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6.2.3 Numerical Simulations of the Collapse Dynamics

In a collaboration with the theory group in Hannover, we have examined more closely

the collapse dynamics of the dipolar condensate in the lattice by means of numerical

simulations. The simulations are based on the time-dependent Gross-Pitaevskii equation

(given by Eq. (6.1)),

∂

∂t

ψ(r, t) =



−

∇

+ V

ext

(r) − i~

|ψ(r, t)|

+ N

4π~

|ψ(r, t)|

+ N

(r − r

) |ψ(r

, t)|



ψ(r, t),

(6.1)

where the non-unitary term proportional to

describes three-body atom losses due

to inelastic collisions.

is the initial number of atoms in the condensate and

ext

(

)

is the combined trapping potential of ODT and lattice with

ext

(

) =

U sin

(

πz/d

lat

) +

i=x,y,z

2. The contact interactions are included via the

-wave scattering length

a and the dipole-dipole interaction potential V

(r − r

) is given by Eq. (2.6).

As a ﬁrst step, the ground-state of the condensate is calculated for a scattering length

= 2

and a lattice depth

init

= 12

. This task is performed by integrating

the GPE in imaginary time, with the loss parameter

set to zero. Then, the simulations

are performed by a real-time evolution of Eq.

(6.1)

, following our experimental sequence

and choosing

100

= 2

−40

. In analogy to the experimental imaging procedure,

the time-evolution of the system, shown in Fig. 6.6(a), is illustrated by integrating the

atomic density along the

-direction. After suﬃciently long expansion of the dBEC in the

TOF, the dilute cloud undergoes a ballistic expansion, since interactions do not play a

role anymore. Calculating the momentum distribution of the system in this regime (see

Fig. 6.6(b)) yields the spatial density distribution of the atomic cloud in the far ﬁeld of

the TOF, corresponding to the absorption images taken in the experiment. Finally, the

time evolution of the number

of coherent atoms is extracted from the simulations, with

the results presented in Fig. 6.6(c). We note that the time origin in the simulations,

= 0,

is set at the end of the lattice ramp. Furthermore, a ﬁxed holding time

hold

= 0

chosen, before starting the TOF.

Collapse dynamics

We split the discussion of the collapse dynamics, shown in Fig. 6.6, into three parts: First,

we discuss the deconﬁnement-induced collapse of a single dBEC (

= 0

). Then, we

consider the stable in-trap conﬁgurations (

= 8

and

= 12

) and ﬁnally we

address the unstable situations at the lattice depths U = 3.2 E

and U = 6.3 E

(i)

By choosing the ﬁnal lattice depth

= 0

, we obtain an unstable in-trap

conﬁguration of the system. At the end of the lattice ramp, the periodic density

100

The loss rate

is chosen equal to the value that is used in the simulations of the

-wave collapse of a

single dBEC [36] (see section 6.1).

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