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the variational calculations match the measured critical scattering length in the regime

of prolate traps (

λ <

1). Thus, the instability mechanism in this regime is identiﬁed

with the phonon instability. At larger trap ratios, however, the variational calculations

predict a more stable situation than it is observed in the experiment. Only the results of

a full numerical solution of the GPE [37] closely recover the observed stability threshold,

as shown also in Fig. 5.5(a). In such type of calculations no assumption is made on

the shape of the wave function. It is suggested in Ref. [37] that the diﬀerence in the

critical scattering length, obtained from numerical and variational calculations, indicates

an instability via roton-like modes. Indeed, the numerical simulations show a decay of

the condensate into a three-peak structure after reducing the scattering length below the

critical value at a trap ratio

= 8 for instance. Furthermore, the numerical calculations

yield a bi-concave density distribution for the condensate close to the stability threshold

(see Fig. 5.5(a)). Such deformation of the atomic cloud corresponds to the structured

ground-states already mentioned in section 5.1.2.

So far, no clear sign of the roton instability has been experimentally observed in a

dipolar BEC. In addition, the exploration of the regime with a strong dipolar stabilization

was hindered by the technical diﬃculty to further increase the trap ratio of the single

trap

. This limitation can be overcome by using the experimental setup of the 1D optical

lattice: here, large (on-site) trap ratios

λ >

100 can be created in the limit of very deep

lattices. In such highly oblate trapping geometry we would expect the system to be stable

down to very negative critical scattering lengths, where a roton minimum is expected to

occur in the excitation spectrum. However, the attractive interactions between dipoles

that are located on diﬀerent lattice sites may destabilize the condensate. The stability

of the dipolar BEC in the lattice is therefore an open question which we have addressed

experimentally, with the results presented in the next section.

5.3 Stability Diagram of a

Cr - BEC in a 1D Lattice

We now present our measurements of the stability of a

Cr BEC in a 1D optical lattice.

We ﬁrst give a detailed description of the measurement procedure and the data evaluation.

Then, we present the stability diagram of the dipolar BEC in the 1D lattice, which displays

the critical scattering length as a function of the lattice depth. We will see that our results

are in good agreement with the stability threshold obtained from numerical calculations,

performed in the group of Luis Santos in Hannover.

Note the logarithmic scale of the horizontal axis (trap ratio

) in the stability diagram shown in

Fig. 5.5(a). To considerably lower the critical scattering length, the trap ratio must be increased much

above λ = 10, which was diﬃcult in the experimental setup of Ref. [35].

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