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

11.3 Nonlinear propagation of sound signals 153
3. Saturation Regime in deep lattices
Provided that the lattice is deep enough to satisfy δ ≪ U/3, shock waves form only in the back
of the signal, as discussed above. If under this condition we further increase the strength of the
external perturbation, the relative phase φℓ+1/2 at some site reaches π (see Fig.11.9, upper
two panels) and there the current starts flowing in the opposite direction (see Eq.(11.10)).
As a consequence, a wake of noise is left behind the sound packets and saturation in the
amplitude of the propagating signal is found (see Fig.11.9, lower two panels). The interesting
feature is that the noise has average zero velocity, since the oscillations of population between
different wells are completely dephased. Hence it never overtakes the sound signal, which is
always able to “escape” from the noise. The system is in an interesting state, where the outer
part is phase coherent, while the part between the two sound signals has lost coherence. We
stress that in the uniform case, in presence of strong nonlinearities, we can observe a strong
deformation of the signal, but in the central region the system is always able to recover the
ground state after the sound wave has passed by (see Fig.11.4 a)).
To demonstrate the saturation of the signal, we let the system evolve for a long time
after the external perturbation and look at the amplitudes ∆n and ∆φ which are defined as
indicated in Fig. 11.2. We find an interesting scaling behavior that helps to distinguish the
three regimes mentioned above in the case of deep lattices and renders evident the saturation
of the signal amplitude. Both in the GPE simulations (for relatively deep lattices) and in the
DNLS simulations, we find a very interesting scaling law of the results, shown in Fig.11.10:
The effect of the barrier is universal when:
• the barrier parameters are combined in the form TP b/w which reflects the capability of
the system to react to an external perturbation;
• the relative density variation is rescaled as ∆n U/δ, while the amplitude of the relative
phase signal ∆φ need not be rescaled.
The phenomenology of the system as summarized in Fig.11.10 shows that for small barrier
parameter Tpb/w, the perturbation produced in the system is small, and depends linearly on
Tpb/w. Increasing the barrier parameter Tpb/w, the signal amplitude ∆n saturates quite
suddenly. The three regions indicated in Fig.11.10 correspond to the three regimes (1-3) listed
above.
Based on the DNLS equation (11.10) one can give a quantitative estimate of the saturation
value for ∆n. Assuming small density and relative phase variations, the first of Eqs.(11.10)
becomes
˙nℓ ≈ nδ
¯h
sin(φℓ+1/2) − sin(φℓ−1/2) ≈ nδ
¯h d∂φ
∂ℓ
. (11.14)
Now, we make the ansatz n =1+∆n[f+(ℓd − ct)+f−(ℓd + ct)] and φ = −∆φ[f+(ℓd − ct)+
f−(ℓd+ct)] for the density and relative phase variation, where f+ and f− describe respectively
the packets moving to the left and to the right. Using that c =(d/¯h) √ δU,we get
∆n =
δ
∆φ.
U
(11.15)