Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
coupling to γ∗ −k by transforming to γ+ = 1
2 (γk + γ−k) and γ− = 1
which both obey the equation of motion
2i (γk − γ−k),
k
iγ = cos(F t)ɛkγ + µ1
2
sin(F t) (γ + γ
ɛk
∗ ) . (6.30)
The unperturbed solution reads γ (0) = γ0 e −iɛk sin(F t)/F and suggests the
ansatz γ (1) = γ1(t) e −iɛk sin(F t)/F for the first-order correction. The increase
of γ1(t) is integrated over one period
k
γ1(T ) = µ1
2
iɛk
� T
0
�
dt sin(F t) γ0 + γ ∗ 0e 2iɛk
�
sin(F t)/F
ɛk
k
= µ1
2 2π
ɛk F J1 (2ɛk/F ) γ ∗ 0 ,
(6.31)
with the Bessel function of the first kind J1. The growth per period is
γ0 → γ0 + γ1(T ) and defines the Lyapunov exponent λT = γ1(T )/γ0. The
factor γ∗ 0/γ0 takes its extremal values +1 and −1 for γ ∈ R and γ ∈ iR,
respectively. Thus, the Lyapunov exponent is given as
�
�
λk = �
� µ1
k2 � ��
2ɛk ���
J1 . (6.32)
F
This result has been derived under the assumption |µ1| ≪ |k 2 + 2µ0| and
holds for both real and imaginary Bogoliubov frequencies. (Imaginary frequencies
appear for µ0 < 0 at k 2 + 2µ0 < 0.) Remarkably, it also holds at
k 2 + 2µ0 = 0, where a calculation similar to the above one yields the result
λ = 2πµ1k 2 /F 2 , which is exactly the limiting value of (6.32). Also in the
limit µ0 = 0, k 2 → 0, the result λ = 0 found from (6.25) is reproduced
correctly by the formula (6.32). In conclusion, the result (6.32) seems to be
valid for all µ1 < max(k 2 , k 2 + µ0), beyond the original validity condition.
6.6.3. Unstable sine
With the Lyapunov exponent (6.32), we understand the growth of the shortscale
perturbations in the case of a sine-like perturbation, figure 6.6(e). With
µ(t) = n0g1 sin(F t), n0 = 1/( √ 2πσ0), the Lyapunov exponent (6.32) reads
λk = |g1n0J1(2k 2 /F )|. Its magnitude is proportional to the strength of the
perturbation g1 and its k dependence is solely determined by J1(2k 2 /F ),
with a maximum λmax ≈ 0.582 g1n0 at k 2 max ≈ 0.921 F . For a quantitative
comparison, we choose a rather wide wave packet and a weak perturbation
(figure 6.10). We find the predicted growth rate λmax of the most unstable
mode in excellent agreement with the results from the numerical integration
of the tight-binding equation of motion (6.5).
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