Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...
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eigenvalues ρ +<br />
k ρ−<br />
k<br />
trace <strong>of</strong> the monodromy matrix ρ ±<br />
k = ∆k ± � ∆2 k<br />
6.6. Dynamical instabilities<br />
= 1, which allows computing the eigenvalues from the<br />
− 1:<br />
∆k = 1 1<br />
tr M =<br />
2 2<br />
The logarithm λk = log[max(ρ +<br />
k , ρ−<br />
k<br />
�<br />
s (a)<br />
k<br />
�<br />
(T ) + d(b) (T ) . (6.27)<br />
)]/T is called Lyapunov exponent and<br />
characterizes the exponentially growing amplitudes sk, dk ∼ eλkt .<br />
For a given g(t), each Fourier component (dk,sk) is integrated over the<br />
common period T <strong>of</strong> g(t) and cos(F t), which yields the Lyapunov exponents<br />
λk. This method turns out to be very efficient for interactions with zero<br />
time average. Unfortunately, it does not work correctly in the case <strong>of</strong> a<br />
finite <strong>of</strong>fset. The constant interaction acts on the width degree <strong>of</strong> freedom,<br />
which is not included in the infinite wave packet.<br />
6.6.2. Bloch periodic perturbations<br />
For perturbations µ(t) = µ0 cos F t + µ1 sin F t modulated with the Bloch<br />
frequency, it is possible to solve the equations <strong>of</strong> motion (6.25) explicitly<br />
and to compute the Lyapunov exponent directly. From the criterion (6.16)<br />
we know that the cosine part alone leads to a periodic time evolution. Contrarily,<br />
the observations in section 6.4, figure 6.6(e) show that the sine part<br />
leads to the growth <strong>of</strong> perturbations. Thus, a perturbation theory in µ1 will<br />
be performed in the following.<br />
Considering for the moment only the part µ (0) (t) = µ0 cos(F t), we can<br />
write the equations <strong>of</strong> motion (6.25) in terms <strong>of</strong> the bounded time η(t) =<br />
sin(F t)/F (6.13)<br />
i ∂d(0)<br />
k<br />
∂η = −�k 2 � (0)<br />
+ µ0 s k<br />
k<br />
i ∂s(0)<br />
k<br />
∂η = k2d (0)<br />
k . (6.28)<br />
The crossed coupling <strong>of</strong> dk and sk is the same as in the homogeneous <strong>Bogoliubov</strong><br />
problem (subsection 2.3.1). We <strong>Bogoliubov</strong> transform the full equation<br />
<strong>of</strong> motion (6.25) using γk = � ɛk/k 2 sk +i � k 2 /ɛkdk and ɛk = � k 2 (k 2 + 2µ0),<br />
cf. equation (2.32)<br />
k<br />
iγk = cos(F t)ɛkγk + µ1<br />
2<br />
sin(F t) (γk + γ<br />
ɛk<br />
∗ −k) . (6.29)<br />
As expected, the solution <strong>of</strong> the unperturbed part γ (0)<br />
k = γ0 e−iɛk sin(F t)/F<br />
is periodic. In order to solve the first-order correction, we eliminate the<br />
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