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2.4.3 TF-Approximation with Contact and Dipolar Interactions

The long-range character of the dipole-dipole interaction signiﬁcantly complicates the

description of a BEC in the TF-approximation. It has been found [110, 111] that in this

regime, the dipolar mean-ﬁeld potential Φ

dip

only contains terms that are either constant

or quadratic in the spatial coordinates. This leads to the remarkable fact that dipolar

BECs in the TF regime also have an inverted parabola shape, just like purely contact

interacting condensates.

To obtain an intuitive understanding how the dipolar interactions modify the properties

of a BEC, we choose a trapping potential with a cylindrical symmetry along the polarization

direction

of the dipoles. The radial and axial coordinates are then deﬁned by

= (

ρ, z

)

with the according trap frequencies

and

, and the trap ratio

def

= ω

/ω

. In a ﬁrst

step, we assume weak dipolar interactions, such that the shape of the BEC still resembles

an inverted parabola, with the cloud aspect ratio

def

= R

deﬁned by the TF-radii

and

in the radial and axial directions. The dipoles then generate a mean-ﬁeld potential

TF,κ

dip

(r) that enters the GPE and which in the cylindrical coordinates reads [110]

TF,κ

dip

(r) = n

−

− f

dip

(κ)

1 −

− 2z

− R

, (2.23a)

with the dipolar anisotropic function f

dip

(κ) given by

dip

(κ) =

1 + 2κ

1 − κ

−

3κ

artanh



√

1 − κ



(1 − κ

)

3/2

. (2.23b)

We illustrate the function

dip

(

), taking numerical values in the interval [

−

1], in

Fig. 2.3(a). As stated before, the terms in Eq.

(2.23a)

are either constant or quadratic

and

, i.e. the general form of the GPE is the same as in the purely contact

interacting case

. Therefore, in the TF-approximation, we obtain the same parabolic

density distribution of the dipolar BEC, as given by Eq.

(2.22)

. However, now the radii of

the condensate depend on both the contact interaction strength

and the relative dipole

strength 

and are given by the following expressions [110, 111]:

15gNκ

4πmω

(

1 + 

dip

(κ)

1 − κ

− 1

!)#

1/5

, (2.24)

and

/κ

. At given trap ratio

, the cloud aspect ratio

may be evaluated via the

implicit equation

3κ

+ 1

dip

(κ)

1 − κ

− 1

+ (

− 1)



− λ



= 0. (2.25)

In Eq.

(2.23a)

, the constant term proportional to

dip

(

) only adds an oﬀset to the chemical potential

and the quadratic terms are of the same form as the trapping potential.

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