SPRINGER BRIEFS IN PHYSICS Carlo F. Parker A Primer on Quantum Fluids 123 SpringerBriefs in Physics Editorial Board Egor Babaev, University of Massachusetts, Massachusetts, USA Malcolm Bremer, University of Bristol, Bristol, UK Xavier Calmet, University of Sussex, Brighton, UK Francesca Di Lodovico, Queen Mary University of London, London, UK Pablo Esquinazi, University of Leipzig, Leipzig, Germany Maarten Hoogerland, Universiy of Auckland, Auckland, New Zealand Eric Le Ru, Victoria University of Wellington, Kelburn, New Zealand Hans-Joachim Lewerenz, California Institute of Technology, Pasadena, USA James Overduin, Towson University, Towson, USA Vesselin Petkov, Concordia University, Montreal, Canada Charles H. Wang, University of Aberdeen, Aberdeen, UK Andrew Whitaker, Queen’s University Belfast, Belfast, UK www.com More information about this series at http://www.com/series/8902 www. Parker • A Primer on Quantum Fluids 123 www.
Parker Joint Quantum Centre Joint Quantum Centre (JQC) Durham-Newcastle (JQC) Durham-Newcastle School of Mathematics and Statistics, School of Mathematics and Statistics, Newcastle University Newcastle University Newcastle upon Tyne Newcastle upon Tyne UK UK ISSN 2191-5423 ISSN 2191-5431 (electronic) SpringerBriefs in Physics ISBN 978-3-319-42474-3 ISBN 978-3-319-42476-7 (eBook) DOI 10.1007/978-3-319-42476-7 Library of Congress Control Number: 2016945839 © The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland www.com Preface This book introduces the theoretical description and properties of quantum fluids. The focus is on gaseous atomic Bose–Einstein condensates and, to a minor extent, superfluid helium, but the underlying concepts are relevant to other forms of quantum fluids such as polariton and photonic condensates.
The book is pitched at the level of advanced undergraduates and early postgraduate students, aiming to provide the reader with the knowledge and skills to develop their own research project on quantum fluids. Indeed, the content for this book grew from introductory notes provided to our own research students. It is assumed that the reader has prior knowledge of undergraduate mathematics and/or physics; otherwise, the concepts are introduced from scratch, often with references for directed further reading. After an overview of the history of quantum fluids and the motivations for studying them (Chap.
1), we introduce the simplest model of a quantum fluid provided by the ideal Bose gas, following the seminal works of Bose and Einstein (Chap. The Gross–Pitaevskii equation, an accurate description of weakly interacting Bose gases at low temperatures, is presented, and its typical time-independent solutions are examined (Chap. We then progress to solitons and waves (Chap. 4) and vortices (Chap.
5) in quantum fluids. For important aspects which fall outside the scope of this book, e. modelling of Bose gases at finite temperatures, we list appropriate reading material. Each chapter ends with key exercises to deepen the understanding.
Detailed solutions can be made available to instructors upon request to the authors. We thank Nick Proukakis and Em Rickinson for helpful comments on this work. Newcastle upon Tyne, UK Carlo F. Barenghi April 2016 Nick G.com Contents 1 Introduction .1 Towards Absolute Zero .1 Discovery of Superconductivity and Superfluidity .2 Bose–Einstein Condensation .2 Ultracold Quantum Gases .1 Laser Cooling and Magnetic Trapping .2 Bose–Einstein Condensate à la Einstein .3 Degenerate Fermi Gases .3 Quantum Fluids Today.
8 2 Classical and Quantum Ideal Gases .3 Ideal Classical Gas.1 Macrostates, Microstates and the Most Likely State of the System .2 The Boltzmann Distribution .2 Bosons and Fermions .3 The Bose–Einstein and Fermi-Dirac Distributions .5 The Ideal Bose Gas .1 Continuum Approximation and Density of States .2 Integrating the Bose–Einstein Distribution.3 Bose–Einstein Condensation .4 Critical Temperature for Condensation .6 Particle-Wave Overlap .com viii Contents 2.10 Ideal Bose Gas in a Harmonic Trap .6 Ideal Fermi Gas. 30 3 Gross-Pitaevskii Model of the Condensate.1 The Gross-Pitaevskii Equation .1 Mass, Energy and Momentum .2 Time-Independent GPE .3 Fluid Dynamics Interpretation .4 Stationary Solutions in Infinite or Semi–infinite Homogeneous Systems .2 Condensate Near a Wall .5 Stationary Solutions in Harmonic Potentials .2 Strong Repulsive Interactions .4 Anisotropic Harmonic Potentials and Condensates of Reduced Dimensionality .6 Imaging and Column-Integrated Density .7 Galilean Invariance and Moving Frames .2 Harmonically-Trapped Condensate. 52 4 Waves and Solitons .1 Dispersion Relation and Sound Waves .2 Landau’s Criterion and the Breakdown of Superfluidity .2 Expansion of the Condensate.1 Dark Soliton Solutions .2 Particle-Like Behaviour .4 Motion in a Harmonic Trap .5 Experiments and 3D Effects .com Contents ix 4.2 Experiments and 3D Effects. 77 5 Vortices and Rotation .3 Classical Versus Quantum Vortices .4 The Nature of the Vortex Core .5 Vortex Energy and Angular Momentum .6 Rotating Condensates and Vortex Lattices.7 Vortex Pairs and Vortex Rings .1 Vortex-Antivortex Pairs and Corotating Pairs .3 Vortex Pair and Ring Generation by a Moving Obstacle .8 Motion of Individual Vortices .1 Three-Dimensional Quantum Turbulence .2 Two-Dimensional Quantum Turbulence .13 Vortices of Infinitesimal Thickness .1 Three-Dimensional Vortex Filaments .2 Two-Dimensional Vortex Points.
109 Appendix A Simulating the 1D GPE .com Acronyms List of Acronyms 1D One-dimensional 2D Two-dimensional 3D Three-dimensional BEC Bose–Einstein condensate GPE Gross–Pitaevskii equation LIA Local induction approximation List of Symbols A Wavefunction amplitude A Vector potential aj Scaling solution velocity coefficients, j ¼ x; y; z a0 Vortex core radius as s-wave scattering length bj Scaling-solution variables, j ¼ x; y; z or j ¼ r; z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B Dark soliton coefficient B ¼ 1 u2 =c2 b Irrotational flow amplitude c Speed of sound CV Heat capacity at constant volume d Average inter-particle distance D System size D Number of dimensions b ej Unit vector. j ¼ x; y; z for Cartesian coordinates or j ¼ r; z; h for cylindrical polar coordinates e Small parameter E Energy E0 Energy per unit mass g Flow angle xi www.com xii Acronyms fj Distribution function, with j ¼ B, BE or FD for the Boltzmann, Bose– Einstein or Fermi–Dirac distributions F Free energy gi Degeneracy of i’th energy level g Density of states, gðEÞ or gðpÞ g GPE nonlinear coefficient R 1 CðxÞ The Gamma function, C ¼ 0 tx1 et dt H0 Cylinder height h Planck’s constant, h ¼ 6:63 1034 m2 kg=s k Wavenumber kB Boltzmann’s constant, kB ¼ 1:38 1023 m2 kgs2 K1 j Quantum of circulation k Trap ratio, xz =xr , of a cylindrically symmetric harmonic trap L Vortex line density Lz Angular momentum about z ‘ Wavepacket size (Chap. 4) ‘ Average inter-vortex distance (Chap. 5) pffiffiffiffiffiffiffiffiffiffiffiffiffi ‘j Harmonic oscillator length ‘j ¼ h=mxj in jth dimension k Wavelength, including de Broglie wavelength kdB m Mass l Chemical potential n Number density N Number of particles, including critical number of particles Nc , and number of particles in i’th level, Ni N ps Number of phase space cells x Angular frequency, e.
of wave or trap x Vorticity X Rotation frequency XðzÞ Complex potential p Momentum (vector p, magnitude p) P Condensate momentum P Pressure, including quantum pressure P0 Pr Probability / Velocity potential w; W Condensate wavefunction q Vortex charge q Mass density r Radial coordinate, r 2 ¼ x2 þ y2 þ z2 or r 2 ¼ x2 þ y2 Rj Thomas–Fermi radius in jth dimension R Local radius of curvature R0 Cylinder radius r Variational width S Phase distribution S Entropy of vortex configuration (Sect.com Acronyms xiii t Time T Temperature, including critical temperature for BEC, Tc u Soliton speed U Internal energy U Inter-atomic interaction potential v Fluid velocity v0 Frame velocity V Trapping potential V Volume W Number of macrostates n Healing length ns Bright soliton lengthscale fðxÞ P 1 The Riemann zeta function, fðxÞ ¼ 1 px p¼1 www.com Chapter 1 Introduction Abstract Quantum fluids have emerged from scientific efforts to cool matter to colder and colder temperatures, representing staging posts towards absolute zero (Fig. They have contributed to our understanding of the quantum world, and still captivate and intrigue scientists with their bizarre properties. Here we summarize the background of the two main quantum fluids to date, superfluid helium and atomic Bose–Einstein condensates.1 Towards Absolute Zero The nature of cold has intrigued humankind. Its explanation as a primordial substance, primum frigidum, prevailed from the ancient Greeks until Robert Boyle pioneered the scientific study of the cold in the mid 1600s.
Decrying the “almost totally neglect” of the nature of cold, he set about hundreds of experiments which systematically disproved the ancient myths and seeded our modern understanding. While working on an air-based thermometer in 1703, French physicist Guillaume Amontons observed that air pressure was proportional to temperature; extrapolating towards zero pressure led him to predict an “absolute zero” of approximately −240 ◦ C in today’s units, not far from the modern value of −273. The implication was profound: the realm of the cold was much vaster than anyone had dared believe. An entertaining account of low temperature exploration is given by Ref.
The liquefaction of the natural gases became the staging posts as low temperature physicists, with increasingly complex apparatuses, raced to explore the undiscovered territories of the “map of frigor”. Chlorine was liquefied at 239 K in 1823, and oxygen and nitrogen at T = 90 and 77 K, respectively, in 1877. In 1898 the English physi- cist James Dewar liquefied what was believed to be the only remaining elementary gas, hydrogen, at 23 K, helped by his invention of the vacuum flask. Concurrently, however, chemists discovered helium on Earth.
Although helium is the second most common element in the Universe and known to exist in the Sun, its presence on Earth is tiny. With helium’s even lower boiling point, a new race was on. A dramatic series of lab explosions and a lack of helium supplies meant that Dewar’s main competitor, © The Author(s) 2016 1 C. Parker, A Primer on Quantum Fluids, SpringerBriefs in Physics, DOI 10.com 2 1 Introduction Fig.1 Timeline of the coldest engineered temperatures, along with some reference temperatures Heike Kamerlingh Onnes, pipped him to the post, liquifying helium at 4 K in 1908.
This momentous achievement led to Onnes being awarded the 1913 Nobel Prize in Physics.1 Discovery of Superconductivity and Superfluidity These advances enabled scientists to probe the fundamental behaviour of materials at the depths of cold. Electricity was widely expected to grind to a halt in this limit. Using liquid helium to cool mercury, Onnes instead observed its resistance to simply vanish below 4 K. Superconductivity, the flow of electrical current without resistance, has since been observed in many materials, at up to 130 K, and has found applications in medical MRI scanners, particle accelerators and levitating “maglev” trains.
Onnes and his co-workers also observed unusual behaviour in liquid helium itself.2 K its heat capacity undergoes a discontinuous change, termed the “lambda” transition due to the shape of the curve. Since such behaviour is char- acteristic of a phase change, the idea developed that liquid helium existed in two phases: helium I for T > Tλ and helium II for T < Tλ , where Tλ is the critical tem- perature.