Bose 2.2 User Manual download pdf (Page 96)

5.4.1 Analysis of Inter-site Eﬀects in the Lattice

In our experiment, both on-site and inter-site interactions are always present and cannot

be “switched oﬀ” selectively to test their relative inﬂuence on the stability of the system.

However, the eﬀect of the inter-site interactions can be revealed by the numerical calcula-

tions (described in the previous section) by introducing a truncated dipolar interaction

potential

box

(

) =

(

)

[Θ((r · ˆz) + d

lat

/2) − Θ((r · ˆz) − d

lat

/2)]

, where Θ(

) is the

Heaviside function. Such cutoﬀ of the dipolar interactions in the lattice direction

eﬀectively amounts to removing the inter-site interactions, still taking into account the

full short-range and the long-range on-site interactions.

In Fig. 5.11, we compare the measured stability threshold to the calculations with

box

(

)

and with the full dipolar potential

(

). The calculations with the full dipolar potential

agree well the experimental data. In contrast, the calculations with the inter-site inter-

actions removed predict a substantially lower stability threshold: for the lattice depth

lat

, the diﬀerence compared to the datapoints is ∆

crit

, which is about

three times our experimental uncertainty. Hence, we identify a destabilizing character of

the dipolar inter-site interactions. Note that in both cases (with and without inter-site

interactions) the stability threshold in the calculations approaches the fundamental limit

−

' −

for very large lattice depths. In this regime, the spatially separated

condensates behave like inﬁnitely extended discs that do not interact, as we have shown

in section 2.5.1.

numerical:

on site + inter-site

on site

variational:

20 30 40 60 80

-25

-20

-15

-10

-5

100

Fig. 5.11, Destabilization of the dBEC by inter-site interactions:

Zoom into the

stability diagram in the deep lattice regime

lat

. Black dots show

the experimental values for the critical scattering length. Red lines show the

results of numerical calculations with the inter-site interactions included (solid

red line) or excluded (dashed-dotted red line). Similar results are obtained

by variational calculations with

= 2000 atoms per lattice site. Here, the

inter-site interactions are included in a calculation with

lat

= 15 lattice sites

(solid blue line) and excluded by using N

lat

= 1 (dashed-dotted blue line).

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