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

n(x, t)/n(x, t=0)
11.3 Nonlinear propagation of sound signals 149
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Figure 11.4: Relative density at t = 480¯h/ER in the GP-simulation with gn =0.5ER at lattice
depths a) s =0,b)s =10and c) s =20yielding the sound velocities included in Fig.11.3.
wake of noise, but still propagates at the sound velocity predicted by Bogoliubov theory.
This regime exists only in the presence of the lattice and requires δ ≪ U/3 to ensure
that shock waves form in the back of the signal as in regime (2).
The signal amplitudes attainable in each regime and the perturbation parameters b, w, TP
needed to reach a certain regime depend on the lattice depth s and on the interaction strength
gn, or equivalently, on δ and U. As a general trend, in the presence of a lattice a stronger
perturbation is needed to obtain the same signal amplitude as without lattice. This reflects
the fact that the condensate is less compressible in a lattice. As already mentioned, regime
(3) exists only in the presence of the lattice and provided that δ ≪ U/3, which for fixed gn or
U can be ensured by making the lattice sufficiently deep.
1. Linear Regime
In Figs.11.5 and 11.6 we present examples for sound signals produced with a weak external
perturbation of the form (11.1) in the uniform case and for a lattice with s =10.Thesignal
amplitudes ∆n and ∆φ are small and do not change during the propagation. The shape of the
signals remains constant and it moves with the sound velocity obtained from Bogoliubov theory.
In fact, all points of the signal move with the same speed. For example, the sound velocity
can be extracted equally well by measuring the position of the signal maximum, minimum or
center of mass as a function of time.