The author presents a tutorial review of some ideas that are basic to our current understanding of the phenomenon of Bose-Einstein condensation (BEC) in the dilute atomic alkali gases, with special emphasis on the case of two or more coexisting hyperfine species. Topics covered include the definition of and conditions for BEC in an interacting system, the replacement of the true interatomic potential by a zero-range pseudopotential, the time-independent and time-dependent Gross-Pitaevskii equations, superfluidity and rotational properties, the Josephson effect and related phenomena, and the Bogoliubov approximation.
CONTENTS
I. Introduction. Scope of the Review 307 II. The Systems 309 A. General 309 B. Trapping potentials 309 1. Laser traps 310 2. Magnetic traps 310 3. Gravity 311 C. The hyperfine and Zeeman interactions 311 D. Imaging 312 E. Orders of magnitude 312 III. The Definition, Origin, and Occurrence of BEC: The Order Parameter 313 A. Definition of BEC 313 B. Why BEC? 314 C. Rigorous results 315 D. The order parameter and the superfluid velocity 316 1. Possible definitions of the order parameter 316 2. The superfluid velocity 317 IV. The Effective Interaction in a Cold Dilute Gas 317 A. Statement of the problem: neglect of l 0 partial waves 317 B. The s-wave scattering length 318 C. The effective interaction 319 D. Effects of indistinguishability 320 E. Effect of the hyperfine degree of freedom 321 F. Time-dependent situations: the MIT hydrogen experiments 322 V. The Gross-Pitaevskii Approximation 307 A. The Gross-Pitaevskii ground state of a spinless system 323 B. The spinless gas: finite-temperature equilibrium 324 C. The spinless gas: time-dependent Gross-Pitaevskii theory 324 D. Effects of the hyperfine degree of freedom 326 E. Applications 329 VI. Rotational Properties: Superfluidity 330 A. Phenomenology of superfluidity in liquid 4 He 330 B. Rotating frames of reference 331 C. Equilibrium of a BEC system in a rotating container 331 D. Metastability of superflow 333 1. General considerations 333 2. A toy model 334 3. Further remarks 335 E. Real-life BEC alkali gases in harmonic traps 335 F. The experimental situation 336 VII. BEC in a Two-State System: Josephson-Type Effects, Phase Diffusion 336 A. General formulation: choice of basis 337 B. Realizations in the BEC alkali gases 338 C. Kinematics of the Josephson effect: the Rabi, Josephson, and Fock regimes 339 D. The Josephson regime: Josephson resonance and macroscopic quantum self-trapping 341 E. The Fock regime: phase diffusion 343 VIII. The Bogoliubov Approximation 347 A. Inconsistency of the Gross-Pitaevskii approximation 347 B. The Bogoliubov ground state in the translationinvariant case 347 C. Properties of the Bogoliubov ground state: elementary excitations 349 D. The inhomogeneous case 350 E. Time-dependent Bogoliubov-de Gennes equations: connection with the timedependent Gross-Pitaevskii equation 351 F. The multicomponent case 352 IX. Further Topics 352 A. Attractive interactions 352 B. Optical properties 352 C. Coexistence of three hyperfine species 353 D. The ''atom laser'' 353 E. Kinetics, damping, relaxation, etc. 353 F. Late-breaking developments 354 Acknowledgments 354 References 354
I. INTRODUCTION. SCOPE OF THE REVIEW
The phenomenon known as Bose-Einstein condensation (hereafter abbreviated BEC) was predicted by Ein-