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

11.3 Nonlinear propagation of sound signals 147
formation time, propagates with a constant speed which we “measure” and compare with the
prediction obtained from Bogoliubov theory. Nonlinearities can lead to a deformation of the
signal and modify the speed at which it propagates. Yet, as we will see below, this need not
be the case in the presence of a lattice, provided that δ ≪ U/3. Moreover, nonlinear effects
lead to the formation of dephased currents in the back of the signal. In correspondence to this
the signal’s amplitude saturates while it keeps propagating at the Bogoliubov sound velocity.
Apart from the relative density, we also look at the evolution of the relative phase φ ℓ+1/2
between neighbouring wells. The quantity φ ℓ+1/2 is defined as
φ ℓ+1/2 = Sℓ+1 − Sℓ
in the case of the DNLS, while in the case of the GP-equation we define
(11.11)
φ ℓ+1/2 = S(x =(l +1)d) − S(x = ld) (11.12)
Note that in a deep lattice the phase distribution within each well is found to be flat to a
good degree. An example for nl(t)/nl(t =0)and φℓ+1/2 is plotted in Fig.11.2. We call ∆n
the amplitude of the sound wave packet in the relative density and ∆φ the amplitude of the
wavepacket in the relative phase, as shown in Fig.11.2.
Our first result is that sound signals of measurable amplitude can be observed also in deep
lattices where the sound velocity is considerably lower than in the uniform system. This is
illustrated in Fig. 11.3 which depicts the sound velocity obtained from the simulation (circles)
with the respective signal amplitudes ∆n together with the Bogoliubov prediction (solid line).
Up to s =20relatively large signal amplitudes can be obtained at this value of g. Note that
the signal amplitudes obtained from the simulation at s =20, 30 correspond to their saturated
values as discussed further below.
1 ↔∆
n
0
x / d
↔
∆φ
Figure 11.2: Example for sound signals as obtained from the DNLS (11.8). The signals are
created in the center with a relatively weak external perturbation of the form (11.1). They
move outward to the left and to the right at the Bogoliubov sound velocity. Upper panel:
Ratio between the density at a long time t after the perturbation and the equilibrium density.
Lower panel: Phase difference between neighbouring wells φ ℓ+1/2 as defined in (11.12). The
signal amplitude in the relative density and in the relative phase is denoted by ∆n and ∆φ
respectively.