Bose 2.2 User Manual download pdf (Page 39)

Refs. [51, 118]),

(2)

inter

lat

) = −

(2π)

3/2

(1 − 3u

)(1 − u

(η + L

))

(1 − ηu

)

5/2

exp

−

2(1 − ηu

)

, (2.33)

where

def

= d

lat

/σ

is the normalized distance between the samples, and

def

− κ

−2

. The

remaining integral in Eq. (2.33) must be computed numerically.

We show the value of the inter-site energy per particle,

(2)

inter

, versus the normalized

distance

for the cases of spherical clouds (

= 1) and pancake-shaped clouds (

= 10)

in Fig. 2.5. Furthermore, we show the overlap of the wave functions

)

def

(

)Ψ

(

)

. This value corresponds to the number of atoms in the region where

the clouds overlap, normalized by the total atom number 2

in the system. For the

calculations, we use typical chromium parameters:

= 3000,

= 6

, and

= 3

0.010.1 1 10 100

200

-2

-4

-6

100

0.1 1 10 100

(a) (b)

Fig. 2.5, Inter-site energy of two Gaussian shaped dipolar clouds:

The ﬁgures

show the inter-site energy per particle (solid blue lines) and the overlap of the

wave functions

) (dashed black lines) versus the normalized distance

lat

/σ

. (a) Case of two spherical clouds (

= 1) and (b) case of two

pancake-shaped clouds with

= 10. We use typical chromium parameters

speciﬁed in the text.

We ﬁrst consider the case of two spherical dipolar clouds, displayed in Fig. 2.5(a). At

large distances

L 

1 the inter-site interaction energy is practically zero. When the clouds

approach each other, the energy becomes more and more negative until the clouds show a

signiﬁcant overlap of

)

∼

5. The minimum value



Min

(2)

inter



∼ h ·

close to

= 2 is small compared to the total energy per particle in the system for our

parameters

E/N ∼ h ·

kHz

. In the limit

L →

0 the clouds entirely overlap and the

For weak interactions, i.e.

N a/a

10, the typical energy per particle is

E/N '

~¯ω

[89, Ch.6.2].

Inserting a typical mean trapping frequency

¯ω

= 2

π ·

500

, we obtain the value

E/N ' h ·

kHz

given in the text.

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