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

130 Macroscopic Dynamics
Large amplitude dipole oscillations
Large amplitude dipole oscillations can be described using the large-current hydrodynamic
equations (9.24,9.25). We consider the velocity field in z-direction to be given by (9.57) as in
the case of small amplitudes, while vMx = vMy =0. The assumption of a density-independent
effective mass is replaced by the more general requirement that the energy ε(k) is independent
of density. This is equivalent to setting it equal to the single particle band
ε(k, gn/ER) ≈ ε(k, gn/ER =0). (9.64)
The hydrodynamic equations for large currents (9.24,9.25) for the center-of-mass as defined
in (9.60) then take the form
∂ 1
Z −
∂t ¯h ∂kε(k) =0, (9.65)
¯h ∂
∂t k + mω2 zZ =0. (9.66)
containing the dispersion ε(k), or more precisely, the momentary group velocity ¯v(t) =
∂kε(k(t)) (see Eq.(6.18)) as a crucial ingredient. Small currents or, equivalently, small group
velocities are associated with small deviations of k from k =0. In this case, the dispersion
is given by ε =¯h 2 k2 /2m∗ and the equations (9.65,9.66) reduce to those for small amplitude
dipole oscillations (9.61,9.62).
In the tight binding regime, ∂kε(k) =δd sin(kd) (see Eq.(4.29)). Thus, the center-of-mass
obeys the equations of motion [73]
∂ δd
Z − sin (kd) =0, (9.67)
∂t ¯h
¯h ∂
∂t k + mω2 zZ =0. (9.68)
They describe a nonlinear pendulum. Its small amplitude oscillation is harmonic with frequency
(9.63), as discussed previously (recall that δ is related to the effective mass through (4.33)).
This regime is reached for
δ
Z ≪ . (9.69)
mω 2 z
Hence, the deeper the lattice the smaller the initial displacement has to be in order to stay in
the regime of harmonic oscillations. For initial values of Z larger than 2δ/mω2 z the solutions
of (9.67,9.68) correspond to a full rotation of k. If the initial value Z = Z0 is much larger
than δ/mω2 z, one can treat the center-of-mass as constant and the solution takes the simple
form
k = −tZ0mω 2 z/¯h, (9.70)
implying a constant increase of the quasi-momentum. This is analogous to the behavior of the
quasi-momentum during Bloch-oscillations driven by a constant force. The time evolution for
small displacement Z − Z0 from is given by
δ
Z = Z0 +
dmω2 zZ0
cos t .
¯h
(9.71)
mω 2 zZ0
This solution corresponds to an oscillation of Z around Z0 with amplitude much smaller than
Z0. Hence, the center-of-mass is always displaced from the center of the trap.