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We end this historical journey with the experiment by ","key":"eBngaxvyJc"},{"type":"cite","identifier":"chevy_transverse_2002","label":"chevy_transverse_2002","kind":"narrative","position":{"start":{"line":14,"column":374},"end":{"line":14,"column":396}},"children":[{"type":"text","value":"Chevy ","key":"p2d9tSHJ70"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"cFIgzKKkG8"}],"key":"bhG5wwnz5w"},{"type":"text","value":" (2002)","key":"GoWLUutkhn"}],"enumerator":"1","key":"kOwQ6rfyQY"},{"type":"text","value":" that observed the un-damped breathing mode. This specific collective mode is the topic of the next section. 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While we neglected the interaction with the non-condensed gas in equation ","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"key":"ZbJ3p7hPF4"},{"type":"crossReference","position":{"start":{"line":28,"column":1},"end":{"line":28,"column":1}},"children":[{"type":"text","value":"(","key":"s3wmFQq7Gm"},{"type":"text","value":"1","key":"YIc3B8Maao"},{"type":"text","value":")","key":"Ghg6hzTafD"}],"identifier":"gpe_with_time","label":"gpe_with_time","kind":"equation","template":"(%s)","enumerator":"1","resolved":true,"html_id":"gpe-with-time","key":"FZpXFKhxzc"},{"type":"text","value":", we can describe the influence of temperature by adding damping to these collective excitations. 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However, it was noted by ","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"lVU86vSuMd"},{"type":"cite","identifier":"fedichev_damping_1998","label":"fedichev_damping_1998","kind":"narrative","position":{"start":{"line":29,"column":508},"end":{"line":29,"column":530}},"children":[{"type":"text","value":"Fedichev ","key":"k6nZUqDqJj"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"NpBX5rREmq"}],"key":"c4LPCGDNIm"},{"type":"text","value":" (1998)","key":"LZNOoysazQ"}],"enumerator":"16","key":"ttXd9ZTnvo"},{"type":"text","value":" that the value of the damping rate “drastically depends on the trapping geometry”.","position":{"start":{"line":29,"column":1},"end":{"line":29,"column":1}},"key":"pqTTgi98ye"}],"key":"ST0YKIno10"},{"type":"listItem","spread":true,"position":{"start":{"line":30,"column":1},"end":{"line":32,"column":1}},"children":[{"type":"text","value":"Beliaev damping refers to the decay of a single excitation into two lower-energy excitations ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"lqGtxk8K0o"},{"type":"emphasis","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"i2z7uo1l7s"}],"key":"tXZygaUabz"},{"type":"text","value":" ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"cO7ht7rvnc"},{"type":"inlineMath","value":"\\nu \\rightarrow \\nu_1 + \\nu_2","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"html":"ν→ν1+ν2\\nu \\rightarrow \\nu_1 + \\nu_2ν→ν1​+ν2​","key":"aB0NPG43wF"},{"type":"text","value":". This damping occurs at zero temperature, as it was originally derived in this context by ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"favpVogfI0"},{"type":"cite","identifier":"beliaev_energy_1958","label":"beliaev_energy_1958","kind":"narrative","position":{"start":{"line":30,"column":223},"end":{"line":30,"column":243}},"children":[{"type":"text","value":"Beliaev (1958)","key":"gb21hGsu1V"}],"enumerator":"17","key":"PXeYj6xISA"},{"type":"text","value":" before being extended to non-zero temperatures by ","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"ZiEwZla2o1"},{"type":"cite","identifier":"popov1972hydrodynamic","label":"popov1972hydrodynamic","kind":"narrative","position":{"start":{"line":30,"column":294},"end":{"line":30,"column":316}},"children":[{"type":"text","value":"Popov (1972)","key":"X9UC3tu4iK"}],"enumerator":"18","key":"Qy10DhZF8J"},{"type":"text","value":".","position":{"start":{"line":30,"column":1},"end":{"line":30,"column":1}},"key":"idKag3Ol0Q"}],"key":"UpUE04yGqR"}],"key":"RpglJQpSFv"},{"type":"comment","value":"It was experimentally investigated by @katz_beliaev_2002 and theoretically applied to BECs by @giorgini_damping_1998. The numerical work by @das_trends_2001 studied both Landau and Beliaev damping mechanisms and their explicit dependence on the temperature, density, and quantum number of the excitation, emphasizing a rich yet complex behavior.","position":{"start":{"line":33,"column":1},"end":{"line":33,"column":1}},"key":"qNUxOAGu7u"},{"type":"paragraph","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"children":[{"type":"cite","identifier":"chevy_transverse_2002","label":"chevy_transverse_2002","kind":"narrative","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":23}},"children":[{"type":"text","value":"Chevy ","key":"A2SWTcQnAs"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"yvoVzfZRlO"}],"key":"BEJhLvPizQ"},{"type":"text","value":" (2002)","key":"R5oesYuNB8"}],"enumerator":"1","key":"o6h2g9u8wT"},{"type":"text","value":" reported the observation of an undamped collective oscillation: the breathing mode (monopole mode) of an elongated ","key":"Nthru62EAg"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"yYBmOftg2y"}],"key":"rsHDBWtjBT"},{"type":"text","value":". In this mode, the transverse radius of the ","key":"r25zniPC00"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"QwR7iezcUV"}],"key":"rJquFwDtFl"},{"type":"text","value":" oscillates at ","key":"NbFx5MSIgH"},{"type":"inlineMath","value":"2\\omega_\\perp","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"html":"2ω⊥2\\omega_\\perp2ω⊥​","key":"d2bOKXfTEW"},{"type":"text","value":", ","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"rFUBe0V2qq"},{"type":"emphasis","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"rln97esBm3"}],"key":"gXevSW4Zl1"},{"type":"text","value":", twice the transverse trap frequency. The authors showed that the damping of this breathing mode is very low compared to others already reported. They also showed that it is independent of the temperature. This “anomalously small measured damping rate” was numerically and theoretically studied by ","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"vyO5HkK5IJ"},{"type":"cite","identifier":"jackson_accidental_2002","label":"jackson_accidental_2002","kind":"narrative","position":{"start":{"line":35,"column":527},"end":{"line":35,"column":551}},"children":[{"type":"text","value":"Jackson & Zaremba (2002)","key":"NAT90UVJU8"}],"enumerator":"19","key":"uPUaBmy8iz"},{"type":"text","value":". They demonstrated that this was due to an “accidental suppression of Landau damping” for this specific mode (transverse breathing) and geometry (elongated). In the usual derivation of a decay rate, the non-condensed cloud is assumed to be in thermal equilibrium. Here, both the ","key":"h6Kqo1nfBK"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"H6Fp3xkZkf"}],"key":"s0aUpdlG6D"},{"type":"text","value":" and the thermal cloud oscillate at ","key":"pGIQxTtUNO"},{"type":"inlineMath","value":"2\\omega_\\perp","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"html":"2ω⊥2\\omega_\\perp2ω⊥​","key":"KwBwXWH86v"},{"type":"text","value":" ","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"mi7j88Pkht"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"children":[{"type":"cite","identifier":"castin_bose_einstein_1996","label":"castin_bose_einstein_1996","kind":"parenthetical","position":{"start":{"line":35,"column":887},"end":{"line":35,"column":913}},"children":[{"type":"text","value":"Castin & Dum, 1996","key":"OSbTler8bz"}],"enumerator":"20","key":"t3QzZcZpmo"},{"type":"cite","identifier":"kagan_evolution_1996","label":"kagan_evolution_1996","kind":"parenthetical","position":{"start":{"line":35,"column":914},"end":{"line":35,"column":936}},"children":[{"type":"text","value":"Kagan ","key":"y1pmFqHdqT"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"OfQ6rBx8do"}],"key":"EmgIlg8H3U"},{"type":"text","value":", 1996","key":"es0LXNOovf"}],"enumerator":"21","key":"xXqWYcCE7Q"}],"key":"RrPlRzA4ja"},{"type":"text","value":", resulting in the suppression of Landau damping. The origin of this damping was already suggested by ","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"inFQZF8pYy"},{"type":"cite","identifier":"pitaevskii_elementary_1998","label":"pitaevskii_elementary_1998","kind":"narrative","position":{"start":{"line":35,"column":1039},"end":{"line":35,"column":1066}},"children":[{"type":"text","value":"Pitaevskii & Stringari (1998)","key":"RquaKRhVXU"}],"enumerator":"22","key":"prUAJW7xHL"},{"type":"text","value":". In their work, they emphasized that this breathing mode “could produce a parametric instability [...] due to decay into two or more axial excitations,” which was further investigated by ","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"gEa4ukqnvV"},{"type":"cite","identifier":"kagan2001damping","label":"kagan2001damping","kind":"narrative","position":{"start":{"line":35,"column":1254},"end":{"line":35,"column":1271}},"children":[{"type":"text","value":"Kagan & Maksimov (2001)","key":"t2KDp6OGTA"}],"enumerator":"23","key":"GIWbb5f9Do"},{"type":"text","value":". Twenty-five years later, it is the subject of this work.","position":{"start":{"line":35,"column":1},"end":{"line":35,"column":1}},"key":"JDmeSiC1cf"}],"key":"DV48WN9cTB"},{"type":"paragraph","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"children":[{"type":"text","value":"In order to study entanglement of the longitudinal ","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"key":"SwxnJmUg4J"},{"type":"inlineMath","value":"(k,-k)","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"html":"(k,−k)(k,-k)(k,−k)","key":"TDWgdhdn7k"},{"type":"text","value":" modes, we will treat the breathing collective oscillation of the ","key":"tdRU0RK9pa"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"ZmADVxs1ey"}],"key":"v0BSZnY7Kr"},{"type":"text","value":" classically, while keeping the collective excitations of the gas along the long axis quantized. We will assume the ","key":"SQjwpLAVyZ"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"OO6aCypF2U"}],"key":"Qp7Mt2wfDx"},{"type":"text","value":" is homogeneous along the ","key":"F1hH3gnbGz"},{"type":"inlineMath","value":"z","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"html":"zzz","key":"aZE5dlDrQD"},{"type":"text","value":"-axis and factor out the transverse profile of the gas","position":{"start":{"line":38,"column":1},"end":{"line":38,"column":1}},"key":"vsDHSOAzAE"}],"key":"NfAWApBE4x"},{"type":"math","identifier":"field_decomposition_eq_bec3","label":"field_decomposition_eq_bec3","value":"\\hat{\\Psi} = \\Psi_0(r,t)[1+\\hat{\\phi}(z,t)].","tight":true,"html":"Ψ^=Ψ0(r,t)[1+ϕ^(z,t)].\\hat{\\Psi} = \\Psi_0(r,t)[1+\\hat{\\phi}(z,t)].Ψ^=Ψ0​(r,t)[1+ϕ^​(z,t)].","enumerator":"2","html_id":"field-decomposition-eq-bec3","key":"W6Z9cSmDcF"},{"type":"paragraph","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"children":[{"type":"text","value":"In the following section, we will study the evolution of ","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"GoIq2fdcix"},{"type":"inlineMath","value":"\\Psi_0(r,t)","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"html":"Ψ0(r,t)\\Psi_0(r,t)Ψ0​(r,t)","key":"jABr9vWqLL"},{"type":"text","value":" when the trap is modulated. Our approach is a special case of the description of ","key":"bMJQgRwSY0"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"acIrUeljOp"}],"key":"I1L2SCYB8f"},{"type":"text","value":"s in time-dependent traps by ","key":"Nc8asgxyUR"},{"type":"cite","identifier":"castin_bose_einstein_1996","label":"castin_bose_einstein_1996","kind":"narrative","position":{"start":{"line":43,"column":185},"end":{"line":43,"column":211}},"children":[{"type":"text","value":"Castin & Dum (1996)","key":"XU8faWyLow"}],"enumerator":"20","key":"KBPe7fla8s"},{"type":"text","value":" and ","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"nKbHYTVShK"},{"type":"cite","identifier":"kagan_evolution_1996","label":"kagan_evolution_1996","kind":"narrative","position":{"start":{"line":43,"column":216},"end":{"line":43,"column":237}},"children":[{"type":"text","value":"Kagan ","key":"Oa0JHbjOwI"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"c0R9P6RK7L"}],"key":"wFKqExmNai"},{"type":"text","value":" (1996)","key":"sHfaR8dKKz"}],"enumerator":"21","key":"dUD03v35iq"},{"type":"text","value":".","position":{"start":{"line":43,"column":1},"end":{"line":43,"column":1}},"key":"Y99oSaKTPB"}],"key":"avaA5Xex8T"},{"type":"comment","value":" \nStill, the decay of those excitations was studied\n%pitaevskii_breathin_1997 %pitaevskii_elementary_1998\n%They also showed that the frequency of the breathing mode was independent of the temperature.\n\n%the sminal results of the damping rate by @hohenberg_microscopic_1965, @popov1972hydrodynamic and also an open question as the Landau damping derived by @hohenberg_microscopic_1965 (that scale with $T^4$) did not seem accurate with the damping reported by @jin_temperature_dependent_1997. It was further studied by @vincent_liu_theoretical_1997. were also deeply studied as the initial Among them, @chevy_transverse_2002 reported the observation of the transverse breathing mode[^footnote_breathin_monopole]. Such resonance \n\n@pitaevskii_breathin_1997\n\n\n[^footnote_breathin_monopole]: This mode is also called the transverse monopole mode. \n\nThis approach is correct for \nIn the last section, we describe the fundamental wave-function of the system *i.e.* the state at zero temperature. When taking into account temperature, one must decribe how lie the quasi-excitations. In the case of homogeneous condensate, the Bogoliubov description. With the development of \nDans Pita et Stringa : The Bogoliubov theory then predicts that the long-wavelength excitations of an interacting Bose gas are sound waves. These excitations can be also regarded as the Goldstone modes associated with breaking of gauge symmetry caused by Bose–Einstein condensation.\nFirst study of collective excitaitons \n\nCheck also @stringari_dynamics_1998 and @stringari_collective_1996\nLe papier de @jackson_accidental_2002 explique la suppression du decay de landau de l'article de @chevy_transverse_2002. Ce dernier explique alors que la principale source de decay pour les particules est due à la création de phonons, comme proposé par @pitaevskii_elementary_1998\nDamping of collective excitation can occur via Landau damping. In a 3D Bose gas, this damping take the form \n\nA vérifier mais askip (RMP Dalfo et Stringa et compagnie) ont fait un Ansatz gaussien avant\nPe ́ rez-Garc ́ıa, V. M., H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, 1996, Phys. Rev. Lett. 77, 5320. \n\nPe ́ rez-Garc ́ıa, V. M., H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, 1997, Phys. Rev. A 56, 1424. ","key":"UulWhvBcxU"},{"type":"heading","depth":2,"position":{"start":{"line":74,"column":1},"end":{"line":74,"column":1}},"children":[{"type":"text","value":"When the laser quenches, it’s time to breathe","position":{"start":{"line":74,"column":1},"end":{"line":74,"column":1}},"key":"TrjSZImasf"}],"identifier":"time_dependent_bec_width","label":"time_dependent_bec_width","html_id":"time-dependent-bec-width","enumerator":"2","key":"zM2QGJlAAa"},{"type":"comment","value":"Dynamics of the BEC width\n breathtaking breathing mode","position":{"start":{"line":75,"column":1},"end":{"line":76,"column":1}},"key":"YHlwltio4c"},{"type":"comment","value":"We aim to describe the BEC response to a time modulation of the transverse trap frequency $\\omega_\\perp $. In particular, we expand the field as the BEC ground state $\\Psi_0$, treated classically, to which we add small longitudinal fluctuations $\\hat{\\phi}(z,t)$. The time dynamics of the small fluctuation will be studied in the next section. Here, we will focus on the time-dependent response of the BEC $\\Psi_0$. We assume the BEC is homogeneous along $z$, of size $L$ and make use of the gaussian Ansatz to describe the gas.","position":{"start":{"line":79,"column":1},"end":{"line":79,"column":1}},"key":"lJyRpGPANF"},{"type":"paragraph","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"We therefore factor out the radial dependence of the ","key":"DHJkFBR2e6"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"vmOtx7QL4F"}],"key":"Fzj2TCJKce"},{"type":"text","value":" wave-function ","key":"wbIspFCL0x"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"(","key":"cVF9HH1ehy"},{"type":"text","value":"2","key":"F5XHNJlEnE"},{"type":"text","value":")","key":"KDrSp5aTxP"}],"identifier":"field_decomposition_eq_bec3","label":"field_decomposition_eq_bec3","kind":"equation","template":"(%s)","enumerator":"2","resolved":true,"html_id":"field-decomposition-eq-bec3","key":"v6HSJ8kaiY"},{"type":"text","value":" and describe the atomic wave-function within the ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"Fjv5jO94VJ"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"Gaussian Ansatz","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"XkxZBwvweH"}],"identifier":"crossover_regime","label":"crossover_regime","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"crossover-regime","remote":true,"url":"/dce-bec","dataUrl":"/dce-bec.json","key":"YfeDkxDrrV"},{"type":"text","value":" seen in section ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"SM6DjaWEOX"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"4","key":"MuRsrD6iev"}],"identifier":"crossover_regime","label":"crossover_regime","kind":"heading","template":"Section %s","enumerator":"4","resolved":true,"html_id":"crossover-regime","remote":true,"url":"/dce-bec","dataUrl":"/dce-bec.json","key":"HAO95f7LRu"},{"type":"text","value":". This choice is justified by the fact that our ","key":"zQmE8zvezG"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"YkmvgpcwLe"}],"key":"dQtdU5WHzj"},{"type":"text","value":" is neither in the 1D Thomas-Fermi regime nor in the 1D mean field regime. We therefore write the transverse profile as","key":"lhoz5u3Uhy"}],"key":"oYdv8haHDr"},{"type":"comment","value":"As we shall see, with this Ansatz, the breathing mode frequency is $2\\omega_\\perp$. In the 1D mean field regime, the breathing mode frequency is shifted and is $\\sqrt{3}\\omega_\\perp$. @fang_quench_induced_2014 showed however that\n time-dependent width $\\sigma(t) $ to describe the BEC transverse profile. We thus describe the field as","position":{"start":{"line":81,"column":1},"end":{"line":82,"column":1}},"key":"nJi7Eb1PM9"},{"type":"comment","value":" ```{math}\n:label: field_decomposition_eq_bec\n\\hat{\\Psi} = \\Psi_0(r,t) \\mathds{1}\n```\nThe BEC is taken homogeneous along $z$, this leads to the simpler wave-function ","key":"rj7hFFSDp3"},{"type":"math","identifier":"initial_transverse_profile_an_chemical_potential","label":"initial_transverse_profile_an_chemical_potential","value":"\\Psi_0 (t,r)=\\sqrt{\\frac{n_1}{\\pi\\sigma_0^2}}e^{-r^2/2\\sigma_0^2} e^{-i\\mu_0t/\\hbar} .","tight":true,"html":"Ψ0(t,r)=n1πσ02e−r2/2σ02e−iμ0t/ℏ.\\Psi_0 (t,r)=\\sqrt{\\frac{n_1}{\\pi\\sigma_0^2}}e^{-r^2/2\\sigma_0^2} e^{-i\\mu_0t/\\hbar} .Ψ0​(t,r)=πσ02​n1​​​e−r2/2σ02​e−iμ0​t/ℏ.","enumerator":"3","html_id":"initial-transverse-profile-an-chemical-potential","key":"rSroCZfci2"},{"type":"paragraph","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"children":[{"type":"text","value":"The subscript ","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"key":"FdZ0LmAXEE"},{"type":"text","value":"0","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"key":"WdR5flVtjS"},{"type":"text","value":" underlines the fact that they are defined at ","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"key":"LGTwSFfT61"},{"type":"inlineMath","value":"t=0","position":{"start":{"line":92,"column":1},"end":{"line":92,"column":1}},"html":"t=0t=0t=0","key":"R5433AeJhl"},{"type":"text","value":", time at which the trap frequencies are constant and the cloud at equilibrium. 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We can therefore see the chemical potential from equation ","key":"i4IqxfNzeB"},{"type":"crossReference","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"children":[{"type":"text","value":"(","key":"H72fe5eJsk"},{"type":"text","value":"17","key":"AJ5vlV4eW6"},{"type":"text","value":")","key":"U1TUcfFTEQ"}],"identifier":"gerbier_mu_equation","label":"gerbier_mu_equation","kind":"equation","template":"(%s)","enumerator":"17","resolved":true,"html_id":"gerbier-mu-equation","remote":true,"url":"/dce-bec","dataUrl":"/dce-bec.json","key":"RDs3aez9Sp"},{"type":"text","value":" as an effective potential for the width ","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"key":"uEHB26xLTZ"},{"type":"inlineMath","value":"\\sigma ","position":{"start":{"line":105,"column":1},"end":{"line":105,"column":1}},"html":"σ\\sigma 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On the plot of ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"SSWHfCQGFk"},{"type":"crossReference","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"children":[{"type":"text","value":"Figure ","key":"hKU24SuGyk"},{"type":"text","value":"1","key":"yLJhVFcEIJ"}],"identifier":"bec-width-response-excitation","label":"bec-width-response-excitation","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"bec-width-response-excitation","key":"UVs3iQjuwd"},{"type":"text","value":", we see that the frequencies of the two quenches differ by a factor 2. 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The excitation form is ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"JsgplLx4fU"},{"type":"inlineMath","value":"\\omega^2_\\perp = \\omega_{\\perp,0}^2[1 +A\\sin(2\\omega_{\\perp,0}t)]","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"ω⊥2=ω⊥,02[1+Asin⁡(2ω⊥,0t)]\\omega^2_\\perp = \\omega_{\\perp,0}^2[1 +A\\sin(2\\omega_{\\perp,0}t)]ω⊥2​=ω⊥,02​[1+Asin(2ω⊥,0​t)]","key":"gS0NT3DtCo"},{"type":"text","value":" with ","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"key":"LXeunwLeSE"},{"type":"inlineMath","value":"A","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"html":"AAA","key":"VkLuiWGjjr"},{"type":"text","value":" = 4-8% during 6 periods. As we excite a system at its resonant frequency, the amplitude of the ","key":"k8r6yTtT95"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"vWnMy6WZeg"}],"key":"YO3yVU1NaY"},{"type":"text","value":" oscillation grows with the excitation duration. ","key":"wdb7XD1e0J"},{"type":"crossReference","position":{"start":{"line":129,"column":1},"end":{"line":129,"column":1}},"children":[{"type":"text","value":"Figure ","key":"c48yB6mc60"},{"type":"text","value":"2","key":"ihvtYFpmLJ"}],"identifier":"bec-width-amplitude-response-excitation","label":"bec-width-amplitude-response-excitation","kind":"figure","template":"Figure %s","enumerator":"2","resolved":true,"html_id":"bec-width-amplitude-response-excitation","key":"Ej2YDbS8zN"},{"type":"text","value":" represents the peak-peak amplitude of the width oscillation as a function of the amplitude of the modulation and its duration. 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To achieve this, we continuously modulate the transverse trap at frequency ","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"key":"wEeBTBb14i"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":145,"column":1},"end":{"line":145,"column":1}},"html":"ωd\\omega_dωd​","key":"B6Aktfb6Bb"},{"type":"text","value":" rather than halting after a certain number of oscillations. 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A second possibility is to rise slowly the modulation frequency, for example with a hyperbolic tangent function as","position":{"start":{"line":151,"column":1},"end":{"line":151,"column":1}},"key":"FL1Fm3QIdQ"}],"key":"AeegHgMdCZ"},{"type":"math","identifier":"excitation_gentle","label":"excitation_gentle","value":"\\omega_{\\perp}^2(t) = \\omega_{\\perp,0}^2[1+A\\sin(\\omega_d t)(1+\\tanh[t/\\tau])/2].","tight":"before","html":"ω⊥2(t)=ω⊥,02[1+Asin⁡(ωdt)(1+tanh⁡[t/τ])/2].\\omega_{\\perp}^2(t) = \\omega_{\\perp,0}^2[1+A\\sin(\\omega_d t)(1+\\tanh[t/\\tau])/2].ω⊥2​(t)=ω⊥,02​[1+Asin(ωd​t)(1+tanh[t/τ])/2].","enumerator":"9","html_id":"excitation-gentle","key":"A3REtT5LU3"},{"type":"comment","value":" We can see that, for this choice of driving frequency $\\omega_d = 3\\omega_{\\perp,0}$, the BEC width follows well the excitation with the sweet modulation while it exhibits a strong response at its natural frequency for the brutal modulation. 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The left column matches the ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"QR47x3KIrh"},{"type":"emphasis","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"brutal","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"OoFiuJxXlQ"}],"key":"mAKy2RTirz"},{"type":"text","value":" excitation ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"n1efItPfw4"},{"type":"crossReference","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"(","key":"krsMMmN0Hl"},{"type":"text","value":"8","key":"GFLAQjWJ8z"},{"type":"text","value":")","key":"tyXYkHoV4X"}],"identifier":"excitation_brutal","label":"excitation_brutal","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"excitation-brutal","key":"Xqn8ebA0dC"},{"type":"text","value":" and the right column the ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"NxrFoj4Sgo"},{"type":"emphasis","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"sweet excitation","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"l4U71OYnuh"}],"key":"fNMZlMwUER"},{"type":"text","value":" ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"HOrI0oeSR2"},{"type":"crossReference","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"(","key":"OAJBHEOhsL"},{"type":"text","value":"9","key":"VXiV4xYism"},{"type":"text","value":")","key":"mbEuddImT2"}],"identifier":"excitation_gentle","label":"excitation_gentle","kind":"equation","template":"(%s)","enumerator":"9","resolved":true,"html_id":"excitation-gentle","key":"YWZF3oyioM"},{"type":"text","value":". The first row represents the trap frequency profile, which is experimentally realized by changing the laser power of the trap. On the left, the excitation starts at ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"uKhoOU3ItL"},{"type":"inlineMath","value":"t=0","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"html":"t=0t=0t=0","key":"Pi4YxVjO5p"},{"type":"text","value":" brutally while the excitation is adiabatically tuned (","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"MvdplkEE2z"},{"type":"inlineMath","value":"\\tau = 2/\\omega_{\\perp,0}","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"html":"τ=2/ω⊥,0\\tau = 2/\\omega_{\\perp,0}τ=2/ω⊥,0​","key":"rzu6q0JroV"},{"type":"text","value":") on the right column. The second row represents the time dependent response of the ","key":"OuYoJS2kJX"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"m2G9dMiVLr"}],"key":"DEvYREG6s6"},{"type":"text","value":" width. The left subplot exhibits a non-sinusoidal behavior while the right one seems ","key":"mG8QcpL03W"},{"type":"emphasis","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"proper","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"G2sPuuBePK"}],"key":"Ci6uvEHiYx"},{"type":"text","value":". The last row depicts the Fourier transform of the width response in steady state regime (","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"oRMsITmD9j"},{"type":"emphasis","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"children":[{"type":"text","value":"i.e.","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"FdAyrlDVai"}],"key":"qIhXezNRjs"},{"type":"text","value":" for ","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"A4nbcJvnHb"},{"type":"inlineMath","value":"t\\omega_{\\perp,0}>6","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"html":"tω⊥,0>6t\\omega_{\\perp,0}>6tω⊥,0​>6","key":"HEJXHso72x"},{"type":"text","value":"). The red are highlights the breathing mode frequency and the green area the driving frequency. The brutal modulation exhibits two peaks: one at the resonance frequency and one at the driving frequency while the sweet modulation has only one frequency. For this figure, the amplitude of the modulation is A = 8%.","position":{"start":{"line":166,"column":1},"end":{"line":166,"column":1}},"key":"ffJnvJeJ5d"}],"key":"jH0LYX70eZ"}],"key":"zUywTo9l1z"}],"enumerator":"3","html_id":"bec-width-oned-response","key":"LxV9jtMX8C"},{"type":"paragraph","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"children":[{"type":"strong","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"children":[{"type":"text","value":"Results:","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"jscCEtNq4b"}],"key":"pjl6e0n7zz"},{"type":"text","value":" The time dependent form of those two excitation functions are represented on the first row of ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"TLysKoFD4h"},{"type":"crossReference","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"children":[{"type":"text","value":"Figure ","key":"Ra90Rx6Khz"},{"type":"text","value":"3","key":"GaeRUg2xXO"}],"identifier":"bec-width-oned-response","label":"bec-width-oneD-response","kind":"figure","template":"Figure %s","enumerator":"3","resolved":true,"html_id":"bec-width-oned-response","key":"tLyFc5BxIu"},{"type":"text","value":". The first column represents the “brutal” modulation in equation ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"aqlWhgmjie"},{"type":"crossReference","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"children":[{"type":"text","value":"(","key":"n1mDKTVgqq"},{"type":"text","value":"8","key":"i1yFEXAQhh"},{"type":"text","value":")","key":"ugL4b0LMOA"}],"identifier":"excitation_brutal","label":"excitation_brutal","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"excitation-brutal","key":"Ll44KWIcOD"},{"type":"text","value":", the second one the “sweet” excitation in equation ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"O3tnQpE8bc"},{"type":"crossReference","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"children":[{"type":"text","value":"(","key":"HIxnipxnDL"},{"type":"text","value":"9","key":"IjInEbouIK"},{"type":"text","value":")","key":"T4rwOZTxTr"}],"identifier":"excitation_gentle","label":"excitation_gentle","kind":"equation","template":"(%s)","enumerator":"9","resolved":true,"html_id":"excitation-gentle","key":"B6FyB7nWVd"},{"type":"text","value":", for which the modulation is raised adiabatically (with respect to ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"tSujiZ38s7"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"Uo0LOk0Pir"},{"type":"text","value":"). The modulation frequency chosen here is ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"hN49rMXvbv"},{"type":"inlineMath","value":"\\omega_d = 3\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"ωd=3ω⊥,0\\omega_d = 3\\omega_{\\perp,0}ωd​=3ω⊥,0​","key":"VJkL3fyzhe"},{"type":"text","value":". The second row depicts the time evolution of the width of the ","key":"W9nPsl1oTy"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"TjXVS5hmYY"}],"key":"kVyyzv32yS"},{"type":"text","value":" ","key":"nvBxjLXkm8"},{"type":"inlineMath","value":"\\sigma(t)","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"σ(t)\\sigma(t)σ(t)","key":"lsFvemSXpD"},{"type":"text","value":". In the case of a brutal modulation (left), the ","key":"xRxW8Uhavf"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"WslCzcqTHJ"}],"key":"hG8DZ555Bf"},{"type":"text","value":" width response does not look like a sine function and exhibits two harmonics. In the case of a sweet modulation, the width response seems much nicer (right). To confirm this intuition, one can look at the third row that represents the Fourier transform ","key":"vWCRG7wdsn"},{"type":"inlineMath","value":"\\tilde{\\sigma}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"σ~\\tilde{\\sigma}σ~","key":"EhMX2Hkzir"},{"type":"text","value":" of the ","key":"fPElkZEkxM"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"uaKwU6iFV6"}],"key":"LXxhtn5HTz"},{"type":"text","value":" width ","key":"cuUFQyvS6h"},{"type":"inlineMath","value":"\\sigma(t)","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"σ(t)\\sigma(t)σ(t)","key":"ZxV8hTEw7B"},{"type":"text","value":". The Fourier transform is computed between ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"XpHJ9tUZj1"},{"type":"inlineMath","value":"6\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"6ω⊥,06\\omega_{\\perp,0}6ω⊥,0​","key":"fapL4IIW6V"},{"type":"text","value":" and ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"qHujYwDi3f"},{"type":"inlineMath","value":"20\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"20ω⊥,020\\omega_{\\perp,0}20ω⊥,0​","key":"Dr0O92BqFW"},{"type":"text","value":" so that the response is in the steady state regime. On the Fourier spectrum, the green and red vertical lines with transparent shading represents respectively the driving frequency ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"EEFmw2yC4U"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"ωd\\omega_dωd​","key":"nt8uI9BYvJ"},{"type":"text","value":" and the natural frequency ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"kc8EbxvU9T"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"q9V9qMKF07"},{"type":"text","value":". The brutal modulation response presents two peaks: one at the resonant frequency ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"jWTxVt64ua"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"IVN1prgho1"},{"type":"text","value":" and the second one at the driving frequency ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"lbsP376Q8j"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"ωd\\omega_dωd​","key":"GGGJ4rHUbU"},{"type":"text","value":". In the case of the “sweet” modulation, we observe only one peak, located at the driving frequency ","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"F7TJnuHVat"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"html":"ωd\\omega_dωd​","key":"TEAAu61Jcm"},{"type":"text","value":".","position":{"start":{"line":169,"column":1},"end":{"line":169,"column":1}},"key":"ymtqKRYF8j"}],"key":"TqoOH8Pdfe"},{"type":"admonition","kind":"seealso","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Conclusion","position":{"start":{"line":171,"column":1},"end":{"line":171,"column":1}},"key":"KbFxoA1TDz"}],"key":"bukqTBHycF"},{"type":"paragraph","position":{"start":{"line":173,"column":1},"end":{"line":173,"column":1}},"children":[{"type":"text","value":"We conclude that, at the excitation frequency ","position":{"start":{"line":173,"column":1},"end":{"line":173,"column":1}},"key":"qLQ8oYJ2t2"},{"type":"inlineMath","value":"3\\omega_{\\perp,0}","position":{"start":{"line":173,"column":1},"end":{"line":173,"column":1}},"html":"3ω⊥,03\\omega_{\\perp,0}3ω⊥,0​","key":"JL3hoaLNfm"},{"type":"text","value":", the response of the ","key":"vsSoQYslef"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"fgGGDf4nPu"}],"key":"qSDb5Z8K5d"},{"type":"text","value":" follows well the “sweet” modulation function ","key":"HXaUIVgg1Q"},{"type":"crossReference","position":{"start":{"line":173,"column":1},"end":{"line":173,"column":1}},"children":[{"type":"text","value":"(","key":"egBJjw8B1s"},{"type":"text","value":"9","key":"abCDsCBa7d"},{"type":"text","value":")","key":"KghyN0g2Y1"}],"identifier":"excitation_gentle","label":"excitation_gentle","kind":"equation","template":"(%s)","enumerator":"9","resolved":true,"html_id":"excitation-gentle","key":"cQ41nHIifd"},{"type":"text","value":". This allows us to control the oscillation frequency of the ","key":"VA9LXAVfTJ"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"uUOVFAx6N1"}],"key":"GtqRgasTIO"},{"type":"text","value":" i.e. we control the collective excitation. That said, this was checked for a particular excitation frequency. This study could be scaled up to examine other excitation frequencies.","key":"VnCgPKdG0O"}],"key":"XNAo5JloZy"}],"key":"N6fuQq2iCm"},{"type":"comment","value":" As the time dependent response inferred, we now have the confirmation that the ","key":"UhVt9UUkyB"},{"type":"container","kind":"figure","identifier":"bec-width-spectrum-response","label":"bec-width-spectrum-response","children":[{"type":"image","url":"/~gondret/phd_manuscript/build/spectrum_response-fcd19e91c5f91b6812486a62849119c3.png","alt":"Response of the BEC transverse profile to an excitation","width":"80%","align":"center","key":"hMjUTYR6UI","urlSource":"images/spectrum_response.png","urlOptimized":"/~gondret/phd_manuscript/build/spectrum_response-fcd19e91c5f91b6812486a62849119c3.webp"},{"type":"caption","children":[{"type":"paragraph","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"children":[{"type":"captionNumber","kind":"figure","label":"bec-width-spectrum-response","identifier":"bec-width-spectrum-response","html_id":"bec-width-spectrum-response","enumerator":"4","children":[{"type":"text","value":"Figure ","key":"IwcdR5Vq2q"},{"type":"text","value":"4","key":"YiwPjmpwrZ"},{"type":"text","value":":","key":"uhq7CY0tew"}],"template":"Figure %s:","key":"hJlkFaLTKH"},{"type":"text","value":"Spectrum of the ","key":"wagmgOd3gG"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"PYCLpLY4oa"}],"key":"ouji4ej3jL"},{"type":"text","value":" width ","key":"pWJKbz4DqW"},{"type":"inlineMath","value":"\\tilde{\\sigma}(\\omega)","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"σ~(ω)\\tilde{\\sigma}(\\omega)σ~(ω)","key":"r9mUubNV4j"},{"type":"text","value":" response to a modulation at frequency ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"U85h4Qp6fT"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"ωd\\omega_dωd​","key":"glZxtqXz3N"},{"type":"text","value":", on the x-axis. The vertical axis represents the Fourier frequency of the function ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"Y00j40axPW"},{"type":"inlineMath","value":"\\tilde{\\sigma}(\\omega)","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"σ~(ω)\\tilde{\\sigma}(\\omega)σ~(ω)","key":"bdHLbNowUm"},{"type":"text","value":" whose amplitude is proportional to the colorscale. A bluer color represents a higher amplitude of the Fourier component. When the transverse trap is modulated with a ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"JpFRrUNHf2"},{"type":"emphasis","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"children":[{"type":"text","value":"brutal modulation","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"cr2aEfKafg"}],"key":"AZ3rtet7H1"},{"type":"text","value":" given by equation ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"JeFshFgdFx"},{"type":"crossReference","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"children":[{"type":"text","value":"(","key":"THgjJlkLOb"},{"type":"text","value":"8","key":"i3bzsn0msW"},{"type":"text","value":")","key":"HiZnrhiOlZ"}],"identifier":"excitation_brutal","label":"excitation_brutal","kind":"equation","template":"(%s)","enumerator":"8","resolved":true,"html_id":"excitation-brutal","key":"rlOoB1AlcG"},{"type":"text","value":", the system does not follow perfectly the driving frequency but tends to oscillate at its own natural frequency ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"hHsmEhqGY4"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"v7UxH9K3Ok"},{"type":"text","value":". This is shown by the strong horizontal blue line. 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For this image, ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"i4MGBGaBpy"},{"type":"inlineMath","value":"A=5\\%","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"A=5%A=5\\%A=5%","key":"oKZXNCLsZb"},{"type":"text","value":" and ","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"key":"AgmwO5pJPH"},{"type":"inlineMath","value":"\\tau = 2/\\omega_{\\perp,0}","position":{"start":{"line":182,"column":1},"end":{"line":182,"column":1}},"html":"τ=2/ω⊥,0\\tau = 2/\\omega_{\\perp,0}τ=2/ω⊥,0​","key":"XtOaTeLZaH"},{"type":"text","value":". 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The Fourier amplitude is normalized so that the maximal Fourier component is 1: each column of the map is normalized. This allows us to see for each driving frequency ","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"mcWEEa7SOw"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"html":"ωd\\omega_dωd​","key":"DK8nWWHKtp"},{"type":"text","value":" what the main frequencies (bluer) are in the ","key":"DIeKW5tYWO"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"fnVkBmG2io"}],"key":"elckRzsEnU"},{"type":"text","value":" width response. If the ","key":"qa4cLOqg0B"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"otXQyc9r9m"}],"key":"xL2DxP4sul"},{"type":"text","value":" oscillates exactly at the driving frequency ","key":"frnp0cEcoy"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"html":"ωd\\omega_dωd​","key":"cbRsvuY34Y"},{"type":"text","value":", one would observe a stronger signal on the diagonal, as the Fourier transform of the ","key":"hcMLbLx10r"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"Eow6xUJhuk"}],"key":"zIEBab7HkD"},{"type":"text","value":" width would exhibit a peak only at ","key":"xCoN6WH6Yf"},{"type":"inlineMath","value":"\\omega = \\omega_d","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"html":"ω=ωd\\omega = \\omega_dω=ωd​","key":"YAwJe4B3AC"},{"type":"text","value":".","position":{"start":{"line":189,"column":1},"end":{"line":189,"column":1}},"key":"NOMbJGCWKz"}],"key":"IUVPiYhF7r"},{"type":"paragraph","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"text","value":"The right panel of ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"T56gP3Ubsm"},{"type":"crossReference","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"children":[{"type":"text","value":"Figure ","key":"v2F9YxwU0G"},{"type":"text","value":"4","key":"RPj07BCmmp"}],"identifier":"bec-width-spectrum-response","label":"bec-width-spectrum-response","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"bec-width-spectrum-response","key":"xda04jH1tK"},{"type":"text","value":" is easier to interpret: we observe a strong signal on the diagonal, which means that the main frequency of the system is the driving frequency. The only exceptions are at ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"CmYNkEEevs"},{"type":"inlineMath","value":"1\\omega_{\\perp,0}","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"html":"1ω⊥,01\\omega_{\\perp,0}1ω⊥,0​","key":"PrXdf8qRag"},{"type":"text","value":" and ","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"hd8jASESZQ"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"iYndtHIkuT"},{"type":"text","value":".","position":{"start":{"line":191,"column":1},"end":{"line":191,"column":1}},"key":"hE0rrWNknt"}],"key":"XEhZrcME56"},{"type":"paragraph","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"children":[{"type":"text","value":"The left panel is a bit more chaotic. For low driving frequencies ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"KpwrjamdU9"},{"type":"text","value":"ω","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"i49utYSF0q"},{"type":"text","value":", that is, on the left of the graph, the system follows the driving frequency: the Fourier transform ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"IyRBO2HlWw"},{"type":"inlineMath","value":"\\tilde{\\sigma}","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"σ~\\tilde{\\sigma}σ~","key":"c7wWYUPyWA"},{"type":"text","value":" is well-peaked on the diagonal. When the driving frequency is greater than ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"i8OPtyz1gg"},{"type":"inlineMath","value":"\\omega_{\\perp,0}","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"ω⊥,0\\omega_{\\perp,0}ω⊥,0​","key":"s28G5FLwCu"},{"type":"text","value":", the main component of ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"odd2osTb8D"},{"type":"inlineMath","value":"\\tilde{\\sigma}","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"σ~\\tilde{\\sigma}σ~","key":"Mswn0fG9wE"},{"type":"text","value":" is no longer the driving frequency but the resonant frequency ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"Xlg7kiCQ8O"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"sMI9sZBgUq"},{"type":"text","value":". The system is oscillating at its natural frequency. We can interpret this as the oscillation at ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"PxswVD8j9Y"},{"type":"inlineMath","value":"2\\omega_{\\perp,0}","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"2ω⊥,02\\omega_{\\perp,0}2ω⊥,0​","key":"MXgT4WBWGP"},{"type":"text","value":" being due to the energy injected at ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"Ap0WDSFDGy"},{"type":"inlineMath","value":"t = 0","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"html":"t=0t = 0t=0","key":"Kk8HkaknNM"},{"type":"text","value":", when the frequency of modulation is not yet defined. When the modulation is applied smoothly, the system is better controlled and keeps oscillating at the driving frequency","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"t2hF4zIzvO"},{"type":"footnoteReference","identifier":"smart_footnote_failed","label":"smart_footnote_failed","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"number":1,"enumerator":"1","key":"H04xqksO79"},{"type":"text","value":". The conclusion of this subsection leaves no doubt: the softer, the better. This is why we titled it with Orlando’s line “","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"cmBlf0VHFO"},{"type":"emphasis","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"children":[{"type":"text","value":"Let gentleness my strong enforcement be.","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"zAxMhA3PIP"}],"key":"Op0Kv4Xd96"},{"type":"text","value":"”, ","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"nN7X5R7J62"},{"type":"link","url":"https://www.folger.edu/explore/shakespeares-works/as-you-like-it/read/2/7/","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"children":[{"type":"emphasis","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"children":[{"type":"text","value":"As you Like It","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"yXnpXeVjUc"}],"key":"JiAsVAygk3"}],"urlSource":"https://www.folger.edu/explore/shakespeares-works/as-you-like-it/read/2/7/","key":"Aks5Jojqth"},{"type":"text","value":", Shakespeare.","position":{"start":{"line":193,"column":1},"end":{"line":193,"column":1}},"key":"L0fIoQPVSH"}],"key":"zxg81CFjPn"},{"type":"comment","value":" \n:::{seealso} Conclusion\nThe conclusion of this subsection leaves no doubt: the softer, the better. This is why we titled it with Orlando's line \"*Let gentleness my strong enforcement be.*\", [*As you Like It*](https://www.folger.edu/explore/shakespeares-works/as-you-like-it/read/2/7/), Shakespeare.\n::: ","key":"hxqoUU8SHl"},{"type":"comment","value":" We can now study the Fourier transform of the BEC width response when we modulate it with equation [](#excitation_brutal) or equation [](#excitation_gentle) with a modulation frequency $\\omega_d$. The Fourier transform of the BEC width response, particularly is computed after the system reached a steady state, *i.e.* the steady state regime (i.e. away from resonance). ","key":"JEct4tN245"},{"type":"comment","value":" For low driving frequencies $\\omega$, the system follows the driving frequency: the Fourier transform $\\tilde{\\sigma}$ is well-peaked on the diagonal. but the resonant frequency $2\\omega_{\\perp,0}$: the system is oscillating at its natural frequency.\\ ","key":"OtnId6H3lH"},{"type":"comment","value":" On the right, the modulation amplitude is tuned adiabatically with a typical duration $\\tau$ of a few $1/2\\omega_{\\perp,0}$ ","key":"QkH54sBbjT"},{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"Summary","position":{"start":{"line":211,"column":1},"end":{"line":211,"column":1}},"key":"KzULg4d1dX"}],"key":"wv7d45unqG"},{"type":"paragraph","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"children":[{"type":"text","value":"This section showed, using the Gaussian Ansatz, that a quench or a small variation of the transverse trap frequency causes the ","key":"Ty6jCVuLht"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"A6N2MjwVqI"}],"key":"cSga0YhblE"},{"type":"text","value":" to ","key":"F45J33qgpS"},{"type":"crossReference","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"children":[{"type":"text","value":"oscillate","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"xqteKJUkuJ"}],"identifier":"time_dependent_bec_width","label":"time_dependent_bec_width","kind":"heading","template":"Section %s","enumerator":"2","resolved":true,"html_id":"time-dependent-bec-width","key":"DHNYUjS0RB"},{"type":"text","value":" at twice the frequency of the trap: that is, the breathing mode. We then proposed a ","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"rjqV1NzhBj"},{"type":"crossReference","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"children":[{"type":"text","value":"protocol","position":{"start":{"line":213,"column":1},"end":{"line":213,"column":1}},"key":"AIalmmnh9z"}],"identifier":"forcing_oscillations","label":"forcing_oscillations","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"forcing-oscillations","key":"sSRk5cDFEo"},{"type":"text","value":" to force the ","key":"CWSl1OoCOV"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"OP6qIuApUj"}],"key":"XQD2bfyKDA"},{"type":"text","value":" to oscillate at any frequency.","key":"M3vkyIRLH1"}],"key":"atlGqXuK0n"}],"key":"W2FUhs6Jvk"},{"type":"comment","value":" \nLet gentleness my strong enforcement be,\nIn the which hope I blush and hide my sword.\nQue la douceur soit ma forte autorité (violence)\nDans cet espoir je rougis et cache mon épée.\n ","key":"WP8HgxAjaC"},{"type":"footnoteDefinition","identifier":"smart_footnote_failed","label":"smart_footnote_failed","position":{"start":{"line":215,"column":1},"end":{"line":215,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"children":[{"type":"text","value":"Before attempting this adiabatically raised modulation, I initially attempted a more sophisticated approach. I aimed to engineer a modulation ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"M5KhAxikIJ"},{"type":"inlineMath","value":"\\omega_{\\perp,0}^2","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"ω⊥,02\\omega_{\\perp,0}^2ω⊥,02​","key":"XZtGaFqmhL"},{"type":"text","value":" so that its response ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"yDkxJ0NmXJ"},{"type":"inlineMath","value":"\\sigma(t)","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"σ(t)\\sigma(t)σ(t)","key":"WHmRz0cCgj"},{"type":"text","value":" would oscillate at frequency ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"RbSS5muWMW"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"ωd\\omega_dωd​","key":"lApzT4iIv0"},{"type":"text","value":". Let ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"DLhUbaqUS8"},{"type":"inlineMath","value":"f","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"fff","key":"uUCf2b3wB8"},{"type":"text","value":" be a function representing this targeted width, behaving like a constant plus a small modulation at frequency ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"j3Ndgr3LOW"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"ωd\\omega_dωd​","key":"EIKu0qjete"},{"type":"text","value":", for instance. By utilizing equation ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"GeTcZcvm8L"},{"type":"crossReference","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"children":[{"type":"text","value":"(","key":"lZp04ryd6q"},{"type":"text","value":"5","key":"NbaBMFU5wO"},{"type":"text","value":")","key":"mRkJUAFnCM"}],"identifier":"sigma_equation","label":"sigma_equation","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"sigma-equation","key":"buqx33RL87"},{"type":"text","value":", one can designed the modulation function ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"m4vvJ5WCTe"},{"type":"inlineMath","value":"\\omega_\\perp^2\\propto 1/f^4 - \\ddot{f}/f","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"ω⊥2∝1/f4−f¨/f\\omega_\\perp^2\\propto 1/f^4 - \\ddot{f}/fω⊥2​∝1/f4−f¨​/f","key":"rsfivtoRFl"},{"type":"text","value":" so that the solution ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"HOs7leQIen"},{"type":"text","value":"σ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"pSiXzOJxaC"},{"type":"text","value":" of ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"pomcWG1F6s"},{"type":"crossReference","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"children":[{"type":"text","value":"(","key":"YVPsXZOkGp"},{"type":"text","value":"5","key":"Qrk0XBR97E"},{"type":"text","value":")","key":"gGLw5odlc3"}],"identifier":"sigma_equation","label":"sigma_equation","kind":"equation","template":"(%s)","enumerator":"5","resolved":true,"html_id":"sigma-equation","key":"P1dE1gIqll"},{"type":"text","value":" would converge to ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"nvqkWAdLRP"},{"type":"inlineMath","value":"f","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"html":"fff","key":"It2cuNznOm"},{"type":"text","value":". However, the results from this method proved no better than those depicted in ","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"l5lxKV2Hll"},{"type":"crossReference","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"children":[{"type":"text","value":"Figure ","key":"GWqJsERvGk"},{"type":"text","value":"4","key":"WVPkM88gJ9"}],"identifier":"bec-width-spectrum-response","label":"bec-width-spectrum-response","kind":"figure","template":"Figure %s","enumerator":"4","resolved":true,"html_id":"bec-width-spectrum-response","key":"RqLuHjCTzi"},{"type":"text","value":" when the modulation was turned on adiabatically, and it performed even worse results near the resonance. The simpler being the better, I preferred to stick with the smooth approach.","position":{"start":{"line":207,"column":1},"end":{"line":207,"column":1}},"key":"kjmqHpqBmH"}],"key":"S9lEBSwTHW"}],"number":1,"enumerator":"1","key":"AA7vOetz2F"}],"key":"xjNlzs6mds"}],"key":"wBhNBZt1IX"},"references":{"cite":{"order":["chevy_transverse_2002","KANAMORI1972346","bogoliubov_theory_1947","bardeen_theory_1957","anderson_1995_observation","davis1995bose","jin_temperature_dependent_1997","stamper_collisionless_1998","stringari_collective_1996","stringari_dynamics_1998","fliesser_hydrodynamic_1997","menotti_collective_2002","hohenberg_microscopic_1965","pitaevskii_landau_1997","vincent_liu_theoretical_1997","fedichev_damping_1998","beliaev_energy_1958","popov1972hydrodynamic","jackson_accidental_2002","castin_bose_einstein_1996","kagan_evolution_1996","pitaevskii_elementary_1998","kagan2001damping","micheli_entanglement_2023","leach_ermakov_2008","robertson_controlling_2017"],"data":{"chevy_transverse_2002":{"label":"chevy_transverse_2002","enumerator":"1","doi":"10.1103/PhysRevLett.88.250402","html":"Chevy, F., Bretin, V., Rosenbusch, P., Madison, K. 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Physical Review Letters, 75(22), 3969."},"jin_temperature_dependent_1997":{"label":"jin_temperature_dependent_1997","enumerator":"7","html":"Jin, D. S., Matthews, M. R., Ensher, J. R., Wieman, C. E., & Cornell, E. A. (1997). Temperature-Dependent Damping and Frequency Shifts in Collective Excitations of a Dilute Bose-Einstein Condensate. Physical Review Letters, 78(5)."},"stamper_collisionless_1998":{"label":"stamper_collisionless_1998","enumerator":"8","doi":"10.1103/PhysRevLett.81.500","html":"Stamper-Kurn, D. M., Miesner, H.-J., Inouye, S., Andrews, M. R., & Ketterle, W. (1998). Collisionless and Hydrodynamic Excitations of a Bose-Einstein Condensate. Phys. Rev. Lett., 81(3), 500–503. 10.1103/PhysRevLett.81.500","url":"https://doi.org/10.1103/PhysRevLett.81.500"},"stringari_collective_1996":{"label":"stringari_collective_1996","enumerator":"9","doi":"10.1103/PhysRevLett.77.2360","html":"Stringari, S. (1996). Collective Excitations of a Trapped Bose-Condensed Gas. Phys. Rev. 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