Contents
Preface page xi
1 Introduction 1
1.1 Bose–Einstein condensation in atomic clouds 4
1.2 Superfluid
4
He 6
1.3 Other condensates 8
1.4 Overview 10
Problems 13
References 14
2 The non-interacting Bose gas 16
2.1 The Bose distribution 16
2.1.1 Density of states 18
2.2 Transition temperature and condensate fraction 21
2.2.1 Condensate fraction 23
2.3 Density profile and velocity distribution 24
2.3.1 The semi-classical distribution 27
2.4 Thermodynamic quantities 29
2.4.1 Condensed phase 30
2.4.2 Normal phase 32
2.4.3 Specific heat close to T
c
32
2.5 Effect of finite particle number 35
2.6 Lower-dimensional systems 36
Problems 37
References 38
3 Atomic properties 40
3.1 Atomic structure 40
3.2 The Zeeman effect 44
v
vi Contents
3.3 Response to an electric field 49
3.4 Energy scales 55
Problems 57
References 57
4 Trapping and cooling of atoms 58
4.1 Magnetic traps 59
4.1.1 The quadrupole trap 60
4.1.2 The TOP trap 62
4.1.3 Magnetic bottles and the Ioffe–Pritchard trap 64
4.2 Influence of laser light on an atom 67
4.2.1 Forces on an atom in a laser field 71
4.2.2 Optical traps 73
4.3 Laser cooling: the Doppler process 74
4.4 The magneto-optical trap 78
4.5 Sisyphus cooling 81
4.6 Evaporative cooling 90
4.7 Spin-polarized hydrogen 96
Problems 99
References 100
5 Interactions between atoms 102
5.1 Interatomic potentials and the van der Waals interaction 103
5.2 Basic scattering theory 107
5.2.1 Effective interactions and the scattering length 111
5.3 Scattering length for a model potential 114
5.4 Scattering between different internal states 120
5.4.1 Inelastic processes 125
5.4.2 Elastic scattering and Feshbach resonances 131
5.5 Determination of scattering lengths 139
5.5.1 Scattering lengths for alkali atoms and hydrogen 142
Problems 144
References 144
6 Theory of the condensed state 146
6.1 The Gross–Pitaevskii equation 146
6.2 The ground state for trapped bosons 149
6.2.1 A variational calculation 151
6.2.2 The Thomas–Fermi approximation 154
6.3 Surface structure of clouds 158
6.4 Healing of the condensate wave function 161
Contents vii
Problems 163
References 163
7 Dynamics of the condensate 165
7.1 General formulation 165
7.1.1 The hydrodynamic equations 167
7.2 Elementary excitations 171
7.3 Collective modes in traps 178
7.3.1 Traps with spherical symmetry 179
7.3.2 Anisotropic traps 182
7.3.3 Collective coordinates and the variational method 186
7.4 Surface modes 193
7.5 Free expansion of the condensate 195
7.6 Solitons 196
Problems 201
References 202
8 Microscopic theory of the Bose gas 204
8.1 Excitations in a uniform gas 205
8.1.1 The Bogoliubov transformation 207
8.1.2 Elementary excitations 209
8.2 Excitations in a trapped gas 214
8.2.1 Weak coupling 216
8.3 Non-zero temperature 218
8.3.1 The Hartree–Fock approximation 219
8.3.2 The Popov approximation 225
8.3.3 Excitations in non-uniform gases 226
8.3.4 The semi-classical approximation 228
8.4 Collisional shifts of spectral lines 230
Problems 236
References 237
9 Rotating condensates 238
9.1 Potential flow and quantized circulation 238
9.2 Structure of a single vortex 240
9.2.1 A vortex in a uniform medium 240
9.2.2 A vortex in a trapped cloud 245
9.2.3 Off-axis vortices 247
9.3 Equilibrium of rotating condensates 249
9.3.1 Traps with an axis of symmetry 249
9.3.2 Rotating traps 251
viii Contents
9.4 Vortex motion 254
9.4.1 Force on a vortex line 255
9.5 The weakly-interacting Bose gas under rotation 257
Problems 261
References 262
10 Superfluidity 264
10.1 The Landau criterion 265
10.2 The two-component picture 267
10.2.1 Momentum carried by excitations 267
10.2.2 Normal fluid density 268
10.3 Dynamical processes 270
10.4 First and second sound 273
10.5 Interactions between excitations 280
10.5.1 Landau damping 281
Problems 287
References 288
11 Trapped clouds at non-zero temperature 289
11.1 Equilibrium properties 290
11.1.1 Energy scales 290
11.1.2 Transition temperature 292
11.1.3 Thermodynamic properties 294
11.2 Collective modes 298
11.2.1 Hydrodynamic modes above T
c
301
11.3 Collisional relaxation above T
c
306
11.3.1 Relaxation of temperature anisotropies 310
11.3.2 Damping of oscillations 315
Problems 318
References 319
12 Mixtures and spinor condensates 320
12.1 Mixtures 321
12.1.1 Equilibrium properties 322
12.1.2 Collective modes 326
12.2 Spinor condensates 328
12.2.1 Mean-field description 330
12.2.2 Beyond the mean-field approximation 333
Problems 335
References 336
Contents ix
13 Interference and correlations 338
13.1 Interference of two condensates 338
13.1.1 Phase-locked sources 339
13.1.2 Clouds with definite particle number 343
13.2 Density correlations in Bose gases 348
13.3 Coherent matter wave optics 350
13.4 The atom laser 354
13.5 The criterion for Bose–Einstein condensation 355
13.5.1 Fragmented condensates 357
Problems 359
References 359
14 Fermions 361
14.1 Equilibrium properties 362
14.2 Effects of interactions 366
14.3 Superfluidity 370
14.3.1 Transition temperature 371
14.3.2 Induced interactions 376
14.3.3 The condensed phase 378
14.4 Boson–fermion mixtures 385
14.4.1 Induced interactions in mixtures 386
14.5 Collective modes of Fermi superfluids 388
Problems 391
References 392
Appendix. Fundamental constants and conversion factors 394
Index 397
Preface
The experimental discovery of Bose–Einstein condensation in trapped
atomic clouds opened up the exploration of quantum phenomena in a qual-
itatively new regime. Our aim in the present work is to provide an intro-
duction to this rapidly developing field.
The study of Bose–Einstein condensation in dilute gases draws on many
different subfields of physics. Atomic physics provides the basic methods
for creating and manipulating these systems, and the physical data required
to characterize them. Because interactions between atoms play a key role
in the behaviour of ultracold atomic clouds, concepts and methods from
condensed matter physics are used extensively. Investigations of spatial and
temporal correlations of particles provide links to quantum optics, where
related studies have been made for photons. Trapped atomic clouds have
some similarities to atomic nuclei, and insights from nuclear physics have
been helpful in understanding their properties.
In presenting this diverse range of topics we have attempted to explain
physical phenomena in terms of basic principles. In order to make the pre-
sentation self-contained, while keeping the length of the book within reason-
able bounds, we have been forced to select some subjects and omit others.
For similar reasons and because there now exist review articles with exten-
sive bibliographies, the lists of references following each chapter are far from
exhaustive. A valuable source for publications in the field is the archive at
Georgia Southern University: http://amo.phy.gasou.edu/bec.html
This book originated in a set of lecture notes written for a graduate-
level one-semester course on Bose–Einstein condensation at the University
of Copenhagen. We have received much inspiration from contacts with our
colleagues in both experiment and theory. In particular we thank Gordon
Baym and George Kavoulakis for many stimulating and helpful discussions
over the past few years. Wolfgang Ketterle kindly provided us with the
xi
xii Preface
cover illustration and Fig. 13.1. The illustrations in the text have been
prepared by Janus Schmidt, whom we thank for a pleasant collaboration.
It is a pleasure to acknowledge the continuing support of Simon Capelin
and Susan Francis at the Cambridge University Press, and the careful copy-
editing of the manuscript by Brian Watts.
Copenhagen Christopher Pethick Henrik Smith
1
Introduction
Bose–Einstein condensates in dilute atomic gases, which were first realized
experimentally in 1995 for rubidium [1], sodium [2], and lithium [3], provide
unique opportunities for exploring quantum phenomena on a macroscopic
scale.
1
These systems differ from ordinary gases, liquids, and solids in a
number of respects, as we shall now illustrate by giving typical values of
some physical quantities.
The particle density at the centre of a Bose–Einstein condensed atomic
cloud is typically 10
13
–10
15
cm
−3
. By contrast, the density of molecules
in air at room temperature and atmospheric pressure is about 10
19
cm
−3
.
In liquids and solids the density of atoms is of order 10
22
cm
−3
, while the
density of nucleons in atomic nuclei is about 10
38
cm
−3
.
To observe quantum phenomena in such low-density systems, the tem-
perature must be of order 10
−5
K or less. This may be contrasted with
the temperatures at which quantum phenomena occur in solids and liquids.
In solids, quantum effects become strong for electrons in metals below the
Fermi temperature, which is typically 10
4
–10
5
K, and for phonons below
the Debye temperature, which is typically of order 10
2
K. For the helium
liquids, the temperatures required for observing quantum phenomena are of
order 1 K. Due to the much higher particle density in atomic nuclei, the
corresponding degeneracy temperature is about 10
11
K.
The path that led in 1995 to the first realization of Bose–Einstein con-
densation in dilute gases exploited the powerful methods developed over the
past quarter of a century for cooling alkali metal atoms by using lasers. Since
laser cooling alone cannot produce sufficiently high densities and low tem-
peratures for condensation, it is followed by an evaporative cooling stage, in
1
Numbers in square brackets are references, to be found at the end of each chapter.
1
2 Introduction
which the more energetic atoms are removed from the trap, thereby cooling
the remaining atoms.
Cold gas clouds have many advantages for investigations of quantum phe-
nomena. A major one is that in the Bose–Einstein condensate, essentially all
atoms occupy the same quantum state, and the condensate may be described
very well in terms of a mean-field theory similar to the Hartree–Fock theory
for atoms. This is in marked contrast to liquid
4
He, for which a mean-field
approach is inapplicable due to the strong correlations induced by the inter-
action between the atoms. Although the gases are dilute, interactions play
an important role because temperatures are so low, and they give rise to
collective phenomena related to those observed in solids, quantum liquids,
and nuclei. Experimentally the systems are attractive ones to work with,
since they may be manipulated by the use of lasers and magnetic fields. In
addition, interactions between atoms may be varied either by using different
atomic species, or, for species that have a Feshbach resonance, by changing
the strength of an applied magnetic or electric field. A further advantage
is that, because of the low density, ‘microscopic’ length scales are so large
that the structure of the condensate wave function may be investigated di-
rectly by optical means. Finally, real collision processes play little role, and
therefore these systems are ideal for studies of interference phenomena and
atom optics.
The theoretical prediction of Bose–Einstein condensation dates back more
than 75 years. Following the work of Bose on the statistics of photons [4],
Einstein considered a gas of non-interacting, massive bosons, and concluded
that, below a certain temperature, a finite fraction of the total number of
particles would occupy the lowest-energy single-particle state [5]. In 1938
Fritz London suggested the connection between the superfluidity of liquid
4
He and Bose–Einstein condensation [6]. Superfluid liquid
4
He is the pro-
totype Bose–Einstein condensate, and it has played a unique role in the
development of physical concepts. However, the interaction between helium
atoms is strong, and this reduces the number of atoms in the zero-momentum
state even at absolute zero. Consequently it is difficult to measure directly
the occupancy of the zero-momentum state. It has been investigated ex-
perimentally by neutron scattering measurements of the structure factor at
large momentum transfers [7], and the measurements are consistent with a
relative occupation of the zero-momentum state of about 0.1 at saturated
vapour pressure and about 0.05 near the melting curve [8].
The fact that interactions in liquid helium reduce dramatically the oc-
cupancy of the lowest single-particle state led to the search for weakly-
interacting Bose gases with a higher condensate fraction. The difficulty with
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