Bose 2.2 User Manual download pdf (Page 69)

For lattice depths

lat

∼

, there is still particle exchange between the lattice

sites. However, the lattice is suﬃciently deep that the localized wave functions show

almost no overlap from one site to the next. This greatly simpliﬁes the description

of an interacting condensate in a 1D optical lattice, which is the topic of the next

section.

4.3 The Interacting BEC in a 1D Optical Lattice

Having discussed the non-interacting condensate in the perfectly periodic lattice potential,

we now come closer to the real experimental conditions: we include an underlying harmonic

trapping potential and also the inter-atomic contact interactions. We here consider the

case of suﬃciently deep lattices, where the BEC is split into a linear chain of spatially

separated atomic samples. Assuming a harmonic trapping of all these “sub-condensates”,

we then apply the so-called tight-binding approximation. This allows us to derive some

basic ground-state properties of the system, including the analytical expression for the

atom number distribution over the lattice sites.

4.3.1 Tight Binding Approximation

In the experiment, we conﬁne the BEC in an harmonic trapping potential

harm

(

), created

by the ODT, which is overlapped with the 1D lattice potential

lat

(

), given by Eq.

(4.2)

Here, we restrict for simplicity the harmonic potential to be cylindrically symmetric along

the lattice direction z. The full external trapping potential then writes

ext

(r) = V

harm

(r) + V

lat

(z)



+ y



+ ω

+ U

lat

sin



πz

lat



(4.14)

with

and

the radial and axial trapping frequencies of the ODT, respectively. When

the lattice potential is deep enough, the initially single condensate is split into several

sub-condensates that are localized at the discrete positions

j d

lat

with integer numbers

. In this case, we may apply the so-called tight-binding approximation (TBA). Here,

we consider the “generalized” TBA [179], where the spatial parts of the on-site wave

functions

(

r, N

) can depend on the local populations

of the lattice sites. The

ground-state wave function of the whole lattice system may then be written in the form

Ψ(r, t)

def

(t) Φ

(r, N

), (4.15a)

with ψ

(t)

def

−iϕ

(t)

, (4.15b)

The on-site wave functions Φ

(r, N

) are normalized to unity,

r |Φ

= 1.

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