Bose 2.2 User Manual download pdf (Page 98)

show the energy spectrum of the collective excitations for

lat

= 15 dipolar condensates,

using parameters close to the experimental ones. The lowest lying mode of the band-

0 0.1 0.2 0.3

0.4

0.5

-22.75

-18.20

-13.75

(a) (b)

homogeneous layers lattice + trap

Fig. 5.12, Inter-site coupling in homogeneous and trapped dBECs:

(a) Band-

like excitation spectrum in a system of

lat

= 15 coupled 2D homogeneous

chromium BECs (blue lines). The red line shows a single site spectrum

with same parameters: trap frequency

lat

= 2

π ·

kHz

, scattering length

−

, density

= 10

−3

, lattice spacing

lat

= 534

. (b) Phonon

instability threshold and regions with structured ground-states (obtained by

numerical calculations), depending on the number of lattice sites

lat

and the

scattering length

, taken from Ref. [28]. The calculations are performed with

a Gaussian distribution of the atom numbers on the diﬀerent lattice sites, with

= 3300 atoms in the central site. Parameters:

lat

= 2

π ·

kHz

, radial

trap frequency

= 2

π ·

143

lat

= 560

, and chromium parameters for

mass and dipole strength.

like excitation spectrum is signiﬁcantly softened, i.e. it is showing lower energies

(

⊥

)

compared to the spectrum of a single populated site. As a consequence, the lattice system

is prone to develop a roton instability already at higher scattering lengths than a single-site

system. In addition, we observe that the quasi-momentum

rot

, corresponding to the local

minimum in the excitation spectrum, is found at smaller momentum values than in the

single-site conﬁguration. This means that the more lattice sites are populated, the longer

is the wavelength of the excitations involved in the roton instability. Essentially, it is the

global extension of the system in the lattice direction, which deﬁnes the wavelength of the

most unstable roton mode.

Numerical calculations allow to translate the eﬀect of the roton softening to a trapped

multi-site system: for instance a bi-concave structure is expected to emerge in the on-site

wave function of a dBEC in a 1D lattice, while a single-site system does not show such

modulation for the same parameters [29]. Furthermore, it was found that such structured

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