html_id":"shift-section","enumerator":"6","key":"Ocw3tFiiAB"},{"type":"paragraph","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"children":[{"type":"text","value":"On the 2D histograms of ","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"key":"F75L5O4ZPS"},{"type":"crossReference","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"children":[{"type":"text","value":"Figure ","key":"s4MzLQLLvh"},{"type":"text","value":"1","key":"yDJ9M0LqfJ"}],"identifier":"exponential_creation_oscillation","label":"exponential_creation_oscillation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"exponential-creation-oscillation","key":"bLED0gNCTq"},{"type":"text","value":" and ","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"key":"Kb0HcfQWQo"},{"type":"crossReference","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"children":[{"type":"text","value":"Figure ","key":"CQP7OsZpZ2"},{"type":"text","value":"4","key":"WFELCrHSzb"}],"identifier":"saturation_effect_growth","label":"saturation_effect_growth","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"saturation-effect-growth","key":"qkZlLJzBKK"},{"type":"text","value":", one can see that the two phonon peaks seem to move apart. This subsection reports on such phenomena. On the left panel of ","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"key":"DUaCFMsi6h"},{"type":"crossReference","position":{"start":{"line":226,"column":1},"end":{"line":226,"column":1}},"children":[{"type":"text","value":"Figure ","key":"MNeOcHC1tr"},{"type":"text","value":"5","key":"MYifLUpWFV"}],"identifier":"displacement_of_the_peaks","label":"displacement_of_the_peaks","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"displacement-of-the-peaks","key":"czBo6Tv3DI"},{"type":"text","value":", we show the momentum distribution at different times, ranging from 2 ms (dark green) to 6 ms (light green). We note that the width of the peaks increases with time and an asymmetry in their broadening. Furthermore, the peaks maximum shift away from the ","key":"t9tClahLqP"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"D7aNF44FuJ"}],"key":"Yac2ScRPjo"},{"type":"text","value":": the momentum of the density maximum is smaller at short time (dark color) than at long time (light green).","key":"opVZWaCYH3"}],"key":"wWQ6Ei7yXE"},{"type":"container","kind":"figure","identifier":"displacement_of_the_peaks","label":"displacement_of_the_peaks","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/density_plot_ecart_p-b575c215c2aee2e97e04e68d04d07bc5.png","alt":"Displacement of the resonant peak","width":"100%","align":"center","key":"MQFe9OFQru","urlSource":"images/density_plot_ecart_peaks.png","urlOptimized":"/~gondret/phd_manuscript/build/density_plot_ecart_p-b575c215c2aee2e97e04e68d04d07bc5.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":236,"column":1},"end":{"line":236,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"displacement_of_the_peaks","identifier":"displacement_of_the_peaks","html_id":"displacement-of-the-peaks","enumerator":"5","children":[{"type":"text","value":"Figure ","key":"yzPfuEwthO"},{"type":"text","value":"5","key":"xHTNCY6Vde"},{"type":"text","value":":","key":"HbDCAB3R8R"}],"template":"Figure %s:","key":"pMKLJXcqUO"},{"type":"text","value":"Left: atomic density at different times, from 2 ms (darkest color) to 6 ms (lightest color). Right: Position of the peak density as a function of time. Each color/symbol corresponds to different growth rate of ","position":{"start":{"line":236,"column":1},"end":{"line":236,"column":1}},"key":"d3shyzmVBn"},{"type":"crossReference","position":{"start":{"line":236,"column":1},"end":{"line":236,"column":1}},"children":[{"type":"text","value":"Figure ","key":"btbfw9Fz3W"},{"type":"text","value":"3","key":"TMjHQNoL28"}],"identifier":"growth_rate_comparison","label":"growth_rate_comparison","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"growth-rate-comparison","key":"W66Mnyiek5"},{"type":"text","value":". ®Data taken on the 25/04/24 (left) and September 2024 (right).","position":{"start":{"line":236,"column":1},"end":{"line":236,"column":1}},"key":"Wmpf3gdtXD"}],"key":"gLPWPvNt7g"}],"key":"iHh8eZfTY4"}],"enumerator":"5","html_id":"displacement-of-the-peaks","key":"JNUgOO3DCT"},{"type":"paragraph","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"children":[{"type":"text","value":"Faraday waves in quantum gases were studied in other groups. The first work conducted by ","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"aTfm5Ak8RB"},{"type":"cite","identifier":"engels_observation_2007","label":"engels_observation_2007","kind":"narrative","position":{"start":{"line":239,"column":90},"end":{"line":239,"column":114}},"children":[{"type":"text","value":"Engels ","key":"XJazeTRVJQ"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"eiZJ2lUk7T"}],"key":"bAbLTmEu5t"},{"type":"text","value":" (2007)","key":"P91rciCC3r"}],"enumerator":"9","key":"c9MRcluqiO"},{"type":"text","value":" studied the wave vector dependance with the excitation frequency, but did not report on a shift in time of the wave vector. This study triggered many theoretical works, especially by ","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"f5zz6EY1z5"},{"type":"cite","identifier":"nicolin_faraday_2007","label":"nicolin_faraday_2007","kind":"narrative","position":{"start":{"line":239,"column":298},"end":{"line":239,"column":319}},"children":[{"type":"text","value":"Nicolin ","key":"PthwyXykfK"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"JkJ83wVClx"}],"key":"OXm7Ishnx2"},{"type":"text","value":" (2007)","key":"DiXc8Qp9KU"}],"enumerator":"10","key":"pkewquHNT5"},{"type":"text","value":" and ","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"TkiYViAaOx"},{"type":"cite","identifier":"nicolin_resonant_2011","label":"nicolin_resonant_2011","kind":"narrative","position":{"start":{"line":239,"column":324},"end":{"line":239,"column":346}},"children":[{"type":"text","value":"Nicolin (2011)","key":"k7APjbJxex"}],"enumerator":"11","key":"L2xV7hg8sJ"},{"type":"text","value":". In their work, the authors performed 3D Gross-Pitaevskii simulation of the system, focusing on the resonant wave vector ","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"RxY6RiwSsa"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"html":"kresk_{res}kres​","key":"DqKfwDwcVf"},{"type":"text","value":". However, neither do they report on a shift in time of ","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"kvALjvlIeL"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"html":"kresk_{res}kres​","key":"wtAUVydM5w"},{"type":"text","value":". They also provide raw images of their numerical simulation at different time: based on these data, no drift is observed.","position":{"start":{"line":239,"column":1},"end":{"line":239,"column":1}},"key":"OKWpPliGmN"}],"key":"cPvZgrZvZw"},{"type":"paragraph","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"text","value":"In his PhD thesis, ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"t5jTuKl2hh"},{"type":"cite","identifier":"groot_2015_excitations","label":"groot_2015_excitations","kind":"narrative","position":{"start":{"line":241,"column":20},"end":{"line":241,"column":43}},"children":[{"type":"text","value":"Groot (2015)","key":"ZDwIAXAO7E"}],"enumerator":"12","key":"KoFQLySbu0"},{"type":"text","value":" reports on such drift of ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"WW1JFoW7mz"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"html":"kresk_{res}kres​","key":"epn9Upl3RM"},{"type":"text","value":". In addition, he reports on the “revival” of the Faraday wave: the spatial modulation appears then disappears before reappearing again. Such revival was further studied by ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"LwfCzcDluH"},{"type":"cite","identifier":"nguyen_parametric_2019","label":"nguyen_parametric_2019","kind":"narrative","position":{"start":{"line":241,"column":252},"end":{"line":241,"column":275}},"children":[{"type":"text","value":"Nguyen ","key":"a6jnwI2S87"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"lvcoxyGBsF"}],"key":"Riu9AlMg9L"},{"type":"text","value":" (2019)","key":"WBcZiLlqq4"}],"enumerator":"13","key":"fIT1VMA8uk"},{"type":"text","value":" who showed it was related to the axial breathing mode of the ","key":"w4vXApqtup"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"jp9iVyO4Eh"}],"key":"OeKh7vruM8"},{"type":"text","value":" (lowest quadrupolar mode), whose frequency is between ","key":"cO2pRPB0e8"},{"type":"inlineMath","value":"\\sqrt{5/2}\\omega_z","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"html":"5/2ωz\\sqrt{5/2}\\omega_z5/2​ωz​","key":"bsID90Q7dw"},{"type":"text","value":" (3D cigar-shape) and ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"fbBnoxhCk4"},{"type":"inlineMath","value":"\\sqrt{3}\\omega_z","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"html":"3ωz\\sqrt{3}\\omega_z3​ωz​","key":"ol9SvqlPeH"},{"type":"text","value":" (1D Thomas-Fermi) ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"FUPEH5ws9Y"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"children":[{"type":"cite","identifier":"stringari_collective_1996","label":"stringari_collective_1996","kind":"parenthetical","position":{"start":{"line":241,"column":475},"end":{"line":241,"column":501}},"children":[{"type":"text","value":"Stringari, 1996","key":"P0eglRyKf1"}],"enumerator":"14","key":"K8V4Jty8DG"}],"key":"oIqS3MpG9n"},{"type":"text","value":". However, the authors do not report on a shift in time. Similar experiment was carried by ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"ciiPiIJ1uE"},{"type":"cite","identifier":"hernandez_rajkov_faraday_2021","label":"hernandez_rajkov_faraday_2021","kind":"narrative","position":{"start":{"line":241,"column":594},"end":{"line":241,"column":624}},"children":[{"type":"text","value":"Hernández-Rajkov ","key":"KTav9VATZn"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Z1hPyV1lSe"}],"key":"oEsTGMIIYO"},{"type":"text","value":" (2021)","key":"qjxuOuF9NU"}],"enumerator":"15","key":"YxhuDf8op7"},{"type":"text","value":" with a molecular Fermi superfluid. In their work, they provide the time evolution of the density Fourier spectrum. Clearly, even though the saturation of the colormap of their figure does not allow a precise measurement, ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"fMSzz59qPQ"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"html":"kresk_{res}kres​","key":"m5bz2rUkkj"},{"type":"text","value":" does not seem to shift in time. Faraday-like patterns were also studied on a 2D setup by ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"bR4h2CoPcF"},{"type":"cite","identifier":"liebster_emergence_2023","label":"liebster_emergence_2023","kind":"narrative","position":{"start":{"line":241,"column":945},"end":{"line":241,"column":969}},"children":[{"type":"text","value":"Liebster ","key":"gxvmsD9wyI"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"Y3jCxKV6Ze"}],"key":"EEDH3Pn3ek"},{"type":"text","value":" (2023)","key":"OvteuWxkZi"}],"enumerator":"8","key":"a5NKWckeXS"},{"type":"text","value":". In their work, the authors provide the Fourier transform of the density for different times: we observe on their setup a shift of ","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"YCVFOcTAzq"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"html":"kresk_{res}kres​","key":"TrRkWq4xx7"},{"type":"footnoteReference","identifier":"observation_shift","label":"observation_shift","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"number":4,"enumerator":"4","key":"Lrk7VOK8kN"},{"type":"text","value":".","position":{"start":{"line":241,"column":1},"end":{"line":241,"column":1}},"key":"d8Qr775C5T"}],"key":"nAekN7laBS"},{"type":"paragraph","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"children":[{"type":"text","value":"In the work of ","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"key":"hk9vf8hhWv"},{"type":"cite","identifier":"groot_2015_excitations","label":"groot_2015_excitations","kind":"narrative","position":{"start":{"line":245,"column":16},"end":{"line":245,"column":39}},"children":[{"type":"text","value":"Groot (2015)","key":"iDPOPc1slt"}],"enumerator":"12","key":"lDzj1t4L0J"},{"type":"text","value":", the author relates the shift of ","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"key":"uxMONNF181"},{"type":"inlineMath","value":"k_{res}","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"html":"kresk_{res}kres​","key":"WPuQCw5PFf"},{"type":"text","value":" to the axial breathing mode of the ","key":"qNqfwqkpwC"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"tFIOExOh2f"}],"key":"eWDQEq8akX"},{"type":"text","value":". In the left panel of ","key":"IjoZtJMppo"},{"type":"crossReference","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"children":[{"type":"text","value":"Figure ","key":"XxIxQoY6C7"},{"type":"text","value":"5","key":"UwgsDu6RV1"}],"identifier":"displacement_of_the_peaks","label":"displacement_of_the_peaks","kind":"figure","template":"Figure %s","enumerator":"5","resolved":true,"html_id":"displacement-of-the-peaks","key":"KmFoADkH4c"},{"type":"text","value":", we plot the speed of the resonant phonon as a function of time. Each color and marker shows a different modulation strength hence growth rate. If the shift is related to the trap frequency, the displacement should not depend on the growth rate. On the data reported, the trap frequency is constant, and we observe a different shift dynamics. Here, the speed of the peak displacement seems to depend on the modulation strength. This pushes for a complementary explanation to this harmonic trapping effect. Furthermore, this shift is also observed on the experiment conducted by ","position":{"start":{"line":245,"column":1},"end":{"line":245,"column":1}},"key":"QIbj69Wxzm"},{"type":"cite","identifier":"liebster_emergence_2023","label":"liebster_emergence_2023","kind":"narrative","position":{"start":{"line":245,"column":754},"end":{"line":245,"column":778}},"children":[{"type":"text","value":"Liebster ","key":"aDHCZVKhUi"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"bETXe8blyL"}],"key":"IDyr0Agtxi"},{"type":"text","value":" (2023)","key":"raHdc3Bw3G"}],"enumerator":"8","key":"DSSfhN20wO"},{"type":"text","value":" in which the authors have a homogeneous ","key":"Vdx3CJMSBK"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"WwXVzGWeg0"}],"key":"pugfZjOlme"},{"type":"text","value":".","key":"iMRO2gsSMN"}],"key":"vrXK1yGNtT"},{"type":"paragraph","position":{"start":{"line":248,"column":1},"end":{"line":248,"column":1}},"children":[{"type":"text","value":"To conclude this section, the shift of these phonon peaks remains un-explained. To further investigate this, it would be interesting to witness the revival of the Faraday pattern reported by ","position":{"start":{"line":248,"column":1},"end":{"line":248,"column":1}},"key":"zjCf6jEqiu"},{"type":"cite","identifier":"nguyen_parametric_2019","label":"nguyen_parametric_2019","kind":"narrative","position":{"start":{"line":248,"column":192},"end":{"line":248,"column":215}},"children":[{"type":"text","value":"Nguyen ","key":"usj8GRhOiU"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"mvEDZhycod"}],"key":"sfik16Q3FZ"},{"type":"text","value":" (2019)","key":"qdViZmsiAv"}],"enumerator":"13","key":"Rs3VlBNgMu"},{"type":"text","value":".","position":{"start":{"line":248,"column":1},"end":{"line":248,"column":1}},"key":"f6yMIpvJzG"}],"key":"lpS5m2xZ5t"},{"type":"comment","value":" So far, we considered only homogeneous BECs while the trapping potential is harmonic. It means that the quasi-particle momentum $k$ is not a good quantum number and that quasi-particle is a superposition of Fourier. ","key":"OXcB25haNm"},{"type":"comment","value":" **Conclusion:** to conclude this section, we (sadly) did not (yet?) understood the shift in momentum of the mainly excited mode. To further investigate this, it would be interesting to witness the revival of the Faraday pattern reported by @nguyen_parametric_2019. ","key":"arQi2jqsWg"},{"type":"comment","value":"This shift is not observed in the available numerics by @nicolin_faraday_2007, and it is not reported by the recent experiment of @nguyen_parametric_2019. It is however also observed on a homogeneous 2D system [@liebster_emergence_2023].","position":{"start":{"line":255,"column":1},"end":{"line":255,"column":1}},"key":"OSQGE9MpHW"},{"type":"comment","value":"### Growth of the phonon number: comparison between theory and experiment\nThe most striking thing when comparing [theory](#controlling_creation_phonons) and [](#exponential_creation_oscillation), is the amplitude of the oscillations. Here, the fit gives 0.48(11) for the negative peak and 0.46(9) for the positive peak. This 50% relative amplitude is much larger than what we expected in the theoretical [section](#controlling_creation_phonons).\nThe large fit uncertainty associated to the fit result makes however this remark pointless.","position":{"start":{"line":257,"column":1},"end":{"line":259,"column":1}},"key":"q2nNs3kWo4"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Summary","position":{"start":{"line":262,"column":1},"end":{"line":262,"column":1}},"key":"NcD8UZyDkZ"}],"key":"QMIDNVjPit"},{"type":"paragraph","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"children":[{"type":"text","value":"In this section, we observe the exponential growth of the phonon occupation number, on almost two decades. We show that the growth dynamics is in agreement with theoretical predictions. 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We also observe the saturation of the growth dynamic. Here again, our result is in agreement with theoretical predictions even though our setup is not well suited to measure such a high atom number. In the last section ","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"key":"FMeWI6qDWb"},{"type":"crossReference","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"children":[{"type":"text","value":"6","key":"Pr5KGa9D7A"}],"identifier":"shift_section","label":"shift_section","kind":"heading","template":"Section %s","enumerator":"6","resolved":true,"html_id":"shift-section","key":"jNNa0mNiCg"},{"type":"text","value":", we report on the shift of the phonon peaks whose origin remains unknown.","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"key":"pkMiHTynZm"}],"key":"IwAGmWHaqk"}],"key":"LkDPPs6Gkz"},{"type":"footnoteDefinition","identifier":"note_obtention_equation","label":"note_obtention_equation","position":{"start":{"line":263,"column":1},"end":{"line":263,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"To obtain this equation, we express the atomic field population in the phonon basis, as in equation ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"ScekRSXbjV"},{"type":"crossReference","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"(","key":"qRQV36Kq21"},{"type":"text","value":"18","key":"WOhRoichbD"},{"type":"text","value":")","key":"F85YvbnKQI"}],"identifier":"jenaimarre","label":"jenaimarre","kind":"equation","template":"(%s)","enumerator":"18","resolved":true,"html_id":"jenaimarre","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"qz5SdZI2Ar"},{"type":"text","value":". We then use the phonon operator ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"myfnl6cDIr"},{"type":"crossReference","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"evolution","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"u53uz2HhDj"}],"identifier":"parametric_amplification_phonons_th","label":"parametric_amplification_phonons_th","kind":"heading","template":"{name}","resolved":true,"html_id":"parametric-amplification-phonons-th","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"hheEtp3jgR"},{"type":"text","value":" during parametric amplification in equation ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"jwCw9V1jbJ"},{"type":"crossReference","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"children":[{"type":"text","value":"(","key":"WWR7GOwcKC"},{"type":"text","value":"12","key":"J2HhbVkxgM"},{"type":"text","value":")","key":"JEaICQOIxS"}],"identifier":"evolution_bk","label":"evolution_bk","kind":"equation","template":"(%s)","enumerator":"12","resolved":true,"html_id":"evolution-bk","remote":true,"url":"/dce-bogoliubov","dataUrl":"/dce-bogoliubov.json","key":"F9toFB2DtS"},{"type":"text","value":". We then assume ","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"key":"eAwPqIhRIB"},{"type":"inlineMath","value":"\\alpha_k \\sim \\beta_k","position":{"start":{"line":64,"column":1},"end":{"line":64,"column":1}},"html":"αk∼βk\\alpha_k \\sim \\beta_kαk​∼βk​","key":"aJeUmEK32o"},{"type":"text","value":" (see their definition in Eq. 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