Bose 2.2 User Manual download pdf (Page 24)

using the reduced mass

red

2. We immediately see that the prefactors in the

contact coupling strength

, deﬁned in Eq.

(2.4b)

, are chosen such that

is the

intersection point of the wave function with the

-axis. Thus, the parameter

may be

identiﬁed with the

-wave scattering length introduced above and the scattering from

the

-shape potential

contact

can be related to the intuitive picture of the hard sphere

scattering. The fact that we can replace the full molecular potential by the contact

interaction potential, determined only by the scalar value of the

-wave scattering length,

will be of great beneﬁt when later, we will turn our attention to the mean-ﬁeld description

of a many-body system.

2.2.2 Dipolar Interactions

We now introduce the interactions between two dipoles which, in contrast to the interactions

considered in section 2.2.1, cannot be described by the pseudo contact interaction potential.

This arises from the long-range nature of the dipole-dipole interaction potential that we

also discuss here.

Throughout this thesis we consider magnetic dipoles polarized by a suﬃciently strong

external magnetic ﬁeld. The dipole-dipole interaction potential then reads [14]

(r, ϑ) =

4π

1 − 3 cos

, (2.6)

where

def

π ·

−7

T m/A

is the permeability in free space,

is the permanent

magnetic dipole moment and

is the angle between the polarization direction and the

relative position of the dipoles

, as illustrated in Fig. 2.2(a). The prime denotes the fact

that we only consider the case

r >

0 which is important, because

diverges when

tends to zero.

The dipole-dipole interaction (DDI) potential has two characteristic features:

(i)

its anisotropy, illustrated in Fig. 2.2: depending on the relative position of two

dipoles, they attract each other (

= 0, head-to-tail conﬁguration), repel each other

(

π/

2, side-by-side conﬁguration) or do not interact at all (

≈

◦

magic angle conﬁguration), and

(ii) its long-range character, which we now discuss in more detail.

We will use two methods to address the question whether a potential has a short-

range or long-range character [99]. Let us ﬁrst consider an homogeneous system from a

thermodynamical point of view. If the particles in the system interact only via short-range

interactions, the energy per particle is intensive, i.e. it depends only on the local properties,

such as the density. In contrast, in systems with long-range interactions, the energy per

particle also depends on global parameters, as for example the total number of atoms. The

classiﬁcation of a potential

int

(

) can be immediately checked by testing the convergence

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