Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

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