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

146 Sound propagation in presence of a one-dimensional optical lattice
valid in the tight binding limit (see chapter 10). In the GPE, the external potential Vtot(x, t) is
givenbythesumofthelatticepotentialV = sER sin2 (πx/d) and the time-dependent barrier
potential VB(x, t) (11.1). In the DNLS (11.8) instead, the potential is just given by the barrier
Vℓ = VB(ld, t), since the presence of the optical potential is included in the assumption that
space is discretized.
The GP-equation (11.7) is governed by two parameters: the optical lattice depth s and
the interaction strength gn (see chapter 5). These two quantities determine the parameters δ
and U in the DNLS, describing respectively the tunneling coupling between two neighboring
wells (corresponding to the height of the lowest Bloch band, see chapter 6.2) and the on-site
interaction U (see chapter 10).
For future use, it is important to rewrite the equations (11.7,11.8) in terms of the density and
the phase of the condensate wavefunction. In continuous space, the condensate wavefunction
can be written as ϕ(x, t) = n(x, t)/n exp[iS(x, t)] and the GP-equation (11.7) becomes
˙n(x) =−∂x n(x) ¯h
m ∂xS(x)
, (11.9)
˙S(x) =− 1
Vtot(x, t)+gn(x) −
¯h
¯h2
2m √ n ∂2 √ ¯h
x n + 2
2
(∂xS(x)) .
2m
In the discrete case, we write ψℓ = nℓ(t)/n exp[iSℓ(t)] and obtain
˙nℓ =
ℓ ′ δ
nℓ(t)nℓ
¯h
=ℓ±1
′(t)sin[Sℓ(t) − Sℓ ′(t)], (11.10)
˙Sℓ = − µℓ
δ nℓ
+
¯h 2¯h
′(t)
nℓ(t) cos[Sℓ(t) − Sℓ ′(t)] ,
ℓ ′ =ℓ±1
where µℓ = Unℓ + V (ℓ). Note that apart from the additional potential V (ℓ), these are the
equations of motion (10.11,10.12) derived in chapter 10 with the density dependent Wannier
function replaced by the single particle solution. Already at this stage, an important difference
between the uniform case and the deep lattice case can be pointed out: The sin and cos-terms
in the DNLS correspond to the terms involving ∂xS and (∂xS) 2 in the GPE for the uniform
case respectively. Hence, the presence of a deep lattice is associated with a dependence of the
current on the gradient of the phase analogous to the one in the absence of the lattice only if
Sℓ(t) − Sℓ ′(t) ≪ π. When this condition is not fulfilled nonlinear effects arise which have no
analogue in the uniform case.
11.3 Nonlinear propagation of sound signals
We solve numerically the GP-equation (11.7) and the DNLS (11.8) for various values of the
barrier height b, the barrier width w and the perturbation time Tp. We also vary the parameters
s and gn in the case of the GP-equation and the parameters δ and U in the case of the DNLS
to explore different regimes of lattice depth and interaction strength. The effect produced by
the perturbation can be made visible by looking at the ratio between the density at a certain
time n(x, t) and the equilibrium density distribution n(x, t =0). A sound signal emerges as a
wavepacket in the ratio n(x, t)/n(x, t =0). It maintains a compact shape and, after an initial