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

138 Array of Josephson junctions
the expression (10.26) coincides with the large-s limit of (10.25) (see Eq.(7.39) and respective
discussion) indicating that short length scale fluctuations of the average density are suppressed
in a deep lattice.
The formalism based on the dynamical equations (10.11,10.12) assigns a simple physical
picture to small groundstate perturbations: They correspond to small amplitude plane wave
variations of the phases Sl and the average densities nl
∆Sl ∝ e i(lqd−ω(q)t) , (10.27)
∆nl ∝ e i(lqd−ω(q)t) . (10.28)
The phase difference of the perturbation at neighbouring sites equals qd. Since the wavelength
of such excitations can’t be smaller than twice the lattice period d, the wavenumber
q corresponding to the quasi-momentum of the Bogoliubov amplitudes used in section 7, has
its maximal physically relevant value at π/d. For this maximal value the phase difference is
π implying that the perturbation at neighbouring sites is exactly out of phase: One site takes
the minimal values of phase and population when the neighbouring sites reach the maxima.
This is just another way of saying that the perturbation has wavelength 2d and that a particle
exchange takes place between neighbouring wells. In contrast, at the minimal value of
q =2π/L the wavelength of the perturbation equals the size of the system and thus, particles
involved in the perturbation can be carried across half of the system length L.
10.3 Josephson Hamiltonian
Let us neglect the density dependence of δl,l′ (10.11,10.12). This corresponds to setting both quantities equal to the time-independent
single particle tunneling matrix element (4.30)
δ l,l′
= δ l,l′
µ = δgn=0 = −2 dz f ∗
gn=0(z) − ¯h2 ∂
2m
2
+ V (z) fgn=0(z − d) , (10.29)
∂z2 and δ l,l′
µ appearing in the dynamical equations
where fgn=0 is the single particle Wannier function of the lowest band. Consistently with this
step, we replace in µl as defined in (10.13) the density-dependent Wannier function by fgn=0.
In this way, we get
µl = ε0sp + nlgd f 4 gn=0dz , (10.30)
where ε0sp is the time-independent term
ε0sp = fgn=0 − ¯h2 ∂2
z
+ V (z) fgn=0dz , (10.31)
2m
which we omit in the following. It is common to write the dynamical equations in terms of
the populations Nl rather than the average densities nl and to replace the one-dimensional
Wannier function fgn=0(z) by the corresponding three-dimensional one fgn=0(r) =f(z)/L.
Note that the latter step affects only the g-dependent term in the equation for ˙ Sl. Weobtain
¯h ˙ Nl =
′) , (10.32)
l ′ =l+1,l−1
¯h ˙ EC
Sl = −Nl
2
δgn=0 NlNl ′ sin(Sl − Sl
+
l ′ =l+1,l−1
δgn=0
2
Nl ′
Nl
cos(Sl − Sl ′) , (10.33)