Bose-Einstein Condensation, Superfluidity and Superconductivity
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Bose-Einstein Condensation, Superfluidity and Superconductivity Prof.dr. Jacques Tempere Universiteit Antwerpen Faculteit Wetenschappen Departement Fysica ii Contents Introduction ix 1 Bose-Einstein Condensation 1 1.1Theexperimentalsearchforthenewphase............ 3 1.2TheidealBosegas.......................... 9 1.2.1 TheideabehindBEC.................... 9 1.2.2 Densityofstates....................... 11 1.2.3 Transitiontemperature................... 13 1.2.4 Condensatefraction..................... 15 1.2.5 Condensatedensityandvelocitydistribution....... 16 1.2.6 Thermodynamicquantities................. 17 1.3ThePenrose-OnsagercriterionforBEC.............. 19 1.4TheGross-Pitaevskiiequation.................... 22 1.4.1 Derivation........................... 23 1.4.2 SolutionfortheidealBosegas............... 25 1.4.3 TheThomas-Fermiapproximation............. 25 1.4.4 Coherencelength....................... 28 1.5Condensatedynamics........................ 30 1.5.1 TimedependentGross-Pitaevskiiequation......... 30 1.5.2 Velocityasaphasegradient................. 31 1.5.3 Hydrodynamicequations.................. 32 1.5.4 Superfluidity......................... 37 1.6Vortices................................ 37 1.6.1 Quantizationofthecirculation............... 37 1.6.2 Vortices............................ 40 1.6.3 Gross-Pitaevskiiinarotatingframe............ 42 1.6.4 Criticalrotationfrequencyinabigbucket......... 45 1.6.5 Abrikosovlattice....................... 47 1.6.6 Hess-Fairbank effect and fieldcooling........... 49 1.7Condensatesatnon-zerotemperatures............... 49 1.7.1 Thermalcloudversuscondensate.............. 49 1.7.2 Bogoliubovexcitations.................... 52 1.7.3 Excitation spectrum for the homogeneous Bose gas . 54 1.7.4 ThermodynamicsoftheinteractingBosegas....... 57 iii iv CONTENTS 2 Superfluid helium 59 2.1 The discovery of superfluidity.................... 59 2.1.1 Helium-II........................... 59 2.1.2 Superflow........................... 62 2.1.3 Andronikashvili torsional oscillator experiment ...... 63 2.1.4 Aretheresupersolids?.................... 64 2.2 The two-fluidmodel......................... 66 2.2.1 Fountain effect........................ 68 2.2.2 Superfluidcreep....................... 69 2.2.3 Secondsound......................... 70 2.3Microscopictheoryofhelium.................... 72 2.3.1 Fluctuationexpansion.................... 72 2.3.2 Equivalencebetweenbosonsandoscillators........ 75 2.3.3 Bogoliubov excitations in helium .............. 76 2.3.4 Thephonon-rotonspectrum................. 78 2.4 The Landau criterion for frictionless flow.............. 81 3 Superconductivity 85 3.1Phenomenologyofsuperconductors................. 85 3.1.1 Perfect conduction and Meissner effect........... 85 3.1.2 Thenon-classicalnatureofsuperconductivity....... 86 3.1.3 The London equations and the penetration depth . 88 3.1.4 Superconducting films.................... 90 3.1.5 ThePippardcorrelationlength............... 93 3.2Keyexperiments........................... 96 3.2.1 Microwaveabsorption/ACresistivity........... 96 3.2.2 Fluxquantization...................... 97 3.2.3 Specificheatanomaly.................... 98 3.2.4 Isotope effect......................... 99 3.3Penrose-Onsager-Yangcriterion...................100 3.3.1 Densitymatrices.......................101 3.3.2 Paircondensationoffermions................102 3.3.3 Energyofthepaircondensate................104 3.4Ginzburg-Landauformalism.....................106 3.4.1 Ginzburg-Landauenergyfunctional............106 3.4.2 Ginzburg-Landauequations.................107 3.4.3 EmpiricaldeterminationoftheGLparameters......109 3.5TypeIandtypeIIsuperconductivity................111 3.5.1 Coherencelength.......................111 3.5.2 Superconductivity at interfaces ...............112 3.5.3 Superconducting Vortices . .................115 3.6BCStheory..............................121 3.6.1 BCSHamiltonian ......................121 3.6.2 Bogoliubov transformation .................125 3.6.3 BCSgroundstate,gapandexcitationspectrum......127 3.6.4 Critical magnetic fieldanddensityofstates........128 3.6.5 Gapequation.........................133 3.6.6 Criticaltemperature.....................134 3.6.7 ResultsandchallengesforBCStheory ..........137 CONTENTS v A Thermodynamics of magnetism 141 A.1GibbsversusHelmholtz.......................141 A.2Ginzburg-LandauGibbsenergy...................143 A.3Vortexenergyandpotential.....................144 B Superfluidity in Helium-3 147 B.1BCSdescriptionofhelium-3.....................147 B.2Singletandtripletorderparameter.................149 B.3 A and B phases of superfluidhelium-3...............150 Bibliography 155 vi CONTENTS Preface In these course notes I attempt to give an overview of the physics of (i) Bose-Einstein condensates in dilute atomic gases, (ii) superfluid helium, and (iii) superconductivity. While the choice of material and presentation is my own, I like to acknowledge hand-picking nice ideas for presentations from a score of excellent textbooks, that I list in the biography. I would encourage you to also consult these textbooks — they can in particular give additional clues to solving the problem sets. vii viii PREFACE Introduction The world around us, observed and experienced on the human scale, is very well described by classical theories (and a bit of Pauli exclusion). While quantum mechanics is (at the moment) the fundamental theory on which we build our insight into the world, its counterintuitive predictions usually remain hidden deep into the microscopic world of atoms. At the macroscopic scale quantum effects usually only appear in the form of subtle refinements of classical theories, brought about by intricate experiments. To bring out the quantum nature of matter these experiments peer at the microscopic scale. This course discusses another way to probe quantum mechanics, namely by transfering the quantum aspect of the microscopic world on to macroscopic objects, making those objects behave in the same manner as the quantum particles they are built up from. This is, condensed in one sentence, what superfluids and superconductors are — in this state of matter, the appropriate description is a macroscopic wave function. Until 1995 there were only two macroscopic quantum states available: superfluid helium and superconducting solids. Since then, a third realization appeared: ultracold dilute atomic gases. The latter offers an unexpected level of experimental control: it is a very pure state whose geometry, interaction strength, and composition can be very accurately tuned by experimentalists. It has since then been used to build systems that correspond to specificmodel Hamiltonians. These Hamiltonians can now be studied and manipulated not only by analytical calculation or by computer simulation, but “in vivo”, and they can be brought in conditions hitherto unavailable. So, not only fundamental tests of quantum mechanics are made possible, but also new territory has been opened up for exploration, in particular regarding the many-body aspects of quantum mechanics. That is why in this course we will spend a lot of attention to this new and exciting system — the first chapter focuses entirely on it. In that chapter we introduce and discuss the concept of Bose-Einstein condensation. While this concept and the properties engendered by it are most clearly learnt from the quantum gases, they are also at the basis of superfluidity in liquid helium and of superconductivity. The difference lies in the nature of the interactions and the building blocks. In superfluid helium-4, the system is no longer dilute, and the role of interactions is more prominent. We discuss this system in the second chapter. Finally, in superconductors, electrons exhibit macroscopic quantum behavior — but in order to do so they have to form pairs first. Here, the interactions are crucial. We investigate superconductivity in the third chapter. ix x INTRODUCTION Chapter 1 Bose-Einstein Condensation Classical physics started out as a study of the familiar world around us, on human length scales. But as our ability to probe larger or smaller scales got better, it became clear that classical physics had to be extended and corrected. Better telescopes shifted the boundary of the largest possible scale, and better microscopes or particle accelators reduced the smallest scales available. At the largest scale, Newtonian physics could not explain the precession of Mercury, and at the smallest scale, black-body radiation did not fitinwithclassical mechanics. The former required general relativity to be developed, and the latter led to quantum mechanics. Here, we focus on another boundary that physicists have been working on: the record of the coldest temperature. It is by moving that boundary ever closer to absolute zero that first superconductivity andthensuperfluidity were discovered. For these phenomena, the laws of classical physics do not apply, and it was found that quantum mechanics —under normal circumstances of temperature and pressure confined to the microscopic world— governs the behaviour of superconductors and superfluids on a macroscopic scale. To understand how low temperatures can bring out the quantum nature of matter, we start from Boltzmann’s classical description of a gas of atoms. The atoms are described as mass points with definite positions and momenta, drawn Figure 1.1: A gas of atoms, from Boltzmann to De Broglie, from classical to quantum mechanics. 1 2 CHAPTER 1. BOSE-EINSTEIN CONDENSATION Figure 1.2: Upon cooling, the wave packets spread out, until they overlap