DOCSLIB.ORG

Hydrodynamics of the Bose-Einstein Condensate According to the Gross-Pitaevskii Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Hydrodynamics of the Bose-Einstein Condensate According to the Gross-Pitaevskii Theory Hydrodynamics of the Bose-Einstein condensate according to the Gross-Pitaevskii theory Author: Agisilaos Papatheodorou Supervisor: Ch.C. Moustakidis, Assosiate Professor Department of Physics, Aristotle University of Thessaliniki February 13, 2020 Abstract We consider a dilute and ultra-cold bosonic gas of particles. Within the framework of statistical physics, we derive the thermodynamic equations that dictate the gas's behavior for non-interacting par- ticles. With the use of quantum mechanics the addition of inter- particle interaction through s-wave scattering is possible obtain- ing the Gross-Pitaevskii equation. By investigating the weakly- interacting boson gas's characteristics we define it's size and shape. Lastly we draw a connection from the quantum mechanical equa- tions to those of fluid dynamics and we utilize these hydrodynamic properties to examine the gas's behavior and attributes. i PerÐlhyh O σκοπός αυτής thc ergasÐac eÐnai h αναπαραγωγή thc sta- tisτικής jewrÐac kai thc jewrÐac Gross-Pitaevskii gia thn pe- rigrafή twn idioτήτwn ενός sumpukn¸matoc Bose-Einstein. Ta sumpukn¸mata Bose-Einstein eÐnai mÐa μορφή ύlhc apoτελούμε- nh από swmatÐdia me akèraia τιμή κβαντικού ariθμού idiostro- φορμής, λεγόμενα μποζόnia, ta opoÐa èqoun yuqjeÐ se jermo- krasÐec kontά se αυτή tou apόlutou μηδενός. Ta swmatÐdia autά, kaj¸c h jermokrasÐa teÐnei na mhdenisteÐ, ωθούντai ma- ζικά sthn βασική touc ενεργειακή katάσtash gegoνός όpou katasteÐ εύφορη mÐa plhj¸ra kbantik¸n fainomènwn upaÐtia gia thn αλλαγή twn idioτήτwn tou ulikoύ. Melet¸ntac thn allhlepÐdrash se upoatομική klÐmaka twn swmatidÐwn όπou apoτελούν to mpozoνικό aèrio se qamhlèc jermokrasÐec sto- χεύουμε sthn παραγωγή twn αξιωμάτwn pou dièpoun tic ιδιόth- tec tou sumpukn¸matoc Bose-Einstein. Αυτή h diadikasÐa apo- dÐdei montèla ta opoÐa diakrÐnontai se aèria mpozonÐwn ta opo- Ða den αλληλεπιδρούν metαξύ touc kai se aèria mpozonÐwn όπου αλληλεπιδρούν metαξύ touc. To montèlo mh-allhlepidr¸ntwn swmatidÐwn me akèraia idiosτροφορμή, to opoÐo eÐnai to pr¸to όπου anaparάγουμε se αυτή thn ergasÐa, αναπτύχθηκε από touc Satyendra Nath Bose kai Albert Einstein en èth 1924, mèsw thc jewrÐac thc statisτικής φυσικής. 'Eqei wc stόqo thn periγραφή thc συμπεριφοράς twn mh-allhlepidr¸ntwn kai mh-diakrit¸n mpozonÐwn katά thn αλλαγή thc jermokrasÐac tou aerÐou. Qrhsimopoi¸ntac statistikèc exis¸sewn gia thn perigrafή thc ενεργειακής katανομής twn mpozonÐkwn sw- matidÐwn, eÐnai δυνατή h εξαγωγή twn jermodunamik¸n exi- s¸sewn pou periγράφουν èna συμπύκνωμα apoτελούμενο a- po tètoia swmatÐdia. H εξάρτηση twn makroskopik¸n poso- τήτwn tou sumpukn¸matoc από thn jermokrasÐa parèqei mia pr¸th εικόna gia thn φύση tou susτήματoc wstόσο παράγει kai asunèqeiec sto shmeÐo metάβασης tou aerÐou από thn fu- sioloγική tou katάσtash se αυτή tou sumpukn¸matoc mpo- zonÐwn. Me σκοπό thn èntaxh twn swmatidiak¸n allhlepi- ii δράσεων, gegoνός όπου εδράζετai sthn pragmatikόthta, a- ναπτύσουμε mèsa apό kbantomhqanikèc exis¸seic thn jewrÐa Gross-Pitaevskii kai thn sunoνόματη exÐswsh. To δυναμικό al- lhlepÐdrashc pou qrhsimopoiούμε eÐnai autό thc skèdashc δύο swmatidÐwn gia μηδενική sτροφορμή, to opoÐo kai upologÐzou- me gia tic αλληλεπιδράσειc metαξύ κάθε mpozonÐou tou sumpu- kn¸matoc me κάθε allό sto Ðdio σύσthma swmatidÐwn. H λύση thc μη-γραμμικής diaforikèc exÐswshc Gross-Pitaevskii, pou προκύπτει, kajistά perÐplokh kai δυσνόητη thn peraitèrw a- νάλυση thc αναλυτικής λύσης tou sumpukn¸matoc. Gia na ξεπεράσουμε autό to εμπόδιo qrhsimopoiούμε prosseggÐseic όπως h Thomas-Fermi prosèggish, gegoνός pou epitrèpei ton prosdioriσμό twn qarakthristik¸n tou sumpukn¸matoc όπως to mègejoc kai thn συνεκτικόthta ston q¸ro tou. Ta axi¸ma- iii ta αυτής thc prosèggishc basÐzontai ston auxhmèno ariθμό swmatidÐwn entός tou sumpukn¸matoc όπως kai thn elaqi- stopoihmènh jermokrasÐa gia na apλοποιήσουν thn exÐswsh Gross-Pitaevskii afair¸ntac ton όρο thc kiνητικής enèrgeiac twn mpozonik¸n swmatidÐwn. 'Eqontac εξάγei tic sqèseic pou kajorÐzoun ta βασικά qarakthrisτηκά megèjh ενός aerÐou al- lhpedr¸ntwn mpozonÐwn, παράγουμε tic aparaÐthtec udrodu- namikèc exis¸seic gia thn periγραφή thc kÐnhshc ενός sumpu- kn¸matoc ston q¸ro wc reustό, mèsw thc kumatοσυνάρτη- shc tou susτήματoc swmatidÐwn όποu to apoτελούν. Basi- ζόμενοι sta παραπάνω meletούμε thn διάδοση twn mhqanik¸n κυμάτwn entός tou sumpukn¸matoc, όπου παρατηρούμε fai- νόμενα upèrreusthc συμπεριφοράς. To apotèlesma to opoÐo mac odhgeÐ se autό to sumpèrasma eÐnai h metaboλή thc ener- γειακής εξάρτησης se sqèsh me to kumatοδιάνυσμα thc ορμής twn swmatidÐwn από τετραγωνική εξάρτηση se gramμική. To sumpèrasma autό brÐsketai se sumfwnÐa me tic peiramatikèc μετρήσειc, όσοn αφορά thn uperreustόthta yugmènwn mpozo- nÐwn, me ta pr¸ta peirάμata se sqèsh me autό to faiνόμενο na eÐnai autό tou Pyotr Kapitsa kai tou John F. Allen oi opo- iv Ðoi xeqwristά παρατήρησαν upèrreusth συμπεριφορά se υγρό ήλιo, en èth 1938. Me thn ανάλυση Gross-Pitaevskii aitiologe- Ðtai αυτή h upèrreusth μακροσκοπικής klÐmakac συμπεριφορά arqÐzontac από upoatομικής klÐmakac swmatidiakèc allhle- πιδράσειc prosjètontac βασικό θεωρητικό υπόβαθρο gia thn πλήρη katανόηση tou fainomènou. Tèloc axioloγούμε thn je- wrÐa Gross-Pitaevskii wc ergaleÐo melèthc συμπυκνωμάτwn a- ntiparajètontάς thn me progenèsterec όπως kai metagenèste- rec jewrÐec kai anadÐdontac thn sumboλή thc sthn θεωρητική kai πειραματική φυσική. v Introduction The Bose-Einstein condensate is a state of matter of bosonic par- ticles cooled to near absolute zero temperature. These conditions drive a large fraction of the subatomic particles to the ground en- ergy state and perplex their wavefunctions. These microscopic quantum phenomena translate into different macroscopic proper- ties as superconductivity and superfluidity. The Gross-Pitaevskii theory, developed by Eugene P. Gross and Lev Petrovich Pitaevskii in 1961, is a theoretical model based on the Hartee-Fock approxi- mation which approximately describes a Bose-Einstein condensate of interacting bosons in their ground state through a pseudopo- tential. This theory is able to produce a non-linear differential equation called the Gross-Pitaevskii equation for the whole con- densate. This allows for hydrodynamic equations to be extracted and further theoretical analysis of the phenomenon. vi CONTENTS Contents Abstract i PerÐlhyh ii Introduction vi 1 General Information 1 1.1 Historical Context . 1 1.2 Bose-Einstein condensation criteria . 3 1.3 Bose-Einstein distribution . 3 1.4 Generalized density of states . 5 1.4.1 Free particle density of states . 5 1.4.2 Harmonic-Oscillator density of states . 6 1.5 Bosonic interaction . 7 1.5.1 Scattering length . 7 1.5.2 Effective interaction . 9 1.6 Hartee-Fock approximation . 11 1.7 Fermion condensates - Cooper pairs . 11 2 Bose-Einstein condensation statistics 13 2.1 Transition temperature . 13 2.2 Condensed fraction . 14 2.3 Thermodynamic Quantities . 15 2.3.1 Condensed phase . 15 2.3.2 Normal phase . 17 2.3.3 Continuity close to the transition temperature . 18 3 Gross-Pitaevskii theory 20 3.1 Gross-Pitaevskii equation . 20 3.2 Cloud of trapped bosons in their ground state . 23 3.2.1 Thomas-Fermi approximation . 23 3.2.2 Healing of the condensate . 26 3.3 Non-Zero temperatures . 28 4 Hydrodynamics of the condensate 30 4.1 Hydrodynamic equations . 30 4.2 Time-dependent Gross-Pitaevskii equation . 30 vii LIST OF FIGURES 4.3 Derivation of the Hydrodynamic equations . 31 4.3.1 Continuity equation . 31 4.3.2 Momentum conservation equation . 32 4.4 Condensate sound-waves . 33 5 Conclusion 37 List of Figures 1 (Provided by: Wikipedia)Electrical impedance of mercury in low temperatures. 2 2 (Provided by: Wikimedia Commons)Distribution functions of Fermi-Dirac and Bose-Einstein statistics. 5 3 (Vladimir V. Meshkov, Andrey V. Stolyarov, Robert J Le Roy, Rapid, accurate calculation of the s-wave scattering length, 2011) The wave-function in comparison to the two particles distance during s-wave scattering. 9 4 (Karmela Padavi´c,Cooper Pairs, Superconductivity and Flash Mobs, 2018) Electrons forming a Cooper pair through a crys- talline lattice. 12 5 (Matthew J. Davis: Microcanonical temperature for a classical field: Application to Bose-Einstein condensation, 2003)The specific heat C as a function of T , for different values of a. 19 Tc 6 (Cornell group, 2010) BEC under rapid rotation according to the Gross-Pitaevskii equations. 24 7 (Carlo F. Barenghi, Introduction to quantised vortices and turbulence) The distance from the rigid wall in comparison to the wave-function, where the healing or coherence length is denoted a0 instead of ξ as in our calculations. 28 8 (Clement Jones, Paul H. Roberts, Motions in a Bose con- densate. IV. Axisymmetric solitary waves, 1982) Energy to momentum-wave vector spectrum, where the dotted line is the small value ~q contribution and the continuous one the high value ~q contribution. 35 viii 1 General Information 1.1 Historical Context The first experimental evidence of BEC1 came from Kammerlingh Onnes2 of Holland's Leiden University. Onnes was the first to liquify Helium at 4.14 kelvin in 1908 following his famous tutor's, Johannes Van der Waals, work on the equation of state of real gasses. He later did further studies involving different properties of metals such as mercury, tin and lead in extremely low temperatures, of around 1 degree kelvin, and especially aimed his attention at electrical impedance which was thought to tend to infinity as the temper- ature drops to zero. This was based on the hypothesis that electrons move through the conductor
Load more

Recommended publications
  • Book of Abstracts
    Book of Abstracts 3 Foreword The 23 rd International Conference on Atomic Physics takes place in Ecole Polytechnique, a high level graduate school close to Paris. Following the tradition of ICAP, the conference presents an outstanding programme of invited VSHDNHUVFRYHULQJWKHPRVWUHFHQWVXEMHFWVLQWKH¿HOGRIDWRPLFSK\VLFVVXFKDV Ultracold gases and Bose Einstein condensates, Ultracold Fermi gases, Fundamental atomic tests and measurements, Precision measurements, atomic clocks and interferometers, Quantum information and simulations with atoms and ions, Quantum optics and cavity QED with atoms, Atoms and molecules in optical lattices, From two-body to many-body systems, Ultrafast phenomena and free electron lasers, Beyond atomic physics (biophysics, optomechanics...). The program includes 31 invited talks and 13 ‘hot topic’ talks. This book of abstracts gathers the contributions of these talks and of all posters, organized in three sessions. The proceedings of the conference will be published online in open access by the “European 3K\VLFDO-RXUQDO:HERI&RQIHUHQFHV´KWWSZZZHSMFRQIHUHQFHVRUJ On behalf of the committees, we would like to welcome you in Palaiseau, and to wish you an excit- ing conference. 4 Committees Programme committee Alain Aspect France Michèle Leduc France Hans Bachor Australia Maciej Lewenstein Spain Vanderlei Bagnato Brazil Anne L’Huillier Sweden Victor Balykin Russia Luis Orozco USA Rainer Blatt Austria Jian-Wei Pan China Denise Caldwell USA Hélène Perrin France Gordon Drake Canada Monika Ritsch-Marte Austria Wolfgang Ertmer Germany Sandro Stringari Italy Peter Hannaford Australia Regina de Vivie-Riedle Germany Philippe Grangier France Vladan Vuletic USA Massimo Inguscio Italy Ian Walmsley UK Hidetoshi Katori Japan Jun Ye USA Local organizing committee Philippe Grangier (co-chair) Institut d’Optique - CNRS Michèle Leduc (co-chair) ENS - CNRS Hélène Perrin (co-chair) U.
    [Show full text]
  • Aleksei A. Abrikosov 1928–2017
    Aleksei A. Abrikosov 1928–2017 A Biographical Memoir by M. R. Norman ©2018 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. ALEKSEI ALEKSEEVICH ABRIKOSOV June 25, 1928–March 29, 2017 Elected to the NAS, 2000 Shortly after the 2003 announcement that Aleksei Abrikosov had won the Nobel Prize in Physics, a number of colleagues took Alex to lunch at a nearby Italian restau- rant. During lunch, one of the Russian visitors exclaimed that Alex should get a second Nobel Prize, this time in Literature for his famous “AGD” book with Lev Gor’kov and Igor Dzyaloshinskii (Methods of Quantum Field Theory in Statistical Physics.) Somewhat taken aback, I looked closely at this individual and realized that he was deadly serious. Although I could imagine the reaction of the Nobel Literature committee to such a book (for a lay person, perhaps analogous to trying to read Finnegan’s Wake), I had to admit that my own copy of this book is quite dog-eared, having been put to good use over the By M. R. Norman years. In fact, you know you have made it in physics when your book gets a Dover edition. One of the most charming pictures I ever saw was a rare drawing in color that Alexei Tsvelik did (commissioned by Andrei Varlamov for Alex’s 50th birthday) that was proudly displayed in Alex’s home in Lemont, IL. It showed Alex with his fingers raised in a curled fashion as in the habit of medieval Popes.
    [Show full text]
  • CURRICULUM VITAE SANDRO STRINGARI Born in Italy on March
    CURRICULUM VITAE SANDRO STRINGARI Born in Italy on March 2nd, 1949 Married with two daughters. Italian citizenship EDUCATION, MAIN APPOINTMENTS AND AWARDS 1972 Laurea in Physics cum laude at the Università di Pisa and Scuola Normale Superiore 1972/1973 Fellowship at the Scuola Normale Superiore 1973-1976 Research assistant at the Università di Trento 1976 Visiting scientist at the Department of Theoretical Physics, Oxford (6 months) 1976-1980 Assistant professor at the Università di Trento 1978/1979 CNRS visitor at the Institute de Physique Nucleaire, Orsay 1980-1990 Associate professor at the Università di Trento 1985/1986 CNRS visitor at the Institute de Physique Nucleaire, Orsay 1990 Visiting scientist at the Laue-Langevin Institute, Grenoble (1 month) 1990 Full professor at the Università di Trento 2002 M.T. Messori Roncaglia and E. Mari prize of the Accademia dei Lincei 2002 Visiting professor at the Ecole Normale Superieure, Paris (1 month) 2004/2005 European Chair at the College de France, Paris 2007 Visiting professor at the Institut Henri Poincaré, Paris (1 month) 2008 Visiting professor at the Center for Ultracold Atoms, MIT-Harvard (2 months) ORGANIZATION ACTIVITIES (selection) 1992-1995 Chairman of the Physics Department, Università di Trento 1996/1997 Vice-Rector of the Università di Trento 1996-2000 National coordinator of the INFM Project "Experimental and Theoretical Investigation of Bose-Einstein Condensation in Alkali Atoms” 2002 Director of the INFM Research and Development BEC Center 2006 Member of the Evaluation Committee
    [Show full text]
  • Arxiv:1610.06882V2 [Astro-Ph.HE] 1 Mar 2017
    International Journal of Modern Physics D c World Scientific Publishing Company NEUTRON STARS IN THE LABORATORY VANESSA GRABER1;2 [email protected] NILS ANDERSSON1 [email protected] MICHAEL HOGG1 1Mathematical Sciences and STAG Research Centre, University of Southampton Southampton SO17 1BJ, United Kingdom 2Department of Physics and McGill Space Institute, McGill University, 3550 rue University, Montreal, QC, H3A 2T8, Canada Received Day Month Year Revised Day Month Year Neutron stars are astrophysical laboratories of many extremes of physics. Their rich phe- nomenology provides insights into the state and composition of matter at densities which cannot be reached in terrestrial experiments. Since the core of a mature neutron star is expected to be dominated by superfluid and superconducting components, observations also probe the dynamics of large-scale quantum condensates. The testing and under- standing of the relevant theory tends to focus on the interface between the astrophysics phenomenology and nuclear physics. The connections with low-temperature experiments tend to be ignored. However, there has been dramatic progress in understanding labo- ratory condensates (from the different phases of superfluid helium to the entire range of superconductors and cold atom condensates). In this review, we provide an overview of these developments, compare and contrast the mathematical descriptions of laboratory condensates and neutron stars and summarise the current experimental state-of-the-art. arXiv:1610.06882v2 [astro-ph.HE] 1 Mar 2017 This discussion suggests novel ways that we may make progress in understanding neutron star physics using low-temperature laboratory experiments. Keywords: PACS numbers: Contents 1. Neutron Stars and Fundamental Physics .
    [Show full text]
  • Sandro Stringari Books and Main Review Papers: Main Edited Books
    Sandro Stringari Books and main review papers: Theory of ultracold atomic Fermi gases Stefano Giorgini, Lev P. Pitaevskii, Sandro Stringari, cond-mat/0706.3360, to appear in Rev. Mod. Phys. (2008) Bose-Einstein Condensation Lev Pitaevskii and Sandro Stringari, Int. Series of Monographs on Physics (Oxford University Press) 2003 Theory of Bose-Einstein condensation in trapped gases Franco Dalfovo, Stefano Giorgini, Lev Pitaesvkii and Sandro Stringari, Reviews of Modern Physics 71 , 463 (1999) Sum Rules and Giant Resonances in Nuclei E. Lipparini and S. Stringari, Physics Reports 175, 103 (1989) Main edited books Bose Einstein Condensation in atomic gases Proceedings of the 1998 E. Fermi Summer School held in Varenna, M. Inguscio, S. Stringari and C. Weiman eds.(IOS Press, Amsterdam, 1999) Bose Einstein Condensation Proceedings of the 1993 international workshop held in Levico, A. Griffin, D. Snoke and S. Stringari eds. (Cambridge University Press, 1995) Publications 2008 Collisional properties of a polarized Fermi gas with resonant interactions G.M. Bruun, A. Recati, C.J. Pethcick, H. Smith and S. Stringari, arXiv:0804.1885 , Phys. Rev. Lett. The role of interactions in spin-polarized atomic Fermi gases at unitarity A. Recati, C. Lobo and S. Stringari, cond-mat/0803.4419, Phys. Rev. A Theory of ultracold atomic Fermi gases Stefano Giorgini, Lev P. Pitaevskii, Sandro Stringari, cond-mat/0706.3360, to appear in Rev. Mod. Phys. (2008) Destroying superfluidity by rotating a fermi gas at unitarity , I. Bausmerth, A. Recati and S. Stringari, Phys. Rev. Lett.. 100 070401 (2008) Tkachenko modes in a superfluid Fermi gas at unitarity Gentaro Watanabe, Marco Cozzini, and Sandro Stringari, Phys.
    [Show full text]
  • Leonid Keldysh Federico Capasso, Paul Corkum, Olga Kocharovskaya, Lev Pitaevskii, and Michael V
    Leonid Keldysh Federico Capasso, Paul Corkum, Olga Kocharovskaya, Lev Pitaevskii, and Michael V. Sadovskii Citation: Physics Today 70, 6, 75 (2017); doi: 10.1063/PT.3.3605 View online: https://doi.org/10.1063/PT.3.3605 View Table of Contents: https://physicstoday.scitation.org/toc/pto/70/6 Published by the American Institute of Physics ARTICLES YOU MAY BE INTERESTED IN Spartak Timofeevich Belyaev Physics Today 70, 72 (2017); https://doi.org/10.1063/PT.3.3602 Mildred S. Dresselhaus Physics Today 70, 73 (2017); https://doi.org/10.1063/PT.3.3603 Anthony Philip French Physics Today 70, 74 (2017); https://doi.org/10.1063/PT.3.3604 Hidden worlds of fundamental particles Physics Today 70, 46 (2017); https://doi.org/10.1063/PT.3.3594 The new Moon Physics Today 70, 38 (2017); https://doi.org/10.1063/PT.3.3593 Magnetic monopole search, past and present Physics Today 70, 12 (2017); https://doi.org/10.1063/PT.3.3571 became the chief lecturer and manager Leonid Keldysh Early in his career, in 1957–58, of a course taken by more than 500 Keldysh predicted an electric-field- MIT students. Subsequently, he became eonid Keldysh, a world-prominent induced shift of a semiconductor’s ab- associate head. L Russian theoretical physicist, passed sorption edge, now known as the Franz– The acronym for French’s course be- away from pneumonia on 11 Novem- Keldysh effect. It turned out to be of came notorious. French said, “I called it, ber 2016 in Moscow. He was one of the paramount importance in the develop- blandly, ‘Physics: A New Introductory most imaginative and influential physi- ment of a powerful spectroscopic tech- Course.’ .
    [Show full text]
  • The Discovery of Superfluidity Sebastien Balibar
    The discovery of superfluidity Sebastien Balibar To cite this version: Sebastien Balibar. The discovery of superfluidity. 2006. hal-00113242 HAL Id: hal-00113242 https://hal.archives-ouvertes.fr/hal-00113242 Preprint submitted on 11 Nov 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The discovery of superfluidity S´ebastien Balibar∗ Laboratoire de Physique Statistique de l’Ecole Normale Sup´erieure associ´eaux Universit´es Paris 6 et 7 et au CNRS 24 Rue Lhomond, 75231 Paris Cedex 05, France Superfluidity is a remarkable manifestation of quantum mechanics at the macroscopic level. This article describes the history of its discovery, which took place at a particularly difficult period of the twentieth century. A special emphasis is given to the role of J.F. Allen, D. Misener, P. Kapitza, F. London, L. Tisza and L.D. Landau. The nature and the importance of their respective contributions are analyzed and compared. Of particular interest is the controversy between Landau on one side, London and Tisza on the other, concerning the relevance of Bose-Einstein condensation to the whole issue, and also on the nature of thermal excitations in superfluid helium 4.
    [Show full text]
  • ¹ 2002 WILEY-VCH Verlag Gmbh & Co. Kgaa, Weinheim 1439-4235/02
    736 ¹ 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1439-4235/02/03/09 $ 20.00+.50/0 CHEMPHYSCHEM 2002, 3, 736 ± 753 When Atoms Behave as Waves: Bose ± Einstein Condensation and the Atom Laser (Nobel Lecture) Wolfgang Ketterle*[a] KEYWORDS: alkali elements ¥ Bose ± Einstein condensates ¥ cooling ¥ low temperature physics ¥ quantum optics The lure of lower temperatures has attracted physicists for the past century, and with each advance towards absolute zero, new and rich physics has emerged. Laypeople may wonder why ™freezing cold∫ is not cold enough. But imagine how many aspects of nature we would miss if we lived on the surface of the sun. Without inventing refrigerators, we would only know gaseous matter and never observe liquids or solids, and miss the beauty of snowflakes. Cooling to normal earthly temper- atures reveals these dramatically different states of matter, but this is only the beginning: Many more states appear with further cooling. The approach into the kelvin range was rewarded with the discovery of superconductivity in 1911 and of superfluidity in Figure 1. Annual number of published papers, which have the words ™Bose∫ and helium-4 in 1938. Cooling into the millikelvin regime revealed ™Einstein∫ in their title, abstracts, or keywords. The data were obtained by the superfluidity of helium-3 in 1972. The advent of laser cooling searching the ISI (Institute for Scientific Information) database. in the 1980s opened up a new approach to ultralow temperature physics. Microkelvin samples of dilute atom clouds were and with mass m can be regarded as quantum-mechanical generated and used for precision measurements and studies wavepackets that have a spatial extent on the order of a thermal of ultracold collisions.
    [Show full text]
  • [Physics.Hist-Ph] 13 Nov 2006 Mn U Olauswoddntwn Ihteeiesi Mind in Ideas Histor for the These with to About Superfluidity
    The discovery of superfluidity S´ebastien Balibar∗ Laboratoire de Physique Statistique de l’Ecole Normale Sup´erieure associ´eaux Universit´es Paris 6 et 7 et au CNRS 24 Rue Lhomond, 75231 Paris Cedex 05, France Superfluidity is a remarkable manifestation of quantum mechanics at the macroscopic level. This article describes the history of its discovery, which took place at a particularly difficult period of the twentieth century. A special emphasis is given to the role of J.F. Allen, D. Misener, P. Kapitza, F. London, L. Tisza and L.D. Landau. The nature and the importance of their respective contributions are analyzed and compared. Of particular interest is the controversy between Landau on one side, London and Tisza on the other, concerning the relevance of Bose-Einstein condensation to the whole issue, and also on the nature of thermal excitations in superfluid helium 4. In order to aid my understanding of this period, I have collected several testimonies which inform us about the work and attitude of these great scientists. PACS numbers : 67.40.-w,01.65.+g 1. Introduction Scientific progress has become a collective process. No physicist can ever pretend that he has achieved something, that he had had a personal idea or made any original discovery independently of his colleagues. Recognizing this situation does not mean that it is impossible to identify the authors of scientific discoveries, but one should do it carefully. Instead, there is some tendency to attribute discoveries to single persons, an attitude which is not arXiv:physics/0611119v1 [physics.hist-ph] 13 Nov 2006 fair enough.
    [Show full text]
  • Bose-Einstein Condensates in a Double Well Potential: a Route to Quantum Interferometry
    RUPRECHT-KARLS-UNIVERSITÄT HEIDELBERG Jens Appmeier Bose-Einstein condensates in a double well potential: A route to quantum interferometry Diplomarbeit HD-KIP-07-06 KIRCHHOFF-INSTITUT FÜR PHYSIK Faculty of Physics and Astronomy University of Heidelberg Diploma thesis in Physics submitted by Jens Appmeier born in Leipzig 2007 Bose-Einstein condensates in a double well potential: A route to quantum interferometry This diploma thesis has been carried out by Jens Appmeier at the Kirchhoff Institute of Physics under the supervision of Prof. Dr. M. K. Oberthaler Bose-Einstein Kondensate im Doppelmuldenpotential: Eine M¨oglichkeit der Quanten-Interferometrie Im Rahmen dieser Arbeit wurde eine neue optische Dipolfalle f¨urdie BEC Apparatur unserer Gruppe geplant und aufgebaut. Der Fallenlaser hat einen minimalen Radius von 5.15 µm und erm¨oglicht deshalb einen starken Einschluß der Atome. Die Konzeption und der Aufbau erfolgten derart, dass der erreichbare Parameterbereich des bereits ex- istierenden Doppelmuldenpotentials erweitert wurde. Ziel dabei war es das Fock-Regime zu erreichen. In diesem sind die Atomzahlschwankungen klein und es sollte m¨oglich sein ein Wiederkehren der relativen Phase zwischen den beiden Mulden des Potentials zu beobachten. Dies schließt die M¨oglichkeit ein, unter das statistische Limit (standard quantum limit) der Schwankungen zu gehen und w¨urdeneue M¨oglichkeiten f¨urhoch- genaue interferometrische Experimente er¨offnen. Infolge des starken Einschlusses der Falle werden hohe Atomdichten erreicht was zu einer hohen 3-K¨orper Verlustrate f¨uhrt. Es wird gezeigt das es durch diesen Verlustmechanismus m¨oglich wird die Atomzahl im Experiment sehr genau zu kontrollieren. Bose-Einstein condensates in a double well potential: A route to quantum interferometry In this work a new dipole trap was designed and added to the BEC apparatus of our group.
    [Show full text]
  • Lev Landau Was Never to Work Again, He Was Awarded the Nobel Prize Later That Year and His Legacy Lives on to This Day Lev Landau: Physicist and Revolutionary
    A near-fatal accident 40 years ago robbed theoretical physics of one its greatest minds. Although Lev Landau was never to work again, he was awarded the Nobel prize later that year and his legacy lives on to this day Lev Landau: physicist and revolutionary Alexei Kojevnikov LANDAU requires no introduction. He is too well known, al- RCHIVES though not quite as well understood. Several generations A ISUAL of theoretical physicists learned their trade by struggling V through the 10 volumes of his famous Course of Theoretical EGRÈ Physics – known colloquially as Landau and Lifshitz – which S MILIO Landau supervised, although he did not write a single word. AIP E Several dozen physicists – some absolutely first-class – contin- ued to identify themselves as members of the Landau school long after they became scientists in their own right. About a dozen landmark results in physics bear his name. If one were to choose his most important breakthroughs, these would probably be his theory of phase transitions (1937), the theory of superfluidity in liquid helium (1941), the Ginzburg– Landau phenomenological theory of superconductivity (1950) and the Landau “Fermi-liquid” theory (1956). But his fame is Lev Davidovich Landau (1908–1968) was one of the most influential certainly much greater than these concrete results alone might theoretical physicists of the 20th century, winning the 1962 Nobel Prize for suggest – even if one were to include other formulae, such as Physics for his pioneering theories of condensed matter. the Landau diamagnetism of free electrons (1930) or Landau damping in plasmas (1946). Landau’s published papers are so capital, to the middle-class Jewish family of a petroleum en- laconic – consisting mainly of formulae and a few categorical gineer.As a child, Landau was a prodigy and enfant terrible, and statements on what is right and what is wrong, without suf- to some extent he retained childish characteristics well into ficient explanation – that even professional physicists often adult life.
    [Show full text]
  • 'Stripe Phase' in a Dipolar Bose-Einstein Condensate
    Universit`adi Pisa DIPARTIMENTO DI FISICA Corso di Laurea Magistrale in Fisica Tesi di laurea Observation of a rotonic ’stripe phase’ in a dipolar Bose-Einstein condensate Relatori: Candidato: Dr. Andrea Fioretti Francesca Fam`a Matricola 507594 Prof. Giovanni Modugno Internal: Prof. Francesco Fuso Anno Accademico 2017-2018 Contents Introduction 5 1 Theory of dipolar Bose-Einstein condensates9 1.1 A consequence of quantum statistics......................... 9 1.2 Trapping potential................................... 10 1.3 An interacting BEC: Short Range Interaction.................... 14 1.3.1 Bogoliubov approximation and Gross-Pitaevskii equation ......... 15 1.3.2 Thomas Fermi Approximation and Solution of G-P stationary equation . 16 1.3.3 Three Body recombination .......................... 18 1.3.4 A tunable interaction: Feshbach resonances................. 19 1.3.5 A competition among energies ........................ 21 1.4 An interacting BEC: Long range and anisotropic interaction............ 22 1.4.1 DDI: Dipole-Dipole Interactions ....................... 24 1.4.2 Roton mode................................... 25 1.4.3 Instabilities.................................... 27 1.4.4 Experimental observation........................... 32 1.5 Beyond Mean-field effects ............................... 34 1.5.1 Quantum fluctuations ............................. 34 1.5.2 Quantum droplets ............................... 35 2 Obtaining a dipolar Bose Einstein condensate 39 2.1 Dysprosium....................................... 39 2.2 Theoretical
    [Show full text]