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

Chapter 9
Macroscopic Dynamics
In this chapter, we show how to describe the long length scale GP-dynamics of a condensate
in a one-dimensional optical lattice by means of a set of hydrodynamic equations for the
macroscopic density and the macroscopic superfluid velocity (see sections 9.2 and 9.3). Within
this formalism, we can account for the presence of additional external fields, as for example
a harmonic trap, provided they vary on length scales large compared to the lattice spacing
d. As an input the equations require the energy and chemical potential Bloch band spectra
calculated in presence of the optical lattice only (see chapter 6.1 above). In the case of a
groundstate condensate in the presence of harmonic trapping, the hydrodynamic equations
reproduce the results obtained in chapter 5.4 for the smoothed density profile.
As an application we derive an analytic expression for the sound velocity in a Bloch state
condensate (see section 9.4, see also discussion in chapter 7.4 above). The sound velocity in
the groundstate condensate is determined by the effective mass and the compressibility. In a
moving condensate the sound velocity also involves information about the chemical potential
band spectrum discussed above in chapter 6.1.
In the combined presence of optical lattice and harmonic trap, the hydrodynamic equations
can be solved for the frequencies of small amplitude collective oscillations (see section 9.5).
Our treatment relies on the assumption that the effective mass is density-independent and that
the change in the compressibility brought about by the lattice can be accounted for by the
effective coupling constant ˜g (see chapter 5). It turns out that in a trap with axial frequency ω
superimposed to the lattice, the condensate oscillates as if it was confined in a harmonic trap
of axial trapping frequency m/m∗ωz with the lattice off. The slow-down of the collective
oscillations observed in the experiment [75] for increasing lattice depth has confirmed our
prediction.
The frequency ω = m/m∗ωz for the small amplitude center-of-mass motion is independent
of the equation of state (see section 9.6), i.e. it can be derived without knowing how the
chemical potential depends on density. The large amplitude center-of-mass motion is described
by a set of equations which does not involve the equation of state and is valid at any lattice
depth. As an input it requires the single particle Bloch band spectrum calculated with the
lattice potential only. Using these equations of motion we show that the deeper the lattice
the harder it is to fulfill the condition on the initial trap displacement for the observation of
harmonic dipole oscillations. In the tight binding regime, these equations also have a simple
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