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

134 Array of Josephson junctions
The formalism developed in this chapter is useful in understanding the propagation of sound
signals in a condensate subject to a lattice potential (see the subsequent chapter 11).
In [102], we presented the dynamical equations for the occupations and phases of each
site, as well as the result for the lowest Bogoliubov band. The discussion of the Josephson
Hamiltonian is added here.
10.1 Current-phase dynamics in the tight binding regime
In section 6.1 we have discussed Bloch state solutions of the stationary GP-equation in the
periodic potential of the optical lattice. We have shown that the condensate Bloch functions
Ψjk(z) can be written in terms of the condensate Wannier functions fj,l(z) in the following
way
Ψjk(z) = √
ndfj,l(z)e ikld . (10.1)
l
Note that Ψjk is normalized to the total number of atoms Ntot and apart from the normalization
coincides with the Bloch function ϕjk (see Eq.(5.2)). In this section we will use
an ansatz similar to (10.1) for the time-dependent condensate wavefunction Ψ(z,t), allowing
for the time-dependence of site occupation and phase. In the tight binding regime the
time-dependent phases and populations of each lattice site emerge as the dynamical variables.
Let us consider a stationary Bloch state of the lowest band (j =1) in the tight binding
regime. For convenience, we will choose the corresponding Wannier functions fl to take only
real values. Since fl is well localized at site l wecansaythatthecondensateatsitelhas phase
Sl = kld − µ(k)
t. (10.2)
¯h
The density of particles at site l is simply given by
nl = n, (10.3)
where, as previously, n is the average density of the system. Hence, in this state the population
of each site is constant across the sample at any time t, while the phases Sl vary with the site
index and have the simple time dependence µ(k)t/¯h. Furthermore, recall that two Wannier
functions fl, fl ′ can be obtained from each other by a simple displacement
and that the set {fl} is found for a given average density n
Ansatz for the time-dependent wavefunction
fl(z) =fl ′(z − (l − l′ )d) (10.4)
fl(z) =fl(z; n) . (10.5)
We are interested in time-dependent solutions Ψ(z,t) of (3.14). We focus on states whose
form is obtained by releasing the restriction of the phases Sl and the average densities nl to