Bose 2.2 User Manual download pdf (Page 82)

0.0 0.1 0.2 0.3 0.4 0.5

0.0

0.5

1.0

1.5

2.0

(-20) a

(-21) a

(-21.9) a

roton-maxon spectrum

momentum dependence DI

0.01 0.1 1 10

-0.5

0.0

0.5

1.0

(a) (b)

Fig. 5.2, Excitations in a 2D homogeneous dipolar BEC:

(a) The function



⊥

√



, characterizing the momentum dependent dipolar interactions

(DI): below

⊥

∼ l

−1

, the DI is eﬀectively repulsive, while above this momentum

value the attraction dominates. (b) Excitation spectrum of the dipolar 2D

homogeneous system for three diﬀerent scattering lengths, speciﬁed in the inset.

The spectra are calculated for chromium (mass / dipole strength) using the

parameters {ω

= 2π · 30 kHz, n

= 10

−3

wavepackets. Their size in the

-direction is given by the width

of the condensate,

which depends mainly on the external conﬁnement. In the radial directions, the extension

of the wavepackets is given by their wavelength, which is inversely proportional to the

quasi-momentum

⊥

. At small momenta

⊥

 l

−1

, the wavepackets are therefore highly

anisotropic, with the dipoles mainly aligned side-by-side. As the dipoles eﬀectively repel

each other in this conﬁguration, they stabilize the system giving a positive contribution

to the excitation spectrum. In the extreme case

⊥

→

0, the dipoles can stabilize

the sample even for negative scattering lengths as long as

g & −

(see Eq.

(5.2)

For increasing momenta

⊥

, the radial size of the wavepackets decreases. When their

overall shape is spherically symmetric, i.e. for

⊥

≈ l

−1

, the dipolar contribution to the

excitation energy vanishes. This behaviour is directly related to the vanishing dipolar

interaction energy in a spherically symmetric BEC, shown in section 2.4.3. Eventually, in

the limit of large momenta (

⊥

 l

−1

), the eﬀective dipolar interaction becomes attractive

since the density wavepackets are cigar-shaped, with the dipoles aligned mainly in the

head-to-tail conﬁguration. The momentum dependence of the excitation spectrum, given

by Eq.

(5.2)

, is therefore induced by the anisotropy of the long-range dipolar interactions

Roton instability

To discuss the phenomenon of the roton and the related instability, we ﬁrst consider very

A roton-type excitation spectrum is also predicted for BECs with isotropic non-local interactions [191,

192]. Here, the momentum of the rotons is related to the characteristic range of the interactions.

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