## Bose-Einstein CondensationThe book is an introductory text to the physics of Bose-Einstein condensation. This phenomenon, first predicted by Einstein in 1925, has been realized experimentally in 1995 in a remarkable series of experiments whose importance has been recognized by the award of the 2001 Nobel Prize in Physics. The condensate is actually a new state of matter, where quantum-mechanical wave functions of atoms behave as coherent matter waves in the same way as coherent light waves in the case of a laser. The authors provide a theoretical presentation of the main concepts underlying the physics of dilute atomic gases in conditions of extremely low temperatures where quantum effects play a crucial role. The main effort is devoted to discussion of the relevant theoretical aspects exhibited by these systems, such as the concept of order parameter, long range order, superfluidity and coherence. The mathematical formalism is presented in a form convenient for practical use. The book develops the theory of Bose gases starting from the pioneering Bogoliubov approach and gives special emphasis to the new physical features exhibited by non-uniform gases which are produced in the recent experiments with magnetic and optical traps. These features include the determination of the equilibrium profiles, the collective oscillations, the mechanism of the expansion of the gas after releasing the trap, the interference patterns obtained by overlapping two condensates, the rotational properties revealing the effects of superfluidity (quantized vortices, behaviour of the moment of inertia), the Josephson-like phenomena associated with the coherence of the phase, the beyond mean field phenomena exhibited by quantum gases in conditions of reduced dimensionality (1D and 2D) etc. The book also discusses the analogies and differences with the physics of "classical" superfluids like liquid helium and introduces some of the major features of trapped Fermi gases at low temperature, pointing out the consequences of superfluidity. |

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### Contents

Introduction | 1 |

Longrange order symmetry breaking and order | 7 |

The ideal Bose gas | 13 |

The weaklyinteracting Bose gas | 26 |

Nonuniform Bose gases at zero temperature | 38 |

Superfluidity | 63 |

Linear response function | 87 |

Superfluid 4He | 108 |

Dynamics of a trapped condensate | 177 |

Thermodynamics of a trapped Bose gas | 222 |

Rotation of a trapped Bose gas | 244 |

Coherence interference and the Josephson effect | 274 |

BoseEinstein condensation in optical lattices | 303 |

BoseEinstein condensation in low dimensions | 323 |

Atomic Fermi gases in traps | 346 |

367 | |

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### Common terms and phrases

angular momentum angular velocity approximation behaviour Bogoliubov Bose gas Bose gases Bose-Einstein condensation calculated Chapter characterized chemical potential classical condensate fraction condition configuration consequence corresponding critical temperature density matrix density profile derived discussed dispersion law dynamic structure factor effects Einstein condensation elementary excitations equilihrium evaluated exhibits experimental Fermi gas finite temperatures fluctuations frequency given GP equation Gross-Pitaevskii equation Hamiltonian harmonic trapping healing length hydrodynamic ideal Bose gas ideal gas important integral Josephson kinetic energy limit linear macroscopic magnetic many-body mean field mode momentum distribution noninteracting normal number of atoms number of particles obtained occupation number one-body density matrix one-dimensional operator optical lattice order parameter oscillations phase phonon prediction quadrupole quantized quantum quasiparticle radial regime response function rotating scattering length Section single-particle solution sound velocity sum rule superfluid theory thermodynamic Thomas-Fermi two-body interactions vortex line vortices wave function zero temperature