Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

136 Array of Josephson junctions
Note that at equilibrium δ l,l′
= δ and δ l,l′
µ = δµ, where δ and δµ have been previously defined
in Eq.(6.31) and in Eq.(6.34). At equilibrium they do not depend on the sites l and l ′ since
the wavefunctions fl are related to each other through (10.4).
In deriving the dynamical equations (10.11,10.12) we have retained contributions from
the nonlinear term involving f 3 l fl±1dz, but discarded terms arising from f 2 l f 2 l±1dz, asin
previous chapters. It can be easily checked that the latter contributions are much smaller than
the former.
An approach equivalent to the Josephson formalism presented in this section, based on the
ansatz
Ψ(r,t)=
ψl(t)fl(r; Nl(t)) (10.15)
l
has been developed in [116, 117]. Here, Nl is the time-dependent occupation of the site l
and fl(r; Nl(t)) is the corresponding Wannier function at site l. In practize, ψl is a discrete
wavefunction which corresponds to our √ nldeiSl. Replacing this nonlinear tight binding ansatz
into the time-dependent GP-equation and integrating out the spatial degrees of freedom, a
discrete nonlinear equation for the ψl(t) is obtained. The authors find that it is justified to
approximate the Wannier functions in (10.15) by their solution at equilibrium. In our case,
this corresponds to setting δl,l′ = δ and δl,l′ µ = δµ in the dynamical equations (10.11,10.12).
Moreover, in this approximation the on-site chemical potential (10.13) depends linearly on the
time-dependent deviation nl − n from the average density.
Lowest chemical potential Bloch band
Let us assume the system to be in a Bloch state of the lowest band. This corresponds to
setting Sl and nl equal to (10.2,10.3) in the equations of motion (10.11,10.12). While the
equation for ˙nl is solved trivially, the equation for ˙ Sl yields
µ(k) =µ0 − δµ cos(kd) , (10.16)
where µ0 is obtained by evaluating (10.13) for nl = n. This is exactly the expression for the
lowest chemical potential Bloch band in the tight binding regime discussed in section 6.2 (see
Eq.(6.32)).
10.2 Lowest Bogoliubov band
In order to recover the tight binding expression (7.38) for the lowest Bogoliubov band in the
tight binding regime we consider small deviations of the phases Sl and the average densities
nl from the groundstate
Sl =∆Sl , (10.17)
nl = n +∆nl , (10.18)
where we used that Sl =0at equilibrium. We then linearize Eqs.(10.11,10.12) in ∆Sl, ∆nl.
The result reads
˙
∆nl = n δ
¯h (2∆Sl − ∆Sl+1 − ∆Sl−1) , (10.19)