Critical Behavior of a Trapped Interacting Bose Gas
Critical Behavior of a Trapped Interacting Bose Gas ∗
T. Donner1, S. Ritter1, T. Bourdel1, A. Öttl1, M. Köhl1,2†, T. Esslinger1
1Institute of Quantum Electronics, ETH Zürich, CH–8093 Zürich, Switzerland
2Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue,
Cambridge CB3 0HE, United Kingdom
†To whom correspondence should be addressed; E-mail: koehl@phys.ethz.ch.
The phase transition of Bose-Einstein condensation is studied in the critical regime, when fluctua-
tions extend far beyond the length scale of thermal de Broglie waves. Using matter-wave interference
we measure the correlation length of these critical fluctuations as a function of temperature. The
diverging behavior of the correlation length above the critical temperature is observed, from which
we determine the critical exponent of the correlation length for a trapped, weakly interacting Bose
gas to be ν = 0.67±0.13. This measurement has direct implications for the understanding of second
order phase transitions.
Phase transitions are among the most dramatic phe-
nomena in nature, when minute variations in the con-
ditions controlling a system can trigger a fundamental
change of its properties. For example, lowering the tem-
perature below a critical value creates a finite magne-
tization of ferromagnetic materials or, similarly, allows
for the generation of superfluid currents. Generally, a
transition takes place between a disordered phase and a
phase exhibiting off-diagonal long-range order which is
the magnetization or the superfluid density in the above
cases. Near a second-order phase transition point the
fluctuations of the order parameter are so dominant that
they completely govern the behavior of the system on all
length scales [1]. In fact, the large scale fluctuations in
the vicinity of a transition already indicate the onset of
the phase on the other side of the transition.
Near a second-order phase transition, macroscopic
quantities show a universal scaling behavior which is
characterized by critical exponents [1] that depend only
on general properties of the system, such as its dimen-
sionality, symmetry of the order parameter or the range
of interaction. Accordingly, phase transitions are classi-
fied in terms of universality classes. Bose-Einstein con-
densation in three dimensions, for example, is in the
same universality class as a three-dimensional XY mag-
net. Moreover, the physics of quantum phase transitions
occurring at zero temperature can often be mapped on
thermally driven phase transitions in higher spatial di-
mensions.
The phase transition scenario of Bose-Einstein conden-
sation in a weakly interacting atomic gas is unique as it
is free of impurities and the two-body interactions are
precisely known. As the gas condenses, trapped bosonic
atoms of a macroscopic number accumulate in a single
∗This is the author’s version of the work. It is posted here by
permission of the AAAS for personal use, not for redistribution.
The definitive version was published in SCIENCE, 315, (16 March
2007), doi:10.1126/science.1138807.
quantum state and can be described by the condensate
wave function, the order parameter of the transition.
However, it has proven to be experimentally difficult to
access the physics of the phase transition itself. In partic-
ular, the critical regime has escaped observation as it re-
quires an extremely close and controlled approach to the
critical temperature. Meanwhile, advanced theoretical
methods have increased the understanding of the critical
regime in a gas of weakly interacting bosons [2, 3, 4, 5].
Yet, the theoretical description of the experimental situ-
ation, a Bose gas in a harmonic trap, has remained un-
clear.
FIG. 1: Schematics of the correlation function and the cor-
relation length close to the phase transition temperature of
Bose-Einstein condensation. Above the critical temperature
Tc the condensate fraction is zero (see right) and for T ≫ Tc
the correlation function decays approximately as a Gaussian
on a length scale set by the thermal de Broglie wavelength
λdB. As the temperature approaches the critical temperature
long-range fluctuations start to govern the system and the
correlation length ξ increases dramatically. Exactly at the
critical temperature ξ diverges and the correlation function
decays algebraically for r > λdB (Eq 1).
We report on a measurement of the correlation length
of a trapped Bose gas within the critical regime just above
the transition temperature. The visibility of a matter-
wave interference pattern gives us direct access to the
http://arxiv.org/abs/0704.1439v1
first order correlation function. Exploiting our experi-
mental temperature resolution of 0.3 nK (corresponding
to 2×10−3 of the critical temperature) we observe the di-
vergence of the correlation length and determine its crit-
ical exponent ν. This direct measurement of ν through
the single particle density matrix complements the mea-
surements of other critical exponents in liquid Helium
[6, 7, 8], which is believed to be in the same universality
class as the weakly interacting Bose gas.
In a Bose gas the physics of fluctuations of the or-
der parameter is governed by different length scales. Far
above the phase transition temperature, classical ther-
mal fluctuations dominate. Their characteristic length
scale is determined by the thermal de Broglie wavelength
λdB and the correlation function can be approximated by
〈Ψ†(r)Ψ(0)〉 ∝ exp(−πr2/λ2dB) with r being the separa-
tion of the two probed locations [9] (Fig 1). Nontriv-
ial fluctuations of the order parameter Ψ close to the
critical temperature become visible when their length
scale becomes larger than the thermal de Broglie wave-
length. The density matrix of a homogeneous Bose gas
for r > λdB can be expressed by the correlation function
[10, 11]
〈Ψ†(r)Ψ(0)〉 ∝ 1
exp(−r/ξ), (1)
where ξ denotes the correlation length of the order pa-
rameter. The correlation length ξ is a function of temper-
ature T and diverges as the system approaches the phase
transition (Fig 1). This results in the algebraic decay of
the correlation function with distance 〈Ψ†(r)Ψ(0)〉 ∝ 1/r
at the phase transition. The theory of critical phenomena
predicts a divergence of ξ according to a power-law
ξ ∝ |(T − Tc)/Tc|−ν , (2)
where ν is the critical exponent of the correlation length
and Tc the critical temperature. The value of the critical
exponent depends only on the universality class of the
system.
While for non-interacting systems the critical expo-
nents can be calculated exactly [1, 12], the presence of
interactions adds richness to the physics of the system.
Determining the value of the critical exponent using Lan-
dau’s theory of phase transitions results in a value of
ν = 1/2 for the homogeneous system. This value is the
result of both a classical theory and a mean field ap-
proximation to quantum systems. However, calculations
initially by Onsager [13] and later the techniques of the
renormalization group method [1] showed that mean field
theory fails to describe the physics at a phase transition.
Very close to the critical temperature – in the critical
regime – the fluctuations become strongly correlated and
a perturbative or mean-field treatment becomes impossi-
ble making this regime very challenging.
We consider a weakly interacting Bose gas with den-
sity n and the interaction strength parameterized by the
s-wave scattering length a = 5.3 nm in the dilute limit
n1/3a ≪ 1. In the critical regime mean-field theory
fails because the fluctuations of Ψ become more domi-
nant than its mean value. This can be determined by
the Ginzburg criterion ξ > λ2dB/(
128π2a) ≃ 0.4µm
[14, 15]. Similarly, these enhanced fluctuations are re-
sponsible for a nontrivial shift of the critical tempera-
ture of Bose-Einstein condensation [2, 3, 4, 17]. The
critical regime of a weakly interacting Bose gas offers an
intriguing possibility to study physics beyond the usual
mean-field approximation [18] which has been observed
in cold atomic gases only in reduced dimensionality be-
fore [19, 20, 21, 22].
In our experiment, we let two atomic beams, which
originate from two different locations spaced by a dis-
tance r inside the trapped atom cloud, interfere. From
the visibility of the interference pattern the first order
correlation function [23] of the Bose gas above the criti-
cal temperature and the correlation length ξ can be de-
termined.
Separation (µm)
1.004 T
1.019 T
1.058 T
0.0 0.5 1.0 1.5 2.0
FIG. 2: Spatial correlation function of a trapped Bose gas
close to the critical temperature. Shown is the visibility of a
matter wave interference pattern originating from two regions
separated by r in an atomic cloud just above the transition
temperature. The gray line is a Gaussian with a width given
by the thermal de Broglie wavelength λdB which changes only
marginally for the temperature range considered here. The
experimental data show phase correlations extending far be-
yond the scale set by λdB . The solid line is a fit proportional
e−r/ξ for r > λdB. Each data point is the mean of on
average 12 measurements, the error-bars are ±SD.
We prepare a sample of 4 × 106 87Rb atoms in the
|F = 1,mF = −1〉 hyperfine ground state in a mag-
netic trap[24]. The trapping frequencies are (ωx, ωy, ωz)
= 2π × (39, 7, 29)Hz, where z denotes the vertical axis.
Evaporatively cooled to just below the critical tempera-
ture the sample reaches a density of n = 2.3× 1013 cm−3
giving an elastic collision rate of 90 s−1. The tempera-
ture is controlled by holding the atoms in the trap for a
defined period of time during which energy is transferred
to the atoms due to resonant stray light, fluctuations of
the trap potential or background gas collisions. From
absorption images we determine the heating rate to be
4.4± 0.8 nK/s. Using this technique we cover a range of
temperatures from 0.001 < (T − Tc)/Tc < 0.07 over a
time scale of seconds.
For output coupling of the atoms we use microwave fre-
quency fields to spin-flip the atoms into the magnetically
untrapped state |F = 2,mF = 0〉. The resonance con-
dition for this transition is given by the local magnetic
field and the released atoms propagate downwards due to
gravity. The regions of output coupling are chosen sym-
metrically with respect to the center of the trapped cloud
and can be approximated by horizontal planes spaced by
a distance r [23]. The two released atomic beams inter-
fere with each other. For the measurement we typically
extract 4 × 104 atoms over a time scale of 0.5 s which
is approximately 1% of the trapped sample. We detect
the interference pattern in time with single atom reso-
lution using a high finesse optical cavity, placed 36mm
below the center of the magnetic trap. An atom entering
the cavity mode decreases the transmission of a probe
beam resonant with the cavity. Due to the geometry of
our apparatus, only atoms with a transverse momentum
(px, py) ≃ 0 are detected which results in an overall de-
tection efficiency of 1% for every atom output coupled
from the cloud. From the arrival times of the atoms we
find the visibility V(r) of the interference pattern [25].
From repeated measurements with different pairs of mi-
crowave frequencies we measure V(r) with r ranging from
0 to 4 λdB where λdB ≃ 0.5µm.
With the given heat rate a segmentation of the ac-
quired visibility data into time bins of ∆t = 72ms length
allows for a temperature resolution of 0.3 nK which cor-
responds to 0.002Tc. The time bin length was chosen
to optimize between shot-noise limited determination of
the visibility from the finite number of atom arrivals and
sufficiently good temperature resolution. For the analysis
we have chosen time bins overlapping by 50%.
Figure 2 shows the measured visibility as a function of
slit separation r very close to the critical temperature
Tc. We observe that the visibility decays on a much
longer length scale than predicted by the thermal de
Broglie wavelength λdB. We fit the long distance tail
r > λdB with Eq 1 (solid line) and determine the correla-
tion length ξ. The strong temperature dependence of the
correlation function is directly visible. As T approaches
Tc the visibility curves become more long ranged and
similarly the correlation length ξ increases. The obser-
vation of long-range correlations shows how the size of
the correlated regions strongly increases as the tempera-
ture is varied only minimally in the vicinity of the phase
transition.
Figure 3 shows how the measured correlation length ξ
diverges as the system approaches the critical tempera-
10-3 10-2
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Reduced temperature (T-T
FIG. 3: Divergence of the correlation length ξ as a function
of temperature. The red line is a fit of Eq 2 to the data with ν
and Tc as free parameters. Plotted is one data set for a specific
temporal offset t0. The error bars are ± SD, according to fits
to Eq 1. They also reflect the scattering between different
data sets. Inset: Double logarithmic plot of the same data.
ture. Generally, an algebraic divergence of the correlation
length is predicted. We fit our data with the power law
according to Eq 2, leaving the value of Tc as a free fit
parameter, which has a typical relative error of 5× 10−4.
Therefore our analysis is independent of an exact cali-
bration of both temperature and heating rate provided
that the heating rate is constant. The resulting value for
the critical exponent is ν = 0.67 ± 0.13. The value of
the critical exponent is averaged over 30 temporal offsets
0 < t0 < ∆t of the analyzing time bin window and the
error is the reduced χ2 error. Systematic errors on the
value of ν could be introduced by the detector response
function. We find the visibility for a pure Bose-Einstein
condensate to be 100% with a statistical error of 2% over
the range of r investigated. This uncertainty of the visi-
bility would amount to a systematic error of the critical
exponent of 0.01 and is neglected as compared to the sta-
tistical error. The weak singularity of the heat capacity
near the λ-transition [1] results in an error of ν of less
than 0.01.
Finite size effects are expected when the correlation
length is large [16, 26] and they may lead to a slight un-
derestimation of ν for our conditions. Moreover, the har-
monic confining potential introduces a spatially varying
density. The phase transition takes place at the center of
the trap and non-perturbative fluctuations are thus ex-
pected within a finite radius R [5]. Using the Ginzburg
criterion as given in [14] we find R ≈ 10µm, whereas the
rms size of the thermal cloud is 58µm. The longest dis-
tance we probe in our experiment is 2µm which is well
below this radius R.
So far, in interacting systems the critical exponent ν
has been determined for the homogeneous system. The
λ-transition in liquid Helium is among the most accu-
rately investigated systems at criticality. One expects to
observe the same critical exponents despite the fact that
the density differs by ten orders of magnitude. In the
measurements with liquid Helium the critical exponent
of the specific heat α has been measured in a spaceborne
experiment [8]. Through the scaling relation α = 2− 3ν
the value of the critical exponent ν ≃ 0.67 is inferred
being in agreement with theoretical predictions [27, 28].
Alternatively, the exponent ζ ≃ 0.67 (which is related
to the superfluid density ρs = |Ψ|2 instead of the order
parameter Ψ) can be measured directly in second sound
experiments in liquid Helium [6, 7]. Due to an argument
by Josephson [29] it is believed that ν = ζ, however, a
measurement of ν directly through the density matrix
has so far been impossible with Helium.
The long-range behaviour of the correlation function
of a trapped Bose gas in the critical regime reveals the
behavior of a phase transition in a weakly interacting sys-
tem. Our measured value for the critical exponent does
not coincide with that obtained by a simple mean-field
model or with that of an ideal non-interacting Bose gas.
The value is in good agreement with the expectations of
renormalization group theory applied to a homogeneous
gas of bosons and with measurements using strongly in-
teracting superfluid Helium.
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for insightful discussions, SEP Information Sciences,
OLAQUI and QSIT for funding. T.B. acknowledges
funding by an EU Marie Curie fellowship under contract
MEIF-CT-2005-023612.
