Embed Size (px)
Citation preview
C-Field and Mean-Field
Treatments of the
Quasi-Two-Dimensional
Harmonically Confined Bose Gas
Russell Bisset
a thesis submitted for the degree of
Master of Science
at the University of Otago, Dunedin,
New Zealand.
August 14, 2009
ii
Abstract
According to the Mermin-Wagner-Hohenberg Theorem, in reduced dimen-
sionality, thermal fluctuations destroy the long-range order characteristic of
most phase transitions. However, in systems that support topological de-
fects, such as vortices, the existence of a quasi-long-range ordered state was
predicted to occur by Berezinskii, Kosterlitz and Thouless (BKT), defining
a new paradigm for phase transitions. There has been significant debate re-
garding the nature of the phase transition that would occur in a 2D trapped
dilute Bose gas, with theoretical predictions varying as to whether a con-
densate transition, BKT transition, or some mixture of both would occur.
Since 2005 four important experiments by the ENS and NIST groups were
published revealing clues about the degenerate phase of the quasi-2D Bose
gas, but also raising several important questions.
We present a c-field theory for the trapped quasi-2D Bose gas. This theory
is non-perturbative and valid in the critical regime. We characterise the
properties of this system over a wide parameter regime. Our results show
that at temperatures well above the BKT transition point (TBKT ) den-
sity fluctuations are suppressed by the formation of a quasicondensate. At
lower temperatures, but still above TBKT , we observe the appreciable devel-
opment of a condensate, which manifests as bimodality in the momentum
distribution of the system. At TBKT algebraic decay of off-diagonal corre-
lations occurs near the trap centre with an exponent of 0.25, as expected
for the uniform system. These results are consistent with observations in
recent experiments.
We develop a high temperature Hartree-Fock mean-field theory which pro-
vides a good description for temperatures well above TBKT . We use our
theory to provide an in-depth analysis of a simplified mean-field theory
proposed by Holzmann, Chevallier and Krauth (HCK) [Europhys. Lett. 82,
30001 (2008)] for predicting the temperature at which the BKT transition
to a superfluid state occurs in the harmonically trapped quasi-2D Bose gas.
Their theory involves an unjustified simplification of the mean-field inter-
action term and a different local density condition to identify the BKT
critical point. We characterise the differences between the predictions of
both theories, and justify that our theory provides a lower bound for TBKT .
iii
Acknowledgements
First, I would like to thank my supervisor Dr. Blair Blakie for being a
great supervisor and for providing the opportunity to attend the KOALA
conference in Brisbane.
Thankyou to Dr. Ashton Bradley, Assoc. Prof. Matthew Davis, Patrick
Ledingham, Sarah Dietrich, Danny Baillie, Chris Foster, Dave McAuslan
and Wynton Moore for highly useful and enjoyable discussions.
Thankyou to my family for always being supportive.
iv
Contents
1 Introduction 1
1.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Phase Transitions in Two-Dimensions . . . . . . . . . . . . . . . . . . . 3
1.3 The Ultra-Cold 2D and quasi-2D Bose Gas . . . . . . . . . . . . . . . . 5
1.4 Recent Quasi-Two-Dimensional Experiments . . . . . . . . . . . . . . . 6
1.4.1 The ENS Experiments . . . . . . . . . . . . . . . . . . . . . . . 6
Observation of Thermally Activated Phase Defects . . . . . . . 8
Measurement of Phase Coherence . . . . . . . . . . . . . . . . . 9
Bimodality and Phase Coherence . . . . . . . . . . . . . . . . . 11
Summary of ENS Results . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 The NIST Experiment . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Research Focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Foundation theory for 2D Systems 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Mermin-Wagner-Hohenberg Theorem . . . . . . . . . . . . . . . . 17
2.3 The XY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.2 Motivation: Relation to the 2D Bose Gas . . . . . . . . . . . . . 20
2.3.3 The High Temperature Limit . . . . . . . . . . . . . . . . . . . 21
2.3.4 The Low Temperature Limit . . . . . . . . . . . . . . . . . . . . 22
2.3.5 The BKT Transition - Critical Temperature . . . . . . . . . . . 24
2.3.6 The BKT Transition - Unbinding of Vortex Pairs . . . . . . . . 26
2.4 Ideal 2D Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Bose Einstein Condensation . . . . . . . . . . . . . . . . . . . . 28
2.4.2 Quasi-Two-Dimensional . . . . . . . . . . . . . . . . . . . . . . 29
v
2.5 Interacting 2D Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 Theoretical Developments for the 2D Bose Gas 33
3.1 Effective 2D Interaction in the Bose Gas . . . . . . . . . . . . . . . . . 33
3.2 Mean-field Theory of the Degenerate Regime . . . . . . . . . . . . . . . 34
3.3 Universality and Critical Physics of the Uniform Bose Gas . . . . . . . 35
3.3.1 Critical Phase Space Density . . . . . . . . . . . . . . . . . . . . 35
3.3.2 Superfluid and Quasicondensate . . . . . . . . . . . . . . . . . . 36
3.3.3 The Extended Critical Region . . . . . . . . . . . . . . . . . . . 38
4 The Degenerate Quasi-2D Trapped Bose gas 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 C-Field Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
The Classical Field Region . . . . . . . . . . . . . . . . . . . . . 42
Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . 43
The Incoherent Region . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Calculation of Macroscopic Parameters and Phases . . . . . . . 45
Atom number, Temperature and Chemical Potential . . . . . . . 45
Quasicondensate . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Coherence and Condensate . . . . . . . . . . . . . . . . . . . . . 46
Superfluid Model of Holzmann et al. . . . . . . . . . . . . . . . 46
4.2.3 Validity Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.1 System Parameters and Procedure . . . . . . . . . . . . . . . . 48
4.3.2 Degenerate Phases . . . . . . . . . . . . . . . . . . . . . . . . . 49
Density Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Degenerate Components . . . . . . . . . . . . . . . . . . . . . . 49
Determination of Tc . . . . . . . . . . . . . . . . . . . . . . . . . 51
Comparing Simulations of Different Atom Numbers . . . . . . . 53
4.3.3 Topological Order and The BKT Transition . . . . . . . . . . . 57
4.3.4 Bimodality and Condensate . . . . . . . . . . . . . . . . . . . . 59
4.3.5 Central Curvature of Density Profiles . . . . . . . . . . . . . . . 59
4.3.6 The Role of Vortices . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.7 Condensate Number Fluctuations . . . . . . . . . . . . . . . . . 68
4.3.8 Simulation Validity . . . . . . . . . . . . . . . . . . . . . . . . . 72
vi
5 High Temperature Mean-field Theory 75
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Semiclassical Quasi-2D Mean-Field Theory . . . . . . . . . . . . . . . . 76
5.2.1 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 Holzmann-Chevallier-Krauth Mean-field Theory . . . . . . . . . 77
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1 2D Phase Space Density . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 BKT Transition in the Trapped System . . . . . . . . . . . . . . 81
Monte Carlo Analysis of the BKT Transition . . . . . . . . . . . 81
Mean-field validity near the transition . . . . . . . . . . . . . . 82
Comparison Between HCK Theory and Monte Carlo Density . . 82
Use of Mean-Field Theory to Predict the BKT Transition . . . . 83
Comparison of Mean-field Predictions for the BKT Transition . 84
5.3.3 Comparison Between C-field and Mean-field Predictions . . . . 87
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Conclusions 91
6.1 Phases of the Quasi-2D Trapped Bose Gas . . . . . . . . . . . . . . . . 91
6.2 Mean-field Description of the Normal Phase . . . . . . . . . . . . . . . 92
6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
References 95
vii
viii
Chapter 1
Introduction
1.1 Bose-Einstein Condensation
Early in the 1920’s, with quantum mechanics still in its infancy, Satyendra Nath Bose
made an unexpected discovery regarding the quantum statistical properties of light.
While giving a lecture at the University of Dhaka on the photoelectric effect and the
ultraviolet catastrophe, Bose intended showing his students that the current theory
was inconsistent with experimental results. However, during this lecture he made an
embarrassing error which, oddly, resulted in agreement with the experiment. This
error, analogous to stating that two fair coins will both produce heads one third of
the time, was understandably considered absurd, but Bose realised that this might not
be an error at all. Initially, Bose encountered much resistance and could not publish
his ideas. This prompted Bose to send a copy of his paper to Einstein in 1924, who
immediately agreed with his idea, thus facilitating the publication of this revolutionary
concept of particle indistinguishability [24]. In subsequent works Einstein generalised
this theory (now called Bose-Einstein statistics) to encompass those atoms that belong
to the group we now refer to as bosons.
The ubiquitous expression resulting from their work is the Bose-Einstein distribu-
tion,
NBE =1
eβ(E−µ) − 1, (1.1)
which relates the mean number of bosons occupying a given mode (NBE) to its energy
(E) and the chemical potential (µ), where β = 1/kBT , kB is Boltzmann’s constant
and T is the temperature. In the period 1924-1925 Bose and Einstein realised that the
Bose-Einstein distribution predicted the existence of a new state of matter, namely the
Bose-Einstein condensate (BEC). According to Eq. (1.1), applied to bosonic atoms, the
1
mean occupation of each mode increases with increasing µ, but to avoid divergence, µ
can only asymptotically approach the ground mode energy (Eg) from below. If µ ≈ Eg
and the combined occupation of all excited modes is finite, then the thermal cloud is
said to be saturated. If a system at a given temperature has a population exceeding
this saturated value, then the excess atoms must exist in the ground mode and the
number is determined by fine adjustments of µ, which asymptotically approaches Eg.
This macroscopic occupation of the ground mode is referred to as the BEC phase.
Alternatively, consider the BEC transition from the viewpoint of a system with fixed
atom number (N) but varying T . Below a critical temperature (Tc) a large fraction
of bosons coalesce into the lowest quantum state. The BEC transition exhibits the
standard features of a 2nd order phase transition from an unordered thermal state
to a state with Long Range Order (LRO). The nature of the ordering in the BEC
state is that phase coherence occurs between spatially distant points in the system
(similar to a laser). This is often referred to as Off-Diagonal LRO (ODLRO), alluding
to G1(r, r′) → const. as |r − r′| → ∞ (where r ≡ x, y, z), where G1 is the first
order correlation function [see Eq. (1.10)]. In contrast, in the unordered state above
Tc, coherence between any two points decays exponentially with separation.
Quantum degeneracy was first observed experimentally via superconductivity of
metals and superfluidity of helium, however these systems are complicated by strong
interactions and a detailed understanding of the many-body state remains illusive. In
the 1980’s, important advances towards realising Bose and Einstein’s original concept
of BEC, in dilute atomic systems, were made with spin-polarised atomic hydrogen (for
example see [52]). Implementing evaporative cooling [38], Doyle et al. [23] were able
to approach Tc within a factor of three. However, progress towards achieving Tc with
hydrogen slowed due to trap losses and heating caused by inelastic collisions. Finally
after a change of tack, in 1995 (70 years after the prediction by Bose and Einstein) BEC
was achieved with the dilute alkali metals rubidium [2], lithium [12] and sodium [16] by
implementing a combined approach of laser cooling and evaporative cooling [1, 17, 43].
S. Chu, C. Cohen-Tannoudji and W. D. Phillips were awarded the 1997 Nobel Prize
in Physics for the development of laser cooling methods. E. A. Cornell, C. E. Wieman
and W. Ketterle received the 2001 Nobel Prize in Physics for the achievement and
fundamental study of BEC.
2
1.2 Phase Transitions in Two-Dimensions
A system’s dimensionality is fundamental for determining the type of phase transitions
that occur. In three dimensions (3D), BEC has been predicted to occur in uniform
systems and has been observed experimentally in trapped systems. For two-dimensional
(2D) systems the situation is not nearly so simple. For uniform 2D systems with
continuous symmetry and short range interactions, Mermin, Wagner and Hohenberg in
1966-1967 [31, 39] showed rigorously that LRO is not possible for non-zero temperature,
hence a BEC transition in the 2D Bose gas is absent. Physically, the lack of an ordered
phase is due to the relatively low energetic cost of long wavelength phonons compared
to the entropy produced by their creation, these phonons destroy LRO (see § 2.2).
However, in 1971-73 Berezinskii, Kosterlitz and Thouless [5, 36] showed that even
though a 2nd order phase transition to an ordered state does not occur, a new type
of phase transition can emerge. The requirement for this Berezinskii, Kosterlitz and
Thouless (BKT) transition to emerge is that the system supports stable topological
defects, and in the low temperature phase the system will exhibit topological order,
in which the two point correlations decay algebraically with the separation distance.
While the canonical system for illustrating this transition is the idealised XY spin model
(see §2.3), the first experimental verification of the BKT transition was by Bishop and
Reppy in 1978 [7], using thin films of helium. Below the transition temperature (TBKT )
they observed superfluidity which sharply vanished at and above TBKT (see Fig. 1.1).
Below TBKT it is energetically favourable for vortices of opposite circulation to be
bound together, referred to as vortex-antivortex pairs (VAPs). At TBKT entropy is
now maximized by vortices unbinding to become free, which destroys the superfluid
density in a discontinuous jump, resulting in a normal phase with exponentially decay-
ing correlations.
The 2D dilute Bose gas belongs to the same universality class as the XY model and,
like the liquid helium system, it is expected that in the uniform 2D case it will undergo
a BKT transition. However, what happens in the experimentally relevant trapped 2D
case is not clear. This issue is complicated further by the prediction of a BEC in the
special case of the ideal trapped 2D regime, this BEC is fragile and simple arguments
suggest this may not survive in the interacting system (see §2.4.1). These scenarios are
summarised in table 1.1.
The possible phase transitions and their nature in the trapped quasi-2D interacting
Bose gas has recently become a hot topic due to new experimental results [14, 29, 37,
3
Figure 1.1: (Figure taken from Ref. [7]). The shift in period, ∆P,
as a function of temperature at the superfluid transition. ”When the
oscillator is cooled below the transition temperature the superfluid
decouples from the torsion pendulum and the period of the oscillator
decreases.”
Ideal Gas Interacting Gas
3D - Homogeneous BEC BEC
3D - Harmonically Trapped BEC BEC
2D - Homogeneous × × (BKT)
2D - Harmonically Trapped BEC ??? ???
Table 1.1: Summary of predicted phase transitions of the Bose gas
for different regimes. (×) No BEC transition.
4
55]. As discussed above, from the current theory it is not clear whether there should
be a BEC transition, BKT transition or some combination of the two. The research
in this thesis is concerned with elucidating the physics of the trapped interacting 2D
Bose gas to help clarify this situation.
1.3 The Ultra-Cold 2D and quasi-2D Bose Gas
Formally, we model the dilute Bose gas by the second quantised Hamiltonian,
H =
∫
d3r؆(r)
H0 +U0
2Ψ†(r)Ψ(r)
Ψ(r), (1.2)
with Ψ(r) being the Bose annihilation field operator and U0 = 4π~2a/m is the interac-
tion strength, where a is the s-wave scattering length and m is the atomic mass. The
single particle Hamiltonian is given by
H0 = − ~2
2m∇2 + V (r), (1.3)
where V (r) is the external trapping potential given as
V (r) =1
2m(ω2
xx2 + ω2
yy2 + ω2
zz2). (1.4)
We now explore how the 2D regime may be attained by a trapped Bose gas. Con-
sider the anisotropic harmonic trap that confines the Bose gas loosely in the x, y (radial)
plane but tightly in the z (axial) direction. If T and the trapping frequencies (ω ≡ 2πf)
satisfy
~ωz ≫ kBT ≫ ~ωx,y, (1.5)
then excited modes of the axial direction are frozen out which results in a 2D harmonic
oscillator density of states, where as usual ~ is the reduced Planck constant. However,
we point out that the purely 2D [Eq. (1.5)] dilute Bose gas has yet to be realised
experimentally, and all relevant experiments [14, 29, 37, 55] (reviewed in §1.4) belong
to regimes which instead satisfy
~ωz ∼ kBT ≫ ~ωx,y, (1.6)
which implies a non-negligible population in the excited axial modes. We hence make
an important distinction. Systems that satisfy (1.5) are referred to as 2D while systems
satisfying (1.6) are quasi-2D (q2D).
5
To distinguish between quasi-2D and 3D, consider the critical mode cutoff energy
(below which the critical modes are confined), this is roughly given by [48]
ǫc ∼ gn, (1.7)
where n is the areal density and g is the effective 2D interaction constant (see §3.1),
thus a further requirement to be quasi-2D is that
gn . ~ωz. (1.8)
This condition ensures the critical modes are confined to the ground axial mode, thus
quasi-2D systems are essentially 2D systems embedded in a 3D thermal cloud. con-
versely, systems for which
gn & ~ωz, (1.9)
are considered to be crossing over to the 3D regime.
1.4 Recent Quasi-Two-Dimensional Experiments
The BKT superfluid transition has been experimentally observed in liquid helium thin
films [7], superconducting Josephson-junction arrays [49] and spin polarised atomic
hydrogen [51], however, the quasi-2D dilute Bose gas remained experimentally illusive
until as late as 2005 (ten years after the realisation of the first 3D BECs). Recent exper-
imental developments make possible the realisation and study of the quasi-2D regime
and, from 2005-2009, four seminal experiments [14, 29, 37, 55] have each revealed clues
along with many questions regarding the degenerate behavior of such systems. This
section will review these experiments.
1.4.1 The ENS Experiments
Late in 2005 the ENS group in Paris successfully created stacks of quasi-2D gases [55].
The general method was to begin in a 3D harmonic trap with an almost pure BEC
consisting of ∼ 4 × 105 87Rb atoms. A 1D optical lattice periodic potential was then
ramped up, slicing the BEC, producing an independent system at each lattice site and
compressing each of these into the quasi-2D regime as shown schematically in fig. 1.2
(a). The final trapping frequencies of each quasi-2D pancake are approximately 11
Hz, 130 Hz and 3.6 kHz along the x, y and z directions respectively. The number of
populated lattice sites is controllable by applying a linear magnetic potential gradient
6
Figure 1.2: (Figure taken from Ref. [29]). (a) Quasi-2D potentials:
The faint ellipsoid represents the 3D BEC before application of the 1D
optical lattice. The lattice (green) is created by two laser beams prop-
agating almost parallel in the y-z plane. The two elongated pankcakes
(red) represent the resulting quasi-2D gases. (b) Expansion and imag-
ing: After equilibrating, the trapping potential is suddenly turned
off and the quasi-2D gases interfere (matter-wave interference fringes
shown in red) as they expand, predominantly in the z-direction. An
imaging beam is directed along the y-axis and the resulting image
captured by a CCD camera.
7
a)
z
x
a)
c)
b)
d)
Figure 1.3: (Figure taken from Ref. [55]). Matter wave interference:
(a) Straight fringes produced by degenerate gases with uniform phase.
(b) Zipper pattern with a single phase dislocation. (c) Comb pattern
with a dislocation and diminished fringe contrast on one side. (d)
Braid pattern with two phase dislocations.
followed by selective evaporation via radio frequency induced spin-flips to untrapped
Zeeman states. This general setup was used by the ENS group for the first three of
the experiments we discuss here.
Observation of Thermally Activated Phase Defects
In their first experiment [55] the ENS group produced stacks of between two and
eight independent quasi-2D clouds, released them by suddenly turning of the trapping
potential, then allowed them to expand for up to 18 ms before imaging the matter-wave
interference pattern (see Fig. 1.2 (b)). Their main result was the observation of phase
dislocations such as zipper, comb and braid structures (see Fig. 1.3). In this work it
was speculated that such structures were due to isolated vortices within at least one of
the clouds prior to expansion. This is significant since vortices are an integral factor
for a BKT transition if it were to occur in the trapped Bose gas (see §2.5). However,
they could not rule out other possibilities, for example a dark soliton in one cloud, with
a π phase change across it, would also produce a phase dislocation of the interference
pattern.
8
Measurement of Phase Coherence
The second experiment [29], published in Nature in 2006, presented strong evidence
for the BKT transition. In this work they studied the matter-wave interference fringes
produced by the release of two independent quasi-2D clouds (see Fig. 1.2) to investigate
the temperature dependence of free vortices and the decay of quasi-long range coher-
ence. To explore different temperature regimes they varied the final radio-frequency
used to evaporatively cool the initial 3D condensate.
Images with phase dislocations were again attributed to the presence of free vortices
prior to expansion, however here they discounted other possible causes such as solitons,
based on theoretical work [53]. The fraction of images with dislocations versus the
average central contrast is shown in Fig. 1.4 (a), where this contrast decreases with
increasing temperature1. Below a contrast of approximately 0.15 the occurrence of free
vortices proliferates sharply.
Coherence of the system is characterized by the first order correlation function
G1(r, r′) = 〈Ψ†(r)Ψ(r′)〉. (1.10)
Information about G1 can be obtained from interference patterns using a heterodyning
method proposed by Polkovnikov, Altman and Demler [45]. In this procedure the
function
F (x, z) = G(x, z) [1 + c(x) cos (2πz/D + φ(x))] (1.11)
was fit to each imaged interference pattern, where G(x, z) is a gaussian envelope, D
is the fringe period, φ(x) characterises the spatial phase (and hence its fluctuations)
and c(x) is the local contrast. The Polkovnikov [45] procedure showed how to extract
from φ(x) and c(x) a decay exponent, α, that reveals the in situ spatial correlations
in the system, for the case of exponential decay of correlations α → 0.5. According to
BKT theory of the uniform system, just below the transition temperature, first order
correlations decay algebraically as (§2.3 and §2.5)
G1(∆r) ∝ (∆r)−0.25, (1.12)
(where ∆r ≡ |∆r| ≡ |r − r′|) for which the Polkovnikov scheme yields α = 0.25.
Figure 1.4 (b) shows the measured behavior of α as a function the average central
contrast (temperature). At cold temperatures, where central contrast is above 0.2,
1Note, central contrast was used because no reliable method was known to quantify the temperature
at the time of these experiments.
9
Figure 1.4: (Figure taken from Ref. [29]). (a) Fraction of images
with phase dislocations. Inset shows a histogram of the phase jumps
∆φ between adjacent CCD pixel columns. Phase jumps greater than
the threshold ∆φ = 2π/3, indicated by the vertical dotted line, are
counted as dislocations. (b) Coherence decay exponent α. Dashed
lines represent the theoretically expected values above and below the
BKT tranistion for the homogeneous system. Error bars indicate
standard deviation due to statistical averages
10
the value of α = 0.25 clearly indicates the algebraic decay of first order correlations
according to Eq. (1.12), i.e. suggesting there is topological order. Whereas, at higher
temperatures the value for α approaches 0.5 indicating a crossover to exponential decay
of G1(∆r). We note that, although it is not explicitly stated, Fig. 1.4 (b) shows that
the algebraic decay of correlations does not occur until the average central contrast is
approximately 0.2, indicating that the onset of interference fringes (coherence) occurs
well in advance of the onset of topological order.
Another result [Fig. 1.4 (a)] is that the loss of topological order and the proliferation
of free vortices both occur qualitatively at similar temperatures, i.e. in the region where
average central contrast decreases from 0.2 to 0.1. These results conclusively show there
is a crossover in the quasi-2D trapped Bose gas that has properties consistent with the
BKT transition of the homogeneous system.
Bimodality and Phase Coherence
In 2007 the results of the third experiment were published [37] by the ENS group.
In this work they examined the density distribution in the x-direction as well as the
interference fringes of the z-direction (see Fig. 1.2). In these experiments they claim to
be able to measure temperature and by keeping the temperature fixed and varying the
total number of atoms between runs, they were able to investigate the density distri-
bution for varying levels of system degeneracy. What they found was the sudden onset
of bimodality at a critical atom number as shown in Fig. 1.5. The two modes are fitted
by a broad Gaussian representing the thermal cloud and a parabolic Thomas-Fermi
profile representing the (central) degenerate component. From these fits the number
of atoms in the Thomas-Fermi portion are plotted, along with the fringe interference
amplitude, in Fig. 1.6 and the point of onset of both appears to coincide.
Furthermore, they claimed that the measured onset of bimodality occurs at a tem-
perature reduced from the ideal gas prediction by approximately a factor of 2. However,
subsequent works put this claim into doubt, the problem being that they compared
their quasi-2D experimental results against a purely 2D ideal gas theory. This error
resulted in the underestimation of the experimental temperature by 30-40% [30] and
the overestimation of the ideal gas critical temperature by approximately 22% [32].
After allowing for the correction of these errors, the temperature at the bimodality
onset and the ideal critical temperature are not too dissimilar and the experimental
uncertainty of temperature is too large for a precise comparison.
11
Figure 1.5: (Figure taken from Ref. [37]). Phase Transition. Large
dots represent the measured line densities for an atom number just
below (left) and just above (right) the critical number. Solid lines
represent bimodal fits.
Figure 1.6: (Figure taken from Ref. [37]). Number of atoms in the
Thomas-Fermi portion (solid dots) and interference amplitude (hol-
low) vs. total atom number, inset shows the corresponding tempera-
ture.
12
Summary of ENS Results
In summary the ENS group observed the following:
(i) The onset of bimodality coincides with the onset of interference fringes (and hence
coherence) [Fig. 1.6].
(ii) Free vortex proliferation and the loss of topological order coincide at a similar
temperature [Fig. 1.4 (a) and (b)].
(iii) The onset of interference fringes (and hence coherence) occurs at a temperature
above that for the onset of topological order [Fig. 1.4 (b)].
Despite observation (iii), the ENS group speculated that their results imply the ex-
istence of a simple BKT-type crossover in which bimodality and topological order
occur together. This interpretation was to be challenged less than a year later by us,
and then subsequently the NIST group as we discuss in the next section.
1.4.2 The NIST Experiment
Most recently the NIST group reported the results on a quasi-2D experiment in 2009
[14]. In contrast to the ENS group, the NIST group trapped sodium atoms by a
cylindrically symmetric, entirely optical, single welled potential with a radial trapping
frequency of 18 Hz and an axial frequency of 1 kHz. The phase space density was
varied at constant temperature by controlling the initial number of trapped atoms.
Two Gaussians were fitted to the bimodal density distribution, the width of the narrow
Gaussian as a function of peak density is shown in Fig. 1.7, the leftmost data points
mark the onset of bimodality. As the density increases the width decreases rapidly and
after a kink, increases again slowly. They identify this minimum width (the kink) as
the BKT transition in the central region of the trap. They hypothesised that the rapid
decrease in width is caused by a decreased number of free vortices, and the gradual
increase is due to repulsive interactions in the absence of free vortices. Furthermore,
the central density sharply peaks at the kink, but only after a large TOF (10 ms) which
infers extended central phase coherence, supporting the absence of free vortices. We
emphasise that between the onset of bimodality (hence onset of interference fringes [37])
and the kink (absence of free vortices hence onset of topological order [29]) the peak
density increases by a factor of 2-3. These results support that a large temperature
range exists where quasi-2D systems exhibit bimodality and extended coherence but
13
Figure 1.7: (Figure taken from Ref. [14]). Width of the narrow part
of the cloud, obtained by fitting 5 ms TOF images with two Gaus-
sians, for two different temperatures. The lines are linear fits to two
portions of the data. Inset: Areal density at the BKT transition
as a function of temperature. The three data points mark the den-
sity at the kink (see text). The dashed line is the theoretical value
nBKTλ2dB = ln(380/g) [Eq. (3.8)] and the solid line is the same value
corrected using a 3D model.
14
are also permeated by free vortices. This is in conflict with the hypothesis proposed
by ENS, of a simple BKT-type crossover in the quasi-2D Bose gas.
1.5 Research Focus
The principle aim of our research is to theoretically investigate the transition to degen-
eracy for the quasi-2D Bose gas, with the goal of resolving the conflicting experimental
interpretations proposed by the ENS and the NIST groups. Due to the critical re-
gion being large, we non-perturbatively treat the degenerate quasi-2D Bose gas using
a c-field method. We investigate the transition from normal to degenerate regime by
calculating a number of parameters simultaneously including condensate and quasicon-
densate fraction, condensate number fluctuations, the decay of G1, momentum space
bimodality, central curvature of position distributions, vortex configurations and peak
phase space density. Some of these results have been published by us in Phys. Rev. A
[9].
Holzmann-Chevallier-Krauth (HCK) [32] developed a high temperature (they set
the condensate to zero) meanfield theory for the quasi-2D Bose gas and extrapolate this
to the critical regime for the prediction of TBKT . We also develop a high temperature
meanfield theory and use it, along with our c-field theory, to critically analyse the HCK
theory. We have published some of the results of this analysis in our second paper in
Phys. Rev. A [8].
1.6 Dissertation Outline
In chapter 2 we bring together a large amount of the background theory for 2D
systems, with particular emphases on the BKT transition and the Bose gas. In chapter
3 we further discuss background theory for the BKT transition and how it relates to
the quasi-2D Bose gas. We also review recent attempts to apply mean-field theory,
and how it is inappropriate for the description of the degenerate quasi-2D Bose gas. In
chapter 4 we outline our formalism and present our main results obtained by our non-
perturbative treatment of the critical regime using c-field methods. In chapter 5 we
develop a high temperature mean-field theory for the quasi-2D Bose gas and use it to
critically analyse the Holzmann-Chevallier-Krauth (HCK) mean-field theory [32]. We
also compare the two mean-field theories with our c-field predictions to demonstrate
the inapplicability of mean-field theories in the degenerate regime. Chapter 6 presents
15
concluding remarks.
16
Chapter 2
Foundation theory for 2D Systems
2.1 Introduction
In this chapter we bring together a large amount of the background theory for the 2D
system, with a particular focus on the Bose gas.
2.2 The Mermin-Wagner-Hohenberg Theorem
In 1966 Mermin and Wagner [39] demonstrated that, for spin systems with continuous
symmetry and short range interactions, the finite temperature phase transition to a
LRO state is not possible for one- or two-dimensions. In the following year, Hohenberg
[31] published a paper which showed this result is more general and extends to ultra-
cold gases, i.e there is no finite temperature LRO for homogeneous Bose and Fermi
systems of one- or two-dimensions. Today, these results are collectively known as the
Mermin-Wagner-Hohenberg Theorem.
Here we give a heuristic derivation of the Mermin-Wagner-Hohenberg theorem. We
assume that a continuous symmetry has been broken, for example resulting in a crystal
(breaking translational symmetry) or a BEC. Such systems support long wavelength
fluctuations in the form of Goldstone bosons, which correspond to long wavelength
(low energy) variations of the order parameter. For definiteness we will refer to these
modes as phonons, which can be labeled by their wavevector k. In the small k limit
(where k ≡ |k|) the occupation of these phonons is given by the equipartition result,
17
so that the density of phonons is given by1
nG ∝∑
k 6=0
kBT
k2. (2.1)
In the infinite volume limit we can replace the sum over k by an integral2,
nG ∝ kBT
∫
dDk1
k2= kBT
∫
dkkD−1
k2, (2.2)
where D is the number of spatial dimensions of the system. For D > 2 the integral
converges, however long wavelength modes cause divergence when D ≤ 2.
This argument implies that systems with broken symmetry are stable against long
wavelength fluctuations for D > 2 at some finite temperature. Conversely for systems
with D ≤ 2, these long wavelength Goldstone bosons destroy LRO for all finite temper-
atures, hence invalidating our original assumption of broken symmetry. We note that
for the marginal case D = 2, the integral in [Eq. (2.2)] diverges only logarithmically.
In this case a quasi-long range order transition, the BKT transition, can occur: To
investigate this further we now review the XY model.
2.3 The XY Model
2.3.1 Introduction
An important class of simplified models for investigating phase transitions are the spin
models. Consider a D-dimensional lattice with a magnetic moment at each lattice site,
n, represented by the classical spin Sn. In general the spin can exist in a space of
dimension s (i.e. S is an s dimensional vector). The interaction Hamiltonian for such
systems is
H = −1
2
∑
n 6=m
JnmSn · Sm, (2.3)
where Jnm is the interaction energy between spins at sites n and m. For systems
where Jnm < 0, energy is minimised when adjacent spins align antiparallel, this is
commonly known as the antiferromagnet. However, we will restrict our attention to
1While we have not justified the ideal-like k2 dispersion of phonons, here, we note that for the
interacting Bose gas the rigorous Bogoliubov result is that the mode occupation obeys nk ≥ −1/2 +
m×n0/n×kBT/k2, where n0/n equals the condensate fraction, for example see Eq. (18) of Hohenberg
[31].2We ignore the possibility of multiple polarizations as this only alters the number of Goldstone
bosons by a constant factor.
18
s = 1 s = 2 s = 3 s = ∞Ising model XY model Heisenberg model Spherical model
D = 1 × × × ×D = 2 X × (BKT) × ×D = 3 X X X X
Table 2.1: Summary of existence (X) or absence (×) of a finite tem-
perature transition to a long range ordered phase, for spin models of
different spatial dimension D and spin dimension s. Note the special
case D = s = 2 where there is no long range order, but there is a
BKT phase exhibiting topological order and superfluidity.
the ferromagnet where Jnm > 0 and adjacent spins prefer to align parallel. Although
many lattice configurations are possible, we will consider the cubic lattice where the
number of nearest neighbours ν = 2D. Also, we now restrict our attention to models
that have a constant interaction J between nearest neighbours only, while the spin
magnitude is constrained to unity (|Sn| = 1), e.g. for s = 2 each spin is isomorphic to
the U(1) group.
Table 2.1 summarises which of these spin models support a finite temperature
transition to a phase with LRO. For systems with D = 3, a phase with LRO is possible
for all spin dimensions, conversely when D = 1, LRO does not occur at any finite
temperature for any s. The intermediate regimes where D = 2 are not so simple, LRO
occurs for s = 13 but is forbidden by the Mermin, Wagner and Hohenberg theorem
for systems with continuous spin symmetry, i.e. s ≥ 2. Of interest is the marginal
regime where D = 2 and s = 2, for which Berezinskii, Kosterlitz and Thouless [5, 36]
showed that even though LRO is not possible, there is another kind of transition to a
superfluid phase which exhibits quasi-long-range order.
An investigation of this marginal regime, D = s = 2, and how it relates to the 2D
Bose gas forms the basis of §2.3.2 through to §2.3.6. For this system the Hamiltonian
can be simplified to
HXY = −J2
∑
〈n,m〉
cos(θn − θm), (2.4)
where 〈n,m〉 indicates the summation over nearest neighbours and θ represents the
3s = 1 is the familiar Ising model for which the D = 2 case was solved by Onsager [41]: As the s = 1
case is discrete (spin up down) there are no Goldstone bosons and the Mermin-Wagner-Hohenberg
theorem does not apply.
19
Figure 2.1: The XY model. Two-dimensional spins of fixed length in
a two-dimensional array. Nearest neighbours tend to align as demon-
strated by the undulating waves. Right: A blown up image of a single
spin site showing how the two dimensional spin may be represented
by a single angle θ.
orientation angle of the spins in the 2-dimensional spin plane, as depicted in Fig. 2.1.
This well studied system is known as the XY-model.
Formally, all systems with 2 spin dimensions are often referred to as the XY model.
However, for simplicity of notation, we will use the term XY model to refer to the 2D
XY model only, i.e. D = s = 2. As we discuss in the next section, this XY model
exhibits properties related to the 2D Bose gas
2.3.2 Motivation: Relation to the 2D Bose Gas
The highly simplified XY model exhibits important features that are related to the
uniform 2D Bose gas. Both systems have continuous symmetry with short range in-
teractions, hence as discussed in §2.2, low energy Goldstone bosons prevent systems
from breaking symmetry and forming an ordered phase. Formally the relationship is
enshrined by both systems being in the same universality class, meaning that many
aspects of their physics are similar in the neighbourhood of a phse transition.
We can intuitively develop this connection between systems by considering the
dependence of energy on the phase variations in the degenerate 2D Bose gas. We
assume the presence of a superfluid phase with density nSF (x) and phase Θ(x) (where
20
x ≡ x, y), which obeys the usual superfluid relation to velocity, given as
v(x) =~
m∇Θ(x), (2.5)
which implies the energy relationship
E ∝∫
d2x nSF (x)|v(x)|2
∝∫
d2x|∇Θ(x)|2. (2.6)
By comparison, for the XY model at low temperatures, θ varies slowly between adjacent
lattice sites (see §2.3.4), this allows us to expand the cosine term in Eq. (2.4) and replace
θi by the field θ(x),
HXY ≈ −J2
∑
〈i,j〉
[
1 − 1
2(θi − θj)
2
]
≈ const. +J
2
∫
d2x | ∇θ(x) |2 . (2.7)
Thus the energy scales with the spin angle (Eq. (2.7)) in the same manner as energy
scales with the superfluid phase (Eq. (2.6)) for the 2D Bose gas. Therefore, in some
sense the behaviour of Θ(x, t) in the 2D complex plane is analogous to θi in the 2D
spin plane. It is worth noting however, even though the superfluid density is analogous
to the spin magnitude, the XY spin magnitude is fixed whereas the density of the 2D
Bose gas can fluctuate. These differences tend to manifest in nonuniversal ways e.g.
differences in the value of TBKT .
2.3.3 The High Temperature Limit
We now investigate the XY model analytically to see how the first order spin correlation
function g1(i, j) behaves in the high temperature limit (we consider the low temperature
limit in the next section). This correlation function is conventionally defined as
g1(i, j) = ℜ〈ei[θi−θj ]〉= 〈cos(θi − θj)〉
=1
Z
∫
DNθ cos(θi − θj) exp[−βHXY (θ)] (2.8)
with β = 1/kBT and∫
DNθ refers to a functional integral over all possible configura-
tions of the spin field i.e.∫
DNθ ≡∫
...
∫ 2π
0
dθ1...dθN
(2π)N, (2.9)
21
with N being the number of lattice sites and θ = [θ1, θ2, ..., θN ]. Using Eq. (2.4) the
partition function for the XY-model is given by the functional integral
Z(β) =
∫
DNθ exp
βJ
2
∑
〈i,j〉
cos(θi − θj)
. (2.10)
In the high temperature limit thermal fluctuations dominate the spin interactions, i.e.
βJ/2 ≪ 1, and we make the expansion
exp
βJ
2
∑
〈i,j〉
cos(θi − θj)
≈ 1 +βJ
2
∑
〈i,j〉
cos(θi − θj) (2.11)
which gives
Z(β) ≈ 1, (2.12)
and we are now in a position to evaluate Eq. (2.8). With i and j being nearest
neighbours one finds that
g1(i, j) ≈ βJ
2(2.13)
and it is not too difficult to show in general that
g1(i, j) ≈(
βJ
2
)rij
= exp
(−rij
ξ
)
,
ξ−1 = ln(2kBT/J), (2.14)
where rij = Rij/a with Rij being the two point separation and a is the lattice spacing.
Therefore in the high temperature limit correlations decay exponentially and LRO is
absent.
2.3.4 The Low Temperature Limit
In the the low temperature limit spin interactions dominate thermal fluctuations, thus
the dominant microstates are those in which the spin angle θ varies slowly across the
lattice. This allows us to replace θi by the field θ(x). For large two point separation
|ρ| ≡ ρ ≫ al, where al is the lattice spacing and ρ is the 2D radial vector4, the
correlation function can now be written as5
g1(ρ) = ℜ〈ei[θ(ρ)−θ(0)]〉
=1
Z
∫
Dθ(x)ei[θ(ρ)−θ(0)]e−βH′
XY . (2.15)
4Here ∆x ≡ x− x′ = ρ because we have defined the origin such that x
′ = 0.5The functional integral is over all possible configurations of the field θ(x).
22
The Hamiltonian is equivalent to Eq. (2.7) but we redefine our energy origin to eliminate
the constant term, i.e.
H ′XY =
J
2
∫
d2x|∇θ|2. (2.16)
Next, we Fourier expand θ(x) in terms of real momentum amplitudes6 φ(k),
θ(x) =
∫
d2k
(2π)2φ(k) exp(ik · x), (2.17)
with k = kx, ky being the wave vector. In the low temperature limit we can forget
that θ(x) is restricted between 0 and 2π and allow the integration with respect to θi
to run from −∞ to ∞ since redundancies are cancelled in the division by Z.
By replacing the integral over all k by the infinite sum,
∫
d2k
(2π)2⇐⇒
∑
ki
≡∑
i
, (2.18)
equation (2.15) can now be evaluated by the following gaussian identity
∫ ∞
−∞
dNy exp
[
−1
2
∑
i,j
yiAijyj +∑
i
Biyi
]
=
(
(2π)N
det[A]
)1/2
exp
[
1
2
∑
ij
BiA−1ij Bj
]
,
(2.19)
where y ≡ [y1, y2, ..., yN ] and A · A−1 ≡ I (I is the identity matrix). For yi we
substitute the amplitude φ(ki) and for A we substitute the square diagonal matrix
A = diag(k21, k
22, ..., k
2N)J/kBT. After these substitutions and some algebra, Eq. (2.15)
now reads
g1(ρ) = exp
[
−kBT
2J
∑
i
1
k2i
exp(iki · ρ) − 12
]
, (2.20)
where |ki| ≡ ki. Making use of the equipartition theorem,
〈φ2(ki)〉 ≈kBT
Jk2i
, (2.21)
and again using relations (2.17) and (2.18) it is not difficult to show that
g1(ρ) = exp
[
−kBT
J
∫
d2k
(2π)2
(1 − eik.ρ)
k2
]
= exp
[
−kBT
2πJ
∫ Λ
0
dk
k
1
2π
∫ 2π
0
dφ(
1 − eik·ρ)
]
, (2.22)
6The approximation that φ(k) be real restricts θ(x) to be symmetric about the origin. This requires
us to keep any two points within the same quadrant of the xy plane when evaluating g1(ρ), this is
always satisfied in our case since one point is fixed at the origin.
23
where Λ ∼ a−1l is the high wave number cutoff due to the lattice spacing and φ is
the polar angle. Using the identity k · ρ ≡ kρ cos φ (where ρ ≡ |ρ|) and making the
substitution kρ→ u one obtains
g1(ρ) = exp
[
−kBT
2πJ
∫ Λρ
0
du
u
1
2π
∫ 2π
0
dφ (1 − cos[u cosφ])
]
. (2.23)
The integral with respect to u converges in the limit k → 0, hence it is not necessary
to worry about infra-red regularisation. Furthermore, in the limit ρ≫ al the cos term
cycles rapidly about zero and thus to leading order it can be ignored, this gives
g1(ρ) ≈(
1
Λρ
)η
, (2.24)
where
η =T
4TBKT,
TBKT =πJ
2kB,
are the decay exponent and characteristic transition temperature respectively. A more
precise numerical treatment of Eq. (2.23) is provided by Chaikin and Lubensky [13],
which gives an expression marginally different to that of Eq. (2.24), i.e.
g1(ρ) = 0.890
(
1
Λρ
)η
. (2.25)
The important result here is that spin-spin correlations decay algebraically as
a function of separation ρ, a phenomenon known as topological order. Although
limρ→∞ g1(ρ) = 0 at all finite temperatures, which implies no LRO, algebraic decay
is much slower than the exponential rate of the high temperature regime, this is why
topological order is often referred to as quasi-long-range order. Significantly, a manifes-
tation of topological order is a large spin rigidity parameter, which is associated with
a finite superfluid component that increases with decreasing temperature [13]. In the
next section we investigate the nature of the phase transition between these low and
high temperature regimes.
2.3.5 The BKT Transition - Critical Temperature
For the low temperature limit we neglected that the angle θ(x) is periodic i.e.
θ = θ + 2πn ∀ n ∈ Z. (2.26)
24
Figure 2.2: Two configurations of a singly quantised vortex. Note that
it is the relative (not the absolute) difference of spin angle between
nearest neighbours which is important, hence both configurations here
are energetically equivalent.
Such an approximation is not appropriate in the intermediate transition region, indeed
this periodicity is crucial for determining the behaviour of the transition from algebraic
to exponential decay of correlations. A manifestation of this periodicity is that the line
integral of the phase gradient around any closed loop must be an integer multiple of
2π i.e.∮
∇θ(x) · dl = 2πq, q ∈ Z, (2.27)
where q is refered to as the topological charge or winding number. Any nontrivial solu-
tion to Eq. (2.27) indicates the presence of at least one vortex. Vortex configurations
are topologically invariant because they cannot be continuously deformed to a spin
wave configuration which has the trivial solution of Eq. (2.27) for all possible path
integrals. Two examples of a singly quantised vortex are displayed in Fig. 2.2.
We now investigate the likelihood for isolated vortices to exist and how this likeli-
hood depends on temperature. First, consider the energy of a single radially symmetric
vortex of charge q centered at the origin. If we define ϕ = tan−1(y/x) then we get
θ = qϕ and it is easy to show that
|∇θ(ρ)| =|q|ρ. (2.28)
Substituting this into Eq. (2.16) we get for the vortex energy,
Eq =q2J
2
∫ L
ac
d2x
r2+ Ec
= πq2J ln(L/ac) + Ec, (2.29)
25
with ac being the lower cutoff, below which is the core that should not be integrated,
L is the system radius and Ec is the energy of the core included separately.
We are now in a position to calculate the free energy F1 of a singly quantised vortex
(q = 1) [35],
F1 = E1 − S1T (2.30)
hence determining the temperature at which isolated vortices becomes favourable. Not-
ing that the system multiplicity Ω1 is approximately the number of ways of positioning
the vortex core within the system, i.e. Ω1 ≈ (L/ac)2 we get
F1 = Ec + (πJ − 2kBT ) ln(L/ac)
= Ec + 2kB ln(L/ac)(TBKT − T ), (2.31)
where TBKT is the critical temperature defined in Eq. (2.24). As the system size
tends to infinity (L → ∞) the energy of the core can be neglected and also since
ln(L/a) → ∞ the value of F1 switches from −∞ to +∞ for a small change in T about
TBKT . According to the 2nd law the system will attempt to minimise its free energy,
this implies that for T < TBKT the single vortex is unfavourable whereas for T > TBKT
it’s existence is favoured and the ground (vortex free) state is no longer stable. The
important relationship for this transition is that both the energy and the entropy scale
with the system size in the same way.
These isolated vortices have the effect of destroying topological order and super-
fluidity above TBKT , and so are responsible for transition to the high temperature
phase where two-point correlations decay exponentially as a function of separation. At
T = TBKT Eq. (2.24) implies that the algebraic decay constant η = 1/4. We also note
that as a result of this vortex proliferation, as the transition is crossed from above the
superfluid density suddenly jumps to a finite value. Superfluidity in the XY model is
associated with spin wave rigidity.
2.3.6 The BKT Transition - Unbinding of Vortex Pairs
It turns out that large systems seldom have just one vortex but instead have many,
and these are usefully described as an interacting gas of vortices behaving analogously
to charged particles in a 2D world.
With the origin of the coordinate axes coinciding with a vortex core Eq. (2.27) can
be rewritten as∮
q
ρρ · n dl = 2πq, (2.32)
26
where dl ≡ |dl|, while ρ and n respectively are the radial and normal unit vectors of the
local frame at each point along the line of integration7. This is analogous to Gauss’s
law in electrostatics for 2D with q behaving like the electric charge and Eelec ≡ qρ/ρ
is like the 2D electric field. Neglecting the core energy in Eq. (2.29) we see that the
energy of a vortex reads
Eq =
∫
J |Eelec|22
d2x, (2.33)
with J |Eelec|2/2 being the analogue of the electrostatic energy density. Hence the
vortices of a spin system can be mapped onto a 2D system of electrostatic charges,
which allows us to obtain the following expression for total interaction energy between
numerous vortices,
Uint = −πJ∑
i6=j
qiqj ln∆xij
ac, (2.34)
where ∆xij is the separation distance between vortices i and j. Thus vortices of like
charge repel and vortices of opposite charge attract. Significantly, this has the con-
sequence that a bound pair of opposite charge has a lower energy than that of an
isolated (free) vortex. In other words, for T ≪ TBKT vortices are absent, but at higher
temperatures T < TBKT vortices are found, but only in tightly bound pairs. As the
critical temperature is approached the average size and number of pairs increases until
T > TBKT , afterwhich pairs unbind and free vortices proliferate, destroying the topo-
logical order and superfluidity. Below TBKT , bound pairs do not destroy the algebraic
decay of correlations (Eq. (2.24)) because each vortex tends to cancel each other, anal-
ogous to the behaviour of an electric dipole, although bound pairs do tend to decrease
the superfluid component but do not destroy it.
To conclude §2.3 we emphasise that although LRO is not possible for the marginal
regime, D = s = 2, a BKT transition to a quasi-long-range ordered phase can occur. A
full renormalisation group treatment [5, 35, 36] is needed to properly quantify properties
such as the superfluid fraction and finite size effects.
7Using this electrostatic analogy one can easily see that in the case of many vortices, Eq. (2.27)
generalises to∮
∇θ(x).dx = 2πQ, where Q =∑
iqi, is the signed sum of all vortex charges confined
inside the closed integration loop.
27
2.4 Ideal 2D Bose Gas
2.4.1 Bose Einstein Condensation
In three-dimensional ideal systems a BEC phase exists regardless of whether a system
is trapped or uniform, however it is not clear that this is true for lower dimensional
cases. Bagnato and Kleppner [3] made predictions for when an ideal BEC is possible in
2D for the case of an algebraic trapping potential U(x) ∼| x |ζ , for power law constant
ζ . They found that BEC occurs for any finite value of ζ , but does not occur in the
uniform system, i.e. in the limit ζ → ∞.
In the 2D uniform system, the absence of BEC in the ideal gas can be understood
if one derives the excited state population Nex as a function of µ, by multiplying the
Bose-Einstein distribution by the density of states and then integrating with respect to
energy. In this case the density of states is a constant and the subsequent integration
yields the excited state phase space density,
nexλ2dB ≡ Nexλ
2dB
A= − ln(1 − eβµ), (2.35)
where A is the area of the system and
λdB ≡ h√2πmkBT
(2.36)
is the thermal deBroglie wavelength. Significantly, the excited state population diverges
i.e. Nex(µ → 0, T ) → ∞, which implies that Nex can never saturate and hence BEC
cannot occur in the 2D uniform system8.
The other special case of interest here is the harmonic potential, i.e. ζ = 2, being
the case most commonly realised in experiments. The density of states is directly
proportional to the energy and a similar integration as above yields,
Nex(µ = 0, T ) =π2
6
(
kBT
~ω
)2
, (2.37)
with ω ≡ √ωxωy and ωx, ωy are the angular trapping frequencies in the x, y directions
respectively. Importantly, Eq. (2.37) is finite which implies that the thermal cloud
becomes saturated when N > Nex(µ = 0, T ) and surplus atoms must exist in the
condensate mode i.e. N0 = N − Nex(µ = 0, T ), hence Nex(µ = 0, T ) is approximately
8This result is just a specific demonstration of the Mermin, Wagner and Hohenberg theorem.
28
the critical number N2DBEC . For later comparison we rearrange Eq. (2.37) in terms of
the 2D critical temperature,
T 2DBEC =
~ω
kBπ
√6N. (2.38)
Even though BEC occurs in the 2D harmonic trap for the ideal gas, we would like to
emphasise this is fragile as we now discuss. In an attempt to perturbatively extend the
ideal prediction to the regime of very weak interactions one might try the local density
approximation. This approximation treats every point as a locally uniform system with
an effective local chemical potential µeff = µ− V (x), which from Eq. (2.35) gives
nex(x)λ2dB = − ln
(
1 − eβ[µ−V (x)])
. (2.39)
Integration of Eq. (2.39), for µ = 0, over all space again yields a finite Nex, given
by Eq. (2.37). The problem arises when one notices that at the origin V (0) = 0
in Eq. (2.39), indicating a divergence of the central density. The existence of any
finite level of interactions would then cause a divergence of the central interaction
energy. Hence, we conclude that interactions cannot be treated perturbatively and the
existence of BEC in the ideal gas does not necessarily imply the existence of BEC in
the interacting regime9.
2.4.2 Quasi-Two-Dimensional
The previous section dealt with the purely 2D regime where ~ωx, ~ωy ≪ kBT and
~ωz ≫ kBT , i.e. excitations in the axial (z) direction are completely frozen out. How-
ever, all relevant experiments to date [14, 29, 37, 55] are in fact quasi-2D (see §1.3),
for example in the recent NIST experiment [14] they report that ~ωz ≈ kBT/2 which
implies a significant fraction of atoms occupy excited axial modes. Although, so long
as Eq. (1.8) is satisfied we expect the critical modes to be confined to 2D, thus these
quasi-2D systems are essentially purely 2D systems embedded within a 3D thermal
cloud. For a relevant comparison with experiment we now focus on finding the quasi-
2D analogue of the 2D critical temperature [Eq. (2.38)], again for the harmonically
trapped regime.
The average occupation of each ideal harmonic oscillator mode is given by the
Bose-Einstein distribution (1.1),
NBE(nx, ny, nz) =1
eβ(nx~ωx+ny~ωy+nz~ωz−µ) − 1, (2.40)
9This should be contrasted with the 3D case where nex(0)λ2
dB= 2.612 at the critical temperature
(Tc).
29
and when µ = 0 the total population of the thermal cloud can be written as
Nex(µ = 0, T ) =∞
∑
nz=0
∞∑
nx,ny=0
NBE(nx, ny, nz)
−NBE(0, 0, 0),
nx, ny, nz ∈ Z, (2.41)
again making a similar identification Nex(µ = 0, T ) ≈ NQ2DBEC is the quasi-2D crit-
ical atom number. Equation (2.41) may be numerically summed by introducing a
high energy cutoff, above which the contribution to NQ2DBEC is negligible. Furthermore,
Eq. (2.41) can be numerically inverted to obtain the ideal quasi-2D condensation tem-
perature T0 as a function of the total number of atoms in the system N .
2.5 Interacting 2D Bose Gas
As discussed in §2.3.2 the interacting uniform Bose gas is of the same universality
class as the XY model. In this section we specialise the XY arguments for the BKT
transition to the Bose gas.
Again we calculate the free energy of an isolated singly quantised vortex [Eq. (2.30)]
but this time in the Bose gas. Noting from Eq. (2.5) that the velocity magnitude reads,
v(ρ) =~
mρ, (2.42)
excluding the core we get for the vortex energy,
E1 =nSF ~
2
2m
∫ L
ξ
2πρ
ρ2dr
=π~
2nSF
mln
(
L
ξ
)
, (2.43)
where nSF is the areal superfluid density and ξ is the vortex core radius10. Analogous to
the XY model, the entropy may be approximated as the number of ways of positioning
the vortex core,
S1 ≈ kB ln
(
L2
ξ2
)
. (2.44)
10Note, the superfluid velocity v(x) = (~/m)∇Θ(x) (Eq. (2.5)) is irrotational since it is proportional
to the gradient of the phase Θ which is a well behaved function, hence the superfluid density must be
zero at the centre of every vortex core.
30
Combining these we arrive at an expression for the free energy11,
F1 ≈ 2kB ln(L/ξ)(TBKT − T ),
TBKT =h2nSF
8πmkB, (2.45)
which is analogous to Eq. (2.31). Exploiting the connection with the XY model, for
T < TBKT , two point correlations should decay algebraically (analogous to Eq. (2.24)),
i.e. the normalised correlation function,
g1(∆x) ≡ 〈Ψ†(x)Ψ(x′)〉√
n(x)n(x′), (2.46)
should obey the relation
g1(∆x) ∝(
1
∆x
)η,
η =T
4TBKT
≡ 1
nSFλ2dB
(2.47)
where n(x) represents the areal density at position x and ∆x = x − x′. For later
comparison it is useful to rewrite Eq. (2.45) as
F1
kBT≈ 1
2ln
(
L
ξ
)
(λ2dBnSF − 4). (2.48)
where now we have the critical phase space density,
λ2dBnSF = 4, (2.49)
that is dependent on both the temperature and superfluid density. Hence, we expect
that for T > TBKT the system consists of a normal phase of free vortices with no
superfluidity and two point phase correlations decay exponentially. At T = TBKT
vortices become paired, there is a discontinuous jump in the superfluid phase space
density to λ2dBnSF = 4 and
g1(∆x) ∝ (∆x)−1/4. (2.50)
For T < TBKT there is an increasing superfluid fraction, vortices are paired and two
point phase correlations decay algebraically. A schematic of the BKT transition is
shown in Fig. 2.3.
It is worth noting that for finite systems, pair stability could be reduced since
the logarithmic dependence of F1 on L [Eq. (2.45)] implies that free vortices are not
11Note that in the limit L → ∞ Eq. (2.45) becomes exact as L dominates ξ in the log term.
31
+
+
+
+
-
-
-
-+
-
++
+
+
+ -
-
- -
TBKTTemperature
+
- +
-
- finite superfluid density
- algebraically decaying correlations
- vortices exist as vortex-antivortex pairs
- normal system
- exponentially decaying correlations
- free vortices
Figure 2.3: Schematic showing the BKT and normal phases versus
temperature for the uniform system.
necessarily that improbable. Hence, we expect smaller systems to exhibit a less well
defined transition. Importantly, we reiterate that these results are for the uniform
system and it is by no means certain to what extent they should be applicable to the
harmonically trapped Bose gas.
32
Chapter 3
Theoretical Developments for the
2D Bose Gas
3.1 Effective 2D Interaction in the Bose Gas
To start: In tightly confining a gas of particles to make a low dimensional system,
special consideration needs to be paid to the effect this may have on the collisions. For
a dilute Bose gas, Petrov et al. [44] showed that when the 3D s-wave scattering length,
a, is much smaller than the confinement length of the axial direction (lz = [~/(mωz)]1
2 )
interactions become energy independent.
They showed: For low energy modes, ka ≪ 1 (k being the wavevector), where
collisions are dominated by s-wave scattering, the system can be described by an
effective 2D interaction parameter given by
g =2√
2π~2
m
1
lz/a+ (1/√
2π) ln(1/πk2l2z). (3.1)
In experiments to date the confinement has typically been in the regime lz ≫ a, for
which the logarithmic term is unimportant, giving the momentum independent 2D
coupling constant
g =
√8π~
2a
mlz, (3.2)
which depends on the 3D s-wave scattering length and the tightness of confinement in
the axial direction. For later discussions it is convenient to define the dimensionless
effective 2D interaction constant as
g ≡ mg
~2=
√8π
a
lz. (3.3)
33
3.2 Mean-field Theory of the Degenerate Regime
As discussed in §2.4.1, the introduction of a harmonic trap modifies the density of states
sufficiently for BEC to form in 2D ideal systems. In the late 1990’s and early 2000’s
the question of whether BEC is possible with the inclusion of interactions became a
contentious topic, with several conflicting predictions made using mean-field theory. In
this section we briefly summarise the main findings of this period.
In 2000, Bhaduri et al. [6] presented a self-consistent Hartree-Fock (HF) theory of
the 2D Bose gas. This semiclassical model describes interactions by using the effective
2D interaction constant given by equation (3.2). One of their main results is that for
any positive value of the interaction constant i.e. g > 0, the uncondensed phase has
a solution for all finite temperatures, and they came to the conclusion “... there is no
strict phase transition for such a system, no matter how weak the repulsion”. In other
words, as the temperature decreases, the chemical potential smoothly increases and
the thermal cloud cannot be saturated for any finite temperature.
In 1998 and 2002 Mullin et al. [25, 40] published works describing results of a
semiclassical Hartree-Fock-Bogoliubov (HFB) theory of the 2D trapped regime. They
found that in the HF approximation the system has two solutions for all temperatures,
one with the condensate included in the model and the other without, this is also
true in the thermodynamic limit. However, the solution for the model that includes
condensate, consistently has the lower free energy at all temperatures, supporting the
existence of a condensate. In contrast, when phonons are included by using the more
complete HFB treatment, only the uncondensed solution remains “Our results confirm
that low-energy phonons destabilise the two-dimensional condensate”.
Later, in 2004 Gies et al. [28] presented results of a HFB model that did not make
the semiclassical simplification. In contrast to the results of Mullin et al. [25], they
did find solutions for the condensed phase in the presence of phonons. They concluded
that “... the presence of the trap stabilizes the condensate against long wavelength
fluctuations”.
In conclusion, the mean-field theoretic description of the 2D Bose gas has made a
range of different predictions for the existence of the BEC phase. An important point
to keep in mind is that if critical fluctuations occur then the mean-field approach is
invalid. We also note that the Mermin-Wagner-Hohenberg theory does not rule out
the possibility of a condensate in the trapped system as the trap constrains the energy
levels to be discrete with an infrared cutoff. The implications of this is that for a
34
system with broken symmetry Eq. (2.1) should not be integrated and the density of
long wavelength Goldstone Bosons does not diverge.
3.3 Universality and Critical Physics of the Uni-
form Bose Gas
In §2.3 we discussed the BKT transition of the XY model and §2.5 considered how many
of the properties of the XY model apply to the homogeneous Bose gas. However there
are important differences, notably that the XY spin magnitude is constant whereas the
density of the Bose gas fluctuates. This section mainly discusses work by Prokof’ev et
al. [48] that quantitatively deals with these specific aspects of the 2D Bose gas using
the concept of universality and numerical Monte Carlo simulations of the classical |ψ|4
model.
3.3.1 Critical Phase Space Density
Equation (2.48) relates the superfluid density and the temperature at the transition
point, however, in practice an important question is how to relate the superfluid density
to the total density. To this end, Refs. [26, 34, 46, 48] suggest that at TBKT the critical
total phase space density of the weakly interacting Bose gas takes the universal form,
nBKTλ2dB = ln
(
ξ
g
)
, (3.4)
where g is the dimensionless interaction strength defined by Eq. (3.3) and ξ is a constant
that is outside existing analytical treatments, since this is a non-universal property of
the system in the critical region, and must be found numerically. This expression
should be compared against Eq. (2.48) for the superfluid phase space density which
implies that at TBKT ,
nSFλ2dB = 4. (3.5)
The relation (Eq. (3.4)) may be obtained by an order of magnitude analysis of the
classical-field |ψ|4 model with the effective long-wavelength Hamiltonian,
H [ψ] =
∫
~2
2m|∇ψ|2 +
g
2|ψ|4 − µ|ψ|2
d2x, (3.6)
where ψ is the complex classical field and µ is the chemical potential. This model
is relevant because all weakly interacting |ψ|4 models, whether they are quantum or
35
classical, continuous or discrete, display the same universal behaviour of the long-
wavelength critical modes [4, 46]. This result follows from the fact that only the
long-wavelength modes are strongly interacting, whereas the short-wavelength modes
are almost free. Hence, the various |ψ|4 models differ only by their high energy ideal
modes, which in turn produces a model dependent nBKT (T ) and ξ for Eq. (3.4).
It follows that the critical density of model A, n(A)BKT , can be related to that of model
B by subtracting and adding the model specific ideal contributions, i.e.
n(A)BKT − n
(B)BKT =
∫ ′ d2k
(2π)2
[
n(A, ideal)k − n
(B, ideal)k
]
, (3.7)
where “′” indicates the integration over all noninteracting modes with momentum k.
With this in mind, Prokof’ev et al. [48] calculated ξ for the classical field Hamiltonian
(3.6) on a simple square lattice model using Monte Carlo simulations. Simulation re-
sults for systems of various size and interaction strength were fitted to scaling relation-
ships predicted by the well known Kosterlitz-Thouless renormalisation group theory,
for the purpose of removing the non-universal finite size effects. Relation (3.7) was
then used to find ξ for the infinite uniform Bose gas resulting in,
nBKTλ2dB = ln
(
ξ
g
)
, ξ = 380 ± 3. (3.8)
3.3.2 Superfluid and Quasicondensate
Even though condensate is absent from 2D uniform systems, density fluctuations are
in fact suppressed for the degenerate regime. A quantity instructive for quantifying
density fluctuations is given by [34, 47, 48]
nQ(x) =√
2〈n(x)〉2 − 〈n(x)2〉, (3.9)
which is referred to as the quasicondensate density. Systems in the normal phase have
Gaussian density fluctuations i.e. they obey Wick’s theorem and hence nQ = 0, at
the opposite extreme, systems without density fluctuations are purely quasi-condensed
with nQ = n. We note that the presence of a condensate implies the presence of
a quasicondensate whereas the reverse is not necessarily true (since quasicondensate
does not necessarily imply ODLRO).
Both quasi-condensation and superfluidity are properties of the strongly interacting
critical modes therefore, as for the critical phase space density, it is not too surprising
that these quantities have also been demonstrated as being universal among weakly in-
teracting |ψ|4 models. This universality was shown by Prokof’ev et al. [47] using Monte
36
0 0.5 1
0.2
0.4
0.6
0.8
T/T
n /ns
c
- 10- 10- 10
- 10
-2
-1
-3
-5
Figure 3.1: (Taken from Ref. [47]). Superfluid fraction of the uniform
system for different dimensionless interaction strengths g. Dashed
lines represent hybrid Monte Carlo-meanfield prediction discussed in
text. Tc ≡ TBKT , nS ≡ nSF .
0 0.5 1 1.5 2
0.2
0.4
0.6
0.8
T/T
n /n0
c
10
10
10
10
-1
-2
-3
-5
Figure 3.2: (Taken from Ref. [47]). Quasicondensate fraction of
the uniform system for different dimensionless interaction strengths
g. Dashed lines represent hybrid Monte Carlo-meanfield prediction
discussed in text. Tc ≡ TBKT , n0 ≡ nQ.
37
Carlo methods, their results completely characterise the superfluid, quasi-condensate
and total density of the infinite uniform system in the vicinity of the critical point.
Diagrams displaying superfluid and quasicondensate density as a function of chemical
potential are shown in Figs. 3.1 and 3.2. Important points to note: First, that Fig. 3.1
displays a sudden jump in the superfluid density at T = TBKT , in agreement with BKT
theory. Second, Fig. 3.2 displays a prominent quasicondensate density at T = TBKT
and a significant quasicondensate tail exists well outside the superfluid region, even
for a dimensionless interaction constant as small as g ∼ 10−5. Quasicondensate has
also been observed experimentally [51], via the reduction of three-body losses in spin
polarised atomic hydrogen on helium films, but it is not clear if it exhibits a tail.
Using Monte Carlo simulations of the uniform system, Prokof’ev et al. [47] showed
that in the highly dengenerate regime (T → 0) the quasicondensate obeys the meanfield
relation,
(nQ + 2n′)g = µ (3.10)
where
n′ = n− nQ (3.11)
is the non-quasicondensate (normal) density. They deduce and their simulations con-
firm that quasicondensate and condensate are indistinguishable at the meanfield level.
Using this idea they construct a combined Monte Carlo-meanfield model by replac-
ing condensate in a conventional meanfield theory by quasicondensate obtained from
Monte Carlo simulations. The results of this hybrid theory are provided (dashed line)
in Figs. 3.1 and 3.2 and show good agreement with the complete Monte Carlo results
for T/TBKT ∼ 1/2 and below. The discrepancy for T/TBKT > 1/2 provides a measure
for the extent of the critical region as we discuss in the next section.
3.3.3 The Extended Critical Region
The meanfield approach has proven to be a useful tool for the analysis of the dilute
Bose gas in 3D, however this theory is inadequate for the description of the critical
region. This problem is much more pronounced in 2D where the temperature range of
the critical region ∆T is of order the critical temperature TBKT , and remains significant
even for minuscule interaction strength, i.e. [26, 47]
∆T
TBKT∼ 1
ln(~2/mg)=
1
ln(1/g). (3.12)
38
Furthermore, Prokof’ev et al. [47] find that the conventional meanfield theory result
nSF
n= 1 − T
TBKT
, (3.13)
which predicts a linear relation between nSF and T , is not applicable in 2D as the
Monte Carlo results reveal a nonlinear relationship (see Fig. 3.1).
The extensive critical region of the 2D Bose gas can be approximately visualised
from Figs. 3.1 and 3.2. The good agreement of the hybrid Monte Carlo-meanfield model
for T/TBKT . 1/2 gives an indication of the lower boundary for the critical region.
Whereas for the upper boundary, extrapolation of this meanfield theory for nSF to
the region T > TBKT finds an intersection with the temperature axis at T/TBKT ≈ 1.5
when g = 10−5, and for larger interactions strengths the upper boundary is significantly
greater. Prokof’ev et al. [47] discuss the critical regime in the 2D trapped Bose gas and
conclude that, due to the vast critical regime, “... when the density at the trap centre
is tuned to the critical point [Eq. (3.8)], practically the whole density profile finds itself
in the fluctuating (critical) region where meanfield equations do not work.”
In conclusion, the critical region of the 2D Bose gas is large and extends significantly,
both below and above the critical temperature. Hence, from a quantum mechanical
perspective meanfield theory is inapplicable to the region of most interest. Quantum
Monte Carlo methods are prohibitively computationally intensive and have only in-
vestigated limited aspects of the 2D Bose gas. This provides significant motivation to
find a method that is both applicable to the critical region while still being computa-
tionally efficient, this is the motivation behind our work in §4. Meanfield theory may
be inapplicable in the degenerate regime but it remains a useful tool for describing
the quasi-2D trapped Bose gas well beyond the critical regime T ≫ TBKT where the
quasicondensate density becomes negligible, this high temperature meanfield theory is
the subject of our work in §5.
39
40
Chapter 4
The Degenerate Quasi-2D Trapped
Bose gas
4.1 Introduction
As discussed in §3.3.3 the 2D Bose gas is particularly difficult to model in the degen-
erate regime, as the critical region extends over a large temperature range on both
sides of the transition point. Meanfield theory is inadequate for the description of
these strongly fluctuating critical modes and Monte Carlo methods are prohibitively
slow to numerically implement. In this chapter we present results for the description
of the quasi-2D harmonically trapped Bose gas, obtained via classical field (c-field)
techniques, and relate these to experiments and other theory. The key to our method
is that we treat the critical modes non-perturbatively by using an inhomogeneous clas-
sical |ψ|4 model. For a weakly interacting system with a given trapping potential, both
quantum and classical |ψ|4 models will give a similar description of the critical modes
provided their occupation is large enough to neglect quantum fluctuations. Impor-
tantly, our method is computationally efficient, allowing for a detailed study of the low
temperature components, and also allows us to simulate system dynamics.
4.2 C-Field Techniques
4.2.1 Formalism
Here we briefly outline the c-field formalism (developed by Davis et al. [11, 18, 21,
22, 27]) applied specifically to the quasi-2D trapped Bose gas by Simula, Blakie and
41
......
Ecut
ClassicalRegion
IncoherentRegion
Energy
Figure 4.1: Schematic representation of the partition of modes by
Ecut. The highly occupied critical modes are contained within the
classical (or c-field) region.
Davis [53, 54]. The general idea is to divide the modes of the system, according to
energy, into two regions separated by an energy cutoff Ecut as depicted in Fig. 4.1. The
low energy (highly occupied) modes are described by a classical field, hence we refer to
the low energy region as the c-field region. Whereas the high energy (sparsely occupied)
modes may be treated by meanfield theory, we refer to this high energy region as the
incoherent region.
The Classical Field Region
Formally, we begin with the second quantised Hamiltonian of the dilute Bose gas given
by Eq. (1.2). We choose to divide the Bose field operator into two parts,
Ψ(r) = ψC(r) + ψI(r), (4.1)
where ψC(r) is the c-field representing the low energy modes, and ψI(r) is the field
operator of the high energy incoherent region. The energy cutoff is implemented via the
division of our numerical basis, the 3D single particle harmonic oscillator eigenvectors
φn(r) with energy En, into low and high energy regions,
ψC(r) ≡∑
n∈C
anφn(r), (4.2)
ψI(r) ≡∑
n∈I
anφn(r), (4.3)
42
where an are Bose annihilation operators and the c-field and incoherent regions are
defined respectively by
C ≡ n : En ≤ Ecut, (4.4)
I ≡ n : En > Ecut. (4.5)
We have replaced the mode operators an by the complex amplitudes cn for the c-field
region, that is set
ψC(r) ≡∑
n∈C
cnφn(r), (4.6)
which is known as the classical field approximation. The value of Ecut is chosen such
that the average occupation of modes at the cutoff, nmin, is of order one or greater.
This ensures that all modes in the c-field region are appreciably occupied, thus averting
an ultraviolet divergence and justifying the classical field approximation. In contrast,
the incoherent region contains many sparsely occupied modes, for which the classical
field approximation would be inappropriate. However, these non-critical modes of the
incoherent region are described well by meanfield theory.
We now focus on the c-field region, the general idea is to treat the c-field region
as an independent system in diffusive and thermal equilibrium with the incoherent
region. By neglecting dynamical couplings between regions the equation of motion for
ψC becomes the projected Gross-Pitaevskii equation (PGPE)
i~∂ψC
∂t= H0ψC + PCU0|ψC|2ψC, (4.7)
where
PCF (r) ≡∑
n∈C
φn(r)
∫
d3r′φ∗n(r′)F (r′), (4.8)
is the projection operator, which constrains particles to the c-field region.
Simulation Procedure
Our approach for obtaining equilibrium microstates relies on the PGPE (Eq. (4.7))
being ergodic. The idea is that all microstates are equally probable and that nonlinear
interactions provide the mode mixing required for microstate exploration. For a given
Ecut, U0 and harmonic trapping potential initial microstates are created with two im-
portant constants of motion, total energy EC and total atom number NC (of the c-field
43
region) given by
EC =
∫
d3rψ∗C
(
H0 +U0
2|ψC|2
)
ψC (4.9)
NC =
∫
d3r|ψC(r)|2. (4.10)
Since the PGPE is ergodic the choice of initial state is not crucial but will affect
the time taken for the system to reach equilibrium. To obtain initial state ηI(x), we
choose to mix a Thomas-Fermi approximation for the 2D ground state ηTF (x) [15] with
a high energy randomised state ηrand(x),
ηI(x) = α1ηTF (x) + α2ηrand(x), (4.11)
where α1 and α2 are parameters to be adjusted to reach the desired EC and NC. We
choose to produce ηrand(x) by generating random complex numbers for cn of Eq. (4.6)
and scaling these such that∑
n∈C
|cn|2 = NC. (4.12)
For practical purposes, ηrand(x) is approximately orthogonal to ηTF (x) allowing us to
make use of the relation
α2 =√
1 − |α1|2. (4.13)
We judge whether a system has thermalised by averaging condensate fraction and
temperature (we discuss the evaluation of these quantities later on) over time windows
and monitoring whether fluctuations of these parameters have settled down about equi-
librium values. The suitability of Ecut may be evaluated a posteriori, with simulations
accepted or rejected based on the requirement that all c-field modes are appreciably
occupied. With a valid c-field region we then use the equilibrium temperature, chemi-
cal potential and density to add the atoms of the sparsely occupied incoherent region
using a Hartree-Fock description.
The Incoherent Region
The high energy atoms above Ecut are relatively easy to deal with as the modes are
sparsely occupied and the particles are almost free, we thus implement a semiclassi-
cal Hartree-Fock model as used by Refs. [53, 54]. Due to the quasi-2D nature of our
system, i.e. ~ωx, ~ωy ≪ kBT . ~ωz, the xy-plane (radial plane) may be treated semi-
classically, but the z-direction (axial direction) must be treated quantum mechanically.
44
The Hartree-Fock expression for the areal density of the incoherent region in the j-th
axial mode is
nj(x) =1
(2π)2
∫
ΩI
d2k1
exp β(ǫj(x,k) − µ) − 1, (4.14)
The Hartree-Fock energies are
ǫj(x,k) =~
2k2
2m+m
2(ω2
xx2 + ω2
yy2) + j~ωz + 2g0jnC(x) + 2
∞∑
k=0
gkjnk(x), (4.15)
where
nC(x) =
∫
dz|ψC(r)|2, (4.16)
is the c-field areal density and
gkj =4πa~2
m
∫
dz|ξk(z)|2|ξj(z)|2, (4.17)
with ξk(z) being the axial bare harmonic oscillator states. The region of integration
for Eq. (4.14) is restricted by the cutoff energy, i.e.
ΩI =
x,k :~
2k2
2m+m
2(ω2
xx2 + ω2
yy2) + j~ωz ≥ Ecut
. (4.18)
We are now in a position to calculate the total areal density of the system,
n(x) = nC(x) + nI(x)
= nC(x) +∞
∑
j=0
nj(x), (4.19)
and hence the total number of particles,
N =
∫
d2xn(x). (4.20)
4.2.2 Calculation of Macroscopic Parameters and Phases
Atom number, Temperature and Chemical Potential
In the previous section we showed that calculation of the total number of particles for
a given simulation is relatively simple, determination of the temperature and chemical
potential for the microcanonical c-field region however, is not so straightforward. In
1997 Rugh [50] proved that ensemble averages of certain quantities constructed from
the Hamiltonian may be used to obtain entropy derivatives that define the temperature
and chemical potential. Application of Rugh’s method to the PGPE is described by
Refs. [19, 20], and is the method employed for our results, this scheme is nonperturba-
tive and quite accurate.
45
Quasicondensate
The quasicondensate density is given by Eq. (3.9), i.e.
nQ(x) =√
2〈nC(x)〉2 − 〈nC(x)2〉, (4.21)
where we use only the density of the c-field region, as the sparsely occupied incoherent
region has gaussian density fluctuations and does not contribute to nQ(x). We evaluate
quantities such as 〈nC(x)〉 by time-averaging, i.e.
〈nC(x)〉 =1
Ns
∑
l
|ψC(x, tl)|2, (4.22)
where tl is the time of each microstate l, Ns is the number of averaged microstates and
ψC(x, tl) =∫
dzψC(r, tl).
Coherence and Condensate
There are many ways to define coherence, especially for an inhomogeneous system, and
here we choose to use a measure based on the Penrose-Onsager criterion [42]. The 2D
one-body density matrix for the c-field region is defined as1
G1(x,x′) = 〈ψ∗C(x)ψC(x′)〉. (4.23)
Normally for the Penrose-Onsager criterion in the thermodynamic limit, the largest
eigenvalue ofG1(x,x′) is equivalent to the condensate population and the corresponding
eigenvector characterises the condensate mode. Here we choose to use the largest
eigenvalue and eigenvector as a measure of coherence, and refer to them as condensate
only for clarity (also see Ref. [11]), we denote the condensate density as nc(x). We also
note that the one-body density matrix is also useful for evaluating the appropriateness
of Ecut, as the smallest eigenvalue is a measure of nmin.
Superfluid Model of Holzmann et al.
Recently Holzmann et al. [33] proposed a model for the superfluid component based
on a local density application of the uniform results [47, 48], of which we discussed in
§3.3. Specifically, they suggest a superfluid areal density of the form
nSF (x) =
m(ωρBKT )2/2g[
1 − (ρ/ρBKT )2] + 4λ2
dB
, ρ ≤ ρBKT
0, ρ > ρBKT
(4.24)
1We ignore the incoherent region as these normal modes have exponentially decaying correlations
and do not contribute to long range coherence or the condensate population.
46
where for cylindrically symmetric traps ω is the radial trapping angular frequency, and
ρBKT is the radius at which the trapped system density equals the critical value for
the uniform 2D Bose gas to undergo the BKT transition (Eq. (3.8)). To obtain the
superfluid density using Eq. (4.24) hence requires a comprehensive calculation of the
system density (e.g. using c-field theory or quantum Monte Carlo methods) from which
rBKT can be determined.
4.2.3 Validity Conditions
The c-field method provides an accurate description of the low energy component of
the quasi-2D trapped Bose gas as long as three conditions can be satisfied (see Ref. [10]
for more details).
(i) Interactions must be weak such that g is small compared to unity, which in-
sures that classical fluctuations dominate quantum fluctuations. This condition is
well-satisfied in our calculations (g = 0.107) and in current experiments (for the ENS
group g ≈ 0.13 [29] and for the NIST group g ≈ 0.02 [14]).
(ii) All modes of the c-field region must be appreciably occupied. In practice this
amounts to making sure that the least occupied (highest energy) mode of the c-field
region has mean occupation (nmin) of order one or greater.
(iii) The numerical basis must describe the modes of the interacting classical region
well. To achieve this, Ecut must be large enough such that
Ecut −E0 & g〈nC(x)〉, (4.25)
where E0 is the energy of the single particle ground state. Condition (4.25) also ensures
that the universal long wavelength (critical) modes are contained within the c-field
region [see Eq. (1.7)], where they are treated non-perturbatively as required (see §3.3.3).
4.3 Results
Recall from §1.4 that the recent ENS experiments [29, 37, 55] indicate a phase tran-
sition (or at least a crossover) of the quasi-2D trapped system similar to the BKT
transition. They showed that the onset of bimodality coincides with the observation
47
fx [Hz] fy [Hz] fz [Hz] g
our parameters 9.4 9.4 1880 0.107
ENS experiment 11 130 3600 0.13
NIST experiment 18 18 1000 0.02
Table 4.1: Comparison between our theory and experiments, of trap
parameters and interaction strength.
of interference fringes (and hence extended coherence). Furthermore, a rapid prolifer-
ation of free vortices and a transition from algebraic to exponential decay of G1(∆x)
was also observed. They speculate that all four of these phenomena are manifestations
of a simple BKT-type crossover and that no BEC-type transition is involved. However,
their interpretation over simplifies their observations (Figs. 1.4 and 1.6) which show
that bimodality and coherence occur, together, at a temperature appreciably above
that where the onset of topological order is observed.
Also, recall the NIST experiment [14] (published 27 April 2009) which investigated
bimodal density profiles, and in particular the width of the narrow fitted Gaussian.
In conflict with the interpretation of the ENS group, the results from the NIST group
support that bimodality and extended coherence occur at phase space densities signifi-
cantly less than that where free vortices become suppressed, suggesting that bimodality
and extended phase coherence significantly precede a BKT-like transition. In this sec-
tion we present results obtained by a c-field method which provides an in depth analysis
of the quasi-2D trapped Bose gas to investigate the aforementioned issues.
4.3.1 System Parameters and Procedure
We simulate a gas of 87Rb atoms in a cylindrically symmetric trap with a radial trapping
frequency of 9.4 Hz and an axial frequency of 1880 Hz. We take the mass of 87Rb to
be 86.9 atomic mass units and the s-wave scattering length as 5.29 nm, the value of g
is given by Eq. (3.3). Table 4.1 compares our trap frequencies and interaction strength
with those of the ENS and NIST experiments.
All simulations have a fixed cutoff energy EC = 125~ω, where ω ≡ ωx = ωy is
the radial angular frequency. The number of particles in the c-field region is fixed, i.e.
NC = 3750, while the initial energy of the c-field region is adjusted. This varies the
equilibrium temperature and number of particles in the incoherent region, which in
turn modifies the total number of particles. For our results, we choose to evolve the
48
initial state for 40 trap periods for thermalisation before ergodically averaging 2000
sampled microstates over a further 40 trap periods. The suitability of Ecut is evaluated
a posteriori, with simulations accepted or rejected based on the requirement that nmin
is of order one or greater, we discuss this further in §4.3.8.
4.3.2 Degenerate Phases
Due to variation of both the total number of particles and temperature, for comparison
between simulations we find it instructive to make use of the reduced temperature
defined as T ′ = T/T0, where T0 is the quasi-2D condensation temperature defined in
§2.4.2.
Density Profile
A time averaged density profile is shown in Fig. 4.2 for T ′ = 1 (T = 50.7 nK). The low
energy modes of the c-field region exhibit a density that is peaked near the trap centre,
whereas the density of the incoherent region is centrally suppressed but extends much
further radially. Note that, the peak density of the c-field region is approximately a
factor of two larger than that for the incoherent region, however the majority of atoms
exist in the incoherent region, i.e. NI = 23990 compared with NC = 3750 for the c-field
region. We find, a posteriori, that nmin = 5.7, which is large enough to satisfy the
validity condition that all modes of the c-field region are appreciably occupied. Also,
we find that g〈nC(x)〉/(Ecut − E0) ≈ 0.32, which satisfies the requirement that the
critical modes are contained within the c-field region.
Degenerate Components
Here we calculate quasicondensate and condensate densities as outlined by §4.2.2, the
superfluid density is obtained from the local total density according to the model by
Holzmann et al. [Eq. (4.24)]. Figure 4.3 shows degenerate component densities for (a)
T ′ = 0.91 (T = 30.6nK) and (b) T ′ = 1 (T = 48.5nK). For the low temperature case
(subplot (a)) the phase space density is slightly greater than that required to satisfy
criticality (Eq. (3.8)), hence the superfluid profile appears as a narrow feature. We also
observe significant but clearly distinct condensate and quasicondensate components.
However, for the high temperature case (subplot (b)) the only degenerate component
present is a quasicondensate.
49
−60 −40 −20 0 20 40 600
1
2
3
4
5
x [µm]
n(x
)λ2 dB
−60 −40 −20 0 20 40 600
1
2
3
4
x [µm]
n(x
)λ2 dB
Figure 4.2: Phase-space areal density for T ′ = 1 as a function of x.
C-field region density nC(x) (dashed), incoherent region density nI(x)
(solid) and total density n(x) (dotted-line).
50
0 10 20 30 400
2
4
6
8
10
n(x
)λ2
x [µm]
(a) T ′ = 0.91
0 10 20 30 400
1
2
3
4
5
6
n(x
)λ2
x [µm]
(b) T ′ = 1.00
Figure 4.3: Component phase-space densities for (a) T ′ = 0.91 and
(b) T ′ = 1: n(x) (dotted-line), nQ(x) (dashed), nc(x) (solid) and
nSF (x) (shaded region). λ ≡ λdB.
We calculated density profiles of these degenerate components for approximately
250 simulation trajectories, each with the same regime parameters but for a varied
energy of the c-field region, which in turn varied the equilibrium temperature and total
number of particles. Fig. 4.4 plots the degenerate component fractions of the sys-
tem over a large temperature range. At the highest temperatures we observe a small
quasicondensate fraction, which increases steeply with decreasing temperature. Below
T ′ ≈ 0.98 the condensate fraction becomes appreciable (indicated as the condensate
temperature Tc in Fig. 4.4). Finally, at TBKT (T ′ ≈ 0.93) the peak density at the
trap center satisfies Eq. (3.8) and the model of Holzmann et al., Eq. (4.24), predicts a
non-zero superfluid fraction. As the temperature decreases further, the relative differ-
ence between quasi-condensate, condensate and superfluid fractions decreases, though
they are always clearly distinguishable. An important result is that the emergence of
coherence occurs at higher temperatures than that where the conditions for uniform
system superfluidity are satisfied, TBKT , i.e. the emergence of spatial coherence in the
trapped system occurs before the peak phase space density satisfies Eq. (3.8)2.
Determination of Tc
To determine the temperature at the onset of condensation, not only did we monitor
the condensed fraction, we also kept track of this fraction relative to the occupation of
higher modes. Figure 4.5 displays the fraction of atoms in both the first and second
2The correct phase space density to use in Eq. (3.8) is that for the ground axial mode only (see
5.3.2), this is in contrast to Holzmann et al. who instead use the total density.
51
0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Tc
TBKT
T/T0
frac
tion
Figure 4.4: Fraction of atoms in the quasicondensate (dots), conden-
sate (squares) and superfluid (triangles) components as a function of
reduced temperature.
52
0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
Tc
T/T0
frac
tion
Figure 4.5: Eigenvector occupations of normalised one-body density
matrix. Largest eigenvalue (i.e. condensate fraction) (squares) and
second largest eigenvalue (dots).
largest eigenvectors of the one-body density matrix. At the highest temperatures we
observe only a small difference in the size of these eigenvalues. As the temperature
decreases, both eigenvalues increase until at T ′ ≈ 0.98 the largest eigenvalue increases
sharply and diverges from the second largest eigenvalue, hence we assign this temper-
ature to be approximately Tc.
Comparing Simulations of Different Atom Numbers
To convey our motivation for using T ′ instead of T in our plots of degenerate compo-
nents, Fig. 4.6(a) shows how the total number of atoms varies as a function of T . For
T ≈ 12nK, the total number of atoms is of order 4× 103 and as the temperature rises
to 80 nK, the total atom number rises by more than an order of magnitude to approx-
imately 6.5 × 104. Figure 4.6(b) shows how this large change in atom number affects
the ideal quasi-2D critical temperature, T0, which increases by nearly a factor of four
53
over the same temperature range. While it is useful to plot the degenerate components
as a function of T , even though N is changing, we find it particularly instructive to
plot these degenerate components verses T ′ instead, which to first order removes the
effects produced by the varying atom number.
Figure 4.7 shows how the temperature varies compared with the energy spacing of
the axial direction. For T ′ < 1 we have that kBT/~ωz is always less than 0.6 which
implies the system is strongly 2D at, and near, the transition region. Well above the
transition region kBT/~ωz surpasses unity and hence the population of excited axial
modes is significant, although we still expect that the critical modes are confined within
the ground axial mode, i.e. that a 2D system is embedded in a 3D thermal cloud.
54
10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
T [nK]
To
tal N
um
be
r
(a)
10 20 30 40 50 60 70 8020
40
60
80
100
T [nK]
T0 [n
K]
(b)
×105
Figure 4.6: (a) Total atom number and (b) Ideal quasi-2D BEC tem-
perature.
55
0.6 0.7 0.8 0.9 10
0.20.40.60.8
11.21.41.6
kB
T/
hωz
T /T0
Figure 4.7: Relative thermal activation of z direction of the system
as a function of reduced temperature
56
4.3.3 Topological Order and The BKT Transition
An important prediction of the Berezinskii, Kosterlitz and Thouless theory for the uni-
form system is that in the superfluid regime off-diagonal correlations decay algebraically
according to Eq. (2.47), i.e.
g1(∆x) ∝(
1
∆x
)η,
η =1
nSFλ2dB
, (4.26)
and that the critical superfluid phase space density is given by Eq. (2.49), i.e.
λ2dBnSF = 4.
Recall the formal definition of the first order correlation function for the uniform
system given by (Eq. (2.46))
g1(∆x) ≡ 〈Ψ†(x)Ψ(x′)〉√
n(x)n(x′).
However, because of spatial inhomogeneity due to the trap, this correlation function
depends on center-of-mass and relative coordinates, we choose to examine these corre-
lations at the trap center. The normalised correlation function we evaluate is
g1(∆x) =〈ψ∗
C(x)ψC(−x)〉
√
n(x)n(−x), (4.27)
where for this case ∆x = 2|x|, i.e. we compare points symmetrically placed about the
origin and average in the radial plane. The denominator contains the total density,
whereas the numerator of Eq. (4.27) neglects the incoherent region contribution. This
approximation is good for ∆x & λdB, as we expect the atoms of the incoherent region
to be thermal with exponentially decaying correlations.
Typical results for this correlation function for a range of temperatures are shown
in Fig. 4.8(a). We least-squares fit a model decay curve of the form cr−α to g1(∆x) over
the spatial range 1.2λdB < ∆x < 5λdB to determine the exponent α (with c a constant
fit parameter). The lower spatial limit excludes the short range contribution of normal-
component atoms, while the upper limit restricts the effects of spatial inhomogeneity
(typical size of the clouds is of order 30λdB). The model fits (shown in Fig. 4.8(a))
are poor in the three highest temperature cases, suggesting that the algebraic fit is
inappropriate. Remarkably however, for T ′ . 0.93 the fits are good indicating that
correlations decay algebraically. By substituting the fitted values of α for η of Eq. (4.26)
57
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
T ′ =0.73T ′ =0.85
T ′ =0.93
T ′ =0.98T ′ = 1.00T ′ =1.03
∆x [µm]
g1(∆
x)
(a)
0.7 0.75 0.8 0.85 0.9 0.95 1 1.050
2
4
6
8
T ′ = T/T0
nsfλ
2
(b)
Figure 4.8: (a) g1(∆x) for various temperatures (solid lines). Fitted
region (light coloured segments of curves) and algebraic fits (dashed)
are also shown. (b) Peak superfluid density determined from fits to al-
gebraic decay (triangles) and from model density (circles), Eq. (4.24).
58
we infer the superfluid density at the trap centre. These results, shown in Fig. 4.8(b),
compare well with the peak of the superfluid density given by the model by Holzmann
et al. Eq. (4.24), for temperatures below TBKT , i.e. T ′ < 0.93. At higher temperatures
where n < nBKT [see Eq. (3.8)], the two results disagree, however in this temperature
range the algebraic fit is poor and an exponential fit appears to be more appropriate.
It is worth emphasising that at TBKT [i.e. where the peak total phase space density
satisfies Eq. (3.8)], the peak superfluid phase space density determined by the algebraic
fit has a value of 4, i.e. Eq. (2.49) is satisfied.
4.3.4 Bimodality and Condensate
In Fig. 4.9 we show the position and momentum density distributions for our system as
a function of temperature. Fig. 4.9(b) shows that for temperatures below Tc a strong
bimodality is apparent in the momentum space density of the system associated with
the extended spatial coherence arising from the condensate.
Figure 4.9(a) shows the position dependent phase-space density smoothly peak as
the temperature decreases, which is associated with quasicondensate (see Fig. 4.3),
this peaking of nλ2dB is enhanced relative to n since λdB increases with decreasing
T . We emphasise that TBKT occurs at a phase space density significantly greater
(approximately 50%) than that for Tc.
4.3.5 Central Curvature of Density Profiles
Time-averaged density profiles are shown in Fig. 4.10 with quadratic curves fitted to
the central region of width 10µm. For all temperatures shown, the density is clearly
peaked at the centre but this peaking is more prominent for the lowest temperature
cases. Corresponding to this peaking, the quadratic fits are good for T ′ = 0.85, 0.93
but become worse as the temperature increases.
Figures 4.11 and 4.12 demonstrate how the central curvature, ∂(n(x))/∂(x2), changes
as a function of temperature. Fig. 4.11 uses a fitting window 10µm across while for
Fig. 4.12 it is 20µm across, the latter is statistically smoother but includes more of the
non-central behaviour. From Fig. 4.11 we see that at low temperatures up to T ′ ≈ 0.8,
the curvature remains approximately constant at a value slightly greater than that of
the Thomas-Fermi profile. Above this temperature the curvature increases and peaks
at T ′ ≈ 0.93 = TBKT , before sharply decreasing to approximately 75% of the maximum
at T ′ = 0.98 = Tc, and falling to approximately 50% of the maximum at T ′ = 1. The
59
0.60.8
1 −60 −40 −20 0 20 40 60
5
10
15
20
25
n(x
)λ2
(a)
x [µm]T ′
0.60.8
1 −1.5 −1 −0.5 0 0.5 1 1.5
500
1000
2000
4000
8000
n(p
x)
[1/hm
ω]
(b)
T ′ p [√
hmω]
Figure 4.9: Density distributions versus temperature. (a) Position
phase-space density and (b) momentum (p) space density distribu-
tions. The black (white) curves mark the densities at TBKT (Tc).
λ ≡ λdB.
60
low temperature discrepancy compared to the Thomas-Fermi curvature is presumably
due to the appreciable contribution of kinetic energy3. The increase in the curvature
near T ′ ≈ 0.93 = TBKT is presumably due to the reduced occupation of the ground
mode along with the slight decrease in total density. The sharp reduction in curvature
above T ′ ≈ 0.93 = TBKT is in part a manifestation of a rapidly increasing thermal
population. Figure 4.12 shows similar behaviour to that of 4.11, but in Fig. 4.12 the
curvature is narrower at cold temperatures since more of the low density region is in-
cluded in the fitting procedure, for which the density profile behaves somewhat more
like an ideal gas.
The quantum Monte Carlo results of Holzmann et al. displayed in Fig. 4.13 agree
qualitatively with our results, although they only provide a limited number of data
points (only one data point near the transition). They claim that the increase in cur-
vature compared to that of the Thomas-Fermi ground mode, κ = mω2π/g, is due to the
widening of the total density profile in the axial direction and hence a reduced g should
be used for the calculation of κ. Fig. 4.13 includes the ground state Thomas-Fermi
curvature and a modified Thomas-Fermi curvature evaluated using a modified g based
on a single particle density distribution in the z-direction (see caption). However they
do not justify this this modified expression for κ. The Thomas-Fermi approximation
assumes there are no density fluctuations, but this conflicts with their calculation of
the modified g which includes excited axial modes that behave thermally with gaus-
sian density fluctuations. Their Monte Carlo results do not agree with either of the
Thomas-Fermi curvatures. Furthermore, our simulation regime is sufficiently 2D such
that all modes in the c-field region belong to the ground axial mode and we observe
this narrower than Thomas-Fermi curvature even when the density of the incoherent
region is excluded. Since our numerical basis constrains the density profile in the z-
direction to be fixed, we conclude that the behaviour seen in Fig. 4.11 is a property of
the radial modes and not a modification of g in the Thomas-Fermi expression, which
was proposed by Holzmann et al.
Our results also qualitatively agree with the experimental results from the NIST
group (shown in Fig. 1.7) even though our simulation parameters differ from their
experiment. At high phase space densities (low temperature) they observe a gradual
reduction in the width of the central mode in the bimodal fit with decreasing phase
space density (presumably due to decreasing role of interactions as density decreases)
3Fits of Fig. 4.10 have a 1/e radius of approximately 10µm whereas the corresponding ideal gas
ground mode has a 1/e radius of 5µm.
61
0
2
4
6
8x 10
12
n(x
)[m
−2]
(a) T ′ = 0 . 85 (b) T ′ = 0 . 93
−50 0 500
2
4
6
8x 10
12
x [µm]
n(x
)[m
−2]
(c) T ′ = 0 . 98
−50 0 50x [µm]
(d) T ′ = 1
Figure 4.10: Time averaged density profiles at (a) T ′ = 0.85, (b)
T ′ = 0.93, (c) T ′ = 0.98 and (d) T ′ = 1: n(x) (solid curve), fitting
region (between vertical lines) and quadratic fits (dashed).
until after a certain point they observe that the bimodal width increases rapidly with
decreasing phase space density (presumably associated with free vortex proliferation).
They propose that this qualitative change of the bimodal width is a manifestation of
the BKT transition, and our results are consistent with this interpretation.
62
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
22
Tc
TBKT
T/T0
−∂n
(x
)/∂(x
2)[m
−4]
Figure 4.11: Central curvature of time averaged density profiles. The
fitting region is centrally positioned with a width of 10 µm, for ex-
ample see Fig. 4.10. The Thomas-Fermi curvature, mω2/2g, is shown
by the horizontal dashed line.
63
0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
3.5
4x 10
22
Tc
TBKT
T/T0
−∂n
(x
)/∂(x
2)[m
−4]
Figure 4.12: Central curvature of time averaged density profiles. The
fitting region is centrally positioned with a width of 20 µm. The
Thomas-Fermi curvature, mω2/2g, is shown by the horizontal dashed
line.
64
0.8
0
50
0 0.2 0.4 0.6 0.8
curv
ature
of n
(r)λ
2
T/T0BEC
groundst. TF
Boltzm. TF
Figure 4.13: (Figure taken from Ref. [33]). Central curvature, κ =
−(λ2dB/kBT )∂n(x)/∂(x2)|x=0, with ~ = m = ω = 1 and here r ≡ x.
Calculation parameters: N = 5.8 × 105, ωz = 325 and g = 0.13.
Also, plotted is the Thomas-Fermi curvature (i.e. κ = mω2π/g) for
g = 0.13 (dashed line) and for g = 4π~2a/m
∫
dz[n(z)]2 (solid line)
where n(z) is ideal gas density of distinguishable particles normalised
to unity.
65
4.3.6 The Role of Vortices
In figure 4.14 we show instantaneous planar densities of four simulation trajectories,
each at a different temperature. For all temperatures a quasicondensate is present, and
is the only degenerate component for the highest temperature result of Fig. 4.14. As the
temperature decreases to T ′ = 0.97 a condensate forms, however in this temperature
regime vortices are prolific and we frequently observe free vortices in the condensate.
For the lower temperature results (i.e. T ′ . 0.93 ≈ TBKT ) [see the two leftmost
columns of Fig. 4.14] we see that in the central (condensate) region density fluctuations
are significantly reduced, vortices are less frequent and those that do occur are mostly
found in vortex and anti-vortex pairs (i.e. in close proximity). The proliferation of
free vortices above TBKT supports the NIST group’s [14] proposal that free vortices are
responsible for the rapid increase in width of the bimodal fit to their density distribution
as discussed in §4.3.5. This also supports that the rapid increase in central curvature
above TBKT for Figs. 4.11, 4.12 and 4.13 is in part due to the proliferation of free
vortices.
For T ′ = 0.87 (Fig. 4.14, first column) a vortex pair is observed within the con-
densate at both microstates t4 and t5, suggesting the lifetime of this pair is of order
0.02 trap periods (2.1 ms). We often found vortex pairs within the condensate region
to have a lifetime of more than 0.2 trap periods (21 ms). Our observations of vortex
behaviour is consistent with the existence of a BKT transition in the trapped Bose gas,
although this transition is somewhat smeared due to spatial inhomogeneity, and as a
function of temperature due to finite size effects. We found that even though vortices
are mostly bound for temperatures below TBKT , occasionally free vortices exist near the
trap centre, even at a few nano-kelvin below this transition temperature. Conversely,
above the transition we find that even though it is usual to find free vortices, there are
periods of time where free vortices are absent from the central region. This smearing
of the BKT transition is consistent with the free energy arguments for a finite system
as discussed in §2.5 (in particular the logarithmic dependence on system size).
66
t1
T′=0.87
−20 0 20
t2
−20 0 20
t3
−20 0 20
t4
−20 0 20
t5
x [µm]−20 0 20
T′=0.94 (≈ TBKT
)
−20 0 20
−20 0 20
−20 0 20
−20 0 20
x [µm]−20 0 20
T′=0.97
−20 0 20
−20 0 20
−20 0 20
−20 0 20
x [µm]−20 0 20
T′=1
y[µ
m]
−20 0 20
−20
−10
0
10
20
y[µ
m]
−20 0 20
−20
−10
0
10
20
y[µ
m]
−20 0 20
−20
−10
0
10
20
y[µ
m]
−20 0 20
−20
−10
0
10
20
y[µ
m]
x [µm]
log10
n(x)λdB2
−20 0 20
−20
−10
0
10
20
−1.5 −1 −0.5 0 0.5 1
Figure 4.14: Instantaneous planar density of the c-field region. In-
ner green (outer black) circle marks the 1/e-boundary of the conden-
sate (quasicondensate) density. Vortices (+) and anti-vortices () are
shown. Each column samples a single simulation with temperature
T ′ and each row labeled t1 to t5 represents equidistant time steps
separated by 2.1 ms (0.02 trap periods).
67
4.3.7 Condensate Number Fluctuations
Here we investigate how the condensate population fluctuates over time which pro-
vides a useful measure of the condensate’s temporal coherence. For each trajectory
we calculate the average condensate mode ψc(x) (as outlined in §4.2.2) using 5 × 104
equilibrium microstates4 sampled over 1000 trap periods (106 seconds), which written
in terms of the c-field basis reads
ψc(x) =∑
n∈C
bnφn(x), (4.28)
where bn are the spectral amplitudes of the condensate mode. Knowing the equilibrium
condensate mode we can obtain its instantaneous condensate amplitude in a c-field
microstate by computing the overlap integral,
αc(t) =
∫
d2xψ∗c (x)ψC(x, t) (4.29)
=∑
n∈C
b∗ncn(t). (4.30)
Hence the condensate occupation at time t is given by
Nc(t) = |αc(t)|2. (4.31)
Figure 4.15 demonstrates how Nc(t) behaves as a function of time for T ′ ≈ TBKT ,
here we see long lived variations in the condensate number (of order seconds) due to the
presence or absence of long lived free vortices in the central region. Figures 4.16 and
4.17 show results for eight trajectories, each at a different temperature, the microstates
are binned according to their corresponding value of Nc(t)5.
From Fig. 4.16 (a) we see that at low temperature the average condensate fraction
is large and fluctuations about the mean value are small. As the temperature rises
[subplots (b) and (c)] the average fraction decreases and condensate fluctuations are
significantly larger which is consistent with the observed increase in vortices (Fig. 4.14).
When T ′ ≈ TBKT [Fig. 4.16 (d) and Fig. 4.17 (e)] there is a significant change in be-
haviour i.e. the condensate now occasionally fluctuates to zero. For TBKT < T ′ < Tc
[Fig. 4.17 (f)] the condensate frequently fluctuates to zero, consistent with the prolif-
eration of centrally positioned free vortices (for example see Fig. 4.14). Furthermore,
4Data from the first 1000 microstates (20 trap periods) is not included in our condensate fraction
histograms presented in this section to ensure the systems have first reached equilibrium.5Atoms of the incoherent region are neglected and the condensate fraction is with reference to the
c-field number.
68
89 90 91 92 93 94 95 96 970
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time [s]
Con
dens
ate
Fra
ctio
n
Figure 4.15: Condensate fraction of the c-field region [Nc(t)] versus
time. This trajectory is for a portion of the same data as for Fig. 4.16
(d). T ′ = 0.93 (≈ TBKT ). Time = 0 indicates the time of the simula-
tion initial state.
69
0
1000
2000
3000
4000F
requ
ency
(a) T ′ = 0 . 62 (b) T ′ = 0 . 85
0 0.2 0.4 0.60
1000
2000
3000
4000
Condensate Fraction
Fre
quen
cy
(c) T ′ = 0 . 91
0 0.2 0.4 0.6Condensate Fraction
(d) T ′ = 0 . 93
Figure 4.16: Histogram showing the number of microstates with each
fraction of c-field atoms in the instantaneous condensate for (a) T ′ =
0.62 (b) T ′ = 0.85 (c) T ′ = 0.91 and (d) T ′ = 0.93 (≈ TBKT ).
when vortex configurations and condensate fractions are monitored simultaneously we
observe a strong correlation between the existence of centrally positioned free vor-
tices and the significant reduction of instantaneous condensate. Once the temperature
reaches T ′ = 0.98 ≈ Tc [Fig. 4.17 (g)] there is another significant change in behaviour,
i.e. the peak of the condensate number distribution shifts to zero, and takes the form
of an incoherent distribution. Slightly above Tc [e.g. see Fig. 4.17 (h) for which T ′ = 1]
the system is now normal and the condensate fraction peaks at zero.
The results of this section support that vortex pairs suppress the condensate while
free vortices do likewise but in a more dramatic manner. These results also add weight
to our identification and location of the two distinct temperatures TBKT and Tc.
70
0
1000
2000
3000
4000
Fre
quen
cy
(e) T ′ = 0 . 94 (f) T ′ = 0 . 96
0 0.2 0.4 0.60
1000
2000
3000
4000
Condensate Fraction
Fre
quen
cy
(g) T ′ = 0 . 98
0 0.2 0.4 0.6Condensate Fraction
(h)
×4
T ′ = 1 . 00
Figure 4.17: Histogram showing the number of microstates with each
fraction of c-field atoms in the instantaneous condensate for (e) T ′ =
0.94 (≈ TBKT ) (f) T ′ = 0.96 (g) T ′ = 0.98 (≈ Tc) and (h) T ′ = 1 (the
×4 indicates that the frequencies have been divided by 4 for subplot
(h) to avoid an inconvenient rescaling of the vertical axis).
71
4.3.8 Simulation Validity
In Fig. 4.18 we show various quantities used to establish the validity of our calculations.
Subplot (a) shows how the temperature varies as a function of energy per particle of
the c-field region. Of the three validity conditions discussed in §4.2.3, only condition
(i) (that g is small compared to unity) can be assured a priori. Conditions (ii) and
(iii) need to be verified a posterori as we now discuss. In Fig. 4.18 (b) the average
occupation of the least occupied mode is shown. For all calculations this is larger
than unity (and near the transition it is typically & 3 − 4) so that the classical field
approach is well justified for these results. In Fig. 4.18 (c) we show that the ratio
gnC/(Ecut − E0) is small, ensuring that our cutoff is large enough to provide a good
description of the spectrum and ensuring that the fluctuating region is well contained
within the classical field. In summary Figs. 4.18 (b) and (c) show that our method
provides a valid representation of the quasi-2D system we have simulated.
72
110 112 114 116 1180.2
0.3
0.4
0.5
Energy p er par t i cl e [ hω ]
gn
C/E
′ cu
t
0
20
40
60
80
Tem
p.
[nK
]
0
2
4
6
8
nm
in
Figure 4.18: Quantities determined from c-field evolution as a func-
tion of the c-field energy per particle. (a) Simulation temperature.
(b) Average occupation of the least occupied mode. (c) Ratio of in-
teraction energy scale to the energy cutoff, where E ′cut ≡ Ecut −E0.
73
74
Chapter 5
High Temperature Mean-field
Theory
5.1 Introduction
It is of course desirable to have a simple mean-field description of the quasi-2D system
[30, 32]. However, mean-field theories are of limited applicability in the critical region,
where density fluctuations are strong, and it is well known that the 2D critical region
is large (discussed in §3.3.3). However, recently Holzmann, Chevallier, and Krauth
(HCK) [32] made a novel proposal to use a high-temperature Hartree-Fock mean-field
theory to extrapolate into the lower-temperature critical regime. They then used this
theory to estimate the transition temperature (THCKBKT ) as that where the peak phase
space density of the system satisfies the critical value [see Eq. (3.8)] known for the
uniform pure-2D Bose gas [47, 48].
However, several aspects of the HCK theory are inconsistent, namely (i) the method
by which they attribute an effective 2D interaction parameter and (ii) the phase space
density they use to identify the transition point. Motivated by the same goal as HCK,
to formulate a simple theory to estimate TBKT , in this chapter we develop a consistent
formalism. In our analysis we use a more complete high-temperature mean-field theory
that avoids the interaction simplification used by HCK. We also show that the correct
generalisation of the pure 2D condition for the BKT transition to the quasi-2D system
involves the areal density of the ground axial mode of the system. Through numerical
calculations we show that our improved treatment of these two aspects leads to sig-
nificant differences in our theoretical predictions from those of HCK. We also discuss
the main limitation of mean-field theory extrapolation into the critical regime, which
75
indicates that our improvements on the HCK theory will provide a lower bound on the
temperature TBKT in the trapped quasi-2D system. We also compare the predictions
of both mean-field theories with the results of our c-field method.
5.2 Semiclassical Quasi-2D Mean-Field Theory
5.2.1 Formalism
Again we begin with the effective low-energy Hamiltonian for ultracold bosonic atoms
given by Eq. (1.2). The quasi-2D system we consider here is realized when the trapping
potential is sufficiently tight in one direction (which we take to be z) such that ~ωx,y ≪kBT ∼ ~ωz.
Our interest is in the thermal properties of the quasi-2D system when there is no
condensate present, a regime for which Hartree-Fock theory is appropriate. If inter-
actions are small compared to ~ωz then the Hartree-Fock modes for the Hamiltonian
[Eq. (1.2)] take the separable form ψkσ(r) = fkσ(x)ξk(z), where the axial modes ξk(z)
are bare harmonic oscillator states. In the quasi-2D regime the xy-plane can be treated
semiclassically, eliminating the need to diagonalize for the modes fkσ(x). However, the
axial modes must be treated quantum mechanically, and the Hartree-Fock expression
for the areal density of the system in the j-th axial mode is
nj(x) =1
(2π)2
∫
d2k1
exp
ǫj(x,k)−µ
kBT
− 1, (5.1)
where the Hartree-Fock energies are
ǫj(x,k) =~
2k2
2m+m
2(ω2
xx2 + ω2
yy2) + j~ωz + 2
∞∑
k=0
gkjnk(x), (5.2)
µ is the chemical potential, and (4.17)
gkj =4πa~2
m
∫
dz |ξk(z)|2|ξj(z)|2, (5.3)
describes the interactions between atoms in the k and i axial modes. Performing the
momentum integration in Eq. (5.1) and adding up the axial mode densities gives the
total areal density
n(x)λ2dB = −
∞∑
j=0
ln [1 − exp (µ− Vj(x)/kBT )] , (5.4)
76
where
Vj(x) =m
2(ωxx
2 + ωyy2) + j~ωz + 2
∞∑
k=0
gkjnk(x), (5.5)
is the effective potential for atoms in the j-th axial mode. We solve Eqs. (5.1) and
(5.2) self-consistently, i.e. by iterating until the solutions converge.
5.2.2 Numerical Implementation
Here we briefly summarise our numerical implementation of this mean-field theory. We
calculate the interaction parameters gki by numerical integration of the bare harmonic
oscillator states in Eq. (5.3). For our results we take advantage of radial symmetry
which allows us to map Eqs. (5.1) and (5.2) onto a 1-dimensional form, i.e. n(x) →n(ρ).
The inputs to the calculation are the trap geometry, scattering length, tempera-
ture and (desired) total number of atoms. We then iterate solving the Hartree-Fock
equations to find the desired total atom number by adjusting the chemical potential.
In each iteration we solve Eqs. (5.1) and (5.2) self consistently (this itself requires
many iterations), to determine n(x) and hence N . In practice we begin by finding one
chemical potential that produces an excessive atom number and another that produces
a deficient atom number, we then use the bisection method to adjust the chemical
potential until the desired atom number is reached to arbitrary precision.
5.2.3 Holzmann-Chevallier-Krauth Mean-field Theory
A central concern of our work is to compare our mean-field theory, as outlined above,
to the mean-field theory used by HCK [32]. In this section we briefly review their
theory.
The HCK theory is a simplification of our mean-field scheme presented above made
by taking the interactions to be axial mode independent, i.e., simplifying the mean-field
interaction term according to
2
∞∑
k=0
gkjnk(x) → 2gHCKn(x). (5.6)
77
The average interaction strength used in the HCK model is given by1
gHCK =4πa~2
m
∫
dz[ρ(z)]2 (5.7)
= a
√
8πωz~3
m
√
tanh
[
~ωz
2kBT
]
, (5.8)
where ρ(z) is the density of a single atom in a harmonic oscillator of frequency ωz
at temperature T . Approximation (5.6) has no rigorous justification, but allows a
closed-form expression for total density.
The density profile can be determined in two steps, first the density as function of
the effective potential Veff is given by
n(ρ) = − 1
λ2dB
∞∑
ν=0
ln 1 − exp [β(µHCK − Veff(ρ) − ν~ωz)]
= n(Veff), (5.9)
where
µHCK = µ− 2gHCKn(0), (5.10)
and then the expression for the effective potential,
Veff(ρ) =mω2ρ2
2+ 2g[n(ρ) − n(0)], (5.11)
is inverted to give
ρ(n(Veff), Veff) =
√
2TkB
mω2
(
Veff −mgnλ2
dB
π~2
)
(5.12)
and hence inputting the function Veff into Eqs. (5.9) and (5.12) simultaneously specifies
both ρ and n analytically and thus n(ρ).
5.3 Results
5.3.1 2D Phase Space Density
In Fig. 5.1 we compare the two mean-field models and the ideal gas model (i.e. Eqs. (5.1)
and (5.2) with a = 0) for three different temperatures. The density profiles shown in
1We note that gHCK reduces to g in the limit T → 0.
78
0
0.5
1
1.5
2
n(x
)λ2 dB
(a)
0
2
4
6
8
n(x
)λ2 dB
(b)
0 0.5 1 1.5 2 2.5 30
4
8
x [×10−4m]
n(x
)λ2 dB
(c)
Figure 5.1: Comparison of areal phase space densities for systems at
T=(a) 221, (b) 172, and (c) 150 nK. Our mean-field model (solid),
HCK model (dotted), and ideal gas (dashed). Ground axial mode
areal densities are shown in gray in (c). Calculation parameters are
N = 104 87Rb atoms with ωx,y = 2π× 59 Hz and ωz = 2π× 2530 Hz.
We note that the parameters (including temperature) in (a) are the
same as those for Fig. 5.2.
79
T [nK] gHCK × m~2 N
(MF )0 N
(HCK)0 N
(Boltz)0 (nλ2)peak (n0λ
2)peak
132 0.0851 7666 7888 6024 10.4 9.38
150 0.0805 6815 7010 5559 7.24 6.08
172 0.0756 5834 5928 5073 3.96 2.76
221 0.0673 4560 4531 4236 1.85 0.91
270 0.0612 3774 3788 3630 1.17 0.46
Table 5.1: Comparison of parameters and theoretical predictions for
the quasi-2D system considered in Fig. 5.1. N = 104 87Rb atoms.
Interaction parameters, values of the ground axial mode occupation
N0 =∫
d2xn0(x) for our theory (N(MF )0 ), HCK theory (N
(HCK)0 ), and
the Boltzmann case (N(Boltz)0 ). The peak (areal) total phase space and
ground axial mode phase space densities of our theory are shown. In
our theory the first few interaction parameters are: g00, g01, g11 =
(~2/m)0.1299, 0.0650, 0.0974.
Fig. 5.1(a) are at the pure-2D condensation temperature[3], i.e.2 T 2DBEC ≡
√6N~
√ωxωy/πkB.
In this regime both mean-field models make similar predictions, and have lower density
than the ideal gas at the trap center due to the effects of repulsive interactions.
The results in Fig. 5.1(b) are at a colder temperature where the ideal gas is almost
saturated. (i.e. approximately at the quasi-2D condensation temperature, T0 < T 2DBEC ,
where the central density begins to diverge). Interaction effects play a more significant
role here, and prevent the density from spiking in both mean-field theories. In this
regime, and at lower temperatures [Fig. 5.1(c)], the differences between the mean-field
theories become clearly apparent. This discrepancy arises from how interactions are
treated in the theories [see Eq. (5.6)] in two ways:
(i) Averaged interaction parameter: The interaction parameter, gH , in the HCK
theory assumes that the various modes follow a Boltzmann distribution. As the system
becomes degenerate, n0λ2 ∼ 1, this approximation is inaccurate as it fails to account
for quantum statistical effects that increase the ground axial mode occupation. A
comparison of both mean-field theories and the Boltzmann prediction of the ground
band occupation are given in Table 5.1, and reveals the increasing difference between
the mean-field and Boltzmann results as the phase space density increases.
2We note that interaction effects and the quasi-2D nature of the trap suppress the expected con-
densation temperature and we only mention the purely 2D Bose-Einstein condensation temperature
to indicate why the ideal gas shows saturation.
80
(ii) Mode independence of interaction: The marked difference between the ideal
and mean-field solutions [e.g. see Fig. 5.1(b)] arises from interaction effects. These
effects are dominated by the atoms in the ground axial mode. Because gH < g00, our
ground mode atoms are more strongly interacting than those in the HCK theory, thus
our theory predicts n0(x) to be lower and broader [see Fig. 5.1(c)], with less atoms in
the ground axial mode [see Table 5.1].
Finally, we comment on validity aspects of our Hartree-Fock approach (also see
§5.3.2). First, the separability of our Hartree-Fock description into planar (fkσ(x))
and axial (harmonic oscillator, ξk(z)) modes requires that interaction effects are small
compared to the axial energy scale, i.e. n0(0)g00/~ωz ≪ 1. For the results presented
in Fig. 5.1, the value of this ratio varies from 0.034 (Fig. 5.1(a)) to 0.16 (Fig. 5.1(c)).
5.3.2 BKT Transition in the Trapped System
Monte Carlo Analysis of the BKT Transition
The BKT transition occurs when the superfluid density, nSF , satisfies nSFλ2 = 4 [36]
[see also Eq. (2.49)]. As discussed in §3.3.1, Monte Carlo calculations by Prokof’ev et
al., [48] have characterized the uniform 2D Bose gas, showing that this condition, in
terms of the total density, is
n2DBKT =
1
λ2dB
ln
(
ξ
g
)
, (5.13)
where ξ = 380 ± 3, g is the 2D interaction strength [47, 48], and the superscript 2D
emphasizes that this result is for the pure-2D Bose gas. An important point discussed
in §3.3.1 is that the long wavelength behavior of all 2D weakly interacting |ψ|4 models
is universal at the transition point. However, differences between models emerge in
high energy modes, which contribute to the total density at the critical point, but do
not affect the strongly fluctuating critical region. Hence, we can add or subtract mean-
field contributions for the high energy modes which modifies the total density and the
parameter ξ used in Eq. (5.13). The value ξ ≈ 380 only applies to the continuous Bose
gas.
The uniform criterion (5.13) can be applied to the quasi-2D trapped gas using a local
density approximation. Within such an approximation the transition will be spatially
dependent, and will occur first at the centre of the trap where the density is highest.
Some care has to be taken in extending the pure-2D theory to the quasi-2D case, since
the additional axial modes correspond to a different mean-field theory. The correct
81
quasi-2D extension of result (5.13) should be applied to the subsystem consisting of
atoms occupying the ground axial mode, with relevant interaction parameter g00 =
mg00/~2, i.e.
ncrit0 =
1
λ2dB
ln
(
ξ
g00
)
, (5.14)
where ξ ≈ 380.
Mean-field validity near the transition
We note that when condition (5.14) is satisfied the mean-field theory must be inap-
plicable as a complete description (See §3.3.3). However, the mean-field theory itself
predicts no transition to BKT (or BEC) phases and varies smoothly through this tem-
perature range. Thus the basis of the HCK approach, the use of mean-field theory
to estimate the system phase space density, would seem to be a reasonable starting
point for estimating TBKT in the trapped system. The main deficiency of the mean-
field approach is that it assumes Gaussian fluctuations, whereas near the transition
these are suppressed via the formation of a quasi-condensate [48] (also see our results
in Fig. 4.4). In such regimes the Hartree-Fock treatment overestimates the energetic
cost due to interactions, and predicts the system to spread out more and hence have a
lower central density. We hence conclude that near the BKT transition our mean-field
density will be less than the actual system density. The simplifying assumption of the
HCK theory (as we describe next) leads to a lower self interaction for the ground mode
which increases the density, though for reasons unrelated to quasi-condensation. Thus
the HCK theory cannot be assured to provide a lower bound for the system density.
Comparison Between HCK Theory and Monte Carlo Density
Density profile predictions of the HCK theory have been compared against those gen-
erate by quantum Monte Carlo simulations. The results given in [32] are reproduced
in Figs. 5.2 and 5.3. The good agreement of these results is not surprising since the
system is quite far from the degenerate regime. In particular the peak phase space den-
sity of the ground mode is 0.93 and 2.7 for the mean-field results in Figs. 5.2 and 5.3
respectively3. Hence these results are in a regime where the quasicondensate influence
is minimal and mean-field theory should work well (note: The results we presented
earlier in Fig. 5.1 (a) are for the same case as in Fig. 5.2, and show that our mean-field
theory works well in this regime). Later in (Fig. 5.7) we consider our mean-field theory
3A large component of the phase space density in Figs. 5.2 and 5.3 is due to excited axial modes.
82
0
0.5
1
1.5
2
0 0.5 1 1.5 2
den
sity
nλ
2
position rβ1/2
ideal q2dmean field q2d
QMC N=105
Boltzmann
0.0001
0.001
0.01
0.1
1
1 8
πk
k
Figure 5.2: (Figure taken from Ref. [32]) Areal phase space density for
Monte Carlo simulation (+) and HCK theory (solid line) for T = 221
nK. Calculation parameters are otherwise the same as for Fig. 5.1.
nλ2 ≡ n(ρ)λ2dB .
in the high phase space density regime of the transition region to more clearly reveal
the breakdown of this theory.
Use of Mean-Field Theory to Predict the BKT Transition
While both Hartree-Fock mean-field theories discussed fail to predict any transition
in the quasi-2D Bose gas, their predictions for the density can be extrapolated4 to
estimate when the quasi-2D condition for the BKT transition is fulfilled [Eq. (5.14)].
This approach was formulated in [32] but using the total areal density, n, and the
averaged interaction parameter, gHCK , in expression (5.13), i.e.
ncrit =1
λ2dB
ln
(
ξ
gHCK
)
. (5.15)
These choices are in contrast to the correct quasi-2D extension of the universal result
that we have given in Eq. (5.14).
83
0
1
2
3
4
5
6
7
0 0.5 1 1.5 2
den
sity
nλ
2
position rβ1/2
mf 2dideal q2d
mf q2d (g)QMC N=105
mf q2d (~g)
0.4
0.6
0.8
0 1 2 3 4
tq2d
KT
~ωz
Figure 5.3: (Figure taken from Ref. [32]) Areal phase space density for
Monte Carlo simulation (+) and HCK theory (solid line) for T = 177
nK. Calculation parameters are otherwise the same as for Fig. 5.1.
nλ2 ≡ n(ρ)λ2dB.
Comparison of Mean-field Predictions for the BKT Transition
We now explore the difference in the mean-field-extrapolated predictions for the BKT
transition temperature (TBKT ) between (i) our mean-field theory using Eq. (5.14) and
(ii) the HCK theory using ncrit = λ−2dB ln (ξ/gHCK), an example is displayed in Fig. 5.4
for the case of N = 3 × 104 atoms. These predictions are summarised in Fig. 5.5
for systems of various total particle number. Over the entire range of atom numbers
considered our TBKT estimates are significantly lower than those of the HCK theory,
indeed, for the system with 8 × 104 atoms, our prediction is approximately 50 nK
lower. However, we note that for the largest atom numbers considered, the ratio of
kBTBKT/~ωz is sufficiently large that the system is crossing over to the 3D regime.
A basic comparison of the two Hartree-Fock theories has already been discussed in
the previous subsection, however two main factors are responsible for the appreciable
differences in their extrapolated-predictions of TBKT :
(i) Smaller interaction parameter: The interaction parameter used by HCK
to determine the critical density (i.e. gHCK) is smaller than ours (g00) [see Table 5.1].
Noting that the interaction parameter only effects the critical density logarithmically
4We use the term extrapolate to emphasize that these theories are not valid at the transition as
quasi-condensation will significantly effect the density at this point
84
200 210 220 230 240 250 2600
2
4
6
8
10
12
14
T MFBKT T HCK
BKT
g
gHCK
T [ nK]
n(x
)λ
2 dB
Figure 5.4: Peak phase space density as a function of T . Our
theory (pluses), HCK theory (triangles) and polynomial fits (cyan
solid lines). Also displayed, as nearly horizontal solid lines, are
the critical phase space densities ln(380/gHCK) and ln(380/g) where
gHCK = mgHCK/~2 and g ≡ g00. Determined by the curve intersec-
tions are the predictions for the transition temperature made by our
theory (TMFBKT ) and the HCK theory (THCK
BKT ). N = 3 × 104 and all
other parameters as in Fig. 5.1.
85
0 1 2 3 4 5 6 7 8100
150
200
250
300
350
400
450
500
N [×10 4]
T[n
K]
T 2DBEC
T0
T MFBKT
T HCKBKT
Figure 5.5: Predictions for the BKT transition temperature as a func-
tion of N . Our theory (squares), HCK theory (triangles), quasi-2D
BEC (points) and pure 2D BEC temperature (crosses). All other
parameters as in Fig. 5.1.
86
(5.14)), this difference leads to a negative shift in the HCK prediction for the critical
temperature.
(ii) Use of mean-field density: Using the total areal density to judge when
the system is critical causes the HCK theory to predict a higher critical temperature
than our approach which instead uses n0(x). Furthermore the HCK mean-field theory
predicts higher densities through the use of the averaged interaction parameter, as
discussed earlier. A comparison of the various areal densities of these two theories in
the critical regime for N = 104 atoms is given in Fig. 5.1(c).
Effect (ii) dominates for the results in Fig. 5.5, increasingly so for larger numbers
of atoms.
5.3.3 Comparison Between C-field and Mean-field Predictions
In figure 5.6 we predict TBKT , using our mean-field theory TMFBKT and the HCK theory
THCKBKT , for the regime we simulated in §4 using our c-field method. For this regime
TMFBKT and THCK
BKT are similar to each other but are both significantly reduced from
TBKT (approximately 15%), here g and gHCK are similar due to the large aspect ratio,
ωz/ωx,y = 200, and hence the system is strongly confined to 2D at these temperatures.
The deficiency of the mean-field models for predicting TBKT can be understood
from Fig. 5.7. At T ′ ≈ TBKT [Fig. 5.7(a)] the phase space density predicted by the
c-field method is much more peaked than that predicted by the mean-field theories,
this is consistent with a significant quasicondensate fraction (shown in Fig. 4.4). As
the temperature rises well above the BKT transition point [Figs. 5.7(c) and 5.7(d)]
the agreement of the mean-field theories with the c-field method improves in line with
the corresponding decrease in quasicondensate fraction. For the highest temperature
case [Fig. 5.7(d)] the quasicondensate fraction is negligible and all models are in good
agreement as expected.
5.4 Summary
In this chapter we developed a consistent mean-field theory, applicable to the high
temperature quasi-2D Bose gas, and we compared this to the HCK theory (which
makes an unjustified simplification of the interaction term). These mean-field theories
may be extrapolated into the critical region to give a first order estimate of TBKT ,
however, the HCK theory uses the incorrect density (total density instead of ground
axial mode density) when extending the uniform system theory to the trapped system.
87
24 26 28 30 32 34 360
2
4
6
8
10
12
TBKTT MFBKT T HCK
BKT
g
gHCK
T [ nK]
n(x
)λ
2 dB
Figure 5.6: Peak phase space density as a function of T . Our mean-
field theory (pluses), HCK theory (triangles) and polynomial fits
(cyan solid lines). Also displayed, as nearly horizontal solid lines,
are the critical phase space densities ln(380/gHCK) and ln(380/g).
Included are the predictions for the transition temperature made by
our mean-field theory (TMFBKT ), the HCK theory (THCK
BKT ) and our c-
field method (TBKT ). Calculation parameters are the same as for §4,
i.e. 11267 87Rb atoms with ωx,y = 2π × 9.4 Hz and ωz = 2π × 1880
Hz
88
0
1
2
3
4
n(x
)λ
2 dB
(a)
×2
T ′ =0 .9 3 (b) T ′ =1.0 2
0 50 1000
1
2
3
4
x [µm]
n(x
)λ
2 dB
(c) T ′ =1.0 3
0 50 100x [µm]
(d) T ′ =1.0 6
Figure 5.7: Phase space density for different temperatures at (a) T ′ =
0.93 ≈ TBKT , (b) T ′ = 1.02, (c) T ′ = 1.03 and (d) T ′ = 1.06. Our
mean-field model (solid), HCK model (dashed), and c-field calculation
(dot-dashed). Calculation parameters are the same as for Fig. 5.6 but
for a varying N. (the × 2 in subplot (a) indicates the phase space
density has been divided by 2 in that figure).
89
As a result, our theory is assured to provide a lower bound for TBKT whereas the HCK
theory is not. We observe large differences between the predictions by our theory and
the HCK theory at low T and large N . The agreement of the HCK model with Monte
Carlo simulations (Figs. 5.2 and 5.3) is unremarkable as these systems are well outside
the critical region and deep into the normal phase. Finally, we compared density
profiles from our c-field simulations with the two mean-field theories. We found that
well outside the critical region all three theories are in good agreement whereas at
(and near) TBKT the mean-field theories grossly underestimate the central density (see
Fig. 5.7) as expected.
90
Chapter 6
Conclusions
6.1 Phases of the Quasi-2D Trapped Bose Gas
For uniform 2D systems with continuous symmetry and short range interactions, Mer-
min, Wagner and Hohenberg in 1966-1967 [31, 39] showed rigorously that LRO is not
possible for non-zero temperature, hence a BEC transition in the 2D Bose gas is ab-
sent. However, in 1971-73 Berezinskii, Kosterlitz and Thouless [5, 36] showed that even
though a 2nd order phase transition to an ordered state does not occur, a new type
of phase transition can emerge. This low temperature BKT phase exhibits topologi-
cal order, in which the two point correlations decay algebraically with the separation
distance.
Using Monte Carlo simulations in 2001-2002, Prokof’ev et al. [47, 48] studied the
BKT superfluid transition in the uniform (pure-2D) Bose gas, and they quantitatively
determined the critical phase space density at the transition point. They also showed
that the critical region of the 2D Bose gas is large and thus mean-field theory is
inadequate in the vicinity of the transition point.
Almost immediately after the observation of BEC in the 3D trapped gas there had
been considerable theoretical debate over the degenerate phase diagram for the 2D
trapped gas. In particular the competition between condensation, quasicondensation
and the BKT transition.
The three ENS experiments from the period 2005-2007 provided the first significant
clues to the behaviour of the quasi-2D trapped regime. They found that the onset of
phase defects (in this case free vortices) occurs at a similar temperature compared to
that where the decay of G1(∆x) changes from algebraic to exponential. Their results
also showed that the onset of bimodality coincided with the onset of phase coherence.
91
The ENS group speculated that their results imply the existence of a simple BKT-type
crossover in which bimodality and topological order occur together.
In 2008 we performed a comprehensive analysis of the low-temperature properties
of the quasi-2D trapped Bose gas (presented in chapter 4) and we elucidated the roles of
condensation and quasicondensation. The results of this c-field analysis form the main
results for this thesis. Above Tc we observe the suppression of density fluctuations (a
quasicondensate) associated with the peaking of the position density. At Tc we observed
the emergence of a condensate which we found to be associated with bimodality of the
momentum distribution. In disagreement with the ENS interpretation, we found Tc
(onset of condensation and momentum bimodality) to occur well above TBKT . Below
TBKT we found G1(∆x) to decay algebraically as a function of ∆x, and at TBKT [where
the peak phase space density satisfies Eq. (3.8)] we found that the algebraic decay
constant satisfies the uniform system criterion [Eq. (2.47)] i.e. η = 1/4. Also, we have
provided evidence that supports that at TBKT and below, free vortices seldom occur
at the trap centre, while above TBKT , central free vortices become more prolific with
increasing temperature. We note that our results contradict the ENS interpretation,
they propose the existence of a simple BKT-type crossover in which bimodality and
topological order occur together. However, our results are consistent with their data,
i.e. Fig. 1.4 (b) implies that coherence occurs at a temperature greater than that for
which G1(∆x) begins to decay algebraically.
Also in 2008, Holzmann et al. [33] presented results from their Monte Carlo simula-
tions of the quasi-2D Bose gas. However, their data resolution near the transition (1-2
data points) is poor and hence a comprehensive analysis could not be made. Further-
more, they incorrectly identify TBKT by applying the total density to Eq. (3.8) instead
of the ground axial mode density. More recently (2009), the NIST group [14] obtained
the quasi-2D regime experimentally. Their results support that coherence onsets at a
phase space density significantly less than that at the BKT transition, also supporting
our results.
We comprehensively analysed the transition from many perspectives and obtained
a consistent picture that describes both the ENS and NIST experimental observations.
6.2 Mean-field Description of the Normal Phase
In chapter 5 we have outlined a systematic Hartree-Fock mean-field theory for the
trapped quasi-2D Bose gas, which should provide a good description above the BKT
92
critical temperature. A central concern of chapter 5 has been to compare our theory
against that of HCK [32], which involves an unjustified simplification of the mean-field
interaction term. We show that the density profiles predicted by these theories disagree
as the temperature decreases.
We have also considered how to extend the BKT critical density condition to the
trapped quasi-2D Bose gas. Using this extension we are able to extrapolate our mean-
field theory to predict the critical temperature. We show that the HCK extrapolation,
which uses a different local density condition to identify the BKT critical point, predicts
a significantly higher value for TBKT .
Finally, we would like to emphasise that our treatment for extrapolating mean-field
theory to predict the BKT transition will likely underestimate TBKT . This arises be-
cause the assumed Gaussian fluctuations of the Hartree-Fock theory reduce the system
density compared to that of the actual system, which forms a quasicondensate as a
precursor to the BKT transition (a striking example of this reduced mean-field density
is shown by the comparison with c-field theory in Fig. 5.7). Thus our theory will pro-
vide a lower bound on where the system will obtain the critical density expected for
the BKT transition to occur, however, the same cannot be assured by the HCK theory.
6.3 Future Work
The understanding of the degenerate quasi-2D Bose gas has come along way since the
first experiments in 2005, however, a number of details remain to be determined, these
include:
(i) Is the coherence at Tc (above TBKT ) a finite size effect or is it analogous to a
conventional BEC transition of a harmonic trap? We note that for our results Tc =
0.98T0 where T0 is the ideal gas quasi-2D BEC temperature. To answer this question,
future experiments or simulations should investigate whether the gap between TBKT
and Tc diminishes in the limit of large systems.
(ii) Do systems with a temperature between TBKT and Tc have a finite superfluid
component, or alternatively, do free vortices destroy this superfluidity?
(iii) Will vortex pairing/unpairing at TBKT occur more suddenly for larger systems?
(iv) How do different trap geometries or interaction strengths affect the behaviour
of TBKT and Tc in the quasi-2D Bose gas?
However, now that the equilibrium phase diagram of this system has been clarified,
93
future work will no doubt examine the dynamical properties of this system. A regime
for which c-field theory is well suited to addressing.
94
References
[1] C. S. Adams, H. J. Lee, N. Davidson, M. Kasevich, and S. Chu, Evaporative cooling
in a crossed dipole trap, Phys. Rev. Lett. 74 (1995), 3577–3580.
[2] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell,
Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269
(1995), 198–201.
[3] V. Bagnato and D. Kleppner, Bose-Einstein condensation in low-dimensional
traps, Phys. Rev. A 44 (1991), 7439–7441.
[4] G. Baym, J.-P. Blaizot, M Holzmann, F. Laloe, and D. Vautherin, The transition
temperature of the dilute interacting Bose gas, Phys. Rev. Lett. 83 (1999), 1703–
1706.
[5] V. L. Berezinskii, Sov. Phys. JETP 32 (1971), 493.
[6] R K Bhaduri, S M Reimann, S Viefers, A Ghose Choudhury, and M K Srivas-
tava, The effect of interactions on Bose-Einstein condensation in a quasi two-
dimensional harmonic trap, J. Phys. B: Atomic 33 (2000), 3895–3903.
[7] D. J. Bishop. and J. D. Reppy, Study of the superfluid transition in two-
dimensional 4He films, Phys. Rev. Lett. 40 (1978), 1727–1730.
[8] R. N. Bisset, D. Baillie, and P. B. Blakie, Analysis of the holzmann-chevallier-
krauth theory for the trapped quasi-two-dimensional bose gas, Phys. Rev. A 79
(2009), 013602.
[9] R. N. Bisset, M. J. Davis, T. P. Simula, and P. B. Blakie, Quasicondensation and
coherence in the quasi-two-dimensional trapped Bose gas, Phys. Rev. A 79 (2009),
033626.
95
[10] P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Dy-
namics and statistical mechanics of ultra-cold Bose gases using c-field techniques,
Adv. Phys. 57 (2008), 363.
[11] P. B. Blakie and M. J. Davis, Projected Gross-Pitaevskii equation for harmonically
confined Bose gases at finite temperature, Phys. Rev. A 72 (2005), 063608.
[12] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Evidence of Bose-
Einstein condensation in an atomic gas with attractive interactions, Phys. Rev.
Lett. 75 (1995), 1687–1690.
[13] P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics, Cam-
bridge University Press, 2000.
[14] P. Clade, C. Ryu, A. Ramanathan, K. Helmerson, and W. D. Phillips, Observation
of a 2D Bose gas: From thermal to quasicondensate to superfluid, Phys. Rev. Lett.
102 (2009), 170401.
[15] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein
condensation in trapped gases, Rev. Mod. Phys. 71 (1999), 463–512.
[16] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M.
Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms,
Phys. Rev. Lett. 75 (1995), 3969–3973.
[17] K. B. Davis, M.-O. Mewes, M. A. Joffe, M. R. Andrews, and W. Ketterle, Evap-
orative cooling of sodium atoms, Phys. Rev. Lett. 74 (1995), 5202–5205.
[18] M. J. Davis, R. J. Ballagh, and K. Burnett, Dynamics of thermal Bose fields in
the classical limit, J. Phys. B: Atomic 34 (2001), 4487–4512.
[19] M. J. Davis and P. B. Blakie, Calculation of the microcanonical temperature for
the classical Bose field, J. Phys. A 38 (2005), 10259–10271.
[20] M. J. Davis and S. A. Morgan, Microcanonical temperature for a classical field:
Application to Bose-Einstein condensation, Phys. Rev. A 68 (2003), 053615.
[21] M. J. Davis, S. A. Morgan, and K. Burnett, Simulations of Bose fields at finite
temperature, Phys. Rev. Lett. 87 (2001), 160402.
[22] M. J. Davis, S. A. Morgan, and K. Burnett, Simulations of thermal Bose fields in
the classical limit, Phys. Rev. A 66 (2002), 053618.
96
[23] J. M. Doyle, J. C. Sandberg, I. A. Yu, C. L. Cesar, D. Kleppner, and T. J.
Greytak, Hydrogen in the submillikelvin regime: Sticking probability on superfluid
4He, Phys. Rev. Lett. 67 (1991), 603–606.
[24] A. Einstein, Quantentheorie des einatomigen idealen gases, Stiz. Presussische
Akademie der Wissenschaften Phys-math. Klasse, Sitzungsberichte 23 (1925), 3–
14.
[25] J. P. Fernandez and W. J. Mullin, The two-dimensional bose-einstein condensate,
J. Low Temp. Phys. 128 (2002), 233–249.
[26] D. S. Fisher and P. C. Hohenberg, Dilute Bose gas in two dimensions, Phys. Rev.
B 37 (1988), 4936–4943.
[27] C. W. Gardiner and M. J. Davis, The stochastic Gross-Pitaevskii equation: II, J.
Phys. B 36 (2003), 4731–4753.
[28] C. Gies, B. P. van Zyl, S. A. Morgan, and D. A. W. Hutchinson, Finite-temperature
theory of the trapped two-dimensional Bose gas, Phys. Rev. A 69 (2004), 023616.
[29] Z. Hadzibabic, P. Krger, M. Cheneau, B. Battelier, and J. Dalibard, Berezinskii-
Kosterlitz-Thouless crossover in a trapped atomic gas, Nature 441 (2006), 1118–
1121.
[30] Z. Hadzibabic, P. Krger, M. Cheneau, S. P. Rath, and J. Dalibard, The trapped
two-dimensional Bose gas: from boseeinstein condensation to BerezinskiiKosterl-
itzThouless physics, New J. Phys. 10 (2008), 045006.
[31] P. C. Hohenberg, Existence of long-range order in one and two dimensions, Phys.
Rev. 158 (1967), 383–386.
[32] M. Holzmann, M. Chevallier, and W. Krauth, Semiclassical theory of the quasi
two-dimensional trapped Bose gas, Europhys. Lett 82 (2008), 30001.
[33] M. Holzmann and W. Krauth, Kosterlitz-Thouless transition of the quasi-two-
dimensional trapped Bose gas, Phys. Rev. Lett. 100 (2008), 190402.
[34] Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Sov. Phys. JETP 66 (1987),
314.
[35] J. M. Kosterlitz, The critical properties of the two-dimensional xy model, J. Phys.
C 7 (1974), 1046–1060.
97
[36] J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions
in two-dimensional systems, J. Phys. C 6 (1973), 1181–1203.
[37] P. Kruger, Z. H., and J. Dalibard, Critical point of an interacting two-dimensional
atomic Bose gas, Phys. Rev. Lett. 99 (2007), 040402.
[38] N. Masuhara, J. M. Doyle, J. C. Sandberg, D. Kleppner, T. J. Greytak, H. F.
Hess, and G. P. Kochanski, Evaporative cooling of spin-polarized atomic hydrogen,
Phys. Rev. Lett. 61 (1988), 935–938.
[39] N. D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism
in one- or two-dimensional isotropic heisenberg models, Phys. Rev. Lett. 17 (1966),
1133–1136.
[40] W. J. Mullin, A study of Bose-Einstein condensation in a two-dimensional trapped
gas, J. Low Temp. Phys. 110 (1998), 167–172.
[41] L. Onsager, Crystal statistics. i. a two-dimensional model with an order-disorder
transition, Phys. Rev. 65 (1944), 117–149.
[42] O. Penrose and L. Onsager, Bose-Einstein condensation and liquid helium, Phys.
Rev. 104 (1956), 576–584.
[43] W. Petrich, M. H. Anderson, J. R. Ensher, and E. A. Cornell, Stable, tightly
confining magnetic trap for evaporative cooling of neutral atoms, Phys. Rev. Lett.
74 (1995), 3352–3355.
[44] D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Bose-Einstein condensation
in quasi-2D trapped gases, Phys. Rev. Lett. 84 (2000), 2551–2555.
[45] A. Polkovnikov, E. Altman, and E. Demler, Interference between independent fluc-
tuating condensates, Proc. Natl. Acad. Sci. USA 103 (2006), 6125–6129.
[46] V. N. Popov, Functional integrals in quantum field theory and statistical physics,
Reidel, Dordrecht, 1983.
[47] N. Prokof’ev and B. Svistunov, Two-dimensional weakly interacting bose gas in
the fluctuation region, Phys. Rev. A 66 (2002), 043608.
[48] N. Prokof’ev, O. Ruebenacker, and B. Svistunov, Critical point of a weakly inter-
acting two-dimensional Bose gas, Phys. Rev. Lett. 87 (2001), 270402.
98
[49] D. J. Resnick, J. C. Garland, J. T. Boyd, S. Shoemaker, and R. S. Newrock,
Kosterlitz-Thouless transition in proximity-coupled superconducting arrays, Phys.
Rev. Lett. 47 (1981), 1542–1545.
[50] H. H. Rugh, Dynamical approach to temperature, Phys. Rev. Lett. 78 (1997),
772–774.
[51] A. I. Safonov, S. A. Vasilyev, I. S. Yasnikov, I. I. Lukashevich, and S. Jaakkola,
Observation of quasicondensate in two-dimensional atomic hydrogen, Phys. Rev.
Lett. 81 (1998), 4545–4548.
[52] I. F. Silvera and M. Reynolds, Spin-polarized hydrogen: A weakly interacting boson
gas, J. Low Temp. Phys 87 (1992), 343–374.
[53] T. P. Simula and P. B. Blakie, Thermal activation of vortex-antivortex pairs in
quasi-two-dimensional Bose-Einstein condensates, Phys. Rev. Lett. 96 (2006),
020404.
[54] T. P. Simula, M. J. Davis, and P. B. Blakie, Superfluidity of an interacting trapped
quasi-two-dimensional Bose gas, Phys. Rev. A 77 (2008), 023618.
[55] S. Stock, Z. Hadzibabic, B. Battelier, M. Cheneau, and J. Dalibard, Observation
of phase defects in quasi-two-dimensional Bose-Einstein condensates, Phys. Rev.
Lett. 95 (2005), 190403.
99
![QUASi-Axisymmetric Research (QUASAR) Experiment · Quasi-axisymmetry is the axial symmetry of the magnitude of the magnetic field, in magnetic coordinates [1, 2], and is a generalization](https://img.dokumen.tips/doc/110x75/5f3b2bb75e220e77f87f44e5/quasi-axisymmetric-research-quasar-experiment-quasi-axisymmetry-is-the-axial-symmetry.jpg)
