Bose 2.2 User Manual download pdf (Page 40)

inter-site energy vanishes. Thus, we recover the result obtained in the Thomas-Fermi

approximation, where we found a vanishing dipolar contribution to the chemical potential

of a spherically symmetric cloud (see section 2.4.3).

The pancake-shaped density distribution leads to an even more complex behaviour for

the inter-site energy, as displayed in Fig. 2.5(b). The minimum value of the inter-site

energy is now



Min

(2)

inter



∼ h ·

. This means that the inter-site interactions

are stronger than in the spherical case, but still remain small when compared to the total

energy per particle. The reason for the increased interaction strength is the following: at

a large cloud aspect ratio

(but constant radial size) the two samples can get much closer

to each other without overlapping, strongly increasing the DDI between the neighbouring

dipoles. For

L →

0, i.e. when the clouds fully overlap, we ﬁnd a ﬁnite positive value for

(2)

inter

. This value corresponds to twice the dipolar energy of a single cloud containing

N atoms, and we can therefore deﬁne the on-site dipolar energy E

on,dip

def

lim

L→0

(2)

inter

= −

2(2π)

3/2

1 − 3u

(1 − ηu

)

3/2

(2.34a)

= −

2(2π)

3/2

κf

dip

(κ), (2.34b)

with the dipolar anisotropic function f

dip

(κ) introduced in section 2.4.3.

We have seen, that the inter-site interaction between two dipolar clouds is rather small

compared to the total energy in the system when using typical parameters for chromium

condensates. However, if we consider more than two clouds in a linear array, we may

enhance the dipolar inter-site interaction energy, as we discuss next.

2.5.3 Interaction Energy in a Linear Chain of Dipolar BECs

Let us consider a stack of

lat

regularly spaced pancake-shaped dipolar clouds, aligned in

the polarization direction

, as illustrated in Fig. 2.6(a). The total inter-site energy

inter,tot

in such system may be calculated by summing over the inter-site energy contributions

(2)

inter

(∆z) of all the possible pairs of condensates,

inter,tot

lat

−1

j=1

lat

− j) · E

(2)

inter

(∆z = j · d

lat

), (2.35)

where we make use of the equal spacing

lat

between the next-neighbours. We also take

care in Eq. (2.35) that each pair of samples is only counted once.

In Fig. 2.6(b), we plot the total inter-site interaction energy divided by the total number

of particles

tot

lat

· N

as a function of the number of samples in the linear array.

We choose parameters close to the experimental ones (see section 5)

{σ

= 3

m, κ

, d

lat

= 534

nm, µ

= 6

}

, and a constant on-site population

= 3000. Comparing

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