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

10.3 Josephson Hamiltonian 141
For both systems the second crucial parameter apart from EC is given by EJ = Nδgn=0
setting an upper bound to the current that can flow through the lattice. Note that also
the analogies between m, Z and Φ, kdgo along with a physical resemblance. Yet, a clear
difference consists in the fact that the “charging energy” in the case of the center-of-mass
motion is not provided by interactions but by the external trapping potential. The plasma
frequency ωP = √ ECEJ/¯h of the single Josephson junction (10.46) corresponds to the dipole
frequency ωD = π2δ/2ERωz of the harmonically confined lattice in the tight binding regime
(see section 9.6 Eq.(9.51) and use (6.41)).
Note that a double well system with a TF-condensate in each well is not well described
within the framework presented here. The Josephson Hamiltonian for such a system should
involve a density-dependent tunneling parameter and a compressibility which is non-linear in
the density. In fact, it is easy to see that if the condensate has a TF-profile in each well,
EC =2∂µ/∂N ∼ N −3/5 indeed depends on N. Moreover, the tails of the wavefunction in
the region of the barrier are obtained by matching the TF-profile to an exponential decaying
function. This matching procedure makes δ sensitive to N. Since for a larger N the TFprofile
becomes steeper at the barrier boundary, δ can in fact decrease for increasing N [57], in
contrast to the behavior of δ discussed in section 7.3 where the wavefunction of the condensate
at each site resembles the single particle wavefunction.
Within GP theory, the phases and occupation numbers are classical quantities. One can
go beyond this description by quantizing the appropriate Josephson Hamiltonian, replacing
the classical dynamical variables with operators (see discussion in [1] chapter 15.6). This
shows that the parameters EJ = Nδ and EC (10.34) play an important role in evaluating the
quantum fluctuations of phases and site occupations: The inequality EC ≪ EJ, corresponding
to strong tunneling, turns out to be the conditon for applying GP-theory. In this limit the
qunatum phase fluctuations are small while the occupation number fluctuations are large. In
the opposite case of weak tunneling EC ≫ EJ the relative phases are distributed in a random
way, while fluctuations in the occupations numbers are small.
Discrete nonlinear Schrödinger equation
The dynamical equations (10.11,10.12) with the quantities δ l,l′
,δ l,l′
µ ,µl evaluated using fgn=0
are equivalent to the Discrete Nonlinear Schrödinger Equation (DNLS) for the discrete wavefunction
ψl
i¯h ˙ ψl = − δgn=0
2
(ψl+1 + ψl−1)+U|ψl(t)| 2 ψl(t), (10.57)
which can be obtained using the ansatz (10.15) with the single particle Wannier functions.
Here, δgn=0 governs the tunneling between neighbouring wells and U =˜gn describes on-site
interactions. The DNLS has been used for example in [147] to investigate nonlinear phenomena
like solitons and breathers. In the next chapter, we will use it to study sound propagation in
the presence of a lattice.