Bose 2.2 User Manual download pdf (Page 55)

actly within the TF approximation

[169]. We are thus able to numerically calculate

the values

and

after TOF as a function of the scattering length, where

(

) is the Thomas-Fermi-radius of the BEC in the

-direction (

-direction). The result

of such calculations are shown in Fig. 3.4(a), for typical experimental parameters. For

a &

both values scale linearly with the scattering length, comparable to the case of a

non-dipolar BEC. In the regime

a <

, however, the linear scaling is only observed in

the

-direction, i.e. transverse to the alignment of the dipoles. Using a cigar-shaped trap

(

, ω

> ω

), the value

tends to zero when

a ≈ a

. Thus, the scaling of

/N with the scattering length is well described by the empirical formula

= σ (a − a

oﬀset

) , (3.2a)

leading to

a = σ

−1

+ a

oﬀset

, (3.2b)

where the parameters

and

oﬀset

can be obtained from a linear ﬁt to the calculated

values of

(

), as shown in Fig. 3.4(b). In principle, we can measure

and

diﬀerent values of the magnetic ﬁeld strength

close to the Feshbach resonance, and

obtain the scattering length

(

) via the relation

(3.2b)

. In practice, it turns out that the

uncertainties on the trapping frequencies, on the BEC radius and on the atom numbers

are too large to perform an accurate calibration in such direct way. Nonetheless, we

have performed measurements of

in low magnetic ﬁeld (far from the Feshbach

resonance) where

/jointfilesconvert/317304/bg

and ﬁnd a deviation of only 6

2% to the value calculated in the

TF approximation (see Fig. 3.4(b)).

Considering the form of Eq.

(3.2b)

, we see that any systematic scaling uncertainty on

may be absorbed by replacing the calculated parameter

by an eﬀective value,

σ →

eﬀ

. Since we know precisely the background value of the scattering length,

/jointfilesconvert/317304/bg

= (102

[144], we obtain

eﬀ

directly from the expansion measurements, with the detailed

procedure given in appendix A.7. Thus, the only required input from the calculations

is the parameter

oﬀset

. With the trap frequencies

(x,y,z)

= 2

π ·

(680

624

270)

, we

obtain the value

oﬀset

= (14

from the calculations in the TF approximation.

Since the parameter

oﬀset

is a crucial input in the calibration of the scattering length,

we compare the TF-calculations to full numerical simulations of the expansion of the

dBEC

, which yield the slightly diﬀerent value

oﬀset

= (9

(see Fig. 3.4(b)). We

are using this latter value of

oﬀset

for the calibration of the scattering length, since the

In Ref. [169] a sign problem occurs in the expansion formulae. The corrected formualae are given

explicitely in Ref. [113, A.5.9].

In calculations without dipolar interactions we ﬁnd the zero crossing at a = 0 a

The simulations are performed by K. Pawłowski from the theory group of K. Rzążewski in Warsaw,

using the numerical methods described in Ref. [118].

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