Bose 2.2 User Manual download pdf (Page 130)

The choice of the Gaussian form for the density distributions

(

) allows for a simple

expression of their Fourier transforms

F{n

} =

(2π)

3/2

exp



−

− ilδ



, (A.21)

where

= 0 and

= 1. We see that the relative distance

between the

two clouds in coordinate space has transformed into a phase shift in Fourier space. Using

the Fourier transform of the dipolar potential [113, Ch. A.5.2]

F{V

}

−g

(1 − 3k

)

we obtain the following expression for the inter-site interaction energy:

inter

= −

(2π)

1 − 3

exp



−

− ilk



. (A.22)

For the next step, we exchange the integration variable

by the dimensionless variable

. We furthermore deﬁne the cloud aspect ratio

/σ

and the dimensionless

distance L = l/σ

between the clouds, such that

inter

= −

(2π)

1 − 3

exp



−

−2

− iLq



. (A.23)

To perform the integration, we use the spherical coordinates (

q, θ, ϕ

): the integration over

the angle

simply yields a factor 2

as the system is cylindrically symmetric. With the

substitution u

def

= cos θ, we obtain

inter

= −

2π · g

(2π)

∞

−1

du q



1 − 3u



exp



−



(1 − u

) + κ

−2



− iqLu



(A.24)

The ﬁnal analytical step is the integration over q, leading to the result

inter

= −

(2π)

3/2

(1 − 3u

)(1 − u

(η + L

))

(1 − ηu

)

5/2

exp

−

2(1 − ηu

)

. (A.25)

For simpliﬁed notation, we have used

def

− κ

−2

and furthermore, we have changed the

integration limits, exploiting the even symmetry in the variable

. The last remaining

integration over

has to be performed numerically. Therefore, Eq.

(A.25)

is the ﬁnal

analytical result for the dipolar interaction energy of two Gaussian-shaped clouds, in

agreement with Ref. [51].

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