Download Chapter 1 BKT physics with two

July 24, 2011
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World Scientific Review Volume - 9.75in x 6.5in
BKT physics with two-dimensional atomic gases
Hadzibabic˙Dalibard
13
compatible with a significant condensed fraction in the trapped gas. The condensed
fraction ⇧0 is defined as the largest eigenvalue of the one-body density matrix ⇢ˆ,
with hr|ˆ
⇢|r 0 i = g1 (|r r 0 |) in a uniform system. When g1 (r) is significantly di↵erent
from zero for values of r comparable to the characteristic radius of the trapped gas R,
then we expect that this gas exhibits a detectable macroscopic quantum coherence.
Let us first address the case where the superfluid threshold (1.14) has been
reached and suppose that R stands for the radius of the disk where the superfluid
fraction is non-zero. Since g1 decays algebraically in the superfluid region with an
exponent that is smaller than 1/4, we have ⇧0 ⇡ g1 (R) & (⇠/R)1/4 . The radius
R can be estimated using the Thomas–Fermi approximation1 : in the limit where
the kinetic energy is negligible, the spatial density n(r) in the superfluid disk is
obtained from the simple relation gn(r) + V (r) = µ, hence R = axy (4g̃N/⇡)1/4 ,
where N is the number of atoms in the disk. Taking as typical values axy = 3 µm,
g̃ = 0.1, N = 2 104 , we find R ⇡ 20 µm. The healing length in the center of the
trap ⇠is ⇡ 0.6 µm, so that ⇧0 & 0.4.
The emergence of a detectable condensed fraction can significantly precede the
point where the critical density (1.9) is reached in the center of the trap. This is due
to the particular behaviour of the correlation length ` characterising the exponential
decay of g1 in the normal (non-superfluid) region. In a uniform 2D fluid, the length
` is predicted to diverge exponentially at the critical point (` ⇠exp[b/(T Tc )1/2 ],
where b is a constant), hence ` may take a value comparable to the size R of the
trapped cloud notably before the threshold (1.14) is reached. One can thus predict
that the transition from a fully thermal to a significantly coherent gas is actually
a crossover, whose relative width T /Tc ranges from 5% to 20% for realistic trap
parameters.13 An evidence for this crossover is provided by classical field Monte
Carlo calculations for a trapped gas.45,46
1.4. Experimental investigation of 2D physics in atomic gases
In ultracold atom research essentially all raw data take the form of images of atomic
clouds. The most commonly used technique is absorption imaging, whereby a laser
beam tuned close to resonance with an atomic transition is passed through the
cloud, and the “shadow” of the cloud is imaged on a camera. From the fractional
attenuation of the laser beam at any point in the image plane, one infers the column
atomic density along the “line of sight” of the laser. While this probe always couples
to the atomic density, what the observed density distribution actually represents
depends on what one does with the atomic cloud just before imaging it. We can
distinguish three typical scenarios which are relevant for our discussion here:
(1) If one takes an image of a trapped atomic cloud then one indeed simply
obtains the (column) density distribution.
(2) In the “time of flight” (TOF) technique, the cloud is released from the trap
some time before the image is taken, and the atoms are allowed to freely expand. For