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Thermodynamics and Magnetism of Antiferromagnetic Spinor Bose-Einstein Condensates Camille Frapolli

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Thermodynamics and Magnetism of Antiferromagnetic Spinor Bose-Einstein Condensates Camille Frapolli Thermodynamics and magnetism of antiferromagnetic spinor Bose-Einstein condensates Camille Frapolli To cite this version: Camille Frapolli. Thermodynamics and magnetism of antiferromagnetic spinor Bose-Einstein con- densates. Other [cond-mat.other]. Université Paris sciences et lettres, 2017. English. NNT : 2017PSLEE006. tel-01515042v2 HAL Id: tel-01515042 https://tel.archives-ouvertes.fr/tel-01515042v2 Submitted on 2 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences Lettres – PSL Research University préparée à l’École normale supérieure Thermodynamics and magnetism of antiferromagnetic spinor Bose-Einstein condensates Thermodynamique et magnétisme dans des condensats de Bose-Einstein de spin 1 avec interactions antiferromagnétiques École doctorale n°564 Spécialité : Physique Soutenue le 29.03.2017 Composition du Jury : Mme. Patrizia Vignolo Université de Nice – Sophia Antipolis par Camille Frapolli Présidente du jury M. Gabriele Ferrari Università degli Studi di Trento dirigée par Fabrice Gerbier Rapporteur & Jean Dalibard M. Bruno Laburthe-Tolra CNRS, Université Paris 13 – Villetaneuse Rapporteur M. Fabrice Gerbier CNRS Directeur de thèse Thermodynamics and magnetism of antiferromagnetic spinor Bose-Einstein condensates Camille Frapolli supervised by Fabrice Gerbier & Jean Dalibard Contents 0 Introduction 9 1 Elements of theory for spinor Bose gases 13 1.1 Single component Bose gas at inite temperature . 13 1.1.1 Ideal gas in a harmonic trap . 14 1.1.2 Finite size efects . 17 1.1.3 Role of interactions . 19 1.1.4 Role of realistic trapping potentials . 21 1.2 Single component Bose-Einstein condensates at T=0 . 29 1.2.1 Ground state of Bose-Einstein condensates . 29 1.2.2 Free expansion of Bose-Einstein condensates . 31 1.3 Spin-1 Bose-Einstein condensate at T=0 . 32 1.3.1 Spinor scattering properties . 32 1.3.2 Zeeman energy . 34 1.3.3 Single mode approximation . 36 1.4 Critical temperatures of ideal spin-1 gases . 39 1.5 Conclusion......................................... 42 2 Production and diagnostics of ultracold gases of sodium atoms 43 2.1 Experimental realization of spinor Bose-Einstein condensates . 44 2.1.1 Overview . 44 2.1.2 Laser cooling and trapping . 45 2.1.3 Optical dipole traps and evaporative cooling . 48 2.1.4 Absorption imaging . 50 2.1.5 Characterization of the trapping potential . 53 2.2 Spin-dependent imaging . 57 2.2.1 Stern-Gerlach Imaging . 57 2.2.2 Imaging noise reduction . 58 2.2.3 Magniication . 66 2.2.4 Determination of the scattering cross-sections . 68 2.3 Imageanalysis ....................................... 71 2.3.1 Fit of images . 71 2.3.2 Extraction of temperatures . 72 2.4 Manipulation of the atomic spin . 73 2.4.1 Rabi oscillations . 73 2.4.2 Adiabaticity in magnetic ield ramps . 76 2.4.3 Diagnostics of the magnetic environment . 77 2.4.4 Preparation of the magnetization . 80 2.4.5 Adiabatic rapid passage . 81 2.5 Conclusion......................................... 82 3 Magnetic phases of antiferromagnetic spinor BEC at low temperatures 85 3.1 Mean-Field description of spinor Bose-Einstein condensates . 86 3.1.1 Alignment and spin-nematic order . 87 3.1.2 Mean-Field description at T=0 . 88 3.1.3 T=0 magnetic phase diagram (D. Jacob et al. PRA86061601) . 89 3.2 Observation of phase locking in spinor BEC . 91 3.2.1 Methods . 91 3.2.2 Observation of phase locking . 93 3.2.3 Efect of a inite kinetic temperature . 96 3.2.4 Efect of a inite spin temperature . 98 3.3 Spin luctuations . 100 3.3.1 Spin luctuations for low magnetization . 101 3.3.2 Experimental measurement of luctuations . 101 3.3.3 Spin thermometry . 103 3.4 Conclusion......................................... 106 4 Thermodynamic phase diagram of a spin 1 Bose gas 109 4.1 Experimental methods . 111 4.1.1 Overview . 112 4.1.2 Boost to the Stern-Gerlach separation . 114 4.1.3 Image Analysis . 115 4.2 Experimental thermodynamic phase diagram of a spin 1 Bose gas . 121 4.2.1 Extraction of the critical temperature . 121 4.2.2 Graphical representation of the data . 124 4.2.3 hermodynamic phase diagram . 126 4.3 Hartree-Fock model . 130 4.3.1 Hartree-Fock description . 130 4.3.2 Semi-ideal “four gasesž formalism . 132 4.3.3 Numerical resolution of the HF model and comparison with data . 136 4.4 Conclusion......................................... 143 5 Conclusion 145 5.1 Summary.......................................... 145 5.2 Prospects .......................................... 147 Appendices 149 A Alternative laser source for laser cooling and Raman transitions 151 A.1 Second Harmonic Generation . 151 A.2 Intracavity frequency doubling . 153 A.3 Iodine spectroscopy . 155 B Light-Atom Interaction in the ground state electronic manifold 157 B.1 Derivation of the lightshit hamiltonian . 157 B.2 Lightshit operator in the hyperine basis . 161 C Raman Schemes 163 C.1 Rabi oscillations with copropagating beams . 163 C.2 Spin Orbit coupling: Spielman scheme ......................... 166 C.3 Spin orbit coupling: 4 photons transitions . 166 D Spin-mixing oscillations 171 D.1 Spin mixing oscillations ater a Rabi Oscillation . 171 D.2 Evolution of the populations during a magnetic ield ramp . 174 E Perturbative development around the point of simultaneous condensation 177 F Remerciements 181 Bibliography 183 0 Introduction any body physics describes the behavior of physical systems with a large number of constituents. MIts applications covers many domains such as quantum chemistry, where it describes the behavior of electrons in atoms or molecules, nuclear physics, condensed matter physics where it describes electrons in a crystal or ultracold atoms physics where it describes small gaseous samples (up to a few billions atoms). he N body problems with N 1 are not exactly solvable despite their ubiquity in physics. Many approximate solutions were developed to circumvent this issue and obtain efective descriptions of these systems that can be≫ used to predict their properties. In almost all cases, a statistical approach (see refs. [1, 2]), allowed by the large number of particles, is used to describe the system by a limited set of macroscopic state functions (temperature T, pressure P, density n...). he relations between these are derived from speciic assumptions on the microscopic system. Perhaps the simplest assumption consists in considering the constituents of the gas to be independent, which leads to the ideal gas model. Ideal gas theory however fails when the interparticle distance becomes small enough such that interactions between atoms or molecules are not negligible. his happens in dense samples (condensed matter, nuclei) and also, perhaps more surprisingly, in dilute atomic gases at ultralow temperatures. In this case, the deviations from ideal theory can be treated perturbatively, for example with a virial expansion. he parameters of such expansion depend on the microscopic properties of the system and are in general challenging to compute for large corrections. A particularly spectacular failure of this type of theories is observed when interactions lead to a particular ordering at the macroscopic level, i.e. to a new state of matter. his new state oten form collective states and are usually observed at “lowž temperature. he ordering characteristic of the collective state washes out due to thermal luctuations above a second order phase transition at a temperature that depends on the interaction strength. Examples of such systems are realized with ferromagnetic materials (see ref. [3]), in which a permanent magnetic moment can be observed even in absence of magnetic ields below the Curie temperature, due to the alignment of the spins within the material. Another example of collective behavior results from the observation of superluid low in liquid helium at low temperatures (see refs. [4, 5]). his originates from a peculiar efect of the quantum statistics of bosonic particles predicted by Bose for photons (see ref. [6]) and generalized by Einstein (see ref. [7]) in 1924 that leads to a large fraction of the liquid composed of atoms being in the lowest energy single particle state (a plane wave at rest in uniform systems). his phenomenon called Bose-Einstein condensation, is predicted even for an ideal 10 0. Introduction gas. he large interactions in liquid helium complicates the theoretical description of its exact properties (see ref. [5]). he advances in laser cooling of atoms (that were awarded by a Nobel prize to S. Chu, C. Cohen- Tannoudji and W. Phillips in 1997, see ref. [8ś10]) enabled the production of extremely cold gaseous samples at temperature in the microKelvin range. Further advances in trapping and evaporative cooling led to the production of even lower temperature clouds and to the irst gaseous Bose-Einstein condensate, where a macroscopic population appears in the ground state of the trapping potential containing the gas. heir realization was awarded by a Nobel prize to E. Cornell, W. Ketterle and C. Wieman in 2001, see refs [11ś13] in 1995. Depending on the
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