Bose 2.2 User Manual download pdf (Page 57)

width and the center of the resonance is obtained by ﬁtting the function

a(I

) = a

/jointfilesconvert/317304/bg

1 −

∆I

− I

FB,0

, (3.3)

to the data, where the width and the center of the FR are expressed in terms of the

Feshbach current. The goal of this two-step ﬁtting procedure is to extract not only

the ﬁtting parameters ∆

and

FB,0

(and their uncertainties), but also the correlation

between them. If we ﬁnd e.g. an anti-correlation, the value of interest (in our case the

scattering length) is better known than the single uncertainties on the ﬁtting parameters

would suggest. We discuss the determination of the uncertainty ∆

on the scattering

length in details in appendix A.7.

The results for the calibration of the scattering length are shown in Fig. 3.5. Using only

the data above the FR (

B > B

) for the calibration, we obtain a small uncertainty ∆

close to the datapoints, i.e. for

a &

. However, in the region

a <

0 the uncertainty

grows signiﬁcantly. This indicates a rather large uncertainty on the ﬁtting parameters

∆

and

FB,0

, combined with a strong anti-correlation between them. If we also use

the data for the magnetic ﬁeld strengths below the resonance, the uncertainties on ∆

and

FB,0

typically decrease by a factor of around four, while the anti-correlation becomes

smaller. As shown in Fig. 3.5(b), we thus obtain a more precise calibration in the region

a < 0, which is an interesting region for the experiments presented in this thesis.

-10

100

300

-5

10 15

-40

scatt. length a (a )

B-B (G)

calibration a

uncertainty

(a) (b)

Fig. 3.5, Calibration of the scattering length:

(a) Scattering length values computed

from the measured values

above the Feshbach resonance (ﬁlled black

dots) and below the FR (open black dots). The black line is a ﬁt with the

theoretically expected behaviour

(

), given by Eq.

(3.1)

. (b) Uncertainty ∆

on the scattering length

. Using only the datapoints above the FR in the

ﬁtting procedure yields a large uncertainty ∆

in the experimentally interesting

region, i.e. when

a .

0 (dashed line). When including datapoints below the FR

(solid line), ∆a decreases in this region, owing to reduced uncertainties on the

ﬁtting parameters.

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