Bose 2.2 User Manual download pdf (Page 135)

strong oscillations of the Feshbach current which are damping out after around 2

These oscillations are reduced when using a linear ramp (with a ramping time of 1

and almost vanish when using an s-shaped ramp of the form

prog

(

t) = B

− (B

− B

) ·

(k + 1) · (1 −

t )

− k · (1 −

t )

k+1

, (A.33)

where

t/T

ramp

is the time normalized by the ramping time

ramp

and

k ≥

2 is the

ramping parameter that characterizes the “steepness” of the s-shaped curve. The particular

form of the ramp given by Eq.

(A.33)

ensures a smooth behaviour of the programmed

ﬁeld, as both the derivatives of

prog

at the beginning (

= 0 ) and at the end (

= 1 ) of

the ramp vanish. For the measurements presented here, we have used the parameters

ramp

= 1 ms and k = 3.

Eddy currents

As mentioned before, eddy currents in the metallic vacuum chamber lead to a delay of

the real magnetic ﬁeld

(

) at the position of the atoms with respect to the programmed

ﬁeld B

prog

(t). The ﬁeld B(t) is well described by the diﬀerential equation [36]

B(t) + B(t) = B

prog

(t) (A.34)

and we can measure its value at diﬀerent times of the s-shaped ramp by performing

Zeeman spectroscopy

118

. As expected for the s-shaped ramp, we see no oscillations in the

measured magnetic ﬁeld and the behaviour of

(

) is well described by the numerical

solution of Eq.

(A.34)

with the parameter

= 0

. At the time

= 1

, i.e. at the

end of the programmed ramp, the real magnetic ﬁeld has accomplished around 90 % of

the ramp. We can therefore experimentally realize well controlled ramps of the Feshbach

magnetic ﬁeld on time scales around 1 ms.

118

Each value of the magnetic ﬁeld shown in Fig. A.2 is the result of a spectroscopy measurement: We

vary the detuning of the probe laser in the absorption imaging and extract the resonance frequency

from a Lorentzian ﬁt to the recorded atom numbers. From the known Zeeman shift of the magnetic

sublevels, we can eventually calculate the magnetic ﬁeld at the position of the atoms.

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Bose 2.2 User Manual Page 135

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