Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
6.6. Dynamical instabilities
Time-dependent mass and interaction provide a source of energy for the
growth of perturbations. In section 6.4, we have seen that in some cases perturbations
on a length scale much shorter than the width of the wave packet
lead to the decay of the wave packet and k-space broadening. These perturbations
cannot be described by the collective coordinates of section 6.5.
Instead, the sudden broadening shown in figures 6.6(e) and 6.6(f) suggests
an exponential growth of small perturbations.
6.6.1. Linear stability analysis of the infinite wave packet
In order to describe the growth of perturbations quantitatively, we perform
a linear stability analysis on top of an infinitely extended wave function with
density |Ψn| 2 = n0
Ψn = [ √ n0 + δΦn] e ip(t)n e −iϕ(t) . (6.24)
Inserting this into the discrete Gross-Pitaevskii equation (6.5), we determine
the parameters p(t) = −F t and ˙ϕ = −2 cos(p) + µ(t), µ(t) = g(t)n0 from
the zeroth order in δΦ.
With this, the first-order equation of motion for δΦn is derived. Similarly
to the procedure in subsection 6.2.2, we use a transformation to the moving
reference frame x(t) = 2 cos(F t)/F in order to eliminate the first derivative
of the Taylor expansion of δΦn±1: s(z) + id(z) = δΦ x(t)+z. The Fouriertransformed
equations of motion then read
˙
dk = − � ɛ 0 k(t) + 2n0g(t) � sk , ˙sk = ɛ 0 k(t)dk , (6.25)
with ɛ 0 k (t) = k2 cos(F t). The equations of motion (6.25) are linear, with real,
time-periodic coefficients (provided, the frequency of the external modulation
µ(t) = n0g(t) is commensurate with the Bloch frequency F ), which
makes them accessible for Floquet theory [143]. The integration of two
linearly independent initial conditions, e.g. s (a)
k (0) = 1, d(a)
k (0) = 0 and
s (b)
k (0) = 0, d(b)
k (0) = 1, over one period T yields all information necessary
for the time evolution over n ∈ N periods:
� �
sk(t + nT )
= M
dk(t + nT )
n
� �
sk(t)
,
dk(t)
�
s
M =
(a)
�
k (T ) s(b)
k (T )
.
(T ) d(b) (T )
(6.26)
The eigenvalues ρ ±
k of the monodromy matrix M determine the growth of
the perturbations. Liouville’s theorem det M = 1 fixes the product of the
130
d (a)
k
k