Bose 2.2 User Manual download pdf (Page 80)

where

denotes the angle between the quasi-momentum of the excitations and the

polarization direction of the dipoles (the contact coupling strength

is deﬁned in Eq.

(2.4b)

and the dipolar coupling strength

is given by Eq.

(2.9b)

). Let us look at the limiting

cases of the excitation spectrum: in the limit of large momenta, we recover the quadratic

dispersion of a free particle

free

(

) =

). In contrast, at low momenta, the

excitations show a linear dispersion, comparable to sound waves, with a speed of sound

ω/q

determined by the density and the strength of the inter-atomic interactions

Note that the dipoles create an anisotropic sound velocity

(

), i.e. the speed of sound

depends on the propagation direction of the excitations. In a quasi-particle description,

we can identify these low-momentum excitations as phonons, in close analogy to the

theoretical description of excitations in solid state systems

stable (á = 0) unstable (á = ð/2)

(a) (b)

Fig. 5.1, Phonons in homogeneous dBECs:

(a) Density waves (phonons) travelling

parallel to the polarization direction of the dipoles (

= 0). Lines of increased

density (dark shaded areas) are generated with the dipoles sitting side-by-side.

(b) For

π/

2, the density maxima correspond to dipoles sitting in a head-to-

tail conﬁguration. The stability of the two conﬁgurations is discussed in the

text.

In the limit of vanishing dipolar interactions (

 g

), the excitation spectrum given

in Eq.

(5.1)

reduces to the well-known Bogoliubov spectrum of purely contact interacting

BECs [190]. In this case, the phonon dispersion does not depend on the direction of motion

of the excitations. For dominant dipolar interactions (DI) (

 g

), however, the dipolar

excitation spectrum clearly reveals the anisotropy of the DI: Phonons travelling parallel

to the orientation of the dipoles (

= 0) yield real and positive excitation frequencies, i.e.

a stable conﬁguration of the system. In contrast, phonons moving in the perpendicular

direction (

π/

2) lead to imaginary excitation frequencies, indicating an unstable

conﬁguration of the dBEC. We illustrate this so-called phonon instability in Fig. 5.1. In

In absence of dipolar interactions, the transition between linear and quadratic dispersion occurs at the

quasi-momentum q ∼ ξ

−1

, where ξ

def

= ~/

√

2mn

g is the so-called healing length of the condensate.

The dipolar excitation spectrum, given in Eq.

(5.1)

, may also be obtained by using quantum ﬁeld theory

which emphasizes the particle character of the excitations [113, 189].

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