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A.6 Excitation Spectrum of a 2D Homogeneous Dipolar BEC

Here, we show a brief derivation of the excitation spectrum in a 2D homogeneous dipolar

gas, discussed in section 5.1.2. A more detailed treatment can be found in Refs. [16, 26,

27]. We start from the excitation spectrum in the 3D homogeneous system, given by

Eq. (5.1),

E(q) = ~ω(q) =

[g + g

(3 cos

α − 1)], (A.26)

where

is the (constant) atomic density,

|q|

is the quasi-momentum of the excitations

and

the angle between their propagation direction and the polarization direction

of the dipoles. We see that in Eq.

(A.26)

, the contact and the dipolar interactions are

represented by the Fourier transform of the two-body interaction potential (see Eq.

(2.8)

)

given by

int

= g + g

(3 cos

α − 1).

In a 2D homogeneous condensate, we may approximate the shape of the wave func-

tion in the

-direction by a Gaussian

(

) = (

√

)

−1/2

exp [−z

/(2l

)]

, with

the

width of the sample. The atomic density along

is then given by

(

) =

|φ

(z)|

(

√

)

−1

exp [−z

]

with its Fourier transform

(

) = (2

)

−1/2

exp [−q

/4]

. With

this ansatz for the wave function, we obtain the eﬀective 2D interaction potential in

Fourier space

(2D)

int

⊥

) via [16]

(2D)

int

⊥

)

def

∞

−∞

n(q

)

int

n(−q

) dq

2π

∞

−∞

−q

g + g

⊥

+ q

− 1

√

2πl

g + 2g

⊥

√

, (A.27)

where we have used

cos

(

⊥

) with

⊥

(

) the radial (axial) component

of the quasi-momentum. The function

(

) that we obtain from the integration is given

by H

(χ) = 1 −

√

π|χ|exp[χ

] erfc[χ], with erfc[χ] the complementary error function.

We furthermore introduce the two-dimensional homogeneous density

√

2π l

Finally, by replacing in Eq.

(A.26)

the expressions

q → q

⊥

→ n

and

int

→

(2D)

int

(

⊥

we obtain the excitation spectrum of a 2D homogeneous dBEC,

E(q

⊥

) =

⊥

2 n

(2D)

int

⊥

)

⊥

g + 2g

⊥

√

. (A.28)

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