Bogoliubov Excitations of Inhomogeneous Bose-Einstein ...

6. Bloch Oscillations and Time-Dependent Interactions
6.3. Periodic solutions
We can construct a class of functions g(t) that allow a strictly periodic time
evolution of the wave packet. Quite generally, a rigid wave packet is by
no means necessary for persistent Bloch oscillations. Already in the linear
case g(t) = 0 we have seen that the wave packet breathes. Also in the
interacting case, one can find non-trivial functions g(t) that are compatible
with the time-reversal idea of the linear Bloch oscillation. Consider the class
of periodic functions
g(t) = cos(F t)P � sin(F t)/F � , (6.12)
in which a factor cos(F t) can be separated from a polynomial P (η) in the
bounded time variable
η(t) = 1
2
� t
0
m(s) −1 ds =
sin(F t)
F
. (6.13)
Because ∂tη(t) = 2m(t) −1 , the explicit time dependence of the mass factorizes
from all terms in the equation of motion (6.8) for A(z, t) = Ã(z, η(t)):
i∂η Ã(z, η) = − ∂2 z Ã(z, η) + P (η)|Ã|2 Ã(z, η). (6.14)
The ensuing dynamics for Ã(z, η) as function of η may be quite complicated.
However, as η(t) itself is a periodic function of time, also the solution A(z, t)
must be periodic: any dynamics taking place in the first quarter of the Bloch
period, while η runs from 0 to 1/F , is exactly reversed in the next quarter,
when η runs back. Figure 6.5 illustrates this argument by showing the
time-dependence of several key quantities over one Bloch cycle, as well as a
k-space density plot with clearly visible breathing dynamics.
The class of functions (6.12) covers all stable modulations that are Bloch
periodic. The above argument can be generalized for periodic functions
with frequencies commensurate with the Bloch frequency. Let us consider
a modulation with the ν2-fold Bloch period. The fundamental frequency
is then F/ν2, which suggests defining the time variable τ = F t/ν2 + τ0,
such that cos(F t) = sin(ν2τ). Trigonometric identities permit this to be
expanded as cos(F t) = sin(τ)M(cos τ), with some polynomial M. Now,
if g(t) is of the form sin(τ)P (cos(τ)), with some other polynomial P , we
can define the bounded time η ′ = −ν2 cos(τ)/F with ∂t = sin(τ)∂η ′. Then,
the factor sin(τ) factorizes from all terms of (6.8) and we find an equation
similar to (6.14)
122
i∂η ′Ã(z, η′ ) = − M(η ′ )∂ 2 z Ã(z, η′ ) + P (η ′ )| Ã|2 Ã(z, η ′ ). (6.15)