Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

Chapter 12
Condensate fraction
The presence of the optical potential may introduce phase fluctuations which reduce the degree
of coherence of the sample. This effect yields spectacular consequences such as number
squeezing [89] and the occurence of a quantum phase transition from the superfluid to the
Mott insulator phase [155, 90, 91]. In the presence of a one-dimensional optical lattice with
large N at relatively low lattice depth one can predict interesting effects on the condensate
fraction. The determination of the quantum depletion also serves as a check for the validity
of GP-theory.
In a shallow lattice, the quantum depletion of the condensate is slightly enhanced (see
section 12.3). It still has the form of the standard Bogoliubov result for the uniform system
(see review in section 12.2), with g replaced by the effective coupling constant ˜g >gand the
mass by the effective mass m∗ .
In the tight binding regime, the situation is very different (see section 12.4): The deeper
the lattice the more the system acquires the character of an array of condensates. In the
thermodynamic limit coherence is maintained across the whole system and the quantum depletion
is only little enhanced with respect to the shallow lattice. However, its dependence
on the scattering length changes with respect to the case of a shallow lattice: Reflecting the
two-dimensional character of each condensate of the array, the expression for the quantum
depletion takes the same form as for a disc-shaped condensate whose motion is frozen in the
tightly confined axial direction. In this configuration, the increase of the quantum depletion
with respect to the shallow lattice is due to its larger value in each disc and fails to capture
the loss of coherence between the discs. The loss of overall coherence has to be demonstrated
in a different way: Considering a large, but finite system we single out the contribution to the
depletion without transverse excitations corresponding to the depletion in a one-dimensional
system. This contribution grows logarithmically with the number of lattice sites. Moreover,
it becomes larger with increasing effective mass and with decreasing number of particles per
site. As the lattice is made deeper this 1D contribution becomes dominant and indicates the
transition to a regime where the condensate fraction is small and the long range order behavior
is modified.
The results included in this chapter were published in [102].
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