Bose 2.2 User Manual download pdf (Page 28)

operators

ˆa

and ˆa

, such that

Ψ (r) =

(r) ˆa

and

Ψ(r) =

(r) ˆa

, (2.11)

with

(

) a set of single-particle states. Assuming a macroscopic population

of the

lowest lying single-particle state

(such that

+ 1

' N 

1) we can replace the

according operators by numbers [91], with

ˆa

√

. We may then rewrite the ﬁeld

operator in the form

Ψ(r) =

√

N φ

(r) +

k>0

(r) ˆa

def

√

N ψ(r ) + δ

Ψ(r), (2.12)

where we have introduced the complex function

(

), deﬁned by the expectation value

of the ﬁeld operator

Ψ(r)

√

Nψ

(

). The function

(

) has the meaning of an

order parameter and exhibits a well deﬁned phase, which is spontaneously chosen at

the phase transition from the normal gas to a Bose-Einstein condensate [91]. Hence, the

phase transition to a BEC manifests itself by a broken gauge symmetry in the many-body

system. In the following, we will call ψ(r) the wave function of the condensate.

In general, the ﬁeld operator is time-dependent with a time evolution described by the

Heisenberg equation,

∂

∂t

Ψ(r, t) =

Ψ,

−

∇

+ V

ext

(r) +

Ψ (r

, t) V

(2)

int

(r − r

)

Ψ(r

, t)

Ψ(r, t).

(2.13)

With the ﬁeld operator given by Eq.

(2.12)

, and neglecting the ﬂuctuations

(

), we

obtain the time-dependent Gross-Pitaevskii equation (GPE) of a dipolar condensate

∂

∂t

ψ(r, t) =



−

∇

+ V

ext

(r)

+ gN |ψ(r, t)|

+ N

(r − r

) |ψ(r

, t)|



ψ(r, t),

(2.14)

where we have inserted the two-body interaction potential

(2)

int

(

r −r

), given by Eq.

(2.8)

Note that the DDI potential

(

r −r

) adds a non-local character to the Gross-Pitaevskii

equation. In contrast, the “standard” GPE of a purely contact interacting BEC depends

only on the local density n(r, t) = N |ψ(r, t)|

of the condensate.

The operators ˆa

and ˆa

obey the usual bosonic commutation relations

ˆa

, ˆa

= δ

α,β

, [ˆa

, ˆa

] = 0, and

ˆa

, ˆa

= 0.

We choose the function ψ(r) to be normalized to unity.

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Bose 2.2 User Manual Page 28

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