Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The non-interacting Bose gas
- 3 Atomic properties
- 4 Trapping and cooling of atoms
- 5 Interactions between atoms
- 6 Theory of the condensed state
- 7 Dynamics of the condensate
- 8 Microscopic theory of the Bose gas
- 9 Rotating condensates
- 10 Superfluidity
- 11 Trapped clouds at non-zero temperature
- 12 Mixtures and spinor condensates
- 13 Interference and correlations
- 14 Optical lattices
- 15 Lower dimensions
- 16 Fermions
- 17 From atoms to molecules
- Appendix: Fundamental constants and conversion factors
- Index
7 - Dynamics of the condensate
Published online by Cambridge University Press: 25 January 2011
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The non-interacting Bose gas
- 3 Atomic properties
- 4 Trapping and cooling of atoms
- 5 Interactions between atoms
- 6 Theory of the condensed state
- 7 Dynamics of the condensate
- 8 Microscopic theory of the Bose gas
- 9 Rotating condensates
- 10 Superfluidity
- 11 Trapped clouds at non-zero temperature
- 12 Mixtures and spinor condensates
- 13 Interference and correlations
- 14 Optical lattices
- 15 Lower dimensions
- 16 Fermions
- 17 From atoms to molecules
- Appendix: Fundamental constants and conversion factors
- Index
Summary
The time-dependent behaviour of Bose–Einstein condensed clouds, such as collective modes and the expansion of a cloud when released from a trap, is an important source of information about the physical nature of the condensate. In addition, the spectrum of elementary excitations of the condensate is an essential ingredient in calculations of thermodynamic properties. In this chapter we treat the dynamics of a condensate at zero temperature starting from a time-dependent generalization of the Gross–Pitaevskii equation used in Chapter 6 to describe static properties. From this equation one may derive equations very similar to those of classical hydrodynamics, which we shall use to calculate properties of collective modes.
We begin in Sec. 7.1 by describing the time-dependent Gross–Pitaevskii equation and deriving the hydrodynamic equations, which we then use to determine the excitation spectrum of a homogeneous Bose gas (Sec. 7.2). Subsequently, we consider modes in trapped clouds (Sec. 7.3) within the hydrodynamic approach, and also describe the method of collective coordinates and the related variational method. In Sec. 7.4 we consider surface modes of oscillation, which resemble gravity waves on a liquid surface. The variational approach is used in Sec. 7.5 to treat the free expansion of a condensate upon release from a trap. Finally, in Sec. 7.6 we discuss solitons, which are exact one-dimensional non-linear solutions of the time-dependent Gross–Pitaevskii equation.
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- Bose–Einstein Condensation in Dilute Gases , pp. 182 - 224Publisher: Cambridge University PressPrint publication year: 2008