Bose 2.2 User Manual download pdf (Page 51)

scattering properties in the system can be tuned by changing the potential diﬀerence

∆

between the open and the closed channel, which is the basic idea of the Feshbach

resonance technique.

Fig. 3.3, Feshbach resonance, principle and tuning of the scattering length:

(a) A Feshbach resonance occurs, when the energy

of a bound-state in the

closed channel is tuned in resonance with the kinetic energy

of the scattering

particles. The energy diﬀerence ∆

between the closed channel (red line) and

the open channel (black line) can be tuned e.g. by an external magnetic ﬁeld

of variable strength

. (b) Scattering length

(blue line) as a function of the

magnetic ﬁeld strength

close to a FR. The FR is located at

and

has a width ∆

. The vertical dashed line indicates that we obtain

= 0 at

+ ∆

. The horizontal dashed line indicates the asymptotic value

a = a

/jointfilesconvert/317304/bg

far away from the center of the FR.

We label the diﬀerent channels by the total spin

of the two colliding

atoms and its projection onto the quantization axis

, with

1,2

and

1,2

the quantum numbers of the single particles. Furthermore, the quantum numbers

(

l, m

) indicate the relative angular momentum of the nuclear motion in the two-particle

system. In the chromium BEC, we prepare all atoms in the energetically lowest state

|s = 3, m

= −3i

, as described in section 3.1.2. Therefore, the open (

-wave) channel is

labelled by the state

|S = 6, M

= −6, l = 0, m

= 0i

. While in alkali systems there are

usually only few channels coupled by the isotropic exchange interactions, the situation is

diﬀerent in chromium: the dipole-dipole interaction couples the open channel to several

other channels, that fulﬁll a certain set of selection rules

[95, 127, 162].

Note, that in chromium

|s, m

|J, m

, as the ground-state

has zero electronic orbital angular

momentum. Here, we choose the |s, m

i representation to be consistent with Refs. [95, 127, 162].

The selection rules for ﬁrst order coupling via DDI are: ∆

= 0

, ±

2; ∆

= 0

, ±

2; and ∆

= 0

, ±

(see appendix A.1.1). There is no coupling from

= 0 to

= 0. Due to the axial symmetry of the DDI

potential, the projection of the angular momentum is conserved, i.e. ∆

+ ∆

= 0. Even more

channels are coupled (weakly) by the DDI at the second order.

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