THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences Lettres – PSL Research University préparée à l’École normale supérieure Phase separation and spin domains in quasi-1D spinor condensates ◦ École doctorale n 564 Spécialité : Physique Soutenue le 09.11.2017 Composition du Jury : Mme. Isabelle Bouchoule CNRS - Institut d’Optique Rapporteuse M. Jan Arlt Andrea Invernizzi Aarhus University par Rapporteur Mme. Anna Minguzzi dirigée par Fabrice Gerbier CNRS - LPMMC Grenoble & Jean Dalibard Examinatrice M. Fabrice Gerbier CNRS Directeur de thèse M. Jean Dalibard Collège de France Co-directeur de thèse Contents 1. Introduction 3 2. Elements of Bose-Einstein condensation 7 2.1. The scalar Bose-Einstein condensate in a 3d harmonic trap. . . . . . . . . 7 2.1.1. The ideal Bose gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2. The role of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3. The mean-field approximation at T 0 . . . . . . . . . . . . . . . . 9 2.1.4. Mean field approximation at T 0 .=. . . . . . . . . . . . . . . . . . 11 2.2. The scalar Bose gas in a 1D harmonic >trap . . . . . . . . . . . . . . . . . . 15 2.2.1. Bose gases in one dimension . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.2. Quasicondensation in 3D anisotropic trap . . . . . . . . . . . . . . . 17 2.2.3. Phase Fluctuations in TOF . . . . . . . . . . . . . . . . . . . . . . . 19 2.3. The spin-1 Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1. Hyperfine structure of Na atoms . . . . . . . . . . . . . . . . . . . . 21 2.3.2. Two-body scattering between Na atoms . . . . . . . . . . . . . . . . 22 2.3.3. The Zeeman shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.4. The Spinor Many-body Hamiltonian . . . . . . . . . . . . . . . . . . 25 2.3.5. Spinor BEC in the single spatial mode. . . . . . . . . . . . . . . . . 26 3. Production and characterization of a spin-1 Bose-Einstein condensate of Sodium atoms 33 3.1. Experimental Setup and cooling techniques . . . . . . . . . . . . . . . . . . 33 3.1.1. UHV chamber and atomic source . . . . . . . . . . . . . . . . . . . . 33 3.1.2. Magneto-optical trap . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.3. The crossed dipole trap and the dimple optical traps . . . . . . . . 36 3.1.4. Stern-Gerlach time of flight . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.5. Imaging after TOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2. Manipulating internal states with spin degrees of freedom . . . . . . . . . 44 3.2.1. Magnetic field control . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2.2. Rabi Oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.3. Adiabatic rapid passage . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.4. Magnetization preparation . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3. Image characterisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1. Magnification characterisation . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2. Calibration of spin-dependent cross sections . . . . . . . . . . . . . 53 3.4. Image analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.1. Noise modelisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 1 Contents 2 3.4.2. Noise reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.5. From the 3d to the 1d geometry . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.1. The adiabatic transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.5.2. Characterisation of the trap frequencies . . . . . . . . . . . . . . . . 63 4. Stepwise Bose-Einstein Condensation in a Spinor Gas 67 4.1. Article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2. Supplementary Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1. Experimental sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.2. Evaporation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2.3. Extracting T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 c 4.2.4. Theoretical models of spinor gases at finite temperatures . . . . . 80 5. Spin-1 BEC in 1D: Spin domains and phase transition 87 5.1. Stable phases of a 1D Spin-1 antiferromagnetic BEC . . . . . . . . . . . . 88 5.1.1. The uniform case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.2. Adding an harmonic potential in the LDA approximation . . . . . 91 5.1.3. The phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.1.4. GP simulation vs LDA solution . . . . . . . . . . . . . . . . . . . . . 95 5.2. 1D-3D crossover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.3. Preparation and study of spin domains . . . . . . . . . . . . . . . . . . . . . 99 5.3.1. Minimisation of magnetic field gradients . . . . . . . . . . . . . . . 100 5.3.2. Fitting the Spin Domains . . . . . . . . . . . . . . . . . . . . . . . . 102 5.3.3. Equation of State and temperature . . . . . . . . . . . . . . . . . . . 105 5.4. 1D Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6. Binary mixtures 117 6.1. Spin-dipole polarisability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.1.1. Polarised cloud response: z . . . . . . . . . . . . . . . . . . . . . . . 119 0 6.1.2. Spin-dipole polarisability vs magnetisation . . . . . . . . . . . . . . 119 7. Conclusions and perspectives 125 Appendices 131 A. Adiabatic Transfer of a quasi-condensate in 1D 133 A.1. Theory of quasi-condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.2. The adiabatic transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 B. Numerical solution of the spin 1 Gross-Pitaevskii equations 139 1. Introduction On June the 5th, 1995, the team led by Eric Cornell and Carl Wieman produced for the first time a Bose-Einstein condensate (BEC). Using the cooling laser techniques developed in the previous years, [27, 29, 143], they cooled down a dilute ensemble of Rb atoms to the quantum degeneracy via evaporative cooling. Slightly later, the team led by Wolfgang Ketterle obtained a condensate of Na atoms. Their discovery, awarded with the Nobel Prize in 2001 [31, 85], gave birth to a new research field, at the crossing point of atomic physics, condensed matter and quantum optics called Quantum gases. The theoretical prediction of BEC condensation dates back to the works [21, 39] by Einstein and Bose. We know from quantum mechanics that the particle-wave duality becomes more and more visible as we lower the temperature. In a system of particles (bosons) at temperature T, we can define for each particle the De Broglie wavelength λ T−1￿2, describing the size of the wave associated to each particle. Lowering the dB temp∝erature, λdB increases and will eventually become comparable to the average inter- particle distance n−1￿3. While fermions tend to avoid each other (Pauli principle), bosons tend to gat∝her into a single state [105]: this phenomenon is BEC. The Bose-Einstein condensate represents an interesting system by itself. Its phase coherence properties were proofed by making two BEC interfere, [5] and by measuring its long range coherence, [18]. The superfluid nature of the BEC was proofed by the observation of quantised vortices, [110, 108, 1] and of superfluid flow, [150]. An "atom laser" was built from a BEC, exploiting the wave nature of this state of matter, [19, 60]. At the same time, the techniques used for producing BEC were extended to fermions, obtaining the first ultracold degenerate Fermi gas [38]. Dilute systems are characterised by weak interactions, but Feshbach resonances, ob- served from the early years [72], can be used to change the interaction strength and to switch from repulsive to attractive interactions and vice-versa. Feshbach resonances were also used to produce molecules in Fermi gases to observe the crossover from a BEC of weakly bound molecules to a superfluid composed of Coopers pairs [22, 11]. 3 4 The first BEC experiments were based on magnetic trapping. A few years later, the development of optical trapping, [170, 57], allowed a great control over the trap geometry. Increasing the confinement along one or two directions permitted the study of systems in lower dimensionality. In 1D traps, the Tonks-Girardeau regime was observed [90, 136] and in 2D traps, the Berezinskii-Kosterlitz-Thouless transition was observed [59]. More recently, a text-book model like the box potential was realised, [49]. In the last years, more complicated trapping potentials have been produced. The light of different laser beams can be made interfere to obtain periodic potentials in 1,2 and 3 dimensions: the optical lattices. Thanks to these potentials different condensed matter models were studied, as the phase transition between superfluid and Mott insulator phase [56, 167, 174, 80] and magnetism on a lattice [177]. Disordered potentials were also produced and quantum localisation phenomena observed [14]. Spinor Bose Gases Opticaltrappingprovidedalsoapotentialindependentfromtheatomicspin. Thisenabled the study of multicomponent gases. In condensed matter superfluid 3He, [185] and some unconventional superconductors with spin-triplet Cooper pairing [127] are examples of multicomponent quantum fluids. In Bose gases, mixtures composed by different isotopes of the same atom [135] and by different atomic species [114] have been studied, alongside with Fermi gases mixtures, [157], and Bose-Fermi gases mixtures, [46, 155]. Degenerate Bose gases with a spin degree of freedom are called spinor gases and consti- tute another example of multicomponent quantum fluids. The interplay between external and internal degrees of freedom in these systems gives rise to phenomena unfamiliar from studies of single-component ("scalar") quantum fluids. The macroscopic occupation of the ground state allows to distinguish energy levels whose energy difference ✏ is small com- pared to the system temperature ✏ k T. This quantum-statistical Bose enhancement, B [169], allows inter-component interactions, which have typical energies 1nK, to order the system just below the Bose-Ei￿nstein condensation temperature and makes spinor condensates a good system to study magnetic phases of matter. ∼ Thanks to spin exchange contact interactions between the internal components, spinor condensates present coherent spin oscillations [104, 194, 25] and parametric spin amplifi- cation [92]. Moreover, depending on the nature of the spin interaction, ferromagnetic or antiferromagnetic, the magnetic ordering of the BEC results in different possible magnetic phases, [169]. Dipolar interactions, thanks to their long range, give an important contribution to the physics of the system. Dipolar effects are clearly visible in high spin atoms, as chromium, erbiumanddysprosium, wheredensitydeformations[98]andinstability[97]wereobserved. Dipolar gases present also spin relaxation phenomena [43, 137, 126] and anisotropic exci- 5 Introduction tations [15]. The anisotropy of dipole interactions favours low-energy states characterised by spin textures [190]. Spinor Gases and Stepwise condensation Part of this thesis is devoted to the study of the thermodynamics of a spin-123Na gas. In particular, we study how the magnetic ordering appear in the system as we lower the temperature and we cross the critical point. Several studies, [188, 79, 40] indicate that without additional constraints, the Bose-Einstein statistics favours ferromagnetism. In our system, however, the longitudinal magnetisation is conserved. This has a deep impact on the thermodynamic phase diagram. From the theoretical point of view the problem has been already studied in a certain number of ways, [73, 192, 82, 100, 182, 145, 83] and the generic solution is that BEC occurs first in one specific component, while magnetic order appears at lower temperatures, when two or more components condense. In the thesis we will present the observation of multi-step condensation in our condensate of sodium atoms. Spin Domains and Phase transition In the second part of the thesis we present the study of the spin-1 condensate in an anisotropic trap. In this configuration, the system presents the formation of domains of spins, [173]. This system was already studied in [173, 117, 171], but the ground state configuration of the system in a uniform magnetic field has not yet been studied in detail. The ground state of spin-1 antiferromagnetic Bose gases presents two magnetic phases, [76]: an antiferromagnetic phase and a transverse magnetised phase. For a system with spin domains, the phase transition between these two different phases corresponds also to the transition, respectively, from the miscible to the immiscible regime. In the thesis we report on the experimental investigation of the ground state of the system in a uniform magnetic field as well as on the observation of the phase transition. The measurement of the response to a magnetic field gradient is also presented. 6 Thesis Outline The manuscript is organised as follow: Chapter 2 is divided in three sections. The first section is a brief introduction to 3D condensates in a harmonic trap. The second section introduces some notions on 1D condensates, we will focus on the phase fluctuations and the concept of quasi-condensate. In the third and final section we introduce the spin-1 23Na spinor condensate. Chapter 3isfocusedontheexperimentaltechniquesusedtotrap,cooldownandimage our spinor condensate. After a first part devoted to the experimental sequence to obtain a 3D condensate, we focus on the methods used to manipulate the internal spin degrees of freedom of the atoms. A description of the analysis method used for the images follows. Thechapterendswithadescriptionofthetransferoftheatomsfromthe3Dtothe1Dtrap. Chapter 4 is devoted to the presentation of the article "Stepwise Bose-Einstein Con- densation in a Spinor Gas", [48], a work performed at the beginning of my thesis and already described in [47]. We report the article without modifications. Chapter 5 is divided into two parts. In the first part we discuss the theory that predicts the ground state of a spin-1 condensate in an anisotropic harmonic trap. In the second part we present the experimental characterisation of the ground state, the measured Equation of State of a polarised cloud (all the atoms in the m 1 Zeeman F sub-level) and the observed phase transition. =+ Chapter 6 presents some measurements on the spin-dipole polarisability of the system. Appendix A contains a theoretical study of the transfer of our condensate from the 3D to the 1D trap. Appendix B presents the algorithm used to solve the Gross-Pitaevskii equations for a spin-1 Bose gas in a unidimensional system. 2. Elements of Bose-Einstein condensation Bose-Einstein condensation (BEC) is a second order transition observed in Bose gases defined by the macroscopic occupation of the ground state of the system. The first time Bose-Einstein condensation was observed, [36, 4], the atoms were trapped in a magnetic trap. Nowadays optical dipole traps, produced by focused laser beams, are widely used and enabled the trapping of atoms in different internal states at the same time. Duringthisthesisworkwestudiedaspin1Bosegasof23Naatomsthatwecooldowntothe quantum degeneracy in optical dipole traps. Optical dipole traps can be approximated, in the neighbourhood of their focus, by a harmonic potential. Hence, we limit our discussion to Bose gases trapped in a potential: 1 V r m ! x2 ! y2 ! z2 (2.1) ext x y z 2 ( )= ￿ + + ￿ where m is the mass of a single atom and ! , with i x, y, z , are the harmonic oscillator i frequencies along the three coordinates axis. The{versati}lity of these kinds of traps allowed us to study the system in two different geo∈metries: in a 3D configuration, where ! ! ! , and in a 1D configuration, where ! ! ! . x y z x y z Thi∼s cha∼pter presents the basics elements of Bose-Ei∼nstei￿n condensation theory in 3D and 1D geometries before focusing on the spin 1 Sodium condensate. The contents introduced here constitute a minimal theory reference necessary to understand the experiments performed on spinor BECs reported in this thesis. We refer the reader to some more general reviews [33, 141, 169]. The content is organised as follow: Section 2.1 and Section 2.2 present the theory of single component scalar BEC in 3D and 1D harmonic traps, respectively. Section 2.3 describes the theory of spin-1 spinor condensates. 2.1. The scalar Bose-Einstein condensate in a 3d harmonic trap This Section is devoted to a brief presentation of the Bose-Einstein condensation in 3D harmonic trap. We introduce the T 0 theory for an ideal Bose gas, before describing = 7 2.1. The scalar Bose-Einstein condensate in a 3d harmonic trap 8 the effects of interactions and the local density approximation. The T 0 theory, with the Bogoliubov and Hartree-Fock approximations follow. > 2.1.1. The ideal Bose gas The ground state wavefunction of the non interacting Bose gas, trapped of a potential like (2.1), corresponds to the ground state of the 3-dimensional harmonic oscillator: 1 1 4 x2 Ψ N ￿ exp − i (2.2) √ ⇡a2 a2 x,y,z￿ i,ho￿ ￿ i,ho￿ = where ￿ ￿ ￿ ￿ h a (2.3) i,ho ￿m! ￿ i is the harmonic oscillator length. We ne=ed a description of the atoms in the excited states; we do so by adopting a semi-classical approximation [147]. We obtain an analytical formula for the density of thermal atoms : 1 n g eβ(µ−Vext) (2.4) th λ3 3￿2 th ( ) = whereλ h 2⇡mk T isthethermalwavelengthand g isapolylogarithmicfunction, th √ B 3￿2 or Bose function of the general form g x +∞ xj. If we integrate over the entire = ￿ ↵ ∑j=1 j↵ system we find that the total number of t(he)rmal atoms is = 3 k T N B g eβµ (2.5) 3 h!¯ ￿ ￿ ( ) = ￿ where!¯ ! ! ! 1￿3isageometrical￿averag￿eofthethreeharmonicoscillatorfrequencies. x y z The chemi(cal poten)tial µ 0 is fixed by the condition Ntot N N0, where N0 is the number =of atoms in the ground state. At a fixed temperature, if we increase the number of atoms in the system, µ g<rows until µ 0 in (2.5). The Bos=e fun+ction g x reaches at 3 x 1 its maximum value and this corresponds to a maximum value for t(he)number of atoms in the excited states N. Bose func=tions are not defined for values x 1, physically th=is means that if we add more particles to the system, they will not populate the already saturated excited states, but they will start to condense in the ground stat>e N N 0. 0 tot This saturation of the excited states marks the onset of Bose-Einstein condensation.≠ ￿ 2.1.2. The role of interactions Even when atomic gases are extremely dilute, they are far from ideal gases. Interactions play very important roles and must be taken into account to predict the experimental observations [33]. As soon as the condensate forms, the density inside the degenerate cloud rises significantly such that the interactions become important to quantitatively