Bose 2.2 User Manual download pdf (Page 25)

(a) (b)

Fig. 2.2, Dipole-dipole interaction (DDI):

(a) Two dipoles polarized by an external

magnetic ﬁeld

along the

-direction. The separation

|r|

and the angle

∠

(

z, r

) enter the DDI potential given by Eq.

(2.6)

. (b) The interaction

between two dipoles is attractive in a head-to-tail conﬁguration (

= 0), repulsive

in a side-by-side conﬁguration (

π/

2) and vanishes at the magic angle

ϑ = ϑ

of the potential energy at large distances. We therefore evaluate the following integral

I =

∞

cutoﬀ

int

D−1

dr, (2.7)

where

cutoﬀ

is some small but ﬁnite cut-oﬀ radius, and

is the dimensionality of the

system. Following this deﬁnition, the potential

int

is short-range if it decays faster

than

−D

dimensions, as the integral

converges in this case. Hence, the DDI

potential

∝ r

−3

is long-range in three dimensions (3D) and short-range in one and

two dimensions (1D and 2D).

The second method to determine the characteristic behaviour of a potential is more closely

connected to the physics of ultracold atomic samples. Referring to the considerations

in section 2.2.1, short-range potentials may be replaced by a hard sphere potential

characterized by one single parameter, the

-wave scattering length

. By solving the

two-body scattering problem for a potential

int

∝ r

−3

, the authors of Ref. [99] show

explicitly that, in one and three dimensions, the resulting wave function (at large distances

) is not compatible with the freely propagating waves obtained in the hard sphere

scattering. This means that the DDI potential shows a long-range character in 1D and 3D.

In two dimensions, it is however possible to deﬁne a 2D scattering length

, reproducing

correctly the wave function for r → ∞.

Interestingly, the two methods deliver diﬀerent answers on the question of the interaction

range of the DDI in one dimension. For all the calculations in this thesis we will use

the exact form of the DDI potential given by Eq.

(2.6)

. In this way, we obtain a correct

description of the interactions, independent of the dimensionality of the system.

This approach is not limited to the three-dimensional case. In one and two dimensions we would call it

a hard-wall potential.

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