UNIVERSITY OF OSLO Department of Physics Bose-Einstein Condensates Numerical solution of the Gross-Pitaevskii equation using nite elements Master thesis Carl Joachim Berdal Haga June 2006 i Contents 1 Introduction 1 2 Background 5 2.1 History of Bose-Einstein condensation . 5 2.1.1 Prediction........................... 5 2.1.2 Superfluidity in liquid helium . 6 2.1.3 The search for Bose-Einstein condensation in dilute gases . 6 2.1.4 BEC in alkali atoms . 7 2.1.5 Further experiments . 8 2.2 Quantum particles . 9 2.2.1 Symmetry........................... 10 2.2.2 The Pauli exclusion principle . 12 2.3 Particle statistics . 12 2.3.1 Maxwell-Boltzmann statistics . 12 2.3.2 Fermi-Dirac statistics . 13 2.3.3 Bose-Einstein statistics . 13 2.3.4 Comparison at T → 0 .................... 14 3 Theory of Dilute Bose Gases 16 3.1 Many-particle theory . 16 3.1.1 The harmonic oscillator . 17 3.1.2 Energy basis quantisation . 17 3.1.3 Canonical quantisation . 19 3.1.4 Fock space representation . 19 3.1.5 A note on quasi-particles . 19 3.2 Homogenous Bose gas . 20 3.2.1 The uniform, ideal gas . 20 3.2.2 The weakly interacting gas . 21 3.2.3 Scattering theory . 22 3.2.4 Canonical quantisation . 24 3.2.5 Canonical (Bogoliubov) transformation . 27 3.2.6 Ground state energy . 28 3.2.7 On the validity of the hard-sphere approximation . 29 3.3 Trapped Bose gases . 30 ii Contents 3.3.1 The Gross-Pitaevskii equation . 30 3.3.2 Properties of the trapped Bose gas . 32 3.3.3 Critical temperature . 33 3.3.4 Healinglength......................... 33 3.3.5 The Thomas-Fermi approximation . 34 3.3.6 Scaling............................. 36 3.4 Dynamics of the trapped Bose gas . 37 3.4.1 Time evolution of the condensate . 37 3.4.2 Hydrodynamical formulation . 38 3.4.3 Vortexstates ......................... 38 4 Numerical Methods 41 4.1 The finite element method . 41 4.1.1 The general approach . 42 4.1.2 Elementwise formulation . 44 4.1.3 Example: The non-interacting Bose gas with linear elements 46 4.1.4 Different element types . 48 4.1.5 Example: The non-interacting Bose gas with quadratic elements . 50 4.1.6 Finite elements in higher dimensions . 50 4.1.7 Boundary conditions . 51 4.1.8 Convergence properties . 52 4.2 Linearalgebra............................. 57 4.2.1 Linear solvers for large sparse systems . 57 4.2.2 Eigensolvers . 64 4.2.3 Matrix representations and properties . 65 4.2.4 Continued example: The non-interacting Bose gas . 67 4.3 Nonlinear equations . 68 4.3.1 The model problem: The interacting Bose gas . 69 4.3.2 Successive substitutions . 69 4.3.3 Inverse power method . 69 4.3.4 Newton’s method . 70 4.3.5 Imaginary time propagation . 73 4.4 Timeevolution ............................ 75 4.4.1 Finite difference methods . 75 4.5 Excited states . 80 4.5.1 Choosing the initial state . 80 4.5.2 Externally applied phase . 80 4.5.3 Rotating lab frame . 81 5 Results 83 5.1 Verification .............................. 83 5.1.1 The linear problem . 83 5.2 Physical applications . 86 5.2.1 A stationary vortex lattice in 2D . 88 Contents iii 5.2.2 Splitting of a double-quantised vortex in 2D . 89 5.2.3 Splitting of a double-quantised vortex in 3D . 90 5.3 Numerical and algorithmic results . 90 5.3.1 Convergence of the nonlinear solvers . 91 5.3.2 Convergence of the linear solvers . 93 5.3.3 Stability of time evolution . 94 5.3.4 Parallel efficiency . 95 5.4 Visualisation.............................. 96 5.4.1 Two-dimensional condensates . 96 5.4.2 Three-dimensional condensates . 99 6 Conclusion 102 A Implementation Notes 105 A.1 Diffpack ................................ 105 A.2 The solver . 108 A.2.1 Numerical integration over the finite elements . 108 A.2.2 Building the linear system . 110 A.2.3 Nonlinear solvers . 112 A.2.4 Parallel execution . 113 A.2.5 Parameters .......................... 113 A.2.6 Arpack integration . 115 A.3 Post-processing . 115 B Auxilliary Results 117 B.1 Error estimation . 117 B.2 Results for linear solvers . 119 C Mathematical and Physical Topics 121 C.1 The virial theorem . 121 iv List of Figures 1.1 Diagram of the properties of particles in a gas . 2 2.1 Observations of BEC in sodium atoms . 9 2.2 Comparison of the Boltzmann, Fermi-Dirac and Bose-Einstein dis- tributions ............................... 14 2.3 Plot of the Boltzmann, Fermi-Dirac and Bose-Einstein distribu- tions relative to their lowest-energy state at successively falling temperatures ............................. 15 3.1 The quasi-particle spectrum . 28 3.2 Relative size of corrections to the energy expansion . 30 3.3 Illustration of the Gaussian and the Thomas-Fermi ground states 35 3.4 Rotation in regular and irrotational fluids . 39 4.1 Approximation of integration by piecewise interpolation . 42 4.2 Different views of linear interpolation . 45 4.3 The global matrix M, assembled from element matrices . 46 4.4 Common element-local basis functions in 1 dimension . 48 4.5 Hermite interpolating functions in 1 dimension . 49 4.6 Example of the global matrix assembly in 2D using square linear elements . 51 4.7 Ratio of nonzero matrix elements for a few different grid choices . 65 4.8 Example of a general sparse matrix representation . 66 5.1 The error in the eigenvalues and the virial theorem for the 1D harmonic oscillator . 84 5.2 Convergence of the nonlinear problem for different element sizes . 85 5.3 Ground state of the condensate in the large gas parameter regime 86 5.4 Single vortex state in the large gas parameter regime . 87 5.5 Density profile of vortex lattices . 88 5.6 Decay of double-quantised vortex in 2D, visualised using density mapping................................ 89 5.7 A decayed double-quantised vortex in 3D, visualised using inverse densitymapping............................ 90 5.8 Convergence of the nonlinear solvers on a weakly nonlinear problem 91 List of Figures v 5.9 Convergence of the nonlinear solvers on a strongly nonlinear problem 92 5.10 Evolution of conserved quantities in the Crank-Nicholson method overlongtime............................. 94 5.11 Maximum deviation from energy and norm . 94 5.12 Parallel scaling on up to 16 processors . 95 5.13 A single vortex in a 2D anisotropic trap . 97 5.14 Phase and density of the vortex lattice in figure 5.5 (p. 88) . 98 5.15 Density and velocity fields of a two-vortex system . 99 5.16 Double vortex in an elongated rotating trap . 100 5.17 Inverse density rendering of the single-vortex condensate . 101 5.18 Decay of double-quantised vortex visualised using phase mapping 101 A.1 Simplified class diagram . 106 B.1 Examples of extrapolation curves . 118 vi List of Tables 2.1 Limits of current cooling techniques . 7 3.1 Lengths scales under typical dilute conditions . 22 4.1 Properties of the matrices used in the Gross-Pitaevskii equation . 58 4.2 Estimated memory requirements for the various matrix represen- tations................................. 67 5.1 Ground state energies compared with reference finite difference calculations .............................. 86 5.2 Single vortex energies compared with reference finite difference calculations .............................. 87 B.1 Extrapolation of the ground state energies of the MGP to h → 0 . 118 B.2 Linear solver convergence for initial state on coarse grid, high initialerror .............................. 119 B.3 Linear solver convergence for initial state on fine grid, high initial error.................................. 119 B.4 Linear solver convergence for initial state on coarse grid, low initial error.................................. 120 B.5 Linear solver convergence for initial state on fine grid, low initial error.................................. 120 B.6 Linear solver convergence for time evolving state on coarse grid . 120 B.7 Linear solver convergence for time evolving state on fine grid . 120 vii Acknowledgments I would like to thank my supervisors for their support throughout my thesis work. Prof. Morten Hjorth-Jensen, my primary supervisor, for his help with physics questions and extensive writing support, as well as unflagging optimism; Prof. Hans Petter Langtangen, who wrote if not the book then at least a book on the finite element method, and whose intimate knowledge of diffpack saved the day on a few occasions; and Dr. Halvor Møll Nilsen, who knows well how to simu- late Bose-Einstein condensates, having done it a lot, and has corrected several of my mistakes, and suggested fruitful methods and avenues of investigation. 1 Chapter 1 Introduction Bose-Einstein condensation describes a peculiar state of matter that occurs when certain gases are cooled to a few billionths of a degree above absolute zero tem- perature.1 It is characterised by all atoms entering the same quantum state,2 thus merging to form what has been described as one super-atom. All matter obey the laws of quantum mechanics. Yet, usually the behaviour that we experience can be described by the simpler classical laws of Newtonian mechanics. A room-temperature gas behaves almost like a swarm of billiard balls, bouncing against each other. At low temperatures, however, the wave nature of the atoms becomes apparent, and the quantum effects start to dominate. This is illustrated in figure 1.1. Atoms that behave as shown in the figure are called bosons, while fermions, the other species of atoms, do not condense in this way. The three quantum effects that cause Bose-Einstein condensation are the indis- tinguishability of particles, the discrete nature of the energy of each particle, and that several particles are allowed to be in the same state.