Bose 2.2 User Manual download pdf (Page 37)

This expression means that two inﬁnite planes of dipoles do not interact [121]. In

this limit, the system is translationally invariant in the

-plane, and therefore the

mean-ﬁeld potential is zero everywhere.

uniform disc

Gaussian

B,z

lat

(a) (b) (c)

0 1

-4

-2

-10

-5

Fig. 2.4, Mean-ﬁeld potential of two distant dipolar samples:

(a) Schematic

drawing of the system. The two ﬂat samples with the dipoles oriented along

are separated by the distance

lat

. Dipoles with angle

ϑ < ϑ

(red cone) give

negative contributions to the inter-site mean-ﬁeld potential at the position

of the probe dipole (tip of the cone). In contrast, for

ϑ > ϑ

the contribution

is positive. (b),(c) Inter-site mean-ﬁeld potential (solid blue line) and density

distribution (dashed black line, normalized to the central density) in homoge-

neous disc-shaped samples and in samples with Gaussian density distribution,

respectively.

Let us now discuss the position dependent inter-site mean-ﬁeld potential Φ

disc

inter

(

). For

the geometry of two discs at a given ratio

R/d

lat

= 20, we numerically obtain the result

for Φ

disc

inter

(

) shown in Fig. 2.4(b). As expected for homogeneous discs, the mean-ﬁeld

potential close to the center (at

= 0) is almost constant and negative (see case (ii)

above). Towards the edges, however, the potential develops ﬁrst a deep minimum and

then a maximum at

ρ & R

, before it eventually approaches the zero value. The behaviour

of Φ

disc

inter

(

) close to the edge may be understood with the following geometrical argument:

only dipoles within the magic angle

, indicated by the red cone in Fig. 2.4(a), give a

negative contribution to the mean-ﬁeld potential since they attract each other. In contrast,

the dipoles just outside the cone give the strongest positive contribution, as the dipolar

potential falls oﬀ with

−3

. Therefore, once the edge of the cone comes closer to the

edge of the neighbouring disc, there are less and less atoms with positive contributions

to the mean-ﬁeld potential which therefore develops a minimum. For the chosen ratio

R/d

lat

= 20, the minimum sits at the position

ρ ' R − d

lat

tan ϑ

∼

. Further

out, the number of dipoles inside the cone reduces and eventually vanishes, which leads

to the maximum in the mean-ﬁeld potential. Far outside the disc, the potential ﬁnally

approaches the zero value due to the increasing distance to the neighbouring sample.

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Bose 2.2 User Manual Page 37

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