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

140 Array of Josephson junctions
For a typical double well system, the m-dependence in front of the cos Φ of the Hamiltonian
(10.39) is not important. The gas must be very dilute for it to be relevant (see [1] chapter 15
for a discussion of this issue). Neglecting this contribution, the Hamiltonian and the equations
of motion read
HJ = − EC
2 m2 + EJ cos Φ , (10.46)
¯h ˙m = EJ sin(Φ) , (10.47)
¯h ˙ Φ=−ECm, (10.48)
with the Josephson tunneling energy
EJ = Nδgn=0 . (10.49)
A characteristic property of such a Josephson junction is its possibility to exhibit plasma
oscillations of frequency
√
ECEJ
ωP =
. (10.50)
¯h
They correspond to harmonic small amplitude oscillations around m = 0, Φ = 0 of the
pendulum described by the equations (10.47,10.48). The existence of such oscillations was
first proposed by Josephson with respect to tunneling currents between superconductors [146].
Using (10.35,10.49) we can rewrite ωP in the form
2δgn=0˜gn
ωP =
. (10.51)
¯h
The junction described by the more complicated Hamiltonian (10.39) exhibits plasma oscillations
with the frequency
δgn=0(δgn=0 + NEC)
δgn=0 (δgn=0 +2˜gn)
ωP =
=
. (10.52)
¯h
¯h
It is interesting to note that the lowest Bogoliubov band excitation with q = π/d is the
physical analogue for a lattice of the plasma oscillations for a single Josephson junction:
As discussed above, the excitation at this value of q involves a particle exchange between
neighbouring sites. The corresponding frequency is given by (see Eq.(10.25))
¯hω(q = π/d) = 2δgn=0 [2δgn=0 +2˜gn] , (10.53)
where consistenly with the discussion of the Josephson Hamiltonian we have set δ = δgn=0
and κ−1 =˜gn. This frequency coincides with (10.52) apart from a factor √ 2. This difference
is due the fact that each site of the lattice exchanges particles with two neighbouring wells
instead of with just one as in the double well case.
A further interesting connection between the lattice and the double well case emerges when
comparing the equations of motion (10.47,10.48) with those for the center-of-mass motion
in the tight binding regime given the combined presence of lattice and harmonic trap (see
Eqs.(9.67,9.68)): The two sets of equations are formally identical if one identifies
m ↔ NZ
, (10.54)
d
Φ ↔ kd , (10.55)
EC ↔ mω2 zd2 . (10.56)
N