Two-component Bose-Einstein condensates with competing interactions

Two-component Bose-Einstein condensates with competing interactions by Julio Sanz S´anchez A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in Photonics Thesis supervisor: Prof. Dr. Leticia Tarruell Ultracold Quantum gases group The Barcelona Institute of Science and Technology - BIST Institut de Ci`encies Fot`oniques - ICFO Universitat Polit`ecnica de Catalunya - UPC c Copyright 2020 by Julio Sanz S´anchez Dedicada a la meva familia, a la meva parella, i als meus amics. “Ojal´a nuestra tecnolog´ıa nunca deje atr´as nuestra filosof´ıa. Ojal´a nuestro poder nunca supere nuestra compasi´on. Y que el motor del cambio no sea el miedo, sino el amor.” Edmond Kirsch Abstract This thesis reports the experimental study of two-component Bose-Einstein condensates with tunable interactions, which are exploited as a platform to perform quantum simulation of many-body quantum systems. To perform this experiments, we have implemented an atomic source consisting on a glass cell 2D MOT vacuum chamber and a high resolution optical system to image and manipulate the atoms. Furthermore, we develop and characterize a polarization phase contrast technique which is able to probe optically dense atomic mixtures at intermediate and high magnetic fields in open transitions. This technique has been used to either probe the total column density of a two-component atomic cloud or the difference in column density between both components. We report on the first observation of composite quantum liquid droplets in an incoherent mixture with residual mean field attraction. Strikingly, this novel phase is stabilized due to the repulsive beyond mean field corrections in a weakly interacting system. Moreover, we have characterized the liquid to gas phase transition which occurs for small atom numbers. Additionally, we have compared two different self-bound states in a quasi1D geometry with incoherent mixtures: quantum droplets and bright solitons. Depending on the atom number and interaction strengths both states can be smoothly connected through a crossover or be distinct entities separated by a transition. We have measured its composition, its phase diagram and mapped out the soliton to droplet transition. Finally, we report on a technique to modify the elastic and inelastic inter- actions in a two-component Bose-Einstein condensate with very unequal and competing interactions under the presence of strong coherent coupling. This technique provides a wide flexibility and has allowed us to observe bright solitons in quasi-1D in a coherently coupled dressed state. We exploit the fast temporal control of the effective interactions to quench them into the attractive regime and study the resulting modulational instability which develops into a bright soliton train. viii Resum Aquesta tesi descriu l’estudi experimental d’una mescla de dos condensats de Bose-Einstein amb interaccions ajustables. Aquest sistema ´es utilitzat com una plataforma per a estudiar sistemes qu`antics formats per moltes part´ıcules a partir de la simulaci´o qu`antica. Per a fer aquests experiments, he constru¨ıt una font at`omica formada per una trampa magneto-`optica en 2D que s’implementa en una cambra de buit feta de vidre. A m´es a m´es, he desenvolupat i caracteritzat una t`ecnica d’imatge de contrast de fase basada en la rotaci´o de la polaritzaci´o de la llum. Aquesta t`ecnica est`a preparada per fer imatges de mescles at`omiques a camps magn`etics intermedis i alts amb una gran densitat `optica i amb transicions `optiques obertes. Hem utilitzat la t`ecnica per a mesurar la densitat integrada total en l’eix `optic aix´ı com la difer`encia entre ambdues components. Es descriu la primera observaci´o de gotes l´ıquides qu`antiques compostes per dues components incoherents amb una atracci´o residual en l’aproximaci´o de camp mitj`a. Sorprenentment, aquesta nova fase est`a estabilitzada a causa de la repulsi´o generada per les correccions de l’energia m´es enll`a de l’aproximaci´o de camp mitj`a en un sistema amb interaccions d`ebils. Tamb´e hem caracteritzat la transici´o de fase l´ıquid-gas que succeeix quan el sistema t´e un nombre d’`atoms redu¨ıt. A m´es a m´es, hem comparat dos estats autoconfinats de diferent natura en una geometria quasi-1D amb una mescla d’`atoms incoherents: les gotes qu`antiques i els solitons brillants. Segons el nombre d’`atoms i la for¸ca de les interaccions aquests estats poden estar connectats o b´e suaument o b´e per una transici´o de fase. Hem mesurat la seva composici´o, el diagrama de fases i hem tra¸cat el mapa ix Later on, I shared time in the lab with the postdoc Bruno Naylor and the PhD student Anika Fr¨olian. Among many things that I learned thanks to Bruno, I finally understood two major things: the level structure of the atom and the ‘level structure’ of society. I thank you a lot for all the useful scientific and political discussions that enriched me both from the professional and personal point of view. At the time, we had a major accident in the experiment. The impact factor of such incident in my learning process at ICFO has been by far greater than that of any published results used to evaluate my success. Among all the team who helped to rebuild the experiment, I want to thank especially Anika. Besides all the pressure that I put you through, you kept pushing and were essential to revive the Fenix from its ashes. Moreover, together with the PhD student Craig Chisholm, who joined few months later, you have made my life extremely easy, becoming the new masters of the machine very fast. With you, Anika and Craig, I feel like the baby is in very good hands for long time. I also owe Craig many tricks about programming tricks and rf sources. Around a year ago, the master student David Jacobs and the PhD student Jonathan H¨oschele started a new lab with Strontium. I wish I could have spent more time with you hand on hand in the lab. Still, it was a pleasure to have you in the team. I must admit that after some time seeing team members leave I was starting to be sad. This feeling changed for me after you arrived. Together with Anika and Craig you have been part of a new team generation who made my time at ICFO very fun. In this new generation, I am also very grateful to have shared time with the postdocs Ram´on Ramos, Elletra Neri and Vasily Makhalov. You have suddenly provided a great deal of expertise to the Potassium and Strontium labs and have become essential members of the QGE family. From ICFO cal¸cotadas, long discussions over lunch to the ICFO parties. . . I will keep great memories from all of you! During all this time, I have also shared a fantastic time working with other people. I keep great memories from the always smiley Lisa Saemisch. Jordi Sastre master of Signadyne. The refined humor of Vincent (the 2nd ) Brunaud. The funny summers with ´ I˜nigo Urtiaga and Alberto Mu˜noz playing with the DMD. The lattice and language master Phillip Thomas. The amazing time with Teo Gil and Daniel Allepuz building the magnetic field stabilization and measuring xvi the aberrations of the DMD and the Raman master Manon Ballu. I also had the pleasure to meet the former team members Vincent (the 1st ) Lienhard and Manel Bosch. I am thankful for all these time at ICFO sharing good memories with you and learning from you. The experimental work that we carried out during this time would not have been possible without the help of Xavier Menino, the mechanical workshop team, Jose Carlos Cifuentes and the electronic workshop team. They have been paramount not only in providing good advice and constructing all the devices that we use in the lab, but on teaching me mechanical and electronics design. I am also thankful to the IT, purchasing, logistics, maintenance and administration department who have made my work at ICFO extremely easy. I will especially remember Mika, Santi, Magda, Carlos, Lu´ıs and N´uria. Especial thanks to the members of the PhD committee Professor Luis Santos, Jordi Boronat and Thomas Bourdel for the very careful reading of the manuscript and the flattering reports. I have also had the pleasure of sharing very inspiring discussions on quantum droplets throughout my PhD with you. I will always be thankful for that. My work during the PhD would not have been possible without my family, partner and friends. Com b´e sabeu, el doctorat ha estat emocionalment com una montanya russa per a mi. A la meva familia, la meva parella i els meus amics us dedico aquesta tesi per tota la vostra estima i paci`encia per a treure’m un somriure en tot moment. Als meus pares els ho dec tot. Vosaltres m’heu educat com s´oc i m’heu portat fins aqu´ı. M’heu ensenyat a ser perseverant, a tenir paci`encia, a treballar dur, a ser humil, a aixecar-me quan he caigut. . . M’heu fet veure quan m’ofegava en un got d’aigua i m’ho heu posat tot f`acil per a ser feli¸c. Infinites gr`acies per tot! A l’Alexandra, la meva parella, li estar´e sempre agra¨ıt. T´u me has acompa˜nado en todo momento en este trayecto de emociones durante estos a˜nos. Has sido mi balanza para mantenerme en pie cuando lo he pasado mal y me has sostenido todo este tiempo. Sin duda, eres la persona con la que m´as he pagado el estr´es injustamente, y aun as´ı has sido siempre paciente y comprensiva conmigo. Me has hecho muy feliz durante todo este tiempo y me has convencido d´ıa a d´ıa que todo este esfuerzo ha valido la pena. Te estar´e eternamente agradecido. Tambi´en xvii le agradezco much´ısimo a tu familia todo el apoyo que me hab´eis dado. Por tratarme como a un hijo m´as. Mult ,umesc mult. Als meus amics del Maresme els agraeixo tots els moments viscuts. Heu sigut fonamentals per a fer-me feli¸c, evadir-me de l’estr´es del doctorat i compartir alegries. Des dels festivals, les Santes, els Pirates, partits de futbol, paddle, barbacoes, aniversaris, divendres de barof i casal, els Z, nits de jocs de taula i vicis i mil i una hist`ories. Moltes gr`acies per tot! Sou molt grans! Quiero agradecer especialmente a todos los amigos que me han apoyado todo este tiempo en Castelldefels. ´ Erase una vez, una madrile˜na, un vasco y un catal´an en un ´atico en avenida Constituci´o. . . evidentemente la historia no pod´ıa terminar m´as que en un cachondeo constante! A Sandra y Ugaitz, siempre en´ergicos y alegres, hab´eis sido un ejemplo de motivaci´on para mi y un apoyo fundamental durante estos a˜nos. Por todas las charlas de desayuno y cena en pijama. ¡Os echar´e much´ısimo de menos! To the best neighbors Pamina and Kavitha, adventurous and charming. For all the affection and support that you gave me and all the fun we had together during this time. Thank you! To ´ Alvaro, Jose, Eduardo, Catherina, Anuja, Vikas and Rinu. Always ready to hang out, gossip and have fun. Thanks a lot! Finalmente, quiero agradecer al Ministerio de Econom´ıa y Competitividad de Espa˜na por concederme la beca de Formaci´on de Personal Investigador que me ha permitido realizar los estudios de doctorado. xviii xix xx Contents Introduction 1 Quantum simulation with ultracold quantum gases . . . . . . . . . . . . 1 Quantum simulation beyond the mean field approximation . . . . . . . . 3 Attractive Bose-Einstein condensates with competing interactions . . . . 4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 The potassium experiment 9 1.1 Introduction............................... 9 1.2 Experimental apparatus andcoolingsequence.......................... 11 1.3 2DMOTupgrade............................ 15 1.3.1 The leaky steel chamber . . . . . . . . . . . . . . . . . . . . 15 1.3.2 New glass cell 2D MOT . . . . . . . . . . . . . . . . . . . . 17 1.4 High resolution microscopes . . . . . . . . . . . . . . . . . . . . . . 25 1.4.1 Experimental constraints . . . . . . . . . . . . . . . . . . . 25 1.4.2 Spatial resolution, depth of focus and field of view . . . . . 28 1.4.3 Design.............................. 29 1.4.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.5 Cooling sequence after the upgrades . . . . . . . . . . . . . . . . . 50 1.5.1 Modifications due to the installation of the 2D MOT . . . . 50 1.5.2 Modifications due to the installation ofthemicroscopes ....................... 50 1.5.3 Summary of new cooling sequence results . . . . . . . . . . 52 xxi 1.6 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . 55 2 In situ imaging of two-component BECs 57 2.1 Introduction............................... 57 2.2 Probing cold atoms with light . . . . . . . . . . . . . . . . . . . . . 60 2.2.1 Semi-classical treatment . . . . . . . . . . . . . . . . . . . . 61 2.2.2 Quantum - mechanical treatment . . . . . . . . . . . . . . . 64 2.3 Experimental characterization of polarization phase contrast imaging . . . . . . . . . . . . . . . . 75 2.3.1 Concept............................. 77 2.3.2 Experimental Faraday set-up . . . . . . . . . . . . . . . . . 79 2.3.3 Imaging analysis of the atomic polarization rotation . . . . 80 2.3.4 Calibration of the Faraday coefficient . . . . . . . . . . . . . 84 2.3.5 Results ............................. 87 2.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 91 3 Quantum liquid droplets 93 3.1 Introduction............................... 94 3.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2.1 Stabilization of composite quantum droplets through quantum fluctuations . . . . . . . . . . . . . . . . . 95 3.2.2 Excitation spectrum . . . . . . . . . . . . . . . . . . . . . . 98 3.2.3 Extended Gross-Pitaevskii equation with quantum fluctuations ..............................100 3.3 State of the art on dipolar droplets . . . . . . . . . . . . . . . . . . 105 3.4 Experimental challenges . . . . . . . . . . . . . . . . . . . . . . . . 107 3.5 Experimental realization of quantum droplets . . . . . . . . . . . . 110 3.5.1 Methods.............................110 3.5.2 Proof of principle observation: Beyond mean field stabilization of quantum droplets . . . . . . . . . . . . . . . . . . 112 3.5.3 Liquid to gas phase transition and phase diagram . . . . . . 113 3.6 Conclusions...............................119 3.7 Discussion on recent related work . . . . . . . . . . . . . . . . . . . 120 3.7.1 Assessing the mismatch . . . . . . . . . . . . . . . . . . . . 121 xxii 3.8 Outlook .................................123 4 Bright solitons and quantum droplets 125 4.1 Introduction...............................126 4.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . 127 4.2.1 Bright solitons in the mean field regime . . . . . . . . . . . 127 4.2.2 Bright solitons and quantum droplets . . . . . . . . . . . . 129 4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.3.1 Methods.............................134 4.3.2 Observation of composite self-bound states . . . . . . . . . 135 4.3.3 Self-bound state composition . . . . . . . . . . . . . . . . . 136 4.3.4 Experimental phase diagram in quasi-1D . . . . . . . . . . 139 4.3.5 Soliton to droplet transition . . . . . . . . . . . . . . . . . . 141 4.4 Conclusions...............................143 4.5 Discussion on related work . . . . . . . . . . . . . . . . . . . . . . . 143 4.6 Outlook .................................145 5 Coherently coupled interacting BECs 147 5.1 Introduction...............................148 5.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2.1 Coherently coupled dressed states . . . . . . . . . . . . . . 153 5.2.2 Scattering of coherently coupled dressed states . . . . . . . 153 5.2.3 The strong coupling limit in composite Bose Einstein condensates.............................162 5.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . . . . 167 5.3.1 Methods.............................168 5.3.2 Modified interactions . . . . . . . . . . . . . . . . . . . . . . 169 5.3.3 Dressed-state bright solitons . . . . . . . . . . . . . . . . . . 176 5.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 179 Conclusions and Outlook 183 A Energy spectrum vs. magnetic field 189 B Faraday laser detuning 191 xxiii C Technical details of polarization phase contrast imaging 193 Bibliography 199 xxiv xxv 6 a similar fashion as in other in cold atoms inelastic collisonal processes [61–64]. Furthermore, we can tune the interactions into the attractive regime. Hence, in a quasi-1D geometry we can produce single bright solitons formed by atoms in a dressed state. Moreover, by means of coherent coupling the interactions can be temporally controlled. We use this feature to quench the effective interactions of dressed atoms into the attractive regime. Consequently, a modulational instability develops due to the exponential enhancement of density fluctuations, ending up in a soliton train. Outline of the thesis This thesis has been carried out on the Ultracold Quantum gases group led by L. Tarruell at the Institute of Photonic Sciences. I arrived to the group as a master student without any experimental experience in October of 2014. I followed the construction of the main experimental apparatus while working on the design of a high resolution objective and learning the basic experimental techniques on electronics, optical design and laser technology. This experiment was mainly designed and constructed by C. R. Cabrera, P.Cheiney, L. Tanzi and L. Tarruell, and the description of the design and development of the experimental apparatus can be found in the thesis of my predecessor C. R. Cabrera [65]. During this period, we achieved the first Bose-Einstein condensate of Spain with 41K. In November of 2015, I started my PhD in the group. In the following, I outline the organization of the manuscript which describes the main research which I have carried out, together with the potassium team, during the completion of this thesis. •In chapter 1 we describe the upgrades which I performed in the experimental apparatus. First of all, I present a brief summary of the experimental cooling sequence that we used to cool down to degeneracy the bosonic mixture of 39K-41K. Then, I present the design and installation of a new glass cell 2D MOT chamber which was installed to remove a previously leaking stainless-steel 2D MOT chamber. Finally, I describe the design, installation and characterization of an optical set-up for imaging and addressing atoms with high resolution. 7 •In chapter 2, we present a polarization phase contrast technique which we have used to image two-component potassium BECs in open transitions at high magnetic field. This technique has been crucial to perform the experiments developed in the following chapters. •In chapter 3, we present the first observation of composite quantum liquid droplets, which are stabilized by quantum fluctuations. We study its stabilization mechanism and characterize its liquid to gas phase transition. •In chapter 4, we study the similarities and differences between composite quantum droplets and bright solitons in a quasi-1D geometry. We measure the composition of the self-bound states and its phase diagram. We distinguish two regions separated by a critical interaction strength, above there is a crossover between the two types of solutions, below we are able to map out the soliton to droplet phase transition. •In chapter 5, we study the modification of the interactions in a twocomponent Bose-Einstein condensate in the presence of strong coherent coupling. We perform direct measurements of the modification of the elastic and inelastic scattering across the resonance. Moreover, we are able to produce dressed state bright solitons and study the modulation instability which occurs when the interactions are quenched, resulting in a soliton train. •Finally, we present the conclusions and discuss future perspectives of the experiment. 8 Chapter 1 The potassium experiment Abstract In this chaper, we introduce the cooling sequence which we use the cool the bosonic isotopes of potassium down to quantum degeneracy. For the first time, we have been able to condense the 39K-41K mixture. During my thesis, we have replaced a leaky stainless steel 2D MOT chamber by a new glass cell chamber which presents better performance. We present the design and developed a custom-made high resolution optical set-up for imaging and addressing the atoms in situ. Finally, we detail the modification of the cooling sequence which were performed due to the experimental upgrades. 1.1 Introduction In our experiment we perform quantum simulation with ultracold potassium atoms. This atom was chosen for several reasons. First of all, potassium is an alkali atom, with only one electron in its valence band [66]. Thus, it is very simple to cool and manipulate potassium as compared to dipolar or two-electron atoms. Indeed, its two bosonic (39K and 41K) and fermionic (40K) isotopes have been cooled down to the quantum degenerate regime [67–69]. Therefore, potassium is a very good candidate to study Bose-Bose, Bose-Fermi and Fermi-Fermi 9 Chapter 1. The potassium experiment 10 mixtures. Moreover, potassium offers the possibility to control the interactions for each isotope and between different isotopes (40K-41K and 39K-41K) [70–75]. Since the mass of the different isotopes is very similar, the gravitational sag between them is very small and we do not need to have species dependent trapping potentials to have a good overlap between them. Initially, my research in the potassium experimental apparatus started by producing a BEC of 39K and dual BECs of 39K and 41K, which was achieved on January of 2016, and characterizing the Feshbach resonances of potassium BoseBose mixtures [75]. The complete description of the cooling sequence that we use to condense all the bosonic isotopes and the study of the Feshbach resonances can be found in the thesis of C. R. Cabrera [65] and in reference [75]. In the mean time, I kept working on the prototyping of the high resolution objective in order to be able to install it in the experiment to include a key tool of our research: the in situ imaging of potassium mixtures. The installation of the imaging objective was done in June of 2016, and it allowed us to proof the existence of quantum droplets [29] and show its interplay with composite bright solitons [30]. This works were completed around the fall of 2017 with the Sword of Damocles on top: a leaky 2D MOT steel chamber. In November of 2017, we upgraded our 2D MOT with a glass cell chamber. Moreover, we installed a second high resolution objective to be able to produce highly resolved optical potentials on the atoms. The experimental chapter is organized as follows. In section 1.2 we briefly introduce the experimental apparatus and cooling strategy for condensing the bosonic isotopes of potassium. In section 1.3 we describe the 2D MOT upgrade. In section 1.4 we show the design, development and installation of two high resolution objectives to image and address potassium mixtures. In section 1.6 we present the conclusions and future experimental upgrades planned for the next years. Chapter 1. The potassium experiment 11 1.2 Experimental apparatus and cooling sequence In this section, we give an overview of the experimental set-up and the experimental routine to cool down the atoms before the main upgrades of the experimental apparatus were performed. The experimental set-up is divided into two different parts: a laser system where we prepare the frequencies required to laser cool, manipulate and probe the atoms, and a vacuum chamber where the production of a quantum degenerate gas takes place, see Fig. 1.1 (a). (a) (b) 1 2 7 3 6 4 52DMOT 3DMOT Magnetic Trap Hybrid trap Dipole trap Imaging Experimental sequence Gray molasses Figure 1.1: Old experimental apparatus. (a) 1. Ion pump. 2. Potassium source. 3. Stainless steel 2D MOT chamber. 4. Mechanical shutter connected to a cross where the 2D MOT (3) connects through a differential pumping tube. 5. Gate valve. 6. 3D MOT/science chamber. 7. Getter pumps. (b) Schematic of the experimental sequence. We perform a dual (39K-41K) laser cooling process consisting of a 2D MOT, 3D MOT and molasses. Both species are optically pumped and evaporatively cooled sequentially in a magnetic, hybrid and dipole trap until quantum degeneracy is reached. 39K is sympathetically cooled in the magnetic trap through the thermal contact and good collision rate with 41K. While 41K can be cooled at zero field due to its positive background scattering length, 39K is evaporatively cooled near a Feshbach resonance in the dipole trap due to its negative background scattering length. The atomic clouds are imaged either in time of flight or in situ. Chapter 1. The potassium experiment 12 The laser system to cool potassium consists on several lasers which are either close to the D2 (766.7007 nm) or D1 (770.7009 nm) transition from potassium 39K. Two external cavity diode lasers1(ECDL) are locked on the D1 and D2 crossover lines of 39K by using saturated absorption spectroscopy in two hot potassium vapor cells [65, 76]. Two Distributed FeedBack2(DFB) lasers are offset locked from the D2 master laser and are used, together with Acusto-Optic Modulators (AOM), as cooler and repumper lasers for the 41K isotope. Their light is amplified with two independent Tappered Amplifiers (TA) and the light is coupled to a 4 by 4 fiber cluster which mixes the light generated from the laser table and sends it to the experimental apparatus table. An ECDL is offset locked from the D1 master laser and together with Electro Optical Modulators (EOM) and AOMs is used to do the molasses of 41K and optical pumping of 41K and 39K. Its light is amplified with a TA. The molasses light is coupled to the fiber cluster and the optical pumping light is coupled to an independent fiber. A DFB and an ECDL laser are offset locked from the D2 and D1 master lasers respectively and combined sequentially in a TA together with EOMs and AOMs in order to produce the cooler and repumper for the MOT and molasses of 39K. Its light is coupled into the fiber cluster. We use two lasers to trap the atoms. A single mode YAG laser3(1064 nm) which is used to produce the red detuned optical dipole trap beams which are used in the evaporation process. And a single mode fiber amplifier laser4which is used to produce blue detuned optical potentials and was set up by P. Thomas [77]. A full description of the laser set-up can be found in the thesis of C. R. Cabrera [65]. The old version of the vacuum chamber set-up consisted mainly on two stainless steel chambers, see Fig. 1.1(a). A 2D MOT chamber and a chamber which is used as a 3D MOT and science chamber. The 2D MOT chamber was connected to the potassium ampoule sources and an ion pump5with an appropriate control of the temperature gradients in order to produce a potassium vapor pressure 1SYST DL PRO 780 - TOPTICA 2EYP-DFB-0767-00050-1500-TOC03-0005 - Eagleyard photonics 3Mephisto MOPA 25 W - Coherent 4ALS-GR-532-10-A-SP - ALS 5TiTanTM 25S Ion Pump - Gamma Vacuum Chapter 1. The potassium experiment 13 of ∼5×10−8mbar. This 2D MOT chamber is connected through a differential pumping tube to the 3D MOT/science chamber. This chamber has two re-entrant viewports with coated windows of 6 mm thickness and 50 mm clear aperture which are separated by 30 mm. This allow us to have currently a Numerical Apperture (NA) of 0.43. The pressure in this chamber was mantained around 10−11mbar by two Non-Evaporable Getter (NEG) pumps6. The pressure of the science chamber was limited by the reading of our vacuum gauge7(10−11 mbar). A full description of the vacuum system can be found in the thesis of C. R. Cabrera [65]. The main ingredients for producing a degenerate quantum gas include MOTs, magnetic traps, optical traps and a magnetic field control for adjusting interactions. The experimental sequence to produce degenerate Bose-Bose mixtures, sketched in Fig. 1.1(b), is divided in different steps. In the following list, we describe them and show typical parameters which where used before the installation of the new 2D MOT and the high resolution objectives. •We sequentially produce cold atomic beams of 41K and 39K in a 2D MOT. This is performed in a vacuum chamber with high vapor pressure of natural potassium. •The atoms are sent from the 2D MOT chamber to the 3D-MOT/science chamber to load a dual species 3D MOT with ∼1.5×109atoms of 41K and 3×108atoms of 39K at a temperature ∼1 mK. In order to increase the density, they are subsequently compressed in a hybrid D1-D2 compressed MOT reaching temperatures of ∼70 µK. •We perform simultaneously gray optical molasses on the D1-line for 39K and 41K to obtain efficient sub-Doppler cooling. We can achieve temperatures of ∼16µK and 8 ×108and 2 ×108atoms of 41K and 39K, respectively. •The atoms are prepared in |F= 2, mF= 2iby optical pumping on the D1 line with almost 100% efficiency, capturing almost all the atoms in a magnetic quadrupole trap. 6NEXTorr D500 (500 l/s) and NEXTorr D200 (200 l/s) - SAES Getters 7UHV-24p extended range ion gauge, dual-thoria-iridium filaments (9715015)- Agilent Chapter 1. The potassium experiment 14 •Sympathetic cooling is performed in a compressed magnetic quadrupole trap by using radio-frequency (RF) evaporation on the hyperfine transition of 41K, while 39K is cooled by thermal contact. The rethermalization is very efficient due to the large 39K - 41K scattering length a39−41 = 177a0. In this process we are able to increase the Phase Space Density (PSD) of 41K and 39K from 2×10−6to 10−3and from 10−7to 2×10−4, respectively. We obtained around 4 ×107and 2.5×107atoms of 41K and 39K at 30 µK. Instead, if we only cooled down 41K we were able to obtain 6 ×107atoms at 25 µK. •A 15% of the atoms are transferred into a hybrid trap formed by the quadrupole magnetic trap and an optical dipole trap focused a waist below the magnetic field minimum, where they are evaporatively cooled from 9µK to ∼1µK. Typically we end up with a total atom number of ∼2×106 atoms in total. Cooling both isotopes we reach similar temperatures with half of the atoms per isotope. •Almost all the atoms are loaded into a crossed optical dipole trap where hyperfine and Zeeman transfers are performed to bring 39K and 41K to |F= 1, mF=−1iand |F= 2, mF=−2irespectively. There we perform evaporative cooling in the vicinity of a 39K Feshbach resonance where a39 = 150a0. Different resonances (33 G, 402 G) have been used, depending on the experiments. At low field, we use a magnetic field gradient in this process in order to distill 41K from 39K, being able to obtain different balances at will. This is possible because the states we use have magnetic susceptibilities of opposite sign. •The largest 41K and 39K pure BECs which we have been able to produce had around 5 ×105and 3.5×105atoms. The dual condensation of both species lead to 1.5×105atoms per species. •In the end, the atomic clouds can be characterized either by absorption imaging in time of flight or by polarization phase contrast imaging in situ. This sequence has been modified due to the upgrade of the 2D MOT and the installation of the two high resolution objectives. In section 1.5 we will detail Chapter 1. The potassium experiment 15 the modifications of the cooling sequence after introducing the 2D MOT upgrade and the installation of the objectives. In the following section we detail the implementation of a new 2D MOT. 1.3 2D MOT upgrade The 2D MOT steel chamber that was installed initially presented several design and manufacturing problems which led to a large leak in one of the viewports after ten months of operation and to a continuous opening of micro-leaks on another viewport since August 2016. This steel chamber was substituted by a glass cell chamber on November 2017. With the new 2D MOT chamber we have been able to reach similar vapor pressures with smaller temperatures and we didn’t have any leak since its installation. In section 1.3.1, we will introduce the problems of the first steel chamber 2D MOT. Then, in section 1.3.2, we will present the design of the new glass cell 2D MOT, explain its installation and show the results. 1.3.1 The leaky steel chamber The steel 2D MOT chamber shown in Fig. 1.2(a) was a home-made design based on the design from ref. [78–80]. This vacuum chamber was thought to provide a large optical access in order to implement large elliptical beams to increase the atomic flux of the 2D MOT. Since the vacuum chamber geometry was rectangular, four custom made fused silica windows with Anti-Reflection (AR) coating8in both sides were used. These windows were sealed using indium wires9. Instead, the back vacuum window consisted of a CF40 view-port10 made of a fused silica window which was brazed with Kovar11 onto a 304 stainless steel flange. The procedure to migrate the potassium from the source to the 2D MOT chamber was very complicated. To do so, the chamber was heated up to apply temperature gradients that brought the potassium into the 2D MOT. However, it was very hard to heat the 2D MOT chamber evenly and the potassium was 8AR 767 nm/0◦- LaserOptik 9Indium wire 99.99%  4mm (1N522407)-Advent research materials 10CF40 Fused Silica view-port 304L / Kovar - Trinos 11Kovar is a nickel-cobalt ferrous alloy that allows building glass-to-metal seals. Chapter 1. The potassium experiment 22 2D MOT is shorter24 we should aim at having a 28% absorption to have similar vapor pressure. During the upgrading process we installed the potassium source on the bottom part of the 2D MOT chamber, see Fig. 1.5(a). The potassium source is contained inside a glass ampoule25 which is placed between two stainless steel cylinders inside a retractable below26 which is used to crack the ampoule. The below is connected to the bottom valve by a CF Tee27. The Tee, below, blanks and stainless steel where cleaned and baked independently before the installation in a similar fashion as we described previously. To install the new 2D MOT we closed the gate valve to the science chamber, and switched off the ion pump (labeled as 1 on Fig. 1.1(a)) and the ion pump of the NEG pump which is below the cross of the atomic shutter (labeled as 4 in Fig. 1.1(a)). The process of removing the old 2D MOT and installing the new one was performed while continuously flushing Nitrogen through the chamber with an over pressure. This was done in order to avoid any contamination from air to facilitate the pumping process. As soon as the new 2D MOT was installed we put it under vacuum using the turbo pumps initially, followed by the activation of the ion pumps as soon as the pressure was below 10−8mbar. The cross of the atomic shutter and the 2D MOT where baked during few days. As done in the pre-baking stage, we kept the temperature close to the indium wire below 110◦C. The small oven we made around the glass cell was also kept below 100◦C to avoid damaging the coating. After the baking process, we activated the NEG pump, turned on its ion pump and cooled down the chamber. The pressure read by the ion and NEG pumps was 1.8×10−9mbar and <10−10mbar respectively. 24Old 2D MOT. Back view-port to gold-coated mirror to bottom view-port: 215 mm + 28 mm. New 2D MOT. Back of glass cell to gold-coated mirror to bottom of glass cell: 84 mm + 15 mm 25Potassium, ingot, 99.95% metals basis. 244856-5G - Sigma-Aldrich 26CF hydroformed bellow, DN16, 1 flange rotatable, stainless steel 304/316L, length 80mm. V-FXB16R80-316 - Vacom. 27CF Tee, DN16, fixed, stainless steel 316L, bolt holes in line, length 76mm. TE16-316 - Vacom. Chapter 1. The potassium experiment 23 Potassium migration After the baking process, we started the potassium migration from the source to the glass cell. The first time that the migration of potassium was done with the old 2D MOT we realized that the ion pump was also pumping the potassium, and eventually we pumped it all. For this reason, during the migration process we regulated the temperature gradients and the conductance between the ion pump and the glass cell in order to help the potassium migrate towards the glass cell without being completely sucked by the ion pump. After several days of adjustment of the gradient, the potassium arrived to the 2D MOT. In this process, we realized that the potassium was getting stuck on the back of the glass cell, as seen in Fig. 1.5(b). Since the glass cell was only heated through the thermal contact with the main body of the chamber, the back of the glass cell was cold and we couldn’t regulate its temperature independently. We tried adjusting the temperature gradient and conductance of the ion pump and the valve from the source to the 2D MOT in order to maintain a good vapor pressure without having potassium stuck in the glass cell. However, this was not possible. (a) (b) 1 2 3 4 5 6 7 9 8 Figure 1.6: 2D MOT oven. (a) 1. Oven frame made of aluminum. 2. Windows from the old 2D MOT. 3. The windows are held against the aluminum frame (1) using a teflon piece, isolating the aluminum from the environment. 4. Thermoresistor bands are used to heat up the aluminum and make to make an oven for the glass cell. 5. A magnetic quadrupole for the 2D MOT is made with 4 permanent magnets. 6. To adjust the center of the magnetic quadrupole we use 4 coils to produce a bias field in the two transverse directions. (b) Oven installed. 7. Teflon cover to isolate the back part of the oven. 8. A 200 N-BK7 window is used to close the oven from the back. 9 . Brass bars are used to held the oven. Chapter 1. The potassium experiment 24 Source : 77.1 ºC Tee - source : 80.7 ºC Valve source : 75.7 ºC Body - top : 54.2 ºC Body - bottom : 60.8 ºC Oven : 45.8 ºC Valve ion pump : 76.7 ºC Cold finger : 47.1 ºC (a) (b) 1.0 0.8 0.6 Figure 1.7: (a) Temperatures of the 2D MOT. (b) Absorption spectroscopy near the D2 transition. The blue and orange lines correspond to saturated and non-saturated spectroscopy. Both have been rescaled to its maximum value for comparison. The width of the absorption profile corresponds to the Doppler profile due to temperature broadening. The absorption is measured from the non-saturated spectroscopy. For this reason, we installed an oven around the glass cell, see Fig. 1.6(a) and (b). The oven is composed by an aluminum frame which is heated up with thermoresistor bands28. It is held by two brass bars and its octogonal apperture (see Fig. 1.6(a)) fits on the main body of the 2D MOT chamber. Four side rectangular AR coated windows29 and a circular back window30 are used to provide optical access for the transversal 2D MOT beams, longitudinal molasses and push beams. The windows are held against the aluminum frame with teflon to thermally isolate the oven. The aluminum frame also holds four permanent magnets31 and four coils which are used to produce and align the magnetic quadrupole which is necessary for the 2D MOT. Using this oven, we have been able to maintain a vapor pressure of potassium of ∼2×10−7mbar without having the potassium stuck on the glass cell. In the final configuration, the valve to the ion pump remains closed. The temperatures of the new 2D MOT on the current configuration are summarized in Fig. 1.7(a). The obtained Doppler profile can be observed in the absorption spectroscopy, see Fig. 1.7(b). In the final configuration we have a resonant absorption on the D2 of 35 %, and thus the vapor pressure should be similar to what we had before. Moreover, the oven is 28HT10K - Thorlabs 29Reused from the old 2D MOT 30WG12012-B - Thorlabs 31Rare earth magnet NdFeB - HKCM Chapter 1. The potassium experiment 25 only at 45.8◦C, whereas for the old 2D MOT the main body was around 75◦C. In conclusion, we have been able to reach similar vapor pressure as we had with the old 2D MOT with lower temperatures. The design of this 2D MOT also allows for the addition of an extra source. In the future, we plan to install enriched potassium 40K in order to produce degenerate Fermi gases. The loading parameters of the 2D MOT and 3D MOT will be summarized in section 1.5. 1.4 High resolution microscopes Every experiment which we carry out ends up by imaging the atomic cloud so as to extract information from it. Therefore, mastering various imaging techniques is paramount to obtain the most insights in our experimental research. Two main techniques are used in cold atomic experiments. Time of flight imaging, which consists on releasing the atomic cloud from its trap in order to extract its momentum distribution after a long free fall, and in situ imaging, which consists on probing the cloud in the trap in order to probe its spatial distribution. Whereas typical sizes of the cloud after time of flight are on the order of ∼ 0.1−1mm, in situ they are on the ∼1−10µm order. Therefore, we need high numerical aperture microscopes in order to spatially resolve the atomic cloud in situ. In the following section, we will describe the main experimental constraints for the design of a high numerical aperture objective which will allow us to image the atomic clouds with high resolution. Additionally, we have installed a second microscope to be able to generate highly resolved optical potentials on the atomic plane. 1.4.1 Experimental constraints There are two main constrains that we should take into account to design the optical system to image the atoms: the physical constraints imposed by the choice of our science chamber and the pixel size of our camera. Chapter 1. The potassium experiment 26 60(50) 30 6 46 54 (1) (1) (2) (2) (a) (b) (c) (d) (e) Figure 1.8: a) Side image of the science chamber showing the two re-entrant viewports. b) Schematic of the re-entrant view-ports including the imaging and addressing microscopes. All distances are in millimeters. The windows are made of synthetic fused silica and are AR coated for the cooling, trapping and imaging wavelengths. They are placed on a metallic frame and welded on the view-port. Although the metallic frame has an inner diameter of 60 mm the clear aperture is only 50 mm. The high resolution objectives are formed by a custom-made meniscus lens (1) and a high numerical aperture aspherical lens (2). The objective lenses are installed on a plastic tube and held by plastic rings in order to avoid the generation of Eddy currents when magnetic fields are switched on and off. (c) Reflectivity from the anti-reflection coating from the re-entrant view-ports vs. wavelength. (d) Detail of the vacuum window and its metallic frame. (e) Reflectivity from the anti-reflection coating from the asphere and meniscucs vs. wavelength for an average polarization. Chapter 1. The potassium experiment 27 For simplicity, our experiment was designed in order to have a single chamber as a 3D MOT and science chamber. Additionally, the laser cooling of potassium requires large beams in order to reduce the light assisted collisions in the 3D MOT [80, 83]. Thus, the optical access in our experimental apparatus cannot be directly compared to other experiments with independent 3D MOT and science chambers and other atomic species. Nevertheless, in order to have a large numerical aperture our science chamber has two re-entrant view-ports, see Fig. 1.8(a). The distance between both view-ports is 30 mm, the windows have a 6 mm thickness and a clear aperture of 50 mm, see Fig. 1.8(b). The windows are made of synthetic fused silica32 and are AR coated for cooling, trapping and imaging lasers at 405, 532, 670, 767, 852, 1064 and 1178 nm at 0◦of incidence 33, see Fig. 1.8(c). They are placed in a metallic frame which is welded on the view-port, see Fig. 1.8(d). Before the welding process the windows were specified to have a transmitted wave-front error (TWE) of λ/10. The distance between view-ports, clear aperture and window thickness are the main constrains which limit the accessible numerical aperture. Therefore, the design of a 200 long working distance microscope is required in order to have a good resolution. Moreover, the window introduces aberrations on the wave-front of the light, which need to be appropriately corrected. In our experiment, we have decided to image the atomic cloud in situ using a polarization phase contrast technique (see chapter 2). This technique requires the use of an Electron Multiplying Charged-Coupled Device camera (EMCCD), which amplifies the signal to noise ratio of the photogenerated electrons. This type of cameras use pixels with a large size. In particular, our camera has a pixel size of 16µm. Therefore, we must adjust the magnification of our imaging system accordingly. If the magnification is too small, the spatial resolution is limited by the pixel size. If the magnification is too large, the spatial resolution is limited by the diffraction limit of our optics at the expense of a smaller signal to noise ratio. In order to have a good compromise between the spatial resolution rDL and signal to noise ratio, the magnification Mof the optical system is normally 32Spectrosil 2000 synthetic fused silica, λ/10 TWE, 20/10 scratch/dig, 6 mm thickness - UKAEA 33Custom anti-reflection coating. B-11666 AR405+532+670+767+852+1064+1178nm/0◦ UHV - LaserOptik. Chapter 1. The potassium experiment 28 designed so that M×rDL is around 2 - 3 times the pixel size. Before showing the design of the high resolution microscope, in the next section, we briefly review the concepts of spatial resolution, depth of focus and field of view. 1.4.2 Spatial resolution, depth of focus and field of view The most common criterion to define the spatial resolution of an optical system is the minimum distance to distinguish two close by point sources. The imaging of a point source through an optical system is called the Point Spread Function (PSF). When the optical system is diffraction limited, the PSF is an Airy disk. To define the resolution of our optical system we will use the Rayleigh criterion. This criterion states that two point sources cannot longer be distinguished if the distance between the central maximum of both PSFs is closer than the distance between the central maximum and the first minimum of the Airy pattern. From this criterion we get that the resolution of a diffraction limited optical system rDL is: rDL = 0.61 λ NA.(1.4.1) An optical system can only resolve the objects close to its optical axis and focal plane. The region along the optical axis in which the objects can be sharply imaged is called the Depth of Focus (DOF). In analogy to gaussian optics, the depth of focus in a diffraction limited system corresponds to the confocal parameter DOF = 2zR, where zR=πr2 DL/λ is the Rayleigh length. Thus, in the limits of the DOF, the spatial resolution of the imaging system is √2rDL. The Field of View (FOV) is the region on the object focal plane in which the objects can be sharply imaged. Not all optical systems are diffraction limited. Besides the limitations imposed by the diffraction of light through an optical system, we also must take into account the imperfections of our optical system. If lenses are not perfect or they are misaligned, an optical system performance will not be limited by diffraction. The difference between the ideal propagation of a wave-front through an optical system and its real propagation is called the wave-front error. The Chapter 1. The potassium experiment 29 non-ideal propagation of light through an optical system results in optical aberrations. Indeed, the propagation of a spherical wave-front through a thick window introduces wave-front aberrations. For this reasons, we need to design an objective which corrects the aberrations introduced by the view-port and is diffraction limited. In the following section we introduce the design of a diffraction limited high resolution objective. 1.4.3 Design To implement a high resolution microscope in our experimental apparatus we need to address both the optical and mechanical design to adapt it to our experimental constraints. The optical design and initial mechanical prototypes of the objective were done by L. Saemisch [84]. The home made objective which we use has a NA = 0.43. Since the imaging transition is at λ= 766.7 nm, we expect to have a resolution rDL = 1.1µm. During my master thesis, I measured the resolution of the imaging system in a replica set-up from the science chamber. The measurement consisted on determining the PSF of a gold dot of 250 nm, and we obtained a resolution of 1.5µm [85]. At the beginning of my PhD, several challenges needed to be overcome in order to be able to install the imaging objective in the main experiment. First of all, the optical design of the full imaging system was not adapted for the pixel size of our EMCCD. Second, the design of the initial prototypes were not mechanically stable and thus we needed to ensure a submicron mechanical stability so that the resolution was not smeared out by vibrations. Indeed, we think that this was the limiting factor of the prototypes which were developed during my Master thesis. Moreover, since the MOT beams pass through the objectives, the MOT optics need to be adapted in order to have the MOT beams collimated in the science chamber. Initially, when we installed the imaging microscope this required the modification of the top MOT optics. Afterwards, we decided to install an addressing microscope to generate optical potentials in the atoms. Thus the optomechanical design of the imaging set-up and the MOT optics were adapted to install an addressing microscope below the science chamber. Chapter 1. The potassium experiment 30 Optical design of the home-made objective The optical design of our home-made objective was made over the premise of simplicity and reduced cost. We decided to use only two lenses: a meniscus which corrects the spherical aberrations introduced by the window and an asphere which provides a high NA. The meniscus was designed with Zemax so that it would correct the spherical aberrations introduced by the window and together with the aspherical lens form a diffraction limited system. It was fabricated by Ross Optical Industries, is made of N-BK7 and has a custom-made broad band AR coating, see Fig. 1.8(e) . The asphere is a commercial asphere from Edmund Optics34 which is made from L-BAL35. It has an Effective Focal Length (EFL) of ∼50 mm. A schematic of the objective can be seen in Fig. 1.8(b). The optimization of the meniscus curvature, thickness and the distances between window and lenses was performed on Zemax by L. Saemisch assuming an exit pupil diameter of 46 mm. One of the criteria to determine whether an optical system is diffraction limited or not is to check whether the root mean square wave-front error (RMS-WE) is smaller than 0.07λ. Therefore, we have used the RMS-WE as a merit function to optimize the objective. Moreover, several tolerances were considered in the optimization process: •Decentering of the window, meniscus and asphere by 0.2 mm. •Tilt of the window, meniscus and asphere by 0.1◦. •0.1% tolerance on the radii of the asphere. •0.25% tolerance on the radii of the meniscus. •0.1 mm tolerance on the thickness of the meniscus, asphere, window and distance in between. •0.3 mm tolerance on the distance between the vacuum window and the atoms. •0.0005 refraction index tolerance of the lenses. 34Aspheric lens, 50mm  , 0.50 Numerical Aperture NIR Coated. Material L-BAL35. RMS surface flatness 0.75 µm. # 66-336 - Edmund Optics Chapter 1. The potassium experiment 31 In order to account for all the listed tolerances, we defined the distance between the meniscus and the window as a compensator with ±1mm. This does not mean that within 1 mm of tolerance the system will be diffraction limited, but that there should be at maximum ±1mm between the optimal distance and all the configurations required to have a diffraction limited microscope in the presence of imperfections within the listed tolerances. The optimum configuration of the overall set-up is sketched in Fig. 1.9, where we obtain a RMS −WE = 0.028λ. The Working Distance (WD) of the objective is 32.2 mm, the effective focal length of the system is 48.7 mm and the Back Focal Length (BFL) is 33 mm. The numerical aperture of the system is 0.43, the resolution is 1.1 µm and the depth of focus is around 10 µm. Moreover we have computed the FOV of the system by finding the object height at which the RMS-WE is 0.07λ. We found that the FOV comprises a circle of  = 330µm. The tolerance of the optimal set-up has been computed by calculating the imperfections which lead the system to an RMS −WE = 0.07λwithin the compensator limits. We obtain that the tolerance on the tilt between the lenses and the tilt of the whole objective are 0.3◦and 0.07◦respectively. The tolerance on the decentering between the lenses is 165 µm. The tolerance on the distance between lenses is 130 µm. The distance between the atoms and the window can always be compensated within the mechanical constraints of the system. The performance of the objective at different wavelengths can be different due to the chromatic dependence of the index of refraction. Since the design of the objective was not optimized for different wavelengths the chromatic shifts at 532nm and 1064nm are −685µm and 415µm respectively. These shifts are far greater than the DOF, and thus the objective is not achromatic. Provided that the complexity of designing an achromatic objective is far greater and the group didn’t have any expertise on high resolution imaging, we decided to deal with the chromatic shifts in order to simplify its complexity. Chapter 1. The potassium experiment 38 for the imaging set-up. The only difference is the orientation of the objective mounts due to the top-bottom asymmetry of the copper bars which bring the current to the main coils of the experiment. Since the addressing objective is upside down, both the 5-axis stage and the linear stage are reversed so that the movable parts rest on the respective micrometric and nanopositioner screws. In this case, the linear stage has been replaced by another model for convenience50. In the following section, we detail the implementation of the imaging and addressing set-ups on the experimental apparatus. 1.4.4 Implementation The installation of the imaging and addressing set-ups on the main experiment was performed in two stages. In a first stage, we installed the imaging set-up using an achromat of EFL = 750 mm instead of 500 mm (lens L3 in Fig. 1.9). As explained before, the distance between the lenses was such that the probe beam was not collimated in the imaging plane. Moreover, in the installation process we took as a reference beam which was perpendicular to the top vacuum-window and was centered on the center of the magnetic quadrupole. Using this procedure we were not able to perfectly align the optical axis of the imaging set-up with the reference beam and ended up moving the last mirror before the camera to be able to image the atomic cloud. Thus, the imaging was presumably not centered on the field of view of the objective. In a second stage, we modified the imaging set-up in order to improve it and understand its limitations. First of all, we removed the top objective, characterized the aberrations introduced by our vacuum view-ports and discovered that they introduce large astigmatism, trefoil and spherical aberrations. Afterwards, we installed two high NA objectives. The imaging one on top and a second objective on the bottom of the science chamber in order to be able to address the atoms with highly resolved optical potentials. Since the installation procedure of the first and second stages was very similar, we will only describe the installation on the second stage. Finally, we characterized the resolution after the second stage 50M-423 : Newport Chapter 1. The potassium experiment 39 of installation, corrected the astigmatism introduced by our vacuum view-ports and improves the performance of the complete high resolution optical set-up. In the following section, we describe how do we characterize the imaging resolution of the imaging objective after the first installation stage. First stage: characterization of the imaging resolution To characterize the resolution of the imaging set-up we should measure its PSF. Ideally, we would trap a single atom on the center of the chamber and measure its fluorescence. However, it is not a trivial task to trap and image a single atom and generate a good point source inside the vacuum chamber. Instead, we measured the size of a bright soliton which has the size of the harmonic oscillator length of the trap aho = (~/mω)1/2, where ~is the reduced Planck constant, m is the mass of 39K and ωis the trapping frequency. In our experiment, typically aho ∼1.6µm and thus it is not a good point source. Taking as a reference the 1.5µm resolution which was measured in the re-entrant view-port replica set-up, we should observe an Airy disk with the first order minimum at ∼2.2µm from its center. After the first installation stage, we measured a distance of 2.9(1) µm, see Fig. 1.12(a), which indicates that the performance of the imaging set-up is worse than what we measured on the replica set-up and that the resolution is not better than 2.9(1) µm. To determine the size of the bright soliton we performed independent measurements of the magnification of the imaging system. The magnification was calibrated by using Kapitza-Dirac diffraction of a BEC [87]. We imprint shortly a periodic modulation of the phase on a BEC (using a red detuned lattice) in order to generate diffraction. In this procedure, we keep the BEC in an optical waveguide so that the diffracted peaks propagate along the object plane. As it is shown in Fig. 1.12(b), side peaks with momentum p= 2h/λ appear, where h is the constant of Planck constant. By comparing the expected distance between the central and side peaks on the object plane with the observed distances on the camera we can extract the magnification of the imaging system. Chapter 1. The potassium experiment 40 tg(ms) 0.25 1.25 2.25 3.25 4.25 (a) (b) Figure 1.12: (a) In situ image of a bright soliton which was used to estimate the resolution after the first installation of the imaging set-up. As it can be seen there is not a ring but side lobes along the vertical direction. This indicates the presence of astigmatism. (b) In situ images of Kapitza-Dirac diffraction on a BEC vs. guide time on an optical waveguide. The magnification of the initial and new set-ups is 49.6(9) and 33.1(6), respectively. The images on (a) and (b) were taken with the polarization phase contrast technique which is explained in chapter 2. In the next section, we show the observations of the aberrations introduced by the vacuum windows. Second stage: aberrations introduced by the vacuum view-ports In order to characterize the aberrations introduced by our vacuum view-ports we removed the imaging system which was installed in the first stage. To measure the aberrations introduced by the view-port we used a Twymann-Green interferometer. We use a standard non-polarizing beam splitter51 with a reference arm and a path which was crossing the two vacuum windows twice, see Fig. 1.13(a). The mirrors which we use are 200 broadband dielectric mirrors52. We used a gaussian beam of a 34 mm waist to crosscheck the biggest possible area of the windows. The interference is measured on a CCD camera by a lens with 300 mm effective focal length. 5150/50 Non-Polarizing Cube Beamsplitter (700 - 1100 nm). Surface flatness λ/4 and wavefront error λ(at 633 nm). BS032 - Thorlabs) 52  200 Broadband Dielectric Mirror, 750 - 1100 nm. Optical flatness λ/10 (at 633 nm) - Thorlabs Chapter 1. The potassium experiment 41 Initially, we tested the interferometer without the vacuum chamber to crosscheck that the elements of the interferometer itself where performing appropriately. In Fig. 1.13(b) we observe the interference without the vacuum chamber. The shape of the beam does not show any appreciable aberrations. Afterwards, we measured the aberrations introduced by the vacuum view-ports, as sketched in Fig. 1.13(a). We observe that the shape of the interference pattern is slightly modified and its elliptical shape might correspond to astigmatism introduced by the vacuum view-ports. Moreover we tested dichroic mirrors in an independent interferometer. Whereas the dielectric mirrors do not introduce any aberrations (see Fig. 1.13(b)), we have observed that the dichroic mirrors in reflection at 45◦ do introduce large astigmatism, see Fig. 1.13(d). For this reason, as depicted in Fig. 1.11, we have used a dichroic in transmission in the addressing set-up in order to mix the Z dipole trap with the probe and addressing beams. NPBS RM M1 M2 VB VT (a) CCD L (b) (c) (d) Figure 1.13: (a) Scheme of the Twymann-Green interferometer. (RM) Reference 200 dielectric mirror. (M1 and M2) 200 dielectric mirror. (VB and VB) Top and bottom vacuum view-ports. (NPBS) Non-polarizing beam splitter 200. (L) Achromatic lens EFL = 300 mm  = 200 (d) Wave-front error tested without the viewports (VT and VB). (c) Wave-front error measured by the set-up described in (a) including the vacuum view-ports. The ellipticity shows astigmatism. (b) Wave-front error tested without the viewports (VT and VB) and replacing the (M1) mirror by a dichroic. It shows a severe astigmatism of several λ. Chapter 1. The potassium experiment 42 (a) (b) (c) (d) Collimated Non-collimated Measured Figure 1.14: (a) Schematic of shearing plate taken from Thorlabs webpage. (b) Collimated and (c) non-collimated interference patterns taken from Thorlabs webpage. (d) Interference pattern formed after collimating a 34 mm beam which crosses through the vacuum windows of the science chamber. A small curvature of the fringes can be appreciated, showing additional aberrations apart from the observed astigmatism. Additionally, we performed tests with a shearing plate interferometer53 which also helped us to adjust the collimation of the 34 mm beam. The shearing plate is formed by a window which is optically very flat, see Fig. 1.14(a). It produces an interference between the reflection of the beam on its front and back faces. When the beam is collimated the interference fringes are parallel to the direction of propagation of the beam, see Fig. 1.14(b). Instead if the beam is defocused the fringes are tilted, see Fig. 1.14(c). We crosschecked how the collimation of the 34 mm beam is modified when it crosses the vacuum windows. Indeed, we realized that if we collimate the beam before the chamber it is slightly defocused after crossing the vacuum view-ports. This means that the optical flatness of the vacuum windows is no longer λ/10. This effect must be due to the pressure difference, which makes the windows work as a bi-concave lens. By re-collimating the beam after the vacuum chamber and crosschecking the interference with the shearing plate in two orthogonal directions, we realized that the divergence of the beam on the two axis is not symmetric. Hence, the windows also introduce astigmatism on the beam. Moreover, we observe that the 53SI500 - Shearing Interferometer : Thorlabs Chapter 1. The potassium experiment 43 fringes of the pattern of the shearing plate are slightly curved and thus we must have additional aberrations, see Fig. 1.14(d). Second stage: Installation of imaging and addressing objectives To install the high resolution objectives we adopted a different strategy. Instead of aligning the reference beam to the center of the magnetic quadrupole, we aligned the reference beam to the center of the vacuum window and then shifted the magnetic quadrupole to be on the field of view of the objectives. Figure 1.15: Interference pattern formed from the reflection of the top reference beam on the imaging objective. Below the biggest patterns, on the left, we can observe the reflection from the bottom viewport. On top of this reflection and also below we can observe the same interference pattern repeated. This secondary reflections come from the back reflection of the wedged window that we use in the addressing set-up. In the aligning procedure, we realized that the vacuum windows are not parallel between each other, there is an angle of 0.2◦between them, see Fig. 1.15. This angle is not negligible since the tolerance alignment of the objective with respect to the window is 0.07◦. Therefore, we used two reference beams: one for the top and one for the bottom window. We crosschecked in Zemax that the best strategy to have a diffraction limited optical system which comprises both objectives is to align each objective to its own window. Thus each reference beam is perpendicular and centered to the respective window. To align the reference beams we used pinholes centered on the view- Chapter 1. The potassium experiment 44 ports and coupled the reflections from the vacuum windows back into the fiber. We installed the top objective and aligned it roughly with the help of a pinhole and a mirror which we installed on top of the 5 axis stage to align the center and angle of the objective. Afterwards we fine tuned the center and tilt of the objective by checking the reflections of the top reference beam on the surfaces of the objective around 1 m far. The centering and tilts of the 5 axis stage can be adjusted until all the reflections from the different surfaces overlap. In Fig. 1.15, we show an image of the interference pattern which is observed when the objective is aligned. An analogous alignment procedure was performed for the bottom objective. In this case, the fine tune alignment of the reflections was more complicated since the reference beam reflected by the first optical surface of the objective diverges faster and the reflections needed to be crosschecked closer to the science chamber. The achromatic lens with EFL = 500 mm was roughly aligned by hand, with the help of a pinhole and a mirror which were temporarily assembled on the lens tube so that the top reference beam crossed the lens through its optical axis. The fine tuning alignment was performed with the help of the mirror mount to adjust its angle so that its own optical axis coincides with the one of the objective. The telescope formed by the EFL = 80 mm and 250 mm lenses was aligned by using an additional reference beam which we installed on the top breadboard. This additional reference beam was aligned so that it crossed the EFL = 500 mm lens through its optical axis. The alignment of the telescope was performed roughly with the help of a pinhole and a mirror which was placed together with the telescope. The fine tuning alignment of the telescope was performed with the help of the previous 2 mirrors (labeled as M2 and M3 in Fig. 1.9) by overlapping the reflections from the lenses. An additional long pass dichroic mirror54 was used to remove the Z dipole trap beam and align the top reference beam on the center of the camera chip. Ideally, we would remove the Z dipole trap using a narrow bandpass filter in transmission. We could put it either in between the objective and first achromat or in between the second telescope so as to avoid the introduction of aberrations. In between the objective and first achromat we would need a 300 filter and in 54DMLP900L : Thorlabs Chapter 1. The potassium experiment 45 between the second telescope we would need a custom mount to adjust the angle of a 100 filter. For this reason, we decided to remove the Z dipole trap using a dichroic as a last mirror so that we can replace it easily in the future. To adjust the focusing between the lenses, we decided to fix the position of the top objective with the planned distances to the vacuum window. The bottom objective was adjusted, with the help of the shearing plate, so that a collimated beam passing through both objectives is collimated at the output. We found the focus of the first achromat by focusing a collimated reference beam from the top breadboard. We placed a camera in the focusing spot which corresponds to the intermediate image (referenced as I1 in Fig. 1.9). M1 BS SPDM M M Pinhole/ USAF target Probe beam WL1 L2 L3 I1/O2 I2/O3 L1L2 L3 L4 L5 I3 W M M M LPDM USAF 1951 (a) (b) (c) BB Figure 1.16: (a) Schematic of the optical set-up used for characterization with the USAF 1951 target (b) and a pinhole (c). All the optical elements in this set-up correspond to the elements described in Fig.1.9 and 1.11. Additionally we use an achromatic lens with an EFL of 400 mm to image either the USAF target or a 1µm  pinhole in the atomic plane. (O1) USAF target/pinhole in the object plane. (I2/O1) Image on the atomic plane. It is also the object plane of the imaging set-up. (I2/O3) Intermediate image of the imaging set-up. (I3) Image on the EMCCD camera. Chapter 1. The potassium experiment 46 USAF 1951 full apperture Pinhole 25 mm apperture Pinhole full aperture (a) (b) (c) (d) (e) (f) (g) (h) Before correcting astigmatism After correcting astigmatism After correcting astigmatism on atoms Figure 1.17: All the images in this figure are taken on the intermediate image camera. The images on the first column (a),(b) and (c) correspond to images of the USAF 1951 target (full aperture) and the pinhole (for 25 mm and full apperture) before the correction of astigmatism. The images on the second column (d),(e) and (f) correspond to images of the USAF 1951 target (full aperture) and the pinhole (for 25 mm and full apperture) after the correction of astigmatism. The images on the third column (g),(e) correspond to images of the pinhole (for 25 mm and full apperture) after the correction of astigmatism by correcting the images of bright solitons imaged on the EMCCD. All the images have been taken in its optimal configuration at each stage except for figure (c), which is an exemplary figure of the first images of the pinhole. In this case, the image was also taken with a slight defocus. Chapter 1. The potassium experiment 47 Second stage: Characterization of imaging and addressing objectives To characterize the imaging set-up we performed several tests. First of all, we produced an image of a USAF 1951 target55 on the atomic plane using an achromat with an EFL = 400 mm together with the addressing objective. We imaged it with the imaging set-up. The set-up which we used is shown in Fig. 1.16(a) and the ideal USAF 1951 target is shown in Fig. 1.16(b). The initial characterization was performed in the intermediate image. Using this set-up we measured the size of the square of the element 2 from group 2 to be 698 µm on the intermediate image. The real size of this element is 556.8 µm. Therefore the magnification of the complete system is 1.25. Given that we are using an EFL 400/50/50/500 mm lens system we also expect a magnification of 1.25. Therefore, we have assumed that the magnification of the addressing and imaging set-ups is 8 and 10 respectively. The smallest elements we were able to discern are the element 6 from group 5, see Fig. 1.17(a). These elements have a spacing of 8.77 µm which means that we are able to see elements of 1.09 µm spacing. This gives us an indication of the resolution of the imaging system but it does not mean that the resolution of the imaging system is 1.09 µm. Moreover we performed an additional test with a 1 µm pinhole56 and image it with the addressing set-up on the atomic plane, see Fig. 1.16(c). A typical picture of the initial PSF that we image of the pinhole using the full aperture of the objective is shown in Fig. 1.17(c). At this point it was crucial to count with the advise of J. Andilla, an expert from the Super-resolution Light and Nanoscopy lab from ICFO, to understand the response of our optical system. Initially, we optimized the point spread function with a 25 mm aperture by slightly tweaking the tilt and centering of the bottom objective (which was the most difficult to align properly). In Fig. 1.17(b) we show the image of the pinhole on the intermediate image after the initial optimization. Besides the central maximum we observe 4 lobes which indicate the presence of strong astigmatism. The additional features surrounding the area with biggest intensity show the presence of high order spherical aberrations. To improve the performance of the optical system, we have corrected the 55R1DS1P - Thorlabs 56Mounted Precision Pinhole, 1(+0.25/−0.10) Pinhole  . P1H - Thorlabs Chapter 1. The potassium experiment 54 0.00 0.25 0.50 0.75 1.00 41K 0.0 0.4 0.8 1.2 0.00 0.25 0.50 0.75 1.00 39K K K Dual (a) (b) Figure 1.21: (a) Condensed fraction N0/N vs. T/Tc, where Tis the temperature of the cloud and Tcis the critical temperature of a non interaction BEC, for 41K (top panel) and 39K (bottom panel). The experimental data refers to the pure condensation of each species. Whereas the solid red line includes the effect of finite size and interactions on the critical temperature the dashed line does not [88]. The bigger shift on 39K as compared to 41K is due to the difference in scattering lengths (154a0vs. 60a0respectively). The insets in the top and bottom panels show the distribution of velocities measured in time of flight. We observe the transition from the thermal to the quantum degenerate regime. (b) Dual condensation of 41K and 39K. From top to bottom we show pictures of the evaporation of the mixture. Since the critical temperatures are different they condense at different times. As soon as there is dual condensation there is phase separation [54]. •The atoms are transferred to the hybrid trap by shifting the magnetic quadrupole to center it on the field of view of the microscopes. In this process we do not longer transfer the 15% of the atoms but a 10% at around 8µK. The hybrid evaporation ends up with a total atom number of 1.7×106atoms at 1µK. Cooling both species we can obtain roughly half the atom number of each. Chapter 1. The potassium experiment 55 •We transfer around 1.2×106atoms into a crossed dipole trap. After successive hyperfine and Zeeman transfers we perform forced evaporation to reach the degenerate regime. If we want to obtain a pure BEC of 39K, its MOT loading time is adjusted to be left without 41K after the evaporation. We do not perform the spin distillation by transferring 41K to |F= 2, mF=−2i. •At the end of the evaporation we can obtain pure BECs of 41K and 39K with 1.7×105atoms and 1.4×104atoms respectively, see Fig. 1.21(a). If we adjust the MOT loading balance we can obtain dual BECs with around 7×104atoms per species, see Fig. 1.21(b). 1.6 Conclusion and outlook In conclusion, we have upgraded our apparatus with a new glass cell 2D MOT and an optical set-up to image and address the atoms with high resolution. The design of the new glass cell 2D MOT tackles the main issues which we encountered during the operation of the old stainless steel 2D MOT. We have avoided the use metal-to-glass transition materials in order to seal the vacuum view-ports and used only a circular indium wire to seal the glass cell. The quartz glass cell not only provides a similar vapor pressure using smaller temperatures on the oven, but also has a reduced out-gassing as compared to the old stainless steel chamber. Hence, we have been able to maintain the pressure of the 2D MOT chamber with the pumping of the NEG pumps through the differential pumping tube. Additionally, the new 2D MOT contains a cold spot which could be used in the future to facilitate the installation of enriched 40K. In summary, we have improved our 2D MOT and maintained similar atom numbers from the loading of the 2D to the 3D MOT using smaller push powers. Moreover, we have designed and developed a custom-made high resolution optical set-up for imaging and addressing the atoms in situ a performance not too far from the diffraction limit. To do so, we have measured that the main source of aberrations of our optical set-up comes from the vacuum view-ports. We have observed that they introduce a small defocusing, large astigmatism, trefoil and high order spherical aberrations. We have corrected the astigmatism introduced by the windows by compensating it with a thick window. Finally, we Chapter 1. The potassium experiment 56 have observed bright solitons with a size of 3 µm, and measured the resolution of the complete optical set-up to be 1.1 µm and 1.5 µm in each direction. In the future, we could correct the aberrations introduced in our imaging set-up by using a deformable mirror. If we introduce it in a Fourier plane of the image, we could modify the phase for each spatial frequency and also correct the higher order aberrations. Moreover, we could install a digital micromirror device in order to project arbitrary potentials on the atomic cloud. Up to now, we have learned how to use it either in the the Fourier plane or with direct imaging [89, 90]. Additionally, we have corrected the aberrations introduced by the own DMD and its window [86, 91]. Therefore, it is only left to install it in the experiment apparatus to test it on the atoms. Chapter 2 In situ imaging of two-component Bose-Einstein condensates Abstract In this chapter, we develop a polarization phase contrast technique which is able to probe optically dense atomic mixtures at intermediate and high magnetic fields, while exploiting open atomic transitions. We have found a method to directly measure the atomic polarization phase shift and calibrate the corresponding Faraday coefficient, finding good agreement with our theoretical predictions. This technique has been used to image the total column density of a two-component atomic cloud at B= 57 G in a dark field. At B= 396 G we have demonstrated that we can measure the difference in column density between two states. 2.1 Introduction In order to extract information of the atomic clouds we probe them with light. There are different techniques which can be used to image an atomic cloud. 57 Chapter 2. In situ imaging of two-component BECs 58 Namely absorptive and dispersive techniques. Absorptive techniques are based on the absorption and re-emission of photons, also called fluorescence. Instead, dispersive techniques, also known as phase contrast techniques, are based on the phase shift introduced on the electric field of light by the presence of atoms. The most commonly used technique is time of flight absorption imaging due to its simplicity. By shining resonant light on the atomic clouds after time of flight, we can compare the shadow cast on the camera with the light in absence of atoms to measure the optical density and extract the density profile of the atoms. In this case, and for long enough time of flight, the measured density profiles are related with the momentum distribution of the cloud in situ. As explained in section 1.4, we have developed a high resolution imaging objective to probe the clouds in situ and extract its spatial distribution in the optical trap. The imaging complexity is increased for in situ techniques. In general there is not a preferred technique over the others, but one has to take into account the particularities of each system to evaluate the advantages and disadvantages of each technique in order to assess its appropriateness. Let us briefly introduce the different techniques which have been used to probe cold atoms in situ. One of the in situ techniques which has been widely used in the cold atoms community is fluorescence imaging. By shining resonant light onto the atoms, they absorb and re-emit photons which are captured by an imaging set-up. Since the re-emission of photons is in general isotropic, only a few collected photons are captured by the imaging system. However, since the probing beam is not aligned with the optical axis of the imaging, fluorescence imaging has the advantage of having a dark background. This technique, has mainly been used to image systems with low optical density, such as atoms trapped in 1D and 2D optical lattices (see reference [92] and references therein) and one-dimensional waveguides [93]. Besides the partial collection of the re-scattered photons, the main limitation of this technique is the blurring of the images due to the photon recoil. In optical lattice experiments, exposure times on the order of a second are needed together with a sub-Doppler cooling mechanism and the use of pinning lattices in order to have single site resolution. To image optically thick atomic gases in situ, such as Bose-Einstein conden- Chapter 2. In situ imaging of two-component BECs 59 sates with optical densities on the order of hundreds, absorptive techniques fail at low saturation. One alternative is to use highly saturated absorption imaging [94–96]. This technique involves a subtle calibration due to the nonlinear response of atoms on the imaging intensity. Moreover very short imaging pulses on the µs scale are needed in order to avoid motional blurring due to photon scattering [97]. It is important to note that absorption imaging has to be performed exactly on resonance. Otherwise the atomic cloud can act as a gradient-index lens due to phase dispersion [98]. Another possibility to image optically dense clouds are phase contrast techniques. The main strategy of these techniques is to reduce the optical crosssection by imaging far from resonance. Hence, photon scattering is reduced to reduce the motional blurring and perform non-destructive imaging at the expense of a lower signal. As a consequence electron multiplying cameras have to be used in order to get a good signal to noise ratio. Moreover the imaging can suffer from lensing effects which need to be taken into account [98]. There are different types of phase contrast techniques. As mentioned already, dispersive techniques exploit the phase shift introduced in the electric field of the probing light. This phase shift is produced by the real part of the atomic polarizability, which is a second order rank tensor. Therefore the shift can have a scalar, vectorial and tensor component. Although the tensorial component has been measured in references [99, 100] it is not usually exploited to image. Instead, the scalar and vector phase shifts have been used to image cold atoms. The scalar phase shift is measured by interfering the non-diffracted probe beam with the diffracted light [101]. The vector phase shift rotates the polarization of the probe beam and it can be measured with a polarization beam splitter which acts as a polarization analyzer [102]. In our experiment, we have developed a polarization phase contrast technique which is based on the work described in reference [103] and references therein. With our technique we are able to measure the column density of optically dense atomic mixtures in situ at intermediate and high magnetic fields in open transitions. To use this technique, we have developed a direct method to measure the polarization phase shift introduced by the atoms in order to calibrate the effective Faraday coefficient. This technique is flexible enough to measure the Chapter 2. In situ imaging of two-component BECs 60 total column density of an atomic cloud as well as to measure the difference in column densities. In the following section we will explain the physical principle of the lightmatter interaction in order to understand how the probing of cold atoms with light works. We show how the atomic polarizability is connected to the index of refraction and how to compute it in the presence of a magnetic field. Afterwards we motivate why polarization phase contrast is particularly suitable for the experiments we performed during the completion of this thesis. Finally we introduce a simple technique to calibrate the vector polarizability and analyse the results obtained at intermediate and high magnetic fields where the experiments carried out during this thesis have been performed. 2.2 Probing cold atoms with light In order to understand how the different imaging techniques work we will describe the interaction between dilute atomic clouds and light. To do so we will first consider the semi-classical Lorentz model, which describes the atom as a damped harmonic oscillator [104]. This model allows us to calculate the electrically induced dipole moment on an atom and connect the macroscopic index of refraction with the atomic polarizability. As a result we can extract the dispersive and absorptive character of light propagating through a dilute cloud. Even though this model captures the main features of light-matter interaction, a quantum-mechanical calculation is needed in order to account for the correct value of the polarizability. Thus, in a second stage, we develop briefly this calculation to be able to compute the polarizability of an atom under the presence of a magnetic field. Depending on the magnetic field strength there are analytic formulas for the polarizability. However, given that we want to be able to extract the polarizability for very different magnetic field regimes, we have performed a numerical calculation. As mentioned previously, from the knowledge of the atomic polarizability we can compute the dispersion of light propagating through an atomic cloud. This dispersion can introduce scalar, vectorial and tensorial shifts depending on the detuning and polarization of the light. In the last part of the chapter we describe Chapter 2. In situ imaging of two-component BECs 61 the three processes and how we can compute the associated polarizabilities from the quantum-mechanical calculation. 2.2.1 Semi-classical treatment The Lorentz model The Lorentz model is a semi-classical model that assumes that the electron is bound to a steady nucleus with a spring-like force ~ Fs=−mω2 0~x, where mis the electron mass, ω0is the resonant frequency and ~x is the distance vector pointing from the nucleus to the electron. The electric field of the light induces a force ~ Fe=~exqeE0cos (ωt +φ) that makes the electron oscillate at a frequency ω, where qeis the electron charge. The acceleration of the electron induces a radiation that produces a damping force ~ Fd=mΓ˙ ~x, where Γ is the linewidth of the optical transition. The resulting equation of motion of the electron position ~x is the following: ¨ ~x + Γ ˙ ~x +ω2 0~x =~ex qeE0 mcos (ωt +φ).(2.2.1) The dipole moment induced on an atom due to the presence of an electric field is proportional to the polarizability: ~ d(ω) = α(ω)~ E(ω). Moreover the electric dipole moment of an atom has a magnitude of ~ d=qe~x. Therefore, by solving the equation (2.2.1) of the Lorentz model it can be seen that the polarizability of an atom is given by: α(ω) = q2 e/m ω2 0−ω2−iΓω.(2.2.2) For a dilute atomic cloud with an atomic density ρprobed with light with a wavelength λsuch that ρλ31, the interaction between many atoms and photons can be treated independently. That is, neglecting collective effects. Under this approximation it can be seen that the index of refraction of an atomic cloud can be approximated to: ˜n(ω) = p1 + χ(ω)≈1 + 1 2χ(ω),(2.2.3) where χ(ω) = α(ω)ρ/0is the electric susceptibility and 0is the permittivity of Chapter 2. In situ imaging of two-component BECs 62 vacuum. It is easy to see that the real part of the index of refraction introduces a phase on the electric field of the light that propagates through an atomic cloud. Instead, the imaginary part is responsible for the absorption of light. E=E0eikz =E0ei˜nk0z=E0eiRe[˜n]k0ze−Im[˜n]k0z.(2.2.4) Thus, the phase index n(ω) = Re[˜n(ω)] and the absorption coefficient a(ω) = 2k0Im[˜n(ω)] for small detunings |ω−ω0|  ω0are: n(ω) = 1 + ρq2 e 2m0 (ω0−ω)/2ω0 (ω0−ω)2+ (Γ/2)2,(2.2.5) a(ω) = ρq2 e mc0Γ (Γ/2)2 (ω0−ω)2+ (Γ/2)2.(2.2.6) The absorption coefficient a(ω) is related to the optical absorption crosssection σ(ω) by: a(ω) = σ(ω)ρ. Therefore the classical optical absorption crosssection on resonance is: σC 0=q2 e m0cΓ.(2.2.7) Although this model is quite simplified, it is able to capture the main features of light-matter interaction for a dilute cloud where atoms are reduced to a two level system. However there are different aspects that the classical model does not capture properly. The quantum mechanical optical cross section is: σQ 0=3λ2 2π.(2.2.8) As compared to the classical optical cross section, the quantum mechanical one is much larger. In particular, for the D2 transition of potassium: σC 0= 3.95 nm2and σQ 0= 0.28 µm2. In general atoms have multiple energy levels that have to be taken into account. As a result the polarizability contains the contribution of all the possible transitions weighted by the absorption oscillator strength feg =σQ 0/σC 0: α(ω) = q2 e mX e6=g feg ω2 eg −ω2−iΓegω,(2.2.9) Chapter 2. In situ imaging of two-component BECs 63 which is the quantum enhancement of the optical cross-section for the transition between the ground and excited states. Here Γeg corresponds to the linewidth of the transition between the excited state |eiand the ground state |gi. Moreover we have considered the polarizability as a scalar quantity, whereas the transitions to spherically-non symmetric orbitals are different depending on the polarization of the light. This is the case for the transitions from S to P orbitals. Thus we should consider the polarizability as a tensor instead of a scalar. In conclusion, to evaluate the correct quantitative value of the polarizability a quantum mechanical approach needs to be considered. Before introducing the quantum mechanical calculation, in the following section, we will introduce the polarizability as a tensor and explain the imaging principle of the Faraday effect which is exploited in the polarization phase contrast imaging technique. The tensor polarizability and the Faraday rotation effect As discussed previously, the atomic charge density may react differently to the application of the electric field in different directions due to the anisotropy of the electronic orbitals. Thus we should consider the polarizability as a tensor in analogy to what is done with anisotropic materials: di=αijEj,(2.2.10) where diis the induced dipole on the i-th direction due to the application of an electric field in the j-th direction. If we consider that the ”i” and ”j” indices indicate principal axes of the system, the polarizability tensor is diagonal. In particular, if the polarizabilities are different for the different axes, the propagation of an electric field through an anisotropic system can introduce a differential phase shift between the electric field polarization of different axes. Therefore the polarization of light can be modified after propagating through an anisotropic system. In particular, we will focus on the polarization rotation induced by an atomic cloud due to the differential phase shift introduced between the σ+and σ−components of a linearly polarized beam. This effect is commonly known as Faraday Chapter 2. In situ imaging of two-component BECs 70 0 100 200 300 400 500 0 0.5 -0.5 0 1 0.5 -0.5 -1 390 410 1.20 1.05 Figure 2.3: Top panel: Energy of the 2P3/2of 39K state versus magnetic field. The energy is referenced with respect to the D2 transition. There are 16 energy levels from the F0= 0,1,2 and 3 manifolds, which at high field separate into four m0 Jbranches composed by four magnetic sublevels as shown in the inset. Bottom panel: Energy of the 2S1/2state of 39K versus magnetic field. The zero of energy corresponds to the 2S1/2state fine structure energy. The three lowest energy levels, |ai,|biand |ci, are superpositions of the corresponding |mJ=−1/2iand |mJ= 1/2ielectronic states. Their composition depends on the magnetic field. The energy level labeled as |diis a streched state. The ground states can be coupled to the excited states via σ+and σ−light as shown by the dashed and solid arrows respectively. For simplicity we will label the three-lowest energy states as: |ai,|biand |ci, Chapter 2. In situ imaging of two-component BECs 71 see Fig. 2.3. At zero magnetic field we can identify them as: |ai≡|F= 1, mF= 1i |bi≡|F= 1, mF= 0i |ci≡|F= 1, mF=−1i whereas at high field they correspond to: |ai≡|I= 3/2, mI= 3/2, J = 1/2, mJ=−1/2i |bi≡|I= 3/2, mI= 1/2, J = 1/2, mJ=−1/2i |ci≡|I= 3/2, mI=−1/2, J = 1/2, mJ=−1/2i At intermediate fields, such as B∼57 G, where the experiments from chapters 3, 4 and 5 are performed, these states contain contributions of different |I, mI, J, mJistates. In particular, for the droplet experiments, we have used the |biand |cistates. At B= 57 G their composition is the following: |bi=−0.815 |3/2,1/2,1/2,−1/2i+ 0.580 |3/2,−1/2,1/2,1/2i |ci=−0.642 |3/2,−1/2,1/2,−1/2i+ 0.767 |3/2,−3/2,1/2,1/2i If we probe these states with linear polarization close to the D2 transition, the σ+(σ−) component will couple them to the excited states with mJ0= 1/2 and 3/2 (mJ0=−1/2 and −3/2). In Fig. 2.4 we plot the scalar and vector polarizability of |ai,|biand |cifor frequencies around the D2 transition. As it can be seen both polarizabilities contain four features corresponding from the mJ0=−3/2 to the mJ0= 3/2 starting from around 0.1 GHz detuning to 0.5 GHz. Hence, the polarizability of these states at a particular frequency has contributions of several excited states as shown in Fig. 2.3. At high field, the states on the F= 1 manifold are in the Paschen-Back regime and they correspond mainly to a single |I, mI, J, mJistate. Therefore the polarizability at high field will not contain as many contributions from the Chapter 2. In situ imaging of two-component BECs 72 00.2 0.4 0.6 0.8-0.2 60 0 -60 60 0 -60 Figure 2.4: Scalar and vector polarizabilities of the 39K2S1/2states at B= 57 G vs frequency around the D2 transition, top and bottom panels respectively. The states |ai,|biand |ciare the ones that connect to the |F= 1, mF= 1i, |F= 1, mF= 0iand |F= 1, mF=−1irespectively at B= 0 G as shown in Fig. 2.3. The arrow points the frequency used to image at low field. excited states. Currently, we are working on experiments on spin-orbit coupling1 at 396 G with states |ai,|bi. In particular the composition of these states is: |ai= 0.990 |3/2,3/2,1/2,−1/2i−0.144 |3/2,1/2,1/2,1/2i |bi= 0.981 |3/2,1/2,1/2,−1/2i−0.196 |3/2,−1/2,1/2,1/2i Therefore the scalar and vector polarizabilities around the D2 transition mainly have contributions of the transition to the mJ0=−3/2 and mJ0= 1/2 1These experiments will be explained in the PhD theses of A. Fr¨olian and C. S. Chisholm. Chapter 2. In situ imaging of two-component BECs 73 0 -60 60 01 -1 0 -60 60 Figure 2.5: Scalar and vector polarizabilities of the 39K2S1/2states at B= 396 G vs frequency around the D2 transition, top and bottom panels respectively. The states |ai,|biand |ciare the ones that connect to the |F= 1, mF= 1i, |F= 1, mF= 0iand |F= 1, mF=−1irespectively at B= 0 G as shown in Fig. 2.3. The black and red arrows point the frequencies used to image at high field. states. As it can be seen in Fig. 2.5 these transitions correspond to the dispersive features around -0.5 GHz and 1 GHz detuning respectively. Hence it becomes clear that the polarizability at high field at a particular frequency has more resemblance to the two-level system dispersion. Now that we have introduced the polarizabilities at B= 57 G and B= 396 G, we have to make a choice between the scalar and vector polarization phase contrast techniques. In the experiments performed so far we decided to measure the vector instead of the scalar phase shift for simplicity. The technical details about this choice will be clarified in section 2.3. Before entering into details, in the following section we will discuss the differences between the scalar and Chapter 2. In situ imaging of two-component BECs 74 vector polarizabilities at intermediate and high fields, and the detunings that we chose to image potassium mixtures with a polarization phase contrast imaging technique. Scalar vs. vector polarizability for imaging mixtures at intermediate and high magnetic fields Whereas the scalar polarizability contains the sum of the σ+and σ−contributions, the vector polarizability contains the difference, as seen from equations (2.2.22) and (2.2.23). Close to resonance, the magnitude of the scalar and vector polarizabilities are very similar, as shown in Fig. 2.4 and 2.5. Far from resonance, the contributions from the different transitions sum up for the scalar polarizability and can cancel each other for the vector polarizability depending on the spin composition of the ground states. As already explained, the composition of the ground states is formed by mJ=−1/2 and mJ= 1/2 electronic states, which couple respectively to the mJ0=−3/2 (mJ0= 1/2) and mJ0=−1/2 (mJ0= 3/2) via σ−(σ+) light as shown in Fig. 2.3. Provided that the Clesbch-Gordan coefficients satisfy the symmetry relation hJ, mJ;l= 1, ml=q|J0, m0 Ji=−hJ, mJ;l= 1, ml=q|J0,−m0 Ji[107], if the composition of the ground state is 50/50 then the vector polarizability vanishes far from resonance, see Fig. 2.4. For this reason, at B = 57 G, where the composition of |biand |ciis almost balanced, we cannot measure the vector phase shift far from resonance. Instead, at B = 396 G, the scalar and vector phase shifts are also similar far from resonance, see Fig. 2.5. In the presence of a mixture the polarization rotation contains the contribution of both components and thus θA=cF1n1+cF2n2(2.2.24) where θAis the polarization phase shift introduced by the atoms and cFi and ni= Rρi(x, y, z)dz are the Faraday coefficient and column density (integrated along the imaging axis) from the i-th component. In general there are two interesting cases: if cF1 =cF2,θAreflects the sum of the column densities whereas if cF1 = −cF2 it reflects the difference. In the experiments that we carried out during the completion of this thesis (chapters 3, 4 and 5) it was crucial to work in a regime Chapter 2. In situ imaging of two-component BECs 75 where cF1 ≈cF2 to measure the total atom number. Additionally, in this system the stability of the droplets depends on the ratio of population between the two components, and it would have been really interesting to measure the difference in population between the two. However, at low field the condition cF1 =−cF2 is achieved very close to resonance. Instead, at high field the energy level splittings are much bigger and the cF1 =−cF2 are met further from resonance. Under the previous considerations, we have chosen to image the mixtures with a frequency such that for B = 57 G the light for |biand |ciis red detuned by 17.5Γ and 12.2Γ from the closest transitions to measure the total atom number (black arrow in Fig. 2.4). Similarly, for B = 396 G, |aiand |biare red detuned by 44.7Γ and 29.7Γ from the closest transitions (black arrow in Fig. 2.5). For these detunings, the imaging is destructive, as it is shown in section 2.3. Additionally, to measure the difference of column densities, at B = 396 G, we performed some experiments with light which was red (blue) detuned from the |ai(|bi)→mJ0= −3/2 transitions by 7.5Γ (red arrow in Fig. 2.5). In the following section we will explain how the polarization phase contrsat technique works, how do we calibrate it, what are the results obtained at intermediate and high magnetic fields and how the technique is used to image a spin mixture in situ. 2.3 Experimental characterization of polarization phase contrast imaging As mentioned previously the motivation to image the clouds in situ with a polarization phase contrast technique is to image optically dense mixtures with column densities on the order of 1014 m−2with good signal to noise ratio and minimal motional blurring. The technique that we use was initially developed in reference [102]. In particular, the implementation we employ is closely based on the work from M. Gajdacz et al. [103]. In section 2.3.1, we will explain the concept of the experimental implementation of the polarization phase contrast technique. The setup that we developed to implement this technique is detailed in section 2.3.2. Afterwards, in section 2.3.3, we explain the imaging analysis procedure Chapter 2. In situ imaging of two-component BECs 76 to extract the atomic polarization phase shift. Then we describe the technique used to calibrate the atomic polarization phase shift in section 2.3.4. Finally, in section 2.3.5, we present the results that we obtained to image the sum and difference of column densities of a potassium mixture. PBS Image beam Probe beam QWP R-HWP atoms EMCCD Pixel Counts (a) (b) (c) Figure 2.6: (a) Scheme of the polarization phase contrast set-up. A linearly polarized beam is sent through the atoms, which rotate its polarization due to its birefringent character. In the middle of the imaging system a polarization beam splitter works as a polarization analyzer which measures the rotation of the polarization induced by the atoms. The images are taken on an EMCCD camera. To adjust the polarization we use true zero order quarter and half waveplates (768.4 nm). The half waveplate is mounted on a motorized rotating mount to probe the atomic cloud with different input polarizations. (b) Images taken with the EMCCD camera in dark field configuration. In the absence of atoms no light goes to the camera. Top left: IA. Bottom left: IB. Top right: IDA. Bottom right: IDB. The color scale shows the pixel counts. (c) Atomic polarization phase shift retrieved from the images in (b). The color scale shows θA. Chapter 2. In situ imaging of two-component BECs 77 2.3.1 Concept How do we implement the polarization phase contrast technique in our experiment? In Fig. 2.6(a) we show a schematic of the experimental implementation. First of all we prepare a laser light beam with linear polarization. The laser setup that we use is described in section 2.3.2. A rotating λ/2 waveplate2is used to adjust the polarization orientation. The imaging beam is sent through the vacuum chamber to image the atoms. Once the imaging beam crosses the atomic cloud the scattered light picks up a scalar and vector phase shift. The imaging beam crosses the imaging system presented in section 1.4. In the middle of the imaging system a polarization beam splitter3acts as a polarization analyzer. By contrast with the non-scattered light, we image the polarization phase shift introduced by the atoms in an electron multiplying charged-coupled device camera4. In the imaging process we take four pictures, see Fig. 2.6(b). First we take an image of the atomic cloud IAfollowed5by a picture without atoms, which we call bright image IB. After 1 second of readout time, we take two more pictures without any light that we use to remove the background from the atoms and bright pictures. They are called respectively dark atoms IDA and dark bright IDB pictures. The signal that we get on the camera, assuming that the polarization is perfectly linear is: IA(i, j) = β(i, j)·sin2(θA(i, j) + θ) + IDA(i, j) (2.3.1a) IB(i, j) = β(i, j)·sin2(θ) + IDB(i, j) (2.3.1b) where βis the probe beam intensity, θis the input polarization angle with respect to the polarization beam splitter axis, θAis the polarization phase shift introduced by the atoms and (i,j) indicate the pixel index. There are two main configurations that can be used to image the atomic 2True zero order λ/2 waveplate from FOCtek (WPF212H 768.4 nm) installed on a motorized precision rotation stage from Thorlabs (PRM1Z8) driven by the K-Cube Brushed DC Servo Motor Controller (KDC101). 3Thin film polarizer from Qioptiq (Size: 25 mm ×25 mm ×25 mm. Extinction ratio 10−4.) 4EMCCD: Andor iXon Ultra 897 5There is a 1.7 ms delay between both pictures due to the frame transfer of the camera. Chapter 2. In situ imaging of two-component BECs 78 clouds. If we set θ= 0, most of the light from the probe beam is discarded and the images are taken in the dark-field configuration. For small polarization phase shifts θAthe signal IAis quadratic on the polarization phase shift. Instead, if we set θ=π/4, the probe beam is split in half in the polarization analyzer and the signal IAis linear for small polarization phase shifts. In our experiment we generally take images in the dark field configuration. In section 2.3.3 we will detail the analysis of the pictures in order to extract the column density of the imaged cloud and introduce the advantages and disadvantages of working in dark field. In the following section we will explain the experimental set-up to image the atomic cloud in situ using the polarization phase contrast technique. Figure 2.7: Faraday laser set-up. Left panel: The DFB Faraday laser beam is shaped using cylindrical lenses and coupled into a single mode polarization maintaining fiber after passing through an optical isolator. Middle panel: The Faraday laser is split in two. The left path goes into the fiber which sends the light into the experiment table. We use an AOM to adjust the probe beam intensity and as a switch together with a mechanical shutter. The right path goes into a polarization maintaining fiber optic 2×2 coupler, which mixes the light from the Faraday laser with the 39K D2 laser to generate the offset lock. The 39K D2 is offset locked with respect to the Master D2 laser, which is locked on a hot potassium vapor cell using frequency modulated spectroscopy (right panel). Chapter 2. In situ imaging of two-component BECs 79 2.3.2 Experimental Faraday set-up The experimental Faraday set-up is divided in two parts. The laser set-up which we use to prepare the laser light to probe the atomic cloud and the imaging setup. Since the imaging set-up has been described in detail in section 1.4, we will focus on the particular details concerning the polarization phase contrast imaging and detail the Faraday laser set-up. Faraday laser set-up The light that we use to probe the atomic clouds with this technique is produced by a DFB laser6, which we name Faraday laser. In contrast to absorption imaging, we do not need a laser with a narrow linewidth because the imaging is performed far from resonance. Whereas for absorption time of flight imaging we employ light with ∼300 kHz linewidth7for these technique we use a ∼1 MHz linewidth DFB laser. The laser setup that we use is shown in Fig. 2.7. The light from the Faraday laser is offset locked to the additional species D2 laser which is offset locked to the D2 master laser [65, 109]. A Voltage Controlled Oscillator (VCO) on the offset lock of the Faraday laser is used to be able to tune the frequency of the laser. We will call this frequency the Faraday beat detuning ∆FAR BEAT. In Fig. 2.8(a) and (b) we plot the Faraday coefficients that will be relevant throughout this thesis with respect to this detuning. The relation between the frequency of the Faraday laser light that we shine on the atoms and ∆FAR BEAT is described in appendix B. Faraday imaging set-up The set-up which we use to image the atomic clouds in situ has been detailed in section 1.4. In particular, for the Faraday imaging set-up we need to ensure that the polarization that we send to the atoms is linear. We designed the optical set-up in order to have the least possible number of components before the probe beam reaches the atoms. Nevertheless the probe beam still crosses several optical elements that could introduce a circular component on the polarization, 6EYP-DFB-0767-00050-1500-TOC03-0005 - Eagleyard photonics 7Generated with an External Cavity Diode Laser (ECDL). SYST DL PRO 780 - TOPTICA Chapter 2. In situ imaging of two-component BECs 86 where µ=1 2(15Na~2√mωxωyωz)2/5is the chemical potential, mis the mass of 39K,ωx,y,z are the trap frequencies. Hence, the atom number is: N=m2ω4 xR5 TFx 15~2aωyωz ,(2.3.7) and we can extract the atom number from the measured trapping frequencies and RTFx. In this case the main relative error on the calibration of the atom number comes from the measurement of ωxand RTFx: σN N2≈4σωx ωx2 +5σRTFx RTFx 2 ,(2.3.8) where typical measurements on ωxand RTFx have relative errors on the order of 1% and 3% respectively. Therefore this technique allows us to measure the atom number with relative uncertainties of ∼16%. Additionally, a systematic uncertainty which underestimates the atom number is introduced due to the Thomas-Fermi approximation. For typical parameters used in the calibration of the Faraday coefficient the radius computed by solving the Gross-Pitaevskii equation numerically (for an equivalent atom number and scattering length) is ∼3% smaller than the Thomas-Fermi radius from equation (2.3.6). We include this systematic correction to calibrate the atom number. We have crosschecked that both calibration methods agree with each other. Although the second technique may seem more precise, it is more prone to systematic errors on the measurement. Thus, we normally attribute a 25% error to the Faraday coefficient unless stated otherwise. In the future, we could improve the calibration on the atom number by using the technique described in reference [96]. In the following section we summarize the errors on the measurement of the column density. Column density errors The column density of an atomic cloud is extracted from ni=θA/cFi. Hence, the relative column density error is given by the following equation: Chapter 2. In situ imaging of two-component BECs 87 σnA nA2 =σθA θA2 +σcF cF2 (2.3.9) Therefore we need to take into account the error on the measurement of the Faraday coefficient as well as the error in the measurement of θA. As introduced previously, the sinusoidal dependence of IAon θAimposes certain limits on the applicability of this imaging technique. On the one hand, θA(IA) is not a singlevalued function. We have restricted the application of this technique to signals such that θA< π/2. On the other hand, the error on the retrieval of θAis really big for θA∼0, π/2, see Fig. 2.10. In summary, in the dark field configuration, the error on the measurement on the column density can be limited either by the 25% error on the calibration of the atom number or by the measurement of θAfor θA≈0, π/2. For reference, if we limit ourselves to σθA/θA< σcF/cF, we are limited to measure column densities on the range (0.1 rad/cF, 1.5 rad/cF). 2.3.5 Results In the previous sections we have developed the methods to retrieve the atomic polarization phase shift, calibrate the Faraday coefficients and extract the column densities using a polarization phase contrast technique. In this section, we will first discuss the measurements performed for a single spin state and then we will show how this technique can be used to image a spin mixture to extract the total column density and the spin column density imbalance. Experimental measurement of the Faraday coefficient The calibration of the Faraday coefficients which have been performed at low (B = 57 G) and high field (B= 396 G) are shown in Fig. 2.12(a) and (b) and are summarized in table 2.1. All the measurements have been performed using a 3 µs imaging pulse with 250 mW/cm2peak intensity. The imaging beam that we have used to probe the |biand |cistates at B= 57 G is red detuned by −17.5Γ and −12.2Γ from the closest transitions to measure the total atom number (black arrow in Fig. 2.4 which corresponds to ∆FAR BEAT = 153 MHz). The imaging beam Chapter 2. In situ imaging of two-component BECs 88 that we have used to probe the |aiand |bistates at B= 396 G is red detuned by −44.7Γ and −29.7Γ from the closest transitions to measure the total atom number (black arrow in Fig. 2.4, which corresponds to ∆FAR BEAT =−515 MHz). For these detunings, the imaging is destructive. As it can be seen in Fig. 2.6(b) and Fig. 2.9(a) there are no atoms in the bright image IB. However, the imaging pulse is so short that the photon recoil does not blur the atoms image. Moreover we have crosschecked that the response of the atomic cloud is linear with the intensity and that no saturation and lensing effects are observed at these detunings (see appendix C for details). Since the transitions that we probe are not closed, atoms may fall into other states during the exposure. This process is called optical depumping and is produced by the spontaneous emission of the excited states to other ground states. The characteristic time τ∼10 µs of this process is very similar for both fields. Hence, 25% of the atoms fall into other ground states and the effective Faraday coefficient may differ from the theoretical expectations (see appendix C for details). Whereas at low field there is a discrepancy between the measured and theoretical Faraday coefficient, at high field the measurements agree well with theory. This is presumably due to the fact that the imaging at high field is farther from resonance than at low field and the fact that at high field the optical transitions have more resemblance with a two-level system. Additionally, we have performed an independent crosscheck at low field by transferring |cito the |di=|mI=−3/2, mJ=−1/2istreched state (see Fig. 2.3) and probing the cycling transition: |mI=−3/2, mJ=−1/2i→|mI0=−3/2, mJ0=−3/2i(2.3.10) This allows us to compare our calibration with the expected two-level system prediction, finding good agreement. Second, by transferring a variable fraction of atoms to |mI=−3/2, mJ=−1/2i, we confirm the linearity of our imaging scheme and rule out the existence of collective atom-light interaction effects in the imaging. In summary we have been able to characterize the effective Faraday coeffi- Chapter 2. In situ imaging of two-component BECs 89 cients at B = 57 G for the |biand |cistates and at B = 396 G for the |aiand |bi states. This allows us to measure the column density of each state and extract its size and atom number. But how does the imaging perform in the presence of a mixture? B (G) ∆FAR BEAT (MHz) |stateicexp F(10−15rad ·m2)cth F(10−15rad ·m2) 57 153 b -1.4 -2.44 57 153 c -1.1 -2.24 396 -515 a -1.3 -1.47 396 -515 b -2.6 -2.21 Table 2.1: Summary of the Faraday coefficients cFwhich have been measured at low (B = 57 G) and high field (B = 396 G) for 39K. The relative error on the Faraday coefficient is 25%. Imaging mixtures in situ In the presence of a mixture the polarization rotation contains the contribution of both components and is described by equation (2.2.24). As explained previously there are two interesting cases: if cF1 =cF2,θAreflects the sum of the column densities whereas if cF1 =−cF2 it reflects the difference. The experiments performed in chapters 3, 4 and 5 are done in a configuration where cF1 ≈cF2 to measure the total atom number. Moreover, we have started to explore the regime where cF1 ≈ −cF2 at high field in order to measure the difference of column densities. We have performed some experiments in the linear configuration θ=π/4 with light whose frequency lies in the middle of the |ai → mJ0=−3/2 and |bi → mJ0= 1/2 transitions where cF a ≈ −cF b. This light is is red (blue) detuned from the |ai(|bi)→mJ0=−3/2(1/2) transitions by 7.5Γ (red arrow in Fig. 2.5). To illustrate the ability to image a mixture insitu we prepared a BEC in state |aiin an optical dipole trap formed by two intersecting beams. We switched off the vertical beam and let the BEC expand in a waveguide9to reduce its density and avoid lensing effects. Then we apply a radio-frequency pulse for a variable 9The waveguide is aligned within the depth of focus of the imaging objective. Chapter 2. In situ imaging of two-component BECs 90 time to transfer atoms to state |biand immediately image the cloud in situ. With this method we ensure that the density profile of both states is identical and only the ratio of population is modified. In Fig. 2.13 we can observe the signal of the Rabi oscillations measured in situ. In this case: θA=n 2[(cFb −cFa)P+ (cFb +cFa)] ,(2.3.11) where nis the total column density n=na+nband Pis the polarization P=nb−na nb+na. The column densities we used are such that θAis small and therefore we can use the linear approximation from equation (2.3.3). If all atoms are in state |ai(|bi) we expect that θA=cFan(cFbn). Since cF a ≈ −cF b we expect that the polarization rotates in opposite directions and in equal amount, as it is observed in Fig. 2.13. Figure 2.13: Rabi oscillations measured in situ in the linear configuration for B = 396 G and ∆FAR BEAT =−292 MHz. (a) Images of the signal vs. pulse time. The colorbar corresponds to the value of IA−IDA 2(IB−IDB)−1 2. (b) IA−IDA 2(IB−IDB)−1 2vs. pulse time. The inset shows the cropped region used to average the signal. Blue points: Measurement. Black line: Fit to A·sin(Ωt) + B In Fig. 2.14 we represent the data plotted in Fig. 2.13 with respect to the polarization P. We can observe that there is a linear relation between θAand P. Chapter 2. In situ imaging of two-component BECs 91 By doing a linear fit we obtain that cFb/cFa =−1.01(25) as expected. Therefore, with this technique we can measure the difference in column densities between both spin states. Figure 2.14: Spin dependent imaging. We plot IA−IDA 2(IB−IDB)−1 2vs. polarization measured at B= 396 G and ∆FAR BEAT =−292 (blue circles). Black line: Fit to A·P+B. Red line: We plot sin2(A·P+B+π/4) for comparison. The insets show exemplary figures for all atoms in |ai, a balanced mixture and all atoms in |bi. 2.4 Conclusions and outlook In conclusion, we have developed an imaging method based on the Faraday effect in order to probe the real space distribution of an atomic cloud in situ at different magnetic fields. In particular, we have focused on the measurement of the effective Faraday coefficient at B = 57 G and B = 396 G in non-closed transitions. We have measured a Faraday coefficient for the |biand |cistates of cFb =−1.4·10−15rad ·m2and cFc =−1.1·10−15rad ·m2at low field and cFa =−1.3·10−15rad ·m2and cFb =−2.6·10−15rad ·m2for the |aiand |bi ground states at high field in fair agreement with the theoretical predictions. Chapter 2. In situ imaging of two-component BECs 92 This configuration allows us to measure the total column density of the cloud, regardless of the spin composition. Finally, we have explored a regime at B = 396 G in which the Faraday coefficients of the |aiand |bistates are opposite and allow to measure the difference in column density. Currently, we are working to extend this technique in order to measure both the sum and difference of column densities. Performing a non-destructive imaging pulse further from resonance would allow to measure the sum. A second imaging pulse in the middle of both transitions would allow to measure the difference. Since lensing effects may play an important role closer to resonance, we will consider performing highly saturated absorption imaging exactly on resonance to measure the composition of a single state and be able to retrieve the difference in column densities. The implementation of this method remains as a future perspective.