{"kind":"Article","sha256":"76ea71fc97679175e03aa43a926eebb6265c8d888d9f0304f54eb828c79100ad","slug":"cosqua-intro","location":"/theory/cosqua_intro.md","dependencies":[],"frontmatter":{"title":"Parametric creation of quasi-particles","short_title":"Parametric creation of quasi-particles","subtitle":"Wait a second ! Look, here, in the vacuum chamber, an analog inflaton!","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/faraday-ca614bceb3bd08ae7948aa14e6815523.png","thumbnailOptimized":"/~gondret/phd_manuscript/build/faraday-ca614bceb3bd08ae7948aa14e6815523.webp","exports":[{"format":"md","filename":"cosqua_intro.md","url":"/~gondret/phd_manuscript/build/cosqua_intro-a5313fc94c62ba7a3b8a299a2f2b68db.md"}]},"mdast":{"type":"root","children":[{"type":"block","position":{"start":{"line":7,"column":1},"end":{"line":7,"column":1}},"children":[{"type":"paragraph","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"children":[{"type":"text","value":"Faraday’s experiment consists in shaking a fluid vertically at a well-defined frequency. Depending on the viscosity of the fluid and the boundary conditions, a specific pattern appears at the liquid interface. After the proposal of ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"pJGMvPvRQz"},{"type":"cite","identifier":"staliunas_faraday_2002","label":"staliunas_faraday_2002","kind":"narrative","position":{"start":{"line":9,"column":233},"end":{"line":9,"column":256}},"children":[{"type":"text","value":"Staliunas ","key":"Fq63EsBBYa"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"B1WFaFbAb2"}],"key":"viY6FgA4N5"},{"type":"text","value":" (2002)","key":"Esjhr679ul"}],"enumerator":"1","key":"JyiDwxzSb7"},{"type":"text","value":" from which we reproduce numerical simulations in ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"DMjxGqILYz"},{"type":"crossReference","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"children":[{"type":"text","value":"Figure ","key":"Lq0RImVRkE"},{"type":"text","value":"1","key":"lB1qVIM6aM"}],"identifier":"faraday","label":"faraday","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"faraday","key":"seKdDTsPIj"},{"type":"text","value":", ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"aTo7jiqAyf"},{"type":"cite","identifier":"engels_observation_2007","label":"engels_observation_2007","kind":"narrative","position":{"start":{"line":9,"column":320},"end":{"line":9,"column":344}},"children":[{"type":"text","value":"Engels ","key":"TW9ozRo868"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"arYgcOG5wn"}],"key":"XJELmoNwXs"},{"type":"text","value":" (2007)","key":"HZzm7rTs48"}],"enumerator":"2","key":"wWR87XeQpO"},{"type":"text","value":" performed similar experiment with a cigar-shape Bose-Einstein condensate. By modulating periodically the transverse potential of the ","key":"wYqGT1yGcb"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"OGCNq5bezK"}],"key":"gbAvH37BqD"},{"type":"text","value":", the authors excite longitudinal collective excitations. In particular, they show that when the frequency of the modulation is ","key":"JW2tKEoymn"},{"type":"inlineMath","value":"\\omega_d","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"ωd\\omega_dωd​","key":"hgkYG6ZiV8"},{"type":"text","value":", they excite the collective mode with wave-vector ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"OJJ2Oym1o2"},{"type":"inlineMath","value":"k","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"kkk","key":"cLsvn4WW4L"},{"type":"text","value":" such that ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"UwPrx3Lhe1"},{"type":"inlineMath","value":"\\omega(k) = \\omega_d/2","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"ω(k)=ωd/2\\omega(k) = \\omega_d/2ω(k)=ωd​/2","key":"uE3l1etjc5"},{"type":"text","value":", where ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"cccYNdWF2l"},{"type":"inlineMath","value":"\\omega(k)","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"ω(k)\\omega(k)ω(k)","key":"WA7KMH3cJI"},{"type":"text","value":" is the ","key":"rRtmPL9Z3p"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"AcvwYxErmU"}],"key":"CisKbXZZZ1"},{"type":"text","value":" dispersion relation. A series of theoretical papers followed this experiment to better model the dispersion relation ","key":"PymNDekVxO"},{"type":"inlineMath","value":"\\omega(k)","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"ω(k)\\omega(k)ω(k)","key":"fIIhfvicf0"},{"type":"text","value":" ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"dodMlA0pm3"},{"type":"citeGroup","kind":"parenthetical","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"children":[{"type":"cite","identifier":"nicolin_faraday_2007","label":"nicolin_faraday_2007","kind":"parenthetical","position":{"start":{"line":9,"column":869},"end":{"line":9,"column":890}},"children":[{"type":"text","value":"Nicolin ","key":"KyHpcavuZ2"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"g1gFp9MJD5"}],"key":"lG9Om7NP0w"},{"type":"text","value":", 2007","key":"WREPzQlugg"}],"enumerator":"3","key":"x2lDnjIRvm"},{"type":"cite","identifier":"nicolin_faraday_2010","label":"nicolin_faraday_2010","kind":"parenthetical","position":{"start":{"line":9,"column":891},"end":{"line":9,"column":912}},"children":[{"type":"text","value":"Nicolin & Raportaru, 2010","key":"gtlkMJ4OLG"}],"enumerator":"4","key":"RsY0hpTTwN"},{"type":"cite","identifier":"nicolin_resonant_2011","label":"nicolin_resonant_2011","kind":"parenthetical","position":{"start":{"line":9,"column":913},"end":{"line":9,"column":935}},"children":[{"type":"text","value":"Nicolin, 2011","key":"wfJjDysMCH"}],"enumerator":"5","key":"yTq9VquG6B"}],"key":"jU17KA9ACv"},{"type":"text","value":". In their work, the authors use a transverse Ansatz to study an effective 1D Gross-Pitaevskii equation. Linearizing the system for a small perturbation with wave-vector ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"PDehaqSrh8"},{"type":"inlineMath","value":"k ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"kk k","key":"OaurNaOYaL"},{"type":"text","value":", they obtain a Mathieu equation for which a Floquet analysis gives access to unstable regions. As for any periodically driven system, it reveals the presence of multiple resonances, at wave-vectors such that ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"YtCr5gvHFy"},{"type":"inlineMath","value":"\\omega(k) = n\\omega_d/2","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"ω(k)=nωd/2\\omega(k) = n\\omega_d/2ω(k)=nωd​/2","key":"DgVxsEbOEy"},{"type":"text","value":", where ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"eQuGEWAIX1"},{"type":"inlineMath","value":"n","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"html":"nnn","key":"Qla7oGIavM"},{"type":"text","value":" is an integer. The unstable regions, shown in ","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"key":"UNG6QkwcfP"},{"type":"crossReference","position":{"start":{"line":9,"column":1},"end":{"line":9,"column":1}},"children":[{"type":"text","value":"Figure ","key":"UsWFG7ifsY"},{"type":"text","value":"1","key":"xPXd8OmI41"}],"identifier":"faraday","label":"faraday","kind":"figure","template":"Figure %s","enumerator":"1","resolved":true,"html_id":"faraday","key":"f9s7ciGzjI"},{"type":"text","value":", are referred to as Mathieu tongues due to their shape. 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The shaded area represents the unstable regions. In panel (a), there is no damping, so all wave-vectors ","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"t45iapFnNq"},{"type":"inlineMath","value":"k_n","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"html":"knk_nkn​","key":"KaVjxNHPuS"},{"type":"text","value":" such that ","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"pkLaLHvFVe"},{"type":"inlineMath","value":"\\omega(k_n) = n\\omega_d/2","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"html":"ω(kn)=nωd/2\\omega(k_n) = n\\omega_d/2ω(kn​)=nωd​/2","key":"ZcqCs1zLD1"},{"type":"text","value":" can be excited, even with very low parametric forcing. In panel (b), a small dissipative term is included. 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Right: Simulation snapshots showing the evolution of the Faraday pattern in real space (top) and Fourier space (bottom). ©Figures from ","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"rxZPG5ZEMx"},{"type":"cite","identifier":"staliunas_faraday_2002","label":"staliunas_faraday_2002","kind":"narrative","position":{"start":{"line":19,"column":598},"end":{"line":19,"column":621}},"children":[{"type":"text","value":"Staliunas ","key":"ANqf9A8UDN"},{"type":"emphasis","children":[{"type":"text","value":"et al.","key":"INlDxCJrZe"}],"key":"l4Y3jPeuqn"},{"type":"text","value":" (2002)","key":"mSHfcmldzb"}],"enumerator":"1","key":"FUStPkSrpF"},{"type":"text","value":".","position":{"start":{"line":19,"column":1},"end":{"line":19,"column":1}},"key":"ux1HaiF1AS"}],"key":"SA9bt11vYU"}],"key":"ytn7ySV7i6"}],"enumerator":"1","html_id":"faraday","key":"GqhCbItcCo"},{"type":"paragraph","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"children":[{"type":"text","value":"These Faraday waves can be interpreted microscopically as pairs of Bogoliubov quasi-particles with opposite momenta, ","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"key":"hw90xqoe3C"},{"type":"inlineMath","value":"k","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"html":"kkk","key":"xHvNH5g5SQ"},{"type":"text","value":" and ","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"key":"wHUiJUSGQw"},{"type":"inlineMath","value":"-k","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"html":"−k-k−k","key":"iSZ2XoVDJt"},{"type":"text","value":". This is our approach in this manuscript: we partition the system in the ","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"key":"mg04kFU4Nb"},{"type":"inlineMath","value":"(k,-k)","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"html":"(k,−k)(k,-k)(k,−k)","key":"T4jtkcgn9M"},{"type":"text","value":" basis and focus on the non-separability of the two-mode state. This perspective is particularly relevant because, when the trap is turned off, the quasi-particle state maps onto the atomic state, yielding a momentum-entangled source of massive particles. To describe the state entanglement dynamics, we must keep the longitudinal modes quantized. With this approach, our treatment and theoretical description align more closely with other pair-creation mechanisms studied in the literature.","position":{"start":{"line":24,"column":1},"end":{"line":24,"column":1}},"key":"enLBcOObTc"}],"key":"jtvtwfhlpi"},{"type":"list","ordered":false,"spread":false,"position":{"start":{"line":25,"column":1},"end":{"line":30,"column":1}},"children":[{"type":"listItem","spread":true,"position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"children":[{"type":"text","value":"Four-wave mixing with two colliding Bose-Einstein condensates by ","position":{"start":{"line":25,"column":1},"end":{"line":25,"column":1}},"key":"kX8eXQy017"},{"type":"cite","identifier":"perrin_observation_2007","label":"perrin_observation_2007","kind":"narrative","position":{"start":{"line":25,"column":66},"end":{"line":25,"column":90}},"children":[{"type":"text","value":"Perrin ","key":"n6jwADUhbe"},{"type":"emphasis","children":[{"type":"text","value":"et 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In the second section we focus on the transverse dynamics of the ","key":"eILfj98IbB"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"IPFhdtMnbg"}],"key":"yXbN4zfGY4"},{"type":"text","value":" when the transverse trap frequency is modulated. The last section is dedicated to the pair creation process: we introduce Bogoliubov transformation and discuss under which conditions the entanglement of the two-mode state can be observed.","key":"kwwpAbpvxu"}],"key":"yjRRrCb5a4"},{"type":"comment","value":"We therefore start this chapter describing the BEC wave-function, which is the background for the collective oscillations.\nThis chapters starts by describing the BEC wave-function, factoring out the transverse dependance. The second section is dedicated to the oscillation of the transverse profile. Finally, in the third section, we introduce","position":{"start":{"line":33,"column":1},"end":{"line":34,"column":1}},"key":"sC6kzO2RTR"},{"type":"comment","value":"The details of this chapter follows. In the first section, we describe the wave-function of our cigar-shape BEC. We describe two regimes of the elongated Bose gas ([cigar shape](#radial_thomas_fermi_section) in [%s](#radial_thomas_fermi_section) and [1D mean field](#oneD_mean_field_regime) regime in [%s](#oneD_mean_field_regime)). Our gas being in the crossover between those two regimes, we introduce a Gaussian Ansatz to better describe our gas in [section](#crossover_regime) [%s](#crossover_regime). In the second section, we investigate the dynamics of the transverse excitation process. After a short introduction on collective excitations in [section](#collective_excitation) [%s](#collective_excitation), we focus on the breathing mode of the BEC in [section](#time_dependent_bec_width) [%s](#time_dependent_bec_width). We finally propose a protocol to better force arbitrary the BEC transverse oscillation frequency in [section](#forcing_oscillations) [%s](#forcing_oscillations).\nThe last section is dedicated to Bogoliubov treatment and recall the main theoretical results on work published after @jaskula_acoustic_2012 experiment. [Section](#bogoliubov_approx_section) [%s](#bogoliubov_approx_section) derives the Bogoliubov-de Gennes equation, that describes the pair creation process. [Section](#bogoliubov_transformation) [%s](#bogoliubov_transformation) introduces Bogoliubov transformations and [section](#controlling_creation_phonons) [%s](#controlling_creation_phonons) focuses on the parametric excitation mechanism. We numerically solve the field equation and discuss under which conditions two-mode squeezing can be observed.","position":{"start":{"line":36,"column":1},"end":{"line":37,"column":1}},"key":"a05cX4vb7y"},{"type":"comment","value":" \n\n\n In these approaches, the quantum field is not quantized. In our experiment, we are interested in the non-separability of the $(k, -k)$ quasi-particles. \n\n\nIn a stability diagram, this series of resonance are known as Mathieu's tong. They then perform a modulation stability analysis, \n\nfor periodically driven systems. \n%In the Thomas-Fermi regime, excitation can be labeled by an integer $n$ and corresponding energy $\\hbar\\omega_z\\sqrt{k(1+k/4)}$ [@menotti_collective_2002].\n\n\nThe first experimental observation of Faraday waves excited by parametric amplification in Bose-Einstein condensates was performed by @engels_observation_2007 after the proposal by @staliunas_faraday_2002. Theoretical studies that followed this experiment mainly focused on the prediction of the resonant wave-vector [@nicolin_faraday_2007;@nicolin_faraday_2010;@nicolin_resonant_2011]. More recently, @nguyen_parametric_2019 pushed further the excitation process of the Faraday wave to study the granulation regime. Faraday waves were also investigated with Fermi superfluid both theoretically [@capuzzi_2008_faraday;@Tang_2011] and experimentally [@hernandez_rajkov_faraday_2021] with a strongly interacting $^6$Li quantum gas. Faraday patterns were also studied with two-dimensional Bose gas [@zhang_pattern_2020;@liebster_emergence_2023]. \n\n\nAs we explained in the introduction, we are interested here in the non-separability of the quasi-particle pairs. We partition our system on the $(k, -k)$ basis and keep quantized the quasi-particles along the elongated axis: we make use of Bogoliubov formalism. Our theoretical treatment is close to other processes experimentally studied in the literature\n* Four-wave mixing with two colliding Bose-Einstein condensates by @perrin_observation_2007, @kheruntsyan_violation_2012 and @hodgman_solving_2017;\n* Collisional de-excitation of a 1D Bose gas by @bucker_twin_atom_2011;\n* Four-wave mixing by changing the dispersion relation in an optical lattice by @campbell_parametric_2006 and @bonneau_tunable_2013;\n* Modulation of the interaction strength by @clark_collective_2017 though a Feshbach resonance.\n\nThis chapter is decomposed as follows. In the first section, we describe the wave-function of our cigar-shape BEC. We describe two regimes of the elongated Bose gas ([cigar shape](#radial_thomas_fermi_section) in [%s](#radial_thomas_fermi_section) and [1D mean field](#oneD_mean_field_regime) regime in [%s](#oneD_mean_field_regime)). Our gas being in the crossover between those two regimes, we introduce a Gaussian Ansatz to better describe our gas in [section](#crossover_regime) [%s](#crossover_regime). In the second section, we investigate the dynamics of the transverse excitation process. After a short introduction on collective excitations in [section](#collective_excitation) [%s](#collective_excitation), we focus on the breathing mode of the BEC in [section](#time_dependent_bec_width) [%s](#time_dependent_bec_width). We finally propose a protocol to better force arbitrary the BEC transverse oscillation frequency in [section](#forcing_oscillations) [%s](#forcing_oscillations). \nThe last section is dedicated to Bogoliubov treatment and recall the main theoretical results on work published after @jaskula_acoustic_2012 experiment. [Section](#bogoliubov_approx_section) [%s](#bogoliubov_approx_section) derives the Bogoliubov-de Gennes equation, that describes the pair creation process. [Section](#bogoliubov_transformation) [%s](#bogoliubov_transformation) introduces Bogoliubov transformations and [section](#controlling_creation_phonons) [%s](#controlling_creation_phonons) focuses on the parametric excitation mechanism. We numerically solve the field equation and discuss under which conditions two-mode squeezing can be observed.\n\n","key":"zpbX7oGP6Q"},{"type":"comment","value":"This chapter describes the pair creation process and reviews the theoretical studies that has investigated the non-separability of the phonon pairs.","position":{"start":{"line":67,"column":1},"end":{"line":67,"column":1}},"key":"EwLG2MLtqR"},{"type":"comment","value":"The [first section](dce_bec) of this chapter discusses the ground state of the Hamiltonian that is the BEC wave-function. The [second section](dce_bec_time) studies the breathing mode of the BEC and how the BEC radius response to a time dependant trap. The [third section](dce_quasi_particles_creation) of this chapter focuses on the parametric excitation of collective excitations (phonons).","position":{"start":{"line":71,"column":1},"end":{"line":71,"column":1}},"key":"hV2rgkvPqZ"}],"data":{"part":"abstract"},"key":"LBoBKBognc"},{"type":"block","position":{"start":{"line":73,"column":1},"end":{"line":73,"column":1}},"children":[{"type":"admonition","kind":"tip","children":[{"type":"admonitionTitle","children":[{"type":"text","value":"What we knew, what is new ?","position":{"start":{"line":78,"column":1},"end":{"line":78,"column":1}},"key":"s4zVnqjWfd"}],"key":"qhfii12cID"},{"type":"paragraph","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"This chapter is mainly a review of the literature. The only contribution is the proposed protocol to better control the oscillations of the ","key":"ego6ULxnMh"},{"type":"abbreviation","title":"Bose-Einstein Condensate","children":[{"type":"text","value":"BEC","key":"hAPstBFhAz"}],"key":"nDhnlWcfxI"},{"type":"text","value":" radius, in ","key":"KaG7VXW6c4"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"section","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"qFPEdzXsC7"}],"identifier":"forcing_oscillations","label":"forcing_oscillations","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"forcing-oscillations","remote":true,"url":"/dce-bec-time","dataUrl":"/dce-bec-time.json","key":"ds34SHf1vV"},{"type":"text","value":" ","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"OsbRCTaUxr"},{"type":"crossReference","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"children":[{"type":"text","value":"3","key":"vV7ucsP4Gs"}],"identifier":"forcing_oscillations","label":"forcing_oscillations","kind":"heading","template":"Section %s","enumerator":"3","resolved":true,"html_id":"forcing-oscillations","remote":true,"url":"/dce-bec-time","dataUrl":"/dce-bec-time.json","key":"aT2e0eWGGp"},{"type":"text","value":".","position":{"start":{"line":80,"column":1},"end":{"line":80,"column":1}},"key":"aDExrj8oJK"}],"key":"iqKRgi2K6S"}],"key":"ky0XaXugta"},{"type":"comment","value":" Rather than solving the many-body Hamiltonian [](#hamiltonian_dilute_bose_gas), we replace the operator $\\hat{\\Psi}(\\mathbf{x}, t)$ by a classical field $\\Psi(\\mathbf{x}, t)$, the *order parameter*, which is the wave-function of the condensate. This approximation is known as the Bogoliubov approximation [@bogoliubov_theory_1947]. The macroscopically populated ground state will then act as a *reservoir* for the collective excitations, treated as a quantum field. \n ","key":"AyBzpB5uFj"}],"key":"uMeDyhqBwE"}],"key":"LGfOuEXleE"},"references":{"cite":{"order":["staliunas_faraday_2002","engels_observation_2007","nicolin_faraday_2007","nicolin_faraday_2010","nicolin_resonant_2011","nguyen_parametric_2019","hernandez_rajkov_faraday_2021","perrin_observation_2007","kheruntsyan_violation_2012","hodgman_solving_2017","bucker_twin_atom_2011","campbell_parametric_2006","bonneau_tunable_2013","clark_collective_2017"],"data":{"staliunas_faraday_2002":{"label":"staliunas_faraday_2002","enumerator":"1","doi":"10.1103/PhysRevLett.89.210406","html":"Staliunas, K., Longhi, S., & De Valcárcel, G. J. (2002). Faraday Patterns in Bose-Einstein Condensates. Physical Review Letters, 89(21), 210406. 10.1103/PhysRevLett.89.210406","url":"https://doi.org/10.1103/PhysRevLett.89.210406"},"engels_observation_2007":{"label":"engels_observation_2007","enumerator":"2","doi":"10.1103/PhysRevLett.98.095301","html":"Engels, P., Atherton, C., & Hoefer, M. A. (2007). Observation of Faraday Waves in a Bose-Einstein Condensate. Physical Review Letters, 98(9), 095301. 10.1103/PhysRevLett.98.095301","url":"https://doi.org/10.1103/PhysRevLett.98.095301"},"nicolin_faraday_2007":{"label":"nicolin_faraday_2007","enumerator":"3","doi":"10.1103/PhysRevA.76.063609","html":"Nicolin, A. I., Carretero-González, R., & Kevrekidis, P. G. (2007). Faraday waves in Bose-Einstein condensates. Physical Review A, 76(6), 063609. 10.1103/PhysRevA.76.063609","url":"https://doi.org/10.1103/PhysRevA.76.063609"},"nicolin_faraday_2010":{"label":"nicolin_faraday_2010","enumerator":"4","doi":"10.1016/j.physa.2010.06.029","html":"Nicolin, A. I., & Raportaru, M. C. (2010). Faraday waves in high-density cigar-shaped Bose–Einstein condensates. Physica A: Statistical Mechanics and Its Applications, 389(21), 4663–4667. 10.1016/j.physa.2010.06.029","url":"https://doi.org/10.1016/j.physa.2010.06.029"},"nicolin_resonant_2011":{"label":"nicolin_resonant_2011","enumerator":"5","doi":"10.1103/PhysRevE.84.056202","html":"Nicolin, A. I. (2011). Resonant wave formation in Bose-Einstein condensates. Physical Review E, 84(5), 056202. 10.1103/PhysRevE.84.056202","url":"https://doi.org/10.1103/PhysRevE.84.056202"},"nguyen_parametric_2019":{"label":"nguyen_parametric_2019","enumerator":"6","doi":"10.1103/PhysRevX.9.011052","html":"Nguyen, J. H. V., Tsatsos, M. C., Luo, D., Lode, A. U. J., Telles, G. D., Bagnato, V. S., & Hulet, R. G. (2019). Parametric Excitation of a Bose-Einstein Condensate: From Faraday Waves to Granulation. Physical Review X, 9(1), 011052. 10.1103/PhysRevX.9.011052","url":"https://doi.org/10.1103/PhysRevX.9.011052"},"hernandez_rajkov_faraday_2021":{"label":"hernandez_rajkov_faraday_2021","enumerator":"7","doi":"10.1088/1367-2630/ac2d70","html":"Hernández-Rajkov, D., Padilla-Castillo, J. E., Río-Lima, A. D., Gutiérrez-Valdés, A., Poveda-Cuevas, F. J., & Seman, J. A. (2021). Faraday waves in strongly interacting superfluids. New Journal of Physics, 23(10), 103038. 10.1088/1367-2630/ac2d70","url":"https://doi.org/10.1088/1367-2630/ac2d70"},"perrin_observation_2007":{"label":"perrin_observation_2007","enumerator":"8","doi":"10.1103/PhysRevLett.99.150405","html":"Perrin, A., Chang, H., Krachmalnicoff, V., Schellekens, M., Boiron, D., Aspect, A., & Westbrook, C. I. (2007). Observation of Atom Pairs in Spontaneous Four-Wave Mixing of Two Colliding Bose-Einstein Condensates. Physical Review Letters, 99(15), 150405. 10.1103/PhysRevLett.99.150405","url":"https://doi.org/10.1103/PhysRevLett.99.150405"},"kheruntsyan_violation_2012":{"label":"kheruntsyan_violation_2012","enumerator":"9","doi":"10.1103/PhysRevLett.108.260401","html":"Kheruntsyan, K. V., Jaskula, J.-C., Deuar, P., Bonneau, M., Partridge, G. B., Ruaudel, J., Lopes, R., Boiron, D., & Westbrook, C. I. (2012). Violation of the Cauchy-Schwarz Inequality with Matter Waves. Physical Review Letters, 108(26), 260401. 10.1103/PhysRevLett.108.260401","url":"https://doi.org/10.1103/PhysRevLett.108.260401"},"hodgman_solving_2017":{"label":"hodgman_solving_2017","enumerator":"10","doi":"10.1103/PhysRevLett.118.240402","html":"Hodgman, S. S., Khakimov, R. I., Lewis-Swan, R. J., Truscott, A. G., & Kheruntsyan, K. V. (2017). Solving the Quantum Many-Body Problem via Correlations Measured with a Momentum Microscope. Physical Review Letters, 118(24), 240402. 10.1103/PhysRevLett.118.240402","url":"https://doi.org/10.1103/PhysRevLett.118.240402"},"bucker_twin_atom_2011":{"label":"bucker_twin_atom_2011","enumerator":"11","doi":"10.1038/nphys1992","html":"Bücker, R., Grond, J., Manz, S., Berrada, T., Betz, T., Koller, C., Hohenester, U., Schumm, T., Perrin, A., & Schmiedmayer, J. (2011). Twin-atom beams. Nature Physics, 7(8), 608–611. 10.1038/nphys1992","url":"https://doi.org/10.1038/nphys1992"},"campbell_parametric_2006":{"label":"campbell_parametric_2006","enumerator":"12","doi":"10.1103/PhysRevLett.96.020406","html":"Campbell, G. K., Mun, J., Boyd, M., Streed, E. W., Ketterle, W., & Pritchard, D. E. (2006). Parametric Amplification of Scattered Atom Pairs. Physical Review Letters, 96(2), 020406. 10.1103/PhysRevLett.96.020406","url":"https://doi.org/10.1103/PhysRevLett.96.020406"},"bonneau_tunable_2013":{"label":"bonneau_tunable_2013","enumerator":"13","doi":"10.1103/PhysRevA.87.061603","html":"Bonneau, M., Ruaudel, J., Lopes, R., Jaskula, J.-C., Aspect, A., Boiron, D., & Westbrook, C. I. (2013). Tunable source of correlated atom beams. Physical Review A, 87(6), 061603. 10.1103/PhysRevA.87.061603","url":"https://doi.org/10.1103/PhysRevA.87.061603"},"clark_collective_2017":{"label":"clark_collective_2017","enumerator":"14","doi":"10.1038/nature24272","html":"Clark, L. W., Gaj, A., Feng, L., & Chin, C. (2017). Collective emission of matter-wave jets from driven Bose–Einstein condensates. Nature, 551(7680), 356–359. 10.1038/nature24272","url":"https://doi.org/10.1038/nature24272"}}}},"footer":{"navigation":{"prev":{"title":"Introduction","url":"/intro","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"},"next":{"title":"Description of the ground state BEC","short_title":"Description of the ground state BEC","url":"/dce-bec","group":"On the entanglement of quasi-particles in a Bose-Einstein condensate"}}},"domain":"http://localhost:3011"}