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2.2 Two-Body Interactions

The atomic density in a BEC is typically around

BEC

= 10

−

−3

, which is very

low compared to solids (

carbon

∼

−3

) or even air (

air

∼

−3

). From such

low density we might deduce that interactions between the particles do not play any

role regarding the physics of BECs. Yet, despite the diluteness of a condensate, the

interactions determine the fundamental properties of a BEC, as for example its shape,

stability, dynamics, and even its decay [8, 89, 91, 93, 94].

In this section we examine the relevant two-body interactions in a dipolar BEC. We will

ﬁrst show that, in the ultracold limit, the combined potential of all the short-range interac-

tions can be replaced by a zero-range pseudo-potential. The only remaining contribution,

that is not incorporated in the pseudo-potential, is the long-range dipolar interaction

which we will treat separately.

2.2.1 Short-range Interactions

At small separations

between two atoms, several interactions contribute to the so-called

molecular potential

(

). This potential is usually unknown, except from some general

properties:

(i)

at very small distances the electron clouds of the atoms overlap, leading to a strong

Coulomb repulsion,

(ii)

at slightly larger distance exchange interaction takes over, causing strong attraction

and a minimum of the potential at r = r

min

(iii)

further out induced dipole-dipole interaction leads to a weak attraction between the

particles, known as the van-der-Waals interaction, with a scaling V

vdW

∝ −1/r

The molecular potential has typically a depth of

|V (r

min

)|/k

∼

. Therefore, a large

amount of energy can be released, if two atoms in the BEC bind together to form a

molecular state. This indicates that even though the temperature of an ultracold atomic

samples is as low as

T ∼

, the BEC is not the true ground state of the system which

corresponds to a solid phase [8, Ch.9.1]. Fortunately, the rate at which molecules are

created in a BEC is usually suﬃciently low to enable (meta-)stable condensates on the

time scale of around one second.

To estimate the spatial range of the molecular potential, we consider Heisenberg’s un-

certainty principle: any conﬁnement of a particle to a region ∆

demands a momentum

uncertainty ∆

p ≈ ~/

∆

. Then, by equating the corresponding kinetic energy (∆

)

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