Bose 2.2 User Manual download pdf (Page 125)

The situation is diﬀerent in atomic systems with permanent magnetic dipoles where

the dipole length

is ﬁxed: here, the dipolar inelastic scattering is only relevant if the

condition

D  r

is fulﬁlled for the given atomic parameters. Since this is not the case

for chromium (see Tab. A.1), dipolar contributions to the inelastic scattering properties

of a

Cr BEC are expected to be small. They may be substantial, however, in ultracold

dysprosium samples. Due to the unknown short-range scattering properties, experimental

studies are required to further investigate the inﬂuence of the inelastic collisions in this

system.

A.2 GPE in Thomas-Fermi Approximation with Contact Inter-

actions

This section describes the ground-state properties of a BEC with dominant contact

interactions, discussed in section 2.4.2. In the TF approximation, and neglecting the

dipolar interactions, the stationary Gross-Pitaevskii equation, given by Eq.

(2.15a)

, writes

µψ(r)

= [V

ext

(r) + g n(r)] ψ(r), (A.7)

with

the chemical potential,

the contact coupling strength and with the harmonic

potential given by

ext

(

) =

2 (

). Hence, the density distribution

n(r) = N |ψ(r)|

of the BEC in each direction has the shape of an inverted parabola:

n(r) =

µ − V

ext

(r)

= n

· max

(

1 −

−

, 0

)

, (A.8)

where

def

= n

(0) = 15

πR

) is the central density of the condensate, with

(

, R

) its Thomas-Fermi radii in the respective directions. The boundary

(

) = 0

of the BEC is determined by the condition

ext

(

) =

, with

= (

, R

). Therefore,

the TF radii are given by

= 2

µ/

(

mω

) with

x, y, z

. The chemical potential is then

obtained via the normalization condition

n(r) = N, with the result [89]

µ =

2/5



¯a



2/5

~¯ω, (A.9)

where

¯ω

(ω

)

1/3

is the mean trapping frequency and

¯a

~/(m¯ω)

the mean

harmonic oscillator length. With this result, we obtain the mean condensate radius

R = (R

)

1/3

in the form that we have used in section 2.4.2:

R = 15

1/5

¯a



¯a



1/5

≈ 1.72 ¯a



¯a



1/5

. (A.10)

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