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10.7 Properties of Bose-condensed gases 239

To observe the condensate experimenters record an image by illumi-
nating the atoms with laser light at the resonance frequency.^43 Typically,^43 Generally, absorption gives a better
signal than fluorescence but the optical
system and camera are similar in both
cases.

the experiments have an optical resolution of about 5μm, so that the
length of the condensate can be measured directly but its width is not
precisely determined. Therefore the magnetic trap is turned off sharply
so that the atoms expand and some time later a laser beam, that passes
through the cloud of atoms onto a camera, is flashed on to record a
shadow image of the cloud. The repulsion between atoms causes the
cloud to expand rapidly after the confining potential is switched off (see
Exercise 10.6). The cigar-shaped cloud expands more rapidly in the ra-
dial direction (xandy) than alongz, so that after several milliseconds
the radial size becomes bigger than that alongz, i.e. the aspect ratio
inverts.^44 In contrast, the uncondensed atoms behave as a classical gas^44 This expansion of the wavefunction is
predicted by including time dependence
in the nonlinear Schr ̈odinger equation.

and expand isotropically to give a spherical cloud, since by definition the
thermal equilibrium implies the same kinetic energy in each direction.
Pictures such as Fig. 10.12 are the projection of the density distribution
onto a two-dimensional plane, and show an obvious difference in shape
between the elliptical condensate and the circular image of the thermal
atoms. This characteristic shape was one of the key pieces of evidence
for BEC in the first experiment, and it is still commonly used as a di-
agnostic in such experiments. Figure 10.13 shows the density profile of
the cloud of atoms released from a magnetic trap for temperatures close
to the critical point, and below.

10.7 Properties of Bose-condensed gases

Two striking features of Bose-condensed systems are superfluidity and
coherence. Both relate to the microscopic description of the condensate
asN atoms sharing the same wavefunction, and for Bose-condensed
gases they can be described relativelysimply from firstprinciples (as
in this section). In contrast, the phenomena that occur in superfluid
helium are more complex and the theory of quantum fluids is outside
the scope of this book.

10.7.1 Speed of sound

To estimate the speed of soundvsby a simple dimensional argument we
assume that it depends on the three parametersμ, Mandω,sothat^4545 The size of the condensateRis not
another independent parameter, see
vs∝μαMβωγ. (10.43) eqn 10.40.

This dimensional analysis gives^4646 Comparing the dimensions of the
terms in eqn 10.43 gives
ms−^1 =[kgm^2 s−^2 ]αkgβs−^2 γ.
Henceα=−β=1/2andγ=0.

vs

√

μ

M

. (10.44)

This corresponds to the actual result for a homogeneous gas (without us
needing to insert any numerical factor), and gives a fairly good approx-
imation in a trapped sample. The speed at which compression waves