Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
The tight-binding equation of motion thus reads
i� ˙ Ψn = −J(Ψn+1 + Ψn−1) + F dnΨn + gTB(t)|Ψn| 2 Ψn . (6.5)
Here, we have added a constant force F . The tight-binding interaction
parameter gTB is obtained from the one-dimensional interaction parameter
g1D by integrating out also the harmonic-oscillator ground state in the longitudinal
direction, gTB = N � mω �/2π�g1D. The factor N comes from the
convention that the discrete wave function Ψn is normalized to one instead
of the particle number N.
Using an appropriate Feshbach resonance, the interaction parameter
gTB(t) can be controlled by external magnetic fields. The validity of
(6.5) is limited by the transverse trapping potential |gTB| ≪ �ω �/n0, with
n0 = maxn |Ψn| 2 .
The dispersion relation of this single-band model reads ɛ(k) = −2J cos(kd).
Its curvature or inverse mass m −1 = 2Jb 2 cos(kd)/� 2 determines the dynamics
of smooth wave-packets. The equation of motion (6.5) can be derived as
i ˙ Ψn = ∂H/∂Ψ ∗ n from the nonlinear tight-binding Hamiltonian
H = �
�
n
−J(Ψn+1Ψ ∗ n + Ψ ∗ n+1Ψn) + F nd|Ψn| 2 + gTB(t)
|Ψn|
2
4
6.2.2. Smooth-envelope approximation
�
. (6.6)
Experimentally, the initial state is prepared by loading a Bose-Einstein condensate
from an optical dipole trap into an optical lattice created by two
counter-propagating laser beams [40]. If this is done adiabatically, the lowest
oscillator states of the lattice are populated according to the profile of
the condensate in the trap (figure 6.4).
In the following, we use J and d as units of energy and length, respectively.
Furthermore, we set � = 1 and omit the subscript TB of the interaction
parameter. We tackle Eq. (6.5) by separating the rapidly varying Bloch
phase e ip(t)n from a smooth envelope A(z, t) comoving with the center of
mass x(t):
Ψn(t) = e ip(t)n A(n − x(t), t)e iφ(t) . (6.7)
With p(t) = −F t, x(t) = x0 + 2 cos(F t)/F , and φ = φ0 + 2 sin(F t)/F , the
envelope is found to obey the nonlinear Schrödinger equation
120
i∂tA = − 1
2m(t) ∂2 zA + g(t)|A| 2 A , (6.8)