Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

|Ψ(z)| 2
z
6.2. Model
Figure 6.4: Typical initial state
for Bloch oscillations. This
density profile was obtained
as the ground state of the
continuous Gross-Pitaevskii
equation (6.1) with V1D =
V0 cos 2 (πz/d) + 1
2 mω2 zz 2 . The
trap ωz is then switched off.
This state is well described
with a tight-binding ansatz.
rameter g3D = 4π� 2 as/m is proportional to the s-wave scattering length as.
In the one-dimensional Gross-Pitaevskii equation
i� ∂
∂t Ψ(z, t) = � − �2
2m ∇2 + V1D(z) � Ψ(z, t) + g1D |Ψ(z, t)| 2 Ψ(z, t), (6.3)
the potential V1D(z) is given as a deep lattice potential V0 cos 2 (πx/d) with
spacing d. Later-on, the lattice is accelerated in order to observe Bloch
oscillations. This is done by tilting the lattice out of the horizontal plane,
or by accelerating the lattice by optical means.
6.2.1. Tight binding approximation
In sufficiently deep lattice potentials, only the local harmonic-oscillator
ground state in each lattice site is populated. This regime is called the
tight-binding regime (figure 6.4). The condensate is represented by a single
complex number Ψn(t) at each lattice site [139]. Neighboring sites are
weakly coupled by tunneling under the separating barrier with tunneling
amplitude
J ≈ 4
√ π (V0/Er) 3
4 exp(−2 � V0/Er) Er, (6.4)
where E = � 2 π 2 /(2md 2 ) is the recoil energy [136]. Here, changes of the local
oscillator function due to the interaction have been neglected. This assumption
holds in deep lattices with �ω � ≫ gTB|Ψn| 2 (see below). Otherwise, the
tunneling amplitude (6.4) gets modified already for slight deformations of
the Wannier functions [140].
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