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

11.2 Current-Phase dynamics 145
[148] was achieved by employing two different excitations methods: The first one consisted
in raising a barrier in the center of a harmonically trapped condensate. The second method
instead consisted in letting the condensate equilibrate in presence of a barrier, which was then
removed. The first method produces a density bump in the center of the trap, which splits
into two bright sound wavepackets; the second method on the contrary gives rise to a dip in
the density which splits into two dark sound wavepackets.
The excitation method we adopt is a combination of these two: The initial condensate
is at equilibrium in a one-dimensional optical lattice superimposed to a simple box potential.
We then switch on and off a gaussian potential barrier in the center. This procedure has the
advantage that the ground states of the initial and final potential are identical. For zero lattice
depth, we get two composed bright-dark sound signals propagating symmetrically outwards.
The potential creating the barrier is written as a product of its spatial and temporal dependence
VB(x, t) =VBx(x)VBt(t) , (11.1)
where
VBx(x) =bER exp −x 2 /(wd) 2
, (11.2)
VBt(t) =sin 4
πERt
. (11.3)
¯hTp
The tunable parameters are the width of the barrier w, its height b and the time scale Tp.
They are subject to the constraints
w ≫ 1 , (11.4)
in order to address only the quasi-momenta in the central part of the Brillouin zone,
to ensure that the produced excitations are phonons, and
wd ≫ ξ, (11.5)
Tp > 1 (11.6)
in order to excite the lowest Bogoliubov band only. Note that for typical densities and lattice
spacings, w ≫ 1 automatically implies wd ≫ ξ.
11.2 Current-Phase dynamics
In order to study sound propagation in presence of the lattice, we use the GP-equation
i¯h ˙ϕ = − ¯h2 ∂2 x
2m + Vtot(x, t)+gnd|ϕ(x, t)| 2
ϕ(x, t) (11.7)
and the discrete nonlinear Schrödinger equation (DNLS)
i¯h ˙ ψℓ = − δ
2 (ψℓ+1 + ψℓ−1)+
Vℓ(t)+U|ψℓ(t)| 2
ψℓ(t), (11.8)