Bose 2.2 User Manual download pdf (Page 27)

We stress here that the dipolar length

does not correspond to a ﬁnite interaction radius

of the dipolar interactions. Such radius cannot be deﬁned for long-range interactions.

With the prefactors given in Eq.

(2.9a)

, the value of

seems chosen arbitrarily at the

moment. But we will see in section 5 that it was chosen such that an homogeneous dipolar

condensate becomes unstable, when

. Finally, we deﬁne the ratio of the dipolar

and the contact coupling strength,



def

12π~

, (2.9c)

which needs to be non-negligible to observe dipolar eﬀects in a BEC.

2.3 Mean-Field Description of Dipolar Bose-Einstein Conden-

sates

While considering the interactions between two atoms only in the previous section, the

description of an interacting Bose-Einstein condensate (containing around 10

atoms)

requires a many-body theory. We therefore start from the many-body Hamiltonian, which

contains the two-body correlations between all the atom pairs. Then, we derive the

Gross-Pitaevskii equation (GPE), representing an eﬀective single-particle description of

the full system in the so-called mean-ﬁeld model. Such reduction in complexity greatly

simpliﬁes the theoretical description of dipolar BECs, but relies on certain validity criteria

which we also discuss.

2.3.1 Gross-Pitaevskii Equation

The fundamental starting point for describing a system of interacting bosonic particles

conﬁned in a external potential

ext

(

) is the general many-body Hamiltonian [8, 89, 91]

H =

Ψ (r)

−

∇

+ V

ext

(r)

Ψ(r)

Ψ (r)

Ψ (r

) V

(2)

int

(r, r

)

Ψ(r)

Ψ(r

(2.10)

where

(2)

int

(

r, r

) is the full two-body interaction potential, given by Eq.

(2.8)

, and

(

)

and

(

) are the bosonic ﬁeld operators creating and annihilating a particle at the position

, respectively. These operators can be expanded in terms of the creation and annihilation

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