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

10.3 Josephson Hamiltonian 139
where we have introduced the charging or capacitative energy
EC =2g fgn=0(r) 4 d 3 r . (10.34)
We note that EC is closely related to the compressibility of the system
κ −1 =˜gn = N EC
2
, (10.35)
where ˜g is the effective coupling constant introduced in section 5.2 and N, aspreviously,is
the number of particles per well at equilibrium.
The equations of motion (10.32,10.33)) can be rewritten in the Hamiltonian form
¯h ˙ ∆Nl = − ∂HJ
∂Sl
, (10.36)
¯h ˙ Sl = ∂HJ
,
∂∆Nl
(10.37)
where ∆Nl = Nl − N is the deviation of the population at site l from its equilibrium value N
(Note that
l ∆Nl =0). These equations are governed by the Josephson Hamiltonian (see
for example [1], chapter 16)
HJ = − EC
(∆Nl)
4
l
2
+ δgn=0 (N +∆Nl+1)(N +∆Nl) cos(Sl+1 − Sl) .(10.38)
l
This Hamiltonian describes an array of Josephson junctions with on-site charging energy EC
arising from 2-body interaction and a next-neighbour tunneling parameter δgn=0 which does
not depend on the site index and is solely determined by the external lattice potential.
A double-well system with just two lattice sites is a physical realization of a single Josephson
junction. The respective Hamiltonian is given by (10.38) with (l =1, 2). It reads
HJ = − EC
2 m2
+ δgn=0 N − m2 cos Φ , (10.39)
where we have introduced the quantities
m = N1 − N2
,
2
(10.40)
Φ=S1 − S2 . (10.41)
In this case, the difference between populations suffices to describe the dynamics of the system
since ∆N1 = −∆N2. The equations of motion for m and Φ can be obtained from this
Hamiltonian
They read
¯h ˙m = δgn=0
¯h ˙m = − ∂HJ
,
∂Φ
(10.42)
¯h ˙ Φ= ∂HJ
.
∂m
(10.43)
N 2 − m2 sin(Φ) , (10.44)
¯h ˙ m
Φ=−ECm − δgn=0 √ cos(Φ) .
N 2 − m
(10.45)