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

11.3 Nonlinear propagation of sound signals 151
2. Shock Wave Regime
The peculiarity of region (2) is the formation of shock waves. In the uniform case, a wave
front emits shock waves in the forward direction (see Fig.11.4). The stronger the external
perturbation, the more the sound signal deforms and spreads due to the shock waves. A
measurement of the position of the signal maximum or minimum yields a velocity which
deviates from the Bogoliubov prediction for the velocity of sound. To obtain the value of the
sound velocity included in Fig.11.3 we have done a convolution of the signal over a few lattice
sites, mimicking the limited resolution of a detection system. In this way the signal is less
distorted by the shock waves. We then determine the center-of-mass position of the signal
as a function of time. This method allows to extract the exact sound velocity thanks to our
specific excitation method. On the contrary, a similar measurement done with a bright (dark)
sound signal would lead to a higher (lower) value for the sound velocity. The formation of
shock waves in front of a bright sound wave packet (positive density variation) in a uniform
system is predicted analytically and numerically [149]. An analytic solution describing shock
waves in a uniform system has been found by [150]. Their formation has also been discussed
in [151].
In a shallow lattice, shock waves form in the front as in the uniform case. This is because
the formation of a gap in the Bogoliubov spectrum does affect only a small range of quasimomenta
close to qB. Hence, the mode–coupling among Bogoliubov excitations leads to the
creation of excitations outside the phononic regime which travel at a speed larger than the
sound velocity.
In a deep lattice, in contrast, shock waves are formed behind the sound packet. (see Fig.
11.7). In fact the lowest band tight binding Bogoliubov dispersion law, given by (see chapter
7.3)
¯hωq ≈
2δsin 2
πq
2qB
2δsin2
πq
+2U , (11.13)
2qB
has a negative curvature for all q as long as δ/U < 1/3. Iftheratioδ/U is larger than 1/3 the
Bogoliubov dispersion has a positive curvature in a small range of quasimomenta and becomes
negative closer to the zone boundary. Only in deep lattices (where δ/U ≪ 1/3), wavepackets
composed by quasi-momenta out of the phononic regime will propagate slower than the sound
packets. In this case shock wave formation takes place both in the front and in the back of the
sound wavepacket. Note that for typical values of the density δ/U lies between zero and one.
In a deep lattice where shock waves are formed behind the sound packet, we observe that the
relative phase distribution can approach φ ℓ+1/2 ∼ π/2, as shown in Fig. 11.8 which refers to an
early stage of the evolution shown in Fig. 11.7. Then, it becomes strongly deformed indicating
that higher orders of the sine–function in (11.10) are important. However this behaviour does
not become critical up to the point where the relative phase of two neighboring lattice sites
φ ℓ+1/2 = π, which defines the onset of regime (3).