Bose 2.2 User Manual download pdf (Page 35)

ratio [114]. It is given by

(

a − a

)



1 for trap ratios

λ .

1, used e.g. in the

measurements for the calibration of the scattering length, presented in section 3.2.3.

Concluding this section, we have seen that the dipolar interactions inﬂuence the ground-

state properties of a BEC. While in the TF approximation the shape of the condensate

remains the same as in the purely contact interacting case, the radii are diﬀerent: a dipolar

BEC elongates along an external magnetic ﬁeld. Furthermore, the chemical potential

the condensate depends on the relative dipole strength



, with

[1 − 

dip

(κ)]

Depending on the aspect ratio

of the atomic cloud, the chemical potential can be lower

or higher than in the case without dipolar interactions.

2.5 Dipolar Interactions between Spatially Separated Conden-

sates

After describing single dipolar BECs in the previous section, we now turn our attention

to systems with multiple spatially separated atomic samples. The distant dBECs can

interact with each other by the long-range dipolar interactions, whereas purely short-

range interacting condensates do not show any inter-site coupling, if particle exchange is

suppressed.

In this section, we give a basic introduction to dipolar multi-site systems, while more

detailed descriptions of their ground-state properties are given in Refs. [28, 29, 115–120].

We ﬁrst investigate the inter-site mean-ﬁeld potential in the minimal system of two dipolar

layers. We then calculate the inter-site energy between two Gaussian-shaped dipolar

BECs. This allows us to examine the relevance of the inter-site interactions with respect

to the on-site interactions in such system. At the end of this chapter we show that, under

realistic experimental conditions, the inter-site energy can be enhanced when adding more

layers to the system.

2.5.1 Mean-Field Potential in a Dipolar Double-Layer System

We consider a system composed of two identical discs of radius

, with the dipoles aligned

perpendicular to the disc plane, as illustrated in Fig. 2.4(a). The separation ∆

lat

the two samples is assumed to be much larger than their “thickness” in the

-direction.

Then, the distance

between two dipoles (belonging to diﬀerent layers) only depends

lat

and their in-plane separation

, and is given by

lat

+ r

. The angle

deﬁning the relative alignment of the dipoles (see Fig. 2.2), is given by

cos ϑ

lat

Therefore, using Eq.

(2.6)

, the dipole-dipole interaction potential of two dipoles belonging

The in-plane separation is the distance between the two dipoles, when the discs are projected onto each

other.

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